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1,116,691,500,415 | arxiv | \section*{Introduction}
\addcontentsline{toc}{section}{Introduction}
General relativity (GR) and quantum mechanics are two of the best verif\/ied theories of modern physics. While general
relativity has been spectacularly successful in explaining the universe at astronomical and cosmological scales, quantum mechanics gives an equally coherent physical picture on small scales. However, one of the biggest unfulf\/illed challenges in physics remains
to incorporate the two theories in the same framework. Ordinary quantum f\/ield theories, which have managed to describe
the three other fundamental forces (electromagnetic, weak and strong), have failed for general relativity because it
is not perturbatively renormalizable.
Loop quantum gravity (LQG) \cite{lqg3,lqg4,lqg2,lqg1} is an attempt to construct a mathematically ri\-go\-rous, non-perturbative, background independent
formulation of quantum general relativity. GR is reformulated in terms of Ashtekar--Barbero variables, namely the densitized triad and the Ashtekar
connection. The basic classical variables are taken to be the holonomies of the connection and the f\/luxes of the triads and these are then promoted
to basic quantum operators. The quantization is not the standard Schr\"odinger quantization but an unitarily inequivalent choice known as
loop/polymer quantization. The kinematic structure of LQG has been well developed. A~robust feature of LQG, not imposed but emergent, is the underlying discreteness of space.
With the aim of obtaining physical implications from LQG, in the last years the application of loop quantization techniques to cosmological models has undergone a notable development. This f\/ield of research is known under the name of loop quantum cosmology (LQC). The models analyzed in LQC are
mini- and midisuperspace models. These models have Killing vectors which reduce the degrees of freedom of full GR. In the case of minisuperspaces, the reduced theories have no f\/ield-theory degrees of freedom remaining. Although there are f\/ield-theory degrees of freedom in the midisuperspace models, their number is smaller than in the full theory. Therefore, these are simplif\/ied systems which provide toy models suitable for studying some aspects of the full quantum gravity theory. Moreover, classical solutions are well known (in fact, we are aware of very few systems which have closed-form solutions of Einstein equations with no Killing vectors) and it is relatively easy to study the ef\/fects of the quantization.
LQC cannot be considered the cosmological sector of LQG because the symmetry reduction is carried out before quantizing, and the results so obtained may not be the same if the reduction is done after quantization. However by adapting the techniques used in the full theory to the symmetry-reduced cosmological models we may hope to capture some of the crucial features of the full theory, as well as to obtain hints about how to tackle them. Indeed, one of the generic characteristics of LQC is the avoidance of the classical singularity. In the present absence of recognized experimental and observational signatures of quantum gravity, this novel and robust result has been increasing the hope that LQG may indeed be the correct theory of quantum gravity.
In this article we will review the progress made in the various cosmological models studied in LQC in the last few years. A recent review \cite{AsS}
emphasizes aspects that are only brief\/ly mentioned here, such as the ``simplif\/ied'' or ``solvable'' LQC framework, the details of ef\/fective dynamics for FRW models with non-zero curvature and/or cosmological constant, and inf\/lationary perturbation theory in LQC. On the other hand, here we focus more on midisuperspaces and discuss lattice ref\/inement parametrizations at some length. Before starting, we shall brief\/ly recall the main features in the kinematic structures of LQG. Similar ingredients are used in the kinematic structure of the LQC models to be discussed later.
\section{Loop quantization}
\subsection{Ashtekar--Barbero formalism}
In the Hamiltonian formulation, the four-dimensional spacetime metric is described by a three-metric $q_{ab}$ induced in
the spatial sections $\Sigma$ that foliate the spacetime manifold, the lapse function~$N$ and the shift vector $N^{a}$ \cite{adm1,adm2}\footnote{Latin indices from the beginning of the alphabet, $a,b,\dots$, denote spatial indices.}. Both the lapse $N$ and the
shift vector $N^{a}$ are Lagrange multipliers accompanying the constraints that encoded the general covariance of general relativity. These constraints
are, respectively, the scalar or Hamiltonian constraint and the dif\/feomorphisms constraint (which is a three-vector). Therefore, the physically
relevant information is encoded in the spatial three-metric and in its canonically conjugate momentum, or equivalently, in the extrinsic curvature
$K_{ab}=\mathcal L_n q_{ab}/2$, where $n$ is the unit normal to~$\Sigma$ and~$\mathcal L_n$ is the Lie derivative along~$n$~\cite{wald}.
LQG is based in a formulation of general relativity as a gauge theory \cite{lqg0c,lqg0ab, lqg0aa,lqg0ba,lqg0bb}, in which the phase space is described
by a $su(2)$ gauge connection, the Ashtekar--Barbero connection $A^i_a$, and its canonically conjugate momentum,
the densitized triad\footnote{Latin indices from the middle of the alphabet, $i,j,\dots$ are $SU(2)$ indices and label new degrees of freedom introduced when passing to the
triad formulation.} $E^a_i$, that plays the role of an ``electric f\/ield''. To def\/ine these objects, f\/irst one introduces the co-triad
$e^i_a$, def\/ined as $q_{ab}=e^i_ae^j_b\delta_{ij}$, where $\delta_{ij}$ stands for the Kronecker delta in three dimensions, and then one def\/ines the
triad, $e^a_i$, as its inverse $e^a_ie^j_b=\delta_i^j\delta^a_b$. The densitized triad then reads $E^a_i=\sqrt{q}e^a_i$, where $q$ stands for the
determinant of the spatial three-metric. In turn, the Ashtekar--Barbero connection reads \cite{bar} $A^i_a=\Gamma^i_a+\gamma K^i_a$, where $\gamma$ is an
arbitrary real and non-vanishing para\-me\-ter, called the Immirzi parameter \cite{gior1,gior2}, $K^i_a=K_{ab}e^b_j\delta^{ij}$ is the extrinsic curvature in
triadic form, and $\Gamma^i_a$ is the spin connection compatible with the densitized triad. Namely, it verif\/ies
$\nabla_b E^a_i+\epsilon_{ijk}\Gamma^j_b E^{ak}=0$, where $\epsilon_{ijk}$ is the totally antisymmetric symbol and $\nabla_b$ is the usual spatial
covariant derivative~\cite{wald}. The canonical pair $(A,E)$ has the following Poisson bracket:
\begin{gather*}
\{A^a_i(x),E^j_b(y)\}=8\pi G\gamma\delta^a_b\delta^j_i\delta(x-y) ,
\end{gather*}
where $G$ is Newton constant and $\delta(x-y)$ denotes the three-dimensional Dirac delta distribution
on the hypersurface $\Sigma$.
Since the internal Euclidean metric $\delta_{ij}$ is invariant under $SU(2)$ rotations, the internal $SU(2)$ degrees of freedom are gauge.
Therefore, in this formulation of general relativity, besides the dif\/feomorphisms constraint $\mathcal{C}_a$ and the scalar (or Hamiltonian)
constraint $\mathcal{C}$, there is a gauge (or Gauss) constraint $ \mathcal{G}_i$ f\/ixing the rotation freedom that we have just introduced.
In the variables $(A^a_i,E^j_b)$, those constraints have the following expression (in vacuum)\footnote{In the presence of matter coupled to the geometry, there is a matter term contributing to each constraint.} \cite{lqg1},
\begin{gather*}
\mathcal{G}_i =\partial_aE^a_i+\epsilon_{ijk}\Gamma^j_a E^{ak}=0,\\
\mathcal{C}_a =F_{ab}^iE^b_i=0,\\
\mathcal{C} =\frac1{\sqrt{|\det(E)|}}\epsilon_{ijk}\left[F^i_{ab}-(1+\gamma^2)\epsilon^i_{mn}
K^m_aK^n_b\right]E^{aj} E^{bk}=0,
\end{gather*}
where $F^i_{ab}$ is the curvature tensor of the Ashtekar--Barbero connection,
\begin{gather*}
F^i_{ab}=\partial_a A^i_b-\partial_b A^i_a+\epsilon_{ijk}A^j_aA^k_b.
\end{gather*}
\subsubsection{Holonomy-f\/lux algebra}
The next step is to def\/ine the holonomies and f\/luxes which will later be promoted to basic quantum variables.
The conf\/iguration variables chosen are the holonomies of $A^i_a$. They are more convenient than the connection itself thanks to their properties under gauge
transformation. The holonomy of the connection $A$ along the edge $e$ is given by
\begin{gather*}
h_e(A)=\mathcal P e^{\int_e dx^a A^i_a(x)\tau_i},
\end{gather*}
where $\mathcal P$ denotes path ordering and $\tau_i$ are the generators of
$SU(2)$, such that $[\tau_i,\tau_j]=\epsilon_{ijk}\tau^k$.
The momentum conjugate to the holonomy is given by the f\/lux of
$E^a_i$ over surfaces $S$ and smeared with a $su(2)$-valued function $f^i$:
\begin{gather*}
E(S,f)=\int_Sf^i E^a_i\epsilon_{abc}dx^bdx^c.
\end{gather*}
The description of the phase space in terms of holonomies and f\/luxes is not only suitable for its transformation properties,
but also because these objects are dif\/feomorphism invariant and their def\/inition is background independent.
Moreover, their Poisson bracket is divergence-free
\begin{gather*}
\{E(S,f),h_e(A)\}=2\pi G\gamma \epsilon(e,S)f^i\tau_i h_e(A),
\end{gather*}
where $\epsilon(e,S)$ represents the regularization of the Dirac delta: it vanishes if~$e$ does not intersect~$S$, as well as if
$e\subset S$, and $|\epsilon(e,S)|=1$ if $e$ and $S$ intersect in one point, the sign depending on the relative orientation between
$e$ and~$S$~\cite{lqg3}.
\subsection{Kinematic Hilbert space}
In LQG, the holonomy-f\/lux algebra is represented over a kinematical Hilbert space that is
dif\/ferent from the more familiar
Schr\"odinger-type Hilbert space. It is given by the
completion of the space of cylindrical functions (def\/ined on the space of generalized connections)
with respect to the so-called Ashtekar--Lewandowski measure
\cite{ALmeasure1,ALmeasure3,ALmeasure2,baez}.
We give a very brief description of this kinematical Hilbert space below, while the details can be
found in \cite{lqg3,lqg4,lqg2,lqg1} (and references therein).
A generalized connection $h_e(A)\equiv\bar A_e$ is an assignment of $\bar A \in$ $SU(2)$ to any analytic path $e \subset \Sigma$.
A graph $\Gamma$ is a collection of analytic paths $e\subset \Sigma$ meeting at most at their endpoints.
We will consider only closed graphs. The point at which two edges meet is called a vertex.
Let n be the number of edges in $\Gamma$. A function cylindrical with respect to $\Gamma$ is given by
\begin{equation*}
\psi_\Gamma(\bar A) := f_\Gamma \big(\bar A_{e_1},\dots, \bar A_{e_n}\big) ,
\end{equation*}
where $f_\Gamma$ is a smooth function on $SU(2)^n$. The space of states cylindrical with respect to $\Gamma$ are denoted by Cyl$_\Gamma$.
The space of all functions cylindrical with respect to some $\Gamma \in \Sigma$ is denoted by Cyl and is given by
\begin{equation*}
\mbox{Cyl} = \bigcup_\Gamma \mbox{Cyl}_\Gamma.
\end{equation*}
Given a cylindrical function $\psi_\Gamma(\bar A) \in {\rm Cyl}$, the Ashtekar--Lewandowski measure, denoted by~$\mu_0$, is def\/ined by
\begin{equation*}
\int_{\overline{{\mathcal A}}}d\mu_0[\psi_\Gamma(\bar A)] :=
\int_{SU(2)^n} \prod_{e \subset \Gamma} d h^e f_\Gamma \big(\bar A_{e_1},\dots, \bar A_{e_n}\big) ,
\qquad \forall\, \psi_\Gamma(\bar A) ,
\end{equation*}
where d$h$ is the normalized Haar measure on $SU(2)$.
Using this measure we can def\/ine an inner product on Cyl:
\begin{gather*}
\langle \psi_\Gamma, \psi_\Gamma '\rangle := \langle f_\Gamma\big(\bar A_{e_1},\dots, \bar
A_{e_n}\big),g_{\Gamma'}\big(\bar A_{e_1},\dots, \bar A_{e_m}\big)\rangle\nonumber\\
\phantom{\langle \psi_\Gamma, \psi_\Gamma '\rangle}{} = \int_{SU(2)^n} \prod_{e \subset \Xi_{\Gamma \Gamma'}} d h^e\overline{f_\Gamma\big(\bar A_{e_1},\dots,
\bar A_{e_n}\big)}g_{\Gamma'}\big(\bar A_{e_1},\dots, \bar A_{e_m}\big),
\end{gather*}
where ${\Xi}_{\Gamma \Gamma'}$ is any graph such that $\Gamma \subset \Xi_{\Gamma \Gamma'}$ and $\Gamma'
\subset \Xi_{\Gamma \Gamma'}$. Then, the kinematical Hilbert space of LQG is the Cauchy completion of
$\mbox{Cyl}$ in the Ashtekar--Lewandowski norm: ${\mathcal H}_{\rm kin}=L^2(\overline{{\mathcal A}},d\mu_0)$.
A basis on this Hilbert space is provided by \emph{spin network} states, which are
constructed as follows. Given a graph $\Gamma$, each edge $e$ is colored by a non-trivial irreducible representation
$\pi_{j_e}$ of $SU(2)$. Spin network states are cylindrical functions with respect to this colored graph. They are denoted by
$T_s := T_{\Gamma,\vec{j}}(\bar A)$ where $\vec{j} = \{ j_e \}$.
Then, every cylindrical function can be expanded in the basis of spin network states.
On $\mbox{Cyl}_\Gamma$ the operators representing the corresponding holonomies act by multiplication, while the operator representing the f\/lux is given by
\begin{gather*}
\hat{E}_\Gamma(S,f)=i2\pi G\hbar\sum_{e\subset\Gamma}\epsilon(e,S)\text{Tr}\left(f^i\tau_i\bar{A}_e\frac{\partial}{\partial \bar{A}_e}\right).
\end{gather*}
To obtain the quantum version of the more general operators, they have to be f\/irst rewritten in terms of the basic holonomy-f\/lux operators.
Note that the quantum conf\/iguration space is not the space of smooth connections but rather the space of holonomies (or generalized connections).
Since the Ashtekar--Lewandowski measure is discontinuous in the connection, there is no well-def\/ined operator for the connection on
${\mathcal H}_{\rm kin}$. Consequently, the curvature must be def\/ined in terms of holonomies before it can be promoted to a quantum operator.
The strategy in the full theory is to def\/ine any general quantum operator via regularization as follows (see~\cite{lqg3, lqg1} for details):
\begin{itemize}\itemsep=0pt
\item the spatial manifold $\Sigma$ is triangulated into elementary tetrahedra;
\item the integral over $\Sigma$ is replaced by a Riemann sum over the cells;
\item for each cell, we def\/ine a regularized expression in terms of the basic operators, such that we get the correct classical expression in the limit
the cell is shrunk to zero;
\item this is promoted to a quantum operator provided it is densely def\/ined on ${\mathcal H}_{\rm kin}$.
\end{itemize}
In the subsequent sections we shall see how the same strategy is applied for def\/ining the quantum operators in LQC. One signif\/icant dif\/ference is that
in the full theory the f\/inal expressions are independent of the regularization, while in the symmetry-reduced models the regularization
(i.e., the size of the cells) cannot be removed and has to be treated as an ambiguity. However, we can f\/ix the form of the ambiguity by taking hints
from the full theory.
One of the most interesting features of LQG is that the spectra of the operators
representing geometrical quantities like area and volume are discrete. Discrete eigenvalues imply that the underlying spatial manifold is also
discrete at least when we are close to the quantum gravity scale. This is a feature of the quantization scheme and it also plays and important role in
the singularity avoidance in LQC minisuperspace models.
This is the kinematical structure of LQG. However we are interested in physical states, i.e.\ states which are annihilated by the all the constraints.
To obtain the physical Hilbert space we now need to solve the quantum constraints. The Gauss constraint is
easy to solve and we can obtain gauge invariant Hilbert space spanned by the gauge invariant spin networks. The inf\/initesimal
dif\/feomorphism constraint cannot be expressed as a self-adjoint operator on~${\mathcal H}_{\rm kin}$. However we can consider f\/inite dif\/feomorphisms
and the solutions to the f\/inite dif\/feomorphism constraint are obtained via \textit{group averaging}.
It turns out that these solutions do not lie in~${\mathcal H}_{\rm kin}$ but in~Cyl$^\star$, the algebraic dual of Cyl.
In the construction of the Hamiltonian constraint operator we face a number of problems (see~\cite{pftreview} and
references therein for details). Although a well-def\/ined Hamiltonian constraint ope\-ra\-tor can be constructed which satisf\/ies an on-shell anomaly-free quantum constraint algebra, the quantization procedure suf\/fers from a number of ambiguities: in the choice of the regulators, in the transcription in terms of basic quantum variables, and in the choice of curvature appro\-xi\-mants. Also the domain of the Hamiltonian constraint operator is not known. Ef\/forts have been made to reduce the ambiguities by studying the of\/f-shell closure of the constraint algebra and by trying to f\/ind the correct semiclassical limit, but no signif\/icant progress has been made so far. So, although we have a well-def\/ined full quantum theory of gravity at the kinematical level, the physical Hilbert-space construction is beset by a number of open problems and is not yet complete.
LQC tries to study some of the features of Loop quantization while avoiding the problems of the full theory. As we shall see later, the programme of LQC tries to closely follow the same steps, as far as possible, in the much simpler case of cosmological models with no (or at most one) f\/ield-theory degrees of freedom. In minisuperspace models it is possible to go beyond the kinematics and construct the physical Hilbert space. Another useful procedure developed to study the ef\/fect of the underlying discreteness is the use of ef\/fective equations to study homogeneous cosmologies and perturbations therein. This has opened up a large number of systems to semiclassical analyses. It is hoped that lessons learned from LQC can give hints about how to tackle the issues being faced in LQG.
\section{Plan of the review}
Signif\/icant progress has been made in the study of a number of cosmologies in LQC. Here, we shall give an overall account of various facets of LQC, outlining technical aspects, reviewing the results achieved and indicating the directions of further research. The rest of the paper is divided into three parts.
In Part~\ref{part1} we discuss LQC minisuperspace models. The simplest cases of minisuperspace are Friedmann--Robertson--Walker (FRW) models, which are homogeneous and isotropic. The kinematical quantization programme followed for these models will be discussed in detail, using the example of f\/lat FRW. We also describe the results obtained in the physical Hilbert space including the dynamical singularity resolution and the bounce.
Open and closed FRW models, with and without a cosmological constant, are brief\/ly discussed. The next level of complication, Bianchi models, consists in removing the assumption of isotropy. In this case, our illustrative example will be the Bianchi~I model but we also indicate the work done so far for Bianchi~II and Bianchi~IX cases.
Then, Part~\ref{part2} focusses on the LQC of midisuperspace models which are neither homogeneous nor isotropic. We describe the only case whose loop quantization has been studied in some detail, the linearly polarized Gowdy $T^3$ model. Two contrasting approaches have been taken in the study of this model. In the f\/irst approach, the degrees of freedom have been separated into homogeneous and inhomogeneous sectors. The homogeneous sector is quantized using
the tools developed in LQC, while the inhomogeneous sector is Fock quantized. In the second approach, the model is studied as a whole mimicking the steps of LQG. We describe and compare both procedures.
Finally, in Part~\ref{part3} we discuss the programme of ef\/fective dynamics developed in LQC. In contrast to the previous two parts, this approach aims to incorporate the ef\/fects of the discrete geometry as corrections to the classical equations. In this way it may be possible to link LQC to phenomenological evidence.
In the end we summarize the current directions of ongoing research. This review is intended as an introduction of the main results achieved in the f\/ield in the past few years, especially in the Hamiltonian formalism, and it does not cover more recent work being done in the area of cosmological perturbations, phenomenology, and spin-foam cosmology. We will comment about these and other lines of research in Sections~\ref{latti} and~\ref{concl}.
\part{Minisuperspaces in loop quantum cosmology}\label{part1}
LQC \cite{lqc2a,lqc2b,lqc1,lqc3} adapts the techniques developed in loop quantum gravity~\cite{lqg3,lqg2, lqg1} to
the quantization of simpler models than the full theory, as minisuperspace models. Minisuperspace models are solutions of Einstein's equations with a high degree
of symmetry, so much so that there are no f\/ield theory degrees of freedom remaining. They lead to homogeneous cosmological solutions
all of which suf\/fer from a singularity where the classical equations of motion break down.
Since, after quantization, these are essentially quantum
mechanical systems, they serve as good toy models for testing the predictions of~LQG.
In LQC, we start from the classically symmetry-reduced phase space and then try to apply the steps followed in LQG to
these systems. Owing to simplif\/ications due to classical symmetry reduction, many technical complications typical of LQG can be avoided, and the quantization
programme can be carried out beyond what has been achieved so far in the full theory. The fact that there is a well-def\/ined full theory which tells us
that the underlying spatial geometry is discrete is a crucial ingredient in the formulation of LQC. A signif\/icant achievement of LQC is the development of
a well-def\/ined quantum theory for cosmological models where the classical singularity is absent. This resolution of the classical singularity is a
robust feature of LQC as it is seen in all the minisuperspace models studied so far, as well as under various choices made in addressing the ambiguities arising
in quantization. In this part we shall review the LQC of various known minisuperspace cosmological scenarios.
\section{Friedmann--Robertson--Walker models}\label{chap:1-flatFRW}
LQC started with the pioneering works by Bojowald~\cite{boj2, boj1a,boj1b,boj1c,boj1d}, that showed the f\/irst attempts of implementing the methods
of LQG to the quantization of the simplest cosmological model: the f\/lat Friedmann--Robertson--Walker (FRW) model
(homogeneous and isotropic with f\/lat spatial sections), whose geometry is described by a single degree of freedom, the scale factor.
This system, even if very simple, is physically interesting since, at large scales, our universe is approximately homogeneous and isotropic.
In addition, cosmological observations are compatible with a spatially f\/lat geometry.
After the early papers by Bojowald, the kinematic structure of LQC was revised and more rigorously established~\cite{abl}, which made it
possible to complete the quantization of the model in presence of a homogeneous massless scalar f\/ield minimally coupled to the geometry,
as well as to study the resulting quantum evolution \cite{acs, aps1,aps2,aps3}.
Classically, this model represents expanding universes with an initial \emph{big bang} singularity, where certain physical observables, such as the
matter density, diverge. Remarkably, the quantum dynamics resolves the singularity replacing it with a \emph{quantum bounce}, while for semiclassical
states it agrees with the classical dynamics far away form the singularity.
Therefore, even though this is the simplest cosmological model, its loop quantization, also called polymeric quantization, already leads to relevant
results, the most important one being the avoidance of the singularity.
Using the example of the f\/lat FRW model coupled to a massless scalar, we shall discuss in detail the basics and the mathematical structure
of LQC, adopting the so-called improved dynamics prescription \cite{aps3}.
\subsection{Classical phase space description}
\subsubsection{Ashtekar--Barbero formalism}
The classical phase space in the presence of homogeneity is much simpler than the general situation described in the introduction.
In homogeneous cosmology, the gauge and dif\/feomorphisms constraints are trivially satisf\/ied, the
Hamiltonian constraint being the only survivor in the model. Moreover, for f\/lat FRW the spin connection vanishes. In this case, the
geometry part of the scalar constraint in its integral version is\footnote{The lapse function $N$ goes out of the integral due to the homogeneity.} $C_\text{grav}(N)=
NC_\text{grav}$, with
\begin{gather}\label{eq:lig-escalar-lqg-hom}
C_\text{grav}=\int_{\Sigma} d^3x\, \mathcal C=-\frac1{\gamma^2}\int_{\Sigma} d^3x
\frac{\epsilon_{ijk}F^i_{ab}E^{aj}E^{bk}}{\sqrt{|\det(E)|}}.
\end{gather}
Since f\/lat FRW spatial sections $\Sigma$ are non-compact, and the variables that describe it are spatially homogeneous, integrals such as~\eqref{eq:lig-escalar-lqg-hom} diverge. To avoid that, one usually restricts the analysis to a f\/inite cell~$\mathcal V$.
Owing to homogeneity, the study of this cell reproduces what happens in the whole universe. When imposing also isotropy,
the connection and the triad can be described (in a convenient gauge) by a single parameter~$c$ and~$p$, respectively, in the form~\cite{abl}
\begin{gather*
A^i_a=c V_o^{-1/3} \,{}^oe^i_a, \qquad E^a_i=p V_o^{-2/3} \sqrt{{}^oq}\,{}^oe^a_i.
\end{gather*}
Here we have introduced a f\/iducial co-triad ${}^oe^i_a$ that we will choose to be diagonal,
${}^oe^i_a=\delta^i_a$, and the determinant $\sqrt{{}^oq}$ of the corresponding f\/iducial metric. The
results do not depend on the f\/iducial choice.
With the above def\/initions, the symplectic structure is def\/ined via,
\begin{gather*
\{c,p\}=\frac{8\pi G\gamma}{3}.
\end{gather*}
The variable~$p$ is related to the scale factor $a$ commonly employed in geometrodynamics through
the expression
$a(t)=\sqrt{|p(t)|} V_o^{-1/3}$. Note that~$p$ is positive (negative) if physical and f\/iducial triads have the same (opposite) orientation.
On the other hand, a (homogeneous) massless scalar f\/ield~$\phi$, together with its momentum~$P_\phi$, provide the canonical pair describing the matter content, with Poisson bracket
$\{\phi,P_\phi\}=1$.
Then, the total Hamiltonian constraint contains a matter contribution beside the
geometry one, given in equation~\eqref{eq:lig-escalar-lqg-hom}, and reads
\begin{gather}\label{eq:ligFRW}
C=C_\text{grav}+C_\text{mat}=-\frac6{\gamma^2}c^2\sqrt{|p|}+8\pi G\frac{P_\phi^2}{V}=0,
\end{gather}
where $V=|p|^{3/2}$ is the physical volume of the cell $\mathcal V$.
\subsubsection{Holonomy-f\/lux algebra}
When def\/ining holonomies and f\/luxes in LQC, and in the particular case of isotropic FRW models, owing to the homogeneity it is
suf\/f\/icient to consider straight edges oriented along the f\/iducial directions, and with oriented length equal to $\mu V_o^{1/3}$,
where $\mu$ is an arbitrary real number. Therefore, the holonomy along one such edge, in the $i$-th direction, is given by
\begin{gather*}
h_i^\mu(c)=e^{\mu c \tau_i}=\cos\left(\frac{\mu c}{2}\right)\mathbbm{1}+2\sin\left(\frac{\mu
c}{2}\right)\tau_i.
\end{gather*}
Then, the gravitational part of the conf\/iguration algebra is the algebra generated by the matrix elements of the holonomies, namely,
the algebra of quasi-periodic functions of $c$, that are the complex exponentials
\begin{gather*}
\mathcal N_\mu(c)=e^{\frac{i}{2}\mu c}.
\end{gather*}
In analogy with the terminology employed in LQG \cite{lqg3, lqg1}, the vector space of these quasi-periodic functions is called the space of
cylindrical functions def\/ined over symmetric connections, and it is denoted by $\text{Cyl}_\text{S}$.
In turn, the f\/lux is given by
\begin{gather*}
E(S,f)=p V_o^{-2/3}A_{S,f},
\end{gather*}
where $A_{S,f}$ is the f\/iducial area of $S$ times an orientation factor (that depends on~$f$). Then, the f\/lux is essentially described by~$p$.
In summary, in isotropic and homogeneous LQC the phase space is described by the va\-riab\-les~$\mathcal N_\mu(c)$ and~$p$, whose Poisson bracket is
\begin{gather*}
\{\mathcal N_\mu(c),p\}=i\frac{4\pi G\gamma}{3}\mu\mathcal N_\mu(c).
\end{gather*}
\subsection{Kinematical structure}
\label{1sec:kin}
Mimicking the quantization implemented in LQG, in LQC we adopt a representation of the algebra generated by the phase space variables
$\mathcal N_\mu(c)$ and $p$ that is not continuous in the connection, and therefore there is no
operator representing $c$ \cite{abl}. More concretely, the quantum conf\/iguration space is the Bohr
compactif\/ication of the real line, $\mathbb{R}_\text{Bohr}$, and the corresponding Haar measure that
characterizes the kinematical Hilbert space is the so-called Bohr measure~\cite{Vel}.
It is simpler to work in momentum representation. In fact, such Hilbert space is isomorphic to the space of functions of $\mu\in\mathbb{R}$
that are square summable with respect to the discrete measure~\cite{Vel}, known as polymeric space.
In other words, employing the kets $|\mu\rangle$ to denote the quantum states~$\mathcal N_\mu(c)$, whose linear span is the space
$\text{Cyl}_\text{S}$ (dense in $\mathbb{R}_\text{Bohr}$), the kinematical Hilbert space is the completion of
$\text{Cyl}_\text{S}$ with respect to the inner product $\langle \mu|\mu'\rangle=\delta_{\mu\mu'}$.
We will denote this Hilbert space by $\mathcal H_{\text{grav}}$. Note that $\mathcal
H_{\text{grav}}$ is non-separable, since the states $|\mu\rangle$ form a non-countable orthogonal basis.
Obviously, the action of $\hat{\mathcal N}_{\mu}$ on the basis states is
\begin{gather*}
\hat{\mathcal N}_{\mu'} |\mu\rangle= |\mu+\mu'\rangle.
\end{gather*}
On the other hand, the Dirac rule
$[\hat{\mathcal N}_\mu,\hat p]=i\hbar \widehat{\{\mathcal N_\mu(c),p\}}$
implies that
\begin{gather*
\hat{p}|\mu\rangle=p(\mu)|\mu\rangle, \qquad p(\mu)=\frac{4\pi l_\text{Pl}^2\gamma}{3} \mu,
\end{gather*}
where $l_\text{Pl}=\sqrt{G\hbar}$ is the Planck length. As we see, the spectrum of this operator is
discrete, as a~consequence of the representation not being continuous in~$\mu$. Due to this lack of
continuity, the Stone--von Neumann theorem about the uniqueness of the representation in quantum
mecha\-nics~\cite{SvN1,SvN2} is not applicable in this context. Therefore, the loop quantization of
this model is inequivalent to the standard Wheeler--DeWitt (WDW) quantization~\cite{witt, whe},
where operators have a typical Schr\"odinger-like representation. In fact, while the WDW
quantization fails in solving the problem of the big bang singularity, the loop quantization is
singularity free~\cite{aps2,aps3}, as we will see later.
For the matter f\/ield, we adopt a standard Schr\"odinger-like representation, with
$\hat{\phi}$ acting by multiplication and $\hat{P}_\phi=-i\hbar\partial_\phi$ as derivative, being both operators def\/ined
on the Hilbert space $L^2(\mathbb{R},d\phi)$. As domain, we take the Schwartz space
$\mathcal{S}(\mathbb{R})$ of rapidly decreasing functions, which is dense in
$L^2(\mathbb{R},d\phi)$.
The total kinematical Hilbert space is then $\mathcal H_{\text{kin}}=\mathcal
H_{\text{grav}}\otimes L^2(\mathbb{R},d\phi)$.\footnote{Note that the basic operators
def\/ined above are in the tensor product of both sectors (geometry and matter), acting as the
identity in the sector where they do not have dependence. For instance, the operator
$\hat{p}$ def\/ined on $\text{Cyl}_\text{S}\otimes\mathcal{S}(\mathbb{R})$ really means
$\hat{p}\otimes\mathbbm{1}$. Nonetheless, for the sake of simplicity we will ignore the tensor product by
the identity.}
\subsection{Hamiltonian constraint operator}
\label{1sec:lig-ham-aps}
\subsubsection{Curvature operator and improved dynamics}
Since the connection is not well def\/ined in the quantum theory, the classical expression of the Hamiltonian constraint, given in
equation~\eqref{eq:ligFRW}, cannot be promoted directly to an operator. In order to obtain the quantum analogue of the gravitational part, we follow the
procedure adopted in the full theory. We start from the general expression \eqref{eq:lig-escalar-lqg-hom} and express the curvature tensor in terms
of the holonomies, which do have a well-def\/ined quantum counterpart.
Following LQG, we take a closed square loop with holonomy
\begin{gather*}
h^\mu_{\square_{ij}}=h_i^\mu h_j^\mu (h_i^\mu)^{-1} (h_j^\mu)^{-1},
\end{gather*}
that encloses a f\/iducial area $A_{\square}=\mu^2 V_o^{2/3}$.
The curvature tensor then reads~\cite{abl}
\begin{align}\label{eq:curvatura-exacta}
F^i_{ab}=-2 \;\lim_{A_{\square}\rightarrow 0}
\text{tr}\left(\frac{h^\mu_{\square_{jk}}-\delta_{jk}}{A_{\square}}\tau^i\right){}^oe^j_a{}
^oe^k_b.
\end{align}
This limit is classically well def\/ined. However, in the quantum theory we cannot contract the area to zero because that limit does not converge.
Since we have a well def\/ined full theory (unlike WDW quantization), we can appeal to the discretization of geometry coming from it.
In LQG, geometric area has a discrete spectrum with a non-vanishing minimum eigenvalue~$\Delta$~\cite{area2, area1}.
This suggests that we should not take the null area limit, but consider only areas larger than~$\Delta$. Then, we contract the
area of the loop till a minimum value
$A_{\square_{\min}}=\bar\mu^2 V_o^{2/3}$, such that the geometric area corresponding to this f\/iducial area, given by the f\/lux
$E(\square_{\min},f=1)=p\bar\mu^2$, is equal to~$\Delta$. In short, the curvature is def\/ined by the regularized expression
\begin{gather}\label{eq:curvatura}
{F}^i_{ab}=-2\,
\text{tr}\left( \frac{h^{\bar\mu}_{\square_{jk}}-\delta_{jk}}{\bar\mu^2
V_o^{2/3}}\tau^i\right){}^oe^j_a{}^oe^k_b,
\end{gather}
where $\bar\mu$, characterizing the minimum area of the loop, is given by the \emph{Ansatz}
\begin{align}\label{mu}
\frac1{\bar\mu}=\sqrt{\frac{|p|}{\Delta}}.
\end{align}
This choice of $\bar\mu$ is usually called \emph{improved dynamics} in the LQC literature \cite{aps3}.
Note that the smaller the value of $\bar{\mu}$ is, or equivalently the bigger the value of $|p|$ is, the better equation~\eqref{eq:curvatura} approximates the classical expression~\eqref{eq:curvatura-exacta}, so that both expressions agree in the regime in which the area of the cell under study is large enough.
Finally, the curvature operator is obtained by promoting equation~\eqref{eq:curvatura} to an operator. Let us remark that there are two kinds of ambiguities in the def\/inition of this operator. On the one hand, the value of the parameter $\bar{\mu}$, that is f\/ixed by the improved dynamics prescription, as we have just explained. On the other hand, we also have the ambiguity in the $SU(2)$ representation we use for calculating the trace. As usual in LQC \cite{lqc1}, we will compute the holonomies in the fundamental representation of spin $1/2$.
Note that terms of the kind ${\mathcal N}_{\bar\mu}=e^{i\bar\mu c/2}$ contribute to
$h^{\bar\mu}_{\square_{ij}}$. In order to def\/ine the operator $\hat{\mathcal
N}_{\bar\mu}=\widehat{e^{i\bar\mu c/2}}$, it is assumed that this operator generates unit
translations over the af\/f\/ine parameter associated with the vector f\/ield $\bar\mu[p(\mu)]\partial_\mu$~\cite{aps3}.
In other words, we introduce a canonical transformation in the geometry sector of the phase space,
such that it is described by the variable $b=\hbar\bar\mu c/2$ and its canonically conjugate
variable $v(p)=(2\pi\gamma l_\text{Pl}^2\sqrt{\Delta})^{-1}\text{sgn}(p)|p|^{3/2}$ ($\text{sgn}$
denotes the sign), with $\{b,v\}=1$. The variable $v(\mu)=v[p(\mu)]$ indeed verif\/ies
$\partial_v=\bar\mu(\mu)\partial_\mu$. Then, we relabel the basis states of $\mathcal
H_{\text{grav}}$ with this new parameter $v$ that, unlike $\mu$, is adapted to the action of
$\hat{\mathcal N}_{\bar\mu}$. In fact, introducing the operator $\hat{v}$ with action
$\hat{v}|v\rangle=v|v\rangle$, it is straightforward to show that $\hat{\mathcal
N}_{\bar\mu}|v\rangle=|v+1\rangle$, so that the Dirac rule
$[\widehat{e^{ib/\hbar}},\hat{v}]|v\rangle=i\hbar
\widehat{\{e^{ib/\hbar},v\}}|v\rangle$ is satisf\/ied. On the other hand, we obtain
$\hat{p}|v\rangle=(2\pi \gamma l_\text{Pl}^2\sqrt{\Delta})^{2/3}\text{sgn}(v)|v|^{2/3} |v\rangle$.
It is worth mentioning that the parameter $v$ has a geometrical interpretation: its absolute value
is proportional to the physical volume of the cell $\mathcal V$, given by
\begin{gather*
\hat{V}=\widehat{|p|}^{3/2}, \qquad \hat{V}|v\rangle= 2\pi \gamma
l_\text{Pl}^2\sqrt{\Delta}|v||v\rangle.
\end{gather*}
The quantization within the prescription~\eqref{mu} meant an important improvement for LQC~\cite{aps3}.
Earlier, it was assumed that the minimum f\/iducial length was just some constant~$\mu_o$ related to~$\Delta$~\cite{abl}. However,
the resulting quantum dynamics was not successful, inasmuch as the quantum ef\/fects of the geometry could be important
at scales where the matter density was not necessarily high. In that case, in the semiclassical regime the physical results deviated signif\/icantly
from the predictions made by general relativity~\cite{aps2}. Improved dynamics solves this problem. Furthermore, it has been proved that it is
the only minisuperspace quantization (among a certain family of possibilities) yielding to a physically admissible model~\cite{cs},
independent of the f\/iducial structures, with a well-def\/ined classical limit in agreement with GR, and giving rise to a scale of
Planck order where quantum ef\/fects are important and solve the singularity problem.
\subsubsection{Representation of the Hamiltonian constraint}
When trying to promote the gravitational part of the scalar constraint~\eqref{eq:lig-escalar-lqg-hom} to an operator, we f\/ind an additional dif\/f\/iculty concerning the
inverse of the volume,
\begin{gather*}
\frac1{V}=\frac{\sqrt{{}^o q}}{\sqrt{|\det(E)|}V_o}.
\end{gather*}
The volume operator has a discrete spectrum with the eigenvalue zero included, so its inverse
(obtained by using the spectral theorem) is not well def\/ined in zero.
Nonetheless, following LQG \cite{inv-vol-lqg1,inv-vol-lqg2}, from the classical identity
\begin{gather}\label{eq:identidad-lig}
\frac{\epsilon_{ijk}E^{aj}E^{bk}}{\sqrt{|\det(E)|}} =\sum_{k=1}^3\frac{\text{sgn}(p)}{2\pi\gamma
GV_o^{1/3}}\frac1{l}\, {}^o e^k_c \,{}^o \epsilon^{abc}\,
\text{tr}\big(h_k^{l}(c)\big\{[h_k^{l}(c)]^{-1},
V\big\}\tau_i\big),
\end{gather}
we can obtain an operator for the left-hand side of this expression by promoting the functions on the right-hand side to the
corresponding operators, and by making the replacement
$
\widehat{\{\;\;,\;\;\}}\rightarrow -(i/\hbar)[\hat{\;\;},\hat{\;\;}]
$.
Note that the parameter $l$ labels a quantization ambiguity. In order not to introduce new scales
in the theory, we take for $l$ the value $\bar\mu=\sqrt{{\Delta}/{|p|}}$~\cite{aps3}.
Plugging this result into the Hamiltonian constraint~\eqref{eq:lig-escalar-lqg-hom},
as well as the curvature given in equation~\eqref{eq:curvatura}, we obtain that the geometry (or
gravitational) contribution to the Hamiltonian constraint operator is~\cite{aps3}
\begin{gather}\label{eq:operador-lig-aps}
\widehat{C}_\text{grav}=
i\frac{3\widehat{\text{sgn}(p)}}{2\pi\gamma^3
l_\text{Pl}^2\Delta^{3/2}}\hat{V}[\widehat{\sin\left(\bar\mu
c\right)}\widehat{\text{sgn}(p)}]^2\big(\hat{\mathcal
N}_{\bar{\mu}}\hat{V}\hat{\mathcal N}_{-\bar{\mu}}- \hat{\mathcal
N}_{-\bar{\mu}}\hat{V}\hat{\mathcal N}_{\bar{\mu}}\big),
\end{gather}
with
\begin{gather*}
\widehat{\sin(\bar\mu c)}=\frac{\hat{\mathcal
N}_{2\bar\mu}-\hat{\mathcal N}_{-2\bar\mu}}{2i}.
\end{gather*}
Let us now deal with the representation of the matter contribution, given in the second term of
equation~\eqref{eq:ligFRW}. To represent the inverse of the volume, we follow the same strategy as before,
now starting with the classical identity
\begin{gather*
\frac{\text{sgn}(p)}{|p|^{1-s}} =\frac{1}{s4\pi\gamma
G}\frac1{l}\text{tr}\left(\sum_i\tau^ih_i^{l}(c)\big\{[h_i^{l}(c)]^{-1},
|p|^s\big\}\right).
\end{gather*}
As before, we take the trace in the fundamental representation and we choose $l$ equal to $\bar\mu$ in the quantum theory. To f\/ix the ambiguity in the constant $s>0$, we choose for simplicity $s=1/2$. We obtain
\begin{gather}
\widehat{\left[\frac{1}{\sqrt{|p|}}\right]}
=\frac{3}{4\pi\gamma
l_\text{Pl}^2\sqrt{\Delta}}\widehat{\text{sgn}(p)}\widehat{\sqrt{|p|}}\left(\hat{\mathcal
N}_{-\bar{\mu}}\widehat{\sqrt{|p|}}\hat{\mathcal
N}_{\bar{\mu}}- \hat{\mathcal
N}_{\bar{\mu}}\widehat{\sqrt{|p|}}\hat{\mathcal
N}_{-\bar{\mu}}\right).
\end{gather}
The action of this operator on the basis states is diagonal and given by
\begin{gather*
\widehat{\left[\frac1{\sqrt{|p|}}\right]}|v\rangle=b(v)|v\rangle,\qquad
b(v)=\frac{3}{2}\frac1{(2\pi\gamma l_\text{Pl}^2\sqrt{\Delta})^{1/3}}|v|^{1/3}
\big||v+1|^{1/3}-|v-1|^{1/3}\big|.
\end{gather*}
While, for large values of $v$, $b(v)$ is well approximated by the classical value
$1/\sqrt{|p|}$, for small values of $v$ they dif\/fer considerably. In fact, the above operator is
bounded from above and annihilates the zero-volume states.
The matter contribution to the constraint is then given by the operator
\begin{gather*}
\widehat{C}_\text{mat}=-8\pi l_\text{Pl}^2\hbar
\widehat{\left[\frac1{V}\right]}\partial_\phi^2, \qquad
\widehat{\left[\frac1{V}\right]}=\widehat{\left[\frac1{\sqrt{|p|}}\right]}^3.
\end{gather*}
In order for the Hamiltonian constraint operator $\widehat{C}=\widehat{C}_\text{grav}+\widehat{C}_\text{mat}$
to be (essentially) self-adjoint, we need to symmetrize the gravitational term~\eqref{eq:operador-lig-aps}. There is an ambiguity in the chosen symmetric factor ordering and
several possibilities have been studied in the literature~\cite{acs,aps3,klp,mmo,mop,ydm} (see~\cite{mop} for a detailed comparison between them). Due to its suitable properties, here we will
adopt the prescription called sMMO in~\cite{mop}\footnote{The acronym ``MMO'' refers to the
model of \cite{mmo}, by Mart\'in-Benito, Mena Marug\'an, and Olmedo.}, that is a simplif\/ied version
of the prescription of \cite{mmo}. Its two main features are:
\begin{enumerate}\itemsep=0pt
\item [i)] decoupling of the zero-volume state $|v=0\rangle$;
\item [ii)] decoupling of states with opposite orientation of the densitized triad, namely states
$|v<0\rangle$ are decoupled from states $|v>0\rangle$.
\end{enumerate}
As we will see, this will give rise to simple superselection sectors with nice properties.
Re\-markably, the behavior of the resulting eigenstates of the gravitational part of the
constraint already shows the occurrence of a generic quantum bounce dynamically resolving the
singularity. Therefore, this prescription ensures that the quantum bounce mechanism is an intrinsic
feature of the theory, independent of the particular physical state considered\footnote{In~\cite{aps3}, the quantum bounce was shown just for particular semiclassical states. Then, with the
factor ordering adopted in~\cite{acs}, it was shown that the quantum bounce is generic, but the
result is only obtained for a specif\/ic superselection sector. The results of~\cite{mmo} are instead
completely general.}.
Then, following \cite{mmo,mop}, we take
\begin{gather}\label{eq:operador-lig-mmo}
\widehat{C}=\widehat{\left[
\frac{1}{V}\right]}^{1/2}\left(-\frac{6}{\gamma^{2}}
\widehat{\Omega}^2+8\pi
G\hat{P}_{\phi}^2\right)\widehat{\left[
\frac{1}{V}\right]}^{1/2},
\end{gather}
where the operator $\widehat\Omega$ is def\/ined as
\begin{gather}\label{eq:operador-grav-mmo}
\widehat\Omega =\frac1{4i\sqrt{\Delta}}\widehat{|p|}^{3/4}
\left[\big(\hat{\mathcal
N}_{2\bar\mu}-\hat{\mathcal
N}_{-2\bar\mu}\big)\widehat{\text{sgn}(p)}
+\widehat{\text{sgn}(p)}\big(\hat{\mathcal
N}_{2\bar\mu}-\hat{\mathcal
N}_{-2\bar\mu}\big)\right]\widehat{|p|}^{3/4}.
\end{gather}
The action of $\widehat{\text{sgn}(p)}$ on the state $|v=0\rangle$ can be def\/ined
arbitrarily, since the f\/inal action of $\widehat\Omega$ is independent of that choice, provided
that $\widehat\Omega|0\rangle=0$.
Thanks to the splitting of powers of~$p$ on the left and on the right, $\widehat{C}$ annihilates
the subspace of zero-volume states and leaves invariant its orthogonal complement, thus decoupling
the zero-volume states as desired. We can then remove the state~$|0\rangle$ and def\/ine the
operators acting on the geometry sector on the Hilbert space
$\widetilde{\mathcal{H}}_{\textrm{grav}}$ def\/ined as the Cauchy completion (with respect to the
discrete
measure) of the dense domain
\begin{gather*}
\widetilde{\textrm{Cyl}}_\text{S}=\text{span}\{|v\rangle;\;v\in\mathbb{R}\setminus\{0\}\}.
\end{gather*}
As a consequence, the big bang is resolved already at the kinematical level, in the sense that
the quantum equivalent of the classical singularity (namely, the eigenstate of vanishing physical
volume) has been entirely removed from the kinematical Hilbert space (see also~\cite{boj}).
In view of the operator~\eqref{eq:operador-lig-mmo}, it is more convenient to work with its
densitized version, def\/i\-ned~as
\begin{gather*
\widehat{{\cal C}}=
\widehat{\left[\frac1{V}\right]}^{-1/{2}}
\widehat{C}\widehat{\left[\frac{1}{V}
\right]}^{-1/{2}}=-\frac{6}{\gamma^{2}}
\widehat{\Omega}^2+8\pi
G\hat{P}_{\phi}^2,
\end{gather*}
since the operators $\widehat{\Omega}^2$ and $\hat{P}_{\phi}^2=-\hbar^2\partial_\phi^2$ become Dirac
observables that commute with the densitized constraint operator $\widehat{{\cal C}}$.
Note that, if we had not decoupled the zero-volume states, zero would be in the discrete
spectrum of $\widehat{[1/V]}$ and the operator $\widehat{\left[1/{V}\right]}^{-1/2}$
(obtained via spectral theorem) would be ill def\/ined. Nonetheless, in
$\widetilde{\mathcal{H}}_{\textrm{grav}}$ (with domain $\widetilde{\text{Cyl}}_\text{S}$) it is
well def\/ined.
Both the densitized and original constraints are equivalent, inasmuch as their solutions are
bijectively related~\cite{mmo}.
\subsection{Analysis of the Hamiltonian constraint operator}
\label{3sec:grav}
With the aim of diagonalizing the Hamiltonian constraint operator $\widehat{{\cal C}}$, let us characterize the spectral properties of the
operators entering its def\/inition. As it is well known, the operator $\hat{P}_{\phi}^2=-\hbar^2\partial_\phi^2$ is
essentially self-adjoint in its domain $\mathcal{S}(\mathbb{R})$, with double degenerate absolutely
continuous spectrum, its generalized eigenfunctions of eigenvalue~$(\hbar\nu)^2$ being the plane waves~$e^{\pm i|\nu|\phi}$. The gravitational operator $\widehat\Omega^2$
is more complicated and we analyze it in detail in the following.
\subsubsection{Superselection sectors}
The action of $\widehat\Omega^2$ on the basis states $|v\rangle$ of the kinematical sector
$\widetilde{\mathcal{H}}_{\textrm{grav}}$ is
\begin{gather*
\widehat\Omega^2|v\rangle =-f_+(v)f_+(v+2)|v+4\rangle+
\left[f_+^2(v)+f_-^2(v)\right]|v\rangle-f_-(v)f_-(v-2)
|v-4\rangle,
\end{gather*}
where
\begin{gather*
f_\pm(v)=\frac{\pi\gamma l_{\textrm{Pl}}^2}{2}
\sqrt{|v\pm2|}\sqrt{|v|}s_\pm(v),\qquad
s_\pm(v)=\text{sgn}(v\pm2)+\text{sgn}(v),
\end{gather*}
so that $\widehat\Omega^2$ is a dif\/ference operator of step four.
In addition, note that $f_-(v)f_-(v-2)=0$ if $v\in(0,4]$ and $f_+(v)f_+(v+2)=0$ if $v\in[-4,0)$.
In consequence, the operator~$\widehat\Omega^2$ only relates states $|v\rangle$ with support in a
particular semilattice of step four of the form
\begin{gather*
{\mathcal L}_{\varepsilon}^\pm=\{v=\pm(\varepsilon+4n),
\,n\in\mathbb{N}\},\qquad \varepsilon\in(0,4].
\end{gather*}
Then, $\widehat\Omega^2$ is well def\/ined in any of the Hilbert subspaces $\mathcal
H^\pm_{\varepsilon}$ obtained as the closure of the respective domains
$\text{Cyl}_{\varepsilon}^{\pm}=\text{lin}\{|v\rangle,\,v\in{\mathcal L}_{\varepsilon}^\pm\}$,
with respect to the discrete inner product. The non-separable kinematical Hilbert space
$\widetilde{\mathcal H}_{\text{grav}}$ can be thus written as a direct sum of separable subspaces
$\widetilde{\mathcal H}_{\text{grav}}=\oplus_{\varepsilon}(\mathcal H^+_{\varepsilon}\oplus\mathcal
H^-_{\varepsilon})$.
The action of the Hamiltonian constraint (and that of the physical observables, as we will see)
preserves the spaces
$\mathcal H^\pm_{\varepsilon}\otimes L^2(\mathbb{R},d\phi)$, which then provide
superselection sectors. Therefore, we can restrict the analysis to any of them, e.g., to
$\mathcal H^+_{\varepsilon}\otimes L^2(\mathbb{R},d\phi)$, for an arbitrary value of
$\varepsilon\in(0,4]$.
The fact that the gravitational part of the Hamiltonian constraint is a dif\/ference
operator is due to the discreteness of the geometry representation, and therefore it is a generic
feature of the theory. Actually, the dif\/ferent factor orderings analyzed within the improved
dynamics prescription (e.g., \cite{acs,aps3,mmo}) display superselection sectors having support in
lattices of step four. The dif\/ference between the superselection sectors considered here \cite{mmo}
and those of \cite{acs, aps3} is that the formers have support contained in a semiaxis of the
real line, whereas the support of the latters is contained in the whole real line.
\subsubsection{Self-adjointness and spectral properties}
Though the gravitational part of the Hamiltonian constraint operator is not a usual dif\/ferential
operator but a dif\/ference operator, there exists a rigorous proof showing that
it is essentially self-adjoint \cite{kale}. Here we sketch that proof for the operator that we are
considering, $\widehat\Omega^2$, but indeed the proof can be extended for the dif\/ferent
orderings explored in the literature (e.g., \cite{acs, aps3})\footnote{$\widehat\Omega^2$ is
analog to the operator $\Theta$ of \cite{aps3}.}.
In \cite{kale} the authors def\/ine certain operator $\widehat{H}'_\text{APS}$,\footnote{The acronym ``APS'' refers to the
model of \cite{aps3} by Ashtekar, Paw{\l}owski and Singh.} which is a dif\/ference operator of step four, and they show that
$\widehat{H}'_\text{APS}$ is unitarily related, through a Fourier transformation, to the Hamiltonian of a
point particle in a one-dimensional P\"oschl--Teller potential, which is a well-known
dif\/ferential operator. In particular, it is essentially self-adjoint, and then so is
$\widehat H'_\text{APS}$ as well.
In our notation, $\widehat{H}'_\text{APS}$ is def\/ined on the Hilbert spaces:
\begin{itemize}\itemsep=0pt
\item $\mathcal H^+_{\varepsilon}\oplus\mathcal H^-_{4-\varepsilon}$, with domain
$\text{Cyl}_{\varepsilon}^{+}\cup
\text{Cyl}_{4-\varepsilon}^{-},\text{ if }\varepsilon\neq4$;
\item $\mathcal H^+_{4}\oplus\mathcal H^-_{4}\oplus\mathcal H_0$, ($\mathcal H_0$ being the
one-dimensional Hilbert space generated by
$|v=0\rangle$), with domain
$\text{Cyl}_{4}^{+}\cup\text{Cyl}_{-4}^{-}\cup\text{lin}\{|0\rangle\},
\text{ if}$ $\varepsilon=4$.
\end{itemize}
Now, one can show that $\widehat\Omega^2$ and $[4/(3 \pi G)]\widehat{H}'_\text{APS}$ (def\/ined on the same Hilbert space) dif\/fer in a~trace class symmetric operator \cite{mmo,mop}. Then, a theorem by Kato and Rellich \cite{kato} ensures that $\widehat\Omega^2$, def\/ined in the same Hilbert space as $\widehat{H}'_\text{APS}$, is essentially
self-adjoint. From this result, it is not dif\/f\/icult to prove also that the restriction of $\widehat\Omega^2$
to $\mathcal H^+_{\varepsilon}$ (the subspace where we have restricted the analysis) is also essentially self-adjoint \cite{mmo}, just by analyzing its def\/iciency index equation \cite{functional1}.
On the other hand, it was shown in \cite{kale} that the essential and the absolutely
continuous spectra of the operator $H'_\text{APS}$ are both $[0,\infty)$. Once again,
Kato's perturbation theory \cite{kato} allows one to extend these results to the operator
$\widehat\Omega^2$ def\/ined in $\mathcal H^+_{\varepsilon}\oplus
\mathcal H^-_{4-\varepsilon}$. In addition, taking into account the symmetry of
$\widehat\Omega^2$ under a f\/lip of sign in $v$ and assuming the
independence of the spectrum from the label $\varepsilon$, we conclude that the essential and
absolutely continuous spectra of~$\widehat\Omega^2$ def\/ined in $\mathcal H^+_{\varepsilon}$
are $[0,\infty)$ as well. Besides, as we will see in next subsection, the (generalized)
eigenfunctions of $\widehat\Omega^2$ converge for large~$v$ to eigenfunctions of the
WDW counterpart of the operator. This fact, together with the continuity of the spectrum in
geometrodynamics, suf\/f\/ices to conclude that the discrete and singular spectra are empty.
In summary, the operator $\widehat\Omega^2$ def\/ined on~$\mathcal H^+_{\varepsilon}$
is a positive and essentially self-adjoint operator, whose spectrum is absolutely continuous and
given by $\mathbb{R}^+$.
\subsubsection{Generalized eigenfunctions}
Let us denote by $|e^{\varepsilon}_{\lambda}\rangle=\sum_{v\in{\mathcal L}^+_{\varepsilon}}
e^{\varepsilon}_{\lambda}(v)|v\rangle$ the generalized eigenstates of $\widehat\Omega^2$, corresponding to the eigenvalue
(in generalized sense) $\lambda\in[0,\infty)$.
The analysis of the eigenvalue equation
$\widehat\Omega^2|e^{\varepsilon}_{\lambda}\rangle=\lambda|e^{\varepsilon}_{\lambda}
\rangle$ shows that the initial datum $e^{\varepsilon}_{\lambda}(\varepsilon)$ completely determines the rest of
eigenfunction coef\/f\/icients $e^{\varepsilon}_{\lambda}(\varepsilon+4n)$, $n\in \mathbb{N}^+$ \cite{mmo}. Therefore, the spectrum of $\widehat\Omega^2$, besides being positive and absolutely continuous, is also non-degenerate.
We choose a basis of states $|e^{\varepsilon}_{\lambda}\rangle$ normalized to the
Dirac delta such that $\langle e^{\varepsilon}_{\lambda}|e^{\varepsilon}_{\lambda'}\rangle=
\delta(\lambda-\lambda')$.
This condition f\/ixes the complex norm of $e^{\varepsilon}_{\lambda}(\varepsilon)$. The only remaining freedom in the choice of this initial datum is then its phase, that we f\/ix by taking $e^{\varepsilon}_{\lambda}(\varepsilon)$
positive. The generalized eigenfunctions that form the basis are then real, a consequence of the
fact that the dif\/ference operator $\widehat{\Omega}^2$ has real coef\/f\/icients. In short, the
spectral resolution of the identity in the kinematical Hilbert space $\mathcal
H^+_{\varepsilon}$ associated with $\widehat{\Omega}^2$ can be expressed as
\begin{gather*
\mathbbm{1}=\int_{\mathbb{R^+}} d\lambda
|e_{\lambda}^{\varepsilon}\rangle \langle
e_{\lambda}^{\varepsilon}|.
\end{gather*}
The behavior of the eigenfunctions $e^{\varepsilon}_{\lambda}(\varepsilon)$ in the limit $v\rightarrow\infty$ allows us to understand the relation between the quantization of the model within LQC and that of the standard WDW theory, where a Schr\"odinger-like representation is employed in the geometry sector, instead of polymeric. Let us study this limit.
In the WDW theory the analog to the operator $\widehat\Omega^2$ is simply given by \cite{mmo}
\begin{gather*}
\widehat{\underline\Omega}^2=-\frac{\alpha^2}{4}\left[1+4v\partial_{v}+
4(v\partial_v)^2\right],
\end{gather*}
where $\alpha=4\pi\gamma l_{\text{Pl}}^2$. $\widehat{\underline\Omega}^2$ is well def\/ined on the Hilbert space $L^2(\mathbb{R}^+,dv)$. Moreover, it is essentially self-adjoint, and its spectrum is absolutely continuous with double degeneracy.
The generalized eigenfunctions corresponding to the eigenvalue $\lambda\in[0,\infty)$ will be labeled with
$\omega=\pm\sqrt{\lambda}\in\mathbb{R}$ and are given by
\begin{gather}\label{eq:wdw-eig-mmo}
\underline{e}_{\omega}(v)= \frac{1}{\sqrt{2\pi\alpha
|v|}}\exp\left({-i\omega\frac{\ln{|v|}}{\alpha}}\right).
\end{gather}
These eigenfunctions provide an orthogonal basis (in a generalized sense) for $L^2(\mathbb{R}^+,dv)$, with normalization
$\langle \underline{e}_{\omega} |\underline{e}_{\omega^{\prime}}\rangle=\delta(\omega-\omega^{\prime})$.
Using the results of \cite{kp-posL}, one can show that the loop basis eigenfunctions $e^{\varepsilon}_{\lambda}(v)$ converge for large $v$ to an eigenfunction of the WDW analog $\widehat{\underline\Omega}^2$. The WDW limit is explicitly given by~\cite{mmo}
\begin{gather*
e^{{\varepsilon}}_{\lambda}(v) \xrightarrow{v\gg 1} r\big\{
\exp\left[{i\phi_{\varepsilon}(\omega)}\right]
\,\underline{e}_{\omega}(v) +
\exp\left[{-i\phi_{\varepsilon}(\omega)}\right]
\,\underline{e}_{-\omega}(v) \big\},
\end{gather*}
where $r$ is a normalization factor. In turn, the phase $\phi_{\varepsilon}(\omega)$ behaves as \cite{kp-posL, mop}
\begin{gather*
\phi_{\varepsilon}(\omega) = T(|\omega|) +
c_{\varepsilon} +
R_{\varepsilon}(|\omega|),
\end{gather*}
where $T$ is a certain function of $|\omega|$, $c_{\varepsilon}$ is a constant, and
$\lim\limits_{\omega\rightarrow0}R_{\varepsilon}(|\omega|)=0$.
\subsection{Physical structure}
\label{1sec:phys}
\subsubsection{Physical Hilbert space}
We are now in a position to complete the quantization of the model. In order to do that, we can
follow two alternative strategies:
\begin{itemize}\itemsep=0pt
\item We can apply the group averaging procedure \cite{gave2,gave1d,gave1c,gave1b, gave1a}.
The physical states are the states invariant under the action of the group generated by the
self-adjoint extension of the constraint operator, and we can obtain them by averaging over that
group. In addition, this averaging determines a natural inner product
that endows the physical states with a~Hilbert structure.
\item We can solve the constraint in the space
$\big(\widetilde{\text{Cyl}}_\text{S}\otimes\mathcal{S}(\mathbb{R})\big)^*$,
dual to the domain of def\/inition of the Hamiltonian constraint operator\footnote{We do not expect the solutions of the constraint to live in the kinematical Hilbert space $\mathcal H^\pm_{\varepsilon}\otimes L^2(\mathbb{R},d\phi)$, which is quite restricted, but rather in the larger space
\[\big(\widetilde{\text{Cyl}}_\text{S}\otimes\mathcal{S}(\mathbb{R})\big)^*\supset\mathcal H^\pm_{\varepsilon}\otimes L^2(\mathbb{R},d\phi)\supset\widetilde{\text{Cyl}}_\text{S}\otimes\mathcal{S}(\mathbb{R}) .
\]}. Namely, we can look for the elements $(\psi|\in\big(\widetilde{\text{Cyl}}_\text{S}\otimes\mathcal{S}(\mathbb{R})\big)^*$ that verify
$(\psi|\widehat{\mathcal C}^\dagger =0$.
Then, in order to endow them with a Hilbert space structure, we
can impose self-adjointness in a complete set of observables. This determines the physical inner product~\cite{red2, red1}.
\end{itemize}
Both methods give the same result (up to unitary equivalence): the physical solutions are given by\footnote{See, e.g., \cite{aps3} for the application of the group averaging method, or~\cite{mmo} as an example of the second method.}
\begin{gather}\label{ga3}
\Psi(v,\phi) =\int_0^\infty d\lambda\,
e^{\varepsilon}_\lambda(v)\big[\tilde\psi_+(\lambda)e^{i\nu(\lambda)\phi}
+\tilde\psi_-(\lambda)e^ {
-i\nu(\lambda)\phi }
\big],
\end{gather}
where
\begin{gather*}
\nu(\lambda) :=\sqrt{\frac{3\lambda}{4\pi l_\text{Pl}^2\hbar\gamma^2}}.
\end{gather*}
In addition, the physical inner product is
\begin{gather*}
\langle \Psi_1|\Psi_2 \rangle_{\text{phys}}= \int_0^\infty d\lambda\,\big[
\tilde\psi_{1+}^*(\lambda)\tilde\psi_{2+}(\lambda)+\tilde\psi_{1-}^*
(\lambda)\tilde\psi_{2-}(\lambda)\big ].
\end{gather*}
Therefore, the physical Hilbert space, where the spectral prof\/iles $\tilde\psi_\pm(\lambda)$ live,
is
\begin{gather*}
\mathcal
H^{\varepsilon}_{\text{phys}}=L^2\left(\mathbb{R}^+,d\lambda\right).
\end{gather*}
\subsubsection{Evolution picture and physical observables}
\label{1subsec:evol}
In any gravitational system, as the one considered here, the Hamiltonian is a linear combination of
constraints, and thus it vanishes. In other words, the time coordinate of the metric is not a~physical time, and provides a~notion of ``frozen'' evolution, unlike what happens in theories, such as usual QFT, in
which the metric is a static background structure. With the aim of
interpreting the results in a time evolution picture, we need to def\/ine what this concept of evolution is.
To do that, we choose a suitable variable or a function of the phase space, and regard it as
internal time \cite{kuchar2}.
In the model that we are describing, it is natural to choose $\phi$ as the physical time. In this way, we
can regard the Hamiltonian constraint as an evolution equation $\phi$. In turn, $\nu$ plays
the role of frequency associated to that time.
As we see in equation~\eqref{ga3}, the solutions to the constraint can be decomposed in
positive and negative frequency components
\begin{gather*
\Psi_\pm(v,\phi) =\int_0^\infty d\lambda\,
e^{\varepsilon}_\lambda(v)\tilde\psi_\pm(\lambda)e^{\pm i\nu(\lambda)\phi},
\end{gather*}
that, moreover, are determined by the initial data $\Psi_\pm(v,\phi_0)$ via the unitary evolution
\begin{subequations}
\label{eq:evol}
\begin{gather}
\Psi_\pm(v,\phi) = U_\pm(\phi-\phi_0)\Psi_\pm(v,\phi_0) ,\\
U_\pm(\phi-\phi_0) = \exp{\left[\pm
i\sqrt{\frac{3}{4\pi l_\text{Pl}^2\hbar\gamma^2} \widehat{\Omega}^2}(\phi-\phi_0)
\right]}.
\end{gather}
\end{subequations}
This allows us to def\/ine Dirac observables ``in evolution'', namely relational observables \cite{bianca-obs1,bianca-obs2, rovelli-obs}, and in turn, to interpret the physical results. Let us note f\/irst that,
in the classical theory, although~$v$ is not a constant of motion, $v(\phi)$ turns out to be a single-valued function of~$\phi$ in
each dynamical trajectory~\cite{aps3}, and then $v|_{\phi=\phi_0}$ is a well-def\/ined observable
for each f\/ixed value~$\phi_0$. It measures the volume at time $\phi_0$. The quantum analogue of that observable is the operator
\begin{gather*}
\widehat{v}|_{\phi_0} \Psi(v,\phi)= U_+(\phi-\phi_0)v \Psi_+(v,\phi_0)+
U_-(\phi-\phi_0)v\Psi_-(v,\phi_0).
\end{gather*}
We see that, given a physical solution $\Psi(v,\phi)$, the action of this operator consists in:
\begin{enumerate}\itemsep=0pt
\item [i)] decomposing the solution in its positive and negative frequency components,
\item [ii)] freezing them at the initial time $\phi=\phi_0$,
\item [iii)] multiplying its initial datum by $v$, and
\item [iv)] evolving through equation~\eqref{eq:evol}.
\end{enumerate}
The result is again a physical solution, and then the operator $\widehat{v}|_{\phi_0}$ constructed in this way is indeed a Dirac observable.
Then, the constant of motion $\hat P_\phi=-i\hbar\partial_\phi$
and the operator $\widehat{v}|_{\phi}$ form a complete set of Dirac (and then physical) observables.
Note that both the physical observables and the physical inner product preserve not only the superselection sectors,
but also the subspaces of positive and negative frequency. Therefore, any of these subspaces provide an irreducible
representation of the observables algebra, and the analysis can be restricted, for instance, to the positive frequency sector.
The operator $\widehat{v}|_{\phi}$ allows to analyze the physical results in evolution.
Namely, one can compute the expectation value of that observable on physical states at dif\/ferent
times. We will carry out that analysis in the next section for semiclassical
states, and see graphically the occurrence of the quantum bounce.
\subsection{Dynamical singularity resolution: quantum bounce}
\label{sec:bounce}
In the classical theory, when the volume of the universe vanishes, the energy density diverges, leading to a big bang singularity.
Now, in the quantum theory, the structure of the superselection sectors, and more specif\/ically the form of the eigenfunctions
$e^{\varepsilon}_\lambda(v)$, ensures that the classical big bang singularity is replaced by a quantum bounce.
Actually, this result is a consequence of the following properties:
\begin{itemize}\itemsep=0pt
\item {Exact standing-wave behavior}:
As we have seen, the eigenfunctions $e^{\varepsilon}_\lambda(v)$ converge in the large $v$ limit to a combination of two eigenfunctions of
the WDW theory. These eigenfunctions, given in equation~\eqref{eq:wdw-eig-mmo}, contract and expand in $v$, respectively, and can be interpreted as
incoming and outgoing waves. These components contribute with the same amplitude to the limit, and in this sense the limit is an exact standing-wave.
\item {No-boundary description}: On the other hand, the eigenfunctions $e^{\varepsilon}_\lambda(v)$ have support in a single semiaxis that, moreover, does not contain the
putative singularity $v=0$. This feature is due just to the functional properties of the gravitational operator $\widehat{\Omega}^2$, and
not derived from imposing any particular boundary condition. In that sense, the eigenfunctions verify a no-boundary description\footnote{In quantum cosmology, the concept of no-boundary has been employed in a dif\/ferent sense of the one discussed here \cite{haw2,haw1,haw3}.}.
\end{itemize}
These features imply that the incoming component must evolve into the outgoing one, and vice versa, since the f\/lux cannot escape through $v=0$.
Therefore, in the physical solution~\eqref{ga3}, restricted for instance to the positive frequency sector, the expanding and contracting components
must lead to two branches of a universe, one in expansion and one in contraction, that meet at some positive expectation value of
$\hat{v}|_{\phi_0}$ forming a quantum bounce. This result is then independent of the considered physical prof\/ile $\tilde\psi_+(\lambda)$.
For semiclassical states in the region of large $v$, the expectation value of $\hat{v}|_{\phi_0}$ is
peaked on trajectories that show the replacement of the classical big bang by a \emph{big bounce},
as depicted in Fig.~\ref{fig:bounce}. This example corresponds to a physical prof\/ile given by a
logarithmic normal distribution of the type
\begin{gather}\label{profile}
\tilde\psi_+(\lambda)=\frac1{(2\pi)^{1/4}\sqrt{\sigma\lambda}} e^{-[\ln(\lambda/\lambda_o)]^2/(4\sigma^2)}
\end{gather}
(as the ones considered in \cite{mop}), where the parameters $\lambda_o$ and $\sigma$ are related to the expectation value of
$\hat P_\phi$ and to its dispersion $\Delta\hat P_\phi$ by the relations~\cite{mop}
\begin{gather*}
\langle \hat P_\phi \rangle=\sqrt{12\pi G}\lambda_o e^{\sigma^2/2},\qquad \frac{\Delta\hat
P_\phi}{\langle \hat P_\phi \rangle}=\sqrt{e^{\sigma^2}-1} .
\end{gather*}
Around the bounce point, the expectation values approach the classical value very fast, so that the
semiclassical limit of the quantum theory agrees with general relativity, as desired. Furthermore,
it has been proven for quite a general class of states that semiclassicality is preserved
through the bounce~\cite{cs2, kp-posL}.
Another analytic result holds independently of the choice of
state, and can be illustrated in the $b$ representation~\cite{acs}. To this
purpose, we choose already classically the densitized Hamiltonian constraint, i.e., the total Hamiltonian such that the lapse function is equal to the volume, $N=\sqrt{|\det(E)|}=V$. Thus, one avoids the need to rewrite inverse powers of the volume in terms of Poisson brackets:
\begin{gather}
C = -\frac{1}{\gamma^2}{E_i^{\alpha}E_j^\gamma}\epsilon^{ij}_{\ \ k}F^k_{\alpha\gamma}+ V C_{\rm
mat} = -24(\pi G)^2\lim_{A_{\square}\to 0}\frac{\bar\mu^2}{A_{\square}}v^2\sin^22 b +V C_{\rm
mat} .\label{SH3cls}
\end{gather}
At the quantum level, one should choose an operator ordering for the Hamiltonian costraint. Dif\/ferent orderings correspond to inequivalent def\/initions of the theory but they may lead anyway to very similar physics. In the absence of a guiding principle selecting one particular ordering over the others, one can make a choice convenient for calculational purposes. As an example illustrating this point, after regularizing equation~\eqref{SH3cls} ($A_\square\to \bar\mu^2$), choosing the superselection sector $v=4n$, and quantizing in the $b$ representation ($v\to \hat V=-i\hbar\partial_b$), the operator ordering can be arranged so that
\begin{gather*}
\left[3\pi l_\text{Pl}^2(\sin2b\partial_b)^2-\partial_\phi^2\right]\Psi[b,\phi]=0
\end{gather*}
Because one has a discrete one-dimensional lattice in $v$ space and the Fourier transform in $b$ space has support on the interval
$b\in(0,\pi/2)$ \cite{acs}, one can def\/ine
\begin{gather*
z:= \frac{1}{\sqrt{12\pi l_\text{Pl}^2}} \ln\tan b ,
\end{gather*}
so that we get
\begin{gather*
\left(\partial_z^2-\partial^2_\phi\right)\Psi[z,\phi]=0 .
\end{gather*}
This expression is formally identical to the Wheeler--DeWitt equation and the ensuing quantization follows step by step \cite{acs}.
A key dif\/ference, however, is that invariance of the wavefunction under parity (frame re-orientation) is not gauge-f\/ixed \emph{ab initio}
and physical states are required to satisfy $\Psi_+[-z,\phi]=-\Psi_+[z,\phi]$. It follows that the left- and right-moving sectors are not
superselected and must be considered together. In particular, we can write
\begin{gather*}
\Psi_+[z,\phi]=\Psi_{+,{\rm L}}[z_+]+\Psi_{+,{\rm R}}[z_-]=\xi(z_+)-\xi(z_-) ,
\end{gather*}
where $z_\pm=z\pm\phi$ and $\xi$ is some function.
This fact is crucial for the resolution of the big bang singularity. The volume operator in the $z$ variable is
\begin{gather*
\hat V=-i v_*\cosh(\kappa_0 z) \partial_z ,
\end{gather*}
where $v_*$ is a positive constant and $\kappa_0=\sqrt{12\pi l_\text{Pl}^2}$. At any time $\phi$ and on any physical state, one can show that
the expectation value of the volume is
\begin{gather}
\langle |\hat V|\rangle = \langle\Psi_{+}|\,|\hat V|\,|\Psi_{+}\rangle= V_*\cosh(\kappa_0\phi) ,\label{finalbounce}
\end{gather}
where $V_*>0$ is the minimal volume at the bounce. Equation~\eqref{finalbounce} completes the proof that the big bang singularity is avoided in
minisuperspace LQC. Further evidence comes from noticing that matter energy density has an absolute upper bound (approximately equal
to~$0.41$ times the Planck density) on the whole physical Hilbert space~\cite{acs}. We can reach the same quantitative conclusion, albeit not as
robustly, when looking at the ef\/fective dynamics on semiclassical states (Section~\ref{hodi}).
\begin{figure}[t]
\centering
\includegraphics[width=10cm]{Calcagni-Fig1}
\caption{Expectation values and dispersions of $\hat{v}|_{\phi}$ (in red, vertical bars) in the superselection
sector with $\varepsilon=1$, corresponding to the
physical prof\/ile given in equation~\eqref{profile} with $\langle \hat P_\phi \rangle=1000$ and
${\Delta\hat P_\phi}/{\langle \hat P_\phi \rangle}=0.1$. The quantum evolution is also compared with
the classical trajectories, one
in expansion (blue curve, increasing from left to right) and the other in contraction (green curve, decreasing). Graph by courtesy of J.~Olmedo.}\label{fig:bounce}
\end{figure}
\subsection{FRW models with curvature or cosmological constant}\label{klam}
In the previous sections, we ignored the contribution both of the intrinsic curvature $\Gamma_a^i=(\textsc{k}/2)\delta_a^i$ and of a
cosmological constant $\Lambda$. Here, we sketch scenarios where the universe is not f\/lat ($\textsc{k}=\pm 1$) and/or $\Lambda\neq 0$.
For more details, consult \cite{AsS}.
\subsubsection{Closed universe}
The case of a universe with positive-def\/inite spin connection, $\textsc{k}=1$, was studied in \cite{APSV,BoT,BoVa,LMNT,MHS,SiT,ViS,SKL}.
Due to the extra term in the connection, the form of the classical Hamiltonian constraint~\eqref{eq:ligFRW} as a function of $c$
(related to metric variables as $c=\gamma \dot a+\textsc{k}$, a~dot denotes derivative with respect to synchronous time) is modif\/ied by the
replacement $c^2\to c(c-V_o^{1/3})+(1+\gamma^2)V_o^{2/3}/4$. In the classical Friedmann equation, this replacement corresponds to
$H^2\to H^2+\textsc{k}/a^2$ with $\textsc{k}=1$, where $H:=\dot a/a$ is the Hubble parameter. The quantum constraint and the resulting dif\/ference
equation are modif\/ied accordingly. There is no arbitrariness in the f\/iducial volume~$V_o$, since it can be identif\/ied with the total volume of the
universe, which is f\/inite and well def\/ined. Then, the choice of elementary holonomy is more natural than in the f\/lat case and, locally, one can
distinguish between the group structure of~$SU(2)$ and $SO(3)$~\cite{SKL}. As in the f\/lat case, the constraint operator is essentially self-adjoint~\cite{SKL} and the singularity at~$v=0$ is removed from the quantum evolution~\cite{APSV, BoVa,SKL}. However, instead of a single-bounce event one now
has a cyclic model~\cite{APSV}. This can be traced back to the fact that the classical and quantum scalar constraint have both contracting and
expanding branches coexisting in closed-universe solutions, while these branches correspond to distinct solutions in the f\/lat case.
\subsubsection{Open universe}
Loop quantum cosmology of an open universe \cite{ViS,Szu07,van06} is slightly more delicate to deal with. In contrast with the f\/lat and closed
cases $\textsc{k}=0,1$, the spin connection is non-diagonal, so that also the connection is non-diagonal and it has two (rather than one) dynamical
components~$c(t)$ and~$c_2(t)$. The Gauss constraint f\/ixes $c_2=1$ and one ends up with the same number of degrees of freedom as usual. The volume
of the universe is inf\/inite as in the f\/lat case, and a~f\/iducial volume must be def\/ined. The classical Hamiltonian constraint is equation~\eqref{eq:ligFRW}
with $c^2\to c^2-V_o^{2/3}\gamma^2$. The quantum constraint is constructed after def\/ining a suitable holonomy loop; the bounce still takes place and the
$v=0$ big-bang state factors out of the dynamics.
\subsubsection{$\Lambda\neq 0$}
Another generalization is to add a cosmological constant term, positive \cite{BoVa,KaP,MHS} or nega\-tive~\cite{BeP,BoT,kale,Szu07}. At the level of the dif\/ference equation, these models have been studied in relation to the self-adjoint property.
For $\Lambda>0$, below a critical value $\Lambda_*$ (of order of the Planck energy), the Hamiltonian constraint operator admits many
self-adjoint extensions, each with a discrete spectrum. Above $\Lambda_*$, the operator is essentially self-adjoint but there are no physically
interesting states in the Hilbert space of the model~\cite{KaP}.
For $\Lambda<0$, the scalar constraint is essentially self-adjoint and its spectrum is discrete \cite{kale} (while, we recall, for
$\Lambda=0$ it is continuous and with support on the positive real line), also when $\textsc{k}=-1$ \cite{Szu07}.
As in the $\Lambda=0$, $\textsc{k}=1$ case, the universe undergoes cycles of bounces \cite{BeP}.
\section{Bianchi I model}
\label{chap:bianchi}
The next step in extending loop quantum cosmology to more general situations consists in the consideration of (still homogeneous but) anisotropic cosmologies. The simplest anisotropic spacetime is the Bianchi I model, since it has f\/lat spatial sections.
This model has been extensively studied, owing to its simplicity and applications in cosmology. In fact, prior to the development of loop quantum cosmology, its quantization employing Ashtekar variables was already analyzed \cite{aspu,neg1,neg2}. The f\/irst attempts of constructing a kinematical Hilbert space and the Hamiltonian constraint operator within a polymeric formalism were done in \cite{boj}. Then, soon after the quantization of the f\/lat FRW model was completed within the improved dynamics scheme \cite{aps3}, the same programme was applied to Bianchi~I, which we shall review now.
\subsection{Classical formulation in Ashtekar--Barbero variables}
\label{4sec:clas}
For simplicity, we will consider the model in vacuo. Unlike the FRW universe, which is static in vacuo,
the vacuum Bianchi I model has non-trivial dynamics. Its solutions are of Kasner type~\cite{kasner},
with two expanding scale factors and the third in contraction, or vice versa.
Moreover, for later convenience, we will consider a spatial three-torus topology. Therefore, it will
not be necessary to introduce any f\/iducial cell, since the model already provides a natural f\/inite
cell, that of the three-torus, described with angular coordinates $\{\theta,\sigma,\delta\}$
running from~0 to~$2\pi$.
Like in the isotropic case, we f\/ix the gauge and choose a diagonal f\/lat co-triad ${}^o
e_a^i=\delta_a^i$. The presence of three dif\/ferent directions requires three variables to describe
the Ashtekar--Barbero connection and three more for the densitized triad, that is\footnote{In the following, we will not use the Einstein summation convention, unless specif\/ied otherwise.}
\begin{equation*
A_a^i=\frac{c^{i}}{2\pi}\delta_a^i, \qquad
E_i^a=\frac{p_{i}}{4\pi^2}\delta_i^a\sqrt{{}^oq},\qquad i=\theta,\sigma,\delta.
\end{equation*}
The Poisson brackets def\/ining the phase space are then $\{c^i,p_j\}=8\pi G\gamma\delta^i_j$. The spacetime metric in these variables reads
\begin{equation*}
ds^2= -N^2
dt^2+\frac{|p_\theta p_\sigma p_\delta|}{4\pi^2}\left(
\frac{d\theta^2}{p_\theta^2}+\frac{d\sigma^2}{p_\sigma^2}+\frac{d\delta^2}{p_\delta^2}\right).
\end{equation*}
In turn, the phase space is constrained by the Hamiltonian constraint
\begin{gather}\label{eq:ligBianchi}
C_\text{BI}=-\frac2{\gamma^2}\frac{c^\theta p_\theta c^\sigma p_\sigma+c^\theta p_\theta c^\delta
p_\delta+c^\sigma p_\sigma c^\delta p_\delta}{V}=0.
\end{gather}
In this expression, $V=\sqrt{|p_\theta p_\sigma p_\delta|}$ is the physical volume of the universe.
\subsection{Quantum representation}
In order to polymerically represent this system, we follow the approach described in
Section~\ref{1sec:kin}~\cite{chio}. Holonomies
$
h_i^{\mu_i}(c^i)=e^{\mu_{i}c^{i}\tau_{i}}
$
are def\/ined along straight edges of f\/iducial length
$2\pi\mu_i\in\mathbb{R}$ and oriented in the f\/iducial directions, here labeled by
$i=\theta,\sigma,\delta$. The f\/luxes of the densitized triad through rectangular surfaces of f\/iducial area
$A_{\square}^i$ and orthogonal to the $i$-th direction, given by
$
E(A_{\square}^i,f=1)=[p_{i}/(4\pi^2)]A_{\square}^i
$, complete the description of the phase space before quantization.
The conf\/iguration algebra is the tensor product of the algebras of quasi-periodic functions of the connection for each f\/iducial direction:
$
\text{Cyl}_\text{S}=\otimes_i\text{Cyl}_\text{S}^i=\text{lin}\{|\mu_\theta,\mu_\sigma,
\mu_\delta\rangle\}
$,
where the kets $|\mu_i\rangle$ denote the quantum states corresponding to the matrix elements of the holonomies
$\mathcal N_{\mu_i}(c^i)=e^{i\mu_{i}c^{i}/2}$ in momentum representation. Hence, the kinematical Hilbert space is the tensor product
$
\mathcal H_{\text{grav}}=\otimes_i\mathcal H_{\text{grav}}^i
$,
where $\mathcal H_{\text{grav}}^i$ is the Cauchy completion of $\text{Cyl}_\text{S}^i$ with respect
to the discrete inner product
$\langle\mu_i|\mu_i^\prime\rangle=\delta_{\mu_i\mu_i^\prime}$.
The basic operators are $\hat p_i$ and
$\hat{\mathcal N}_{\mu_i^\prime}$. Their action on the basis states
$|\mu_i\rangle$ is
\begin{gather*
\hat p_i|\mu_i\rangle =p_i(\mu_i)|\mu_i\rangle,\qquad p_i(\mu_i)=4\pi\gamma l_\text{Pl}^2\mu_i,\qquad
\hat{\mathcal N}_{\mu_i^\prime}|\mu_i\rangle =|\mu_i+\mu_i^\prime\rangle,
\end{gather*}
such that $[\hat{\mathcal N}_{\mu_i},\hat p_j]=i\hbar \widehat{\{\mathcal N_{\mu_i}(c^i),p_j\}}$.
\subsection{Improved dynamics}
\label{4sec:improved}
The most involved aspect that one encounters when trying to adapt the quantization of the isotropic
case to the anisotropic case lies in the implementation of the improved dynamics, explained in Section~\ref{1sec:lig-ham-aps}. In the presence of anisotropies, we need to introduce three minimum f\/iducial
lengths $\bar\mu_i$, when def\/ining the curvature tensor in terms of a loop of holonomies.
Originally, a naive \emph{Ansatz} was chosen, given by
\begin{gather}\label{eq:mubarraA}
{\frac1{\bar\mu_i'}}=\frac{\sqrt{|p_i|}}{\sqrt{\Delta}}.
\end{gather}
This is the simplest generalization of the ansatz of the isotropic case, equation~\eqref{mu}. As a~consequence, the operators entering the Hamiltonian constraint have the same form as those of the
f\/lat FRW model. Furthermore, operators corresponding to dif\/ferent directions commute among one
another. This allows to complete the quantization obtaining the physical Hilbert space~\cite{mmp}, in
the same way as for the FRW model. However, when the topology is non-compact and then a f\/inite
f\/iducial cell is introduced, the physical results depend on this f\/iducial choice~\cite{luc2}.
This drawback led to the revision of the def\/inition of~$\bar\mu_i$, and another \emph{Ansatz} free of these problems was proposed, this time given by%
\footnote{Whenever the three indices $i$, $j$, $k$ appear in the same expression, we will consider $\epsilon_{ijk}\neq0$, so that they are dif\/ferent.}
\begin{gather}\label{eq:mubarraB}
{\frac1{\bar\mu_i}}=\frac1{\sqrt{\Delta}}\sqrt{\left|\frac{p_j p_k}{p_i}\right|}.
\end{gather}
This choice is geometrically better justif\/ied (for a discussion about its derivation see~\cite{AsW}). Furthermore, this prescription is the only one verifying a remarkable property: For all
the f\/iducial directions, the exponents $\bar\mu_i c^i$ of the matrix elements $\mathcal
N_{\bar\mu_i}(c^i)$ have a constant and f\/ixed (up to a sign) Poisson bracket with the variable
\begin{gather}\label{v}
v=\text{sgn}(p_\theta p_\sigma p_\delta)\frac{\sqrt{|p_\theta p_\sigma p_\delta|}}{2\pi\gamma
l_\text{Pl}^2\sqrt{\Delta}} ,
\end{gather}
which is proportional to the volume. Note that it coincides with the parameter $v(p)$ of the isotropic case if we identify the three
f\/iducial directions. As a consequence, as we will see, the volume will suf\/fer constant shifts in the quantum theory, as in the isotropic
case. Thanks to this property, the improved dynamics prescription \eqref{eq:mubarraB} nicely implements the interplay between the anisotropies
and the volume. Instead, within the naive prescription given by equation~\eqref{eq:mubarraA}, there is no interplay between the degrees of freedom
associated with dif\/ferent directions, because of the commutation between the operators acting on dif\/ferent f\/iducial directions. Thus, apart
from giving dependencies on f\/iducial choices, it also seems less physically motivated.
Because of these reasons, today it is generally accepted that the more correct improved dynamics
prescription is equation~\eqref{eq:mubarraB}, which we shall consider in this paper.
\subsection{Hamiltonian constraint operator}
As in the isotropic case, in order to obtain the Hamiltonian constraint operator we cannot
represent directly its classical form \eqref{eq:ligBianchi}, but its expression in terms of
the curvature tensor. For homogeneous models with vanishing spin connection, as Bianchi I, this expression was given in
equation~\eqref{eq:lig-escalar-lqg-hom}.
For simplicity, we will densitize the Hamiltonian constraint classically, by simply multiplying it by the volume $V$.
In this way we avoid the appearance of inverse powers of the volume that make the quantum theory complicated.
In any case, as seen in the f\/lat FRW model, the densitization could be carried out with no problem in the quantum theory.
In analogy with the isotropic case, but now taking into consideration that the three f\/iducial directions are dif\/ferent,
the curvature operator is the quantum counterpart of the classical expression
\begin{gather}\label{eq:curvatura-bianchi}
{F}^i_{ab}=-2\sum_{j,k}
\text{tr}\left(
\frac{h^{\bar\mu}_{\square_{jk}}-\delta_{jk}}{4\pi^2\bar\mu_j\bar\mu_k}\tau^i\right)\delta^j_a
\delta^k_b,\qquad h^{\bar\mu}_{\square_{jk}}=h_j^{\bar\mu_j} h_k^{\bar\mu_k} (h_j^{\bar\mu_j})^{-1}
(h_k^{\bar\mu_k})^{-1}.
\end{gather}
Taking into account equation~\eqref{eq:curvatura-bianchi}, the expression of the densitized triad and the def\/inition of~$\bar\mu_i$, the densitized Hamiltonian constraint for the Bianchi~I model reads
\begin{gather}\label{eq:ligadura-bianchi-B}
C_\text{BI}=\frac{2}{\gamma^2\Delta}V^2\sum_{i,j,k}\epsilon^{ijk}\text{sgn}
(p_j)\, \text{sgn}(p_k)\,\text{tr}\big(\tau_i h^{\bar\mu}_{\square_{jk}}\big).
\end{gather}
In order to represent this constraint as an operator, we f\/irst need to def\/ine the operators~$\hat{\mathcal N}_{\bar\mu_i}$, which represent the matrix elements of the holonomies~$h_i^{\bar\mu_i}$. To def\/ine them we follow a~similar strategy to that adopted in the isotropic
case. We start by reparametrizing $p_i(\mu_i)$ with a~pa\-ra\-me\-ter~$\lambda_i(p_i)$ such that the
vectorial f\/ield $\bar\mu_i\partial_{\mu_i}$ produces constant translations in~$\lambda_i$.
The solution is
$\lambda_i(p_i)={\text{sgn}(p_i)\sqrt{|p_i|}}/{(4\pi\gamma
l_\text{Pl}^2\sqrt{\Delta})^{1/3}}$ \cite{AsW}.
The dif\/ference with respect to the isotropic case is that the translations produced by
$\bar\mu_i\partial_{\mu_i}$, being constant with respect to the dependence on~$\lambda_i$, do depend
on the parameters~$\lambda_j$ and $\lambda_k$ associated with the other two directions. In fact, we
have $\bar\mu_i\partial_{\mu_i}=(2|\lambda_j\lambda_k|)^{-1}\partial_{\lambda_i}$.
As in the isotropic case, we def\/ine the operator $\hat{\mathcal N}_{\bar\mu_i}$ such that its action on the basis states~$|\lambda_i\rangle$ is the same as the transformation generated by $\bar\mu_i\partial_{\mu_i}$ on the parameter~$\lambda_i$, that is
\begin{gather*
\hat{\mathcal
N}_{\pm\bar\mu_\theta}|\lambda_\theta,\lambda_\sigma,
\lambda_\delta\rangle=\bigg|\lambda_\theta\pm\frac1 {
2|\lambda_\sigma\lambda_\delta|},\lambda_\sigma,\lambda_\delta\bigg\rangle,
\end{gather*}
and similarly for $\hat{\mathcal N}_{\pm\bar\mu_\sigma}$ and $\hat{\mathcal N}_{\pm\bar\mu_\delta}$.
Moreover, inverting the change of variable, we obtain
\begin{gather*
\hat{p}_i|\lambda_\theta,\lambda_\sigma,\lambda_\delta\rangle=(4\pi\gamma
l_\text{Pl}^2\sqrt{\Delta})^{2/3}\text{sgn}(\lambda_i)\lambda_i^2|\lambda_\theta,\lambda_\sigma,
\lambda_\delta\rangle.
\end{gather*}
As explained in the f\/lat FRW case, one can always choose a suitable factor ordering for
the Hamiltonian constraint that allows to remove the kernel of the volume operator, generated by the
states with
$\lambda_\theta\lambda_\sigma\lambda_\delta=0$. This is what we will consider. Therefore, the
operators $\hat{\mathcal N}_{\bar\mu_i}$ are well def\/ined.
The action of $\hat{\mathcal N}_{\pm\bar\mu_i}$ can be slightly simplif\/ied by introducing the variable
$v$ def\/ined in equation~\eqref{v}, that in terms of $\lambda$'s is given by
$v=2\lambda_\theta\lambda_\sigma\lambda_\delta$.
Indeed, making the change from, e.g., the states
$|\lambda_\theta,\lambda_\sigma,\lambda_\delta\rangle$ to the states
$|v,\lambda_\sigma,\lambda_\delta\rangle$, one can check that, under the action of $\hat{\mathcal
N}_{\bar\mu_i}$, $v$~suf\/fers a constant shift equal to~$1$ or~$-1$ depending on the orientation of
the densitized triad coef\/f\/icients. On the other hand, the variables~$\lambda_\sigma$ and $\lambda_\delta$ suf\/fer a dilatation or contraction that only depends on their
own sign and on $v$ (see~\cite{AsW} for the details).
The variable $v$ is proportional to the volume
\begin{gather*}
\hat{V}=\widehat{\sqrt{|p_\theta p_\sigma p_\delta|}}, \qquad
\hat{V}|v,\lambda_\sigma,\lambda_\delta\rangle=2\pi\gamma
l_{\text{Pl}}^2\sqrt{\Delta}|v||v,\lambda_\sigma,\lambda_\delta\rangle.
\end{gather*}
Therefore, as happened in the isotropic case, in this scheme the volume undergoes constant
translations. The other two variables measure the degree of anisotropy of the system.
Once we know how to represent the matrix elements of the holonomies, we can promote the Hamiltonian
constraint to an operator. When symmetrizing it, we will adopt the prescription of~\cite{gow-B},
whose factor ordering is analog to that considered in Section~\ref{chap:1-flatFRW} (see
equation~\eqref{eq:operador-grav-mmo}). Explicitly, it is given by~\cite{gow-B,mmw}
\begin{gather}\label{densCB}
\widehat{C}_{\text{BI}} =-\frac1{\gamma^2}(\widehat{\Omega}_\theta\widehat{\Omega}_\sigma+
\widehat{\Omega}_\sigma\widehat{\Omega}_\theta+\widehat{\Omega}_\theta\widehat{\Omega}_\delta+
\widehat{\Omega}_\delta\widehat{\Omega}_\theta+\widehat{\Omega}_\sigma\widehat{\Omega}_\delta+
\widehat{\Omega}_\delta\widehat{\Omega}_\sigma),
\end{gather}
where
\begin{gather}\label{cp-quantum}
\widehat\Omega_i =\frac1{4i\sqrt{\Delta}}\widehat{\sqrt{V}}
\left[\big(\hat{\mathcal
N}_{2\bar\mu_i}-\hat{\mathcal
N}_{-2\bar\mu_i}\big)\widehat{\text{sgn}(p_i)}
+\widehat{\text{sgn}(p_i)}\big(\hat{\mathcal
N}_{2\bar\mu_i}-\hat{\mathcal
N}_{-2\bar\mu_i}\big)\right]\widehat{\sqrt{V}}.
\end{gather}
This operator dif\/fers from that of \cite{AsW} in the treatment applied to the signs of $p_i$ when
symmetrizing.
Then, $\widehat{{C}}_{\text{BI}}$ not only decouples the zero-volume states, but also it does not
relate states with opposite orientation of any of the triad coef\/f\/icients,
namely, states $|v,\lambda_\sigma,\lambda_\delta\rangle$ with opposite sign in any of their quantum
numbers. Therefore, $\widehat{{C}}_{\text{BI}}$ leaves invariant all the octants in the
tridimensional space def\/ined by $v$, $\lambda_\sigma$ and $\lambda_\delta$. Hence, we can restrict
the study to any of them. We will restrict ourselves to the subspace of positive densitized triad
coef\/f\/icients, given~by
\begin{gather*
{\text{Cyl}}_\text{S}^+ =\text{lin}\{|v,\lambda_\sigma,\lambda_\delta\rangle;\,
v,\lambda_\sigma,\lambda_\delta>0\}.
\end{gather*}
The action of $\widehat{{C}}_{\text{BI}}$ on the states of
${\text{Cyl}}_\text{S}^+$ turns out to be
\begin{gather*
\widehat{{C}}_{\text{BI}}|v,\lambda_\sigma,\lambda_\delta\rangle =\frac{(\pi
l_{\text{Pl}}^2)^2}{4}\big[x_-(v)|v-4,\lambda_\sigma,\lambda_\delta\rangle_-
-x^-_0(v)|v,\lambda_\sigma,\lambda_\delta\rangle_{-}\nonumber\\
\phantom{\widehat{{C}}_{\text{BI}}|v,\lambda_\sigma,\lambda_\delta\rangle =}{} -
x^+_0(v)|v,\lambda_\sigma,\lambda_\delta\rangle_{+}+x_+(v)|v+4,\lambda_\sigma,
\lambda_\delta\rangle_+\big],
\end{gather*}
where we have introduced the coef\/f\/icients
\begin{subequations}\label{coeficientes}
\begin{alignat}{3}
& x_-(v)=2\sqrt{v}(v-2)\sqrt{v-4}[1+\text{sgn}(v-4)],\qquad
&& x_+(v)=x_-(v+4), & \label{coefficient1}\\
& x^-_0(v)=2(v-2)v[1+\text{sgn}(v-2)],\qquad &&x^+_0(v)=x^-_0(v+2), &\label{coefficient3}
\end{alignat}
\end{subequations}
and the following linear combination of states
\begin{gather}
|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm =\bigg|v\pm n,\lambda_\sigma,\frac{v\pm n}
{v\pm2}\lambda_\delta\bigg\rangle+\bigg|v\pm n,\frac{v\pm n}
{v\pm2}\lambda_\sigma,\lambda_\delta\bigg\rangle\nonumber\\
\hphantom{|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm =}{}
+\bigg|v\pm n,\frac{v\pm2}
{v}\lambda_\sigma,\lambda_\delta\bigg\rangle+\bigg|v\pm n,\lambda_\sigma,
\frac{v\pm2} {v}\lambda_\delta\bigg\rangle\nonumber\\
\hphantom{|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm =}{}
+\bigg|v\pm n,\frac{v\pm2}
{v}\lambda_\sigma,\frac{v\pm n} {v\pm2}\lambda_\delta\bigg\rangle+\bigg|v\pm n,\frac{v\pm n}
{v\pm2}\lambda_\sigma,\frac{v\pm2}{v}\lambda_\delta\bigg\rangle.\label{eq:combi}
\end{gather}
Note that, in fact, the operator $\widehat{{C}}_{\text{BI}}$ is well def\/ined
in ${\text{Cyl}}_\text{S}^+$, since $x_-(v)=0$ if $v\leq4$, and $x^-_0(v)=0$ if $v\leq2$. Since there is no $v=0$ state, the singularity has no longer
analog in the kinematical Hilbert space, and then is resolved already kinematically.
\subsection{Superselection sectors}\label{superselection}
The analysis of the action of $\widehat{C}_{\text{BI}}$ on a generic basis state
$|v,\lambda_\sigma^\star,\lambda_\delta^\star\rangle$ shows that:
\begin{itemize}\itemsep=0pt
\item [i)] Concerning the variable $v$, it suf\/fers a constant shift equal to $4$ or $-4$, the
latest only if $v>4$. Therefore,
$\widehat{{C}}_{\text{BI}}$ preserves the subspace of states whose quantum number $v$ belongs to
any of the semilattices of step four
\begin{equation*}
\mathcal
L_{\varepsilon}^+=\{\varepsilon+4k,k=0,1,2,\dots\},\qquad\varepsilon\in(0,4].
\end{equation*}
\item [ii)] Concerning the anisotropy variables, $\lambda_\sigma$ and $\lambda_\delta$,
the ef\/fect upon them does not depend on the initial quantum numbers $\lambda_\sigma^\star$ and
$\lambda_\delta^\star$, but only on $v=\varepsilon+4k$. Moreover, this dependence occurs
via fractions whose denominator is two units bigger or smaller than the numerator.
As a consequence, the iterative action of the constraint operator on
$|v,\lambda_\sigma^\star,\lambda_\delta^\star\rangle$, only relates this state with states whose
quantum numbers
$\lambda_\sigma$ and $\lambda_\delta$ are of the form
$\lambda_a=\omega_{\varepsilon}\lambda_a^\star$, with~$\omega_{\varepsilon}$
belonging to the set
\begin{gather}
\mathcal
W_{\varepsilon}=\bigg\{\left(\frac{\varepsilon-2}{\varepsilon}\right)^z\prod_{m,
n\in\mathbb{N}}\left(\frac{\varepsilon+2m}{\varepsilon+2n}\right)^{k_n^m};\nonumber\\
\hphantom{W_{\varepsilon}=\bigg\{}{}
k_n^m\in\mathbb{N},\; z\in\mathbb{Z}\text{ if } \varepsilon>2,\;\;z=0\text{ if }
\varepsilon<2\bigg\}.\label{W-set}
\end{gather}
\end{itemize}
The set $\mathcal W_{\varepsilon}$ is inf\/inite and, moreover, one can prove that it is dense in
the positive real line \cite{gow-B}. Nonetheless, it is countable.
Therefore, while the variable $v$
has support in simple semilattices of constant step, the variables $\lambda_a$ take
values belonging to complicated sets, but they also provide separable subspaces. As a concrete
example,
we see that, if both $\varepsilon$ and
$\lambda_a^\star$ are integers, then $\lambda_a$ take values in the positive rational numbers.
In conclusion, the operator $\widehat{{C}}_{\text{BI}}$ leaves invariant the Hilbert
subspaces $\mathcal H^+_{\varepsilon,\lambda_\sigma^\star,\lambda_\delta^\star}$,
def\/ined as the Cauchy completion of
\begin{gather*
\text{Cyl}^+_{\varepsilon,\lambda_\sigma^\star,\lambda_\delta^\star}=\text{lin}\big\{
|v,\lambda_\sigma,\lambda_\delta\rangle;\;
v\in\mathcal L_{\varepsilon}^+,\;\lambda_a=\omega_{\varepsilon}\lambda_a^\star,\;
\omega_{\varepsilon}\in\mathcal W_{\varepsilon},\;\lambda_a^\star\in\mathbb{R}^+\big\},
\end{gather*}
with respect to the discrete inner product
$
\langle
v,\lambda_\sigma,\lambda_\delta|v',\lambda'_\sigma,\lambda'_\delta\rangle=\delta_{vv'}\delta_{
\lambda_\sigma \lambda'_{\sigma}}\delta_{\lambda_\delta \lambda'_{\delta}}
$.
As we will see, physical observables also preserve these separable subspaces
$\mathcal H^+_{\varepsilon,\lambda_\sigma^\star,\lambda_\delta^\star}$, and therefore they
provide sectors of superselection, and we can restrict the study to any of them.
\subsection{Physical Hilbert space}
The Hamiltonian constraint operator $\widehat{{C}}_{\text{BI}}$ is quite complicated and,
unlike the isotropic case (and unlike previous quantizations of the model~\cite{mmp}),
its spectral properties have not been determined. Consequently, it has not been diagonalized
either. Therefore, the group averaging approach is not useful in this situation, and to analyze the
physical solutions one has to impose the constraint directly on the dual space
$(\text{Cyl}^{+}_{\varepsilon,\lambda_\sigma^\star,\lambda_\delta^\star})^*$. The elements
$(\psi|$ of that space have the formal expansion
\begin{gather*}
(\psi| =\sum_{v\in\mathcal
L_{\varepsilon}^+}\sum_{\omega_{\varepsilon}\in\mathcal
W_{\varepsilon}}\sum_{\bar\omega_{\varepsilon}\in\mathcal W_{\varepsilon}}
\psi(v,\omega_{\varepsilon}\lambda_\sigma^\star,\bar\omega_{\varepsilon}
\lambda_\delta^\star)
\big\langle
v,\omega_{\varepsilon}\lambda_\sigma^\star,\bar\omega_{\varepsilon}
\lambda_\delta^\star\big|.
\end{gather*}
From the action of $\widehat{C}_\text{BI}$, one obtain that the constraint
$\big(\psi\big|\widehat{\mathcal
C}_\text{BI}^\text{B}{}^\dagger=0$ leads to the following recurrence relation,
\begin{gather*
\psi_+(v+4,\lambda_\sigma,\lambda_\delta)=\frac1{x_+(v)}
\big[{x^-_0(v)}\psi_{-}(v,\lambda_\sigma,\lambda_\delta)+{x^+_0(v)}\psi_{+}(v,
\lambda_\sigma,\lambda_\delta)\nonumber\\
\hphantom{\psi_+(v+4,\lambda_\sigma,\lambda_\delta)=\frac1{x_+(v)}
\big[}{} -{x_-(v)}\psi_-(v-4,\lambda_\sigma,\lambda_\delta)\big].
\end{gather*}
In this expression, in order to simplify the notation, we have introduced the projections of
$(\psi|$ on the linear combinations of six states def\/ined in equation~\eqref{eq:combi}, namely,
\begin{gather*}
\psi_\pm(v\pm n,\lambda_\sigma,\lambda_\delta)=
(\psi|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm .
\end{gather*}
Owing to the property $x_-(\varepsilon)=0$, the above recurrence relation, that is of order 2 in
the variable $v$, becomes a f\/irst-order equation if
$v=\varepsilon$:
\begin{gather*
\psi_+(\varepsilon+4,\lambda_\sigma,\lambda_\delta)=\frac1{x_+(\varepsilon)}
\big[{x^-_0(\varepsilon)}\psi_{-}(\varepsilon,\lambda_\sigma,\lambda_\delta)+{
x^+_0(\varepsilon)}\psi_{+}(\varepsilon,\lambda_\sigma,\lambda_\delta)\big].
\end{gather*}
Therefore, if we know all the data in the initial section $v=\varepsilon$, we obtain all the combinations of six terms given by
\begin{gather*
\psi_+(\varepsilon+4,\lambda_\sigma,\lambda_\delta)
=\psi\left(\varepsilon+4,\lambda_\sigma,\frac{\varepsilon+4}{\varepsilon+2}
\lambda_\delta\right)
+\psi\left(\varepsilon+4,\frac{\varepsilon+4}{\varepsilon+2}\lambda_\sigma,
\lambda_\delta\right)
\nonumber\\
\hphantom{\psi_+(\varepsilon+4,\lambda_\sigma,\lambda_\delta)=}{}
+\psi\left(\varepsilon+4,\lambda_\sigma,
\frac{\varepsilon+2}{\varepsilon}\lambda_\delta\right)
+\psi\left(\varepsilon+4,\frac{\varepsilon+2}{\varepsilon}\lambda_\sigma,
\lambda_\delta\right)
\nonumber\\
\hphantom{\psi_+(\varepsilon+4,\lambda_\sigma,\lambda_\delta)=}{}
+\psi\left(\varepsilon+4,\frac{\varepsilon+2}{\varepsilon}\lambda_\sigma,
\frac{\varepsilon+4}{\varepsilon+2} \lambda_\delta\right)
+\psi\left(\varepsilon+4,\frac{\varepsilon+4}{\varepsilon+2}\lambda_\sigma,
\frac{\varepsilon+2}{\varepsilon} \lambda_\delta\right).
\end{gather*}
From the combinations $\psi_+(\varepsilon+4,\lambda_\sigma,\lambda_\delta)$, it is possible to
determine any of the individual terms
$\psi(\varepsilon+4,\lambda_\sigma,\lambda_\delta)$ that compose them, since it has been shown that
the system of equations that relate the formers with the latters is formally invertible~\cite{mmw}. This has been proven not only for $v=\varepsilon+4$ but also for all
$v\in\mathcal L_{\varepsilon}^+$. In conclusion, the physical solutions of the
Hamiltonian constraint are completely determined by the set of initial data
\begin{gather*
\{\psi(\varepsilon,\lambda_\sigma,
\lambda_\delta)=\psi(\varepsilon,\omega_\varepsilon\lambda_\sigma^\star,
\bar\omega_\varepsilon\lambda_\delta^\star),\
\omega_\varepsilon,\bar\omega_\varepsilon\in\mathcal W_{\varepsilon},\
\lambda_\sigma^\star,\lambda_\delta^\star\in\mathbb{R}^+\} ,
\end{gather*}
and we can identify solutions with this set. We can also characterize the physical Hilbert space as the Hilbert space of the initial data.
In order to endow the set of initial data with a Hilbert structure, one can take a complete set of
observables forming a closed algebra, and impose that the quantum counterpart of their
complex conjugation relations become adjointness relations between operators. This determines a
unique (up to unitary equivalence) inner product.
\looseness=-1
Before doing that, it is suitable to change the notation. Following \cite{mmw}, let us
introduce the variables $x_a=\ln(\lambda_a)=\ln(\lambda_a^\star)+\rho_{\varepsilon}$. Note that
$\rho_{\varepsilon}$ takes values in a dense set of the real line, given by the logarithm of the
points in the set $\mathcal W_{\varepsilon}$. We will denote that set by $\mathcal Z_{\varepsilon}$.
For each direction $a=\sigma$ or $\delta$, we consider the linear span
$\text{Cyl}_{\lambda_a^\star}$ of the states whose support is just
one point $x_a$ of the superselection sector def\/ined by taking the product of
$\lambda_a^\star$ with all the points in the set $\mathcal Z_{\varepsilon}$.
We call $\mathcal H_{\lambda_a^\star}$
the Hilbert completion of this vector space with the discrete inner product.
Then, a set of observables acting on the initial data
$\tilde\psi(x_\sigma,x_\delta):=\psi(\varepsilon,x_\sigma,
x_\delta)$ is that formed by the operators
$\widehat{e^{ix_a}}$ and $\widehat{U}_a^{\rho_a}$, with $\rho_a\in\mathcal
Z_{\varepsilon}$ and $a=\sigma,\delta$, def\/ined as
\begin{gather}\label{conjunto-completo}
\widehat{e^{ix_\sigma}}\tilde\psi(x_\sigma,x_\delta)={e^{ix_\sigma}}\tilde\psi(x_\sigma,x_\delta),
\qquad
\widehat{U}_\sigma^{\rho_\sigma}\tilde\psi(x_\sigma,x_\delta)=
\tilde\psi(x_\sigma+\rho_\sigma,x_\delta),
\end{gather}
and similarly for $\widehat{e^{ix_\delta}}$ and $\widehat{U}_\delta^{\rho_\delta}$.
These operators provide an overcomplete set of observables and are unitary in
$\mathcal H_{\lambda_\sigma^\star}\otimes\mathcal H_{\lambda_\delta^\star}$, according with their
reality conditions.
Therefore, we conclude that this Hilbert space is precisely the
physical Hilbert space of the vacuum Bianchi I model.
Owing to the complicated form of the solutions, together with the fact that a basis of
eigenfunctions of the Hamiltonian constraint operator is not known, the evolution picture of this
model has not been studied yet. It is expected that, as in the isotropic models, a quantum bounce
solves dynamically the big bang singularity. Indeed, this has been already checked using an
ef\/fective dynamics~\cite{chi2}, in the case of the model coupled to a massless scalar f\/ield, as in
the FRW case. In this respect, it is worth commenting that in analyses where a massless
scalar f\/ield is introduced, the latter serves as internal time to describe the notion of evolution, as
seen in Section~\ref{1subsec:evol}. This f\/ield (unlike the geometric degrees of freedom)
is quantized adopting a standard Schr\"odinger-like representation, which makes straightforward the construction of a~family of unitarily related
observables parametrized by the internal time. However, in vacuum cases such as the Bianchi~I model we have just described, where
a suitable matter f\/ield is not at hand, it would be interesting to describe the evolution regarding as
internal time one of the geometry degrees of freedom. There is a complication because of the fact that such internal time is
polymeric, since the geometry degrees of freedom are polymerically quantized.
Although such description has not been carried out for the Bianchi~I model within the current improved dynamics, it has been
nonetheless constructed for the vacuum Bianchi~I model quantized within the naive
improved dynamics given in equation~\eqref{eq:mubarraA}. That was done in~\cite{mmp2}, and an
analog construction could just as well serve to describe the evolution of the current vacuum Bianchi~I
model (quantized within the scheme given in equation~\eqref{eq:mubarraB}), using
either the volume variable~$v$ or its momentum as internal time.
\subsection{Loop quantization of other Bianchi models}
Bianchi models are characterized by possessing three spatial Killing f\/ields. In the case of the
Bianchi I model, the Killing f\/ields commute and, then, it is the simplest of the Bianchi models.
This is the reason why it has been extensively studied in the literature, and in particular in the
framework of LQC. Nevertheless, loop quantization has been also extended to other Bianchi
models (with non-commuting Killing f\/ields), in particular to Bianchi~II and Bianchi~IX. A~preliminary loop quantization of these models was already considered just after the
birth of LQC~\cite{boj,boj-date}. More recently, their quantization has been achieved implementing
the improved dynamics developed for Bianchi~I, in~\cite{AsW2} for Bianchi~II and in~\cite{we} for Bianchi~IX. In the following we summarize the main
characteristics of these works. We refer the reader to those references for further details.
Both the Bianchi II and the Bianchi IX models possess non-commuting Killing f\/ields. As a~consequence, the f\/iducial triad and co-triad cannot be chosen to be diagonal, a f\/irst feature that
complicates the analysis in comparison with Bianchi~I. Moreover, the spin connection of
those models is non-trivial. After choosing a suitable gauge and appropriate parameterizations for
the Ashtekar--Barbero variables, the Hamiltonian constraint of
both Bianchi~II and Bianchi~IX consists in that of the (analog) Bianchi~I model plus an extra term,
of course dif\/ferent for each model, but that in both cases involves components of the connection and
inverses of the densitized triad coef\/f\/icients.
The kinematical Hilbert space of the Bianchi II and IX models is identical to that of
Bianchi~I. The main dif\/f\/iculty lies in the representation of the components of the connection
that appear in the extra term of the Hamiltonian constraint\footnote{The inverse of the components
of the densitized triad can be regularized using commutators with holonomies, in an analog way as
that employed in the FRW model.}. To tackle this issue, the strategy adopted is to def\/ine an
operator representing the connection~\cite{AsW2,ydm2}, choosing a suitable loop of holonomies and
implementing the ideas employed when constructing the curvature operator for Bianchi~I. As
a result, the components $c^i$ of the connection turn out to be represented by the polymeric
operator
\begin{gather*}
\hat c_i=\widehat{\frac{\sin \bar \mu_i c^i}{\bar \mu_i}},
\end{gather*}
where $\bar\mu_i$ is the minimum length def\/ined in equation~\eqref{eq:mubarraB}.
Once the above operator is def\/ined, the representation of the Hamiltonian constraint as a~symmetric
operator follows straightforwardly, in the same way as for the Bianchi~I model. Once again,
classical singularities are avoided in both models, since the kernel of the volume operator
can be removed from their quantum theories.
\part{Midisuperspace models in loop quantum gravity}\label{part2}
In the previous part we have discussed the loop quantization of several homogeneous cosmological
models. In all of them the classical cosmological singularity is avoided. Nonetheless, it is
natural to ask whether the resolution of the singularity is an intrinsic feature of the loop
quantization or, on the contrary, if it is a result due to the high symmetry of the homogeneous
models. In order to answer this question, it seems inevitable to extend the loop quantization to inhomogeneous
systems. Furthermore, it is essential to make this step in order to develop a realistic theory of
quantum cosmology. Indeed, as the results obtained in modern cosmology indicate, in the early
universe the inhomogeneities played a fundamental role in the formation of the cosmic structures
that we observe today.
The quantization of inhomogeneous models is technically more complicated than that of minisuperspaces, since they possess f\/ield degrees of freedom, as
the full theory. Hopefully, facing the loop quantization of midisuperspaces we will get insights about the open problems present in loop quantum gravity.
In this section we will review the status of loop quantization of midisuperspace models. Classically, these models have Killing vectors which reduce
the degrees of freedom of the metric but the number of Killing vectors are low enough to ensure that the remaining
degrees of freedom are local. The metrics are therefore parametrized by functions of time and spatial coordinate(s).
Thus, unlike minisuperspace scenarios, midisuperspace models are f\/ield theories (for a comprehensive review, see~\cite{midisuperspacereview}). Questions beyond the reach of minisuperspace models can be addressed in the study of midisuperspaces both classically
as well as in the context of a~particular quantization scheme. In the context of LQG, these issues may include the construction of Dirac
observables or of quasilocal observables, the closure of the quantum constraint algebra and, most importantly, whether the singularity resolution
mechanism of LQC continues to be valid in the f\/ield theory context.
There exist a number of midisuperspace models but the loop quantization procedure has been attempted only in a few of them so far.
One important class of models is obtained by symmetry reduction of GR.
\begin{itemize}\itemsep=0pt
\item {\em Spherical symmetry.} In $3+1$ dimensions, a spacetime is called spherically symmetric if its isometry group contains a subgroup isomorphic to $SO(3)$,
and the orbits of this subgroup are 2-spheres such that the induced metric thereon is Riemannian and proportional to the unit round
metric on $S^2$. These are, in a sense, midway between the minisuperspace models and models with an inf\/inite number of physical degrees of freedom.
Spherically symmetric spacetime metrics depend on the radial coordinate, and therefore these models have to be treated as f\/ield theories. However, in
vacuum, the physical solutions are characterized by only a single parameter according to Birkhof\/f's theorem. In that respect they are
{\em dynamically trivial}, although the gauge-f\/ixing procedure is extremely non-trivial. To make them into physical f\/ield theories, we need to add
matter in the form of dust shells.
Here we are mainly interested in cosmological models while the spherical symmetric models are mostly black hole solutions, in particular
the Schwarzschild solution. Hence, we will brief\/ly mention the progress made in the context of LQG for completeness.
In \cite{martinsph1} the kinematical framework for studying spherically symmetric models in LQG was introduced. The volume operator was constructed in
\cite{martinsph2} and it was shown that the volume eigenstates are not eigenstates of the f\/lux operator. Consequently, the standard prescription of
constructing the Hamiltonian operator cannot be used. This problem was circumvented in
\cite{martinsph3}, where the Hamiltonian constraint operator was constructed in terms of non-standard variables which mix the connection and the
extrinsic curvature. This formulation was extended to explore the question of singularity resolution of Schwarzschild black holes in
\cite{martinsph4} and the Lema\^itre--Toleman--Bondi collapse of a spherical inhomogeneous dust cloud in \cite{martinsph5}. The choice of
variables made in the above programme is similar to the polymer quantization of the Gowdy $T^3$ model, which will be described in detail later.
Since the basic quantum variables used in these constructions are dif\/ferent from the basic quantum variables of the full
theory, another approach in loop quantization of the spherically symmetric models was explored in~\cite{pullinsph1}. In this approach, the dif\/feomorphism
constraint is f\/ixed leaving the Gauss and the Hamiltonian constraints. The latter is then applied to the Schwarzschild solution. The exterior solutions
agree with the ones obtained in geometrodynamics. The interior solution was studied in~\cite{pullinsph2} where it was shown that, after a partial gauge
f\/ixing, it can be mapped to the minisuperspace Kantowski--Sachs model. After loop quantization, the singularity is replaced by a bounce.
In \cite{pullinsph3} the issue of the residual dif\/feomorphism invariance of loop-quantized spherical symmetric models has been investigated.
Although the quantization programme is still incomplete, the studies done so far indicate that the singularity resolution mechanism described in the
previous section may be a robust feature of loop quantization.
\end{itemize}
Another class of midisuperspace models can be roughly classif\/ied on the isometry group of the metric. In most cases, this is equivalent to a classif\/ication
based on the number of Killing vectors of the metric.
\begin{itemize}\itemsep=0pt
\item {\em Spacetimes with one Killing vector.}
These models are obtained by symmetry reduction corresponding to one-dimensional spatial isometry groups taken to be ${\mathbb R}$ or $U(1)$.
Some of the important features of GR are retained, such as dif\/feomorphism invariance and the f\/ield-theoretic non-linear nature of the physical
degrees of freedom. It was shown in \cite{geroch} that, locally, it is possible to interpret these $3+1$ dimensional models as \mbox{$2+1$} dimensional
general relativity coupled to two matter f\/ields, a scalar f\/ield and a one-form f\/ield corresponding, respectively, to the norm and the twist of the
four-dimensional Killing vector f\/ield. However, not much work has been done so far in loop quantizing this reduced system apart from the preliminary
analysis carried out in \cite{husainpullin} in terms of complex Ashtekar variables.
\item {\em Spacetimes with two Killing vectors.}
The next level of simplif\/ication is to consider two-dimensional spacelike isometry groups. Two types of group action and spatial topologies have been
studied in LQG:
\begin{itemize}\itemsep=0pt
\item Isometry group of ${\mathbb R} \times U(1)$ with the spatial topology being ${\mathbb R}^3$. These correspond to Einstein--Rosen cylindrical
waves. An preliminary attempt has been made to construct the kinematic Hilbert space and def\/ine the volume operator (see Section~5.11 of~\cite{lqc1}),
although there has not been much progress so far.
\item Isometry group of $U(1) \times U(1)$ with the spatial topology being $T^3$. These correspond to Gowdy models.
\end{itemize}
If we impose the additional condition that the Killing vectors are mutually orthogonal, we obtain {\em polarized} models.
These are the simplest midisuperspace models with only one f\/ield theory degree of freedom. An example is the
polarized Gowdy $T^3$ model. This has been studied in some detail in LQG from two dif\/ferent perspectives, both of which will
be described in detail in the next two sections.
\end{itemize}
\section{Hybrid quantization of the polarized Gowdy $T^3$ model}\label{hybrid-gowdy}
Gowdy models are among the best known inhomogeneous cosmologies \cite{gowdy1,gowdy2}. They represent
globally hyperbolic vacuum spacetimes, with compact spatial sections and with two spatial Killing
f\/ields. The simplest example is the linearly polarized Gowdy $T^3$
model. Indeed, its classical solutions are exactly known \cite{ise, mon1, mon2}. They represent
gravitational waves propagating in a closed expanding universe. Its standard quantization was already
considered since the 70's \cite{berger1,berger2,berger3,ccq,men0,misner,pierri,torre}. Moreover, a Fock quantization of the model, in which the dynamics is implemented unitarily, was
achieved~\cite{men1b, men1a}, and it has been shown that this quantization is essentially
unique~\cite{men2,men3} (in a sense that we will explain later).
In order to apply the quantization employed in LQC to this model, the simplest possibility is to
carry out a hybrid quantization, that combines the polymeric quantization of the degrees of freedom
that parametrize the homogeneous solutions, with the Fock quantization for the inhomogeneities.
This hybrid quantization was developed in~\cite{gow-B,gow-let,gow-ijmp}. Here we
summarize its construction and main results.
\subsection{Classical description of the Gowdy $T^3$ model}
The Gowdy $T^3$ model represents vacuum solutions to the Einstein equations, with two spatial
Killing f\/ields that commute, and with spatial hypersurfaces homeomorphic to a three-torus. As said
before, we will consider the linearly polarized model, that possesses an additional symmetry: the
Killing vectors are orthogonal to hypersurfaces and, therefore, are mutually orthogonal everywhere.
Letting $\partial_\sigma$ and $\partial_\delta$ be the Killing vectors, the model admits global
coordinates $\{t,\theta,\sigma,\delta\}$ adapted to the symmetries, with $\theta,\sigma,\delta\in
S^1$.
After a 3+1 decomposition, we can describe the spacetime metric in terms of the three-metric~$q_{ab}$ induced in the spatial sections foliating the four-dimensional manifold, the densitized
lapse ${N_{_{_{\!\!\!\!\!\!\sim}}\;}}=N/\sqrt{q}$, and the shift vector $N^{a}$, with
$a,b\in\{\theta, \sigma,\delta\}$. Owing to the isometries~$\partial_\sigma$ and~$\partial_\delta$,
the Gowdy model verif\/ies $q_{\theta\sigma}=0=q_{\theta\delta}$. This condition f\/ixes the gauge
freedom associated with the momentum constraint in those directions, and implies
$N^\sigma=0=N^\delta$~\cite{man}. As a~consequence, the metric components only depend on $t$ and
$\theta$ and are periodic in the latter. This periodicity allows us to decompose the metric
components in Fourier modes\footnote{We adopt the following convention to
def\/ine the Fourier modes $\phi_{m}$ of a generic f\/ield $\phi(\theta)$:
\begin{gather*
\phi(\theta)=\sum_{m\in\mathbb{Z}}\frac1{\sqrt{2\pi}}\phi_{m}e^{im\theta},
\qquad \phi_{m}=\frac1{\sqrt{2\pi}}\oint d\theta
\phi(\theta)e^{-im\theta}.
\end{gather*}}.
On the other hand, the condition of linear polarization imposes
$q_{\sigma\delta}=0$. Therefore, the three-metric is diagonal and can be described by three f\/ields
$(\tau,\xi,\bar\gamma)$, that essentially characterize the area of the isometry group orbits, the
norm of one of the Killing vectors, and the scale factor of the metric induced on the set of group
orbits. The phase space is then parametrized by those f\/ields and by their momenta
$(P_{\tau},P_{\xi},P_{\bar\gamma})$, and constrained by the $\theta$-momentum constraint and by the
Hamiltonian constraint~\cite{men1b}.
In order to prepare the model for its quantization, the gauge is further reduced. One
imposes that the generator of the conformal transformations,~$P_{\bar\gamma}$, and the area of
the isometry group orbits,~$\tau$, are homogeneous functions. These conditions f\/ix the gauge freedom
associated with the non-zero Fourier modes of the $\theta$-momentum constraint and of the
densitized Hamiltonian constraint, and imply that the functions $N_\theta$ and
${N_{_{_{\!\!\!\!\!\!\sim}}\;}}$ are homogeneous \cite{men1b,gow-let,gow-ijmp}.
Then, two global constraints remain in the model, the spatial average of the
$\theta$-momentum constraint, generating rigid rotations in the circle, and the spatial average
of the densitized Hamiltonian constraint. We will denote them by~$C_\theta$ and~$C_{\text{G}}$,
respectively.
The classically reduced phase space can be split into homogeneous and inhomogeneous sectors.
The homogeneous sector coincides with the phase space of the Bianchi~I spacetime with three-torus
topology. This sector will be quantized \emph{\`a la} LQC, and
therefore is parametrized by the variables $\{(c^i,p_i),i=\theta,\sigma,\delta\}$, with Poisson
bracket $\{c^i,p_j\}=8\pi G\gamma\delta^i_j$, as described in Section~\ref{chap:bianchi}.
The inhomogeneous sector is given by the non-zero (inhomogeneous) modes of the f\/ields unaf\/fected by
the gauge f\/ixing, namely
$\{(\xi_{_m},P_{\xi_{_m}}),m\in\mathbb{Z}\setminus \{0\}\}$. This sector will be quantized employing the
Fock quantization of \cite{men1b}.
To employ this Fock representation, the above inhomogeneous modes are in turn described by
annihilation and creation variables~$(a_m,a_m^*)$, def\/ined as those related to a free
massless scalar f\/ield.
This quantization is preferred as long as is the only Fock quantization of the
deparametrized system in which the dynamics is unitary and with a vacuum invariant under the~$S^1$
translations (the remaining symmetry after gauge f\/ixing)~\cite{men2,men3}.
The three-metric in terms of the chosen variables reads~\cite{gow-B}
\begin{gather}
ds^2 =-q_{\theta\theta}\left(\frac{|p_\theta|}{4\pi^2}\right)^2
{N_{_{_{\!\!\!\!\!\!\sim}}\;}}^2dt^2+q_{\theta\theta}d\theta^2+q_{\sigma\sigma}d\sigma^2
+q_{\delta\delta}d\delta^2,\nonumber\\
q_{\theta\theta} =\frac1{4\pi^2}\left|\frac{p_\sigma
p_\delta}{p_\theta}\right|\exp\left\{\frac{2\pi}{\sqrt{|p_\theta|}}\frac{c^\delta p_\delta-c^\sigma
p_\sigma}{c^\sigma p_\sigma+c^\delta p_\delta}\tilde\xi(\theta)-\frac{\pi^2}{|p_\theta|}[
\tilde\xi(\theta)]^2 -\frac{8\pi G\gamma}{c^\sigma
p_\sigma+c^\delta p_\delta} \zeta(\theta)\right\},\nonumber\\
q_{\sigma\sigma} =\frac1{4\pi^2}\left|\frac{p_\theta
p_\delta}{p_\sigma}\right|\exp\left\{-\frac{2\pi}{\sqrt{|p_\theta|}}\tilde\xi(\theta)\right\},
\nonumber\\
q_{\delta\delta} =\frac1{4\pi^2}\left|\frac{p_\theta
p_\sigma}{p_\delta}\right|\exp\left\{\frac{2\pi}{\sqrt{|p_\theta|}}\tilde\xi(\theta)\right\},\label{newmetric}
\end{gather}
where the inhomogeneities are encoded in the terms
\begin{gather*}
\tilde\xi(\theta) =\frac1{\pi}\sum_{m\neq0}\sqrt{\frac{G}{|m|}}(a_m+a_{-m}^*)e^{im\theta},
\\
\zeta(\theta) =i\sum_{m\neq0}\sum_{\tilde m\neq0}\text{sgn}(m+\tilde m)\frac{\sqrt{|m+\tilde
m||\tilde m|}}{|m|}(a_{-\tilde {m}}-a^*_{\tilde m}\big)\big(a_{m+\tilde m}+a^*_{-(m+\tilde
m)}\big)e^{im\theta}.
\end{gather*}
On the other hand, the remaining constraints have the following form:
\begin{gather}
C_\theta =\sum_{m=1}^\infty
m(a_m^*a_m-a_{-m}^*a_{-m})=0,\nonumber \\
C_{\text{G}} = C_\text{BI}+C_\xi=0,\quad C_\xi={G}\left[\frac{(c^\sigma
p_\sigma+
c^\delta p_\delta)^2}{\gamma^2|p_\theta|}
H_\text{int}^\xi+32\pi^2|p_\theta|
H_0^\xi\right].\label{Cclas}
\end{gather}
In the above expression,
\begin{gather} \label{HIclas}
H_\text{int}^\xi
=\sum_{m\neq
0}\frac{1}{2|m|}\left[2a^*_ma_m+
a_ma_{-m}+a^*_ma^*_{-m}\right], \qquad H_0^\xi=\sum_{m\neq
0}|m|a^*_ma_m,
\end{gather}
and $C_\text{BI}$ is the (densitized) Hamiltonian constraint of the Bianchi~I model given in equation~\eqref{eq:ligadura-bianchi-B}.
In the Hamiltonian constraint, the inhomogeneities appear in the term $H_0^\xi$, that corresponds
to the Hamiltonian of a free massless scalar f\/ield, and in the term~$H_\text{int}^\xi$, that
represents an interaction term. The inhomogeneities are coupled to the homogeneous sector in a
non-trivial way, so that the feasibility of the hybrid quantization is not straightforward \emph{a priori}.
\subsection{Fock quantization of the inhomogeneous sector}\label{Fphysi}
Once the inhomogeneous sector is described with the appropriate annihilation and creation like
variables,~$a_m$ and~$a^*_m$, it is straightforward to get its Fock quantization.
With that aim, we promote the variables~$a_m$ and~$a^*_m$ to annihilation and creation operators,~$\hat a_m$ and~$\hat a^\dagger_m$ respectively, such that
$[\hat a_m,\hat a^\dagger_{\tilde m}]=\delta_{m\tilde m}$. From the vacuum state
$|0\rangle$, characterized by the equations
\begin{equation*}
\hat a_m|0\rangle=0,\qquad \forall \, m\in\mathbb{Z},
\end{equation*}
we construct the one-particle Hilbert space, and the associated symmetric Fock space
$\mathcal{F}$ \cite{wald2}.
The annihilation and creation operators are densely def\/ined in the
subspace of $\mathcal{F}$ given by f\/inite linear
combinations of $n$-particle states
\begin{gather*}
|\mathfrak n \rangle:=|\dots,n_{-2},n_{-1},n_1,n_2,\dots\rangle,
\end{gather*}
such that $\sum_mn_m<\infty$, being $n_m\in\mathbb{N}$ the occupation number (or number of
particles) of the $m$-th mode. We will denote that space by $\mathcal
S$. Note that the $n$-particle states provide a basis for the Fock space, orthonormal with respect
to the inner product $\langle\mathfrak n^\prime|\mathfrak n\rangle=\delta_{\mathfrak
n^\prime\mathfrak n}$.
The action of $\hat a_m$ and $\hat a^\dagger_m$ on these states is
\begin{gather*}
\hat a_m |\dots,n_m,\dots\rangle=\sqrt{n_m}|\dots,n_m-1,\dots\rangle ,\\
\hat a_m^\dagger|\dots,n_m,\dots\rangle=\sqrt{n_m+1}|\dots,n_m+1,\dots\rangle .
\end{gather*}
\subsubsection{Generator of translations in the circle}
The constraint that generates translations in the circle, $C_\theta$, does not af\/fect the
homogeneous sector, and then it is represented on the above Fock space. Taking normal ordering, the
corresponding operator is
\begin{equation*
\widehat C_\theta=\hbar\sum_{m>0}^\infty m(\hat
a^\dagger_m \hat a_m-\hat a^\dagger_{-m} \hat a_{-m}).
\end{equation*}
This operator is self-adjoint in the Fock space $\mathcal F$.
The $n$-particle states annihilated by $\widehat C_\theta$ are those that satisfy the condition
\begin{equation*
\sum_{m>0}^\infty m X_m=0,\qquad X_m=n_m-n_{-m}.
\end{equation*}
They provide a basis for a proper subspace of the Fock space, that we will denote by $\mathcal
F_f$.
\subsection{Hamiltonian constraint operator}
Physical states must be annihilated as well by the quantum counterpart of the Hamiltonian
constraint $C_\text{G}$, given in equation~\eqref{Cclas}, which involves both homogeneous and
inhomogeneous sectors.
In the previous section, we have already described the representation of the inhomogeneous sector,
with basic operators $\hat a_m$ and $\hat a^\dagger_m$ acting on the Fock space $\mathcal F$, which
thus constitutes the inhomogeneous sector of the kinematical Hilbert space.
On the other hand, the homogeneous sector is quantized following LQC, namely, it is given by the loop
quantization of the Bianchi~I model.
As we discussed in Section \ref{chap:bianchi}, in the
literature two dif\/ferent implementations of the improved dynamics has been applied to the Bianchi~I
model. Therefore, there exist also two dif\/ferent descriptions for the hybrid Gowdy model, one
adopting the naive {\em Ansatz}~\eqref{eq:mubarraA} \cite{gow-B,gow-let,gow-ijmp}, and another
adopting the improved {\em Ansatz}~\eqref{eq:mubarraB} \cite{gow-B,mmw}. Here we will just explain the
second description, which adopts the quantization of the Bianchi~I model described in Section~\ref{chap:bianchi} when representing the homogeneous sector. This sector of the
kinematical Hilbert space will be the kinematical Hilbert space
$\mathcal H^+_{\varepsilon,\lambda_\sigma^\star,\lambda_\delta^\star}$, def\/ined in Section~\ref{superselection}.
The f\/irst term of the Hamiltonian constraint operator, $\widehat C_\text{G}=\widehat
C_\text{BI}+\widehat C_\xi$, is thus the Bianchi~I operator~\eqref{densCB}. We just need to construct the operator $\widehat C_\xi$ that couples homogeneous and inhomogeneous sector.
Let us f\/irst focus on the inhomogeneous terms.
In order to represent the free Hamiltonian~$H_0^\xi$ and the
interaction term $H_\text{int}^\xi$, def\/ined in equation~\eqref{HIclas}, we choose normal ordering. Then,
their quantum analogs are given by
\begin{gather*
\widehat{H}_0^\xi =\sum_{m>0}^\infty m\hat N_m,\qquad\hat N_m=
\hat{a}^{\dagger}_m \hat{a}_m+\hat{a}^{\dagger}_{-m} \hat{a}_{-m},
\\
\widehat{H}_\text{int}^\xi
=\sum_{m>0}^\infty\frac{\hat N_m+\hat Y_m}{m},\qquad\hat Y_m=\hat{a}_m
\hat{a}_{-m}+\hat{a}^{\dagger}_m\hat{a}^{\dagger}_{-m},
\end{gather*}
both densely def\/ined in the space $\mathcal S$ of $n$-particle states.
The operator $\widehat{H}_0^\xi$ acts diagonally on the $n$-particle states, and then it is
well-def\/ined in the Fock space $\mathcal F$. On the contrary, $\widehat{H}_\text{int}^\xi$ does
not leave invariant the domain $\mathcal S$. Indeed, the operator $\hat Y_m$
annihilates and creates pairs of particles in modes with the same wavenumber~$|m|$, and then
$\widehat{H}_\text{int}^\xi$ creates an inf\/inite number of particles. However, one can prove~\cite{gow-B} that the norm of $\widehat{H}_\text{int}^\xi|\mathfrak n\rangle$ is f\/inite for all
$\mathfrak n\in\mathcal S$, and therefore this operator, with domain $\mathcal S$, is also well
def\/ined in the Fock space~$\mathcal F$.
For the homogeneous terms, we recall that the operator $\widehat\Omega_i$, def\/ined in equation~\eqref{cp-quantum}, is the loop quantum analogue of the classical term~$c^ip_i$, and that the inverse
powers of~$|p_i|$ can be regularized taking commutators of $p_i$ with holonomies.
In view of these prescriptions, $C_\xi$ can be represented by the symmetric operator~\cite{gow-B,mmw}
\begin{gather*
\widehat{\mathcal{C}}_{\xi}
=l_{\text{Pl}}^2\left\{
\widehat{\left[\frac{1}{|p_\theta|^{\frac1{4}}}\right]}^2
\frac{(\widehat\Omega_\sigma+\widehat\Omega_\delta)^2}{\gamma^2}
\widehat{\left[\frac{1}{|p_\theta|^{\frac1{4}}}\right]}^2
\widehat{H}_\text{int}^\xi+32\pi^2\widehat{|p_\theta|}
\widehat{H}_0^\xi\right\},
\end{gather*}
where
\begin{gather
\widehat{\left[\frac1{|p_\theta|^{\frac1{4}}}\right]}|v,\lambda_\sigma,\lambda_\delta\rangle
=\frac{b_\theta^\star(v,\lambda_\sigma,\lambda_\delta)}{(4\pi\gamma
l_{\text{Pl}}^2\sqrt{\Delta})^{\frac1{6}}}|v,\lambda_\sigma, \lambda_\delta\rangle,\nonumber\\
b_\theta^\star(v,\lambda_\sigma,\lambda_\delta) =
\sqrt{2|\lambda_\sigma\lambda_\delta|}\left|\sqrt{|v+1|}-\sqrt{|v-1|}\right|.\label{btheta}
\end{gather}
The operator $\widehat{\mathcal{C}}_{\xi}$, so constructed, leaves the sectors of
superselection of the Bianchi~I model~in\-variant, and then it is in fact well def\/ined on the separable
kinematical Hilbert space $\mathcal
H^+_{\varepsilon,\lambda_\sigma^\star,\lambda_\delta^\star}\otimes\mathcal F$.
\subsection{Physical Hilbert space}
In order to impose the Hamiltonian constraint
$\big(\psi\big|\widehat{\mathcal{C}}_{\text{G}}^\text{B}{}^\dagger=0$,
we expand a general state $(\psi|$ in the basis of states
$|v,\omega_{\varepsilon}\lambda_\sigma^\star,
\bar\omega_{\varepsilon}
\lambda_\delta^\star\rangle$ of the homogeneous sector.
That is,
\begin{gather*}
(\psi| =\sum_{v\in\mathcal
L_{{\varepsilon}}}\sum_{\omega_{\varepsilon}\in\mathcal
W_{\varepsilon}}\sum_{\bar\omega_{\varepsilon}\in
\mathcal W_{\varepsilon}}\langle
v,\omega_{\varepsilon}\lambda_\sigma^\star,\bar\omega_{\varepsilon}
\lambda_\delta^\star|\otimes(
\psi(v,\omega_{\varepsilon}\lambda_\sigma^\star,
\bar\omega_{\varepsilon}\lambda_\delta^\star)| ,
\end{gather*}
where, let us recall, $\mathcal W_{\varepsilon}$ is the set~\eqref{W-set}.
In the above expression, \[
(\psi(v,\lambda_\sigma,\lambda_\delta)|=
(\psi(v,\omega_{{\varepsilon}}\lambda_\sigma^{\star},
\bar\omega_{{\varepsilon}}\lambda_\delta^{\star})|
\] is the projection of $(\psi|$ on the
state
$|v,\lambda_\sigma,\lambda_\delta\rangle=
|v,\omega_{\varepsilon}\lambda_\sigma^\star,\bar\omega_{\varepsilon}
\lambda_\delta^\star\rangle$ of the homogeneous sector and, in principle, it
must belong to the dual space of some appropriate dense domain of the Fock space~$\mathcal F$.
If we substitute the above expansion in the constraint, and take into account the action of the
operators af\/fecting the homogeneous sector, we obtain that the projections
$(\psi(v,\lambda_\sigma,\lambda_\delta)|$ satisfy dif\/ference equations in~$v$ that, generically,
relate data on the section $v+4$ with data on the sections $v$ and $v-4$, as it happened in the
Bianchi~I model. Following~\cite{gow-B}, to simplify the notation of the resulting equation,
we introduce the projections of $(\psi|$ on the linear combinations given in equation~\eqref{eq:combi}.
Namely, we def\/ine $(\psi_\pm(v\pm n,\lambda_\sigma,\lambda_\delta)|
=(\psi|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm$.
Similarly, it is convenient to introduce the combinations of states
\begin{gather*}
|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm' =\bigg|v\pm n,\lambda_\sigma,\frac{v\pm n}
{v}\lambda_\delta\bigg\rangle+\bigg|v\pm n,\frac{v\pm n}
{v\pm2}\lambda_\sigma,\frac{v\pm2}
{v}\lambda_\delta\bigg\rangle\nonumber\\
\phantom{|v\pm n,\lambda_\sigma,\lambda_\delta\rangle_\pm' =}{}
+\bigg|v\pm n,\frac{v\pm n}{v}\lambda_\sigma,\lambda_\delta
\bigg\rangle+\bigg|v\pm n,\frac{v\pm2}
{v}\lambda_\sigma,\frac{v\pm n}
{v\pm2}\lambda_\delta\bigg\rangle,
\end{gather*}
and to def\/ine the projections of $(\psi|$ on them:
$ (\psi'_\pm(v\pm n,\lambda_\sigma,\lambda_\delta)|=(\psi|v\pm n,\lambda_\sigma,
\lambda_\delta\rangle_\pm'$.
With this notation, the solutions of the Hamiltonian constraint satisfy the explicit relation
\begin{gather}
(\psi_+ (v+4,\lambda_\sigma,\lambda_\delta)|-\eta
[b_\theta^\star(v,\lambda_\sigma,\lambda_\delta)b
_\theta^\star(v+4,\lambda_\sigma,\lambda_\delta)]^2\frac{
v+4}{v}(\psi{}_+'(v+4,\lambda_\sigma, \lambda_\delta)
|\widehat{H}_\text{int}^\xi\nonumber\\
\qquad{} =-\frac{1}{\eta}\frac{32v^2}{\lambda_\sigma^2
\lambda_\delta^2x_+(v)}(\psi(v,
\lambda_\sigma,\lambda_\delta)|\widehat{H}_0^\xi
+\frac{x^-_0(v)}{x_+(v)}(\psi_{-}
(v,\lambda_\sigma,\lambda_\delta)|+
\frac{x^+_0(v)}{x_+(v)}(\psi_{+}
(v,\lambda_\sigma,\lambda_\delta)|
\nonumber\\
\qquad{} -\frac{x_-(v)}{x_+(v)}
(\psi_-(v-4,\lambda_\sigma,\lambda_\delta)|
+\eta[b_\theta^\star(v,\lambda_\sigma,
\lambda_\delta)]^4\bigg\{
\left[\frac{b_\theta^\star(v-4,\lambda_\sigma,\lambda_\delta)}{b_\theta^\star(v,\lambda_\sigma,
\lambda_\delta)}\right]^2
\frac{v-4}{v}\frac{x_-(v)}{x_+(v)}\nonumber\\
\qquad{} \times(\psi_-'(v-4,\lambda_\sigma,
\lambda_\delta)|-
\bigg[\frac{x_{0}^{-}(v)}{x_+(v)}(\psi'_{-
}(v,\lambda_\sigma,\lambda_\delta)|+
\frac{x_{0}^{+}(v)}{x_+(v)}(\psi'_{+
}(v,\lambda_\sigma,\lambda_\delta)|\bigg]\bigg\}
\widehat{H}_\text{int}^\xi,\label{solutionB}
\end{gather}
where $\eta=\left(\frac{l_{\text{Pl}}}
{4\pi\gamma\sqrt{\Delta}}\right)^{2/3}$ is a dimensionless parameter,
$b_\theta^\star(v,\lambda_\sigma,\lambda_\delta)$ is the function~\eqref{btheta}, and the
coef\/f\/icients $x_\pm(v)$ and $x^\pm_0(v)$ were def\/ined in equation~\eqref{coeficientes}.
Similarly to the analysis done in the Bianchi~I model, it has been investigated whether the
solution is totally determined (at least formally) by the data in the initial section
$v={\varepsilon}$. The presence of the interaction term in the left-hand side of equation~\eqref{solutionB} complicates a direct demonstration of the above statement. However, it is
possible to obtain such result in terms of an asymptotic analysis of the solutions. Note that the
model provides a dimensionless parameter $\eta$ that can be used to develop an asymptotic
procedure, without the need to introduce any external parameter by hand. This analysis was carried
out in~\cite{gow-B}, and we refer to it for the details. The main result of this analysis is
that, in fact, the initial data $(\psi({\varepsilon},\lambda_\sigma,\lambda_\delta)|$
(where $\lambda_\sigma$ and $\lambda_\delta$ run over all possible values in their corresponding
superselection sectors) completely determine the solution. The solutions turn out to be formal,
in the sense that the states $(\psi(v+4,\lambda_\sigma,\lambda_\delta)|$ do not belong in general
to the dual space of~$\mathcal S$, owing to the presence of $\widehat{H}_\text{int}^\xi$ in their
expression.
The physical Hilbert space can be characterized, even though the solutions are formal. Indeed, once
we justify that the set of initial data
$
\{\left(\psi({\varepsilon},\omega_{{\varepsilon}}\lambda_\sigma^{\star},
\bar\omega_{\varepsilon}\lambda_\delta^{\star})\right|;\;
\omega_{{\varepsilon}},\bar\omega_{\varepsilon}\in\mathcal W_{{\varepsilon}}\}
$
specif\/ies the solution, we can identify solutions with their corresponding initial data, and the
physical Hilbert space with the Hilbert space of such initial data, exactly as we proceeded with
the Bianchi I model.
Once again, the reality conditions over a complete set of observables, acting on the initial data,
univocally determines the inner product that provides the Hilbert structure. Such observables are
given, for instance, by the overcomplete set of observables of the Bianchi I model, given in equation~\eqref{conjunto-completo}, together with a suitable complete set of observables for the
inhomogeneous sector, given by~\cite{gow-B}
\begin{gather*
\big\{(\hat a_m+\hat a_m^\dagger)\pm(\hat a_{-m}+\hat a_{-m}^\dagger), \
i[(\hat a_m-\hat a_m^\dagger)\pm(\hat a_{-m}-\hat a_{-m}^\dagger)]; \ m\in\mathbb{N}^+\big\}.
\end{gather*}
These operators represent the real Fourier coef\/f\/icients of the non-zero modes of the f\/ield
$\xi(\theta)$ and of its momentum $P_\xi(\theta)$, and in fact they are self-adjoint in the Fock
space $\mathcal F$.
Finally, imposing the remaining symmetry of translations on $S^1$, the result is that the physical
Hilbert space of the Gowdy model is \cite{gow-B}
\begin{gather*
\mathcal H_{\text{phys}}=\mathcal H_{\lambda_\sigma^\star,\lambda_\delta^\star}
\otimes\mathcal F_f .
\end{gather*}
Namely, it is the tensor product of the physical Hilbert space of the Bianchi I model times the
physical Fock space for the inhomogeneities (def\/ined in Section~\ref{Fphysi}). We note that~$\mathcal F_f$ is unitarily equivalent
to the physical space of the Fock quantization of the deparametrized system~\mbox{\cite{men1b, men1a}}.
Therefore, the standard quantum f\/ield theory for the inhomogeneities is recovered, and they can be
seen as propagating over a polymerically quantized Bianchi~I background. This result supports the
validity of the hybrid quantization, since this should lead to the standard quantization of the
system in the limit in which the ef\/fects coming from the discreteness of the geometry are
negligible. This result is not trivial, since the hybrid quantization is introduced in the
kinematical setting, and the relation between kinematical and physical structures cannot be
anticipated before the quantization is completed.
\subsubsection{Singularity resolution}
The classical solutions of the linearly polarized Gowdy $T^3$ model generically display a
cosmological singularity~\cite{mon2}. In the parametrization employed for the hybrid quantization
of the model, this classical singularity corresponds to vanishing values of the coef\/f\/icients $p_i$.
In the quantum theory, the kernel of the operators~$\hat p_i$ is removed and, as a consequence,
there is no analog of the classical singularity. This resolution of the singularity is kinematical
and, therefore, independent of the dynamics. It persists in the Hilbert space of the physical
states since they do not have projection on the kernel of the operators~$\hat p_i$. Moreover, they
only have support in a sector with positive orientation of the coef\/f\/icients~$p_i$ and, then, they
do not cross the singularity towards other branches of the universe corresponding to dif\/ferent
orientations.
A description of the evolution picture of the model is missing, owing to its high complication. It
is worthy to note that, at least for the choice of the original naive improved dynamics, the
ef\/fective dynamics of the model has been thoroughly analyzed~\cite{eff,eff1}. In particular, it has
been studied how the inhomogeneities af\/fect the dynamics of the Bianchi~I background. Numerical
simulations show that the ef\/fect of the inhomogeneities does not destroy the bounce. For the improved dynamics discussed here, a similar analysis has not been done yet,
but we can expect similar results, since the bounce mechanism appears for both improved schemes.
\subsection{The Gowdy $T^3$ model coupled to a massless scalar f\/ield}
So far we have discussed the hybrid quantization of the linearly polarized Gowdy model in vacuo.
This model allows, almost straightforwardly, for the introduction of a minimally coupled free
massless scalar f\/ield with the same symmetries as the metric \cite{gow-matter}. Indeed, after a suitable rescaling of matter modes, these contribute to
the constraints $C_\theta$ and $C_\text{G}$ exactly in the same manner as the gravitational f\/ield
$\xi$. Also, the Fock quantization of that system (after a complete deparametrization) enjoys the
same uniqueness results as that of the model in vacuo and, hence, there is a preferred Fock
description also for the inhomogeneities of the matter f\/ield. Therefore, the hybrid approach
follows exactly in the same way as for the vacuum case.
The interest in considering the model f\/illed with matter lies in the fact
that FRW-type solutions are then allowed. Indeed, as we saw before, the vacuum Gowdy model can be
seen as a~vacuum Bianchi~I background f\/illed with inhomogeneities propagating in one direction, and
the subclass of isotropic solutions of the vacuum Bianchi I model represent trivial Minkowski
spacetimes rather than f\/lat FRW universes. Nonetheless, in the presence of matter, the f\/lat-FRW becomes
the isotropic sector of the Bianchi~I model. In that sense, there is a subclass of solutions of the
Gowdy model coupled to matter that can be regarded as a f\/lat FRW background f\/illed with
inhomogeneities propagating in one direction. Therefore, the linearly polarized Gowdy~$T^3$~model
coupled to a massless scalar f\/ield provides a simple
laboratory where to study, at the quantum level (by means of the hybrid quantization), interesting physical
phenomena such as the backreaction of the (quantum) inhomogeneities on
(polymerically quantized) f\/lat FRW cosmologies or, vice versa, the ef\/fect of the
quantum background geometry on the propagation of the inhomogeneities~\cite{gow-matter2}. This
analysis is intended to be a f\/irst step towards a quantum theory of FRW plus inhomogeneities. Its character is quite preliminary, since the inhomogeneities of Gowdy are just a subclass of the
inhomogeneities that one would introduce in the FRW model to account for the inhomogeneities that we observe in our universe. Nonetheless, a complete quantization of such a
system has not been yet achieved, and the hybrid Gowdy model of\/fers a~suitable setting to start
with. Actually, by employing the same hybrid procedure, the f\/lat FRW model plus perturbations is being
analyzed~\cite{frw-matter}. The hybrid quantization also applies in this more realistic system as
long as a unique Fock quantization for the inhomogeneities is at hand as well~\cite{frw-uni2,frw-uni3, frw-uni1}.
\section{Polymer quantization of the polarized Gowdy $T^3$ model}
In the previous section, we saw a successful quantization scheme of the linearly polarized Gowdy
$T^3$ model where the degrees of freedom were split into homogeneous
and inhomogeneous sectors. The homogeneous sector was quantized using the LQC techniques, while the inhomogeneous part was Fock quantized. One of the
signif\/icant advantages of the hybrid quantization is that the calculations are tractable and the tools developed and studied in LQC can be used to
address questions even in the midisuperspace context. While it is an extremely useful f\/irst step in quantization of midisuperspace models, it crucially
depends on the fact that the inhomogeneous degrees of freedom can be treated perturbatively, and it
is assumed that there exits a regime in which the most important ef\/fects emerging from the discretization of the geometry are those that af\/fect the
homogeneous subsystem. Ideally, we would like to loop quantize the full polarized Gowdy $T^3$ model without separating the
degrees of freedom. In this section we review the work that has been done so far in that direction, which has been carried out in~\cite{kinjal1,kinjal2}.
\subsection{Classical theory}
The variables chosen in this section are signif\/icantly dif\/ferent from the ones used in the rest of the review so far. We shall therefore indicate the
steps followed in obtaining these variables.
\subsubsection{Gowdy $T^3$ model in Ashtekar variables}
In order to loop quantize, we f\/irst need to rewrite the Gowdy $T^3$ model in terms of real Ashtekar
variables. Canonical quantization of the {\em unpolarized} Gowdy $T^3$ model in terms of the complex
Ashtekar variables has been given in \cite{HusainSmolin,Menamarugan} which we will
brief\/ly sketch below in terms of the real Ashtekar variables.
Recall that, owing to global hyperbolicity, spacetime can be decomposed as \mbox{${\mathcal M} = \Sigma_t \otimes {\mathbb R}$}, where $\Sigma_t$
is homeomorphic to a three-tours. As in the previous section, let the angular coordinates of $\Sigma_t$ be ($\theta,\sigma,\delta$), and
the two commuting Killing vectors be $\xi_1^a = \partial_\sigma$ and $\xi_2^a = \partial_\delta$.
These isometries imply that the Lie derivatives along these two Killing vectors vanish, i.e.
\begin{gather*}
{\mathcal L}_{\xi_1}A_a^i = 0 = {\mathcal L}_{\xi_1}E^a_i,\\
{\mathcal L}_{\xi_2}A_a^i = 0 = {\mathcal L}_{\xi_2}E^a_i .
\end{gather*}
The phase-space variables are therefore only functions of $\theta$. The Gauss and the dif\/feomorphism constraint reduce to
\begin{gather*}
G_i = \partial_\theta E^\theta_i + \epsilon_{ij}^k A_a^j E^a_k, \\
V_a = (\partial_a A_b^i)E^b_i - (\partial_\theta A_a^i) E^\theta_i + \epsilon_{jk}^i A_a^j A_b^k E^b_i .
\end{gather*}
The vector constraint given by $C_a = A_a^i G_i - V_a $ generates spatial dif\/feomorphisms.
We now impose the following gauge-f\/ixing conditions:
\begin{gather*}
E^{\theta}_I = 0 = E^{\rho}_3 ,\qquad \rho = \sigma, \delta ,\qquad I = 1, 2 .
\end{gather*}
The constraints $G_I$ and $C_\rho$ are then solved by $A_{\theta}^I = 0 = A_{\rho}^3$.
Thus, only one component of the Gauss constraint ($G_3 $) and one of the dif\/feomorphism constraint along the $\theta$ direction
($C_{\theta} =: C$) survive together with the Hamiltonian constraint. Since none of the quantities depend on $\sigma$ or $\delta$, we can
integrate over the torus $T^2$ and write the symplectic structure as\footnote{In this section, sum over repeated indices is understood.}
\begin{gather}
\Omega = \frac{4\pi^2}{\kappa\gamma}\int {d}\theta\big( {d}A_{\theta}^3 \wedge {d}E^{\theta}_3 +
{d}A_{\rho}^I \wedge {d}E^{\rho}_I \big). \label{symplectic1}
\end{gather}
This is the classical phase phase in terms of real Ashtekar variables. One important observation is that this is basically a
one-dimensional theory. This is useful because in one dimension, under orientation-preserving coordinate transformations, a tensor density of
contravariant rank~$p$, covariant rank~$q$ and weight~$w$, can be thought of as a scalar density of weight $= w + q - p$. Hence, under a~$\theta$ coordinate transformation, $E^{\theta}_3$ transforms as a scalar, $E^{\rho}_I$'s transform as scalar densities of weight 1,
$A_{\theta}^3$ transforms as a scalar density of weight~1, and $A_{\rho}^I$'s transform as scalars.
\subsubsection{Choice of new variables}
It turns out that these variables are not suitable for loop quantization and we need to make
canonical transformations similar to those performed
for the spherical symmetric case in~\cite{martinsph1}. Note that, for each $\rho$, the $A_{\rho}^I$ and $E_I^{\rho}$ rotate among themselves
under the $U(1)$ gauge transformations generated by the Gauss constraint. These suggest that we can perform canonical transformations to def\/ine the
following variables:
\begin{alignat*}{3}
& E^{{\sigma}}_1= E^\sigma \cos \beta , \qquad && E^{{\sigma}}_2 = E^\sigma \sin \beta, & \\
& E^{{\delta}}_1 = - E^\delta \sin \bar\beta,\qquad && E^{{\delta}}_2 =
E^\delta \cos \bar\beta, & \\
& A_{{\sigma}}^1 = A_{{\sigma}} \cos ( \alpha + \beta) , \qquad &&
A_{{\sigma}}^2 = A_{{\sigma}} \sin ( \alpha + \beta) , & \\
& A_{{\delta}}^1 = - A_{{\delta}} \sin ( \bar\alpha + \bar\beta) , \qquad &&
A_{{\delta}}^2 = A_{{\delta}} \cos (\bar\alpha +\bar\beta). &
\end{alignat*}
The angles for the connection components are introduced in a particular fashion for later convenience.
The radial coordinates, $E^\sigma$, $E^\delta$, $A_\sigma$, $A_\delta$, are gauge invariant and always strictly positive (va\-nishing radial
coordinates correspond to a trivial symmetry orbit which is ignored).
In terms of these variables, the symplectic structure (\ref{symplectic1}) gets expressed as
\begin{gather*}
\Omega=\frac{4\pi^2}{\kappa\gamma}\int {d}\theta\left[ {d}A^3_{\theta} \wedge {d}E_3^{\theta} +
{d} X \wedge {d} E^{\sigma} + {d} Y \wedge {d} E^{\delta} + {d} \beta
\wedge {d} P^{\beta} +
{d} \bar\beta \wedge {d} \bar P^{\beta}\right] ,
\end{gather*}
where
\begin{alignat*}{3}
& X:= A_\sigma \cos (\alpha), \qquad && Y := A_\delta \cos (\bar\alpha ), & \\
& P^{\beta} := - E^\sigma A_\sigma \sin(\alpha), \qquad &&
\bar P^{\beta} := - E^\delta A_\delta \sin ( \bar\alpha ).
\end{alignat*}
It is convenient to make a further canonical transformation:
\begin{alignat*}{3}
& \xi = \beta - \bar\beta, \qquad && \eta = \beta + \bar\beta, & \\
& P^{\xi} = \frac{P^{\beta} -\bar P^{\beta}}{2}, \qquad && P^{\eta} = \frac{P^{\beta} + \bar P^{\beta}}{2}. &
\end{alignat*}
The constraints are greatly simplif\/ied and their detailed expressions can be found in~\cite{kinjal1}. This completes the description of the
unpolarized Gowdy $T^3$ model in the variables we have def\/ined. The number of canonical f\/ield variables is 10 while there is a threefold
inf\/inity of f\/irst-class constraints. There are therefore 2 f\/ield degrees of freedom. We now need to impose two second-class constraints such that
the number of f\/ield degrees of freedom are reduced from two to one (as it should be in the polarized case).
\subsubsection{Reduction to polarized model}
In terms of the variables def\/ined above, the spatial three-metric is given by
\begin{gather*}
{d}s^2= \cos\xi \frac{E^\sigma E^\delta}{E_3^\theta} {d}\theta^2 +
\frac{E_3^\theta}{\cos\xi} \frac{E^\delta}{E^\sigma}{d}\sigma^2 +
\frac{E_3^\theta}{\cos\xi} \frac{E^\sigma}{E^\delta}{d}\delta^2 - 2
\frac{E_3^\theta}{\cos\xi}\sin\xi \ {d}\sigma {d}\delta.
\end{gather*}
For the Killing vectors $\partial_\sigma$ and $\partial_\delta$ to be
orthogonal to each other, the ${d}\sigma {d}\delta$
term in the metric should be zero. This implies that the polarization condition is implemented by restricting to
the $\xi = 0$ sub-manifold of the phase space of the unpolarized model\footnote{Actually there are two possible choices, $\xi = 0$ and $\xi = \pi$.
We shall take the constraint to be $\xi = 0$, which implies $E^{\theta}_3 > 0$.}.
In order to get a non-degenerate symplectic structure, we need one more condition. We expect the two conditions to reduce a f\/ield degree of freedom.
This turns out to be
\begin{gather*}
\chi(\theta) := 2 P^{\xi} + E_3^\theta \partial_\theta \ln\frac{E^\delta}{E^\sigma} ~ \approx 0 .
\end{gather*}
Thus, the reduction to the polarized model is obtained by imposing the two {\em polarization constraints}
\begin{equation*}
\xi \approx 0 ,\qquad \chi \approx 0 ,\qquad
\{\xi(\theta), \chi(\theta')\} = 2 \kappa\gamma\delta(\theta - \theta') .
\end{equation*}
We can solve the polarization constraints strongly and use Dirac brackets. Since the polarization constraints weakly commute with all the other
constraints, the constraint algebra in terms of Dirac brackets is same as that in terms of the Poisson brackets and remains unaf\/fected.
Furthermore, equations of motions for all the variables other than~$\xi$,~$P_{\xi}$ also remain unaf\/fected. We can thus set the polarization
constraints strongly equal to zero in all the expressions and continue to use the original Poisson brackets.
It also turns out that the basic variables~$X$,~$Y$, unlike in the full theory, are actually the extrinsic curvature components in the~$\sigma$ and $\delta$ direction, respectively.
The above construction can be carried out equivalently using $SU(2)$ variables by using $\eta$-dependent $\tau$ matrices:
\begin{gather}
\tau_\sigma(\theta) := \cos\eta(\theta) \ \tau_1 + \sin\eta(\theta) \tau_2 , \nonumber\\
\tau_\delta(\theta) := -\sin\eta(\theta) \ \tau_1 + \cos\eta(\theta) \tau_2 , \nonumber\\
\tau_3(\theta) := \tau_3 .\label{taumatrix}
\end{gather}
The $SU(2)$ formulation is useful in the quantum theory especially while constructing the Hamiltonian constraint operator.
Let us review the classical phase space we have constructed. For convenience of notation we rename $E_3^{\theta} := {\mathcal E}$ and
$A^3_{\theta}:= {\mathcal A}$. The basic conf\/iguration variables are $X$, $Y$, ${\mathcal A}$, $\eta$ and the momentum variables are
$E^\sigma$, $E^\delta$, ${\mathcal E}$, $P^{\eta}$, with Poisson brackets of the form $\{X, E^\sigma\}
= (2 G /\pi) \gamma \delta(\theta - \theta')$.\footnote{Only in this section we will absorb the $4\pi^2$ and use $\kappa':= 2G/\pi$.}
The spatial metric is given by
\begin{gather}
{d}s^2= \frac{E^\sigma E^\delta}{\mathcal E} {d}\theta^2 +
{\mathcal E} \frac{E^\delta}{E^\sigma} {d}\sigma^2 +
{\mathcal E}\frac{E^\sigma}{E^\delta} {d}\delta^2 .
\label{polymergowdymetric}
\end{gather}
The expressions of the constraints are greatly simplif\/ied:
\begin{gather}
G_3 = \frac{1}{\kappa'\gamma}\left[\partial_{\theta}{\mathcal E} + P^{\eta}\right], \label{ch4gauss} \\
C_{\theta} = \frac{1}{\kappa'\gamma}\big[E^\sigma \partial_{\theta}X + E^\delta
\partial_{\theta}Y -
{\mathcal A} \partial_{\theta}{\mathcal E} + P^{\eta} \partial_{\theta}{\eta} \big], \label{ch4diffeo} \\
H = - \frac{1}{\kappa'}\frac{1}{\sqrt{E}} \left[ \frac{1}{\gamma^2} \big( X E^\sigma Y
E^\delta + {\mathcal A} {\mathcal E}( X E^\sigma + Y E^\delta)
+ {\mathcal E} \partial_\theta \eta (X E^\sigma + Y E^\delta) \big) - E^\sigma\Gamma_\sigma E^\delta\Gamma_\delta\right] \nonumber \\
\phantom{H =}{}
+ \frac{1}{2\kappa'}\partial_\theta\left\{\frac{2 {\mathcal E} \left(\partial_\theta
{\mathcal E} \right) }{\sqrt{E}}\right\} - \frac{\kappa'}{4} \frac{G^2}{\sqrt{E}} -
\frac{\gamma}{2} \partial_{\theta}\left(\frac{G}{\sqrt{E}}\right) , \label{ch4hamiltonian}
\end{gather}
where $E=|{\mathcal E}| E^\sigma E^\delta$.
It is obvious from these def\/initions that $X$, $Y$, ${\mathcal E}$, $\eta$ are scalars while $E^\sigma$,
$E^\delta$, ${\mathcal A}$, $P^{\eta}$ are
scalar densities of weight~1. The Gauss constraint shows that~${\mathcal A}$ transforms as a~$U(1)$ connection, while~$\eta$ is {\em translated} by the gauge parameter. All other variables are gauge invariant.
This completes the process of symmetry reduction from the unpolarized to the polarized case. This is a consistent symmetry reduction as can be checked
by verifying the constraint algebra. It is also possible to show that the solutions of the equations of motion are equivalent to the standard Gowdy
solutions. In this construction, the goal has been to express the polarized Gowdy $T^3$ in terms of
variables which are suitable for loop quantization. In particular, they allow a simpler choice of edge and point holonomies, a~simpler form for the volume operator, and also a more tractable expression for the Hamiltonian constraint. Although it may not be possible to make the same
choice of variables for other midisuperspace models, similar variables have been used in the preliminary steps of loop quantization of another
midisuperspace model which we describe in brief below.
{\bf Plane gravitational waves.}
The dif\/ference between the polarized Gowdy $T^3$ model and the plane polarized~(pp) gravitational waves is in the global topology. While the Gowdy
model has a compact topology, pp waves have the global topology of Minkowski space. The coordinates are no longer angular but due to the homogeneity
it is possible to choose an arbitrary f\/inite area from the plane wavefronts and consider only f\/inite wave packets. The classical phase space in
Ashtekar variables is constructed in \cite{ppwavesloops,ppwavesloops1} in a similar way as done for the polarized Gowdy $T^3$ model described above.
Waves travelling only in one direction are considered to avoid the problem of wave collision. Finite pulses of pp waves travelling in the positive or
in the negative $z$ direction are characterized by a null Killing vector $k_\mu$ satisfying $\nabla_{(\mu}k_{\nu)} = 0$.
This gives rise to two new constraints:
\begin{gather*
U_+ = E^\sigma K_\sigma + E^\delta K_\delta - \partial_z{\mathcal E} , \qquad
U_- = E^\sigma K_\sigma - E^\delta K_\delta - {\mathcal
E} \ln\frac{E^\delta}{E^\sigma} .
\end{gather*}
It can be also shown that the constraint $U_-$ is identically zero on the constraint surface and has weakly vanishing Poisson brackets with all the
other constraints, i.e.\ it is gauge invariant and conserved under spatial dif\/feomorphisms and time evolutions. The constraint~$U_+$ Poisson commutes with
the Gauss, dif\/feomorphism and Hamiltonian constraints. This constraint can be added as a new f\/irst-class constraint and the standard
constraint algebra can be enlarged. This is an additional ingredient in the analysis of pp waves and the system can now be loop quantized.
\subsection{Quantum theory}
In this section, we review the loop quantization of the Gowdy model~\cite{kinjal2}. The methods and steps used here closely follow those used in LQG and
are to be viewed as f\/irst steps towards constructing a quantum theory of the Gowdy model where all the gravitational degrees of freedom are loop
quantized.
\subsubsection{Basic states}
Since this is a one-dimensional theory, the graphs are just $n$ arcs with $n$ vertices. The conf\/iguration variable ${\mathcal A}$ is a $U(1)$
connection 1-form, so we integrate it along an edge (an arc along~$S^1$) and by taking its exponential we def\/ine the (edge) holonomy variable
valued in $U(1)$:
\begin{equation*}
h^{(k)}_e({\mathcal A}) := \exp\left(i \frac{k}{2} \int_e {\mathcal A} \right) ,\qquad k \in \mathbb{Z} .
\end{equation*}
The conf\/iguration variables $X, Y \in \mathbb{R}$ and $\eta \in \mathbb{R}/\mathbb{Z}$ are scalars and hence no smearing is
needed. For these we def\/ine the point holonomies (at points $v$)
\begin{gather*}
h_v^{(\mu)}(X) := \exp\left[i\frac{\mu}{2} X(v)\right] , \qquad
h_v^{(\nu)}(Y) := \exp\left[i\frac{\nu}{2} Y(v)\right] , \qquad
h_v^{\lambda}(\eta) := \exp\left[i\lambda \eta(v)\right] ,
\end{gather*}
where $\mu,\nu \in \mathbb{R}$ and $\lambda \in \mathbb{Z}$. The $X$, $Y$ point holonomies are interpreted as unitary representations of the
compact Abelian group $\mathbb{R}_{\mathrm{Bohr}}$, which is the Bohr compactif\/ication of the additive group of real numbers $\mathbb{R}$.
The kinematical Hilbert space is thus a tensor product of the Hilbert spaces constructed for~${\mathcal A}$,~$X$, $Y$, $\eta$ variables.
For ${\mathcal A}$, the Hilbert space can be constructed using $U(1)$ holonomies in a procedure similar to full LQG.
For $X$, $Y$ and~$\eta$, we can use the point holonomies as in minisuperspace LQC, where the quantum conf\/iguration space is taken to be the
Bohr compacti\-f\/i\-cation $\mathbb{R}_{\mathrm{Bohr}}$. By contrast, $\eta$ is an angle variable, so the
corresponding point holonomy is valued in~$U(1)$.
An orthonormal basis on the tensor-product Hilbert space is provided by the ``charge network functions''.
They are labelled by a close, oriented graph $\Gamma$ with $n$ edges $e$ and $n$ vertices~$v$, a~$U(1)$~representation~$k_e$
for each edge, a~$U(1)$~representation $\lambda_v \in \mathbb{Z}$ for each vertex, and $\mathbb{R}_{\mathrm{Bohr}}$
representations $\mu_v$, $\nu_v$ for each vertex:
\begin{gather}
T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda}({\mathcal A}, X, Y, \eta) := \prod_{e\in \Gamma} k_e[h^{(e)}]~
\prod_{v \in {V}(\Gamma)} \mu_v [h_v(X)] \nu_v [h_v(Y)] \lambda_v [h_v(\eta)] \label{BasisStates} \\
\hphantom{T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda}({\mathcal A}, X, Y, \eta)}{}
= \prod_{e\in \Gamma} \exp \left(i \frac{k_e}{2} \int_e {\mathcal A} \right) \prod_{v \in {V}(\Gamma)}
\left[\exp \left(i \frac{\mu_v}{2} X \right) \exp \left(i \frac{\nu_v}{2} Y \right) \exp \left( i \lambda_v
\eta \right) \right] ,\nonumber
\end{gather}
where $V(\Gamma)$ represents the set of vertices belonging to the graph~$\Gamma$. Functions where any of the labels are dif\/ferent are
orthogonal~-- in particular, two graphs must coincide for non-zero inner product. These basis states provide an orthogonal decomposition
for the kinematical Hilbert space when all the representation labels are non-zero.
Note that, unlike in the full theory, in this model we have both point and edge holonomies. This construction is also signif\/icantly dif\/ferent
from the hybrid quantization of the previous section. There, the loop Hilbert space has only the homogeneous part represented by point holonomies
similar to LQC, while the inhomogeneous part is Fock quantized. In that case, the full Hilbert space is a tensor product of the two.
\subsubsection{Flux operators}
The conjugate variables are represented as
\begin{equation*}
E^\sigma (\theta) \sim -i \gamma l_{\rm Pl}^2 \frac{\delta h_{\theta}(X)}{\delta
X(\theta)}\frac{\partial}{\partial h_{\theta}(X)},
\end{equation*}
where $l_{\rm Pl}^2 := \kappa'\hbar$.
The f\/lux variables corresponding to $E^\sigma$, $E^\delta$, $P^{\eta}$ are def\/ined by integrating these
densities on an interval
${\mathcal I}$ of the circle, eg ${\mathcal F}_{\sigma,{\mathcal I}} := \int_{\mathcal I} E^\sigma,
{\mathcal F}_{\delta,{\mathcal I}} := \int_{\mathcal I} E^\delta$.
${\mathcal E}$, being a scalar, is already a suitable variable. Their actions on the basis functions~(\ref{BasisStates}) are
\begin{gather*}
\hat{\mathcal E}(\theta) T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda} = \frac{\gamma l_{\rm Pl}^2}{2}
\frac{k_{e^+(\theta)} + k_{e^-(\theta)}}{2} T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda}, \\
\int_I \hat{E}^\sigma T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda} = \frac{\gamma l_{\rm Pl}^2}{2} \sum_{v \in V(\Gamma) \cap
{\mathcal I}} \mu_v T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda}, \\
\int_I \hat{E}^\delta T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda} = \frac{\gamma l_{\rm Pl}^2}{2} \sum_{v \in V(\Gamma) \cap
{\mathcal I}} \nu_v T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda}, \\
\int_I \hat{P}^{\eta} T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda} =
\gamma l_{\rm Pl}^2 \sum_{v \in V(\Gamma) \cap {\mathcal I}} \lambda_v T_{\Gamma, \vec k, \vec \mu, \vec \nu, \vec \lambda} ,
\end{gather*}
where ${\mathcal I}$ is an interval on $S^1$. The symbols $e^{\pm}(\theta)$ either refer to the two oriented edges of the graph $\Gamma$, meeting
at $\theta$ if there is a vertex at $\theta$, or they denote two parts of the same edge if there is no vertex at $\theta$. In such a case,
the $k$ labels are the same. In case a vertex is an end-point of the interval, there is an additional factor of $1/2$ for its contribution to the sum\footnote{This follows from
\begin{gather*}
\int_a^b dx\, \delta(x - x_0) = \begin{cases}
1 & {\rm if} \ x_0 \in (a, b); \\
\frac{1}{2} & {\rm if} \ x_0 = a \ \mathrm{or} \ x_0 = b;
\\
0 & {\rm if} \ x_0 \notin [a, b].
\end{cases}
\end{gather*}}.
Note that classically the triad components $E^\sigma$ and $E^\delta$ are positive. Fluxes, however, can take both signs since they
involve integrals which depend on the orientation. We have thus constructed the kinematical Hilbert space together with the representation of the
basic variables. Next, we turn to the construction of composite operators.
\subsubsection{Construction of more general operators}
The dif\/feomorphism covariance requires that all operators of interest are integrals of expressions in terms of the basic operators.
They also involve products of elementary operators at the same point (same $\theta$) and thus need a ``regularization''. As in LQG, the
general strategy to def\/ine such operators is:
\begin{enumerate}\itemsep=0pt
\item replace the integral by a Riemann sum using a ``cell-decomposition'' (or partition) of $S^1$;
\item for each cell, def\/ine a regulated expression choosing suitable ordering of the basic operators, and evaluate the
action on basis states;
\item check ``cylindrical consistency'' of this action so that the (regulated) operator can be densely def\/ined on
the kinematical Hilbert space via projective limit;
\item f\/inally, one would like to remove the regulator.
\end{enumerate}
Since our model is one-dimensional, both the cell-decomposition and the graphs underlying the basis states are characterized by f\/initely many
points and the arcs connecting the consecutive points. Adapting the techniques used in LQG~\cite{lqg3, lqg1}, the products of elementary variables are
regulated by using a point splitting and then expressing the f\/ields in terms of the appropriate holonomies and f\/luxes.
A regulator, for each given graph $\Gamma$, then consists of a family of partitions
$\Pi^{\Gamma}_{\epsilon}$ such that, for each $\epsilon$, each vertex of $\Gamma$ is contained in exactly one cell. There is also a
choice of representation labels $k_0$, $\mu_0$, $\nu_0$, $\lambda_0 $ which can be taken to be the same for all $\epsilon$. Since each $\Pi^\Gamma$ can
also be thought of as being def\/ined by a set of points such that each vertex is f\/lanked by two points, any orientation-preserving dif\/feomorphism
will automatically preserve the order of the vertices and cell boundaries. Every suf\/f\/iciently ref\/ined partition then automatically becomes a~dif\/feomorphism-covariant regulator. We shall assume that the parameter $\epsilon$ denoting a family of partitions are suf\/f\/iciently ref\/ined and also
plays the role of a dif\/feo-covariant regulator. The regulated expressions depend on $\epsilon$ and we recover the classical expressions as
$\epsilon \to 0$.
As in LQG, the issue of cylindrical consistency is automatically sorted out by referring to the orthogonal decomposition of
${\mathcal H}_{\mathrm{kin}}$, i.e., by specifying the action of the operators on basis states with all representation labels
being non-zero. A few comments about the subsequent construction:
\begin{enumerate}\itemsep=0pt
\item We have assumed the ``length of the intervals'' to be same and equal to $\epsilon$. This corresponds to a ``cubic'' partition and
is chosen for convenience only.
\item The charges $\mu_v$, $\nu_v$ can take both signs depending on the orientation of the interval. However, the eigenvalues of the volume operator
must have explicit absolute values.
\item ${\mathcal I}_i$ denotes the $i$-th cell of the partition. For a given graph, the partition is so chosen that each vertex is included in one
and only one interval ${\mathcal I}_i$. The intervals which do not contain any vertex of the graph, do not contribute to the summation
owing to the property of f\/lux operators. Hence, the sum collapses to contributions only from the vertices, independent of
the partition. The action is manifestly independent of $\epsilon$ and even though the number of intervals go to inf\/inity as
$\epsilon \to 0$, the action remains f\/inite and well def\/ined.
\item Because of this property of the f\/luxes, we can choose the $\bar \theta_i$ point in a cell to coincide with a vertex of a graph if
${\mathcal I}_i$ contains a vertex, or with an arbitrary point if ${\mathcal I}_i$ does not contain a vertex.
\item The measure of the integrals in this section is sometimes suppressed for brevity of notation but can be clearly understood from the context.
\end{enumerate}
{\bf Volume operator.}
In the classical expression for the Hamiltonian constraint, powers of $E := |{\mathcal E}|E^\sigma E^\delta$ occur in the same manner
as in the full theory. It is therefore natural to consider the expression for the volume of a region ${\mathcal I}\times T^2$
and construct the corresponding operator. The classical volume operator written in terms of basic variables is
\begin{gather}
\mathcal{V} \big(\mathcal{I} \times T^2\big) = \int_{\mathcal{I} \times T^2} d^3 x \sqrt {g}
= 4 \pi ^2 \int_\mathcal{I} d\theta \sqrt { | {\mathcal E} | E^\sigma
E^\delta}. \label{volumeexpression}
\end{gather}
To obtain the quantum operator, we f\/irst rewrite equation~\eqref{volumeexpression} as a Riemann sum of volume of the cells, which we denote as
\begin{gather*}
\mathcal{V}_{\epsilon} (\mathcal{I}) \approx \sum_{i = 1}^n \int_{\theta_i}^{\theta_{i} + \epsilon} d\theta
\sqrt{|\mathcal{E}|E^\sigma E^\delta} .
\end{gather*}
This has to be written in terms of the f\/lux variables:
\begin{gather*}
\mathcal{V}_{\epsilon} (\mathcal{I}) \approx \sum_{i = 1}^n \epsilon \sqrt{|\mathcal{E}(\bar \theta_i)|}
\sqrt{E^\sigma(\bar \theta_i)} \sqrt{E^\delta(\bar \theta_i)} \\
\hphantom{\mathcal{V}_{\epsilon} (\mathcal{I})}{}
\approx \sum_{i = 1}^n \sqrt{|\mathcal{E}|} \sqrt{\epsilon E^\sigma} \sqrt{\epsilon E^\delta}
\approx \sum_{i = 1}^n \sqrt{|\mathcal{E}|(\bar \theta_i)}
\sqrt{\left|\int_{\theta_i}^{\theta_i + \epsilon} d\theta E^\sigma\right|}
\sqrt{\left|\int_{\theta_i}^{\theta_i + \epsilon} d\theta E^\delta\right|} .
\end{gather*}
The right-hand side is now expressed in terms of f\/lux variables. The regulated volume operator can be def\/ined as:
\begin{gather*}
\hat{\mathcal{V}_{\epsilon}} (\mathcal{I}) := \sum_{i = 1}^n \sqrt{\hat {|\mathcal{E}|}(\bar \theta_i)} \sqrt{\widehat
{\left|\int_{ {\mathcal I}_i} E^\sigma\right|} } \sqrt{\widehat {\left|\int_{ {\mathcal I}_i}
E^\delta\right|}}.
\end{gather*}
Clearly, this is diagonal in the basis states and its action on a basis state
$T_{\Gamma, \vec{k}, \vec{\mu}, \vec{\nu}, \vec{\lambda} }$ yields the eigenvalue
\begin{equation}
V_{ \vec k, \vec \mu, \vec \nu, \vec \lambda} = \frac{1}{\sqrt{2}} \left(\frac{\gamma l_{\text{Pl}}^2}{2}\right)^{3/2}
\sum_{v \in {\mathcal I} \cap V(\Gamma)}\bigg[|\mu_v|\ |\nu_v|\ | k_{e^{+}(v)} + k_{e^{-}(v)}|\bigg]^{\frac{1}{2}}.
\label{volumeeigenvalues}
\end{equation}
Thanks to our choice of basic variables, the eigenstates of the f\/lux operators are also volume eigenstates.
{\bf Gauss constraint.}
Consider the Gauss constraint (\ref{ch4gauss}):
\begin{gather*}
G_3 = \int_{S^1} d\theta (\partial_{\theta} {\mathcal E} + P^{\eta})
\approx \sum_{i = 1}^n \left[\int_{{\mathcal I}_i} P^{\eta} + {\mathcal E}(\theta_i + \epsilon) - \mathcal{E}(\theta_i)\right], \\
\hat {G_3^{\epsilon}} := \sum_{i = 1}^n \left[\widehat{\int_{ {\mathcal I}_i } P^{\eta}} + \hat {\mathcal E}(\theta_i +
\epsilon) - \hat{\mathcal E}(\theta_i)\right].
\end{gather*}
Again, this is easily quantized with its action on a basis state $T_{\Gamma, \vec{k}, \vec{\mu}, \vec{\nu}, \vec \lambda}$, giving the eigenvalue
\begin{equation*}
{\gamma l_{\rm Pl}^2}\sum_{v \in V(\Gamma)} \left[ \lambda_v + \frac{k_{e^{+}}(v) - k_{e^{-}}(v)}{2}\right]
\end{equation*}
Notice that in the limit of inf\/initely f\/ine partitions, for a given graph, if there is a vertex $v \in {\mathcal I}_i$, then
there is {\em no vertex} in the adjacent cells. As a result, ${\mathcal E}(\theta_{i + 1})$ gives $k_{e^+}(v)/2$ and
$- {\mathcal E}(\theta_i)$ gives $- k_{e^-}(v)/2$, since $\theta_i$ divides the same edge and so does $\theta_{i + 1}$.
Once again, the eigenvalues are manifestly independent of $\epsilon$ and the action is dif\/feo-invariant. Imposition of the Gauss constraint can be done
simply by restricting to basis states with labels satisfying $\lambda_v = - (k_{e^+(v)} - k_{e^-(v)})/2$, $\forall\, v \in V(\Gamma)$.
Since $\lambda_v \in \mathbb{Z}$, the dif\/ference in the $k$ labels at each vertex must be an {\em even} integer. We will assume these restrictions on the
representation labels and from now on deal with gauge-invariant basis states. Explicitly,
\begin{gather*}
T_{\Gamma,\vec k,\vec \mu,\vec \nu}= \prod_{e\in \Gamma} \exp \left\{ i \frac{k_e}{2} \int_e \left[ {\mathcal A} (\theta) -
\partial_{\theta} \eta \right] \right\} \prod_{v \in {V}(\Gamma)}\left\{ \exp \left[i \frac{\mu_v}{2} X(v) \right]
\exp \left[ i \frac{\nu_v}{2} Y(v) \right] \right\}.
\end{gather*}
We have also used $\eta(v^+(e)) - \eta(v^-(e)) = \int_e \partial_{\theta} \eta$, where $v^{\pm}(e)$ denote the tip and tail of the edge~$e$.
\subsubsection{Hamiltonian constraint}
{\bf Preliminaries.}
The Hamiltonian constraint is much more complicated. After some manipulation, we can write~(\ref{ch4hamiltonian})
as a sum of a kinetic and a potential term,
\begin{gather*}
H := - \frac{1}{\kappa'}[ H_K + H_P ], \\
H_K := \frac{1}{\gamma^2} \int_{S^1} {d} \theta N (\theta) \frac{1}{\sqrt{E}} \big[ X
E^\sigma Y E^\delta + \left({\mathcal A} +
\partial_{\theta}\eta\right) {\mathcal E}\big( X E^\sigma + Y E^\delta\big) \big], \\
H_P := -\int_{S^1} {d} \theta N (\theta) \frac{1}{\sqrt{E}} \left[ -\frac{1}{4} \left( \partial_\theta {\mathcal E}
\right)^2 + \frac{({\mathcal E})^2}{4} \left(\frac{\partial_\theta E^\sigma}{ E^\sigma} -
\frac{\partial_\theta E^\delta}{ E^\delta}\right)^2 \right]
\\
\hphantom{H_P :=}{} -\int_{S^1} {d} \theta N (\theta) \frac{1}{2}\partial_\theta\left[\frac{2 {\mathcal E} \left(\partial_\theta {\mathcal E}
\right) }{\sqrt{E}}\right]
\end{gather*}
Here it is more convenient to use $SU(2)$-valued holonomies using the $\eta$-dependent $\tau$ matrices def\/ined equation~(\ref{taumatrix}):
\begin{gather}
h_\theta (\mathcal{I}) := \exp \left[\tau_3 \int_{\mathcal{I}} d \theta' {\mathcal A} (\theta') \right] =
\cos\left(\frac{1}{2}\int_{\mathcal{I}} \mathcal{A}\right) + 2 \tau_3 \sin\left(\frac{1}{2}\int_{\mathcal{I}}
\mathcal{A}\right), \nonumber \\
h_\sigma(\theta) := \exp \left[ \mu_0 X(\theta) \tau_\sigma (\theta) \right] =
\cos\left[\frac{\mu_0}{2} X(\theta)\right] + 2\tau_\sigma(\theta) \sin\left[\frac{\mu_0}{2}
X(\theta)\right], \label{holonomies} \\
h_\delta(\theta) := \exp \left[ \nu_0 Y(\theta) \tau_\delta (\theta) \right] =
\cos\left[\frac{\nu_0}{2} Y(\theta)\right] + 2\tau_\delta (\theta) \sin\left[\frac{\nu_0}{2}
Y(\theta)\right]. \nonumber
\end{gather}
Each of the $SU(2)$-valued holonomies, as well as the sine and cosine, are well def\/ined on the kinematical Hilbert space. The
interval~$\mathcal{I}$ will typically be a cell of a partition, $(\theta_i, \theta_i + \epsilon)$. The parameters~$\mu_0$,~$\nu_0$ are the chosen and f\/ixed representations of $\mathbb{R}_{\mathrm{Bohr}}$, $k_0 = 1$ is the f\/ixed
representation of the~$U(1)$. As before the parameter $\epsilon$ which denotes a family of partitions, also plays the role of a dif\/feo-covariant
regulator. We brief\/ly describe the steps in obtaining a~well-def\/ined quantum operator; the details can be found in \cite{kinjal2}.
Consider an expression of the form Tr($h_ih_jh_i^{-1}h_j^{-1}h_k\{h_k^{-1},\sqrt{E}\}$), for distinct $i$, $j$, $k$ taking
values $\theta$, $\sigma$, $\delta$. For small values of $X$, $Y$, $\int_{\mathcal{I}}\mathcal{A}$, we can make the following approximations:
\begin{gather*}
h_\sigma(\theta)\big\{h_\sigma(\theta)^{-1}, V(\mathcal{I})\big\} =
-\frac{\kappa'\gamma}{2}\mu_0\tau_\sigma \frac{ \mathcal{E}(\theta)
\int_{\mathcal{I}}E^\delta }{V(\mathcal{I})} \approx -\frac{\kappa'\gamma}{2}\mu_0\tau_\sigma
\frac{E^\delta(\theta)
\mathcal{E}(\theta)}{\sqrt{E(\theta)}}, \nonumber \\
h_\delta(\theta)\big\{h_\delta(\theta)^{-1}, V(\mathcal{I})\big\} =
-\frac{\kappa'\gamma}{2}\nu_0\tau_\delta \frac{ \mathcal{E}(\theta)
\int_{\mathcal{I}}E^\sigma }{V(\mathcal{I})} \approx -\frac{\kappa'\gamma}{2}\nu_0\tau_\delta
\frac{E^\sigma(\theta)
\mathcal{E}(\theta)}{\sqrt{E(\theta)}}, \nonumber \\
h_{\theta}\big\{h_{\theta}^{-1}, V(\mathcal{I})\big\} = -\frac{\kappa'\gamma}{2}\tau_3
\frac{\int_{\mathcal{I}}E^\sigma
\int_{\mathcal{I}}E^\delta}{V(\mathcal{I})} \approx -\frac{\kappa'\gamma}{2}\epsilon\tau_3
\frac{E^\sigma(\theta) E^\delta(\theta)}
{\sqrt{E(\theta)}}, \\
\int_{\mathcal{I}}\mathcal{A} \approx \epsilon \mathcal{A}(\theta) , \qquad
\int_{\mathcal{I}}E^\sigma ~ \approx ~ \epsilon E^\sigma(\theta) ,\qquad
\int_{\mathcal{I}}E^\delta ~ \approx ~ \epsilon E^\delta(\theta)
\end{gather*}
where $V(\mathcal{I})$ is the volume of the interval $\mathcal{I}$.
Then, the holonomies can be expanded in a power series. Because of the trace, it is enough to expand each holonomy up to f\/irst order. The surviving
terms are quadratic terms arising from products of the linear ones and a linear term coming from~$h_k$. If one interchanges the
$i \leftrightarrow j $ holonomies, the linear term retains the sign while the quadratic one changes the
sign. Thus, taking the dif\/ference of the two traces leaves us only with the quadratic terms, which are exactly of the form
needed in~$H_K$.
There are derivatives of $\eta$ which arise from the position dependence of the $\tau_\sigma$,
$\tau_\delta$ matrices:
\begin{gather*}
\tau_\sigma(\theta + \epsilon) - \tau_\sigma(\theta) \approx
\epsilon\partial_{\theta}\tau_\sigma =
\epsilon\partial_{\theta}\eta \tau_\delta(\theta) , \nonumber \\
\tau_\delta(\theta + \epsilon) - \tau_\delta(\theta) \approx
\epsilon\partial_{\theta}\tau_\delta
= - \epsilon\partial_{\theta}\eta \tau_\sigma(\theta) .
\end{gather*}
In the quantization of the $H_P$, we also need to use the following identities repeatedly (in the form LHS/RHS = 1):
\begin{gather}
\mathcal{Z}(\mathcal{I}) := \epsilon^{abc}\mathrm{Tr}\left[ h_a\{h_a^{-1}, V(\mathcal{I})\}
h_b\{h_b^{-1}, V(\mathcal{I})\} h_c\{h_c^{-1}, V(\mathcal{I})\} \right] \nonumber \\
\hphantom{\mathcal{Z}(\mathcal{I})}{}
= \frac{3}{2} \left(\frac{\kappa'\gamma}{2}\right)^3 \mu_0\nu_0 V(\mathcal{I}) , \label{Identity1} \\
\mathcal{Z}_{\alpha}(\mathcal{I}) := \epsilon^{abc}\mathrm{Tr}\left[ h_a\{h_a^{-1},({V(\mathcal{I}))^{\alpha}}\}
h_b\{h_b^{-1},({V(\mathcal{I}))^{\alpha}}\} h_c\{h_c^{-1},({V(\mathcal{I}))^{\alpha}}\} \right] \nonumber \\
\hphantom{\mathcal{Z}_{\alpha}(\mathcal{I})}{}
= \frac{3}{2} \left(\frac{\kappa'\gamma}{2}\right)^3 \mu_0\nu_0 \alpha^3 [V(\mathcal{I})]^{3 \alpha - 2}
= \alpha^3 [V(\mathcal{I})]^{3(\alpha - 1)} \mathcal{Z}(\mathcal{I}). \label{Identity2}
\end{gather}
These are essentially versions of the identity $ 1 = (|\det(e_a^i)|/\sqrt{E})^n$ \cite{QSD5}.
It is also convenient to def\/ine the following families of operators:
\begin{gather*}
\hat {\mathcal O}_{\alpha}^\sigma({\mathcal I},\theta) := \left\{ \cos \left[ \frac{1}{2} \mu_0 X(\theta)\right] \hat V^{\alpha}
({\mathcal I}) \sin \left[\frac{1}{2} \mu_0 X(\theta)\right] \right. \nonumber \\
\left. \hphantom{\hat {\mathcal O}_{\alpha}^\sigma({\mathcal I},\theta) := }{}
-\sin\left [\frac{1}{2} \mu_0 X(\theta) \right] \hat V^{\alpha}({\mathcal I}) \cos \left[
\frac{1}{2} \mu_0 X(\theta) \right] \right\}, \nonumber \\
\hat {\mathcal O}_{\alpha}^\delta({\mathcal I},\theta) := \left\{ \cos \left[ \frac{1}{2} \mu_0 Y(\theta)\right] \hat V^{\alpha}
({\mathcal I}) \sin \left [\frac{1}{2} \mu_0 Y(\theta)\right] \right. \nonumber \\
\left. \hphantom{\hat {\mathcal O}_{\alpha}^\delta({\mathcal I},\theta) :=}{}
-\sin\left [\frac{1}{2} \mu_0 Y(\theta) \right] \hat V^{\alpha}({\mathcal I}) \cos \left[ \frac{1}{2} \mu_0
Y(\theta) \right] \right\} , \nonumber \\
\hat {\mathcal O}_{\alpha}^{\theta}({\mathcal I},\theta) := \left[ \cos \left( \frac{1}{2} \int_{ {\mathcal I} }{\mathcal A} \right)
\hat V^{\alpha}({\mathcal I}) \sin \left (\frac{1}{2} \int_{ {\mathcal I} }{\mathcal A} \right)
-\sin\left (\frac{1}{2} \int_{ {\mathcal I} }{\mathcal A} \right) \hat V^{\alpha}({\mathcal I}) \cos
\left( \frac{1}{2} \int_{ {\mathcal I}}{\mathcal A} \right) \right] .
\end{gather*}
Above, $\theta$ is a point in the interval ${\mathcal I}$ and $\alpha > 0$ is the power of the volume operator. Again,
for simplicity of notation we will suppress the $\theta$ labels in the above operators.
The operators ${\mathcal O}_{\alpha}^{a} := [\cos(\cdots) \hat V^{\alpha} \sin(\cdots) - \sin(\cdots) \hat V^{\alpha}
\cos(\cdots)]$, $a =\theta,\sigma,\delta $ appear in all the terms and are functions of both
holonomies and f\/luxes. To see that
this is actually diagonal in the charge network basis, write the cos and sin operators as sums and dif\/ferences of the
exponentials (i.e., holonomies). It then follows that
\begin{gather}
\cos(\cdots) \hat V^{\alpha} \sin(\cdots) - \sin(\cdots) \hat V^{\alpha}
\cos(\cdots) = \frac{1}{2i}\big[ e^{-i(\cdots)} \hat V e^{+i(\cdots)} - e^{+i(\cdots)} \hat V e^{-i(\cdots)} \big].
\label{diagonalO}
\end{gather}
It is now obvious that the operators are diagonal and thus commute with all the f\/lux operators.
Finally, the operator form of ${\mathcal Z}_{\alpha}({\mathcal I})$ can be obtained as
\begin{gather*}
\hat{\mathcal Z}_{\alpha}({\mathcal I}) := \epsilon^{abc} \,\mbox{Tr}
\big\{ \hat{h}_a [ \hat{h}_a^{-1} , {\hat{V}(\mathcal I)}^{\alpha} ]
\hat{h}_b [\ \hat{h}_b^{-1} , {\hat{V}(\mathcal I)}^{\alpha} ]
\hat{h}_c [\ \hat{h}_c^{-1} , {\hat{V}(\mathcal I)}^{\alpha} ] \big\} \\
\hphantom{\hat{\mathcal Z}_{\alpha}({\mathcal I})}{}
= -12 \hat {\mathcal O}_{\alpha}^\sigma({\mathcal I}) \hat {\mathcal O}_{\alpha}^\delta({\mathcal I})
\hat {\mathcal O}_{\alpha}^{\theta}({\mathcal I}).
\end{gather*}
Having noted the ingredients common to the quantization of the dif\/ferent pieces of the Hamiltonian constraint, we turn to
each one in some detail.
{\bf Quantization of $\boldsymbol{H_K}$.}
Choosing a partition of $S^1$ with a suf\/f\/iciently large number of $n$ points at
$\theta_i$, $i = 1, \dots, n$, $\theta_n = 2\pi$, $\epsilon = \theta_{i + 1} - \theta_i $, we write the integral as a sum,
\begin{gather*}
H_K \approx \frac{1}{\gamma^2} \sum_{i = 1}^n \epsilon N (\bar \theta_i) \frac{1}{\sqrt{E}(\bar \theta_i)}
\left[ X E^\sigma Y E^\delta +
\left({\mathcal A} + \partial_{\theta}\eta\right) {\mathcal E}( X E^\sigma + Y E^\delta)
\right](\bar \theta_i) \nonumber\\
\phantom{H_K }{} =
\frac{1}{\gamma^2} \sum_{i = 1}^n N (\bar \theta_i) \frac{1}{V(\mathcal{I}_i)} \left\{ X(\bar \theta_i)
\left(\int_{\mathcal{I}_i}E^\sigma\right) Y(\bar \theta_i) \left(
\int_{\mathcal{I}_i}E^\delta\right) \right. \\
\left. \phantom{H_K \approx }{} +\left(\int_{\mathcal{I}_i}{\mathcal A} + \partial_{\theta}\eta\right) {\mathcal E}(\bar \theta_i)
\left[ X(\bar \theta_i) \int_{\mathcal{I}_i}E^\sigma + Y(\bar \theta_i)
\int_{\mathcal{I}_i}E^\delta \right] \right\} .
\end{gather*}
For small values of the extrinsic curvature components ($ \sim X, Y$, classical regime) and suf\/f\/iciently ref\/ined partition
($\epsilon \ll 1$, continuum limit), the $i$-th term in the sum can be written in terms of the traces
of the $SU(2)$-valued holonomies. The expression in terms of holonomies and f\/luxes goes over to the
classical expression in the classical regime. It can be promoted to an operator by putting hats on the holonomies and
f\/luxes and replacing Poisson brackets by $(i\hbar)^{-1}$ times the commutators. Here, the standard choice of putting the holonomies on the left
is made. Then, we use the expressions for the holonomies in terms of the trigonometric operators given in equation~(\ref{holonomies}),
and evaluating the traces we get the quantum operator as
\begin{gather*}
\widehat H_K^{\mathrm{reg}} =
- i\frac{4}{l_{\rm Pl}^2\gamma^3}\frac{1}{\mu_0\nu_0} \sum_{i = 1}^n N (\bar \theta_i) \Bigg( \left\{ \sin\left[\mu_0 X(\bar \theta_i)
\right]\sin \left[\nu_0 Y(\bar \theta_i) \right] \right\} \times {\mathcal O}_1^{\theta}({\mathcal I}_i)\\
\phantom{\widehat H_K^{\mathrm{reg}} = }{}
+\left\{2 \sin \left [\frac{1}{2} \nu_0 Y(\bar \theta_i + \epsilon) \right]\cos \left[\frac{1}{2} \nu_0
Y(\bar \theta_{i})\right]\sin \left( \int_{\mathcal{I}_i} {\mathcal A} -\Delta_i\right)\right\}\times
{\mathcal O}_1^{x}({\mathcal I}_i)\\
\phantom{\widehat H_K^{\mathrm{reg}} = }{}
+\left\{2 \sin \left [\frac{1}{2} \mu_0 X (\bar \theta_i + \epsilon) \right]\cos \left[\frac{1}{2} \mu_0
X(\bar \theta_{i})\right]\sin \left( \int_{\mathcal{I}_i} {\mathcal A} -\Delta_i\right)\right\} \times
{\mathcal O}_1^{y}({\mathcal I}_i) \Bigg) ,
\end{gather*}
where $\Delta_i := \eta(\bar \theta_i) - \eta(\bar \theta_i + \epsilon)$ is outside the integrals.
{\bf Quantization of $\boldsymbol{H_P}$.}
All the three terms of~$H_P$ are functions only of the momenta, but there are a couple of obstacles in a straightforward transcription of~$H_P$.
These have to be expressed in terms of basic variables, i.e., in terms of f\/luxes and holonomies. Also, the power(s) of momenta in the denominators
will make the action on some states singular. The f\/irst part is easy to take care of thanks to the density weight~1. For the second part, we use the
identities~(\ref{Identity1}) and~(\ref{Identity2})\footnote{In this one-dimensional model, this procedure is equivalent to the
point-splitting procedure of \cite{QSD5}.}.
The common strategy followed for these terms is:
\begin{enumerate}\itemsep=0pt
\item introduce a suf\/f\/iciently large number $k > 0$ of positive powers of
\[
1 = 16[ 3 (\kappa'\gamma)^3 \mu_0\nu_0]^{-1}\mathcal{Z}(\mathcal{I})/V(\mathcal{I}),
\]
and express $\mathcal{Z}$ in terms of $\mathcal{Z}_{\alpha}$. This introduces further powers of the volume;
\item choose $\alpha(k)$ such that explicit multiplicative factors of the volume become 1 and further choose $k$.
\end{enumerate}
Now the expression can be promoted to an operator.
{\underline{\it First term of $H_P$}.}
We f\/irst rewrite this in terms of the basic variables:
\begin{gather*}
-\int_{S^1} d \theta N (\theta) \frac{1}{\sqrt{E(\theta)}} \left[ -\frac{1}{4} \left( \partial_\theta {\mathcal E} \right)^2 \right]
\approx \frac{1}{4} \sum_{i = 1}^n N(\bar \theta_i) \frac{ \left[ \mathcal{E}(\bar \theta_i + \epsilon) - \mathcal{E}(\bar \theta_i)
\right]^2}{\sqrt{\mathcal{E}(\bar \theta_i) \int_{ {\cal I}_i} E^\sigma \int_{ {\cal I}_i}
E^\delta }} .
\end{gather*}
We now follow the strategy mentioned for the terms in the denominator. After some manipulation, the right-hand side becomes
\begin{gather*}
\left. \frac{1}{4} \left[\frac{16}{3 (\kappa'\gamma)^3\mu_0\nu_0 \alpha^3}\right]^k \sum_{i = 1}^n N(\bar \theta_i)
\left[\mathcal{E}(\bar \theta_i + \epsilon) - \mathcal{E}(\bar \theta_i)\right]^2 \left[\mathcal{Z}_{\alpha}
(\mathcal{I}_i)\right]^k \right|_{\alpha := \frac{2}{3} - \frac{1}{3 k}} .
\end{gather*}
The choice $\alpha := 2/3 - 1/(3 k)$ removes explicit factors of the volume.
The choice of $k > 0$ is limited by $\alpha > 0$ (being it a power of the volume appearing in $\mathcal{Z}_{\alpha}$). Some
convenient choices would be $k = 1$ $(\alpha = 1/3)$, $k = 2$ $(\alpha = 1/2)$, and so on. For all of them, the
above expression can be promoted to a well-def\/ined operator.
{\underline{\it Second term of $H_P$}.}
To begin with, one observes that $E^\delta/E^\sigma$ is a scalar and
$\partial_{\theta}\ln(E^\delta/E^\sigma)$ is a scalar density. This term is then
manipulated as:
\begin{gather*}
- \frac{1}{4} \int_{S^1} N (\theta) \frac{[{\mathcal E}(\theta)]^2}{\sqrt{E(\theta)}} \left(
\frac{\partial_\theta E^\sigma}{E^\sigma} -
\frac{\partial_{\theta} E^\delta}{E^\delta} \right)^2
= - \frac{1}{4} \int_{S^1} N (\theta) \frac{[{\mathcal E}(\theta)]^2}{\sqrt{E(\theta)}} \left(\partial_\theta \ln
\frac{E^\delta}{E^\sigma}\right)^2.
\end{gather*}
We write the right-hand side in terms of f\/lux variables as
\begin{gather*}
\mathrm{RHS} = - \frac{1}{4} \sum_{i = 1}^n N(\bar \theta_i) \frac{[{\mathcal E}(\bar \theta_i)]^2}{V(\mathcal{I}_i)} \left[ \frac{\int_{
{\mathcal I}_i} E^\sigma}{\int_{ {\mathcal I}_i} E^\delta}\ \left( \frac{\int_{ {\mathcal I}_{i +
1} } E^\delta}{\int_{ {\mathcal
I}_{i + 1} } E^\sigma} - \frac{\int_{ {\mathcal I}_{i} } E^\delta}{\int_{ {\mathcal I}_{i} }
E^\sigma} \right) \right]^2.
\end{gather*}
Now we have the f\/luxes in the denominator which can be def\/ined exactly as the inverse triad operators of LQC.
Denoting the f\/luxes as ${\mathcal F}_{\sigma,{\mathcal I}} := \int_{\mathcal I}E^\sigma, {\mathcal
F}_{\delta,{\mathcal I}} := \int_{\mathcal I} E^\delta$,
\begin{gather*}
{\mathcal F}^{-1}_{\sigma,{\mathcal I}} = \left(\frac{1}{\kappa'\gamma l} \right)^{\frac{1}{1 -
l}} \big\{ X(v), {\mathcal F}^l_{\sigma,{\mathcal I}}
\big\}^{\frac{1}{1 - l}} \\
\phantom{{\mathcal F}^{-1}_{\sigma,{\mathcal I}}}{}
= \left(\frac{2 i}{\kappa'\gamma l \mu_0} \right)^{\frac{1}{1 - l}} \left(h_v^{(\mu_0/2)}(X)\big\{ h_v^{(- \mu_0/2)}(X),
{\mathcal F}^l_{\sigma,{\mathcal I}} \big\}\right)^{\frac{1}{1 - l}} , \qquad l \in (0, 1) ,
\end{gather*}
and similarly for ${\mathcal F}^{-1}_{\delta,{\mathcal I}}$. These can be promoted to well-def\/ined
operators. Then, following our strategy for the inverse
volume factors, we get
\begin{gather*}
\mathrm{RHS} = - \frac{1}{4} \sum_{i = 1}^n N(\bar \theta_i) \frac{[{\mathcal E}(\bar \theta_i)]^2}{V(\mathcal{I}_i)}
\left[ {\mathcal F}^{-1}_{\delta,{\mathcal I}_i} {\mathcal F}_{\sigma,{\mathcal I}_i} \big(
{\mathcal F}^{-1}_{\sigma,{ {\mathcal I}_{i + 1}}}
{\mathcal F}_{\delta,{\mathcal I}_{i + 1}} - {\mathcal F}^{-1}_{\sigma,{\mathcal I}_{i}} {\mathcal
F}_{\delta,{\mathcal I}_{i}} \big) \right]^2
\\
\phantom{\mathrm{RHS}}{}
= - \frac{1}{4} \left[\frac{16}{3 (\kappa'\gamma)^3\mu_0\nu_0 \alpha^3}\right]^k \sum_{i = 1}^n N(\bar \theta_i)
[{\mathcal E}(\bar \theta_i)]^2 \\
\phantom{\mathrm{RHS}=}{}
\times
\left[ {\mathcal F}^{-1}_{\delta,{\mathcal I}_i} {\mathcal F}_{\sigma,{\mathcal I}_i} \big(
{\mathcal F}^{-1}_{\sigma,{ {\mathcal I}_{i + 1}}}
{\mathcal F}_{\delta,{\mathcal I}_{i + 1}} - {\mathcal F}^{-1}_{\sigma,{\mathcal I}_{i}} {\mathcal
F}_{\delta,{\mathcal I}_{i}} \big) \right]^2
\left. \left[\mathcal{Z}_{\alpha}(\mathcal{I}_i) \right]^k \right|_{\alpha = \frac{2}{3} - \frac{1}{3 k}}.
\end{gather*}
The choice of $\alpha$ would be same as that in the f\/irst term.
{\underline {\em Third term of $H_P$}.} We can rewrite
\begin{gather*}
H_T = - \int_{S^1} N(\theta)\partial_\theta\left[\frac{\mathcal{E}\partial_{\theta}\mathcal{E}}{\sqrt{E(\theta)}}\right]
\approx - \sum_{i = 1}^n N(\bar \theta_i) \epsilon \partial_{\theta}\left[\frac{\mathcal{E}(\bar
\theta_i)\partial_{\theta}\mathcal{E}}{\sqrt{E(\bar \theta_i)}}\right]
\end{gather*}
as
\begin{gather*}
\mathrm{RHS} = - \left[\frac{16}{3 (\kappa'\gamma)^3\mu_0\nu_0 \alpha^3}\right]^k
\sum_{i = 1}^n N(\bar \theta_i) \left\{
{\mathcal{E}(\bar \theta_i + \epsilon) \left[ \mathcal{E}(\bar \theta_i + 2\epsilon) - \mathcal{E}(\bar \theta_i +
\epsilon) \right] }[\ \mathcal{Z}_{\alpha}(\mathcal{I}_{i + 1})\ ]^{k} \right. \\
\left. \left.\hphantom{\mathrm{RHS} =}{} - \mathcal{E}(\bar \theta_{i}) \left[ \mathcal{E}(\bar \theta_i + \epsilon) -
\mathcal{E}(\bar \theta_{i})~ \right] [\ {\mathcal Z}_{\alpha}(\mathcal{I}_{i})\ ]^k \right\}
\right|_{\alpha = \frac{2}{3} - \frac{1}{3 k}},
\end{gather*}
where the choice of $\alpha$ is as before.
We have expressed $H_P$ in terms of the holonomy-f\/lux variables. Quantization can be carried out
simply via the replacement $({\mathcal Z}_{\alpha})^{k} \to (-i/\hbar)^{3k} (\hat {\mathcal Z}_{\alpha})^{k}$.
We have thus managed to write the Hamiltonian constraint as a well-def\/ined operator on the kinematic Hilbert space. Obviously, there are operator
ordering ambiguities in the quantization of~$H_K$. However, because of equation~(\ref{diagonalO}), there are no ordering ambiguities in~$H_P$.
It is also straightforward to verify that the action of the operator is well def\/ined on the states in the Hilbert space~\cite{kinjal2}.
\subsubsection{Ambiguities in the quantization scheme}
In the course of the above construction, there have been obvious issues of quantization ambiguities. These are in fact ambiguities in the ordering of
the operators, in the transcription in terms of basic quantum variables as well as in the choice of partitions.
Let us review them and the choices we made.
\begin{enumerate}\itemsep=0pt
\item In $H_K$ we chose to keep the holonomies to the left. Then, the term containing the volume operator acts f\/irst on the states
and the pieces which do not have any vertices give zero.
\item In the regularization of $H_K$ we used the inverse volume and plaquette holonomies. We could have
introduced inverse f\/lux operators and $\hat {\mathcal E}$ operators to replace $1/\sqrt{E}$ and also replaced the
$X$, $Y$, $\int_{ {\mathcal I}_i}{\mathcal A}$ by $\sin(\mu_0X)/\mu_0$, and similarly for the others. Such a replacement
would still give the classical expression back in the limit of small $X$, $Y$, $\epsilon$. The quantum operator, however, would
be dif\/ferent.
\item The second term in the $H_P$ could be manipulated in terms of inverse powers of $\sqrt{E}$ instead of introducing
inverse f\/lux operators (e.g., by replacing $1/E^\sigma = {\mathcal E} E^\delta/(\sqrt{E})^2$). This
would lead to
${\mathcal E}^2\left({\mathcal F}_{\sigma, {\mathcal I}_i}{\mathcal F}_{\delta, {\mathcal I}_{i +
1}} - {\mathcal F}_{\delta, {\mathcal
I}_i}{\mathcal F}_{\sigma, {\mathcal I}_{i + 1}}\right)^2$ and $\alpha(k) = 2/3 - 5/(3k)$.
In the limit of inf\/inite ref\/inement, each cell would contain {\em at most} one vertex and the cells adjacent to such a~cell
would always be empty. Consequently, the second term of $H_P$, regulated in the above manner, would always give a zero action.
\item Over and above these dif\/ferent transcriptions, we also have the ambiguities introduced by the arbitrary positive power
$k$ (and $\alpha(k)$) and by the arbitrary power $l \in (0, 1)$ in the def\/inition of inverse f\/lux
operators, which is similar to the one in the minisuperspace models described before.
\end{enumerate}
There are also issues related to the choice of partitions, the subsequent $\epsilon \to 0$ limit, and the presence/absence of
local degrees of freedom. This is most dramatically brought out by the second term of~$H_P$. Classically, this is the term
which reveals spatial correlations in a~solution spacetime through $\partial_{\theta} \ln
(E^\delta/E^\sigma)$~\cite{kinjal1} and
ref\/lects the presence of inf\/initely many physical solutions. In the (vacuum) spherically-symmetric case, such a term is absent and
so are local physical degrees of freedom. We would like to see if there is a quantization of this term which ref\/lects these
correlations. This can only be ensured if we chose a partition such that every cell has {exactly} one vertex.
Then, the contributions will explicitly depend upon $\mu$, $\nu$ labels of adjacent vertices and, in this sense, spatial correlations will
survive in the constraint operator. However, the price to pay for that is that we cannot take inf\/inite ref\/inement~($\epsilon \to 0$).
An even more restrictive choice would be to pick the partition def\/ined by the graph itself~-- cells def\/ined by the edges and the
boundary points of cells as vertices. In this case, the new vertices created by~$H_K$ would be the already present vertices
and the constraint equation would lead to a (partial) dif\/ference equation among the labels. The~$\epsilon \to 0$ limit may then be
thought to be relevant when states have support on graphs with a very large (but f\/inite) number of vertices; heuristically, for semiclassical states.
However, more work needs to be done to determine the validity of this proposal.
This completes the kinematic framework of the polymer quantization of this model. Due to the complicated nature of the expressions,
not much progress has been made beyond that so far. A few directions in which there is ongoing further research are:
\begin{itemize}\itemsep=0pt
\item verif\/ication of the quantum constraint algebra,
\item obtaining the spectrum and checking the self-adjointness of the Hamiltonian constraint operator,
\item construction of Dirac and quasi-local energy observables, at least on the kinematic Hilbert space,
\item exploring the possibility of $\bar{\mu}$-type quantizations in the homogeneous directions,
\item checking whether the Bianchi~I cosmological model can be viewed as a sector of this model.
\end{itemize}
\section{Comparison with the hybrid quantization}
As we have seen in the previous two sections, the polarized Gowdy $T^3$ model has been quantized in two dif\/ferent ways within the framework of LQG. While
it would be good to have a procedure to compare the physical results of the two frameworks, it is not possible currently.
Although both quantization schemes start from the classically-reduced phase space of the model, there are signif\/icant dif\/ferences in the treatment,
both in the classical and the quantum theory.
The motivation of the hybrid quantization programme is to utilize the tools of LQC on midi\-superspace models. It tries to determine whether the
singularity resolution in LQC is a feature of the quantization scheme or an artifact of the high degree of symmetry of minisuperspace.
Moreover, it provides a suitable arena to analyze the back-reaction between inhomogeneities and quantum background geometry.
On the other hand, the polymer approach tries to construct a~loop-quantized theory \textit{ab initio}, trying to mimic the procedures of the
full theory. In addition to the fate of the classical singularity in polymer quantization, its aim is to provide a toy model
where some problems of the full theory, such as the verif\/ication of the quantum constraint algebra and the construction of observables, can be explored.
Let us review the progress made in the two quantization schemes so far.
\begin{itemize}\itemsep=0pt
\item The hybrid quantization scheme employs the machinery developed in minisuperspace LQC described before, in order to study the midisuperspace model.
After a partial gauge f\/ixing, it can be easily seen that the polarized Gowdy $T^3$ model can be thought of as a Bianchi I model f\/illed with
inhomogeneities propagating in one direction. This fact is exploited to break up the degrees of freedom into a homogeneous Bianchi I part and an
inhomogeneous scalar f\/ield. In addition, as a consequence of the partial gauge f\/ixing, only two global constraints remain in the model.
The polymer quantization programme, on the other hand, aims to implement the loop quantization programme by def\/ining suitable Ashtekar variables for the
entire model. Unlike in the hybrid approach, the system is symmetry reduced, but no further gauge is f\/ixed at the classical level, so
that the constraints are not global but depend on the point.
The dif\/ference between the two approaches can be seen, on the one hand, in the dif\/ferent way the spatial metric is parametrized, by
comparing equations~(\ref{newmetric}) and (\ref{polymergowdymetric})\footnote{Note that there is
no relation between the scalar f\/ield $\tilde\xi$ in~(\ref{newmetric}) and the angle~$\xi$ in~(\ref{polymergowdymetric}).} and, on the other hand,
in the dif\/ferent constraints surviving in the classically-reduced model.
\item The quantization that is subsequently carried out is also dif\/ferent. In hybrid quantization, the homogeneous Bianchi I is loop quantized while the
inhomogeneous scalar f\/ield is Fock quantized using creation/annihilation operators. The full kinematic Hilbert space is a~tensor product of the two.
There is a non-trivial interaction term in the Hamiltonian which couples the homogeneous and inhomogeneous modes. However, all the constraints
can be expressed as densely def\/ined operators on the tensor product Hilbert space.
In the polymer quantization, all the degrees of freedom are loop quantized but, unlike the full theory, there are both point and edge holonomies. The
techniques of full LQG are used to def\/ine the kinematic polymer Hilbert space. Subsequently, more general operators are constructed including the
Hamiltonian constraint operator and it can be shown that they are well def\/ined on the kinematic Hilbert space and do not depend on the regulator.
Thus, we have two dif\/ferent quantum theories of the same classical system, both of which have a well-def\/ined action of the constraint
operators. However, these operators are def\/ined on very dif\/ferent Hilbert spaces, and the quantum
theories may not be unitarily equivalent. While the polymer quantization scheme is closer in spirit to LQG, progress has stalled
beyond this point because of the extremely complicated nature of the expressions. On the other hand, signif\/icant progress has been made in the
hybrid quantization programme.
\item The construction of the physical Hilbert space has not been carried out so far in the polymer scheme. On the other hand, in the hybrid approach
the physical Hilbert space has been constructed, which turns out to be tensor product of the physical Hilbert space of
the Bianchi I model and the physical Fock space for the inhomogeneities, which are not neglected in this model. Rather, it is possible to
view this system as some inhomogeneous scalar f\/ield on a polymer-quantized Bianchi I space. The classical singularity is absent in the physical
Hilbert space even in the presence of inhomogeneities.
\end{itemize}
\part{Ef\/fective dynamics}\label{part3}
In the last part, we discuss two aspects of loop quantum cosmology which play an important role in
the connection between theory and phenomenology. Section \ref{hodi} presents the ef\/fective FRW
dynamics obtained by evaluating the scalar constraint on semiclassical states. The choice of
parametrization in minisuperspace is determined by rather robust arguments, but the latters undergo
several modif\/ications in the context of inhomogeneous models; this is the subject of Section
\ref{latti}, where the lattice ref\/inement framework is introduced.
\section{Homogeneous ef\/fective dynamics}\label{hodi}
The exact and numerical discrete dynamics stemming from the quantum Hamiltonian constraint provides important information about the singularity resolution in LQC, but it does not yield itself to manipulations suitable for the extraction of inf\/lationary dynamics and observables in a~semiclassical limit. This can be achieved by evaluating the Hamiltonian constraint on semiclassical states, resulting in \emph{continuous} Friedmann equations corrected by quantum terms. At this point, standard analysis techniques developed in classical FRW cosmology can be applied to these equations. The study of linear perturbations (which shall not be reviewed here) requires, however, some extra ef\/fort.
Ef\/fective cosmological equations of motion are derived from the expression of the Hamiltonian constraint on a semiclassical state. The latter is typically decomposed into a gravitational and matter sector, $|\Psi_{\rm sc}\rangle=\sum_{A,B}|{\rm grav}\rangle_A\otimes |{\rm mat}\rangle_B$. In general, geometrical and matter operators do not act separately on physical states because solutions to the Hamiltonian constraint already incorporate correlations between the two sectors. So operators on such states are in general complicated, entangled observables. However, on a semiclassical state geometrical and matter operators commute and they can be treated separately.
Before discussing how semiclassical states determine an ef\/fective dynamics, it is convenient to generalize the Hamiltonian constraint and introduce some ambiguity parameters which were previously kept f\/ixed. This is done in order to accommodate results which will be later obtained in an inhomogeneous setting. In this part we set $\hbar=1$.
\subsection{Parametrization of the Hamiltonian constraint}
Let $\kappa^2=8\pi G$. As before, we def\/ine a pair of variables
\begin{gather*}
b:= \frac{\bar\mu c}2 ,\qquad v:= \frac{6}{(1+n)\gamma\kappa^2}\frac{p}{\bar\mu} ,
\end{gather*}
where $\bar\mu$, however, is now an arbitrary dimensionless function of the densitized triad:
\begin{gather}\label{bu2}
\bar\mu=\left(\frac{p_*}{p}\right)^n = \left(\frac{a_*}{a}\right)^{2n} ,
\end{gather}
where $n\in\mathbb{R}$ (until now it was f\/ixed to $n=1/2$) and $p_*$ and $a_*$ are, respectively, constants of dimension $[p_*]=-2$ and $[a_*]=0$. Then,
\begin{gather*
\{b,v\} =1 .
\end{gather*}
In a purely homogeneous model, there is no reason in favour of (and, in fact, there are some against) taking $n\neq 1/2$, but for the time being we do not attempt to justify this generalization. Another ambiguity parameter~$q\in\mathbb{R}$ can arise when writing down Thiemann's identity
\begin{gather*}
\epsilon^{ijk}\frac{E_{i}^{a}E_{j}^{b}}{\sqrt{|\det
E}|} = 2\epsilon^{abc}\frac{V^{1-q}}{q}\frac{\delta V^q}{\delta
E^c_k}
\end{gather*}
Denote with $\ell_0^2=A_\square$ the area of an elementary plaquette. For a f\/lat homogeneous background, the classical scalar constraint becomes \cite{cqc}
\begin{gather}
C = -\frac{1}{\gamma^2}\frac{E_i^{a}E_j^b}{\sqrt{|\det E|}}\epsilon^{ij}_{\ \ k}F^k_{ab}+C_{\rm mat}\nonumber\\
\phantom{C}{} = \frac{8(1+n)}{\gamma^2q} \left[\frac{(1+n)\gamma\kappa^2}{3}\right]^{\frac{1-2n}{2(1+n)}}
\lim_{\ell_0\to 0}\frac{1}{\ell_0^3}\left(\frac{p_*^{n}}{2}\right)^{\frac{3}{2(1+n)}} v^{\frac{3(1-q)}{2(1+n)}}\sin^22 b\nonumber\\
\phantom{C=}{} \times\Big[\sin b \big\{\cos b,v^{\frac{3q}{2(1+n)}}\big\}-\cos b \big\{\sin b,v^{\frac{3q}{2(1+n)}}\big\}\Big]+C_{\rm mat} .\label{SH3cl}
\end{gather}
The gravitational sector is only a function of $b$ and $v$. The scalar f\/ield part (with potential $U(\phi)$) only contains volume factors,
\begin{gather*
C_{\rm mat}=\kappa^2\frac{\Pi_\phi^2}{p^{3/2}}+p^{3/2}U(\phi) .
\end{gather*}
As before, the quantum constraint is regularized by assuming that holonomy plaquettes cannot be shrunk indef\/initely, replacing the limit $\ell_0\to0$ in equation~\eqref{SH3cl} with $\ell_0\to V_o^{1/3}\bar\mu$. With this substitution, the quantum Hamiltonian operator corresponding to equation~\eqref{SH3cl} is well def\/ined:
\begin{gather}\label{cig}
\hat C = -4 \widehat{\sin2b} \hat A \widehat{\sin2b}+\hat C_{\rm mat} ,
\end{gather}
where
\begin{gather*}
\hat A =
\frac{i(1+n)}{4q V_o\gamma^2}\left[\frac{(1+n)\gamma\kappa^2}{3}\right]^{\frac{1+4n}{2(1+n)}}
\left(\frac{p_*^n}{2}\right)^{-\frac{3}{2(1+n)}} \nonumber\\
\phantom{\hat A = }{}
\times\widehat{|v|^{\frac{3(1+2n-q)}{2(1+n)}}}\bigg[\widehat{\cos b} \widehat{|v|^{\frac{3q}{2(1+n)}}}\widehat{\sin b}-\widehat{\sin b} \widehat{|v|^{\frac{3q}{2(1+n)}}}\widehat{\cos b}\bigg]\nonumber\\
\phantom{\hat A }{}
=\frac{1+n}{8qV_o\gamma^2}\left[\frac{(1+n)\gamma\kappa^2}{3}\right]^{\frac{1+4n}{2(1+n)}}\left(\frac{p_*^n}{2}\right)^{-\frac{3}{2(1+n)}} \nonumber\\
\phantom{\hat A = }{}
\times\widehat{|v|^{\frac{3(1+2n-q)}{2(1+n)}}}\bigg[\widehat{e^{-i b}} \widehat{|v|^{\frac{3q}{2(1+n)}}}\widehat{e^{i b}}-\widehat{e^{i b}} \widehat{|v|^{\frac{3q}{2(1+n)}}}\widehat{e^{-i b}}\bigg].
\end{gather*}
We continue to use the notation $|v\rangle$ as the eigenstates of $\hat v$ upon which holonomies act as translations,
\begin{gather*
\hat v |v\rangle =v |v\rangle ,\qquad \widehat{e^{i v' b}}|v\rangle=|v+v'\rangle .
\end{gather*}
These states are also eigenstates of $\hat A$,
\[
\hat A|v\rangle=A_v|v\rangle ,
\]
with eigenvalues
\begin{gather*}
A_v =
\frac{1+n}{8qV_o\gamma^2}\left(\frac{p_*^n}{2}\right)^{-\frac{3}{2(1+n)}}
\left[\frac{(1+n)\gamma\kappa^2}{3}\right]^{\frac{1+4n}{2(1+n)}}
|v|^{\frac{3(1+2n-q)}{2(1+n)}}\left(|v+1|^{\frac{3q}{2(1+n)}}-|v-1|^{\frac{3q}{2(1+n)}}\right) .
\end{gather*}
\subsection{Minisuperspace parametrization}
While in LQG the area spectrum is bounded from below by the minimum area $\Delta$, due to the symmetry reduction the same property is not shared by loop quantum cosmology. Nonetheless, we have seen that one may draw inspiration from the full theory and \emph{assume} that the kinematical area of any loop inside the comoving volume $V_o$ is bounded by the area gap for the gauge invariant states which are likely to be realized in a homogeneous context. This value is (twice) the LQG area gap \cite{AsW},
\begin{gather}\label{minar2}
\Delta= 4\sqrt{3}\pi \gamma l_{\text{Pl}}^2 ,
\end{gather}
so that
\begin{gather}\label{ineqal}
(a \ell_0)^2\geq \Delta .
\end{gather}
This step is rather speculative inasmuch as it borrows a result of the background-independent framework and forces it into the symmetry-reduced model. It is necessary, however, because the quantum scalar constraint in minisuperspace would be singular if one maintained the limit $\ell_0\to0$. Moreover, the semiclassical limit and the Wheeler--DeWitt equation are reproduced correctly.
In a general background, the edges of a spin-network state would intersect a given cell only once. By symmetry, the edges of a spin-network state in minisuperspace should traverse the f\/iducial cell, rather than intersecting it from one side; hence the factor of two in equa\-tion~\eqref{minar2}~\cite{AsW}. Notice, however, that there is no unique way of f\/ixing the value of $\Delta$, and calculations accounting for dif\/ferent details can produce dif\/ferent numerical prefactors. At any rate, these dif\/ferences are not so large as to give qualitatively inequivalent physical ef\/fects.
If the inequality \eqref{ineqal} is saturated (smallest possible holonomy path), the comoving cell area is also the comoving area gap, that is, the smallest non-vanishing eigenvalue of the area operator measuring comoving surfaces. In particular,
\begin{gather}\label{buimp}
\frac{\ell_0^2}{V_o^{2/3}}=\frac{\Delta}{p} = \left(\frac{p_*}{\Delta}\bar\mu\right)^{1/n} .
\end{gather}
One has $\bar\mu=\ell_0/V_o^{1/3}$ if $p_*=\Delta$ and
\begin{gather}\label{impqs}
n=\frac12 ,
\end{gather}
a choice corresponding to the improved quantization scheme~\cite{acs,aps3,APSV}. The set~$\{|v\rangle\}$ becomes the eigenstate basis of the volume operator, $v\propto p^{3/2}=V$. As the Universe expands, the comoving area gap shrinks to zero and the geometry is better and better described by classical general relativity, while near the big bang quantum ef\/fects become important.
Originally, the variables $p$ and $c$ were used instead of $v$ and $b$, corresponding to $\bar\mu=1$ ($n=0$). In this ``old quantization scheme'', the states $|v\rangle=|\mu\rangle$ coincide with the basis eigenstates of the momentum operator~$\hat p$, with eigenvalues~$v\propto p$~\cite{abl,boj7}. This case leads to severe restrictions of the matter sector if the wavefunctions solving the Hamiltonian constraint are required to be normalizable and to reproduce the classical limit at large scales~\cite{NeSa}. Also for such reason, the improved quantization scheme seems to be the most natural and, as we have already seen, the most reasonable in a purely homogeneous context. However, later motivations lead us to keep~$n$,~$p_*$ and the other free parameters of the model as general as possible. In this case, $p_*$ is some physical squared length determined by the theory which may dif\/fer from the mass gap~$\Delta$.
\subsection{Ef\/fective equations of motion}
A semiclassical state $|\Psi_{\rm sc}\rangle$ is peaked around some point~$(v,b)$ in the classical phase space.
One can compute the expectation value of the Hamiltonian constraint operator thereon, using an appropriate inner product.
Accordingly, for the gravitational part of the Hamiltonian operator~\eqref{cig} we approximate its expectation value as
$\langle\Psi_{\rm sc}|\widehat{\sin \bar\mu c} \hat A \widehat{\sin \bar\mu
c}|\Psi_{\rm sc}\rangle \approx A_v\sin^2 \bar\mu c$, and we may write (e.g.,~\cite{lqc1,Din08,Tav08})
\begin{gather}\label{Hg}
\langle\Psi_{\rm sc}|\hat C|\Psi_{\rm sc}\rangle \approx -\frac{6}{\gamma^2} \alpha\sqrt{p} \frac{\sin^2 \bar\mu c}{\bar\mu^2}+2\kappa^2 p^{3/2}\rho ,
\end{gather}
where the scalar f\/ield energy density is
\begin{gather}\label{rhola}
2\kappa^2\rho := \langle v|\hat C_{\rm mat}|\lambda\rangle = \frac{\nu P_\phi^2}{2p^{3/2}}+p^{3/2}V .
\end{gather}
We have two correction functions, $\alpha$ and $\nu$. The f\/irst is
\begin{gather}
\alpha=\frac{\sigma}{12q}v\left(\left|1+\frac1v\right|^\frac{6q}{\sigma}-\left|1-\frac1v\right|^\frac{6q}{\sigma}\right) ,\label{alpha}
\end{gather}
where
\begin{gather}\label{cmini}
\sigma=4(1+n) ,\qquad \frac13<q\leq 1 .
\end{gather}
The matter correction function is
\begin{gather}
\nu := (\langle v|\widehat{v^{1-l}}\widehat{v^{l-1}}|v\rangle)^{\frac{6}{(1-l)\sigma}} = \left[\frac{v}{2l}\left(\left|1+\frac1v\right|^l-\left|1-\frac1v\right|^l\right)\right]^{\tfrac{6}{(1-l)\sigma}},\label{nudef}
\end{gather}
where the ambiguity $l$ lies in the range \cite{boj12, QSD5}
\begin{gather
\frac12\leq l <1 .
\end{gather}
When $\alpha=1$ and the matter sector is a massless free scalar f\/ield, equation~\eqref{Hg} is exact~\cite{bo10}. In general, however, the evolution of a f\/initely-spread semiclassical state will produce quantum f\/luctuations leading to additional corrections to equation~\eqref{Hg}~\cite{bo11, BHS}. Assuming that the semiclassical wave-packet of the Universe does not spread appreciably, we can stick with equation~\eqref{Hg} also in the presence of a nontrivial scalar potential. Then, the matter energy density~$\rho$ is given by equation~\eqref{rhola}.
The Hamilton equation of motion for the densitized triad gives the Hubble parameter
\begin{gather}\label{hub}
H = \alpha \frac{\sin 2\bar\mu c}{2\gamma a\bar\mu} .
\end{gather}
In the classical limit, $c\to\gamma\dot a$ and the right-hand side tends to $\dot a/a$ for small $\bar\mu c$. Combining equations~\eqref{Hg} and \eqref{hub}, one gets the Friedmann equation
\begin{gather}\label{frwlqc}
H^2=\frac{\kappa^2}{3} \rho\left(\alpha-\frac{\rho}{\rho_*}\right) ,
\end{gather}
where
\begin{gather}\label{rho*}
\rho_* \equiv \frac{3}{\gamma^2\kappa^2\bar\mu^2 p} .
\end{gather}
The equation of motion of the scalar f\/ield is
\begin{gather*}
P_\phi =p^{3/2}\frac{\dot\phi}{N\nu}
\end{gather*}
while the equation for $\dot P_\phi$ leads to the ef\/fective Klein--Gordon equation
\begin{gather}\label{kglqc}
\ddot\phi+\left(3H-\frac{\dot\nu}{\nu}\right)\dot\phi+\nu U_{,\phi}=0 .
\end{gather}
As $\nu\geq 0$ has a maximum at $v=1$ and then decreases down to unity for large $v$, the friction term in equation~\eqref{kglqc} changes sign during the evolution of the universe, the f\/irst stage being of superacceleration.
Setting $\alpha=1=\nu$ in the equations of motion~\eqref{frwlqc} and~\eqref{kglqc}, one ignores inverse-volume corrections. On the other hand, in the limit $\sin 2\bar\mu c\to 2\bar\mu c$ one neglects holonomy corrections and the second term in equation~\eqref{frwlqc} is dropped.
The left-hand side of equation~\eqref{frwlqc} is positive def\/inite and, if $\rho>0$ ($\alpha>0$ if $n>-1$), the energy density is \emph{bounded from above}:
\begin{gather}\label{lowbo}
\rho\leq\alpha\rho_* .
\end{gather}
When $\rho_*\propto a^{2(2n-1)}$ varies with time, there is no constant absolute upper bound. This is avoided in the improved quantization~\eqref{impqs}, where the critical density is constant:
\[
\rho_* = \frac{3}{\gamma^2\kappa^2 p_*} .
\]
For the particular choice $p_*=\Delta$, the critical density is less than half the Planck density,
\begin{gather}\label{rho*im}
\rho_* = \frac{\sqrt{3}}{32\pi^2\gamma^3}\rho_{\text{Pl}}\approx 0.41 \rho_{\text{Pl}} ,
\end{gather}
where we used the value $\gamma\approx 0.238$ \cite{Mei04} from the computation of the entropy of non-rotating black-hole isolated horizons~\cite{ABCK,ABK,Mei04}. The numerical prefactor depends on equations~\eqref{minar2} and~\eqref{ineqal} and it could change in a more complete formulation of the model, but not in a way leading to qualitative dif\/ferences.
If the ambiguity $q$ is set equal to~1, $\alpha=1$ and the lower bound~\eqref{lowbo} is the f\/ixed constant~\eqref{rho*im}~\cite{aps3,sin06,SVV}. Thus, the avoidance of the big bang singularity in LQC is conf\/irmed at the kinematical level (via the spectrum of the inverse volume operator), by the full quantum Hamiltonian dynamics and through the ef\/fective dynamical equations. The big bang is replaced by a bounce at~$H=0$,
where the energy density is about half the Planck energy.
These results are encouraging but, of course, insuf\/f\/icient to establish a solid solution of the big bang issue.
First, the minisuperspace quantization is a toy model. Second, even choosing the improved quantization scheme, the critical density
$\alpha\rho_*$ may be non-constant if $q\neq 1$, in which case one might think that the neat bounce interpretation is lost.
Nevertheless, we have seen how loop quantum gravity could resolve the big bang singularity on one hand, and how far we are from a complete
understanding of the full theory from the other.
\subsection{Inverse-volume corrections in minisuperspace models}
We discuss now the correction functions $\alpha$ and $\nu$ from the point of view of their asymptotic limits, later stressing an interpretational issue.
On a semiclassical state the eigenvalues of $\widehat{|v|^{l-1}}$ are approximated by the classical variable~$v^{l-1}$ itself.
Consistently, the classical limit corresponds to a large-volume approximation where $v\gg 1$, while in the near-Planck regime
(``small volumes''; the reason for quotation marks will be soon clear) $v\ll1$. Since the momentum operator is
$\hat v= 2\widehat{p/\bar\mu}$, the total $p$-dependence of~$v$ is ef\/fectively
\[
v =\frac{12\sqrt{3}}{\sigma}\frac{p_*}{\Delta}\left(\frac{p}{p_*}\right)^{\frac{\sigma}{4}} .
\]
``Near the Planck scale'' ($v\ll1$), the correction functions read
\begin{gather}
\alpha \approx v^{2-\frac{6q}{\sigma}}=: \alpha_1\delta_{\rm Pl}^{-q_\alpha},\label{qa}\\
\nu \approx v^{\frac{6(2-l)}{(1-l)\sigma}}=: \nu_1\delta_{\rm Pl}^{-q_\nu},\label{qn}
\end{gather}
where
\begin{gather*}
q_\alpha = 1-\frac{3q}{\sigma} ,\qquad\alpha_1=\left(\frac{12\sqrt{3}}{\sigma}\frac{p_*}{\Delta}\right)^{2q_\alpha} ,\qquad
q_\nu = \frac{3(2-l)}{(1-l)\sigma} ,\qquad \nu_1=\left(\frac{12\sqrt{3}}{\sigma}\frac{p_*}{\Delta}\right)^{2q_\nu}
\end{gather*}
and
\begin{gather}\label{dp}
\delta_{\rm Pl}:= \left(\frac{p_*}{p}\right)^{\frac{\sigma}{2}}=\left(\frac{a_*}{a}\right)^\sigma .
\end{gather}
From the calculation leading to $\alpha$ and $\nu$, one can argue that the ``natural'' choice of the ambiguities $l$ and $q$ can be set at
the middle of their range:
\begin{gather}\label{natuiq}
l=\frac34 ,\qquad q=\frac12 .
\end{gather}
In the minisuperspace parametrization, the old quantization scheme corresponds to $\sigma=4$ and
\begin{gather*}
q_\alpha = \frac{5}{8} ,\qquad\alpha_1=3^{\frac{15}{8}}=O(10) ,\qquad
q_\nu = \frac{15}{4} ,\qquad \nu_1=3^{\frac{45}{4}}=O(10^5) ,
\end{gather*}
while the improved scheme has
\[
\sigma=6
\]
and
\begin{gather*}
q_\alpha = \frac{3}{4} ,\qquad\alpha_1=2^{3/2}3^{3/4}\approx 6 ,\qquad
q_\nu = \frac{5}{2} ,\qquad \nu_1=2^5 3^{5/2}\approx 500 .
\end{gather*}
In homogeneous models with $n=0$, the duration of this regime depends on the spin representation of the holonomies, small $j$ implying a very short super-inf\/lationary period and, actually, almost no intermediate stage between the discrete quantum regime and the continuum classical limit~\cite{boj12}. Since small-$j$ representations are theoretically favoured, this constitutes a problem. It will be relaxed in a dif\/ferent parametrization when inhomogeneities are taken into account.
In the quasi-classical limit (large volumes), equations~\eqref{alpha} and \eqref{nudef} can be approximated as
\begin{gather}
\alpha \approx 1+\alpha_0\delta_{\rm Pl} ,\label{ca}\\
\nu \approx 1+\nu_0\delta_{\rm Pl} ,\label{cn}
\end{gather}
where
\begin{gather*}
\alpha_0 = \frac{(3q-\sigma)(6q-\sigma)}{6^4}\left(\frac{\Delta}{p_*}\right)^2 ,\qquad
\nu_0 = \frac{\sigma(2-l)}{6^3}\left(\frac{\Delta}{p_*}\right)^2 .
\end{gather*}
For the natural choice \eqref{natuiq}, the old and improved quantization schemes in minisuperspace parametrization correspond, respectively, to
\[
\sigma=4 ,\qquad \alpha_0=\tfrac{5}{2^5 3^4}\approx 0.002 ,\qquad\nu_0=\tfrac{5}{6^3}\approx 0.02 ,
\]
and
\[
\sigma=6 ,\qquad \alpha_0=\tfrac{1}{96}\approx 0.01 ,\qquad\nu_0=\tfrac{5}{144}\approx 0.03 .
\]
Taking $q=1$ instead, one gets a negative $\alpha_0=-1/648$ for $\sigma=4$ and $\alpha_0=0$ for $\sigma=6$.
Although one can resort to dif\/ferent quantization schemes, equations~\eqref{ca}, \eqref{qa}, \eqref{cn} and~\eqref{qn} maintain the same structure, where the coef\/f\/icients $\sigma$, $q_\alpha$, and $q_\nu$ are robust in the choice of the parameters, inasmuch as their order of magnitude does not change appreciably~\cite{BHKS0}. All these parameters can be set to their ``natural'' values, which are dictated by the form of the Hamiltonian or other considerations.
Now we examine an interpretational issue, already mentioned earlier, related to any para\-met\-ri\-za\-tion in pure minisuperspace. On an ideal FRW background, open and f\/lat universes have inf\/inite spatial volume and the super-Hamiltonian constraint is formally ill
def\/ined because it entails a~divergent integration of a spatially constant quantity over a comoving spatial slice $\Sigma$,
\[
\int_\Sigma d^3 x=+\infty .
\]
To make the integral f\/inite, it is customary to def\/ine the constraint
on a freely chosen f\/inite region of size $V=a^3 V_o$, where $V_o$ is the corresponding comoving volume:
\[
\int_\Sigma d^3 x\to\int_{\Sigma(V_o)} d^3 x=V_o<+\infty .
\]
The volume appears in the correction function \eqref{dp} as $\delta_{\rm Pl}\sim
a^{-\sigma}\sim V^{-\sigma/3}$. To make $\delta_{\rm Pl}$ adimensional, one can use
the Planck length $l_{\text{Pl}}$ to write
\begin{gather*
\delta_{\rm Pl} \sim
\left(\frac{l_{\text{Pl}}^3}{V_o}\right)^{\frac{\sigma}{3}} a^{-\sigma}.
\end{gather*}
Physically, the parameter $\sigma$ is related to how the number of plaquettes of an underlying discrete state
changes with respect to the volume as the universe expands. The latter is a phenomenological prescription for the area of holonomy plaquettes,
but ideally it should be an input from the full theory \cite{bo609}. For phenomenology
at the current level of precision, the most signif\/icant parameter among $\{\alpha_0,\nu_0,\sigma\}$ is~$\sigma$, which is not as much af\/fected by dif\/ferent
choices of the minisuperspace scheme.
Since $\delta_{\rm Pl}$ is $V_o$-dependent, inverse-volume corrections cannot strictly be made sense of in a~pure minisuperspace treatment.
To cast the problem in other words, the conformal invariance of the scale factor $a$ in a non-closed universe make statements such as $a_*/a\ll 1$
independent of any physical length scale. One could interpret $V_o$ as a regulator and send $V_o\to\infty$ at the end of calculations, so that in the
quasi-classical limit there are no inverse-volume corrections at all. However, the full theory does contain these corrections, and one should explain
why they do not appear in a cosmological setting. At best, this highlights some tension in the theoretical construction of the homogeneous LQC ef\/fective
dynamics. To get a clearer picture, we should include inhomogeneities already at the fundamental level. The study of midisuperspace models, mentioned
in the previous part, is a step in that direction.
\subsection{Models with $\textsc{k}\neq 0$ and $\Lambda\neq 0$}
The f\/lat ef\/fective dynamical model has been extended to cases with curvature and a cosmological constant.
For a closed universe, $\textsc{k}=1$, there is no f\/iducial volume problem, as mentioned in Section \ref{klam}, and inverse-volume corrections are meaningful also in a pure homogeneous and isotropic setting. The cyclic bounces appearing in the dynamics of the dif\/ference evolution equation \cite{APSV} exist also at the ef\/fective level \cite{BoT,LMNT,MHS}; in particular, the big crunch of classical closed universes can be avoided \cite{SiT}. The bounce persists in an open universe, $\textsc{k}=-1$ \cite{van06}. In general, all past and future strong curvature singularities are resolved in $\textsc{k} = \pm1$ isotropic models; for the closed model, weak singularities in the past evolution may also be resolved \cite{ViS}.
There is evidence that a cosmological constant, if suitably tuned, does not spoil the singularity resolution. When $\Lambda>0$ and $\textsc{k}=1$ \cite{MHS}, the bounce is preserved if the cosmological constant is suf\/f\/iciently small. Above a certain critical value, however, periodic oscillations take place. When $\Lambda<0$, recollapse of the universe is possible, even cyclically \cite{BeP, BoT}. Whatever the sign of the cosmological constant, the ef\/fective Friedmann equation is equation~\eqref{frwlqc}, with the critical density $\rho_*$ shifted by a constant, $\Lambda$-dependent term.
\section{Inhomogeneous models}\label{latti}
So far we have not given any motivation for taking $\bar\mu\propto p^{-n}$. This is the next subject and it resides in a framework which does not enjoy the symmetries of a purely FRW background.
\subsection{Lattice ref\/inement}
In loop quantum gravity, the classical continuum of general relativity is replaced by the appearance of discrete
spatial structures. It is often expected that the scale of the discreteness is determined by the Planck length $l_{\text{Pl}}$, but if discreteness is fundamental, its scale must be set by the dynamical parameters of some underlying state. Such states are spin networks, graphs in an embedding space whose edges $e$ are labeled by spin quantum numbers $j_e$. The quantum number determines the area of an elementary plaquette intersecting only one edge $e$, given by ${\cal
A}=\gamma l_{\text{Pl}}^2 \sqrt{j_e(j_e+1)}$. The geometrical size of the plaquette changes only when the latter intersects another edge, thus increasing in quantum jumps. The scale is determined by the Planck length for dimensional reasons, but the actual size is given by the spin quantum number. Its values in a specif\/ic physical situation have to be derived from the LQG dynamical equations, a task which remains extremely dif\/f\/icult. However, given the form in which
$j_e$ appears in the dynamical equations, its implications for physics can be understood in certain phenomenological situations, such as cosmological scenarios. Then, instead of using the spin labels $j_e$, it is useful to refer to an elementary quantum-gravity length scale $L$, which needs not be exactly the Planck length.
The scale $L$ naturally arises if translation invariance is broken, e.g., by clustering matter or inhomogeneous perturbations. The comoving volume $V_o$ of the system can be discretized as a~lattice whose ${\cal N}$ cells or patches are nearly isotropic, have characteristic comoving size $\ell_0^3$, and correspond to the vertices of the spin network associated with $V_o$. The proper size of a cell is
\begin{gather}\label{Nf}
L^3:=a^3\ell_0=\frac{V}{{\cal N}} .
\end{gather}
To calculate the curvature at the lattice sites within $V_o$, we need to specify closed holonomy paths around such points. A generic holonomy plaquette is given by the composition of elementary holonomies over individual plaquettes. Therefore we set the length of the elementary holonomy to be that of the characteristic lattice cell. In other words, the elementary loops of comoving size~$\ell_0$ we have talked about until now def\/ine the cells' walls, while in a pure FRW background there is only one cell of volume $V_o$ (the number ${\cal N}$ is arbitrary). We naturally identify the previously ad-hoc function $\bar\mu(p)$ as the ratio of the cell-to-lattice size, under the requirement that the lattice be \emph{refined} in time:
\begin{gather}\label{N13}
\bar\mu={\cal N}^{-1/3} .
\end{gather}
The patch size $\ell_0^3$ is independent of the size of the f\/iducial region, since both $V_o$ and ${\cal N}$ scale in the same way when the size of the region is changed. Physical predictions should not feature the region one chooses unless one is specif\/ically asking region-dependent questions (such as: What is the number of vertices in a given volume?). This addresses the issue of conformal invariance brief\/ly mentioned above in minisuperspace. In the presence of inhomogeneities there is no conformal freedom and, on the other hand, f\/luxes are determined by the inhomogeneous spin-network quantum state of the full theory associated with a given patch~\cite{boj11}. This implies that to change the f\/iducial volume~$a^3V_o$ would change the number of vertices of the underlying physical state. Therefore, there is no scaling ambiguity in the equations of motion~\cite{boj11,BH2}, although the physical observables will depend on the choice of spin-network state.
The spin-network state described by the lattice can be (and usually is) excited by the action of the Hamiltonian operator on the spin vertices, increasing their number and changing their edge labels \cite{RS3, inv-vol-lqg2}. This process has not yet been established univocally in the full theory, so it is convenient to parametrize the number of vertices as in equation~\eqref{Nf} \cite{bo609}, where the length $L(t)$ is state dependent and, by assumption, coordinate independent; its time dependence is inherited from the state itself. As the kinematical Hilbert space is usually factorized into gravitational and matter sectors, the problem here emerges of how to def\/ine a natural clock when matter does not enter in the def\/inition of a (purely geometrical) spin network. This issue will require a much deeper understanding of the theory. So, as unsatisfactory as equation~\eqref{Nf} may be, we take it as a phenomenological ingredient in the present formulation of inhomogeneous LQC.
The general form \eqref{bu2} of $\bar\mu(p)$ is obtained if $L(t)$ scales as
\begin{gather*
L\sim a^{3(1-2n)} .
\end{gather*}
Homogeneous models adopting equation~\eqref{buimp} feature holonomies which depend on triad variables; in other words, curvature components are constrained by the area operator although this does not appear in the full constraint. On the other hand, in inhomogeneous models the dependence of the parameter $\bar\mu$ on $p$ is implemented at state (rather than operatorial) level, in closer conformity with the full theory \cite{bo609}.
As a side remark, the patches of volume $L^3$ f\/ind a most natural classical analogue in inhomogeneous cosmologies, in particular within the separate universe picture \cite{WMLL}. For quantum corrections, the regions of size $L^3$ are provided by an underlying discrete state and thus correspond to quantum degrees of freedom absent classically. However, the discrete nature of the state implies that inhomogeneities are unavoidable and no perfectly homogeneous geometry can exist. Given these inhomogeneities and their scale provided by the state, one can reinterpret them in a classical context, making use of the separate universe picture. There, the volume~$V$ can be regarded as a region of the universe where inhomogeneities are non-zero but small. This region is coarse grained into smaller regions of volume~$L^3$, each centered at some point~${\bf x}$, wherein the universe is~FRW and described by a ``local'' scale factor $a(t,{\bf x})=a_{\bf x}(t)$. The dif\/ference between scale factors separated by the typical perturbation wavelength $|{\bf x}'-{\bf x}|\sim \lambda\ll V^{1/3}$ def\/ines a spatial gradient interpreted as a metric perturbation. In a perfectly homogeneous context, $L^3\sim V$ and there is no sensible notion of cell subdivision of $V$; this is tantamount to stating that only the f\/iducial volume will enter the quantum corrections and the observables, ${\cal N}={\cal N}_0$. On the other hand, in an inhomogeneous universe the quantity $L^3$ carries a time dependence which, in turn, translates into a momentum dependence. The details of the cell subdivision (number of cells per unit volume) are intimately related to the structure of the small perturbations and their spectrum. Thus, lattice ref\/inement is better suitable in the cosmological perturbation analysis. As long as perturbations are linear and almost scale invariant, the size of the volume within which the study is conducted is totally irrelevant.
\subsubsection{Critical density and quantum corrections}
From equations~\eqref{rho*}, \eqref{N13} and \eqref{Nf}, the critical density is
\begin{gather}\label{rhol}
\rho_* =\frac{3}{\gamma^2\kappa^2} \left(\frac{{\cal N}}{V}\right)^{2/3}= \frac{3}{\gamma^2\kappa^2 L^2}.
\end{gather}
In all quantization schemes but the improved one ($n=1/2$), $\rho_*$ is not constant and depends on the dynamical patch size $L$. In any case, the critical density is a number density which depends neither on the size of the f\/iducial volume nor on coordinates, so it is physically well def\/ined even outside the improved quantization scheme.
Similar considerations hold for the quantum correction~$\delta_{\rm Pl}$. In a purely homogeneous universe, the only way to write down equation~\eqref{dp} is $\delta_{\rm Pl}\propto (l_{\text{Pl}}/V^{1/3})^\sigma$, which is volume dependent. On the other hand, in the lattice interpretation
\begin{gather}\label{main}
\delta_{\rm Pl}=\left(\frac{\ell_{\text{Pl}}}{L}\right)^{\tilde\sigma} ,
\end{gather}
and the same quantity is determined by the inhomogeneous state through the patch size $L$. Notice that $\tilde\sigma>0$ is not the parameter $\sigma$ determined by equation~\eqref{cmini}; $n=1/2$ will not imply $\tilde\sigma=6$. The inverse-volume corrections~\eqref{main} do not depend on holonomies due to the use of Thiemann's trick (such as equation~\eqref{eq:identidad-lig})~\cite{BCT2}. Another reason to understand this fact is that~$L^2$ is nothing but the expectation value of the f\/lux operator $\hat{F}_S=\int_S d^2y\,E^a_in_a$ (through a surface $S$ with co-normal~$n_a$) on a semiclassical state~\cite{BCT2}. In inverse-volume as well as holonomy corrections, one refers to elementary building blocks of a discrete state, respectively, the plaquette areas and the edge lengths. A pure minisuperspace quantization makes use of macroscopic parameters such as the volume of some f\/iducial region, and f\/luxes are calculated on comoving areas $\sim V_o^{2/3}$. On the other hand, in the lattice-ref\/inement formulation of loop quantum cosmology one uses the microscopic volume of a cell, and f\/luxes are def\/ined on comoving areas $\sim \ell_0^2$. This leads to equation~\eqref{main}, with some phenomenological parameter $\tilde\sigma$.
Intuitively, holonomy corrections become large when the Hubble scale $H^{-1}= a/\dot{a}\sim \gamma L$ is of the size of the discreteness scale, an extreme regime in cosmology. In terms of the classical energy density $\rho=3H^2/\kappa^2$, holonomy corrections can be quantif\/ied by the parameter
\[
\delta_{\rm hol}:= \frac{\rho}{\rho_*}= (\gamma HL)^2 .
\]
These are small when $\delta_{\rm hol}\ll 1$. In order to compare inverse-volume with holonomy corrections, we notice that
\begin{gather}\label{dpl}
\delta_{\rm Pl}=\big(\gamma l_{\text{Pl}} H \delta_{\rm hol}^{-1/2}\big)^{\tilde\sigma} .
\end{gather}
For a universe of causal size $H^{-1}\sim l_{\text{Pl}}$, inverse-volume corrections are considerable and behave very dif\/ferently from what is normally expected for quantum gravity. For small densities, holonomy corrections are small, but inverse-volume corrections may still be large because they are magnif\/ied by an inverse power of $\delta_{\rm hol}$. As the energy density decreases in an expanding universe, holonomy corrections fall
to small values, while inverse-volume corrections increase. For instance, in an inf\/lationary regime with a
typical energy scale of $\rho\sim 10^{-10}\rho_{\rm Pl}$, we can use equation~(\ref{dpl}) with $\tilde\sigma=4$ to write $\delta_{\rm hol} \sim 10^{-9}/\sqrt{\delta_{\rm Pl}}$. Small holonomy corrections of size $\delta_{\rm hol}<10^{-6}$ then require inverse-volume correction larger than $\delta_{\rm Pl}>10^{-6}$. This interplay of holonomy and inverse-volume corrections can make loop quantum cosmology testable, because it leaves only a f\/inite window for consistent parameter values, rather
than just providing Planckian upper bounds. It also shows that inverse-volume corrections become dominant for suf\/f\/iciently small densities (eventually, of course, they are suppressed as the densities further decrease).
\subsubsection{Lattice parametrization}
The lattice ref\/inement picture allows us to reinterpret minisuperspace quantization schemes in a~dif\/ferent language.
Equation \eqref{main} replaces the total lattice f\/iducial volume $V$ as the ``patch'' (i.e., cell) volume $L^3$ \cite{bo08}. This means that one makes the formal replacement $V\to V/{\cal N}$ everywhere in minisuperspace expressions, which can be also justif\/ied as follows. At the kinematical level, internal time is taken at a f\/ixed value but the geometry still varies on the whole phase space. In this setting, we must keep ${\cal N}$ f\/ixed to some constant ${\cal N}_0$ while formulating the constraint as a~composite operator. Since the vertex density does not depend on the choice of f\/iducial volume, it is physically reasonable to expect the ${\cal N}_0$ factor to be hidden in the kinematical quantity~$a_*$ (or~$p_*$). The net result is the Hamiltonian constraint operator of the previous sections.
However, when one solves the constraint or uses it for ef\/fective equations, one has to bring in the dynamical nature of ${\cal N}$ from an underlying full state. This is the motivation for promo\-ting~${\cal N}$ to a time-dependent quantity. For some stretches of time, one can choose to use the scale factor~$a$ as the time variable and represent ${\cal N}(a)$ as a power law (equation~\eqref{N13}),
\begin{gather}\label{m}
{\cal N}={\cal N}_0a^{6n} .
\end{gather}
Overall, quantum corrections are of the form \eqref{main},
\begin{gather}\label{13}
\delta_{\rm Pl}= \left(\ell_{\text{Pl}}^3\frac{{\cal N}}{V}\right)^\frac{\tilde \sigma}{3}= \left(\ell_{\text{Pl}}^3\frac{{\cal N}_0}{V_o}\right)^{\frac{\tilde \sigma}{3}} a^{(2n-1)\tilde \sigma} ,
\end{gather}
where $\tilde \sigma>0$. This equation cannot be obtained in a pure minisuperspace setting.
The parameter $a$ plays two roles, one as a dynamical geometric quantity and the other as internal time. While writing down the semiclassical Hamiltonian with inverse-volume (and holonomy) corrections, one is at a non-dynamical
quantum-geometric level. Then, internal time is taken at a f\/ixed value but the geometry still varies on the whole phase space. In this setting, we must keep~${\cal N}$ f\/ixed while formulating the constraint as a composite operator. The net result is the Hamiltonian constraint operator of the basic formulation of loop quantum cosmology \cite{boj2,boj7} not taking into account any ref\/inement, corresponding to $n=0$ and $\tilde \sigma=\sigma$. On the other hand, equation~\eqref{m} captures operator as well as state properties of the ef\/fective dynamics. The parametrization of ${\cal N}$ as a power law of the scale factor is simply a way to encode the qualitative (yet robust) phenomenology of the theory. The general viewpoint is similar to mean-f\/ield approximations which model ef\/fects of underlying degrees of freedom by a single, physically motivated function.
Comparing with the earlier minisuperspace parameterization, equation~\eqref{13} gives $\sigma=(1-2n)\tilde \sigma$. Since $\partial{\cal N}/\partial V\geq 0$, one has $n\geq 0$: the number of vertices ${\cal N}$ must not decrease with the volume, and it is constant for $n=0$. Also, $\ell_0\sim a^{1-2n}$ is the geometry as determined by the state; in a discrete geometrical setting, this has a lower non-zero bound which requires $n\leq 1/2$. In particular, for $n=1/2$ we have a constant patch volume as in the improved minisuperspace quantization scheme \cite{aps3}. In contrast with the minisuperspace parametrization \eqref{cmini}, in the ef\/fective parametrization of equation~\eqref{13} we have $\sigma=0$ for the improved quantization scheme $n=1/2$.
The range of $n$ is then
\begin{gather*
0<n\leq \frac12 .
\end{gather*}
The critical density $\rho_*\propto a^{6(2n-1)}$, equation~\eqref{rhol}, is still constant for $n=1/2$.
The exponent $\tilde \sigma$ in equation~\eqref{m} can be taken as a small positive integer. In fact, the correction function $\delta_{\rm Pl}$ depends on f\/lux values, corresponding to $p$ for the isotropic background. Since $p$ changes sign under orientation reversal but the operators are parity invariant, only even powers of $p$ can appear, giving $\tilde \sigma=4$ as the smallest value. Therefore we set $\tilde \sigma\geq 4$.
To summarize, $\sigma$ is a time-independent parameter given by the quasi-classical theory and with range
\begin{gather*
\sigma\geq 0 .
\end{gather*}
$\sigma$ may be dif\/ferent in $\alpha$ and $\nu$ for an inhomogeneous model, but we assume that the background equations~\eqref{ca} and~\eqref{cn} are valid also in the perturbed case. The coef\/f\/icients $\alpha_0$ and $\nu_0$ become arbitrary but positive parameters. In fact, from the explicit calculations of inverse-volume operators and their spectra in exactly isotropic models and for regular lattice states in the presence of inhomogeneities~\cite{boj6,boj8, BHKS0}, correction functions implementing inverse-volume corrections approach the classical value always from above. This implies that
\[
\alpha_0\geq 0 ,\qquad \nu_0\geq 0 .
\]
The lattice parametrization replaces the one for homogeneous LQC. In fact, strictly speaking, the use of one parametrization instead of the other is not a matter of choice. A perfectly homogeneous FRW background is an idealization of reality which, in most applications, turns out to be untenable. The study of cosmological perturbations with inverse volume corrections~\cite{BC,BCT1,BCT2,BH1, BH2,BHKS,BHKS2,CH} is an example in this respect. In that case, therefore, the lattice ref\/inement parametrization is not only useful, but also required for consistency. Ef\/fective linearized equations in the presence of holonomy corrections are under development, but we do not have complete control over them yet.
For vector and tensor modes a class of consistent constraints with a closed algebra is known~\cite{BH1,BH2,MCBG}, and therefore inspections of cosmological holonomy ef\/fects have been analyzed for gravitational waves~\cite{CMNS2,GB1,GB3,Mie1,GB2}. On the other hand, anomaly cancellation in the scalar sector has been worked out only recently~\cite{CaBa,CMBG, ed-scalar}.
We conclude with some comments on the superinf\/lationary phase of loop quantum cosmo\-lo\-gy. In the near-Planckian regime, the small-$j$ problem in the homogeneous parametrization is reinterpreted and relaxed in terms of the lattice embedding. The volume spectrum depends on the quadratic Casimir in $j$ representation: $ \bar\mu^{-n}\sim V^{2/3}\sim \sqrt{C_2(j)}\sim j$. A higher-$j$ ef\/fect can be obtained as a ref\/inement of the lattice (smaller $\bar\mu$) \cite{BCK}, thus allowing for long enough superacceleration. A change in $\bar\mu(p)$ can be achieved by varying the comoving volume $V_o$. This is an arbitrary operation in pure FRW, while in inhomogeneous models $\bar\mu$ is a physical quantity related to the number of vertices of the underlying reduced spin-network state. As long as a calculation of this ef\/fect from the full theory is lacking, we will not be able to predict the duration of the small-volume regime. More importantly, the closure of the constrained algebra is a wide open issue outside the approximation \eqref{ca}, \eqref{cn}, and early works on superinf\/lationary cosmological perturbations \cite{boj12,CaC,CMNS0,hos04,ShH, TSM} have not received a rigorous conf\/irmation.
\section{Conclusions}\label{concl}
The kinematical structure of LQG is well def\/ined although there are some technical dif\/f\/iculties in the construction of the physical theory.
One way of increasing our understanding of loop quantization is to apply it to simple systems. Here we look at applications to cosmological systems,
also with the hope of making progress in developing a realistic theory of quantum cosmology.
Owing to the presence of an underlying full theory, this procedure has been much more successful than the earlier WDW quantization.
Trying to follow the steps of LQG, using the fact that the underlying geometry is discrete and making physically well-motivated {\em Ans\"atze} about
the size of the f\/iducial cells, we have obtained a quantization scheme which ensures the resolution of the classical singularity as well as the
correct semiclassical limit. This is a generic feature of LQC of all minisuperspace models. We have described two examples
of minisuperspace LQC in detail:
\begin{itemize}\itemsep=0pt
\item The f\/lat FRW model is the simplest and the most rigorously studied model in LQC. We have used it to explain the kinematic structure of LQC, and to
describe the construction of the physical Hilbert space and of observables. These provide an evolution picture of the universe with respect to a massless
scalar f\/ield playing the role of a clock variable, which in turn serves to illustrate the mechanism of singularity resolution.
\item Bianchi I model serves as an example for the complications which arise in LQC, from the correct choice of the quantization scheme to
the construction of the physical Hilbert space. In this model the evolution picture is not complete yet as long as a basis of states diagonalizing the
Hamiltonian constraint remains unknown.
The Bianchi~I model also plays a crucial role in the hybrid quantization of Gowdy $T^3$ model.
\end{itemize}
Once we have got some handle on the construction of quantum theories of minisuperspace models, we need to consider models with f\/ield theory degrees
of freedom, i.e., midisuperspace models. They can serve as good toy models for testing the f\/ield-theoretical features of the full theory. Also a study of the fate of the classical singularity in these models is needed to ensure that the singularity resolution mechanism in LQC is generic and not an
artifact of the symmetry reduction. However, only one model, the polarized Gowdy $T^3$ model, has been studied so far in LQC. We have described and
compared two approaches, the hybrid quantization and the polymer quantization procedures. The hybrid quantization scheme,
although very successful, quantizes a part of the geometry in the Fock way, while the polymer quantization scheme is still incomplete. We feel the
need to study a complete loop quantization of this and other midisuperspace models for a better understanding of LQG/LQC.
The path we have followed in the f\/irst two parts is to bring more and more complicated systems under the ambit of loop quantization. We have started with f\/lat FRW, which is homogeneous and isotropic. Then we have removed the isotropy condition and studied Bianchi~I. Finally, we have tried to lift even the homogeneity condition (although retaining some other symmetries) and studied the Gowdy $T^3$ model.
Instead of looking at various models, one can take another path and look at the generic features of the theory and try to incorporate these as corrections to the classical equations. This ef\/fective equation approach, presented in Part~\ref{part3}, is a setting where the ef\/fects
of LQC can be incorporated in a way suitable for phenomenology and observations. We have described the progress made in both homogeneous and inhomogeneous contexts. In the latter, the main complication is the closure of the ef\/fective constraint algebra, whose study has been completed for inverse-volume corrections but is still ongoing when holonomy corrections are switched on. While verifying this property, we have to ensure that we retain the features which resolve the singularity while giving the correct large-scale behaviour. Once this is satisf\/ied, we can use this framework to make predictions and constrain the parameters of the model, at least in the semiclassical regime.
A number of issues are under active investigation in Loop quantization of cosmological models. Here we mention some interesting problems in the areas of LQC covered in this review. At the homogeneous level, in solvable LQC we are still trying to understand the nature of the bounce replacing the classical singularity and what are the characteristics of the universe at late times after the bounce. This has been extensively explored in the simple models where an evolution picture has been developed, but it remains a challenge to control the complete evolution even in FRW models with massive scalar f\/ields and in the Bianchi I models. Other Bianchi cosmologies need to be analyzed in greater detail because the non-trivial topologies and properties of the connections introduce additional subtleties in the quantization. Finding methods for solving these issues may indicate ways to address their counterparts in the full theory. Also, it would be important to understand whether some of the features of classical evolution (for example, the chaotic approach to singularity in Bianchi IX models) are retained after Loop quantization. Although we now have a good understanding of the big-bang type singularity from the LQC perspective, the nature and the possible resolutions of big-rip type singularities require further investigation.
Here we concentrated on the Hamiltonian formalism and on cosmological models constructed via the imposition of quantum constraints. Quantum cosmology can be also def\/ined via dif\/ferent sum-over-histories approaches. One is to recast the dynamics of LQC in a form resembling the spin-foam formulation of the full theory, and express it in terms of a ``vertex expansion'' \cite{ACH2,ACH1,CGO1,HRVW,HMQ}. This corresponds to a minisuperspace symmetry reduction of the path integral of the full theory. Another possibility, also dubbed ``spin-foam cosmology'' and attempting to get in closer contact with the full theory, is to evaluate spin-foam amplitudes on graphs interpreted as homogeneous and isotropic geometries \cite{BTRV,BRV,Hel11,LiMa,Rok10,Vid11,Vid10}. A third line of research, inspired by group f\/ield theory, consists in writing down a minisuperspace version of a~scalar f\/ield theory action where wavefunctions are promoted to f\/ields and the Hamiltonian plays the role of kinetic operator \cite{CGO2, GiO}. Interaction terms, possibly nonlocal, are responsible for topology changes and can give ef\/fective contributions to the Hamiltonian constraint in a linear mean-f\/ield approximation. All these approaches are at their f\/irst stages of development.
Much work needs to be done in midisuperspace models where only one cosmological example has been studied so far. Although the hybrid quantization
scheme has made signif\/icant progress, one should still fully justify the approach and check whether similar methods are viable for the other Gowdy models. Again, that may depend on the development of the other Bianchi setups. The polymer quantization scheme provides a genuine toy model suitable for trying to solve the problems facing the full theory, and for this reason it should be further ref\/ined. We also need to look
at other midisuperspace scenarios, particularly 1-Killing vector models which can be mapped to $2+1$ gravity with scalar f\/ields. Finally, it is not
clear how to obtain minisuperspace LQC as a sector of LQG and at what level we expect such an embedding, and whether it is possible at all. However, a similar exercise can be attempted in LQC by checking whether one can embed minisuperspace models into midisuperspace models.
The power and importance of the ef\/fective-dynamics technique have been slowly gaining momentum. Ef\/fective dynamics and ef\/fective quantum constraints provide the means to ask questions such as whether the LQC corrections are consistent with the observed universe and whether the underlying discreteness coming from LQC has any appreciable ef\/fect on the inf\/lationary imprint. All these challenges are currently being undertaken. Confrontation of LQC with experiments is one of the major open problems to be addressed. Detailed investigations on linear cosmological perturbations showed that the size of quantum corrections could be consi\-de\-rab\-ly larger than what previously expected. Both theoretical considerations and observations of the primordial spectra in the microwave sky can set limits on the ambiguity parameters in the model and highlight how the parameter range can change upon performing a symmetry enhancement from pure FRW to perturbed FRW backgrounds. Many of these results have been achieved when taking only inverse-volume corrections into account, but control over holonomy corrections is now suf\/f\/icient for completing the picture at the ef\/fective-constraints level (see Section~\ref{latti} and references therein).
The purpose of this review was to show the dif\/ferent facets of LQC and to describe the work done in various directions. These indicate that, as we move towards studying the LQC of models with lesser degrees of symmetry, a more holistic understanding of all these developments is needed. This is essential for further improving our hold on this quantum theory of gravity.
\subsection*{Acknowledgements}
K.B.\ would like to thank NSFC (Grant No.\ 10975017), China Postdoctoral Science Foundation (Grant No.\ 20100480223)
and the Fundamental Research Funds for the Central Universities for f\/inancial support.
M.M.B.\ is partially supported by the Spanish MICINN Projects No.\ FIS2008-06078-C03-03 and No.\ FIS2011-30145-C03-02.
\addcontentsline{toc}{section}{References}
|
1,116,691,500,416 | arxiv | \section{Introduction}
Perhaps the most prominent application of optimization in ML is the empirical risk minimization problem. However, inspired by the success of GANs~\cite{NIPS2014_5ca3e9b1}, ML practictioners have developed more complicated min-max and adversarial optimization formulations
\cite{yu2021fast,kuhn2019wasserstein,shafieezadeh2015distributionally,sinha2018certifiable,
lin2020gradient,NIPS2016_4588e674, huang2017context,wadsworth2018achieving,zhang2018mitigating,edwards2015censoring,celis2019improved}. Solving these multi-player games leads to issues not seen when minimizing a single loss function.
The competitive nature of a game leads to rotational dynamics that can cause intuitive gradient-based methods to fail to converge \cite{gidel2018a,daskalakis2018training,NEURIPS2020_ba9a56ce}.
A mathematical framework underlying both convex optimization and \col{saddle-point problems} is the \textit{monotone inclusion problem} (See \cite{ryu2016primer} for an introduction).
Methods developed for monotone inclusions will converge for \col{convex-concave} games as they are explicitly designed to handle such problems' governing dynamics.
Nevertheless, monotone inclusion methods and theory are not well known in the ML community,
although there has been recent interest in monotone variational inequalities, which form a special case of monotone inclusions \cite{antonakopoulos2019adaptive,gidel2018a,daskalakis2018training,NEURIPS2020_ba9a56ce,
mertikopoulos2018optimistic}.
The most prevalent methods for solving min-max games in ML are
variants of \textit{gradient descent-ascent} (GDA).
This method alternates between a gradient-descent step for the minimizing player and a gradient-ascent step for the maximizing player.
Unfortunately, GDA requires additional assumptions to converge on convex-concave games, and it even fails for some simple 2D bilinear games \cite[Prop.~1]{gidel2018a}.
While there have been several approaches to modify either GDA \cite{chavdarova2021taming,grnarova2021generative,balduzzi2018mechanics} or the underlying game objective \cite{mescheder2018training,NIPS2017_7e0a0209,NIPS2017_4588e674} to ensure convergence,
this paper instead develops a method for solving monotone inclusions that can naturally handle game dynamics.
Our approach builds upon the recently proposed projective splitting (PS)
method with forward steps~\cite{johnstone2020projective}.
PS is designed specifically for solving monotone inclusions, thus does not
fall prey to the convergence issues that plague GDA, at least for \col{convex-concave} games.
PS is within the general class of projective splitting methods invented in \cite{eckstein2008family} and developed further in \cite{eckstein2009general,alotaibi2014solving,combettes2016async,eckstein2017simplified,johnstone2018convergence,
johnstone2021single,johnstone2020only}.
These methods work by
creating a separating hyperplane between the current iterate and the solution
and then moving closer to the solution by projecting the current iterate onto this hyperplane (see Section \ref{secProjSplit} for an overview).
Other than being able to natively handle game dynamics,
the primary advantage of PS is that it \textit{fully splits} problems involving an arbitrary number of regularizers and constraints.
``Full splitting''
means that the method can handle multiple regularizers and constraints through
their respective individual proximal and projection operators,
along with the smooth terms via gradients.
What makes this useful is that many of the regularizers used in ML have proximal operators that are easy to compute \cite{parikh2013proximal}.
Despite these advantages, the preexisting PS framework has a significant
drawback: it requires deterministic gradient oracles. This feature
makes it impractical for application to large datasets for which
stochastic oracles may be the only feasible option.
\paragraph{Contributions}
The primary contribution of this work is a new projective splitting algorithm that allows for a stochastic gradient oracle. We call the method \textit{stochastic projective splitting} (SPS). It is the first stochastic method to fully split the monotone inclusion problem
\begin{align}\label{mono1}
\text{Find }z\in\rR^d
\,\,\text{ s.t. }\,\,
0 \in \sumin A_i(z) + B(z)
\end{align}
where $B$ is monotone and $L$-Lipschitz and each $A_i$ is maximal monotone and
typically set valued, usually arising from a constraint or a nonsmooth
regularizer in the underlying optimization problem or game (see for example
\cite{ryu2016primer} for definitions). It interrogates the Lipschitz operator $B$
through a stochastic oracle. Previous methods splitting this inclusion have
either required a deterministic oracle for $B$, or have made far more
restrictive assumptions on the noise or the operators \cite{briceno2011monotone+,combettes2012primal,malitsky2020forward,bot2019forward,van2021convergence}. Our proposal is the
first stochastic method that can solve min-max problems under reasonable
assumptions, while easily handling multiple regularizers and constraints.
When moving away from a deterministic gradient oracle in projective splitting, a key difficulty is that the generated hyperplanes do not guarantee separation between the solution and the current point. We solve this issue by relaxing the projection: we only update each iterate in the \textit{direction} of the noisy projection and scale its movement by a decreasing stepsize that allows for control of the stochastic error.
Using the framework of \textit{stochastic quasi-Fej\'{e}r monotonicity} \cite{combettes2015stochastic}, we prove almost-sure convergence of the final iterate and do not require averaging of the iterates (Theorem \ref{thmMain}, Section \ref{secMainResults}). We also provide a non-asymptotic convergence rate for the approximation residual (Theorem \ref{thmConvR}, Section \ref{secMainResults}).
A special case of SPS is the recently-developed Double Stepsize Extragradient Method (DSEG) \cite{NEURIPS2020_ba9a56ce}. When
only $B$ is present in \eqref{mono1}, DSEG and SPS coincide. Thus, our method extends DSEG to allow for regularizers and constraints.
Our analysis also provides a new interpretation for DSEG as a special case of projective splitting.
Our nonasymptotic convergence rate for SPS also applies to DSEG under no additional assumptions. In contrast, the original convergence rate analysis for DSEG requires either strong monotonicity or an error bound.
We close with numerical experiments on a distributionally robust sparse logistic regression problem. This is a nonsmooth convex-concave min-max problem which can be converted to \eqref{mono1} with $n=2$ set-valued operators. Owing to its ability to use a stochastic oracle, SPS performs quite well compared with deterministic splitting methods.
\paragraph{Non-monotone problems} The work \cite{NEURIPS2020_ba9a56ce} included a local convergence analysis for DSEG applied to locally monotone problems. For min-max problems, if the objective is locally convex-concave at a solution and DSEG is initialized in close proximity, then for small enough stepsizes it converges to the solution with high probability. It is possible to extend this result to SPS, along with our convergence rate analysis. This result is beyond the scope of this work, but the appendix provides a proof sketch.
\section{Background on Monotone Inclusions}
\label{secBackG}
Since they are so important to SPS, this section provides some background
material regarding monotone inclusions, along with their connections to convex
optimization, games, and ML. The appendix discusses their connections to
variational inequalities. For a more thorough treatment, we refer to
\cite{bauschke2011convex}.
\paragraph{Fundamentals}
Let $f:\rR^d\to\rR\cup\{\infty\}$ be closed, convex, and proper (CCP). Recall that
its \emph{subdifferential} $\partial f$ is given by
$
\partial f(x) \triangleq \{g:f(y)\geq f(x)+g^\top (y-x)\}.
$
The map $\partial f$ has the property
\begin{align*}
u\in \partial f(x),v\in \partial f(y)\implies (u - v)^\top(x - y) \geq 0,
\end{align*}
and any point-to-set map having this property is called a \emph{monotone
operator}. A minimizer of $f$ is any $x^*$ such that $0\in\partial f(x^*)$.
This is perhaps the simplest example of a \textit{monotone inclusion}, the
problem of finding $x$ such that $0 \in T(x)$, where $T$ is a monotone
operator. If $f$ is smooth, then $\partial f(x) = \{\nabla f(x)\}$ for all $x$, and
the monotone inclusion $0\in\partial f(x)$ is equivalent
to the first-order optimality condition $0 = \nabla f(x)$.
Next, suppose that we wish to minimize the sum of two CCP functions $f,g:\rR^d\to\rR\cup\{\infty\}$. Since under certain regularity conditions (\cite[Thm.~16.47]{bauschke2011convex}) it holds that
$
\partial (f+g) = \partial f + \partial g,
$
minimizing $f + g$ may be accomplished by solving the monotone inclusion
$
0\in\partial f(x) + \partial g(x).
$
The ``+" here denotes the Minkowski sum (also known as the \emph{dilation},
the set formed by collecting the sums of all pairs of points from the two sets); sums of monotone operators formed in this way are also monotone.
Constrained problems of the form $\min_{x\in\cC} f(x)$ for a
closed convex set $\cC$ are equivalent to the above formulation with $g(x) =
\iota_{\cC}(x)$, where $\iota_{\cC}(x)$ denotes the \textit{indicator
function} returning $0$ when $x\in\cC$ and $+\infty$ otherwise. The
subdifferential of the indicator function, $\partial\iota_{\cC}$, is known as
the \emph{normal cone map} and written as $N_{\cC}$. For closed convex sets,
the normal cone map is a maximal~\cite[Def.~20.20]{bauschke2011convex}
monotone operator~\cite[Example 20.26]{bauschke2011convex}
Under certain regularity conditions~ \cite[Cor.~16.5] {bauschke2011convex}, minimizing a sum of CCP
functions $f_1,\ldots,f_n$ is equivalent to solving the monotone inclusion
formed from the sum of their subdifferentials:
\begin{align*}
x^*\in \underset{x\in\rR^d}{\arg\min} \sumin f_i(x)
\iff
0 \in\sumin \partial f_i(x^*).
\end{align*}
Multiple constraints of the form $x \in \cap_{i=1}^c \cC_i$, where each
set $\cC_i \subseteq \rR^d$ is closed and convex, may be imposed by adding a
sum of indicator functions $\sum_{i=1}^c\iota_{\cC_i}$ to the objective. Under standard
regularity conditions \cite[Cor.~16.5]{bauschke2011convex}), we thus have
\begin{align}\label{eqMultiReg}
x^*\in\underset{x \in \left(\bigcap_{i=1}^c \cC_i\right)}{\arg\min} \sumin f(x)
\iff
0 \in\sumin\partial f_i(x^*) + \sum_{j=1}^c N_{\cC_j}(x^*).
\end{align}
\paragraph{ML applications}
The form~\eqref{eqMultiReg} can be used to model ML problems with multiple
constraints and/or nonsmooth regularizers, including
sparse and overlapping group lasso \cite{jacob2009group},
sparse and low-rank matrix estimation problems \cite{savalle2012estimation}, and
rare feature selection~\cite{yan2020rare}. See \cite{pedregosa2018adaptive} for an overview.
\paragraph{Games}
Consider a two-player noncooperative game
in which each player tries to selfishly minimize its own loss, with each loss depending on the actions of both players. Typically, the goal is to find a Nash equilibrium, in which neither player can improve its loss by changing strategy:
\begin{align}\label{defNash1}
x^* \in \underset{x\in\Theta}{\arg\min}\; F(x,y^*)
\quad\text{and}\quad
y^* \in \underset{y\in\Omega }{\arg\min}\; G(x^*,y).
\end{align}
Assuming that the admissible
strategy sets $\Theta\subseteq \rR^{d_x}$ and $\Omega\subseteq \rR^{d_y}$ are closed and convex and
that $F$ and $G$ are differentiable, the first-order necessary conditions for solving
the Nash equilibrium problem are
\begin{align}\label{gameMono}
0 \in
\left[
\begin{array}{c}
\nabla_x F(x^*,y^*)\\
\nabla_y G(x^*,y^*)
\end{array}
\right]
+
\big(
N_\Theta(x^*) \times
N_\Omega(y^*)
\big).
\end{align}
\col{If $G=-F$, then \eqref{defNash1} is a min-max game. If in addition,
$F$ is convex in $x$ and concave in $y$} then
$B: (x,y) \mapsto (\nabla_x
F(x,y),-\nabla_y F(x,y))^\top$ is monotone\footnote{\col{Sufficient conditions for the monotonicity of \eqref{gameMono} in the case where $G\neq-F$ are discussed in e.g.~\cite{scutari2014real,briceno2013monotone}}}
on
$\rR^{d_x+d_y}$ \cite{rockafellar1970monotone}.
\col{In many applications, $B$ is also Lipschitz continuous.}
In this situation, \eqref{gameMono} is a monotone inclusion
involving two operators $B$ and $N_{\Theta \times \Omega}$, with $B$ being
Lipschitz. Using the simultaneous version of GDA on~\eqref{defNash1} is
equivalent to applying the forward-backward method (FB)
\cite[Thm.~26.14]{bauschke2011convex} to
\eqref{gameMono}. However, convergence of FB requires that the operator $B$ be
\textit{cocoercive} \cite[Def.~4.10]{bauschke2011convex}, and not merely
Lipschitz \cite[Thm.~26.14]{bauschke2011convex}. Thus, simultaneous GDA fails
to converge for~\eqref{defNash1} without additional assumptions (see \cite[Prop.~1]{gidel2018a} for a simple counterexample).
Regularizers and further constraints may be imposed by adding more
operators to~\eqref{gameMono}. For example, if one wished to apply a (nonsmooth)
convex regularizer $r:\rR^{d_x} \rightarrow \rR \cup \{+\infty\}$ to the $x$
variables and a similar regularizer $d:\rR^{d_y} \rightarrow \rR \cup \{+\infty\}$
for the $y$ variables, one would add the operator $A_2 : (x,y) \mapsto
\partial r(x) \times \partial d(y)$ to the right-hand side
of~\eqref{gameMono}.
\paragraph{ML applications of games}
Distributionally robust supervised learning (DRSL) is an emerging framework
for improving the stability and reliability of ML models in the face of
distributional shifts
\cite{yu2021fast,kuhn2019wasserstein,shafieezadeh2015distributionally,sinha2018certifiable,
lin2020gradient,NIPS2016_4588e674}. Common approaches to DRSL formulate the
problem as a min-max game between a learner selecting the model parameters and
an adversary selecting a worst-case distribution subject to some ambiguity
set around the observed empirical distribution. This min-max problem is often
further reduced to either a finite-dimensional saddlepoint problem or a convex
optimization problem.
DRSL is a source of games with multiple constraints/regularizers. One such
formulation, based on \cite{yu2021fast}, is discussed in the experiments
below. The paper \cite{NIPS2016_4588e674} uses an amiguity set based on
$f$-divergences, while \cite{sinha2018certifiable} introduces a Lagrangian
relaxation of the Wasserstein ball. When applied to models utilizing multiple
regularizers \cite{jacob2009group,savalle2012estimation,yan2020rare}, both of
these approaches lead to min-max problems with multiple regularizers.
Other applications of games in ML, although typically nonconvex, include
generative adversarial networks
(GANs)~\cite{NIPS2014_5ca3e9b1,pmlr-v70-arjovsky17a}, fair
classification~\cite{wadsworth2018achieving,zhang2018mitigating,edwards2015censoring,celis2019improved}
, and adversarial privacy \cite{huang2017context}.
\paragraph{Resolvents, proximal operators, and projections}
A fundamental computational primitive for
solving monotone inclusions is the \textit{resolvent}. The resolvent of a monotone operator
$A$ is defined to be
$
J_A \triangleq (I+A)^{-1}
$
where $I$ is the identity operator and the inverse of any operator $T$ is
simply $T^{-1} : x \mapsto \{y:Ty \ni x\}$. If $A$ is maximal monotone, then
for any $\rho>0$, $J_{\rho A}$ is single valued, nonexpansive, and has domain
equal to $\rR^d$~\cite[Thm. 21.1 and Prop. 23.8]{bauschke2011convex}.
Resolvents generalize proximal operators of convex functions: the proximal
operator of a CCP function $f$ is
\begin{align*}
\text{prox}_{\rho f}(t) \triangleq \arg\min_{x\in\rR^d}\left\{\rho f(x) + (1/2)\|x - t\|^2\right\}.
\end{align*}
It is easily proved that $\prox_{\rho f} = J_{\rho\partial f}$. In turn,
proximal operators generalize projection onto convex sets: if $f = \iota_\cC$,
then $\prox_{\rho f} = \proj_\cC$ for any $\rho>0$.
In many ML applications, proximal operators, and hence resolvents, are
relatively straightforward to compute. For examples, see
\cite[Sec.~6]{parikh2013proximal}.
\paragraph{Operator splitting methods}
\emph{Operator splitting methods} attempt to solve monotone inclusions such
as~\eqref{mono1} by a sequence of operations that each involve only one of the
operators $A_1,\ldots,A_n,B$. Such methods are often presented in the context
of convex optimization problems like \eqref{eqMultiReg}, but typically apply
more generally to monotone inclusions such as \eqref{mono1}. In the specific
context of~\eqref{mono1}, each iteration of such a method ideally handles
each $A_i$ via its resolvent and the Lipschitz operator $B$ by explicit (not
stochastic) evaluation. This is a feasible approach if the original problem
can be decomposed in such a way that the resolvents of each $A_i$ are
relatively inexpensive to compute, and full evaluations of $B$ are possible.
Although not discussed here, more general formulations in which matrices couple the
arguments of the operators can broaden the applicability of operator splitting
methods.
\section{The Projective Splitting Framework}\label{secProjSplit}
Before introducing our proposed method, we give a brief introduction to the projective splitting class of methods.
\paragraph{The extended solution set}
Projective splitting is a primal-dual framework and operates in an extended space of primal and dual variables. Rather than finding a solution to \eqref{mono1}, we find a point in the \textit{extended solution set}
\begin{align}\label{Sdef}
\cS \triangleq \left\{
(z,w_1,\ldots,w_{n+1})\in\rR^{(n+2)d}
\;\Big|\;
w_i\in A_i(z)\, \forall\, i=1,\ldots,n,
w_{n+1}=B(z),
\sum_{i=1}^{n+1} w_i=0\right\}.
\end{align}
Given $p^*=(z^*,w_1^*\ldots,w_{n+1}^*)\in\cS$, it is straightforward to see
that $z^*$ solves \eqref{mono1}. Conversely, given a solution $z^*$ to
\eqref{mono1}, there must exist $w_1^*,\ldots,w_{n+1}^*$ such that
$(z^*,w_1^*,\ldots,w_{n+1}^*)\in\cS$.
Suppose $p^*=(z^*,w_1^*\ldots,w_{n+1}^*)\in\cS$.
Since $z^*$ solves \eqref{mono1}, $z^*$ is typically referred to as a \textit{primal solution}. The vectors $w_1^*,\ldots,w_{n+1}^*$ solve a dual inclusion not described here, and are therefore called a \textit{dual solution}.
It can be shown that
$\cS$ is closed and convex; see for example \cite{johnstone2020projective}.
We will assume that a solution to \eqref{mono1} exists, therefore the set $\cS$ is nonempty.
\paragraph{Separator-projection framework}
Projective splitting methods are instances of the general
\emph{separator-projection} algorithmic framework for locating a member of a
closed convex set $\cS$ within a linear space $\cP$. Each iteration $k$
of algorithms drawn from this framework operates by finding a set $H_k$ which
separates the current iterate $p^k \in \cP$ from $\cS$, meaning that $\cS$ is
entirely in the set and $p^k$ typically is not. One then attempts to ``move closer" to
$\cS$ by projecting the $p^k$ onto $H_k$.
In the particular case of projective splitting applied to the
problem~\eqref{mono1} using~\eqref{Sdef}, we select the space $\cP$ to be
\begin{align}\label{subspaceP}
\mathcal{P} &\triangleq
\left\{(z,w_1,\ldots,w_{n+1})\in\rR^{(n+2)d}
\;\Big|\;
\suminp w_i = 0\right\},
\end{align}
and each separating set $H_k$ to be the half space
$\{p\in\cP\;|\;\varphi_k(p)\leq 0\}$ generated by an affine function
$\varphi_k : \cP \to \rR$. The general intention is to construct $\varphi_k$ such that
$\varphi_k(p^k)>0$, but $\varphi_k(p^*)\leq 0$ for all $p^*\in\cS$. The construction employed for $\varphi_k$ in the case of~\eqref{mono1} and~\eqref{Sdef} is of the form
\begin{align}\label{sepForm}
\varphi_k(z,w_1,\ldots,w_{n+1})
&\triangleq
\sum_{i=1}^{n+1}\langle z - x_i^k,y_i^k - w_i\rangle
\end{align}
for some points $(x_i^k,y_i^k)\in\rR^{2d}$, $i=1,\ldots,n+1$, that must be
carefully chosen (see below). Note that any function of the
form~\eqref{sepForm} must be affine when restricted to $\cP$. As mentioned
above, the standard separator-projection algorithm obtains its next iterate
$p^{k+1}$ by projecting $p^k$ onto $H_k$. This calculation involves the
usual projection step for a half space, namely
\begin{align}\label{projStepUpdate}
p^{k+1} = p^k - \alpha_k\nabla\varphi_k,
\quad\text{ where }\quad \alpha_k = {\varphi_k(p^k)}/{\|\nabla\varphi_k\|^2},
\end{align}
where the gradient $\nabla\varphi_k$ is computed relative to $\cP$, thus
resulting in $p^{k+1} \in \cP$ (over- or under-relaxed variants of this step
are also possible).
\section{Proposed Method}
\label{secProposed}
The proposed method is given in Algorithm \ref{algSPS} and called
\textit{Stochastic Projective Splitting} (SPS). Unlike prior versions of
projective splitting, SPS does not employ the stepsize $\alpha_k$
of~\eqref{projStepUpdate} that places the next iterate exactly on the
hyperplane given by $\varphi_k(p)=0$. Instead, it simply moves in the
\textit{direction} $-\nabla\varphi_k$ with a pre-defined stepsize
$\{\alpha_k\}$. This fundamental change is required to deal with the
stochastic noise on lines \ref{lineNoise1} and \ref{lineXYend}. This noise
could lead to the usual choice of $\alpha_k$ defined in
\eqref{projStepUpdate} being unstable and difficult to analyze.
In order to guarantee convergence, the parameters $\alpha_k$ and $\rho_k$ must be
chosen to satisfy certain conditions given below.
Note that the gradient is calculated with respect to the subspace $\cP$ defined in
\eqref{subspaceP}; since the algorithm is initialized within $\cP$, it remains
in $\cP$, within which $\varphi_k$ is affine. Collectively, the updates on lines
\ref{lineProj1}-\ref{lineProj2} are equivalent to
$
p^{k+1} = p^k - \alpha_k\nabla\varphi_k,
$
where $p^k = (z^k,w_1^k,\ldots,w_{n+1}^k)$.
\begin{algorithm}[b]
{
\DontPrintSemicolon
\SetKwInOut{Input}{Input}
\Input{$p^1 = (z^1,w_1^1,\ldots,w_{n+1}^1)$ s.t. $\suminp w_i^1 = 0$, $\{\alpha_k,\rho_k\}_{k=1}^\infty$, $\tau>0$}
\For{$k=1,2,\ldots$}
{
\For{$i=1,\ldots,n$}
{
$t_i^k = z^k + \tau w_i^k$\label{lineXYone}\;
$x_i^k = J_{\tau A_i}(t_i^k)$\label{xupdate}\;
$y_i^k = \tau^{-1}(t_i^k - x_i^k)$\label{yupdate}\;
}
$r^k = B(z^k) + \epsilon^k$
\tcp*[r]{$\epsilon^k$ is unknown noise term}\label{lineNoise1}
$x_{n+1}^k = z^k - \rho_k(r^k - w_{n+1}^k)$ \label{xupdateLip} \;
$y_{n+1}^k = B(x_{n+1}^k) + e^k$\tcp*[f]{$e^k$ is unknown noise term}
\label{lineXYend}
$z^{k+1} = z^k - \alpha_k\suminp y_i^k$ \label{lineProj1}\;
$w_i^{k+1} = w_i^k - \alpha_k(x_i^k - \frac{1}{n+1}\suminp x_i^k)\quad i=1,\ldots,n+1$ \label{lineProj2}
}
}
\caption{Stochastic Projective Splitting (SPS)}
\label{algSPS}
\end{algorithm}
Note that SPS does not explicitly evaluate $\varphi_k$, which is only used in
the analysis, but it does keep track of $(x_i^k,y_i^k)$ for $i=1,\ldots,n+1$.
The algorithm's memory requirements scale linearly with the number of
nonsmooth operators $n$ in the inclusion~\eqref{mono1}, with the simplest
implementation storing $(3n + 5)d$ working-vector elements. This requirement
can be reduced to $(n + 7)d$ by using a technique discussed in the
appendix. In most applications, $n$ will be small, for example $2$ or $3$.
\paragraph{Updating $(x_i^k,y_i^k)$}
The variables $(x_i^k,y_i^k)$ are updated on lines
\ref{lineXYone}-\ref{lineXYend} of Algorithm \ref{algSPS}, in which $e^k$ and
$\epsilon^k$ are $\rR^d$-valued random variables defined on a probability
space $(\Omega,\mbF,P)$. For $B$ we use a new, noisy version of the
two-forward-step procedure from \cite{johnstone2020projective}. For each
$A_i$, $i=1,\ldots,n$, we use the same resolvent step used in previous
projective splitting papers, originating with \cite{eckstein2008family}. In the case $\epsilon^k = e^k = 0$,
the selection of the $(x_i^k,y_i^k)$ is identical to that
proposed in~\cite{johnstone2020projective}, resulting in the hyperplane $\{p:\varphi_k(p) = 0\}$ strictly separating
$p^k$ from $\cS$.
SPS achieves full splitting of \eqref{mono1}. Each $A_i$ is processed
separately using a resolvent and the Lipschitz term $B$ is processed via a
stochastic gradient oracle. When the $A_i$ arise from regularizers or constraints, as
discussed in Section \ref{secBackG}, their resolvents can be readily computed so
long as their respective proximal/projection operators have a convenient
form.
\paragraph{Noise assumptions}
Let $\mbF_k\triangleq\sigma(p^1,\ldots,p^k)$ and
$\mbE_k \triangleq\sigma(\epsilon^k)$. The stochastic estimators for the gradients,
$r^k$ and $y_{n+1}^k$, are assumed to be \textit{unbiased}, that is, the noise has
mean $0$ conditioned on the past:
\begin{align}
\E[\epsilon^k|\mbF_k]=0,\quad \E[e^k|\mbF_k]=0\quad a.s.\label{unbiasedAss}
\end{align}
We impose the following mild assumptions on the variance of the noise:
\begin{align}\label{noiseBound1}
\E\left[ \|\epsilon^k\|^2|\mbF_k\right] &\leq N_1+N_2\|B(z^k)\|^2\quad a.s.
\\\label{noiseBound2}
\E\left[ \|e^k\|^2|\mbF_k,\mbE_k\right]&\leq N_3+N_4\|B(x_{n+1}^k)\|^2\quad a.s.,
\end{align}
where $0\leq N_1, N_2, N_3, N_4 <\infty$.
We do not require $e^k$ and $\epsilon^k$ to be independent of one another.
\paragraph{Stepsize choices}
The stepsizes $\rho_k$ and $\alpha_k$ are assumed to be deterministic. A
constant stepsize choice which obtains a non-asymptotic convergence rate will be
considered in the next section (Theorem \ref{thmConvR}). The stepsize conditions we will impose to
guarantee almost-sure convergence (Theorem \ref{thmMain}) are
\begin{align}\label{stepRuleSumInf}
\sumk \alpha_k\rho_k = \infty,\quad
\sumk \alpha_k^2 <\infty,\quad
\sumk \alpha_k\rho_k^2 <\infty,
\,\,
\text{ and }
\,\,
\rho_k &\leq \orho <\frac{1}{L}.
\end{align}
For example, in the case $L=1$, a particular choice which satisfies these constraints is
\begin{align*}
\alpha_k = k^{-0.5 - p} \,\,\text{ for }\,\,0<p<0.5,\,\, \text{ and }\,\,
\rho_k = k^{-0.5+t} \,\,\text{ for }\,\, p \leq t < 0.5p+0.25.
\end{align*}
For simplicity, the stepsizes $\tau$ used for the resolvent updates in lines
\ref{lineXYone}-\ref{yupdate} are fixed, but they could be allowed to vary
with both $i$ and $k$ so long as they have finite positive lower and
upper bounds.
\section{Main Theoretical Results}\label{secMainResults}
\begin{theorem}\label{thmMain}
For Algorithm \ref{algSPS}, suppose \eqref{unbiasedAss}-\eqref{stepRuleSumInf} hold.
Then with probability one it holds that $z^k\to z^*$, where $z^*$ solves \eqref{mono1}.
\end{theorem}
\paragraph{Proof sketch}
Theorem \ref{thmMain} is proved in the appendix, but we provide a brief sketch here.
The proof begins by deriving a simple recursion inspired by the analysis of SGD \cite{robbins1951stochastic}.
Since
$
p^{k+1} = p^k - \alpha_k\nabla\varphi_k,
$
a step of projective splitting can be viewed as GD applied to the affine hyperplane
generator function $\varphi_k$. Thus, for any $p^*\in\cP$,
\begin{align}
\|p^{k+1} - p^*\|^2
&=
\|p^k - p^*\|^2 - 2\alpha_k\langle \nabla\varphi_k,p^k - p^*\rangle + \alpha_k^2\|\nabla\varphi_k\|^2
\nonumber\\\label{eqStart}
&=
\|p^k - p^*\|^2 - 2\alpha_k(\varphi_k(p^k) - \varphi_k(p^*)) + \alpha_k^2\|\nabla\varphi_k\|^2,
\end{align}
where in the second equation we have used that $\varphi_k(p)$ is affine on $\cP$.
The basic strategy is to show that, for any $p^*\in\cS$,
\begin{align*}
\E[\|\nabla\varphi_k\|^2|\mbF_k] \leq C_1\|p^k - p^*\|^2 + C_2 \quad a.s.
\end{align*}
for some $C_1, C_2 > 0$. This condition allows one to establish stochastic
quasi-Fej\'{e}r monotonicity (SQFM) \cite[Proposition
2.3]{combettes2015stochastic} of the iterates to $\cS$. One consequence of
SQFM is that with probability one there exists a subsequence $v_k$ such that
$\varphi_{v_k}(p^{v_k}) -
\varphi_{v_k}(p^*)$ converges to $0$.
Furthermore, roughly speaking, we will show that
$
\varphi_{k}(p^{k}) -
\varphi_{k}(p^*)$
provides an upper bound on the
following ``approximation residual" for SPS:
\begin{align}
O_k
\triangleq
\sumin \|y_i^k - w_i^k\|^2
+\sumin \|z^k - x_i^k\|^2
+ \| B (z^k) - w_{n+1}^k\|^2
.\label{Okdef}
\end{align}
$O_k$ provides an approximation error for SPS, as formalized in the following lemma:
\begin{lemma}
For SPS, $p^k=(z^k,w_1^k,\ldots,w_{n+1}^k)\in\cS$ if and only if $O_k=0$.\label{lemOk}
\end{lemma}
\vspace{-1.5ex}
Since $y_i^k\in A_i(x_i^k)$ for $i=1,\ldots,n$, having $O_k=0$ implies that $z^k =
x_i^k$, $w_i^k = y_i^k$, and thus $w_i^k\in A_i(z^k)$ for $i=1,\ldots,n$.
Since $w_{n+1}^k = B(z^k)$ and $\sum_{i=1}^{n+1}w_i^k =
0$, it follows that $z^k$ solves \eqref{mono1}. The reverse direction is proved
in the appendix.
The quantity $O_k$ generalizes the role played by the norm of the gradient in
algorithms for smooth optimization. In particular, in the special case where
$n=0$ and $B(z)=\nabla f(z)$ for some smooth convex function $f$, one has $O_k
= \|\nabla f(z^k)\|^2$.
Combining the properties of $O_k$ with other results following from SQFM (such as
boundedness) will allow us to derive almost-sure convergence of the iterates
to a solution of \eqref{mono1}.
\paragraph{Convergence rate}
\label{secConvRate}
We can also establish non-asymptotic convergence rates for the approximation
residual $O_k$:
\begin{theorem}\label{thmConvR}
Fix the total iterations $K\geq 1$ of Algorithm \ref{algSPS} and
set
\begin{align}\label{step1}
\forall k=1,\dots, K: \rho_k=\rho\triangleq
\min
\left\{
K^{-1/4},\frac{1}{2L}
\right\}
\quad\text{ and }\quad \alpha_k =
C_f \rho^2
\end{align}
for some $C_f>0$.
Suppose
\eqref{unbiasedAss}-\eqref{noiseBound2} hold.
Then
$$
\frac{1}{K}\sum_{j=1}^K
\E[O_j]
=
\bigO(K^{-1/4})
$$
where the constants are given (along with the proof) in the appendix.
\end{theorem}
Theorem \ref{thmConvR} implies that if we pick an iterate $J$ uniformly at random
from $1,\ldots,K$, then the expected value of $O_J$ is $\bigO(K^{-1/4})$.
As far as we know, this is the first convergence rate for a stochastic full-splitting method solving \eqref{mono1}, and it is not clear whether it can be reduced, either by a better analysis or a better method. Faster rates are certainly possible for deterministic methods; Tseng's method obtains $\bigO(K^{-1})$ rate \cite{monteiro2010complexity}.
Faster rates are also possible for stochastic methods under \textit{strong} monotonicity and when $n=0$ \cite{kannan2019optimal,NEURIPS2020_ba9a56ce}. Faster \textit{ergodic} rates for stochastic methods have been proved for special cases with $n=1$ with a compact constraint \cite{juditsky2011solving}.
What is needed is a tight lower bound on the convergence rate of any first-order splitting method applied to \eqref{mono1}.
Since nonsmooth convex optimization is a special case of \eqref{mono1}, lower bounds for that problem apply \cite{nemirovskij1983problem}, but they may not be tight for the more general monotone inclusion problem.
\section{Related Work}
Arguably the three most popular classes of operator splitting algorithms are
forward-backward splitting (FB) \cite{combettes2011proximal}, Douglas-Rachford
splitting (DR) \cite{lions1979splitting}, and Tseng's method
\cite{tseng2000modified}. The extragradient method (EG) is similar to Tseng's
method, but has more projection steps per iteration and only applies to
variational inequalities
\cite{korpelevich1977extragradient,nemirovski2004prox}. The popular
Alternating Direction Method of Multipliers (ADMM), in its standard form, is a
dual application of DR \cite{gabay1983chapter}. FB, DR, and Tseng's method
apply to monotone inclusions involving two operators, with varying assumptions
on one of the operators. It is possible to derive splitting methods for the
more complicated inclusion \eqref{mono1}, involving more than two operators,
by applying Tseng's method to a product-space reformulation
\cite{briceno2011monotone+,combettes2012primal} (for more on the product-space
setting, see the appendix). The recently developed forward-reflected-backward
method \cite{malitsky2020forward} can be used in the same way. The
three-operator splitting method \cite{davis2015three} can only be applied to
\eqref{mono1} if $B$ is cocoercive rather than merely Lipchitz, and thus its
usefulness is mostly limited to optimization applications and not games.
The above-mentioned methods are all deterministic, but stochastic operator
splitting methods have also been developed. The preprint
\cite{bot2019forward} develops a stochastic version of Tseng's method under
the requirement that the noise variance goes to $0$. In ML, this could be achieved with
the use of perpetually increasing batch sizes,
a strategy that is impractical in many scenarios. The
stochastic version of FRB proposed in \cite{van2021convergence} has more
practical noise requirements, but has stronger assumptions on the problem
which are rarely satisfied in ML applications: either uniform/strong
monotonicity or a bounded domain. The papers \cite{NIPS2016_5d6646aa} and
\cite{pedregosa2019proximal} consider stochastic variants of three-operator
splitting, but they can only be applied to optimization problems. The methods of
\cite{zhao2018stochastic} and \cite{bohm2020two} can be applied to simple
saddle-point problems involving a single regularizer.
There are several alternatives to the (stochastic) extragradient method that
reduce the number of gradient evaluations per iteration from two to one
\cite{NEURIPS2019_4625d8e3,malitsky2020forward,gidel2018a}. However, these
methods have more stringent stepsize limits, making it unclear \emph{a priori}
whether they will outperform two-step methods.
DSEG is a stochastic version of EG \cite{NEURIPS2020_ba9a56ce}. The primary
innovation of DSEG is that it uses different stepsizes for the
extrapolation and update steps, thereby resolving some of the convergence
issues affecting stochastic EG. As noted earlier, DSEG is the special case of
our SPS method in which $n=0$, that is, no regularizers/constraints are present in
the underlying game. The analysis in \cite{NEURIPS2020_ba9a56ce} also did not
consider the fixed stepsize choice given in Theorem \ref{thmConvR}.
\section{Experiments}\label{secExps}
We now provide some numerical results regarding the performance of SPS as
applied to distributionally robust supervised learning (DRSL). We follow the
approach of \cite{yu2021fast}, which introduced a min-max formulation of
Wasserstein DRSL. While other approaches reduce the problem to convex
optimization, \cite{yu2021fast} reduces it to a finite-dimensional min-max
problem amenable to the use of stochastic methods on large datasets. However,
unlike our proposed SPS method, the variance-reduced extragradient method that
\cite{yu2021fast} proposes cannot handle multiple nonsmooth regularizers or
constraints on the model parameters.
Consequently, we consider distributionally robust sparse logistic regression
(DRSLR), a problem class equivalent to that considered in \cite{yu2021fast},
but with an added $\ell_1$ regularizer, a standard tool to induce
sparsity. We solve the following convex-concave min-max problem:
\renewcommand{\arraystretch}{1.4}
\begin{align}
\begin{array}{rl}
\displaystyle{\min_{\substack{\beta\in\rR^d \\ \lambda\in\rR\,\,\,}}} \;\;
\displaystyle{\max_{\gamma\in\rR^m}}
&
\displaystyle{
\left\{
\lambda(\delta - \kappa) +
\frac{1}{m}\sum_{i=1}^m\Psi(\langle \hat{x}_i,\beta\rangle)
+
\frac{1}{m}
\sum_{i=1}^m
\gamma_i(
\hat{y}_i\langle\hat{x}_i,\beta\rangle - \lambda\kappa
)
+
c\|\beta\|_1
\right\}
}
\\
\,\text{s.t.} &
\|\beta\|_2\leq \lambda/(L_\Psi+1) \qquad \|\gamma\|_\infty\leq 1.
\end{array}
\label{drslr}
\end{align}
This model is identical to that of~\cite[Thm. 4.3]{yu2021fast} except for the
addition of the $\ell_1$ regularization term $c\|\beta\|_1$, where $c\geq 0$
is a given constant. The goal is to learn the model weights $\beta$ from a
training dataset of $m$ feature vectors $\hat{x}_i$ and corresponding labels
$\hat{y}_i$. Rather than computing the expected loss over the training set,
the formulation uses, for each $\beta$, the worst possible distribution within
a Wasserstein-metric ball around the empirical distribution of the
$\{(\hat{x}_i,\hat{y}_i)\}$, with the parameter $\delta\geq 0$ giving the
diameter of the ball and the parameter $\kappa\geq 0$ specifying the relative
weighting of features and labels. The variables $\gamma$ and $\lambda$
parameterize the selection of this worst-case distribution in response to the
model weights $\beta$. Finally, $\Psi$ is the logistic loss kernel $t \mapsto
\log(e^t+e^{-t})$ and $L_\Psi=1$ is the corresponding Lipschitz constant.
We converted~\eqref{drslr} to the form~\eqref{mono1} with $n=2$, with the
operator $A_1$ enforcing the constraints, $A_2$ corresponding to the objective
term $c\|\beta\|_1$, and $B$ being the vector field corresponding to the gradients of the remaining elements of the
objective. More details of the formulation are provided in the appendix.
We compared our SPS method to some deterministic methods for solving
\eqref{drslr} for a collection of real datasets from the LIBSVM repository (released under the 3-clause BSD license)
\cite{CC01a}.
In all the experiments, we set $\delta=\kappa=1$ and $c=10^{-3}$. We
implemented SPS with
$
\alpha_k = C_d k^{-0.51}
$
and
$
\rho_k = C_d k^{-0.25}
$
and called it \textit{SPS-decay}. We also implement SPS with the fixed stepsize given in
\eqref{step1} and called it \textit{SPS-fixed}.
We compared the method to deterministic projective splitting
\cite{johnstone2020projective}, Tseng's method
\cite{tseng2000modified,combettes2012primal}, and the
forward-reflected-backward method \cite{malitsky2020forward} (FRB). To the
best of our knowledge, there is no stochastic method besides SPS capable of
solving \eqref{drslr} under standard assumptions. We show results for three
LIBSVM standard datasets: \textit{epsilon}\footnote{original data source
\url{http://largescale.ml.tu-berlin.de/instructions/}} ($m=4\cdot 10^5$,
$d=2000$), \textit{SUSY} \cite{baldi2014searching,Dua:2019} ($m=2\cdot
10^6$, $d=18$), and \textit{real-sim}\footnote{Original data source
\url{https://people.cs.umass.edu/~mccallum/data.html}} ($m=72,\!309$,
$d=20,\!958$). For SPS-fixed, we tuned $C_f$, arriving at $C_f=1$ for
epsilon and real-sim, and $C_f=5$ for SUSY.
For SPS-decay, we tune $C_d$ arriving at $C_d=1$ for epsilon and SUSY, and $C_d=0.5$ for real-sim.
For SPS, we use
a batchsize of $100$. All methods are initialized at the same random point.
To measure the progress of the algorithms, we used the ``approximation residual''
\begin{align}\label{defRk}
R_k &\triangleq
\textstyle{
\sumin \|z^k - x_i^k\|^2 + \big\| B(z^k) + \sumin y_i^k \big\|^2.
}
\end{align}
This measure is related to $O_k$ but does not involve the dual iterates
$w_i^k$.
As with $O_k$, having $R_k=0$ implies that $z^k$ solves \eqref{mono1}.
We use $R_k$ instead of $O_k$ because it is also possible to compute essentially the
same measure of convergence from the iterates of the other tested algorithms,
providing a fair comparison. The appendix provides the details of the derivation
of the residual measure from each algorithm and explores the relationship
between $R_k$ and $O_k$.
Figure \ref{fig} plots the approximation residual versus running time for all
five algorithms under consideration. The computations were performed using
Python 3.8.3 and \texttt{numpy} on a 2019 MacBook Pro with a 2.4GHz 8-core Intel I9 processor and 32GB of RAM . Being a stochastic method, SPS-decay seems to outperform the
deterministic methods at obtaining a medium-accuracy solution quickly.
Overall, SPS-decay outperforms SPS-fixed.
\begin{figure}
\centering
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\linewidth]{eps}
\label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\linewidth]{susy}
\label{fig:sub2}
\end{subfigure}
\begin{subfigure}{.33\textwidth}
\centering
\includegraphics[width=\linewidth]{real_sim}
\label{fig:sub3}
\end{subfigure}
\vspace{-3ex}
\caption{Approximation residual versus running time for three LIBSVM benchmark
datasets, with the markers at 10-iteration intervals. Left: epsilon, middle:
SUSY, right: real-sim. Since SPS is stochastic, we plot the median results
over $10$ trials, with unit standard deviation horizontal error bars for the
running time and the vertical error bars displaying the min-to-max range of
the approximation residual. }
\label{fig}
\end{figure}
\section{Conclusions and Future Work}
We have developed the first stochastic splitting method that can handle
min-max problems with multiple regularizers and constraints. Going
forward, this development should make it possible to incorporate regularizers
and constraints into adversarial formulations trained from large datasets. We
have established almost-sure convergence of the iterates to a solution, proved
a convergence rate result, and demonstrated promising empirical performance on a
distributionally robust learning problem.
Recent versions of deterministic projective
splitting~\cite{combettes2016async,johnstone2020projective} allow for
asynchronous and incremental operation, meaning that not all operators need to
be activated at every iteration, with some calculations proceeding with stale
inputs. Such characteristics make projective splitting well-suited to
distributed implementations. Many of our SPS results may be extended to allow
for these variations, but we leave those extensions to future work.
\section{Broader Impact} \label{secBI}
This work does not present any foreseeable societal consequence.
\bibliographystyle{spmpsci}
|
1,116,691,500,417 | arxiv | \section{Introduction}
Real time processes at finite temperature play an essential role in
the physics of the early universe and of heavy ion collisions. A key
quantity in scenarios of baryogenesis~\cite{kuzmin,rs} is the rate for
electroweak baryon number violation (the sphaleron rate). In the broken
phase the sphaleron rate can be computed with semiclassical
methods~\cite{kuzmin,ar,khlebnikov} but in the symmetric phase
\cite{p} they are not reliable. Unfortunately, a direct
non-perturbative lattice determination of the hot sphaleron rate is
not available, either.
The most promising approach to this problem~\cite{grigoriev} is to
compute the sphaleron rate in a classical real time simulation since
the relevant thermal transitions are essentially
classical. Considerable work has been done in this
direction~\cite{ambjorn91}--\cite{smit}.
Treating the dynamics of a classical gauge field system one
is nevertheless faced
with severe difficulties~\cite{nielsen}--\cite{arnold97}.
The high momentum modes with
$k\mathop{\gsi} T$ which do not behave classically, do not decouple from the
dynamics. In general, these modes lead to ultraviolet divergences in
the classical correlation functions which cannot be removed by
introducing local counterterms in the classical
theory~\cite{bodeker,arnold97}.
There is
another question related to the classical approach which
has hardly been
considered so far:
under which conditions is the classical
approximation for the low momentum modes
reliable? One systematic way
of investigating the validity of the classical approximation is to
compute the first quantum corrections in the $\hbar$-expansion. So
far, the expressions have been derived only for quantum mechanics and
scalar field theories~\cite{b}. However, these simple cases should
already teach us something in spite of the fact that topological
observables and the associated rate do not exist. In these models
relevant observables might be related for instance to the damping
rate~\cite{aarts,smit}.
The purpose of the present paper is to evaluate the quantum
corrections in the simplest non-trivial case, the
quantum mechanical anharmonic oscillator. This study serves to
estimate the feasibility of similar studies in field theories.
Moreover, we believe that some of the general results might
be carried over to that context.
We find that while at small times the classical approximation
is reliable, it breaks down at large enough times. The reason
is that the functional form of the quantum corrections is qualitatively
different from that of the classical answer, in a way which
cannot be accounted for by modifying the parameters of the
classical result.
The paper is organized as follows.
In Sec.~\ref{Formulation} we discuss the formulation of the problem.
In Sec.~\ref{sec:ho} we briefly discuss the harmonic oscillator
and in Sec.~\ref{aho} the anharmonic oscillator. The ``symmetric''
and ``broken'' cases of the latter are analyzed
in more detail in Secs.~\ref{symmetric}, \ref{broken},
and we conclude in Sec.~\ref{concl}.
\section{The formulation of the problem}
\label{Formulation}
We consider one bosonic degree of freedom $q$ with
conjugate momentum $p$ and the Hamiltonian
\begin{equation}
H=\frac{p^2}{2}+U(q),
\end{equation}
where
\begin{equation}
U(q)=\left\{\begin{array}{l}
+\frac{1}{2}\omega^2q^2+\frac{1}{4}g^2 q^4
\\
-\frac{1}{2}\omega^2q^2+\frac{1}{4}g^2 q^4+\frac{\omega^4}{4 g^2}
\end{array}\right. .
\label{U}
\end{equation}
We refer to the two cases of a positive and of a negative quadratic
term as the symmetric and the broken case, respectively.
Quantum mechanical (Heisenberg) operators are denoted by capital letters,
for example
\begin{equation}
Q(t)=e^{\frac{i}{\hbar} H t} Q(0) e^{-\frac{i}{\hbar} H t}.
\end{equation}
The finite temperature correlator we consider is
\begin{equation}
C(t)=
\left\langle
\frac{1}{2}\Big[
Q(t)Q(0)+Q(0)Q(t)
\Big]
\right\rangle= \frac{1}{Z} \mathop{\rm Re}\mathop{\rm Tr}
\left[e^{-\beta H(P,Q)} Q(t)Q(0)\right], \la{c}
\end{equation}
relevant for the time dependence of
\begin{equation}
\left\langle
\Big[Q(t)-Q(0)\Big]^2
\right\rangle.
\end{equation}
Here $Z= \mathop{\rm Tr}\left[\exp(-\beta H)\right]$ and
$\beta$ is the inverse temperature.
Note that $C(t)$ is an even function of $t$.
In~\cite{b}, the expansion
\begin{eqnarray}
C(t) = C_{\rm cl}(t) + C_\hbar(t) + C_{\hbar^2}(t) +
{\cal O}(\hbar^3)
\end{eqnarray}
was derived for $C(t)$. The classical result
is~\cite{dolan,bochkarev,bodeker}
\begin{eqnarray}
C_{\rm cl}(t)= Z_{\rm cl}^{-1} \int \frac{dp dq}{2\pi\hbar}
e^{-\beta H(p,q)} q q_{\rm c}(t) ,
\la{c0}
\end{eqnarray}
where $Z_{\rm cl} = \int \frac{dp dq}{2\pi\hbar} e^{-\beta H(p,q)}$
and $q_c(t)$ is the solution of the classical equations of motion
with the initial conditions $q_c(0)=q,$ $ \dot{q}_c(0)=p$.
This expression corresponds to the prescription suggested
by Grigoriev and Rubakov~\cite{grigoriev}.
As for the quantum corrections,
the contribution $C_\hbar(t)$ vanishes. The result to
order $\hbar^2$ is then~\cite{b}
\begin{eqnarray}
C(t) &=& Z^{-1} \int \frac{dp dq}{2\pi\hbar} e^{-\beta H(p,q)}
\biggl\{ \biggl[ 1 - \frac{\hbar^2\beta^2}{24} U''(q) +
\frac{\hbar^2\beta}{24} \left(
\partial_q^2
+ U''(q) \partial_p^2
\right) \biggr] q q_{\rm c}(t) \nonumber \\
& & \hspace*{2cm} - \frac{\hbar^2}{24}
q \int_0^t dt' U'''\big(q_{\rm c}(t')\big)
\{q_{\rm c}(t'),q_{\rm c}(t)\}_3 \biggr\} + {\cal O}(\hbar^3), \la{ch2}
\end{eqnarray}
where $\{,\}$ denotes the Poisson bracket
\begin{eqnarray}
\{f,g\} = \partial_p f \partial_q g - \partial_p g
\partial_q f,
\end{eqnarray}
and
\begin{eqnarray}
\{f,g\}_0= g ,\qquad \{f,g\}_{n+1} = \{f,\{f,g\}_n\}.
\la{poisson_n}
\end{eqnarray}
Similarly, the expression for $Z$ to order $\hbar^2$ is
\begin{equation}
Z= \int \frac{dp dq}{2\pi\hbar} e^{-\beta H(p,q)}
\left[ 1 - \frac{\hbar^2\beta^2}{24} U''(q) \right]. \la{z}
\end{equation}
There are thus three kinds of terms
in the $\hbar^2$-correction to $C(t)$,
denoted by $C_{\hbar^2}^{(i)}(t)$, $i=a,b,c$:
\begin{eqnarray}
C_{\hbar^2}(t) = C_{\hbar^2}^{(a)}(t) + C_{\hbar^2}^{(b)}(t) +
C_{\hbar^2}^{(c)}(t),
\end{eqnarray}
where
\begin{eqnarray}
C_{\hbar^2}^{(a)}(t) & = & Z_{\rm cl}^{-1}
\biggl(\frac{\hbar^2\beta^2}{24}\biggr)
\int \frac{dp dq}{2\pi\hbar}
e^{-\beta H(p,q)} U''(q)
\biggl[ C_{\rm cl}(t) - q q_{\rm c}(t) \biggr], \la{cha} \\
C_{\hbar^2}^{(b)}(t) & = & Z_{\rm cl}^{-1}
\biggl(\frac{\hbar^2\beta}{24}\biggr)
\int \frac{dp dq}{2\pi\hbar}
e^{-\beta H(p,q)}
\biggl[\partial_q^2
+ U''(q) \partial_p^2
\biggr] q q_{\rm c}(t), \la{chb} \\
C_{\hbar^2}^{(c)}(t) & = & Z_{\rm cl}^{-1}
\biggl(\frac{- \hbar^2}{24}\biggr)
\int \frac{dp dq}{2\pi\hbar}
e^{-\beta H(p,q)}
q \int_0^t dt' U'''\big(q_{\rm c}(t')\big)
\{q_{\rm c}(t'),q_{\rm c}(t)\}_3. \la{chc}
\end{eqnarray}
The term $C_{\hbar^2}^{(a)}(t)$ is a sum of the
$\hbar^2$ correction to the partition function
when it combines with the classical result $C_{\rm cl}(t)$,
and of the corresponding term in the numerator of eq.~\nr{ch2}.
Eqs.~\nr{cha}--\nr{chc} are the corrections we will evaluate below.
One of the key issues of the present
problem is the following: In the
case of static {\it time-independent} correlators, it is
possible (in a weakly coupled theory)
to reproduce
the results of the full quantum theory
from a classical theory with a high
accuracy, provided that the parameters of the
classical theory are modified appropriately. This is called
dimensional reduction~\cite{ginsparg,kajantie}.
The question is then
whether such a resummation might also work
in the time-dependent case. Indeed,
it has been proved that the resummation used in the
time-independent context is sufficient
for making the time-dependent two-point function
in the scalar $\phi^4$ theory finite
to two-loop order in perturbation theory
and even for giving
the corresponding damping rate
the right leading order numerical
value~\cite{aarts}. General arguments
in the same direction were also given in~\cite{mt}. The expansion
in eq.~\nr{ch2} is, in contrast, non-perturbative:
each term involves contributions from all orders in the coupling constant.
Let us therefore
discuss the effects of the resummation in the present
context (see also~\cite{b}).
Of course, the problem of divergences
does not occur unlike in field theory.
First, consider dimensional reduction.
Let us take as an example
the ``symmetric case'' anharmonic oscillator,
\begin{equation}
U(q)=\frac{1}{2}\omega^2 q^2+\frac{1}{4}g^2q^4.
\end{equation}
The starting point is then a 1-dimensional Euclidean field theory
defined by
\begin{equation}
{\cal L}=\frac{1}{2}(\partial_\tau q)^2+\frac{1}{2}\omega^2q^2+
\frac{1}{4}g^2q^4,\quad
Z=\int {\cal D}q
\exp(-\frac{1}{\hbar}\int_0^{\beta\hbar}\!d\tau {\cal L}).
\end{equation}
According to dimensional reduction, this can be written as
\begin{equation}
Z={\rm const.}\times \int \! dq_0 \exp(-S_{\rm eff}),
\end{equation}
where $q_0$ is the zero Matsubara mode. The parameters
in $S_{\rm eff}$ are modified by the non-zero modes. The
non-zero mode propagator is
\begin{equation}
\langle q_n q_m\rangle=
\frac{\delta_{n+m,0}}{\omega^2+(2\pi n T/\hbar)^2}.
\end{equation}
To order $\hbar^2$ (which is a good approximation
as long as $\beta \hbar\omega\ll\pi$), one can then easily
calculate how the mass parameter in the effective theory
is modified:
\begin{equation}
\omega_{\rm eff}^2=\omega^2+3g^2 T\sum_{n\neq 0}
\frac{1}{(2\pi nT/\hbar)^2}=\omega^2+\frac{1}{4}g^2\hbar^2 \beta. \la{effw}
\end{equation}
The change in the coupling constant is of order $\hbar^4$
and thus does not contribute in the present
$\hbar^2$-calculation.
Consider, on the other hand,
eqs.~\nr{ch2}, \nr{z}. In eq.~\nr{z},
$U''=\omega^2+3 g^2q^2$. The constant $\omega^2$-part
of this expression
does not contribute in eq.~\nr{ch2} since it is cancelled
by a similar part in the numerator,
see eq.~\nr{cha}.
The $q^2$-part, on the other hand, can be exactly
reproduced by calculating the classical partition
function $Z_{\rm cl}$ with $\omega^2$ modified according
to eq.~\nr{effw}:
\begin{equation}
\exp\left(-\beta\frac{1}{2}\omega^2q^2\right)
\left(1-\frac{\hbar^2\beta^2}{24}3g^2q^2
\right)=
\exp\left(-\beta\frac{1}{2}
\omega_{\rm eff}^2q^2\right) + {\cal O}(\hbar^4).
\end{equation}
Similarly, the $-\beta^2 U''(q)$-term in the square brackets
in eq.~\nr{ch2} is accounted for by the change in $\omega^2$
according to eq.~\nr{effw}. Thus the term
$C_{\hbar^2}^{(a)}(t)$ in eq.~\nr{cha} is directly related to
changing the parameters of the classical theory.
However, there remain the terms
$C_{\hbar^2}^{(b)}(t)$ and $C_{\hbar^2}^{(c)}(t)$.
On the other hand, $q_c(t)$ is still a solution to the
original Hamilton equations of motion. Hence the question is
whether the $\hbar^2$-effects can be taken into account by determining
$q_c(t)$ form the equations of motion with the
modified parameter $\omega_{\rm eff}^2$
rather than $\omega^2$.
This issue will be discussed below and we find that, in general,
such a resummation does not take place.
Finally, it should be noted that in the field theory
case one is usually interested in a ``rate'' observable:
a time independent constant determining the time dependence
of some Green's function,
for example the sphaleron rate or the damping rate.
We are not aware of such an observable related to $C(t)$
in the present context. We thus consider the
general large-time functional behaviour of $C(t)$.
\section{Harmonic oscillator}
\la{sec:ho}
In order to show in a simple setting how the $\hbar$-expansion
works and to see what the structure of the perturbative solution
is, let us start by considering briefly the harmonic oscillator.
The classical Hamiltonian is
\begin{equation}
H=\fr12 p^2+\frac{1}{2}\omega^2 q^2.
\end{equation}
In this trivial case, the correlation function in eq.~\nr{c}
can be calculated exactly, with the result
\begin{equation}
C(t)=\frac{\hbar}{2 \omega}
\biggl(\tanh\!\frac{\beta\hbar\omega}{2}\biggr)^{-1}
\cos \!\omega t. \la{ho}
\end{equation}
Expanding in $\hbar$, one gets
\begin{equation}
C_{\rm cl}(t)+
C_{\hbar^2}(t)=\frac{\cos \!\omega t}{\beta\omega^2}
\left[ 1+\frac{1}{12}(\beta\hbar\omega)^2
\right]. \la{hoh2}
\end{equation}
The fact that it is the symmetric
combination of $Q(t)Q(0)$
which appears in eq.~\nr{c},
removes the term linear in $\hbar$ from the result.
It is seen that
the quantum corrections change the amplitude of $C_{\rm cl}(t)$,
but not the frequency
since $\omega$ is independent of energy.
The classical $\hbar^0$-term
is reliable in the limit $\beta\hbar\omega \ll 1$,
that is, at high temperatures. At low temperatures, in contrast,
the $T=0$ result (with $\tanh =1$ in eq.~\nr{ho}) is reliable.
How is this result reproduced by eqs.~\nr{c0}, \nr{ch2}?
The solution of the classical equations of motion is
\begin{equation}
q_c(t)=q\cos\!\omega t+\frac{p}{\omega}\sin\!\omega t.
\la{hoqct}
\end{equation}
Substituting this into eq.~\nr{c0}, one sees that
the term proportional to $p$ in $q_c(t)$ does
not contribute due to antisymmetry in $p$, and
one gets directly
\begin{equation}
C_{\rm cl}(t)=\frac{1}{\hbar Z_{\rm cl}}
\frac{\cos\!\omega t}{\beta^2\omega^3}
=\frac{\cos\!\omega t}{\beta\omega^2},
\label{CclHO}
\end{equation}
where it was used that $Z_{\rm cl}=(\beta\hbar\omega)^{-1}$.
As for the quantum corrections,
the last term in eq.~\nr{ch2}
is proportional to the third derivative of the potential
and thus does not contribute, $C_{\hbar^2}^{(c)}(t)=0$.
The term $C_{\hbar^2}^{(a)}(t)$ in eq.~\nr{cha} does not contribute
either, since $U''(q)$ is just a constant. There
remains a contribution from $C_{\hbar^2}^{(b)}(t)$ in eq.~\nr{chb},
reproducing eq.~\nr{hoh2}.
\section{Anharmonic oscillator}
\la{aho}
Let us then move to the less trivial case of the anharmonic oscillator.
Here and in the following we use $\omega$, $g$ and
\begin{equation}
V_0=\frac{\omega^4}{4g^2}
\end{equation}
to introduce the dimensionless variables $\hat q,\hat p,\hat t,\hat \beta,\hat E$:
\begin{equation}
\hat q=\frac{g}{\omega}q, \quad
\hat p=\frac{g}{\omega^2}p,\quad \hat t=\omega t,
\quad \hat \beta = \beta V_0,\quad \hat E = \frac{E}{V_0}.
\la{dimless}
\end{equation}
This rescaling serves to show the parameter
dependence of the final non-perturbative result more clearly.
At the same time, it makes the coupling constant equal to
unity so that if one wants to compare with perturbation
theory, one should go back to the original variables.
In terms of the rescaled variables the potential in eq.~(\ref{U})
reads
\begin{equation}
U(q)=\left\{\begin{array}{lll}
& V_0 (\hat q^4+2\hat q^2), & \mbox{\rm the ``symmetric'' case} \\
& V_0 (\hat q^2-1)^2, & \mbox{\rm the ``broken'' case}
\end{array}\right. .
\end{equation}
A dimensionless combination to which $\hbar$ can
be attached is
\begin{equation}
\epsilon = \frac{g^2\hbar}{\omega^3}.
\end{equation}
The quantity naively
governing the semiclassical expansion is hence $\epsilon^2$.
This is multiplied by some dimensionless function $f(\hat \beta,\hat t)$ which
may scale approximately with some power of $\hat \beta$ for given $\hat t$.
For instance, in the case of the harmonic oscillator, $f(\hat \beta,\hat t)$
scales as $\hat \beta^2$ so that the real expansion parameter is
\begin{equation}
(\epsilon \hat \beta)^2 \propto (\beta\hbar\omega)^2.
\end{equation}
One of the issues below is how the function $f(\hat \beta,\hat t)$
behaves in the anharmonic case
as a function of $\hat \beta$.
With the rescaling performed, one can also write $C(t)$
in a dimensionless form. Factoring out the scale $\omega^2/g^2$,
the classical correlation function is
\begin{equation}
C_{\rm cl}(t)=\left(\frac{\omega^2}{g^2}\right)
\left(\frac{\omega^3}{\pi g^2\hbar}\right)\frac{1}{Z_{\rm cl}}
\int_{-\infty}^{\infty}\!d\hat p \int_0^{\infty}\!d\hat q
e^{-\hat \beta \hat E}\hat q\hat q_c(\htt) \equiv
\left(\frac{\omega^2}{g^2}\right)\hat Z_{\rm cl}^{-1}(\hat \beta)
\hat C_{\rm cl}(\hat \beta,\hat t), \la{cct}
\end{equation}
where
\begin{equation}
Z_{\rm cl} = \left(\frac{\omega^3}{\pi g^2\hbar}\right)
\hat Z_{\rm cl}(\hat \beta),\quad
\hat Z_{\rm cl}(\hat \beta) = \int_{-\infty}^{\infty}\!d\hat p \int_0^{\infty}\!d\hat q
e^{-\hat \beta \hat E}. \la{cz}
\end{equation}
Here we utilized the symmetry of the integrand in
$\hat q\to -\hat q,\hat p\to -\hat p$.
We can also write the quantum corrections in a dimensionless form,
\begin{equation}
C_{\hbar^2}(t)=\epsilon^2
\left(\frac{\omega^2}{g^2}\right)\hat Z_{\rm cl}^{-1}(\hat \beta)
\Bigl[
\hat C_{\hbar^2}^{(a)}(\hat \beta,\hat t)+
\hat C_{\hbar^2}^{(b)}(\hat \beta,\hat t)+
\hat C_{\hbar^2}^{(c)}(\hat \beta,\hat t)
\Bigr]. \la{cq}
\end{equation}
We will then discuss the ``symmetric'' and ``broken'' cases
separately.
\section{The symmetric case}
\la{symmetric}
\subsection{Numerical results}
The detailed form of the classical solution $q_c(t)$
and of the integrals appearing
in the symmetric case
is discussed in \ref{appA}.
The expressions to be evaluated are in
eqs.~\nr{cct}, (\ref{sfzc})--(\ref{sfchc}). We have done
the evaluation numerically, as well as
analytically in certain regimes of $\hat \beta,\hat t$.
Let us discuss the numerical result first. The curves are
displayed in Figs.~\ref{sccl}--\ref{schc}.
\begin{figure}[tb]
\vspace*{-1.0cm}
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{sccl.eps}}
\vspace*{-6cm}
\caption[a]{
The classical correlator $C_{\rm cl}(t)$ in the symmetric case.
The thin line represents the analytic approximation of
Sec.~\ref{ltl}. The difference between the thin and thick lines
in the regime $1\mathop{\lsi} \omega t\mathop{\lsi} \beta V_0$ is due
to higher order perturbative corrections in $1/\beta V_0
\sim g^2/\beta\omega^4$.}
\la{sccl}
\end{figure}
\begin{figure}[tb]
\vspace*{-1.0cm}
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{ssum.eps}}
\vspace*{-6cm}
\caption[a]{
The quantum correction
$C_{\hbar^2}^{(a)}(t) + C_{\hbar^2}^{(b)}(t)$ in the symmetric case.
We have divided out the naive expansion parameter
$(\epsilon\beta V_0)^2=(\fr14 \beta\hbar\omega)^2$.}
\la{ssum}
\end{figure}
\begin{figure}[tb]
\vspace*{-1.0cm}
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{schc.eps}}
\vspace*{-6cm}
\caption[a]{
The quantum correction
$C_{\hbar^2}^{(c)}(t)$ in the symmetric case,
compared with the analytic approximation of
Sec.~\ref{ltl}.}
\la{schc}
\end{figure}
The qualitative features of the solution are the following:
Both the classical solution $C_{\rm cl}(t)$ and the quantum
correction $C_{\hbar^2}^{(a)}(t)+C_{\hbar^2}^{(b)}(t)$ approach
zero at large times. The time scale it takes for the amplitude
to diminish depends on $\hat \beta$, being roughly
proportional to $\hat \beta$, and being somewhat larger for the
quantum corrections. The reason for the attenuation is the
destructive interference of the continuum of classical
solutions with different frequencies.
This feature does not fully persist
in the quantum case where the energy levels are
discrete: rather the behaviour is ``almost periodic''~\cite{dolan}
on a larger time scale. Indeed, already
the term $C_{\hbar^2}^{(c)}(t)$ in Fig.~\ref{schc}
behaves in a manner qualitatively different
from $C_{\hbar^2}^{(a)}(t)$, $C_{\hbar^2}^{(b)}(t)$: it has a constant
amplitude at large times.
Let us discuss these features and their implications
in more quantitative terms.
\subsection{The large time limit}
\la{ltl}
We are mainly interested in the
large time behaviour of the correlation function.
Ordinary perturbation theory breaks down
for large times and is therefore excluded.
This is due to the secular terms in the perturbative series:
at lowest order the solution to the classical equations
of motion is proportional to $\cos(\omega t + \alpha)$ while the next
order contains a term proportional to $g^2 t \sin(\omega t + \alpha)$.
Thus, by dimensional analysis,
straightforward perturbation theory only works for
\begin{eqnarray}
\hat t = \omega t \ll \frac{\omega^4}{g^2} \beta = 4\beta V_0.
\label{tPerturbative}
\end{eqnarray}
The way to avoid the
secular terms is to use the exact
frequency $\Omega(E)$ inside the trigonometric functions appearing
in the perturbative series. The perturbative series
for the classical solution of eq.~\nr{symmqct} is obtained
from eq.~\nr{series}.
In the phase space integration
one then has to compute
(after a change of variables according to eq.~\nr{intvars})
the dimensionless integrals
\begin{eqnarray}
J_{n m} (\beta, t) &=&
(\mbox{$\frac{g^2}{\omega^4}$})^{n + 1}\int_0^\infty dE e^{-\beta E} E^n
\cos(m\Omega(E) t), \nonumber \\
\overline{J}_{n m} (\beta, t) &=& (\mbox{$\frac{g^2}{\omega^4}$)}^{n + 1}
\int_0^\infty dE e^{-\beta E}
E^n \sin(m\Omega(E) t).
\label{Jn}
\end{eqnarray}
In general, this is difficult to do.
Fortunately, an exact evaluation
is not necessary if one is interested in the
large time limit $\omega t\gg 1$. Then it is sufficient to keep
only the first two terms of the low energy expansion
of the exact frequency,
\begin{equation}
\Omega(E) = \omega \Bigl[ 1 + \fr14 c_1 \hat E
+ {\cal O} (\hat E^2 ) \Bigr],
\label{OmegaExpansion}
\end{equation}
where $\hat E=E/V_0$ and $c_1 = 3/4$.
In this approximation we find
\begin{eqnarray}
J_{n 1} (\beta, t) &\approx& n!
\frac{\cos(\omega t + (n + 1)\varphi)}
{\Big((\frac{\omega^4}{g^2}\beta )^2 +
(c_1 \omega t)^2\Big)^{(n+1)/2}},\nonumber \\
\overline{J}_{n 1} (\beta, t) &\approx & n!
\frac{\sin(\omega t + (n + 1)\varphi)}
{\Big((\frac{\omega^4}{g^2}\beta )^2 + (c_1 \omega t)^2\Big)^{(n+1)/2}},
\label{JnExpansion}
\end{eqnarray}
where
\begin{eqnarray}
\varphi = \arcsin\left(
\frac{c_1 \omega t}{\sqrt{(\frac{\omega^4}{g^2}\beta)^2 + (c_1\omega t)^2}}
\right).
\end{eqnarray}
{}From these expressions one can see that
in the region $\omega t\gg \frac{\omega^4}{g^2}\beta$
the terms which have been neglected in eq.\ (\ref{JnExpansion})
are suppressed
by at least one power of $1/(\omega t)$.
The reason is that
each power of $E$ in the phase space integrand of
eq.\ (\ref{Jn}) gives one
power of $1/(\omega t)$. In the region
$1\ll\omega t \ll \beta V_0$, the terms neglected
are suppressed by $g^2\beta/\omega^4$, corresponding
to higher order perturbative corrections.
Note that the approximation
(\ref{JnExpansion}) is valid also for $\omega t \gg \beta V_0 $
where the perturbative expansion breaks down.
Thus the large time expansion for $C(t)$ can be obtained from
the low energy expansion of the phase space integrand.
Using this expansion, we find
for $C_{\rm cl}(t)$ for $\omega t \gg 1$,
\begin{eqnarray}
C_{\rm cl}(t) \approx
\frac{1}{\hbar Z_{\rm cl}}
\frac{\omega^5}{g^4} J_{11}(\beta, t).
\label{CclAnalytic}
\end{eqnarray}
If $\hat \beta=\frac{\omega^4}{4g^2} \beta \gg 1$, there is an overlap of the
``perturbative region'' of eq.~(\ref{tPerturbative}) and the large time
region:
for moderately large times $1 \ll \omega t \ll \frac{\omega^4}{g^2} \beta$ we
recover the leading order perturbative result.
If $\omega t \gg \frac{\omega^4}{g^2}\beta $,
in contrast, eq.\ (\ref{CclAnalytic})
simplifies to
\begin{eqnarray}
C_{\rm cl}(t) \approx - \frac{16}{9} \frac{1}{\hbar Z_{\rm cl}}
\frac{\omega^5}{g^4} \frac{\cos\omega t}{(\omega t)^2}.
\label{CclAsymptotic}
\end{eqnarray}
That is, for large times, the classical correlation function
oscillates with the 'tree level' frequency $\omega$ and with an
amplitude which decreases as $1/t^2$. Comparing eq. (\ref{CclAsymptotic})
with the corresponding result for the harmonic oscillator,
eq.\ (\ref{CclHO}), we see that eq.~(\ref{CclAsymptotic})
is non-perturbative since its functional form cannot be obtained
by adding corrections multiplied by positive powers of $g^2$
to the harmonic oscillator result.
Let us now
note that if a resummation according to eq.~\nr{effw}
would take place, then the $\hbar^2$ quantum result should be
obtained by replacing $\omega^2\to\omega_{\rm eff}^2$
in eq.~\nr{CclAsymptotic}, that is
\begin{equation}
C_{\hbar^2}^{\rm resummed}(t) \approx
\biggl[
1+b_1 \frac{g^2\hbar^2\beta}{\omega^2}\biggr] C_{\rm cl}(t)+
\fr29
\frac{1}{\hbar Z_{\rm cl}}
\frac{\hbar^2\beta\omega^3}{g^2}
\frac{\sin\omega t}{\omega t},
\la{provh2}
\end{equation}
where $b_1$ is some number. We show below that the true
$C_{\hbar^2}(t)$ is not of the form in eq.~\nr{provh2}.
We start with $C_{\hbar^2}^{(a)}(t)$.
It was pointed out already in
Sec.~\ref{Formulation} that this term is related to
the replacement $\omega^2\to\omega_{\rm eff}^2$ in the
Hamiltonian appearing in the Boltzmann factor. To be more specific,
the term $\omega^2$ in $U''(q)$ cancels in eq.~\nr{cha}
and in the limit $\omega t\gg1$ we find
\begin{eqnarray}
\label{CaAsymptotic}
C_{\hbar^2}^{(a)} (t) \approx
\frac{1}{8} (\hbar g \beta)^2
\langle q^2 \rangle_{\rm cl}
C_{\rm cl}(t).
\end{eqnarray}
The contribution proportional to $\langle q^3 q_c(t)\rangle_{\rm cl}$,
on the other hand, has one additional power of $E$ in the phase space
integrand compared with the classical case and is
thus suppressed by a factor $1/(\omega t)$. From
eq.~(\ref{CaAsymptotic}) it is obvious that the quantum correction
$C_{\hbar^2}^{(a)} (t)$ shows the
qualitative behaviour indicated in the first
term in eq.~\nr{provh2}: it is small compared with the classical result
if $\beta\hbar\omega \ll 1$ and this holds even for arbitrarily
large times.
Next we consider the quantum corrections containing the derivatives
$\partial^2_q$, $\partial^2_p$ which we have denoted by
$C_{\hbar^2}^{(b)} (t)$. These derivatives acting on
the trigonometric functions in $q q_c(t)$ give extra
factors of $t$.
When expanding the integrand in powers of energy one has
to count $t$ as $E^{-1}$.
For $\omega t\gg 1$ we find
\begin{eqnarray}
\label{CbAsymptotic}
C_{\hbar^2}^{(b)} (t) \approx \frac{1}{48} \frac{1}{\hbar Z_{\rm cl}}
\frac{\hbar^2\beta \omega^3}{g^2}
\left\{ 4 J_{01}(\beta, t)
- 9 \omega t \overline{J}_{11}(\beta, t)
- \frac94 (\omega t)^2 J_{21}(\beta, t) \right\}.
\end{eqnarray}
The individual terms in the curly brackets behave as
$\sin(\omega t)/(\omega t)$ for large times, which is the
expected behaviour in eq.~\nr{provh2}. Such a result would
at the same time indicate that without resummation, the
semiclassical expansion breaks down for
\begin{eqnarray}
\omega t\mathop{\gsi} \frac{\omega^3}{\hbar g^2}\frac{1}{\beta\hbar\omega},
\end{eqnarray}
when the correction term in eq.~\nr{provh2}
is as large as the leading term.
However, we find that this does not occur: in the limit
$\omega t \gg 1$ the individual terms in the curly brackets in
eq.\ (\ref{CbAsymptotic}) cancel at leading order in
$1/(\omega t)$. Therefore the amplitude of $C_{\hbar^2}^{(b)} (t) $
decreases as $1/(\omega t)^2$ for large times. Thus
$C_{\hbar^2}^{(b)} (t)$ is small compared with the classical result
at high temperatures. The corresponding suppression factor, however,
is not in general given by $(\beta\hbar\omega)^2 $.
There are terms proportional to
$1/(\omega t)^2$ having different dependences on
the temperature: expanding eq.\ (\ref{CbAsymptotic}) gives terms
$\propto \beta^2$ while the subleading terms in the low energy expansion
are proportional to $\beta$. We have not calculated these terms analytically.
The numerical result for the sum of $C_{\hbar^2}^{(a)} (t)$ and
$C_{\hbar^2}^{(b)} (t)$ is shown in Fig.\ \ref{ssum}.
Finally we consider the correction $C_{\hbar^2}^{(c)} (t)$.
The result for $\omega t\gg 1$ is
\begin{eqnarray}
C_{\hbar^2}^{(c)} (t) \approx \frac{9}{64} \frac{1}{\hbar Z_{\rm cl}}
\frac{\hbar^2}{\omega^4} (\omega t)^2 \{ J_{21}(\beta, t)
- \frac14 \omega t \overline{J}_{11}(\beta, t)
\},
\end{eqnarray}
which for $\omega t \gg \frac{\omega^4}{g^2}\beta $ becomes
\begin{eqnarray}
\label{CcAsymptotic}
C_{\hbar^2}^{(c)} (t) \approx - \frac{1}{12}
\frac{1}{\hbar Z_{\rm cl}}
\frac{\hbar^2}{\omega} \cos\omega t.
\end{eqnarray}
Thus at large times $ C_{\hbar^2}^{(c)} (t)$ oscillates with the
``tree
level frequency'' but with a time independent amplitude.
This behaviour is qualitatively different from the classical case.
Comparing eqs.\ (\ref{CclAsymptotic}), (\ref{CcAsymptotic})
we see that $ C_{\hbar^2}^{(c)} (t)$ becomes as large as the classical
correlator for $t\sim t_*$ where
\begin{eqnarray}
\omega t_* = \frac{\omega^3}{\hbar g^2 }=\frac{1}{\epsilon},
\la{t*}
\end{eqnarray}
and for $t>t_*$ the semiclassical approximation breaks down.
The correction in eq.~\nr{CcAsymptotic} is clearly not of the
form allowed by eq.~\nr{provh2}. Since there is a term of a functional
form not allowed and the allowed $\sin \omega t/(\omega t)$-term
does not emerge, we conclude that a resummation according to
eq.~\nr{effw} does not take place in the large time limit.
Neither can one understand the result as a resummation
with a correction factor different from that in eq.~\nr{effw}.
Since a resummation cannot be made, the semiclassical
expansion breaks down at the time given by eq.~\nr{t*}.
It can be checked from
Fig.\ \ref{schc} that for $\omega t \mathop{\gsi} 10$ the analytic approximation
for $C_{\hbar^2}^{(c)} (t)$
indeed gives quite an accurate estimate of the exact numerical result.
To conclude, let us point out that the qualitative features found,
together with the ``almost periodic'' behaviour~\cite{dolan}
at time scales $t\mathop{\gsi} t_*$, can be reproduced with the
following approximation. Writing the full quantum result
in eq.~\nr{c} in the energy basis, one gets
\begin{equation}
C(t)=Z^{-1} \mathop{\rm Re}
\sum_{m,n}e^{-\beta E_m}e^{\frac{i}{\hbar} t(E_m-E_n)}
|\langle m|Q|n\rangle|^2. \la{enbasis}
\end{equation}
Approximating the energy levels to first order in $g^2$,
\begin{equation}
E_n =\hbar\omega\left(n+\fr12\right)
+\fr38\frac{g^2\hbar^2}{\omega^2}\left(n^2+n+\fr12\right),
\end{equation}
and the eigenstates to zeroth order, one gets
\begin{equation}
C(t)\simeq
Z^{-1}\frac{\hbar}{2\omega}
\sum_{m=0}^{\infty}
(m+1)
\left(e^{-\beta E_{m+1}}+e^{-\beta E_{m}}\right)
\cos\! \left(\frac{E_{m+1}-E_m}{\hbar}\right)t.
\end{equation}
The behaviour of this solution for small $\epsilon=\hbar g^2/\omega^3$
follows
the classical
solution in Fig.~\ref{sccl}
until the time scale is of order $t\sim 4 t_*=4/\epsilon$,
but then the periodicity sets in so that
at the time scale $t\sim 8t_*$, the structure around $t=0$ in the
classical solution is repeated. This is the reason for
the breakdown of the classical approximation.
\section{The broken case}
\la{broken}
\subsection{Preliminaries}
In the broken case, the classical Hamiltonian is
\begin{equation}
H=V_0\Bigl[2\hat p^2+(\hat q^2-1)^2\Bigr]. \la{bH}
\end{equation}
There exists, of course, an enormous
literature on this system.
In the present finite temperature context, it has been
previously studied by Dolan and Kiskis~\cite{dolan}
and by Bochkarev~\cite{bochkarev}.
Quite a lot is known about the qualitative
behaviour of $C(t)$. In general,
the solution can be written as in eq.~\nr{enbasis}.
Since the solution is a sum
of periodic contributions corresponding to the different
energy levels that can be excited, $C(t)$ is
``almost periodic''~\cite{dolan}. In particular,
the lowest frequency appearing is determined by
\begin{equation}
\Delta E = E_1-E_0
\propto \hbar\omega \exp \left(-\frac{2\sqrt{2}}{3}
\frac{\omega^3}{g^2\hbar}\right),
\la{pol2}
\end{equation}
implying that the symmetry is restored
already at $T=0$~\cite{polyakov} in the sense that the
correlator averaged over a long enough time period vanishes.
In contrast, the
classical result $C_{\rm cl}(t)$ has a non-zero limiting value
for $t\to \infty$, in which the symmetry is only partially
restored and all the oscillations die out~\cite{dolan}. The oscillations
die out, like in the symmetric case, due to the destructive interference of
the continuum of classical solutions with different frequencies.
The fact alone that the classical result does not show
the expected qualitative behaviour of the full result,
indicates that the classical result is not generically applicable.
We study this problem in more concrete terms below
by evaluating the $\hbar^2$-corrections.
Note that, in seeming contrast to what was just pointed out,
the system in eq.~\nr{bH} has also been used
to illustrate that the classical
approximation {\it is} applicable
to some real time problems~\cite{rs}.
The reason for the difference is that
the situation we consider is
different from the one in~\cite{rs}: we have a
strict equilibrium situation
at a finite temperature $\beta^{-1}$, which is also what is considered
in~\cite{dolan,bochkarev} and which occurs
in the real time sphaleron rate simulations. The
consideration in~\cite{rs}, in contrast,
concerns a non-equilibrium symmetry-restoring rate obtained
by taking an initial state where the system is prepared in one
of the minima.
In the strict equilibrium case, one cannot define such a rate.
Still, the problem of the general applicability
of the classical approximation
to real-time problems remains.
\subsection{Numerical Results}
The form of the classical solution $q_c(t)$ for the broken case
is discussed in~\ref{appB}.
The numerically evaluated classical correlator $C_{\rm cl}(t)$
is shown in Fig.~\ref{bccl}, and the quantum correction
$C_{\hbar^2}^{(a)}(t)+C_{\hbar^2}^{(b)}(t)$ in Fig.~\ref{bsum}.
The most notable difference with respect to the symmetric case
is that there is a constant part in the broken case results.
The energy integrand for $C_{\rm cl}(t)$ is for illustration
shown in Fig.~\ref{encl} where
the emergence of the constant part from $\hat E<1$ can be seen.
It is evident from Fig.~\ref{bccl} that
the partial symmetry restoration in the classical result
is the stronger
the higher the temperature is~\cite{dolan}, and
from Fig.~\ref{bsum} that there is a further
symmetry restoring effect from the quantum corrections.
It is seen in Fig.~\ref{bsum} that at high temperatures
($\beta V_0\sim 0.5-2.0$) the quantum corrections are
roughly proportional to the naive expansion parameter
$(\epsilon\beta V_0)^2=(\fr14\beta\hbar\omega)^2$
which has been factored out.
\begin{figure}[tb]
\vspace*{-1.0cm}
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{bccl.eps}}
\vspace*{-6cm}
\caption[a]{
The classical correlator $C_{\rm cl}(t)$ in the broken
case (thick lines), together with the analytic approximation
of Sec.~\ref{bltt} (thin lines).}
\la{bccl}
\end{figure}
\begin{figure}[tb]
\vspace*{-1.0cm}
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{bsum.eps}}
\vspace*{-6cm}
\caption[a]{
The quantum correction
$C_{\hbar^2}^{(a)}(t)+C_{\hbar^2}^{(b)}(t)$ in the broken case.}
\la{bsum}
\end{figure}
\begin{figure}[tb]
\vspace*{-1.0cm}
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{encl.eps}}
\vspace*{-6cm}
\caption[a]{
The energy integrands for $C_{\rm cl}(t)$
and $C_{\hbar^2}^{(c)}(t)$
in the broken symmetry case for $\beta V_0=2$.
In the limit $\omega t\to\infty$ only the
region $E/V_0<1$ contributes in $C_{\rm cl}(t)$.
The energy integrand for
$C_{\hbar^2}^{(c)}(t)$
has been shown on a logarithmic scale. It
involves essentially the second derivative
of the classical integrand, which is why it has very
high peaks ($\sim 10^{10}$ already at $\omega t\sim 15$) around
$E/V_0=1$.}
\la{encl}
\end{figure}
\begin{figure}[tb]
\hspace{1cm}
\epsfysize=18cm
\centerline{\epsffile{bchc.eps}}
\vspace*{-6cm}
\caption[a]{
The quantum correction
$C_{\hbar^2}^{(c)}(t)$ in the broken case,
divided by $(\epsilon\beta V_0)^2=(\fr14 \beta\hbar\omega)^2$.}
\la{bchc}
\end{figure}
Let us then discuss $\hat C_{\hbar^2}^{(c)}(\hat{\beta},\hat t)$.
Its numerical evaluation turns out to be very difficult
for large $\hat t$. The reason is that
the energy-integrand is highly peaked
and oscillatory around unity.
To see this, note first that
at $\hat t=0$, the integrand
in eq.~\nr{sfchc} vanishes.
Moreover, the integrand involves
terms $\sim\sin \hat\Omega(\hat E) \hat t$,
in analogy with eq.~\nr{intdndn} below.
Hence a particular energy region will contribute provided
that
\begin{equation}
\hat\Omega(\hat E) \hat t \mathop{\gsi} 1. \la{est}
\end{equation}
Let $y=|\hat E-1|\ll 1$. Since $K(k)\sim \ln(4/k')$ close
to $k=1$ ($k'=\sqrt{1-k^2}$),
one gets from eq.~\nr{bfreq} that
\begin{equation}
\hat\Omega(\hat E)=\left\{
\begin{array}{ll}
\frac{2\pi}{\ln(64/y)}, & \hat E\mathop{\lsi} 1 \\
\frac{\pi}{\ln(64/y)}, & \hat E\mathop{\gsi} 1
\end{array}
\right. . \la{omegaatone}
\end{equation}
Eq.~\nr{est} shows then that the energy-integrand
can be large in the region
\begin{equation}
y \mathop{\gsi} e^{-\hat t}.
\la{large}
\end{equation}
On the other hand, the second partial derivative in eq.~\nr{sfchc}
will involve
\begin{equation}
\partial_{\hat p}^2 \Omega(\hat E) =
( \partial_{\hat p} {\hat E} )^2
\partial_{\hat E}^2 \Omega(\hat E)+\ldots,
\end{equation}
where $\partial_{\hat p}\hat E=4\hat p$.
Hence according to eqs.~\nr{omegaatone}, \nr{large},
\begin{equation}
\frac{\partial^2\hat q_c(\hat t')}{\partial \hat p^2} \sim \frac{1}{y^2}\mathop{\lsi}
e^{2 \hat t}.
\end{equation}
Thus the height of the peaks around $\hat E =1$ grows exponentially
with time, and the peaks move closer to $\hat E =1$. The width
of the peaks is diminishing, but their height is growing faster
so that they give an increasing contribution.
In fact, the highest peak's contribution from $\hat E < 1$
(where the peak gives a positive contribution) and from
$\hat E>1$ (where it gives a negative one) to a large extent
cancel, but the cancellation is not complete and one has
to account for it very precisely in the numerics to get
the remaining contribution correctly. This is
why we cannot go to large $\hat t$.
In practice, we can reliably calculate
$\hat C_{\hbar^2}^{(c)}(\hat \beta,\hat t)$ only up to $\hat t=15$,
when the highest peaks in the energy integrand are
of height $\sim 10^{10}$ (for $\beta V_0\sim 2$)
at $\delta\hat E \sim 10^{-5}$
around unity.
The energy integrand is shown in Fig.~\ref{encl}
and the result of the integration
in Fig.~\ref{bchc}.
\subsection{The large time limit}
\la{bltt}
Consider first the classical correlation function.
The form of the solutions in eq.~\nr{bct} can
be read off from eq.~\nr{series}. It is seen that
for $\hat E<1$, $q_c(t)$ contains a constant part in addition to
the cosines.
The $\phi$-integral
obtained with the change of variables in eq.~\nr{intvars},
gives then
\begin{equation}
\int_{-K(k)}^{K(k)} \!d\phi\, {\rm dn}_k(\phi){\rm dn}_k(w\hat t+\phi)=
\frac{\pi^2}{2 K(k)}\biggl\{
1+8\sum_{n=1}^\infty \frac{q^{2n}}{(1+q^{2n})^2}
\cos\!\left[{n\Omega(E)t}\right]
\biggr\}.
\la{intdndn}
\end{equation}
The cosines in eq.~\nr{intdndn} give contributions
which vanish in the limit $t \to\infty$
as in eq.~\nr{JnExpansion}, see below. Hence one gets
from eqs.~\nr{cct}, \nr{intvars} that
the constant part surviving is
\begin{equation}
C_{\rm cl}(t\to\infty) =\frac{\pi}{4} \frac{1}{\hbar Z_{\rm cl}}
\frac{\omega^5}{g^4}
\int_0^1 d\hat E e^{-\hat \beta\hat E}\frac{w}{K(k)}. \la{tlim}
\end{equation}
One may also try to compute the time dependent
part for $\omega t\gg 1$ in the same
way as in the symmetric case. There we saw that the limiting behaviour
for large times can be obtained from a suitable low energy expansion
of the solution to the equations of motion. In the present case it is
obvious that this expansion cannot be convergent
when $\hat E$ approaches unity.
One may argue, however, that for large times only the solutions with small
energies are relevant and that this expansion still works.
We find that
\begin{eqnarray}
C_{\rm cl}(t) = C_{\rm cl}(t\to\infty) +
\frac{1}{16 \sqrt{2} } \frac{1}{\hbar Z_{\rm cl}}
\frac{\omega^5}{g^4}
\int_0^1 d\hat E e^{-\hat \beta\hat E}\Bigg( \hat E
\cos [\Omega(\hat E) t] +
{\cal O}(\hat E^2)
\Bigg).
\end{eqnarray}
Replacing the upper integration limit
by $\infty$ and keeping
only the first two terms of the low energy expansion of
\begin{eqnarray}
\Omega(\hat E) = \sqrt{2}\omega(1 - d_1 \hat E + {\cal O}(\hat E^2))
\end{eqnarray}
where $d_1 = 3/16$, we obtain for $\omega t \gg 1$,
\begin{eqnarray}
C_{\rm cl}(t) \approx C_{\rm cl}(t\to\infty) +
\frac{1}{16\sqrt{2} } \frac{1}{\hbar Z_{\rm cl}}
\frac{\omega^5}{g^4} \,
\frac{\cos(\sqrt{2}\omega t - 2 \varphi)}{(\beta V_0)^2 +
(d_1 \sqrt{2}\omega t)^2},
\la{bAnCcl}
\end{eqnarray}
where now
\begin{eqnarray}
\varphi = \arcsin\left(\frac{d_1 \sqrt{2}\omega t}{\sqrt{(\beta V_0)^2 +
(d_1 \sqrt{2}\omega t)^2}}\right).
\end{eqnarray}
It can be seen in Fig.~\ref{bccl} that eq.~\nr{bAnCcl} is indeed a good
approximation at large times.
The integrals appearing in the quantum corrections
$C_{\hbar^2}^{(a)}(t)$ and $C_{\hbar^2}^{(b)}(t)$
are qualitatively quite similar to that appearing
in $C_{\rm cl}(t)$.
In particular,
there is a constant part in these corrections
which can be evaluated in the same way
as eq.~\nr{tlim}.
It is seen that the constant part
tends to further restore the symmetry
compared with the classical result, see Fig.~\ref{bsum}.
For the quantum correction $C_{\hbar^2}^{(c)}(t)$,
in contrast, the ``small energy expansion'' does not seem
to be applicable.
We have computed the solution but it does not agree
with Fig.~\ref{bchc}. However, this need not be a surprise since,
as discussed,
it is not guaranteed that the small energy expansion
works in the broken case due to the singular nature of the
point $\hat E=1$:
the energy integration extends beyond the radius of convergence of the
small energy expansion.
Moreover, the integrand in $C_{\hbar^2}^{(c)}(t)$ is qualitatively
different from that in $C_{\rm cl}(t)$.
A simple example where the small energy
expansion would not work is given by
\begin{equation}
f(\hat t)=\int_0^\infty d\hat E e^{-\hat \beta\hat E}
\frac{\sin \hat t(1-\hat E)}{\pi (1-\hat E)}.
\end{equation}
At $\hat t\to\infty$ the integrand makes a delta-function, and
$f(\hat t\to\infty)\to \exp(-\hat \beta)$. Yet an expansion
in $\hat E$ of the denominator
around $\hat E=0$ and an integration term by term,
gives a result which oscillates around zero.
We could not find any other analytic way of evaluating the
energy integral for $C_{\hbar^2}^{(c)}(t)$,
either. The integrand is very complicated
around $\hat E\sim 1$. Thus we can only mention some
general features of the solution.
First, note that
the numerical result in Fig.~\ref{bchc} shows
that there is a growing
negative contribution at large $\hat t$ in $C_{\hbar^2}^{(c)}(t)$.
This seems to arise from $\hat E$ a bit larger than unity.
To estimate very roughly when
this kind of a contribution can be important, note
that then the peak heights must be
such that the exponential
suppression cannot hide them any more, that is
\begin{equation}
e^{-\hat \beta} e^{ 2 \hat t} \mathop{\gsi} 1.
\end{equation}
Hence one starts to get an effect at $\hat t \mathop{\gsi} \hat \beta$.
As to the functional form of the solution,
it looks roughly like $-\hat t^4$ at large $\hat t$.
It is easy to see that a linear in $\hat t$ behaviour
cannot occur, since it follows directly
from the definition in eq.~\nr{c} that $C(t)$
is symmetric in $t$.
For $\beta V_0=0.5,1.0$ in which case the asymptotic
behaviour is obtained
earliest, the leading term of
$C_{\hbar^2}^{(c)}(t)$ can be
fitted at $\omega t\sim 8 \ldots 15$ for instance as
\begin{equation}
C_{\hbar^2}^{(c)}(t)\sim \frac{\omega^2}{g^2}\epsilon^2
\Bigl[-0.01 (\omega t)^4 \Bigr].
\la{fit}
\end{equation}
The conclusions one can draw from the broken case
seem rather similar to those from the symmetric
case. The quantum correction $C_{\hbar^2}^{(c)}(t)$
behaves in a manner qualitatively different form
what was observed for $C_{\rm cl}(t)$. Moreover, the
difference is such that it cannot be accounted for
by a simple resummation of the mass parameter~$\omega^2$.
As the classical result in Fig.~\ref{bccl} is of
order unity and the fit in eq.~\nr{fit} would suggest
the behaviour $\epsilon^2 (\omega t)^4$ for the
quantum correction, one would expect that the
semiclassical expansion breaks down at
\begin{equation}
\omega t \sim \omega t_* = \frac{1}{\sqrt{\epsilon}} =
\biggl(\frac{\omega^3}{g^2\hbar}\biggr)^{1/2}.
\end{equation}
In eq.~\nr{t*} in the symmetric case it was rather
observed that $\omega t_* = 1/\epsilon$. However,
the fit in eq.~\nr{fit} should not be taken very
seriously as the interval is very small, and the main
point is that the time scale for the
breakdown seems to be determined by~$1/\epsilon$.
Finally, let us point out that
from the general form of eq.~\nr{enbasis},
one might have expected that at finite temperature
the asymptotic values of $C(t)$ are
oscillating between positive and negative values.
At zero temperature the time scale would be
$\sim\exp[2\sqrt{2}/(3\epsilon)]$ according to eq.~\nr{pol2}.
Thus the quantum correction $C_{\hbar^2}^{(c)}(t)$
seems to restore some of the qualitative features
missing in $C_{\rm cl}(t)$, in the sense that the
behaviour in Fig.~\ref{bchc} looks like the beginning
of an oscillation with a large time scale. The difference
from the zero temperature case,
however, is that the time scale associated with
the oscillation is not exponential.
\section{Summary and Conclusions}
\la{concl}
We have studied the classical
finite temperature real time two-point correlation
function and its first quantum corrections
for the anharmonic oscillator.
The expansion around the classical limit
is made in powers of $\hbar$, so that each order
contains all orders in the coupling constant $g^2$.
One can identify three different
time scales in the results. In the
symmetric case (Section \ref{symmetric}), these are
\begin{equation}
\omega t \sim 1,\quad \omega t \sim \hat \beta\equiv \frac{\omega^4}{4g^2}
\beta,\quad \omega t \sim \omega t_* = \frac{1}{\epsilon}\equiv
\frac{\omega^3}{g^2\hbar}.
\end{equation}
As long as $\omega t \ll
\hat \beta$, perturbation theory works and the correlation function
oscillates with period $\omega t \sim 1$. In the non-perturbative
region $\omega t \mathop{\gsi} \hat \beta$, the correlation function approaches its
asymptotic form. We have developed a large time expansion which
allows to address also the time scales $\omega t\gg \hat \beta$. In this
regime the amplitude of the oscillations in the classical result
attenuates due to the destructive interference of solutions to the
equations of motion with different energies. This attenuation
cannot be associated with a damping rate. Finally, the time scale
$t_*$ is associated with the quantum corrections and becomes infinity in the
formal limit $\hbar\to 0$. There is a hierarchy $\omega t_* \gg
\hat \beta$ provided that $\beta\hbar\omega\ll 1$.
The general result of our study is that at the
non-perturbative time scales $\omega t\mathop{\gsi} \hat \beta$, the
form of the quantum corrections differs qualitatively
from that of the classical result. The
semiclassical expansion breaks down at $t \sim t_*$
when the quantum corrections become as large as the
classical result. Moreover, we found that these large corrections
cannot be resummed by modifying the parameters of the classical
theory.
On the other hand,
the first quantum corrections to the classical correlation
function are small for $t \ll t_*$. From
this we would expect that in this region the
classical limit gives a good approximation for the full
quantum mechanical correlation function.
The expansion parameter for the quantum corrections
in this region is not
just the naive one $(\beta\hbar\omega)^2$, but
$\epsilon (\beta\hbar\omega)$ and $\epsilon^2$ appear, as well.
An essential question is then which of the discussed
features might be carried over to field theory.
Unfortunately, we cannot say very much about this.
However, certainly the present study does not encourage
one to believe in the generic applicability of the
classical approximation in the high temperature limit
for time-dependent quantities at arbitrarily large times.
On the other hand, there are also obvious features which
cannot hold in a four-dimensional field theory: for instance, we found that
the time $t_*$ does not depend on the temperature. This
is unlikely to be true in the pure SU(2) theory, say;
dimensionally, the classical time scale not involving
$\hbar$ is $(g^2 T)^{-1}$ in that case and the time scale
proportional to $\hbar^{-1}$ is $(\hbar g^4 T)^{-1}$.
It would be interesting to extend the present
type of an analysis to field theory to be able to make
more concrete conclusions. Unfortunately, a straightforward
evaluation of the quantum correction $C_{\hbar^2}^{(c)}(t)$
was numerically quite demanding even in the present case,
in particular for the ``broken'' case where the modes
with $E/V_0\sim 1$ are rather singular. In the field theory
case, the partial derivatives of the classical solution
with respect to the initial conditions would be replaced by
functional derivatives, making things even more complicated.
Still, one might hope that the scalar
field theory analogue of the symmetric
case would allow a non-perturbative investigation of the
quantum corrections in the damping rate.
Finally, let us point out that as it appears
that the classical approximation does not describe the
large time behaviour at least in the present case,
it would perhaps be useful to consider the feasibility of
other approaches. In principle
the problem can be solved non-perturbatively
using Euclidean simulations and spectral function
techniques. The anharmonic oscillator considered
in this paper might be a suitable toy model for developing
techniques for such studies, since it appears
that there is some non-trivial structure even in this case.
\section*{Acknowledgements}
D.B is grateful to M.Shaposhnikov and
M.L to K.Kajantie for discussions.
|
1,116,691,500,418 | arxiv | \section{Introduction and Motivation}
Exact solutions of Einstein's Field Equations
\begin{equation}
G_{\mu\nu} = \kappa T_{\mu\nu} \label{ee}
\end{equation}
are, of course, of interest for various purposes. Since the equations are very complicated, to find solutions one often makes simplifying assumptions about the left-hand-side and/or the right-hand-side. Popular simplifying assumptions about the left-hand-side include staticity and spherical symmetry. As is well known, the use of both assumptions together leads to the ansatz~\cite[Sect.23.2]{mtw}
\begin{equation}
ds^{2} = -B(r) dt^{2} + A(r) dr^{2} + r^{2} d\Omega^{2} \label{ansatz}
\end{equation}
for the metric.
Most-often used simplifying assumptions about the right-hand-side of (\ref{ee}) are that $T_{\mu\nu}$ represents vacuum (i.e. vanishes) or an electromagnetic field or a perfect fluid. For example, the vacuum assumption, together with the ansatz (\ref{ansatz}) gives uniquely the Schwarzschild metric, the simplest and best-known black hole solution.
The perfect fluid form of $T_{\mu\nu}$, the stress-energy-momentum tensor, is
\begin{equation}
T_{\mu\nu} = (\rho + p) u_{\mu} u_{\nu} + p g_{\mu\nu} \label{pfemt}
\end{equation}
where $\rho$ and $p$ are the energy density and pressure, respectively, as measured by an observer moving with the fluid, and $u_{\mu}$ is its four-velocity. The use of this $T_{\mu\nu}$ together with ansatz (\ref{ansatz}) describes the interiors of static spherically symmetric stars, for example. But the description (\ref{pfemt}) is not complete: $\rho$ and $p$ should also be specified as functions of particle number density, temperature, etc. One further simplifying assumption, justified under most circumstances, is that there is a relation, called an {\em equation of state} $f(p,\rho)=0$ between $p$ and $\rho$. In cosmology, one usually assumes that the equation of state is a proportionality,
\begin{equation}
p=w\rho, \label{eos}
\end{equation}
with e.g. $w=0$ describing the matter-dominated (or "pressureless dust") case, \mbox{$w=1/3$} the radiation-dominated case, \mbox{$w<-1/3$} dark energy, and \mbox{$w<-1$} phantom energy. The latter two concepts have been introduced into cosmology in the last decade \cite{de,phantom}, after the discovery of the acceleration of the expansion of the universe \cite{acceleration-hiZsst,acceleration-SCP}.
Now that a good case exists that the universe might be dominated by dark energy, even phantom energy, one should look for exact solutions with these sources. In particular, static spherically symmetric solutions would be the easiest to find and might be relevant in the contexts of black holes or static stars. These solutions can be found starting from the ansatz (\ref{ansatz}), which for ``static" perfect fluid source, (i.e. $u^{\mu} = u^{0} \delta_{0}^{\mu}$) leads to the well-known Oppenheimer-Volkoff (OV) equation~\cite{ov}
\begin{equation}
p' = - \frac{(\kappa p r^{3} + F)}{2 r (r-F)} (\rho + p) \label{ov}
\end{equation}
where
\begin{equation}
F(r) = \kappa \int \rho r^{2} dr, \label{FDefNS}
\end{equation}
and prime denotes $r$-derivative. $F(r)$ can be recognized as $\kappa/4\pi$ times the "mass function" defined in the literature. Into the OV equation (\ref{ov}) one must put $p$ in terms of $\rho$ via the equation of state, then $\rho$ in terms of $F'$, via (\ref{FDefNS}), eventually getting a differential equation for $F$. After solving for $F$, the metric functions can be found via
\begin{eqnarray}
A(r) & = & \frac{r}{r-F(r)} \label{EEns4} \\
\frac{B'(r)}{B(r)} & = & \frac{\kappa p r^{2} + 1}{r-F(r)} - \frac{1}{r} . \label{EEns5}
\end{eqnarray}
The solutions can be interpreted as static only for positive $A(r)$ and $B(r)$, however. In general, the ansatz (\ref{ansatz}) admits four classes of solutions, called NS (the standard case), TD, ND (corresponding to Kantowski-Sachs~\cite[Sect.15.6.5]{exsols}, \cite{ks} case) and TS in~\cite{revisited}. The ND and TD solutions are not static, hence the quotes on ``static" in the title and abstract. For each class, one gets a different OV-like equation.
The OV equation is valid in case NS. For equation of state (\ref{eos}), it becomes
\begin{equation}
(w+1) F' (wrF'+F) + 2 w (rF''-2F')(r-F) = 0 \label{MFeq-NS}
\end{equation}
where $F(r)$ is written as $F$ for brevity, and we put no constraint on $w$ other than that it is a constant. This is a nonlinear equation whose general solution is difficult to find. One can attempt a series solution
\begin{equation}
F(r)=\sum_{n=0}^{\infty} a_{n} r^{n} \label{series}
\end{equation}
but the recursion expression one gets for $a_{n}$ involves all of $a_{0} \dots a_{n-1}$ and it seems not possible to even show that (\ref{series}) converges, let alone find a closed expression for $a_{n}$.
We can, however, find {\em all} of the {\em finite-polynomial} solutions of (\ref{MFeq-NS}). This we do in the next section. In fact, we find all finite {\em Laurent} polynomials, i.e. we consider also negative powers\footnote{In the rest of this work, we will use ``power'' also when we really mean ``order of the power''. It should be clear from the context which meaning is intended.} of $r$, but find none in case NS. Four of the found solutions are valid for particular values of $w$, and two for general $w$. While none of the solutions is totally original, the procedure shows that there are no other finite-polynomial solutions; and in Section \ref{sect:NSdisc} we discuss properties of the spacetimes.
In Section \ref{sect:NonStdSols} we discuss similar solutions, derived in the appendix, for the TD, ND(KS) and TS cases. We also ask if we can find any solutions with finite-polynomial $A(r)$.
\section{All finite-polynomial solutions for the mass function from the standard OV equation}
In case NS, any power of $r$ less than 3 in $F(r)$ means a diverging density at the origin; in particular, a constant term corresponds to a point mass there, while negative powers mean diverging mass function, and therefore seem unnatural. On the other hand, the meaning of $F(r)$ is different in the TD, ND(KS) and TS cases, therefore negative powers are more acceptable.
The highest and lowest powers of $r$ in $F(r)$ we will call $m$ and $\tilde{m}$. The second-highest, third-highest, second-lowest and third-lowest powers of $r$ in $F(r)$ we will call $n$, $p$, $\tilde{n}$ and $\tilde{p}$ respectively, when they exist; and $A$, $B$, $C$, $\tilde{A}$, $\tilde{B}$, $\tilde{C}$ will be the respective coefficients. We will substitute the polynomial into the left-hand-side of (\ref{MFeq-NS}) and set coefficients of all powers of $r$ equal to zero.
For $m > 1$, the highest power of $r$ in (\ref{MFeq-NS}) is $2m-1$, but the expression for its coefficient will change if $m=1$, because of the $r$-term in $(r-F)$ (and similar first power terms in TD, ND(KS) and TS cases). For $m=0$, the highest power will be $n$ (if it exists), and for $m<0$, it will be $m$. Similar considerations apply to the lowest power of $r$ in (\ref{MFeq-NS}) as well. Therefore we are led to the matrix of cases shown in Table \ref{table:casematrix}. \\
\begin{table}
\begin{center}
\begin{tabular}{ c | c | c | c | c | }
& $\tilde{m}>1$ & $\tilde{m}=1$ & $\tilde{m}=0$ & $\tilde{m}<0$ [eq.(\ref{mtilde_lowest}) or (\ref{ND-2mt-1})] \\ \hline
{\bf $m>1$} [eq.(\ref{m_from_w}) or (\ref{ND-2m-1})] & 1 & 2 & 3 & 4 \\ \hline
$m=1$ & - & 5 & 6 & 7 \\ \hline
$m=0$ & - & - & 8 & 9 \\ \hline
$m<0$ & - & - & - & 10 \\ \hline
\end{tabular}
\end{center}
\caption{Matrix of cases for solution of equations (\ref{MFeq-NS}) and (\ref{MFeq-ND}) by finite Laurent polynomials with highest power $m$ and lowest power $\tilde{m}$. In the first row and last column, an important equation valid for that row/column is indicated.}
\label{table:casematrix}
\end{table}
For cases 1-4, that is, for $m > 1$, the highest power of $r$ in eq.(\ref{MFeq-NS}) is $2m-1$, with coefficient
\begin{equation}
(w+1)m (wm+1) A^{2} - 2wm(m-3) A^{2} = 0 \label{2m-1_coeff}
\end{equation}
therefore in these cases $A$ is arbitrary and
\begin{equation}
m = \frac{7w+1}{w(1-w)}. \label{m_from_w}
\end{equation}
If this had been an integer, we would have found the order of the polynomial for arbitrary $w$. Since it is not, we conclude that in these cases finite polynomial solutions exist for certain values of $w$ only.
Of course, one can also solve for $w$ in terms of $m$:
\begin{equation}
w = \frac{m-7 \pm \sqrt{(m-7)^2-4m}}{2 m}, \label{w_from_m}
\end{equation}
or write
\begin{equation}
w^2 = \frac{(m-7)w-1}{m}. \label{wsq}
\end{equation}
Similarly, for $\tilde{m} < 0$ (that is, for cases 4,7,9 and 10), we can consider lowest power of $r$ in eq.(\ref{MFeq-NS}) and find
\begin{equation}
\tilde{m} = \frac{7w+1}{w(1-w)}. \label{mtilde_lowest}
\end{equation}
Now, we can start the separate consideration of the cases in Table \ref{table:casematrix}:\\
\noindent {\em Case 1. $m>1, \; \tilde{m}> 1$}\\
For $\tilde{m} > 1$, the lowest power of $r$ in eq.(\ref{MFeq-NS}) is $\tilde{m}$, the contribution coming from the right part. Its coefficient is
\begin{equation}
2w [\tilde{m}(\tilde{m}-1) - 2 \tilde{m}] \label{mtildeCoeff}
\end{equation}
$w=0$ is incompatible with eq.(\ref{2m-1_coeff}), therefore
\begin{equation}
\tilde{m} = 3 .
\end{equation}
We would like to consider the second-highest power of $r$ in eq.(\ref{MFeq-NS}) now. The form of that coefficient will depend on the existence of $n$:\\
\noindent {\em Case 1.1. $m>3$, $\tilde{m} = 3$.}\\
In this case, $n$ exists and the second-highest power of $r$ in eq.(\ref{MFeq-NS}) is $m+n-1$, with coefficient
\begin{equation}
(w+1)[m (wn+1)+ n (wm+1)] A B - 2w [m(m-3) + n(n-3)] A B = 0. \label{m+n-1_coeff}
\end{equation}
giving arbitrary $B$ and, after elimination of $w^{2}$ by using (\ref{wsq}), the equation
\begin{equation}
(4mn + 7m - 7n - 2 m^{2} - 2 n^{2})w +m-n=0 \label{n_from_m}
\end{equation}
which not only gives $n$ in terms of $m$ and $w$, but also means that $w$ is rational.
A careful inspection of (\ref{w_from_m}) shows that there are only three values of $m$ giving rational $w$: 18, 15 and 3. For each $m$, there are two $w$ values, making a total of four subcases of subcase {\em 1.1}, since $m \neq 3$:\\
\noindent {\em Case 1.1.1. $m = 18$, $n>1$, $w=\frac{1}{2}$}.\\
In this case, solving (\ref{n_from_m}) for $n$ gives the values 18 and 27/2, both of which are unacceptable; the former because we should have $m>n$, the latter because it is not an integer. Hence, this case fails.
One can also see this failure using a `brute force' approach: If one puts a general $18^{\rm th}$ order polynomial (in effect, extending the argument down to $\tilde{m}=0$) into the left-hand-side of (\ref{MFeq-NS}) and sets the coefficients of powers of $r$ to zero, starting from the highest ($35^{\rm th}$) power, one gets $a_{18}=A$, $a_{17}=a_{16}=...=a_{2}=0$, $a_{1}=\frac{108}{85}$, $a_{0}=0$ by the time one arrives at the $17^{\rm th}$ power. But when this polynomial is put afresh into the left-hand-side of (\ref{MFeq-NS}), one gets $\frac{108}{25} r$ instead of zero, so lower powers don't cancel entirely. This is not surprising, since there are 36 powers of $r$ in (\ref{MFeq-NS}), but 19 coefficients to be found.\\
\noindent {\em Case 1.1.2. $m = 18$, $n>1$, $w=\frac{1}{9}$}.\\
This time, for $n$ we get 18 and 10, so we should take the latter. We could then continue, separating out the third highest power, but the 'brute force' approach is more straightforward, especially since it can be executed with software. We find that this case also fails.
Similarly, we find that\\
\noindent {\em Case 1.1.3. $m = 15$, $n>1$, $w=\frac{1}{3}$}\\
and\\
\noindent {\em Case 1.1.4. $m = 15$, $n>1$, $w=\frac{1}{5}$}\\
fail too, finishing subcase {\em 1.1.}\\
\noindent {\em Case 1.2. $m=\tilde{m} = 3$.}\\
This subcase gives us two solutions,
\begin{eqnarray}
{\rm {\bf Solution \; 1:}} & \;\; & w =-1, \;\; F(r) = A r^{3} \label{Sol1} \\
{\rm {\bf Solution \; 2:}} & \;\; & w =-\frac{1}{3}, \;\; F(r) = A r^{3}, \label{Sol2}
\end{eqnarray}
which finish case 1.\\
\noindent {\em Case 2. $m>1, \; \tilde{m} = 1$}\\
The lowest power in (\ref{MFeq-NS}) is now 1, the vanishing of whose coefficient gives
\begin{equation}
\tilde{A} = \frac{4w}{w^{2}+6w+1}. \label{Atilde}
\end{equation}
{\em unless} $w=-3\pm 2\sqrt{2}$ (For these values, the coefficient cannot vanish at all).
For more information, we consider the second-highest power in (\ref{MFeq-NS}), $m+n-1$. $n$ exists, but it may or may not be equal to $\tilde{m} = 1$. This necessitates consideration of two subcases:\\
\noindent {\em Case 2.1. $m>1, \; \tilde{m} = 1, n>1$}\\
In this subcase, eq.(\ref{m+n-1_coeff}) is again valid, therefore the same chain of arguments can be followed ending with rational $w$ and allowed $m$ values of 18, 15 and 3. Subcases {\em 2.1.1} - {\em 2.1.4} ($m=18,15$) are covered by subcases {\em 1.1.1} - {\em 1.1.4} in the `brute force' approach, giving no solutions. This leaves\\
\noindent {\em Case 2.1.5. $m = 3$, $n>1$, $w=-1$,}\\
which fails because the solutions for $n$ are 3 and 0,\\
\noindent {\em Case 2.1.6. $m = 3$, $n>1$, $w=-\frac{1}{3}$,}\\
which fails because solutions for $n$ are 3 and 1; this finishes subcase {\em 2.1.} \\
\noindent {\em Case 2.2. $m>1, \; \tilde{m} = 1 = n$}\\
Putting $F(r) = A r^{m} + \tilde{A}r$ into the OV eqn. (\ref{MFeq-NS}), and using (\ref{m_from_w}) and (\ref{Atilde}), we get
\begin{equation}
(1+3w)(w^{2}+6w+1)= 0. \label{case22}
\end{equation}
As pointed out after eq.(\ref{Atilde}), $(w^{2}+6w+1)$ cannot vanish , so we get
\begin{equation}
{\rm {\bf Solution \; 3:}} \;\;\;\; w=-\frac{1}{3}, \;\; F(r) = A r^3 + \frac{3}{2}r. \label{Sol3}
\end{equation}
This finishes case {\em 2}. Solution 3 does {\em not} include Solution 2 as a special case.\\
\noindent {\em Case 3. $m>1, \; \tilde{m} = 0$}\\
This time, the lowest power in (\ref{MFeq-NS}) is $\tilde{n}-1$, with coefficient
\begin{equation}
[(w+1) \tilde{n} - 2w \tilde{n}(\tilde{n}-3)]\tilde{A} \tilde{B} \label{nt-1_coeff}
\end{equation}
hence
\begin{equation}
\tilde{n}=\frac{7w+1}{2w}
\end{equation}
which again means that $w$ is rational and leads to the same $m-w$ pairs as in subcase {\em 1.1}. The subcases {\em 3.1} - {\em 3.4} ($m=18,15$) are again covered by subcases {\em 1.1.1} - {\em 1.1.4} in the `brute force' approach with no solutions, leaving the $m=3$ cases.\\
\noindent {\em Case 3.5. $m=3, \; \tilde{m} = 0, \; w = -1$.}\\
This leads to $ \tilde{n} = 3$, therefore to
\begin{equation}
{\rm {\bf Solution \; 4:}} \;\;\;\; w=-1, \;\; F(r) = A r^3 + C \label{Sol4}
\end{equation}
which includes solution 1 as a special case.\\
\noindent {\em Case 3.6. $m=3, \; \tilde{m} = 0, \; w=-\frac{1}{3}$.}\\
This gives $\tilde{n} = 2$, but also $n=1$ (see subcase {\em 2.1.6} or eq.(\ref{n_from_m})), an impossibility. This concludes case {\em 3}.\\
\noindent {\em Case 4. $m>1, \; \tilde{m} < 0$}\\
In this case, eqs. (\ref{m_from_w}) and (\ref{mtilde_lowest}) mean that $m=\tilde{m}$, hence this case fails.\\
\noindent {\em Case 5. $m=1=\tilde{m}$}\\
Very straightforwardly, one can derive
\begin{equation}
{\rm {\bf Solution \; 5:}} \;\;\;\; w {\rm \; arbitrary,} \;\; F(r) = \frac{4w}{w^{2}+6w+1}r \label{Sol5}
\end{equation}
Note that the coefficient $A$ was arbitrary in solutions 1-4, but it is determined in terms of $w$ in solution 5. Also, solutions 1-4 required certain values of $w$, while solution 5 is valid for arbitrary $w$.\\
\noindent {\em Case 6. $m=1, \; \tilde{m}=0$}\\
It turns out that for one value of $w$, one can add a constant term to the above solution:
\begin{equation}
{\rm {\bf Solution \; 6:}} \;\;\;\; w=-\frac{1}{5}, \;\; F(r) = 5 r + B . \label{Sol6}
\end{equation}
\noindent {\em Case 7. $m=1, \; \tilde{m}<0$}\\
In a sense, this case is a mirror image of case {\em 2}. We have (\ref{mtilde_lowest}), and consideration of highest power of $r$ in (\ref{MFeq-NS}) gives
\begin{equation}
A = \frac{4w}{w^{2}+6w+1}. \label{case7}
\end{equation}
As in case {\em 2}, $(w^{2}+6w+1)$ cannot vanish.
For more information, we consider the second-lowest power $\tilde{m}+\tilde{n}-1$, distinguishing if $\tilde{n}$ is equal to 1 or not.\\
\noindent {\em Case 7.1. $m=1, \; \tilde{m}<0, \tilde{n}<1$}\\
The coefficient of $r^{\tilde{m}+\tilde{n}-1}$ is given by the same expression as eq.(\ref{m+n-1_coeff}) with $m \rightarrow \tilde{m}$, $n \rightarrow \tilde{n}$, $A \rightarrow \tilde{A}$ and $B \rightarrow \tilde{B}$. This makes again $w$ rational, but now $\tilde{m}$ must be 18 or 15 or 3, unacceptable because they are positive.\\
\noindent {\em Case 7.2. $m=1, \; \tilde{m}<0, \tilde{n}=1$}\\
$F(r)$ consists of two terms, $F=Ar+\tilde{A} r^{\tilde{m}}$ now. The vanishing of the coefficient of $r^{\tilde{m}}$ in (\ref{MFeq-NS}) reduces upon the substitutions (\ref{mtilde_lowest}) and (\ref{case7}) to (\ref{case22}) again, giving the unacceptable (positive) $\tilde{m}$ value 3.\\
\noindent {\em Case 8. $m=0,=\tilde{m}$}\\
This case is trivial:
\begin{equation}
{\rm {\bf Solution \; 7:}} \;\;\;\; w {\rm \; arbitrary,} \;\; F(r) = A. \label{Sol7}
\end{equation}
\noindent {\em Case 9. $m=0, \; \tilde{m}<0$}\\
The highest power in (\ref{MFeq-NS}) is $n$ now, with coefficient
$2wn(n-3)B$, similar to lowest power in case {\em 1}. This cannot vanish; hence there is no solution in this case.\\
\noindent {\em Case 10. $m, \tilde{m} < 0$}\\
The same argument as above is valid here for $n \rightarrow m$, so again there is no solution.
This completes all finite polynomial solutions of equation (\ref{MFeq-NS}). Since Solution 1 is a special case of Solution 4, we will not consider it separately in the following section.
\section{Discussion of the solutions found from the standard (NS) OV equation} \label{sect:NSdisc}
To finalize the solutions, we calculate the metric functions $A(r)$ and $B(r)$ by using (\ref{EEns4}), (\ref{EEns5}), (\ref{eos}) and (\ref{FDefNS}). The calculation of $B(r)$ involves an arbitrary multiplicative constant at the last stage, the change of which is usually interpreted as a rescaling of $t$, therefore physically irrelevant. But such rescaling cannot change the sign of that constant, so we consider the two choices of sign as two separate solutions, unless the requirement of correct signature forces a choice upon us. This happens for solutions 3, 4 and 7, whereas for solutions 2, 5 and 6 we have consider both signs. The results are shown in Table \ref{table:NSsols}, where the well-known solutions are indicated in italics.
\begin{table}
\begin{tabular}{ | c | p{20 mm} | c | c | c | p{42 mm} |} \hline
{\bf Sol.No.} & \multicolumn{1}{c|}{{\bf $w$}} & {\bf $F(r)$} & {\bf $B(r)=-g_{tt}$} & {\bf $A(r)=g_{rr}$} & {\bf Comments} \\ \hline \hline
2 a,b & \multicolumn{1}{c|}{\large $-\frac{1}{3}$} & $Ar^{3}$ & $\pm 1$ & \Large $\frac{1}{1-Ar^{2}}$ & \mbox{2a, $A>0$ : {\em ESU}}; \mbox{2a, $A<0$ : open, static}; \mbox{2b : type TD} \\ \hline
3 & \multicolumn{1}{c|}{$-\frac{1}{3}$} & $Ar^{3}+\frac{3}{2}r$ & \Large $ -\frac{\frac{1}{2}+Ar^{2}}{r^{2}}$ & \Large $-\frac{1}{\frac{1}{2}+Ar^{2}}$ & \mbox{$A>0$ : type TD}; \mbox{$A<0$ : BH-like}\\ \hline
4 & \multicolumn{1}{c|}{-1} & $Ar^{3}+C$ & \large $1-\frac{C}{r}-Ar^{2}$ & \Large $\frac{1}{1-\frac{C}{r}-Ar^{2}}$ & {\em K\"{o}ttler (SdS)} \\ \hline
5 a,b & arbitrary, except $-1$, $-3\pm2\sqrt{2}$ & \large $\frac{4w}{w^{2}+6w+1}r$ & $\pm$ \Large $(\frac{r}{r_{0}})^{\frac{4w}{w+1}}$ & \large $1+\frac{4w}{(w+1)^{2}}$ & \mbox{5a: type NS,} \mbox{incl. static phantom}; \mbox{5b: type TD}\\ \hline
6 a,b & \multicolumn{1}{c|}{$-\frac{1}{5}$} & $5r+B$ & \large $\pm \frac{r_{0}}{r}$ & \Large $\frac{1}{\frac{C}{r}-4}$ & \mbox{6a: type NS}; \mbox{6b: type TD}\\ \hline
7 & arbitrary & $A$ & \large $(1- \frac{A}{r})$ &\Large $\frac{1}{1-\frac{A}{r}}$ & {\em Schwarzschild} \\ \hline
\end{tabular}
\caption{All finite-polynomial solutions of the equation (\ref{MFeq-NS}) for the mass function in the standard (NS) OV case, together with the corresponding metric functions. Although we started with the NS OV equation, some of the solutions belong to class TD, as defined in~\cite{revisited}. In Solutions 2, 5 and 6, the upper signs in $B(r)$ apply to solutions a and lower signs to solutions b. The well-known solutions are indicated in {\em italics}.}
\label{table:NSsols}
\end{table}
When the metric functions are negative, the spacetime cannot be supported by normal perfect fluid, the source fluid must be tachyonic. In other words, such a spacetime is of type TD in the terminology of \cite{revisited}. In that case, the OV equation, (\ref{ov}), is not valid, but still, $A(r)$-$B(r)$ pairs satisfy the same equation of pressure isotropy for cases NS and TD. Therefore negative metric functions found from NS-equations represent a valid TD solution, but not with the equation of state that one has started with. If the NS equation of state is (\ref{eos}), the corresponding TD equation of state becomes $p=-\frac{w}{1+2w}\rho$.
Solution 7 is the Schwarzschild solution.
It may at first seem surprising that there is no restriction on $w$. But since $\rho$ vanishes, the value of $w$ does not matter. In other words, it corresponds to a situation where all the fluid --whatever its equation of state parameter is-- has already collapsed to the origin. Also, here we do not apply the usual restriction that $A$ must be positive. If $A$ is negative, the spacetime will give a naked singularity.
Solution 2a with positive $A$ is also well-known: it is the Einstein static universe, with intimate historical connection to the cosmological constant $\Lambda$, equivalent to $w=-1$. But this universe also contains matter ($w=0$), whose attraction is precisely balanced by the repulsion of $\Lambda$. So the matter density is proportional to $\Lambda$ and the net effect is equivalent to a single fluid with $w=-\frac{1}{3}$. Of course, ``in the universe" $Ar^{2}<1$, so $A(r)$ is positive and the signature correct.
For negative $A$, Solution 2a represents an open static universe, albeit with negative energy density, and no coordinate restriction.
Noting that the third well-known solution in Table \ref{table:NSsols} is Solution 4, the K\"{o}ttler (aka Schwarzschild-de Sitter) solution, the de Sitter part sometimes being called anti-de Sitter if $A$ is negative, we turn to the discussion of less well-known solutions; changing the order slightly in the interest of presentation.\\
\noindent {\em Solution 2b} : \\
This solution has correct signature only for $Ar^{2}>1$, which means $A$ must be positive. It is a dynamic spacetime, $r$ being timelike, (it is solution TD1 of \cite{revisited}) and describes a spacetime that first contracts, then expands in angular directions, while distances in the orthogonal spacelike direction stay fixed\footnote{The KS-like form of the metric is $ds^{2}=-d\tau^{2}+d\rho^{2}+\frac{1}{A} \cosh^{2}(\sqrt{A}\tau) \, d\Omega^{2}$.}. Even though we found this solution for $w=-\frac{1}{3}$, the equation of state is actually $p=\rho$. The solution can be identified with the $n=0$ choice of Tolman V~\cite{Tolman}, if const=-1 is chosen\footnote{But Tolman chooses const=$B^{2}$ and later literature reports this form (e.g. \cite{delgaty&lake}).}, with identification $\frac{1}{R^{2}} \rightarrow A$.\\
\noindent {\em Solution 5a} : \\
Correct signature means positivity of $A(r)$ in this solution, which in turn means that the solution is valid except for $-3-2\sqrt{2} < w < -3+2\sqrt{2}$ (and it is of type NS). The cases $w < -3-2\sqrt{2}$, for example, $w=-6$, represent static (ultra)phantom solutions. The $w=\frac{1}{3}$ case is well-known~\cite[Prob. 23.10]{mtw}; the $w \rightarrow \infty$ limit, meaning zero density but nonzero pressure, is the metric called S1 in \cite{s1}; other valid cases with integer power of $r$ in $B(r)$ are $w=1$ and $w=3$.
The density is proportional to $\frac{1}{r^{2}}$, but this is a mild singularity because the mass function goes to zero as $r \rightarrow 0$, i.e. there is no mass point at the origin. Of course, there is no event horizon, so the singularity is naked.
The origin is attractive to test particles for $w < -3-2\sqrt{2}$ and for $w > 0$; repulsive for $-3+2\sqrt{2} < w < 0$. The sign of attraction correlates with the sign of $\rho + 3p$, the so-called "density of active gravitational mass" (e.g.~\cite{groundup}) for the fluid. The pressure is positive for all $w$ ranges, and since $p \propto \rho \propto \frac{1}{r^{2}}$, the pressure gradient is always towards the origin. So, we cannot understand the balance of a fluid element as in terms of $\rho$ (both forces would accelerate the fluid element towards the origin in the ultraphantom case), but in terms of $\rho + p$, the "density of inertial mass" (e.g.~\cite{groundup}).
These static ultraphantom solutions constitute a counterexample to the impression in the literature (e.g. see~\cite{no_stat_phantom}) that everywhere-phantom static spherically symmetric solutions cannot exist.
This solution can be identified with the $n=\frac{2w}{w+1}$, $R \rightarrow \infty$ and $B=\frac{1}{r_{0}}$ (or const=$r_{0}^{-\frac{4w}{w+1}}$) choice of Tolman V~\cite{Tolman}.\\
\noindent {\em Solution 5b} : \\
This is a TD solution (a subcase\footnote{Which subcase it is depends on the sign of $w+1$.} of TD2 of \cite{revisited}) valid for $-3-2\sqrt{2} < w < -3+2\sqrt{2}$, except $w=-1$. Assuming $r$ is future-directed, this spacetime expands in the angular directions, and either expands (for $w < -1$) or contracts (for $w > -1$) in the orthogonal spacelike direction\footnote{Metric in KS-like form: $ds^{2}=-d\tau^{2}+\left(\frac{\tau^{2}}{|A|r_{0}^{2}} \right)^{\frac{2w}{w+1}} d\rho^{2} + \frac{\tau^{2}}{|A|} \, d\Omega^{2}$, where \mbox{$A=\frac{w^{2}+6w+1}{(w+1)^{2}}$}.}. An infinite number of $w$-values, crowding -1, exist that give integer power of $r$ in $B(r)$.
The equation of state is $p=-\frac{w}{1+2w}\rho$. The density is still proportional to $\frac{1}{r^{2}}$, but because of the timelike nature of $r$, $F(r)$ cannot be interpreted as the mass function, and therefore we cannot make the same claim as to the mildness of the singularity as in solution 5a.
This solution can be identified with the almost same subcase of Tolman V~\cite{Tolman} as solution 5a, except\footnotemark[3] const=$-r_{0}^{-\frac{4w}{w+1}}$.\\
\noindent {\em Solution 3} : \\
For positive $A$, this solution is also of type TD, contracting in the angular directions and expanding in the orthogonal spacelike direction\footnote{The KS-like form of the metric is $ds^{2}=-d\tau^{2} + A \coth^{2}(\sqrt{A}\tau) d\rho^{2}+\frac{1}{2A} \sinh^{2}(\sqrt{A}\tau) \, d\Omega^{2}$.} as $r \rightarrow 0$.
For negative $A$, both metric functions switch sign at $r=r_{H}=\sqrt{-\frac{1}{2A}}$, so that the spacetime is static (NS) for $r>r_{H}$ and dynamic (TD) for $r<r_{H}$. As far as test particle motion is concerned, this spacetime would be that of a black hole; but it must be supported by normal matter in the NS region, and tachyonic matter (with $p=\rho$) in the TD region. As unreasonable as this may seem, it is the only possible perfect fluid interpretation \cite{revisited}.
As the origin of $r$ is approached, the density again diverges like $\frac{1}{r^{2}}$, but again $r$ is timelike near the origin, and the same (non)conclusion applies to the singularity as in Solution 5b.
For positive $A$, this solution can be identified with the $a=1$, $b=-1$, $m=0$, $\frac{1}{R^{2}}=A$ (and the trivial $B=1$ or const=1) choice of Tolman VIII~\cite{Tolman}. \\
\noindent {\em Solution 6a} : \\
This solution is type NS. $C$ must be positive and $r<\frac{C}{4}$. Interestingly, radially moving free particles oscillate between a minimum radius and $\frac{C}{4}$ , which may be understood in terms of the repulsion of the negative mass point at the origin ($C=-B$ and $F(r)$ is the mass function) versus the attraction of the fluid, whose ``enclosed active gravitational mass" (e.g.~\cite{groundup}) grows with $r$ (here, both $\rho$ and $\rho + 3p$ are positive).
The origin is a naked singularity, and not only due to the negative point mass there: The scalar curvature is $\frac{8}{r^{2}}$, that is, it diverges without containing $C$. But, after all, the scalar curvature does not contain $M$ in the Schwarzschild case, either (in fact, it vanishes). $r=\frac{C}{4}$ is a type of boundary, it is a turning point for all radial timelike geodesics.
This solution can be identified with the $n=-\frac{1}{2}$, $R \rightarrow -C$ and $B^{2}=r_{0}$ choice of Tolman V~\cite{Tolman}.\\
\noindent {\em Solution 6b} : \\
This TD solution (with $p=\rho/3$) can be identified with the $n=-\frac{1}{2}$, $R \rightarrow -C$ and const=-$r_{0}$ choice\footnotemark[3] of Tolman V~\cite{Tolman}. There is no coordinate restriction for negative $C$, but $r$ must be larger than $\frac{C}{4}$ for positive $C$.
In the latter case, again $r=\frac{C}{4}$ is a turning point for timelike radial geodesics, but $r$ is timelike, so this spacetime first contracts in the angular directions while expanding in the orthogonal spacelike direction, then the evolution reverses.
On the other hand, for negative $C$, the spacetime expands in the angular directions while contracting in the orthogonal spacelike direction\footnote{KS-like form of the metric is $ds^{2}=-d\tau^{2}+\frac{1}{r(\tau)} d\rho^{2} + r^{2}(\tau) \, d\Omega^{2}$, where $\frac{dr}{d\tau}=\pm \sqrt{4-\frac{C}{r}}$}, assuming $r$ is future-directed.
\section{Discussion of solutions found from the OV-like equations in the other cases and another attempt} \label{sect:NonStdSols}
As discussed in the previous section, the TD solutions satisfy the same equation of pressure isotropy as the NS solutions, therefore a solution derived from the OV(like) equations of one class may in fact belong to the other class. Moreover, the same is valid for the ND and TS classes.
It turns out that for the equation of state $p=w\rho$, the OV-like equations of the TD and TS cases do not give any solutions not already covered by NS and ND cases, except for the special $w$ value $-\frac{1}{2}$. This should not be taken as an indication that the TD and TS solutions are trivial relabelings; for more complicated equations of state, there {\em will} be different solutions.
The proof of the above statement, the application of the procedure of Sect.2 to the ND case to find all ND and TS solutions with finite-polynomial $F(r)$ for $w \neq -\frac{1}{2}$, and the TD and TS solutions for $w = -\frac{1}{2}$ are given in the appendix.
In this section, we calculate the metric functions $A(r)$ and $B(r)$ for each solution from the appendix by using the relevant formulae, and discuss the solutions. We also show that no nontrivial solution with finite-polynomial $A(r)$ exists.
\subsection{The TD case}
For Solution 8, we get
\begin{equation}
A = \frac{1}{1-C/r} \label{Soltd3A}\\
\end{equation}
which for $r < C$ (only possible if $C > 0$) gives
\begin{equation}
B = - r_{1}^{-4} \left[ (2 r^{2}+5Cr-15 C^{2})+\sqrt{\frac{C-r}{r}} \left(C_{1}-15 C^{2} \tan^{-1}\sqrt{\frac{C-r}{r}}\right) \right]^{2}, \label{SolTd3B}
\end{equation}
i.e. Solution TD3 of \cite{revisited}.
On the other hand, for $r > C$ we find
\begin{equation}
B = r_{1}^{-4} \left[ (2 r^{2}+5Cr-15 C^{2})+\sqrt{\frac{C-r}{r}} \left(C_{1}+15 C^{2} \ln(\frac{\sqrt{r-C}+\sqrt{r}}{\sqrt{|C|}})\right)\right]^{2} , \label{SolNSB}
\end{equation}
the solution called NS1 in \cite{revisited}, found in \cite{Kuch68I} and named Kuch68 I in~\cite{delgaty&lake}. It describes a spacetime where pure pressure is in static equilibrium with its own gravitational attraction.
\subsection{The ND(KS) and TS cases}
The solutions found for the ND(KS) and TS cases, together with their metric functions, are shown in Table \ref{table:NDsols} (Solution 9 does not appear because it is a special case of Solution 11). As in Sect.3, the sign of $B(r)$ is arbitrary, unless forced by the signature requirement.
\begin{table}
\begin{tabular}{ | c | p{17 mm} | c | c | c | p{35 mm} |} \hline
{\bf Sol.No.} & \multicolumn{1}{c|}{{\bf $w$}} & {\bf $F(r)$} & {\bf $B(r)=-g_{tt}$} & {\bf $A(r)=g_{rr}$} & {\bf Comments} \\ \hline \hline
10 & \multicolumn{1}{c|}{ $-3$} & $Ar^{9}+\frac{9}{8}r$ & \Large $ -\frac{1+8Ar^{8}}{r^{6}}$ & \Large $-\frac{8}{1+8Ar^{8}}$ & \mbox{$A>0$: type ND(KS)}, phantom-filled dynamic universe; \mbox{$A<0$ : BH-like} \\ \hline
11 & \multicolumn{1}{c|}{$-1$} & $Ar^{3}+B$ & \large $1-\frac{B}{r}-Ar^{2}$ & \Large $\frac{1}{1-\frac{B}{r}-Ar^{2}}$ & {\em K\"{o}ttler (SdS)} \\ \hline
12 a,b & arbitrary, except $-\frac{1}{3}$ and $1$ & \large $\frac{4w^{2}}{3w^{2}-2w-1}r$ & $\pm$ \Large $ (\frac{r}{r_{0}})^{-\frac{4w}{w+1}}$ & \Large $-\frac{(w-1)(3w+1)}{(w+1)^{2}}$ & \mbox{$-\frac{1}{3}<w<1$:} type TS; otherwise: \mbox{type ND(KS)}, incl. DE, incl. phantom\\ \hline
13 a,b & \multicolumn{1}{c|}{$\frac{1}{3}$} & $-\frac{1}{3}r+B$ & \large $\pm \frac{r_{0}}{r}$ & \large $\frac{3r}{4r-3B}$ & \mbox{13a: type TS;} \mbox{13b: type ND(KS)}\\ \hline
14 & arbitrary & A & \large $(1- \frac{A}{r})$ &\Large $\frac{1}{1-\frac{A}{r}}$ & {\em Schwarzschild}\\ \hline
15 a,b & \multicolumn{1}{c|}{$1$} & $\frac{C}{r}$ & $\pm 1$ & \Large $\frac{1}{1-\frac{C}{r^{2}}}$ & \mbox{15a: type TS;} \mbox{15b: type ND(KS)}\\ \hline
16 & \multicolumn{1}{c|}{$-\frac{1}{2}$} & $C$ & \large $(1- \frac{C}{r})$ & \Large $\frac{1}{1-\frac{C}{r}}$ & {\em Schwarzschild} \\ \hline
17 & \multicolumn{1}{c|}{$-\frac{1}{2}$} & $\frac{4}{3}r$ & \Large $ -(\frac{r}{r_{0}})^{-4}$ & $-3$ & type ND(KS)\\ \hline
\end{tabular}
\caption{All finite-polynomial solutions for $F(r)$ in the ND(KS) and TS cases as defined in~\cite{revisited}; together with the corresponding metric functions. In Solutions 12, 13 and 15, the upper signs in $B(r)$ apply to solutions a and lower signs to solutions b. The well-known solutions are indicated in {\em italics}.}
\label{table:NDsols}
\end{table}
The Schwarzschild and K\"{o}ttler (SdS) solutions, which appeared in Table \ref{table:NSsols}, are found in this table as well, because they cannot really be classified in this scheme. Our classification is based upon the nature and direction of motion of the fluid, but for these solutions, the stress-energy-momentum tensor is independent of the fluid four-velocity: The $u_{\mu}u_{\nu}$ term in $T_{\mu\nu}$ is multiplied by $p+\rho$; and $p+\rho=0$ for the K\"{o}ttler solution, $p=\rho=0$ for Schwarzschild. Hence, these solutions satisfy the equations for all four cases.
The other solutions in the table are less well-known: \\
\noindent {\em Solution 10} : \\
For positive $A$, this solution is type ND (KS), representing a dynamic spacetime filled with a phantom perfect fluid. Assuming $r$ is future-directed, the spacetime expands in the angular directions; in the perpendicular spacelike direction, it first contracts, reaches a minimum, then expands\footnote{The KS form of the metric is $ds^{2}=-d\tau^{2}+\frac{1+8Ar^{8}(\tau)}{r^{6}(\tau)} d\rho^{2} + r^{2}(\tau) \, d\Omega^{2}$, where \mbox{$\frac{dr}{d\tau}=\pm \sqrt{Ar^{8}+\frac{1}{8}}$.}}. It is singular at both ends of the evolution, that is, at $r=0$ and as $r \rightarrow \infty$, the first singularity being in the finite past, the second in the infinite future. Of course, these attributes switch if $r$ is past-directed.
For negative $A$, Solution 10, like Solution 3, represents a black hole spacetime, as far as test particle motion is concerned; but it must be supported by two different fluids on the two sides of the horizon: tachyonic fluid in the outside, static region and normal fluid in the dynamic region inside/in the future.\\
\noindent {\em Solutions 12a,b} : \\
Solution {\em 12a} is a TS solution, valid for $-\frac{1}{3}<w<1$. For positive $w$, radially incoming test particles are reflected near the origin back to infinity, whereas for negative $w$, the origin constitutes a potential well from which they cannot escape.
Solution {\em 12b}, however, is valid for $w$ values {\em other than} $-\frac{1}{3}<w<1$, and is identical to the $C_{1}=0$ special case of Solution ND2 of \cite{revisited}. If $r$ is future-directed, it expands in the angular directions, and it expands or contracts in the perpendicular spacelike direction, if the sign of $\frac{w}{w+1}$ is negative or positive, respectively\footnote{The KS form of the metric is $ds^{2}=-d\tau^{2}+\left(\frac{\tau^{2}}{r_{0}^{2}|A|} \right)^{-\frac{2w}{w+1}} d\rho^{2} +\frac{\tau^{2}}{|A|} \, d\Omega^{2}$, where \mbox{$A=-\frac{(w-1)(3w+1)}{(w+1)^{2}}$}.}. Note that this means expansion for non-phantom dark energy ($-1<w<-\frac{1}{3}$) and ``radial'' contraction for phantom energy.\\
\noindent {\em Solutions 13a,b} : \\
Solution {\em 13a} is a TS solution, where we must have $4r>3B$, i.e. we have a restriction on $r$ if $B$ is positive. Either way, the equation of motion for test particles shows that tachyonic $w=\frac{1}{3}$ fluid is repulsive, consistent with the solution {\em 12a}.
Solution {\em 13b} is an ND (KS) solution, where $4r<3B$. It represents a radiation-filled universe that expands and recollapses in angular directions while contracting and reexpanding in the perpendicular spacelike direction\footnote{The KS form of the metric is $ds^{2}=-d\tau^{2}+\frac{r_{0}}{r(\tau)} d\rho^{2} + r^{2}(\tau) \, d\Omega^{2}$, where \mbox{$\frac{dr}{d\tau}=\pm \sqrt{\frac{B}{r}-\frac{4}{3}}$}. The arbitrary $r_{0}$ must be chosen as $\frac{3}{4}B$ for agreement with \cite{kantowskiTH}, p.1684.}; first found in \cite{kantowskiTH}. \\
\noindent {\em Solutions 15a,b} : \\
Solution {\em 15a} is solution TS1 of \cite{revisited}, where we must have $r^{2}>C$. For positive $C$, $r_{0}=\sqrt{C}$ is a turning point for radial geodesics; for negative $C$, there are no such turning points.
Solution {\em 15b} is solution {\em ND1} of \cite{revisited}, apparently first found in \cite{thorneTH}, describing a finite-lifetime universe containing stiff matter, expanding and recollapsing in the angular directions\footnote{The KS form of the metric is $ds^{2}=-d\tau^{2}+ d\rho^{2} + (C-\tau^{2}) \, d\Omega^{2}$.}.\\
\noindent {\em Solution 17} : \\
This solution is the $C_{1}=0, A=-3$ special case of solution {\em ND2} of \cite{revisited}, describing a spacetime containing pressure, but no density (because it is an ND (KS) solution found from the TS equations, its equation of state is {\em not} $p=-\frac{1}{2}$); expanding in angular directions while contracting in the perpendicular spacelike direction\footnote{The KS form of the metric is $ds^{2}=-d\tau^{2}+\left(\frac{\tau_{0}}{\tau}\right)^{4} d\rho^{2} + \frac{\tau^{2}}{3} \, d\Omega^{2}$.}, if $r$ is taken to be future-directed.
\subsection{Finite-polynomial $A(r)$?}
Another possible way to look for solutions is to work in terms of $A(r)$ rather than $F(r)$ by using equation (\ref{EEns4}). This leads to an equation with terms second to fourth order in $A(r)$ and/or its derivatives. In trying to find a finite-polynomial solution for $A(r)$, if the highest power of $r$ in $A(r)$ is $m$, the highest power of $r$ in the equation is $4m$; but it is multiplied by $A^{2} (w+1)^{2}$ in cases NS and TD, and $-A^{2} (w+1)^{2}$ in cases ND and TS. Setting the trivial $w=-1$ case aside, therefore, the highest possible value for $m$ is zero. A similar argument shows that the lowest power in the $A(r)$ polynomial must be zero or higher. Hence the only finite polynomial $A(r)$ can be for equation of state (\ref{eos}) is a constant.
\section{Summary and Conclusions}
We have considered spherically symmetric perfect fluid solutions in General Relativity and
found {\em all} finite-polynomial solutions -including negative powers- of the equation satisfied by the so-called "mass function" and its mathematical analogs for the equation of state $p=w\rho$; and discussed the associated spacetimes.
The equation for the mass function follows from the Oppenheimer-Volkoff (OV) equation in the standard case where the fluid is static and normal (i.e. non-tachyonic, $u^{\mu} u_{\mu} = -1$). However, the metric ansatz used in that analysis can also accomodate cases where the spacetime is dynamic in a certain way, or the fluid is tachyonic; as discussed in \cite{revisited}. In these other cases analogous, but different functions exist, satisfying their own equations.
The solutions we found for the standard case, NS, are mathematically not very original; they are either some limiting cases of solutions found long ago by Tolman \cite{Tolman} or simple modifications thereof. Some aspects of the physical nature of these solutions can be seen in new light however, considering the classification in \cite{revisited} and newly cosmologically relevant concepts of dark energy and phantom energy.
The solutions (Table \ref{table:NSsols}) include dynamic spacetimes supported by tachyonic fluids ({\em 2b, 5b, 3} with $A>0$, {\em 6b}) and a static spacetime containing a $w=-\frac{1}{5}$ fluid around a negative point mass ({\em 6a}).
The TD case gives two extra solutions, one describing a spacetime where pure pressure is in static equilibrium with its own gravitational attraction.
Some interesting solutions are also found from the ND(KS) and TS cases (Table \ref{table:NDsols}): There are {\em static} solutions supported by {\em tachyonic} fluids ({\em 12a, 13a, 15a}), the first two presumably original. Some solutions ({\em 10} for positive $A$, {\em 12b, 13b, 15b, 17}) are of the Kantowski-Sachs (KS) class: Solutions {\em 13b, 15b} and {\em 17} describe dynamic KS-universes containing radiation, stiff matter and pure pressure, respectively.
We would like to particularly point out the following solutions:
\begin{itemize}
\item Solution {\em 5a} for $w<-3-2\sqrt{2}$ represents, perhaps unexpectedly, a family of {\em static} ``ultraphantom'' solutions.
\item Solution {\em 3} for negative $A$ is a black hole-like spacetime, which must be supported by normal matter outside the horizon and tachyonic fluid on the inside.
\item Solution {\em 10} for positive $A$ is a {\em phantom} KS solution, probably new.
\item Solution {\em 12b} can also be valid for dark energy, including phantom, exhibiting anisotropic expansion for non-phantom dark energy.
\item Solution {\em 10} for negative $A$ is similar to Solution {\em 3}, a black hole-like spacetime, supported by segregated normal and tachyonic matter, except in this solution, the tachyonic fluid is outside and normal fluid is inside. It was concluded in \cite{revisited} that black holes supported by perfect fluids cannot be ``simple''.
\end{itemize}
There are {\em no other solutions} where $F(r)$ is a finite polynomial of $r$ for the assumed equation of state. One can also express the problem(s) in terms of $A(r)$, and then try to find finite polynomial solutions. The only such solution is $A(r)$=constant.
|
1,116,691,500,419 | arxiv | \section{Introduction}
\label{sec:intro}
The production of bottom quarks has been extensively studied at hadron colliders. Early measurements were already carried out by the UA1 collaboration at the CERN S$p{\bar p}$S~\cite{Albajar:1986iu,Albajar:1988th}
and, subsequently,
by the CDF~\cite{Abe:1995dv,Acosta:2001rz,Acosta:2004yw,Abulencia:2006ps} and D0~\cite{Abachi:1994kj,Abbott:1999se} collaborations at the Fermilab Tevatron. The most recent measurements were performed by the ALICE~\cite{Abelev:2014hla,Abelev:2012sca}, ATLAS~\cite{Aad:2012jga, ATLAS:2013cia}, CMS~\cite{Khachatryan:2011mk,Chatrchyan:2011pw,Khachatryan:2016csy,Chatrchyan:2012hw}, and LHCb~\cite{Aaij:2010gn,Aaij:2012jd,Aaij:2013noa,Aaij:2016avz} collaborations at the CERN LHC in $pp$ collisions at centre-of-mass energies $\sqrt{s}=2.76,7,8$ and 13 TeV.
At the theoretical level, heavy-quark production at hadron colliders is one of the most classic tests of perturbative QCD.
The cross section to produce a pair of heavy quarks with mass $m_Q$ is computable as a power series expansion in the QCD coupling $\as(\mu_R)$,
where the renormalisation scale $\mu_R$ has to be chosen of the order of $m_Q$.
In the case of the bottom ($b$) quark the relatively low mass, $m_b\sim 4-5$ GeV, leads to a slow convergence of the perturbative expansion and, therefore, to large theoretical uncertainties.
Theoretical predictions for $b{\bar b}$ production at next-to-leading order (NLO) in QCD have been available
for a long time~\cite{Nason:1987xz,Nason:1989zy,Beenakker:1988bq},
including calculations~\cite{Mangano:1991jk} of $b{\bar b}$ correlations and generic
infrared safe (IR) observables.
NLO studies based on different schemes for the renormalisation of the
bottom-quark mass are presented in Ref.~\cite{Garzelli:2020fmd}.
At high transverse momenta $p_T$ of the bottom quark, large logarithmic terms of the form $\ln (p_T/m_b)$
need to be resummed to all perturbative orders~\cite{Cacciari:1993mq}.
In the case of single-particle ($b$ quark or antiquark) inclusive cross sections
the resummation can be performed by introducing the perturbative fragmentation function~\cite{Mele:1990cw}
of the bottom quark (which can also be supplemented with all-order soft-gluon effects~\cite{Cacciari:2001cw}).
Predictions obtained by matching such resummed computations to the NLO calculation (the so called ``FONLL'' prediction)~\cite{Cacciari:1998it, Cacciari:2001td,
Cacciari:2002pa, Cacciari:2012ny}
have become the standard reference for the comparison with experimental data. Perturbative predictions for the inclusive production of bare bottom quarks can then be folded with non-perturbative functions
\cite{Kartvelishvili:1977pi, Peterson:1982ak}
describing the fragmentation into the triggered $b$-hadrons.
The parameters that control such fragmentation functions are typically extracted from LEP data (see, e.g., Ref.~\cite{Cacciari:2005uk}).
The variable-flavour-number scheme~\cite{Kniehl:2004fy, Kniehl:2005mk} is another
procedure that is used to combine the NLO calculation with high-$p_T$ resummation
effects for single-inclusive $b$-hadron production, and ensuing data--theory comparisons have been presented in the literature (see, e.g.,
Refs.~\cite{Kramer:2018vde,Benzke:2019usl}).
Phenomenological studies on the impact of bottom
production measurements on parton distribution functions have been presented in Refs.~\cite{Zenaiev:2015rfa,Cacciari:2015fta,Zenaiev:2019ktw}.
At the next-to-next-to-leading order (NNLO) in QCD, theoretical predictions for $b{\bar b}$ production are available only for the total cross section.
Indeed, the $b\bar b$ results can be directly derived by exploiting the corresponding NNLO theoretical calculation~\cite{Baernreuther:2012ws,Czakon:2012zr,Czakon:2012pz,Czakon:2013goa} of the total cross section for top-quark pair production.
NNLO values of the $b{\bar b}$ total cross section at several collider energies are presented in Refs.~\cite{Mangano:2016jyj,dEnterria:2016ids}.
The calculation of the $b{\bar b}$ total cross section at NNLO is implemented in the numerical program {\sc Hathor}~\cite{Langenfeld:2009wd,Aliev:2010zk}.
In this paper we report on the first fully differential NNLO QCD calculation for $b{\bar b}$ production at hadron colliders, and we also present comparisons with some inclusive $b$-hadron data obtained at the Tevatron and at the LHC.
Our computation, which follows the analogous calculation carried out for top-quark pair production~\cite{Catani:2019iny,Catani:2019hip,Catani:2020tko}, is implemented within the {\sc Matrix} framework~\cite{Grazzini:2017mhc} and allows us to evaluate arbitrary IR safe observables for the production of on-shell $b$ and ${\bar b}$ quarks at hadron colliders.
The NNLO calculation requires tree-level, one-loop and two-loop contributions.
We compute the tree-level and one-loop scattering amplitudes by using {\sc OpenLoops}~\cite{Cascioli:2011va,Buccioni:2017yxi,Buccioni:2019sur}.
We use the numerical result of Ref.~\cite{Baernreuther:2013caa} to evaluate the two-loop amplitudes.
The IR divergences that appear at intermediate stages of the computation are handled and cancelled, analogously to Refs.~\cite{Catani:2019iny,Catani:2019hip}, by using the $q_T$-subtraction formalism~\cite{Catani:2007vq}, which was properly extended to deal with heavy-quark production in Refs.~\cite{Catani:2014qha,Bonciani:2015sha}.
The paper is organised as follows. In Section~\ref{sec:matrix} we briefly review the {\sc Matrix} framework applied to heavy-quark production.
In Section~\ref{sec:resu} we present our numerical results: we first discuss the total cross section in Section~\ref{sec:resu:tota}, and then we report results for differential distributions
at the Tevatron in Section~\ref{sec:resu:tev} and at the LHC in Section~\ref{sec:resu:lhc}. Our results are summarised in Section~\ref{sec:summa}.
We devote \ref{app:eta-y} to a detailed discussion of the shape differences between
rapidity and pseudorapidity distributions.
In \ref{app:FONLL} we present a comparison of our NNLO results with FONLL predictions for the transverse-momentum and (pseudo)rapidity distributions of the bottom quark.
\section[{\sc Matrix} framework for heavy-quark production]{M{\normalsize ATRIX} framework for heavy quark production}
\label{sec:matrix}
The results presented in this work are obtained by using the $q_T$-subtraction formalism~\cite{Catani:2007vq} to handle and cancel IR singularities.
Specifically, we use the implementation of the formalism within the computational framework {\sc Matrix}.
In its public version, {\sc Matrix} permits the evaluation of differential distributions at NNLO in QCD for a wide class of processes in which the triggered final state is formed by colourless particles.
Recently, the computation of the last ingredients (namely NNLO soft gluon contributions) needed to extend $q_T$ subtraction to heavy-quark production was completed by some of us~\cite{inprep}.
An independent computation of these contributions is presented in Ref.~\cite{Angeles-Martinez:2018mqh} for the case of top-quark pair production.
With the results of Ref.~\cite{inprep} we were able to carry out a new calculation of top-quark pair production at NNLO in QCD~\cite{Catani:2019iny},
completing a previous work~\cite{Bonciani:2015sha} that was limited to the flavour off-diagonal production channels.
Their integration in the {\sc Matrix} framework allowed us to perform an efficient evaluation of single- and multi-differential distributions for stable top quarks~\cite{Catani:2019hip}.
For the present work we have generalised this implementation to arbitrary heavy-quark mass values and light-flavour numbers.
We apply a scheme with $n_f=4$ light-quark flavours and a massive bottom quark, while the top quark is decoupled from the process, to
obtain differential NNLO results for bottom-quark pair production.
The NNLO differential cross section for bottom-pair production, $d{\sigma}^{b{\bar b}}_{\rm NNLO}$, is obtained within the $q_T$-subtraction method according to the following main formula,
\begin{equation}
\label{eq:main}
d{\sigma}^{b{\bar b}}_{\rm NNLO}={\cal H}^{b{\bar b}}_{\rm NNLO}\otimes d{\sigma}^{b{\bar b}}_{\rm LO}
+\left[ d{\sigma}^{b{\bar b}+\rm{jet}}_{\rm NLO}-
d{\sigma}^{b{\bar b}, \, {\rm CT}}_{\rm NNLO}\right],
\end{equation}
where $d{\sigma}^{b{\bar b}+\rm{jet}}_{\rm NLO}$ represents the $\ensuremath{b {\bar b}}\xspace$+jet cross section at NLO accuracy, which we
evaluate by using the dipole subtraction method~\cite{Catani:1996jh,Catani:1996vz,Catani:2002hc}.
As is customary in the {\sc Matrix} framework, the NLO cross section $d\sigma^{b{\bar b}}_{\rm NLO}$ is also computed by using dipole subtraction, while $q_T$ subtraction is actually used only to evaluate
the NNLO correction $d\sigma^{b{\bar b}}_{\rm NNLO}-d\sigma^{b{\bar b}}_{\rm NLO}$.
The expression in Eq.~(\ref{eq:main}) is completely analogous to the \ensuremath{t {\bar t}}\xspace case~\cite{Catani:2019hip}.
We remind the reader that the term in the square bracket of Eq.~(\ref{eq:main}) is formally finite in the limit $q_T\rightarrow 0$ ($q_T$ is the transverse momentum of the $b\bar b$ pair),
but each of the two contributions in the square bracket is individually divergent.
Therefore a technical cut is introduced in the dimensionless quantity $r=q_T/m_{\ensuremath{b {\bar b}}\xspace}$ ($m_{\ensuremath{b {\bar b}}\xspace}$ is the invariant mass of the $b{\bar b}$ pair).
The $r_\text{cut}\rightarrow 0$ extrapolation is performed following the procedure of Ref.~\cite{Grazzini:2017mhc}, and it is also applied in the computation of differential distributions on a bin-by-bin basis.
The core of {\sc Matrix} is the Monte Carlo program {\sc Munich}\footnote{{\sc Munich}, which is the abbreviation of “MUlti-chaNnel Integrator at Swiss (CH) precision”, is an automated parton-level NLO generator by S. Kallweit.}, which contains a fully automated implementation of the dipole subtraction method for massless~\cite{Catani:1996jh,Catani:1996vz} and massive~\cite{Catani:2002hc} partons and a general implementation of an efficient phase space integration.
The required spin- and colour-correlated tree-level and one-loop amplitudes are obtained by using {\sc OpenLoops}~\cite{Cascioli:2011va, Buccioni:2017yxi,Buccioni:2019sur}, with the exception of the four-parton tree-level colour correlators for which we rely on an analytic implementation.
The two-loop amplitudes are obtained via an interpolation routine, based on the numerical results presented in Refs.~\cite{Czakon:2008zk, Baernreuther:2013caa}.
\section{Results}
\label{sec:resu}
In the following we present our results for total cross sections and differential distributions for $b{\bar b}$ production at the Tevatron ($\sqrt{s}=1.96$ TeV) and at the LHC ($\sqrt{s}=7$ and $13$ TeV).
We use the NNPDF31 parton distribution functions~(PDFs)~\cite{Ball:2017nwa} with $n_f=4$ massless-quark flavours and the value of the QCD coupling $\as(m_Z)=0.118$.
The pole mass of the bottom quark is fixed to $m_b=4.92$~GeV, consistently with the value that is used in the chosen PDF set.
Predictions at the $n$-th perturbative order are obtained by using the corresponding N$^n$LO PDF set and the evolution of $\as$ at $(n+1)$-loops.
Since the NNPDF31 set is not available at LO with $n_f=4$, we instead use the corresponding NNPDF30 set~\cite{Ball:2014uwa} for our LO predictions.
Perturbative uncertainties are estimated with the customary 7-point variation of the renormalisation ($\mu_R$) and factorisation ($\mu_F$) scales by a
factor of two around a common central value $\mu_0$,
with the constraint $0.5 \leq \mu_R/\mu_F \leq 2$.
The value of $\mu_0$ is chosen of the order of the characteristic hard-scattering scale of the process.
The total cross section is controlled by scales of the order of $m_b$, while each differential distribution is characterised by a different hard-scattering scale that has to be specified accordingly.
The fully differential nature of our calculation also allows us to use dynamic scales.
We point out that starting from NNLO the inclusive production of a $b{\bar b}$ pair receives contributions from tree-level diagrams with an additional $b\bar b$ pair in the final state.
Since the bottom-quark mass is kept non vanishing, these four-bottom contributions are separately finite and may be included or not in the perturbative calculation,
according to the actual (theoretical and experimental) definition of the inclusive cross section.
We find that these contributions are generally rather small. They are completely negligible at the Tevatron while
at the LHC they typically contribute at the per mille level, reaching about $0.5\%$ at large transverse momenta of the bottom quarks.
Owing to their small size, these contributions have no visible quantitative effects on the results that we are going to present.
Before presenting our numerical predictions, we discuss the impact of the two-loop virtual corrections on our NNLO results. The IR finite part of the two-loop contribution
is obtained by using the numerical results of Ref.~\cite{Baernreuther:2013caa}, which are provided through a $80\times 21$ dimensional grid in the Born level kinematical variables $\beta$ and $\cos\theta$ ($\beta$ and $\theta$ are the heavy-quark velocity and scattering angle in the partonic centre-of-mass frame).
The grid is then interpolated by using splines, and the ensuing result is supplemented with the analytical results at small and large $\beta$~\cite{Beneke:2009ye,Baernreuther:2013caa}.
The grid of Ref.~\cite{Baernreuther:2013caa} has, to date, been used only for top-quark production.
Since the bottom-quark mass is significantly smaller than the top-quark mass, one may wonder whether the small-angle (collinear) region is sampled sufficiently well in our calculation.
We have studied the effect of the IR finite part of the two-loop contribution on our calculations of the total cross section and several differential distributions for $b{\bar b}$ production.
Moreover, to quantify the impact of having a discrete grid,
we have repeated our calculations by using only half of the available grid points.
We find that the two-loop virtual contribution~\cite{Baernreuther:2013caa} is below $1\%$ in all cases. The differences obtained by reducing the number of points in the grid by a factor of two are
typically at the per mille level for all the distributions we have considered.
We conclude that we can safely use the results of Ref.~\cite{Baernreuther:2013caa} to carry out our fully differential calculation of \ensuremath{b {\bar b}}\xspace production.
\subsection{Total cross section}
\label{sec:resu:tota}
We start the presentation of our results by considering the total cross section.
The total cross section for the production of a pair of bottom quarks is controlled by scales of the order of the bottom mass $m_b$.
Accordingly, we will consider the two central scales $\mu_0=m_b$ and $\mu_0=2m_b$.
As discussed in Sec.~\ref{sec:matrix}, our NNLO results are obtained through an $r_{\rm cut}\rightarrow 0$ extrapolation procedure.
The $r_{\rm cut}$ dependence at the Tevatron with $\sqrt{s}=1.96$ TeV and at the LHC with $\sqrt{s}=13$ TeV is shown in Fig.~\ref{fig:r_cut} in the case $\mu_0=m_b$.
The ensuing NNLO cross section with its extrapolation uncertainty is compared with the corresponding result obtained with the numerical program {\sc Hathor}~\cite{Aliev:2010zk},
and the results of the two NNLO calculations are in good agreement. Similar results are obtained for all scale combinations
of the applied 7-point variation, and also separated into partonic channels as in Ref.~\cite{Catani:2019iny}.
Fig.~\ref{fig:r_cut} shows that the extrapolation uncertainties are larger than the corresponding uncertainties in the case of top-pair production~\cite{Catani:2019iny}, but, remarkably, still at the level of about $0.5\%$.
This larger uncertainty in the NNLO result is simply due to the larger relative size of the ${\cal O}(\as^4)$ contribution when considering a lower quark mass. Nevertheless, this level of uncertainty is perfectly acceptable as it is well below other sources of theoretical (and experimental) uncertainties affecting bottom-quark hadroproduction.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/qt_cut_NNLO.m_b_TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/qt_cut_NNLO.m_b_LHC13}.pdf}
\vspace*{-2ex}
\caption{The dependence on $r_{\rm cut}$ of the total cross section for $b\bar{b}$ production at different hadron colliders.}
\label{fig:r_cut}
\end{figure}
The NNLO results at the Tevatron and the LHC obtained with $\mu_0=m_b$ and $\mu_0=2m_b$ together with their scale uncertainties are reported in Table~\ref{table:total_vs_hathor}, compared to those obtained using {\sc Hathor}~\cite{Aliev:2010zk}. For both central-scale choices the agreement is excellent, including scale uncertainties.
We point out that the two computations are performed by using fully independent methods.
\input{tableHATHOR}
In Table~\ref{table:totalXS} we report the LO, NLO and NNLO results for $\mu_0=m_b$ and $\mu_0=2m_b$.
As in Table~\ref{table:total_vs_hathor}, the cross sections are presented with their perturbative uncertainties estimated through scale variations.
\input{tableTH}
By inspecting the results presented in Table~\ref{table:totalXS}, we immediately see that QCD corrections are very large.
In order to quantify their impact, we introduce $K$-factors, $K_\text{(N)NLO} = \sigma_\text{(N)NLO}/\sigma_\text{(N)LO}$, defined as the ratios of the cross section predictions at two subsequent orders.
The NLO $K$-factor $K_\text{NLO}$ tends to increase as the collider energy decreases. Specifically, the value of $K_\text{NLO}$
ranges between 1.29 (LHC, $\sqrt s =13$ TeV) and 1.74 (Tevatron) for the different energies and central scales under consideration.
The NNLO corrections are still sizeable, but weakly depend on the collider energy, with values of $K_\text{NNLO}$ ranging from 1.25 to 1.34.
These large QCD corrections, considerably larger for instance than the ones observed in top-quark production, are associated to the relatively low energy scales involved, which lead to large values of the strong coupling.
The poor perturbative behaviour is also reflected by the large scale uncertainties that are observed in Table~\ref{table:totalXS}.
It is important to remark, however, that in all cases the inclusion of the NNLO contribution allows us to strongly reduce the theoretical uncertainties.
In addition, $K_\text{NNLO}$ is always smaller than $K_\text{NLO}$, providing a sign of (slow) convergence of the perturbative series
We also note that the values of $\sigma_{\text{NNLO}}$ and $\sigma_{\text{NLO}}$ are consistent within their scale uncertainites for each energy and choice of $\mu_0$ (the values of $\sigma_{\text{NLO}}$ and $\sigma_{\text{LO}}$ are consistent as well).
We now turn to the discussion of the scale dependence of our results.
Due to the overall proportionality of the cross section to the factor $\as^2(\mu_R)$ at LO, the cross section typically decreases as $\mu_R$ increases.
On the contrary, the cross section generally increases as $\mu_F$ increases.
This is due to the fact that bottom-quark hadroproduction is sensitive to relatively low momentum fractions of the colliding partons, and in this kinematical region PDF scaling violations are typically positive.
When performing scale variations, the dominant effect is given by variations of the renormalization scale $\mu_R$.
Comparing the predictions corresponding to the two different scale choices, we observe that the results obtained with $\mu_0 = 2m_b$ present smaller scale uncertainties. Such difference is evident at LO and NLO, and less noticeable at NNLO, especially at the Tevatron.
The choice $\mu_0=2m_b$ leads to slightly smaller NNLO $K$-factors at the LHC ($K_\text{NNLO}=1.27(1.26)$ to be compared with $K_\text{NNLO}=1.31(1.34)$ at $\sqrt{s}=7(13)$ TeV for $\mu_0=m_b$), and
to larger $K$-factors at the Tevatron ($K_\text{NNLO}=1.30$ to be compared with $K_\text{NNLO}=1.25$ for $\mu_0=m_b$).
Despite the differences between the results obtained with these two central scales, both choices provide predictions that are fully compatible within scale uncertainties, and are equally acceptable on general grounds.
The use of $\mu_0 = m_b$ as central scale choice leads to larger uncertainties due to the lower values of $\mu_R$ involved, but we do not observe a particularly worrisome perturbative behaviour for this scale choice, in spite of the low value of $m_b$.
Besides the uncertainties related to missing higher-orders in the perturbative expansion, additional uncertainties on the bottom-quark cross section arise from the errors in the determination of the pole mass $m_b$, the parton distributions, and the strong coupling $\as(m_Z)$.
In Table \ref{table:totalXSwithunc} we report our result for the NNLO cross section computed with $\mu_0=m_b$ including these additional uncertainties.
As for the bottom mass, we follow Ref.~\cite{deFlorian:2016spz}, and we vary $m_b$ between $m_b=4.79$ GeV and $m_b=5.05$ GeV, corresponding to $m_b=4.92\pm 0.13$ GeV.
The effect of changing the bottom mass
is below $10\%$ and slightly decreases with increasing collider energy. The PDF uncertainties are much smaller and increase with the collider energy.
This is not unexpected: indeed,
$b{\bar b}$ production at the Tevatron and the LHC is mainly sensitive to PDFs with momentum fractions $x\sim 2m_b/\sqrt{s}$ in the range ${\cal O}(5\cdot 10^{-3})-{\cal O}(10^{-3})$. In this region the uncertainty in the gluon distribution increases as $x$ decreases.
As for the QCD coupling, we compute the corresponding uncertainty by evaluating the NNLO cross section with NNPDF31 NNLO PDFs obtained with $\as(m_Z)=0.119$ and $\as(m_Z)=0.117$.
Naively, one could expect relatively large effects since the process starts at ${\cal O}(\as^2)$
and small variations on $\as(m_Z)$ induce relatively large variations on $\as(m_b)$. However, it is well known that
the value of $\as(m_Z)$ and the gluon density are correlated. Correspondingly,
the ensuing effect on the NNLO cross section reported in Table~\ref{table:totalXSwithunc} is rather small and roughly independent on the collider energy.
In summary, from the results in Table~\ref{table:totalXSwithunc} we conclude that, despite the inclusion of the NNLO corrections and the consequent reduction of the scale uncertainties, the missing higher orders in the QCD perturbative expansion still represent the dominant source of theoretical uncertainty in $b{\bar b}$ production.
\input{tableTHuncertainties}
\subsection{Differential distributions: Tevatron}
\label{sec:resu:tev}
We start the presentation of our differential results by showing selected distributions in $p{\bar p}$ collisions at the Tevatron ($\sqrt{s}=1.96$ TeV).
Throughout this paper we always consider observables obtained by averaging the corresponding bottom and antibottom quark distributions. In particular we consider the average of the transverse-momentum, rapidity and pseudorapidity distributions of the (anti)bottom quark, which are denoted as $p_{T,b_{\rm av}}$, $y_{b_{\mathrm{av}}}$ and $\eta_{b_{\mathrm{av}}}$, respectively.
These observables are the parton-level equivalent of the corresponding $b$-hadron distributions.
In Fig.~\ref{fig:pT_av.Tev} we present our LO, NLO and NNLO predictions for the transverse-momentum distribution.
The calculation is carried out without applying any kinematical cut on the final-state partons.
The transverse-momentum distributions of the bottom and antibottom quark are controlled by hard-scattering scales of the order of the corresponding transverse mass,
\begin{equation}
m_{T,b/\bar b}=\sqrt{m_b^2+p_{T,b/\bar b}^2}\;.
\end{equation}
Therefore, in Fig.~\ref{fig:pT_av.Tev} (left and central panels) we show predictions with the scale choices $\mu_0=m_T$ and $\mu_0=2m_T$, i.e. by using $\mu_0=m_{T,b(\bar b)}$ or $\mu_0=2m_{T,b(\bar b)}$ for the (anti)bottom transverse-momentum distribution used to
compute the average.
In addition to these two `natural' scales, we also show predictions with the dynamic scale $\mu_0=H_T/2$ (right panel in Fig.~\ref{fig:pT_av.Tev}), defined as the average of the two transverse masses:
\begin{equation}
\frac 12 H_T=\frac{m_{T,b}+m_{T,\bar b}}2\;.
\end{equation}
Note that we have $H_T/2 = m_T$ at LO.
Differential results at each order in the perturbative expansion are shown in the upper panels of Fig.~\ref{fig:pT_av.Tev}, while in the lower panels the ratio of each perturbative order to our NNLO prediction is presented.
From Fig.~\ref{fig:pT_av.Tev} we see that the distribution is peaked at $p_{T,b_{\rm av}}\sim 3$ GeV. The average transverse momentum of the (anti)bottom quark is $4.6$ GeV (at both NLO and NNLO),
i.e., as expected, quite close to the value of the bottom-quark mass. As a consequence, the total cross section receives a small contribution from the large-$p_{T,b_{\rm av}}$ region.
For instance, the region with $p_{T,b_{\rm av}}<20$ GeV gives about $99\%$ of the total cross section.
\begin{figure}[t]
\centering
\includegraphics[width=0.32\textwidth]{{plots/distributions/pT_av.mT.Tevatron}.pdf}\hfill
\includegraphics[width=0.32\textwidth]{{plots/distributions/pT_av.2mT.Tevatron}.pdf}\hfill
\includegraphics[width=0.32\textwidth]{{plots/distributions/pT_av.HT_2.Tevatron}.pdf}
\vspace*{-2ex}
\caption{Transverse momentum distributions at the Tevatron, for the scale choice $\mu_0=m_T$ (left), $\mu_0=2m_T$ (central) and $\mu_0=H_T/2$ (right). The lower panels show the ratio to the NNLO predictions.}
\label{fig:pT_av.Tev}
\end{figure}
For all the considered central-scale choices LO and NLO predictions are consistent within uncertainties only in the low-$p_{T,b_{\mathrm{av}}}$ region, where the bulk of the cross section is located, while they present very different shapes in the tail of the distribution, where the NLO corrections become more sizeable. In this region the LO and NLO scale uncertainty bands do not overlap.
The inclusion of the NNLO corrections leads to a nice stabilisation of the perturbative result, analogously to what we have observed for the total cross section. In particular, the uncertainty at NNLO is smaller than at NLO, and the NLO and NNLO bands overlap in the entire region of transverse momenta.
By comparing the left and central panels in Fig.~\ref{fig:pT_av.Tev} we see that the choice of a smaller scale $\mu_0=m_T$ leads to a better overlap between the NLO and NNLO uncertainty bands, similarly to what was observed for the total cross section.
In particular, the scale choice $\mu_0=2m_T$ leads to a significantly worse perturbative convergence in the tail of the distribution.
The dynamic scale $\mu_0=H_T/2$ presents similar features to those observed for $\mu_0=m_T$, consistently with the fact that both scales are equivalent at LO, with a good overlap of the NLO and NNLO bands for all values of $p_{T, b_\mathrm{av}}$.
Perturbative predictions for bottom-quark production can be compared to experimental data for the inclusive production of $b$-hadrons.
A precise comparison in the region of large transverse momenta of the bottom quark (i.e. $p_T\gg m_b$)
would require the resummation of the logarithmically enhanced terms at large transverse momenta~\cite{Cacciari:1993mq}.
Such higher-order contributions are included up to next-to-leading logarithmic accuracy in the resummed predictions
of Refs.~\cite{Cacciari:1998it,Cacciari:2001td}. The non-perturbative effects of the fragmentation of the bottom quark into the triggered $b$-hadron
should eventually be accounted for by folding the perturbative result with an appropriate non-perturbative fragmentation function.
In this paper we limit ourselves to considering perturbative predictions up to NNLO, and a thorough comparison with experimental data is beyond the scope of this work.
In \ref{app:FONLL} we present a comparison of our NLO and NNLO results with the
FONLL prediction from Ref.~\cite{webpage}.
Such comparison shows that in the region of transverse momenta considered in this paper the resummation effects have a limited impact on the NLO transverse-momentum and (pseudo)rapidity distributions, and
that our NNLO results have smaller perturbative uncertainties than the FONLL results and can thus be considered more accurate.
The impact of non-perturbative fragmentation is typically rather small on $p_T$-inclusive observables, definitely smaller than the NNLO perturbative uncertainties.
Having in mind the above issues and limitations, our fixed-order perturbative predictions can be compared with experimental measurements for inclusive $b$-hadron production.
In Ref.~\cite{Acosta:2004yw}, the CDF collaboration performed a measurement of the $J/\psi$ cross section in the rapidity range $|y|<0.6$.
The total $b$-hadron production cross section in the same rapidity region was extracted by using a Monte Carlo simulation of the decay kinematics of $b$-hadrons to the final states containing a $J/\psi$ hadron.
The result is
\begin{equation}\label{eq:tot_tev}
\sigma^{H_b} (|y_{H_b}|<0.6)=17.6\pm 0.4(\text{stat})^{+2.5}_{-2.3}(\text{syst})\, \mu \text{b}\, .
\end{equation}
In order to study the possible effect of a rapidity cut in our parton-level calculation,
we have repeated the computation of the transverse-momentum distribution reported in Fig.~\ref{fig:pT_av.Tev} by applying the constraint $|y_{b/\bar{b}}|<0.6$, see Fig.~\ref{fig:rap.pT_av.Tev}.%
\footnote{The rapidity cut is applied only to the corresponding parton (bottom or antibottom) before computing the average between the two transverse momentum distributions.}
\begin{figure}[t]
\centering
\includegraphics[width=0.32\textwidth]{{plots/distributions/rap.pT_av.mT.Tevatron}.pdf}\hfill
\includegraphics[width=0.32\textwidth]{{plots/distributions/rap.pT_av.2mT.Tevatron}.pdf}\hfill
\includegraphics[width=0.32\textwidth]{{plots/distributions/rap.pT_av.HT_2.Tevatron}.pdf}
\vspace*{-2ex}
\caption{Transverse momentum distributions at the Tevatron with the rapidity cut $|y|<0.6$, for the scale choice $\mu_0=m_T$ (left), $\mu_0=2m_T$ (central) and $\mu_0=H_T/2$ (right). The lower panels show the ratio to the NNLO predictions.}
\label{fig:rap.pT_av.Tev}
\end{figure}
By comparing the results in Figs.~\ref{fig:pT_av.Tev} and \ref{fig:rap.pT_av.Tev} we observe that the inclusion of the rapidity cut does not significantly modify the behaviour of the perturbative series, and that the shapes of the distributions remains rather similar.
We have computed the parton-level analogue of the $b$-hadron cross section in Eq.~(\ref{eq:tot_tev}).
In order to do so, we compute two independent cross sections, with cuts on the rapidity of the bottom \emph{or} the antibottom quark, respectively. These two cross sections are then averaged.
Since the characteristic scale for this observable is the bottom-quark mass $m_b$, we have chosen $\mu_0=m_b$ as the central scale.
We find
\begin{align}
\sigma^{b/\bar b}_{\text{LO}} (|y_{b/\bar{b}}|<0.6) &= 7.840(3) ^{+51\%}_{-34\%}\;\mu\text{b,}\nonumber\\
\sigma^{b/\bar b}_{\text{NLO}} (|y_{b/\bar{b}}|<0.6) &= 14.282(6) ^{+53\%}_{-28\%}\;\mu\text{b,}\\
\sigma^{b/\bar b}_{\text{NNLO}} (|y_{b/\bar{b}}|<0.6) &= 17.87(12) ^{+22\%}_{-21\%}\;\mu\text{b}\nonumber
\end{align}
and compare these results with the inclusive predictions presented in Table~\ref{table:totalXS}. The cross section turns out to be about 4 times smaller in the presence of this rapidity cut.
The NLO and NNLO $K$-factors are close to those for the total cross section, and also the scale uncertainties are rather similar.
This is a consequence of the fact that the impact of QCD radiative corrections is rather uniform in rapidity (see e.g. Fig.~\ref{fig:y_av.LHC}).
Comparing our perturbative predictions with the CDF measurement in Eq.~(\ref{eq:tot_tev}), we find that
NNLO corrections considerably improve the agreement with the data. Scale uncertainties are, although still sizeable, largely reduced at NNLO,
and only at this order they approach the size of the experimental uncertainties.
\subsection{Differential distributions: LHC}
\label{sec:resu:lhc}
We start the presentation of our differential results for the LHC by considering
the transverse momentum distribution of the bottom quark. Using different central scales, such as $\mu_0=m_T$, $\mu_0=2m_T$ and $\mu_0=H_T/2$, we obtain relative differences that are similar to those
predicted at the Tevatron (Fig.~\ref{fig:pT_av.Tev}). Therefore, we set $\mu_0=m_T$, and in Fig.~\ref{fig:pT_av.LHC} we show our LHC results at $\sqrt s = 7$~TeV (left panel) and $13$~TeV (right panel).
The shape of the $p_{T,b_\mathrm{av}}$ distribution is rather similar to what was observed at the Tevatron, and the peak is located at $p_{T,b_\mathrm{av}}\sim 4$ GeV.
The average transverse momentum is only slightly larger at LHC energies, being (at both NLO and NNLO) 5.5 GeV and 5.9 GeV at $\sqrt s = 7$ and $13$~TeV, respectively.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/distributions/pT_av_2.mT.7TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/distributions/pT_av_2.mT.13TeV}.pdf}
\vspace*{-2ex}
\caption{Transverse momentum distribution at the LHC for the scale choice $\mu_0=m_T$, for $\sqrt s = 7$~TeV (left) and $13$~TeV (right). The lower panels show the ratio to the NNLO prediction.}
\label{fig:pT_av.LHC}
\end{figure}
We see that, as already observed at the Tevatron, LO and NLO predictions are consistent within uncertainties only in the low-$p_{T,b_{\mathrm{av}}}$ region, while they present very different shapes in the tail of the distribution, where the NLO corrections become very large. In this region the LO and NLO scale uncertainty bands do not overlap.
The inclusion of NNLO corrections leads to a nice stabilisation of the perturbative result, analogously to what we have observed for the total cross section. In particular, the uncertainty band at NNLO is smaller than at NLO, and it overlaps with the latter over the entire region of transverse momenta.
At small transverse momenta the NNLO scale uncertainty is larger than at the Tevatron, consistently with our observation for the corresponding total cross sections. On the contrary, at large transverse momenta the NNLO band is smaller at the LHC (note that the plots in Fig.~\ref{fig:pT_av.LHC} extend to $p_{T,b_\mathrm{av}}=50$~GeV, while the Tevatron result in Fig.~\ref{fig:pT_av.Tev} is shown up to $25$~GeV).
\begin{figure}[p]
\centering
\includegraphics[width=0.48\textwidth]{{plots/distributions/y_av.m_b.7TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/distributions/y_av.m_b.13TeV}.pdf}
\vspace*{-2ex}
\caption{Rapidity distribution of the bottom quark at the LHC for $\sqrt s = 7$~TeV (left) and $13$~TeV (right). The lower panels show the ratio to the NNLO prediction.}
\label{fig:y_av.LHC}
\vspace*{5ex}
\centering
\includegraphics[width=0.48\textwidth]{{plots/distributions/eta_av.m_b.7TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/distributions/eta_av.m_b.13TeV}.pdf}
\vspace*{-2ex}
\caption{Pseudorapidity distribution of the bottom quark at the LHC for $\sqrt s = 7$~TeV (left) and $13$~TeV (right). The lower panels show the ratio to the NNLO prediction.}
\label{fig:neweta_av.LHC}
\end{figure}
The rapidity distribution of the bottom quark, computed with $\mu_0=m_b$, is shown in Fig.~\ref{fig:y_av.LHC} for $\sqrt s = 7$~TeV (left) and $13$~TeV (right).
The two distributions at different LHC energies show a similar behaviour. The impact of QCD radiative corrections is uniform in rapidity and, therefore,
it does not produce sizeable changes in the shape\footnote{The shape of
$d\sigma/dy$ at different perturbative orders also depends on the PDFs
at the corresponding order.
In general, shape differences of the PDFs at different orders can
produce ensuing shape variations of $d\sigma/dy$.}
of the distribution (even from LO to NLO).
Since the radiative corrections are relatively flat, they are of the same size as those for the total cross section.
The NNLO results almost completely overlap with the NLO results, and
they show smaller scale uncertainties, which are about $\pm 30\%$ over the whole spectrum.
In Fig.~\ref{fig:neweta_av.LHC} we report analogous results to those of
Fig.~\ref{fig:y_av.LHC}, but we consider the pseudorapidity ($\eta$) distribution of the bottom quark rather than its rapidity distribution. The most striking effect that we observe in going from Fig.~\ref{fig:y_av.LHC} to Fig.~\ref{fig:neweta_av.LHC}
is the different shape (independently of the perturbative order) of the rapidity and pseudorapidity distributions. The rapidity cross section is maximal at $y=0$ and monotonically decreases as $y$ increases. The pseudorapidity cross section has a maximum at a finite (non-vanishing) value of $\eta$ and a local minumum at $\eta=0$,
where it shows a `central-pseudorapidity dip' (the value of $d\sigma/d\eta$ at
$\eta=0$ is smaller than the value of $d\sigma/dy$ at $y=0$).
These shape differences are a well-known fact: they have a general kinematical origin
and are due to the non-vanishing mass of the produced particle. We recall the origin of these shape differences in \ref{app:eta-y}.
Examining the NLO and NNLO radiative corrections, we note that they have a highly similar effect (at both the qualitative and the quantitative levels) on the rapidity and pseudorapidity distributions of the $b$ quark (see the ratio plots in
Figs.~\ref{fig:y_av.LHC} and~\ref{fig:neweta_av.LHC}). This feature is a consequence of the fact that $d\sigma/dy$ and $d\sigma/d\eta$ mainly probe the same underlying dynamics, and that their relative differences are basically of kinematical origin (see the discussion in \ref{app:eta-y}).
Despite the inclusion of NNLO corrections, the results in Figs.~\ref{fig:pT_av.LHC} and \ref{fig:y_av.LHC} show that perturbative uncertainties in transverse-momentum and rapidity distributions are still relatively large.
A possible attempt to reduce scale uncertainties is to consider {\it normalised} distributions, as discussed in the following.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/normalised/pT_av_2.mT.7TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/normalised/pT_av_2.mT.13TeV}.pdf}
\vspace*{-2ex}
\caption{Normalised transverse momentum distribution of the bottom quark at the LHC for $\sqrt s = 7$~TeV (left) and $13$~TeV (right). The lower panels show the ratio to the NNLO prediction.}
\label{fig:pT_av.LHC.norm}
\end{figure}
In Fig.~\ref{fig:pT_av.LHC.norm} we consider the normalised $p_{T,b_\mathrm{av}}$ distribution at LO, NLO and NNLO.
The scale uncertainty bands are obtained as follows: for each scale choice needed to study the 7-point scale variation the corresponding distribution is normalised to unity, and the envelope of each
normalised distribution is constructed. The scale uncertainties in the peak region are strongly reduced, whereas at high transverse momenta the NLO and NNLO uncertainty bands are slightly larger than
those observed in Fig.~\ref{fig:pT_av.LHC}. This is not unexpected,
since the low-$p_{T,b_\mathrm{av}}$ region gives the dominant contribution to the total cross section.
Therefore, the scale uncertainties of the total and differential cross sections are strongly correlated at low $p_{T,b_\mathrm{av}}$, and increasingly decorrelated as $p_{T,b_\mathrm{av}}$ increases.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/normalised/y_av.m_b.7TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/normalised/y_av.m_b.13TeV}.pdf}
\vspace*{-2ex}
\caption{Normalised rapidity distribution at the LHC for $\sqrt s = 7$~TeV (left) and $13$~TeV (right). The lower panels show the ratio to the NNLO prediction.}
\label{fig:y_av.LHC.norm}
\end{figure}
We repeat the same procedure for the rapidity distribution. In Fig.~\ref{fig:y_av.LHC.norm} we show the normalised rapidity distribution of the bottom quark,
which is constructed in the same manner as for Fig.~\ref{fig:pT_av.LHC.norm}.
In this case, since the impact of QCD radiative corrections is rather uniform, the normalised distribution is quite stable and shows reduced scale uncertainties, except for the large-rapidity region.
An alternative strategy to reduce theoretical uncertainties is to consider ratios between distributions computed at different collider energies.
In the context of heavy-quark production, the use of such ratios was proposed in Ref.~\cite{Cacciari:2015fta}, to the purpose of constraining the gluon PDF.
Assuming that the scale variations at different energies are fully correlated (i.e.\ the same values of the renormalisation and factorisation scales can be used in the numerator and the denominator when constructing the ratio), the ratios of differential distributions at different energies exhibit a smaller sensitivity to scale variations.
The validity of this assumption can be tested by comparing the ratios at different perturbative orders, and checking the reliability of the (reduced) uncertainty bands obtained assuming such correlation.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/distributions/pT_av_2.mT.ratio}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/distributions/y_av.m_b.ratio}.pdf}
\vspace*{-2ex}
\caption{Ratios of 13 TeV to 7 TeV predictions for the transverse momentum (left) and rapidity (right) distributions at the LHC. The lower panels show the result normalised to the NNLO prediction.}
\label{fig:ratio.LHC}
\end{figure}
In Fig.~\ref{fig:ratio.LHC} (left) we show the ratio of transverse-momentum distributions. As the centre-of-mass energy increases, the $p_{T,b_{\mathrm{av}}}$ distribution becomes (slightly) harder, and, therefore, the ratio increases as $p_{T,b_{\mathrm{av}}}$ increases. Considering perturbative uncertainties,
we notice a strong reduction of the uncertainty bands: while the width of the NNLO band in the original distributions ranges from about $\pm 30\%$ at low $p_{T,b_{\mathrm{av}}}$ to about $\pm 10\%$ in the tail, in the ratio the scale uncertainties are reduced to about $\pm 5\%$.
In Fig.~\ref{fig:ratio.LHC} (right) we show the corresponding ratio of rapidity distributions. We see that the ratio increases as the rapidity increases,
consistently with the fact that at 13 TeV the bottom quarks have a stronger tendency to be produced in the forward direction.
Also in this case the LO, NLO and NNLO bands overlap and their size decreases as the order increases, leading to ${\cal O}(\pm 5\%)$ uncertainties at NNLO.
Comparing the results at different perturbative orders for both the transverse-momentum and rapidity distributions, we observe an overlap between the LO, NLO and NNLO uncertainty bands over the full kinematical range. This remarkable stability of the perturbative expansion tends to confirm the approach of computing the ratio by using correlated scale variations at different collider energies.
We close this section with an investigation of bottom-quark production in the forward region.
In Ref.~\cite{Aaij:2016avz}, the LHCb Collaboration has presented results for the measurement of the $b$-hadron production cross section at the LHC.
This measurement used semileptonic decays of $b$-hadrons into a charmed hadron associated with a muon.
The data were collected at $\sqrt{s}=7$~TeV and $13$~TeV, in the pseudorapidity interval $2<\eta<5$, and inclusively in the $b$-hadron transverse momentum.
Measurements of the ratio between the $7$~TeV and $13$~TeV rates were provided as well.
Having in mind the limitations of a comparison between theoretical predictions for bottom-quark production and $b$-hadron production data (as already mentioned in Sec.~\ref{sec:resu:tev}),
in Fig.~\ref{fig:eta_av.LHC} we present our perturbative predictions for the pseudorapidity distribution of the bottom quark and
compare them with the experimental data of Ref.~\cite{Aaij:2016avz}.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/distributions/LHCb_eta_av.m_b.7TeV}.pdf}\hfill
\includegraphics[width=0.48\textwidth]{{plots/distributions/LHCb_eta_av.m_b.13TeV}.pdf}
\vspace*{-2ex}
\caption{Pseudorapidity distribution at the LHC for the scale choice $\mu_0=m_b$, for centre of mass energy $\sqrt s = 7$~TeV (left) and $13$~TeV (central). The lower panels show the ratio to the NNLO predictions. The theoretical predictions are compared with the LHCb data of Ref.~\cite{Aaij:2016avz}.}
\label{fig:eta_av.LHC}
\end{figure}
We first comment on the theoretical results.
As previously noticed, the higher order corrections to the pseudorapidity distributions present similar features to those observed for the rapidity spectrum.
The corrections are almost independent of the value of $\eta_{b_{\rm av}}$ and quantitatively very similar to those
affecting the total cross sections at both collider energies. At NNLO the scale uncertainty is smaller than at NLO, and the corresponding uncertainy bands largely overlap.
The NNLO $K$-factor is reduced with respect to its NLO equivalent.
The effects due to the hadronization of the bottom quarks into $b$-hadrons are expected to be relatively small (see \ref{app:FONLL}),
at the few-percent level in the current range of pseudorapidity and, therefore, well below the perturbative uncertainties at NNLO.
We now comment on the comparison with data. The measured distributions at both collider energies, shown in Fig.~\ref{fig:eta_av.LHC}, are consistent with our NNLO results in the entire pseudorapidity range.
While there is an overlap between NLO prediction and data in almost all cases, the inclusion of the NNLO corrections generally improves the agreement with the central predictions and always strongly reduces the scale uncertainties, making the comparison with data more significant. We note, however, that the
NNLO scale uncertanties are still considerably larger than the experimental errors.
At both collider energies and independently on the perturbative order, we observe that the shapes of the predicted distributions are different from the measured spectra,%
\footnote{Apparently, the data exhibit a maximum at $\eta\sim3$, which is shifted by about one pseudorapidity unit with respect to the theoretical prediction (see Fig.~\ref{fig:neweta_av.LHC}).}
while all individual points are compatible with the data:
at low pseudorapidity values the measurement is below the predicted central value, whereas in the tail the data are closer to the upper bound of the scale uncertainty band. We also observe that the agreement between the NNLO predictions and the data
is slightly worse in the tail of the distribution at $13$~TeV, although
the theory prediction is compatible with the data in each bin.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{{plots/distributions/LHCb_eta_av.m_b.ratio}.pdf}
\vspace*{-2ex}
\caption{Ratio of 13 TeV to 7 TeV predictions for the pseudorapidity distribution in the forward region. The lower panels show the result normalised to the NNLO prediction. The theoretical predictions are compared with the LHCb data of Ref.~\cite{Aaij:2016avz}.}
\label{fig:ratio_eta}
\end{figure}
In Fig.~\ref{fig:ratio_eta} we show the ratio of the 13 TeV and 7 TeV predictions for the pseudorapidity distribution, computed as for the results in Fig.~\ref{fig:ratio.LHC}.
As expected, the ratio features a strong reduction of the scale uncertainties at NNLO, from about $\pm 30\%$ to only about $\pm 5\%$. Moreover, the
LO, NLO and NNLO uncertainty bands overlap, thereby supporting again the use of correlated variations of the renormalisation and factorisation scales.
Considering the data--theory comparison, we observe that
the reduced NNLO scale uncertainties of the ratio predictions are significantly smaller than the experimental errors.
Comparing the NNLO prediction with the experimental data, we observe that, with the exception of the first bin, the data are systematically above the central NNLO result. In two bins the data points lie outside of the NLO and NNLO uncertainty bands.
We note, however, that the $\eta$-independent systematic errors are the dominant source of uncertainty in the measurement of the ratio: while the total (statistical plus systematic) uncertainty ranges between $\pm 8.2\%$ and $\pm 10.5\%$ for the different bins, the $\eta$-independent uncertainty is $\pm 7.4\%$ (see Tables~3 and 4 of Ref.~\cite{Aaij:2016avz}).
This $\eta$-independent uncertainty is of the same order as the observed data--theory discrepancy.
In the data--theory comparison of Fig.~\ref{fig:eta_av.LHC} we had observed similar shape differences at $7$~TeV and $13$~TeV, which largely cancel out in the ratio.
The pseudorapidity distribution can be integrated over the range $2<\eta_{b_{\rm av}}<5$ to obtain the accepted cross section.
\input{tableLHCb}
In Table \ref{table:eta.LHC} the corresponding NNLO predictions at $\sqrt{s}=7$ and 13 TeV
as well as their ratio are compared with the LHCb data. The NNLO result is stated for two values of central scales, $\mu_0=m_b$ and $\mu_0=2m_b$, and we find the same ratio of the 13 and 7 TeV predictions
for these two scale choices. The result for $\mu_0=m_b$ suggests larger (and thus more conservative) perturbative uncertainties.
Comparing the NNLO predictions for the integrated cross section to the LHCb measurement, we find that both scale choices lead to predictions that are consistent with the data,
the choice $\mu_0=m_b$ leads to a better agreement though. The NNLO prediction for the ratio is in good agreement with the experimental measurement.
Considering the $b{\bar b}$ total cross section at NNLO,
in Table~\ref{table:totalXSwithunc} we have reported its PDF uncertainty
($\Delta_{\rm PDFs}$) and scale variation uncertainty
($\Delta_{\rm scale}$), showing that $\Delta_{\rm PDFs}$ is definitely smaller than $\Delta_{\rm scale}$. In this paper we do not present a study of PDF uncertainties
on differential cross sections. We note that, in the very-forward region,
PDF uncertainties on (pseudo)rapidity differential cross sections (and on their ratio at different energies) can be larger than the corresponding uncertainties on the total cross section, and their size can be similar to the size of the scale variation uncertainties that we find at NNLO.
We remark on this fact by quoting, for instance, some values of $\Delta_{\rm PDFs}$
from the NLO study of Ref.~\cite{Cacciari:2015fta}. The PDF uncertainties tend to slightly increase as $\sqrt s$ increases (see accompanying comments to Table~\ref{table:totalXSwithunc}). In the case of $d\sigma/d\eta$ at ${\sqrt s}=13$~TeV, the
NLO value of $\Delta_{\rm PDFs}$ is about $\pm 9\%$ at $\eta \sim 4$ and increases to about $\pm 18\%$ at $\eta \sim 5$~\cite{Cacciari:2015fta}
(we recall that we find $\Delta_{\rm scale} \sim \pm 30\%$ at NNLO, see
Fig.~\ref{fig:eta_av.LHC}). In the case of the ratio of $d\sigma/d\eta$
at 13~TeV and 7~TeV, the NLO value of $\Delta_{\rm PDFs}$ is of ${\cal O}(1\%)$
at $\eta \raisebox{-.4ex}{\rlap{$\,\sim\,$}} \raisebox{.4ex}{$\,<\,$} 3$ and increases to about $\pm 5\%$ ($\pm 8\%$) at $\eta \sim 4$
($\eta \sim 5$)~\cite{Cacciari:2015fta} (at NNLO we find that $\Delta_{\rm scale}$
varies in the range between $\pm 5\%$ and $\pm 3\%$, see Fig.~\ref{fig:ratio_eta}).
\section{Summary}
\label{sec:summa}
In this paper we have presented the first fully differential NNLO calculation of bottom-quark pair production at hadron colliders.
The calculation was carried out by using the $q_T$ subtraction formalism to handle and cancel IR divergences. It extends the corresponding calculation for top-quark pair production~\cite{Catani:2019iny,Catani:2019hip}.
The computation has been implemented in the {\sc Matrix} framework, which allows us to evaluate single- and multi-differential distributions with arbitrary IR safe selection cuts.
We have presented results for the total cross section at the Tevatron and the LHC and compared them with predictions obtained by using the numerical code {\sc Hathor}, finding excellent agreement.
We have studied different sources of theoretical uncertainty and observed that, despite the inclusion of NNLO corrections,
perturbative uncertainties are still sizeable and dominant over other sources of uncertainties.
We have presented predictions for single-differential distributions at the Tevatron ($\sqrt{s}= 1.96$ TeV) and at LHC ($\sqrt{s} = 7$ TeV and $\sqrt{s} = 13$ TeV).
As a general feature, we observed that the inclusion of NNLO corrections suggests a (slow) convergence of the perturbative series, with a good overlap between NLO and NNLO bands and a significant reduction of perturbative uncertainties, which are estimated as usually through scale variations.
Since perturbative uncertainties at NNLO are still large also for differential distributions, we have investigated possible ways to reduce them.
We considered both normalised distributions and ratios of distributions at different energies, and we assumed scale uncertainties to be fully correlated in the respective ratios.
In both cases we showed that the ensuing results are perturbatively stable and that LO, NLO and NNLO uncertainty bands overlap, suggesting that this approach indeed provides reliable predictions with reduced perturbative uncertainties.
We have also compared our predictions with experimental measurements for $b$-hadron production from the CDF Collaboration at the Tevatron and the LHCb Collaboration at the LHC, finding reasonably good agreement.
Further studies in the high-$p_T$ region require the resummation of large logarithmic contributions of the form $\ln p_T/m_b$, while more detailed data--theory comparisons could benefit from the inclusion of fragmentation effects.
Such studies are left for future work.
\noindent {\bf Acknowledgements}
\noindent
We thank Matteo Cacciari, Katharina M{\"u}ller and Giovanni Passaleva for useful discussions and comments on the manuscript.
We are indebted to Federico Buccioni, Jean-Nicolas Lang, Jonas Lindert and Stefano Pozzorini for their ongoing support with {\sc OpenLoops~2}, and in particular for making specific amplitudes available to us.
We are also grateful to Emanuele Nocera and the NNPDF collaboration for providing us with the PDF grids needed to evaluate $\as$ uncertainties.
This work is supported in part by the Swiss National Science Foundation (SNF) under contracts 200020\_188464 and IZSAZ2\_173357.
The work of SK is supported by the ERC Starting Grant 714788 REINVENT.
|
1,116,691,500,420 | arxiv | \section{Introduction}
Dark matter is a crucial ingredient in the cosmological history of the universe and accounts for about 27\% of the energy budget in the universe today.
As its existence is supported by galactic-scale to cosmological-scale gravity-based evidence, various experiments were performed, are operational, and are planned to detect dark matter via its hypothetical non-gravitational interactions with Standard Model (SM) particles.
While no conclusive observations have been made thus far, the XENON Collaboration has recently reported an excess of electron recoil events over known backgrounds with an exposure of 0.65~ton$\cdot$year~\cite{Aprile:2020tmw}.
The excess is shown below 7~keV and most of the events populate at $2-3$~keV.
The XENON1T detector is designed to have an extremely low rate of background events, so this excess could be considered as a sign of new physics.
The XENON Collaboration has claimed that while the unresolved $\beta$ decays of tritium can explain the excess at 3.2$\sigma$ significance, the solar axion model and the neutrino magnetic moment signal can be favored at 3.5$\sigma$ and 3.2$\sigma$ significance, respectively.
It is expected that confirmation or rejection of these hypotheses will be done with more statistics in the near future.
By contrast, the interpretation with conventional dark matter is less favored, essentially because of its non-relativistic nature.
For dark matter sufficiently heavier than electron, the scale of electron recoil (kinetic) energy is $\sim m_e \times(10^{-3}c)^2 \approx \mathcal{O}({\rm eV})$ with $m_e$ and $10^{-3}c$ being the mass of electron and the typical velocity of dark matter near the earth, respectively.
In other words, the energy deposition by conventional dark matter is not large enough to accommodate the excessive events of $\mathcal{O}({\rm keV})$.
However, this issue may be avoided by envisioning non-conventional dark-sector scenarios involving a mechanism to exert a sufficient boost on a dark matter component, rendering the dark matter hypothesis plausible enough to explain the excess.
In particular, upon confirmation, the XENON1T anomaly can be the first signal to indicate that the associated dark sector is non-conventional, opening a new pathway toward dark matter phenomenology.
Indeed, the authors in Ref.~\cite{Giudice:2017zke} pointed out, for the first time, that the XENON1T experiment would be sensitive enough to the fast-moving $\chi_1$ -- which arises in the two-component boosted dark matter (BDM) scenario -- interacting with electrons.
Along this line, we entertain a class of non-conventional dark-sector scenarios to explain the XENON1T excess in this paper, in particular, focusing on the impact of the particle mediating the dark-matter--electron interactions in the context of the BDM scenario as a concrete example.
\section{Dark Matter Interpretation}
As mentioned previously, it is challenging to accommodate the XENON1T anomaly using the ordinary halo dark matter since its typical velocity is too small to invoke keV-scale energy deposition on target electrons.
Bosonic dark matter (e.g., axion-like particle and dark photon) of keV-scale mass could be absorbed, depositing its whole mass energy in the XENON1T detector.
However, this is likely to give rise to a line-like signature, so that this possibility is less preferred by the observed recoil energy spectrum.
Indeed, the XENON Collaboration found that no bosonic dark matter of mass within 1 and 210~keV shows more than 3$\sigma$ significance unlike the other interpretations, so they simply set the limits for relevant dark matter candidates.
The upshot of this series of observations is that dark matter (or more generally, a dark matter component) should acquire a sizable enough velocity to transfer keV-scale kinetic energy to a target electron.
This approach has been investigated in Ref.~\cite{Kannike:2020agf} where the authors claimed that fast-moving dark matter with velocity of $\mathcal{O}(0.1 c)$ can fit in the XENON1T excess.
An important implication of this way of dark matter interpretation is that the dark matter (candidate) responsible for the excess is not the (cold) galactic halo dark matter, i.e., it is a subdominant fast-moving component and the underlying dark matter scenario is non-conventional.~\footnote{See also Refs.~\cite{Su:2020zny, Bally:2020yid, Paz:2020pbc, Primulando:2020rdk, Jho:2020sku, An:2020tcg, Ge:2020jfn, Bhattacherjee:2020qmv} for other explanations of the excess.}
Furthermore, it requires a certain mechanism to ``boost'' this dark matter component in the universe today.
There are several mechanisms and scenarios to serve this purpose, which were originally proposed for other motivations; semi-annihilation~\cite{DEramo:2010keq}, (two-component) boosted dark matter scenarios~\cite{Belanger:2011ww,Agashe:2014yua,Kim:2016zjx}, models involving dark-matter-induced nucleon decays inside the sun~\cite{Huang:2013xfa}, and energetic cosmic-ray-induced dark matter~\cite{Yin:2018yjn, Bringmann:2018cvk, Ema:2018bih}.
Of them, we discuss the BDM scenario as also considered in Ref.~\cite{Fornal:2020npv}, focusing on implications of the XENON1T anomaly on the spins of BDM and mediator particles.
The standard two-component BDM scenario~\cite{Agashe:2014yua} assumes two different dark matter species; one (say, $\chi_0$) is heavier than the other (say, $\chi_1$).
Their stability is often protected by separate unbroken symmetries such as $Z_2\otimes Z_2'$ and ${\rm U}(1)'\otimes{\rm U}(1)''$.
One of the species (usually the heavier one $\chi_0$) has no direct coupling to SM particles, but communicates with the other species $\chi_1$.
By contrast, $\chi_1$ can interact with SM particles with a sizable coupling.
Therefore, $\chi_0$ is frozen out via the indirect communication with the SM sector with the ``assistance'' of $\chi_1$ (a.k.a. ``assisted'' freeze-out mechanism)~\cite{Belanger:2011ww}.
In other words, $\chi_0$ pair-annihilates to $\chi_1$ while $\chi_1$ pair-annihilates to SM particles.
The relatively sizable coupling of $\chi_1$ to SM particles renders it the negligible dark matter component while keeping $\chi_0$ dominant in the galactic halo.
In most of the well-motivated parameter space, conventional dark matter direct detection experiments do not possess meaningful sensitivity to relic $\chi_0$ and $\chi_1$ because of tiny coupling and negligible statistics, respectively.
A phenomenologically intriguing implication of this model setup, particularly relevant to the XENON1T excess, is that $\chi_1$ can acquire a sizable boost factor, which is simply given by the ratio of the $\chi_1$ mass to the $\chi_0$ mass, in the universe today.
Therefore, one may look for the signal induced by such boosted $\chi_1$.
Due to the small $\chi_1$ flux (see also Eq.~\eqref{eq:flux}), it is usually challenging for small-volume detectors to have signal sensitivity, but ton-scale dark matter direct detection experiments can be sensitive to the boosted $\chi_1$ signal~\cite{Cherry:2015oca,Giudice:2017zke}.
As mentioned earlier, Ref.~\cite{Giudice:2017zke} has performed the first sensitivity study for the boosted $\chi_1$ interacting with electrons in XENON1T, LUX-ZEPLIN, and DEAP3600 experiments.
Motivated by the proposal in Ref.~\cite{Giudice:2017zke}, the COSINE-100 Collaboration has conducted the first search for BDM-induced signals as a dark matter direct detector\footnote{Note that Super-Kamiokande, a 10 kton-scale neutrino detector, performed a dedicated search for BDM interacting with electrons~\cite{Kachulis:2017nci}.} and reported the results~\cite{Ha:2018obm} including limits on the models of inelastic BDM~\cite{Kim:2016zjx}.
Denoting the $\chi_0$ and $\chi_1$ mass parameters by $m_0$ and $m_1$ correspondingly, we find that if $m_1$ is given by approximately $99.0 - 99.9\%$ of $m_0$, $\chi_1$ coming from the pair-annihilation of $\chi_0$ in the present universe can be as fast-moving as $0.04-0.14c$.
While this simple consideration determines the ``desired'' mass relation between $\chi_0$ and $\chi_1$, not all mass values are favored by the excess aside from the various existing limits.
More importantly, as will be discussed later, the ``favored'' velocity range can be significantly altered, depending on the underlying mass spectrum and particle types.
To investigate these points more systematically, we first consider the number of signal events $N_{\rm sig}$.
As well known, it is given by
\begin{equation}
N_{\rm sig} = \mathcal{F}_1\,\sigma_{1e}\, N_{e,\,{\rm tot}}^{\rm eff} \,t_{\rm exp}\,, \label{eq:Nsig}
\end{equation}
where $\mathcal{F}_1$, $\sigma_{1e}$, $N_{e,\,{\rm tot}}^{\rm eff}$, and $t_{\rm exp}$ are the flux of boosted $\chi_1$ near the earth, the total scattering cross-section of $\chi_1$ with an electron, the number of effective target electrons in the fiducial volume of the XENON1T detector, and the total exposure time, respectively.
Here $\sigma_{1e}$ could be affected by the threshold and/or detection efficiencies for recoiling electrons if a significant number of events are populated in the region where the recoil electron energy is near the threshold and/or the associated efficiencies are not large enough.
The last two factors are experimentally determined and their product can be easily deduced from 0.65 ton$\cdot$year.
Regarding $N_{e,\,{\rm tot}}^{\rm eff}$, we remark that the binding energy of electrons in the xenon atom is not negligible given the keV scale of recoiling electron kinetic energy.
While the outermost electron (in the $O$ shell) needs 12.1~eV~\cite{1978ps1..book.....C} to get ionized, the innermost electron (in the $K$ shell) requires an ionization energy of 34.6~keV~\cite{Bearden:1967gqa}.
Therefore, only some fraction of electrons can be target electrons for the BDM mostly inducing keV-scale energy deposition.
Some works considered form factors to calculate the dark matter event rate to explain the XENON1T excess.
For example, Ref.~\cite{Kannike:2020agf} used the atomic excitation factor with relativistic corrections and Ref.~\cite{Cao:2020bwd} considered the dark matter and ionization form factors, restricting to the $N$-shell and $O$-shell electrons. We here take a shortcut scheme, reserving a dedicated analysis for future work~\cite{AKKMPS}.
As a conservative approach, we consider electrons from three outermost orbitals ($5p$,$5s$ and $4d$), which are known to be the dominant contribution~\cite{Essig:2011nj,Lee:2015qva,Cao:2020bwd}, i.e., the number of target electrons in a single xenon atom $N_e^{\rm eff}$ is taken to be 18 throughout our analysis.\footnote{For 1 ton of liquid xenon, $N_{e,{\rm tot}}^{\rm eff} = 4.59 \times 10^{27} \, N_e^{\rm eff}$.}
Note that the largest ionization energy among the electrons belonging to the three orbitals is $\sim76$~eV~\cite{1978ps1..book.....C} which would induce less than 5\% uncertainty in estimating $2-3$~keV energy deposition.
Since we will consider energy resolution of $\sim 450$~eV~\cite{Aprile:2020yad},
we expect that the $\lesssim0.1$~keV level uncertainty is buried in the detector resolution.
We also note that each of the $N$-shell and $O$-shell electrons gets excited with a different weight.
We expect that this would make an $\mathcal{O}(1)$ effect, so our findings and conclusions in the analysis would remain valid.
We will revisit this aspect before we conclude our study.
\begin{figure}[t]
\centering
\includegraphics[width=11cm]{model_indep_fig1}
\caption{Maximum recoil energy of electrons scattered off by BDM, $E_r^{\rm max}$ (solid colored curves), and required BDM-electron scattering cross-sections to have 100 recoil events with the 0.65 ton$\cdot$year exposure, $\sigma_{1e}^{100}$ (orange lines), in the $(m_1, E_1)$ plane.
The gray-shaded lower-right area is disfavored because the expected maximum electron recoil energy is less than the typical energy associated with the observed excess, i.e., $E_r^{\rm max} < 2$ keV.
The upper region requires large cross-sections which can result in too small mean free paths ($\bar{\ell}_1 \propto 1/\sigma_{1e}$) inside the earth to reach the XENON1T detector.
We show a mean free path value at $E_1=100$~MeV for reference.
Three diagonal lines represent the velocity of BDM for a given choice of the $(m_1, E_1)$ pairs.
}
\label{fig:m0-m1}
\end{figure}
The estimate of flux $\mathcal{F}_1$ depends on the source of BDM, and we consider here the $\chi_1$ coming from the galactic halo for illustration.
Assuming that the $\chi_0$ halo profile follows the Navarro-Frenk-White profile~\cite{Navarro:1995iw,Navarro:1996gj}, we see that $\mathcal{F}_1$ from all sky is given by~\cite{Agashe:2014yua}
\begin{eqnarray}
\mathcal{F}_1 = 1.6~{\rm cm}^{-2}{\rm s}^{-1} \times \left(\frac{ \langle \sigma_{0\to 1}v\rangle }{5\times 10^{-26}~{\rm cm}^3{\rm s}^{-1}} \right)\left(\frac{10~{\rm MeV}}{m_0} \right)^2\,, \label{eq:flux}
\end{eqnarray}
where the velocity-averaged annihilation cross-section $\langle \sigma_{0\to 1}v\rangle$ is normalized to $5\times 10^{-26}~{\rm cm}^3{\rm s}^{-1}$ to be consistent with the observed relic abundance.
Note that the flux is proportional to inverse mass square, so roughly speaking a large (small) $m_0$ prefers a large (small) value of $\sigma_{1e}$ to reproduce the excessive number of events of XENON1T.
It is instructive to investigate the BDM parameter space to (potentially) accommodate the XENON1T anomaly in a model-independent fashion.
In figure~\ref{fig:m0-m1}, we present the maximum recoil energy of electrons scattered off by BDM,
\begin{equation}
E_r^{\rm max}=\frac{2m_ep_1^2}{s}\,, \label{eq:maxE}
\end{equation}
where $p_1^2=E_1^2-m_1^2$ and $s=m_1^2+m_e^2+2m_eE_1$ with $E_1$ being the total energy of boosted $\chi_1$, and required BDM-electron scattering cross-sections to have 100 recoil events at the XENON1T detector with the 0.65 ton$\cdot$year exposure, $\sigma_{1e}^{100}$, in the $(m_1, E_1)$ plane.
Note that although the number of excessive events is about 50, the nominal number of signal events can be a few times larger due to detector efficiency and resolution, depending on the underlying model details.
In the two-component annihilating BDM scenario that we consider here, $E_1$ is simply identified as $m_0$.
$E_r^{\rm max}$ must be at least 2 keV because the observed excess is pronounced most at $2-3$ keV.
The disfavored region of $E_r^{\rm max} < 2$ keV is gray-shaded.
From Eqs.~(\ref{eq:Nsig}) and (\ref{eq:flux}), $N_{\rm sig} \propto \mathcal{F}_1 \sigma_{1e} \propto \sigma_{1e}/E_1^2$, so the required cross-section increases quadratically in $E_1$.
One should keep in mind that too large $\sigma_{1e}^{100}$ is constrained by too short a mean free path and (potentially) by various experimental bounds on the mediator mass and the associated coupling.
We will discuss these issues in the context of specific benchmark points later.
\begin{table}[t]
\centering
\resizebox{\columnwidth}{!}{
\begin{tabular}{c|c|c|c|c}
\hline \hline
Case & Mediator & Dark matter & $\mathcal{L}_{\rm int}$ & $\overline{|\mathcal{A}|}^2$ \\
\hline
VF & $V_\mu$ & $\chi_1$ & $(g_e^V \bar{e} \gamma^\mu e + g_\chi^V \bar{\chi}_1\gamma^\mu\chi_1)V_\mu$ & $8m_e\left\{ m_e(2E_1^2-2E_1E_r+E_r^2)-(m_e^2+m_1^2)E_r\right\}$ \\
VS & $V_\mu$ & $\varphi_1$ & $(g_e^V \bar{e}\gamma^\mu e+g_\varphi^V \varphi_1^*\partial^\mu \varphi_1+ {\rm h.c.})V_\mu$ & $8m_e\left\{ 2m_eE_1(E_1-E_r)-m_1^2E_r \right\}$ \\
PF & $a$ & $\chi_1$ & $(ig_e^a\bar{e} \gamma^5 e+ig_\chi^a \bar{\chi}_1\gamma^5 \chi_1)a$ & $4m_e^2E_r^2$ \\
PS & $a$ & $\varphi_1$ & $(ig_e^a\bar{e}\gamma^5 e+ ig_\varphi^a m_1\varphi^*\varphi )a$ & $8 m_e m_1^2E_r$ \\
SF & $\phi$ & $\chi_1$ & $(g_e^\phi \bar{e}e+g_\chi^\phi \bar{\chi}_1 \chi_1)\phi$ & $4m_e(E_r+2m_e)(2m_1^2+m_eE_r)$ \\
SS & $\phi$ & $\varphi_1$ & $(g_e^\phi \bar{e}e + g_\varphi^\phi m_1 \varphi^*\varphi)\phi$ & $8m_e m_1^2(E_r+2m_e)$ \\
\hline \hline
\end{tabular}
}
\caption{Example scenarios associated with the interaction between BDM and electron that we consider in this study.
$V_\mu$, $a$, and $\phi$ denote vector, pseudo-scalar, scalar mediators, respectively, while $\chi_1$ and $\varphi_1$ denote (Dirac-)fermionic and (complex-)scalar dark matter.
For the PS and SS cases, the scale of mediator couplings to dark matter is normalized to the mass of BDM for convenience.}
\label{tab:models}
\end{table}
To study the model-dependence of the BDM scattering cross-section, we consider a vector mediator $V_\mu$, pseudo-scalar mediator $a$, and scalar mediator $\phi$ together with (Dirac-)fermionic BDM $\chi_1$ and (complex-)scalar BDM $\varphi_1$; the six different cases in total are summarized in Table~\ref{tab:models} with the relevant interaction terms and coupling constants.
For the PS and SS cases, the scale of mediator couplings to dark matter is normalized to $m_1$.
Assuming that the incoming $\chi_1$ is much faster than the electrons in xenon atoms, we find that the spectrum in the kinetic energy of recoiling electrons $E_r$ with incoming BDM energy $E_1$ has the form of
\begin{equation}
\frac{d\sigma_{1e}}{dE_r}=\frac{(g_j^ig_e^i)^2m_e}{8\pi \lambda(s,m_e^2,m_1^2)(2m_eE_r+m_i^2)^2} \overline{|\mathcal{A}|}^2 \label{eq:xs}
\end{equation}
where $i\in \{V,\,a,\,\phi\}$, $j\in \{\chi,\,\varphi\}$, and $\lambda(x,y,z)=(x-y-z)^2-4yz$.
Here $\overline{|\mathcal{A}|}^2$ is the spin-averaged amplitude squared, in which the denominator from the propagator contribution is factored out, and the expressions for the six cases are also tabulated in Table~\ref{tab:models}.
\section{Case Studies}
We are now in the position to look into the aforementioned cases, starting with ($a$) the vector mediator case, followed by ($b$) the pseudo-scalar mediator case and ($c$) the scalar mediator case.
Since the two dark matter components are assumed to be thermally produced, we assume that the mass of the heavier component (i.e., dominant relic) is larger than, at least, a few MeV.
To develop the intuition on this differential spectrum, we consider three different regions of mass space:
\begin{eqnarray}
&(i)& m_0\approx m_1 \gg m_e,\quad m_i \gg m_e,\nonumber \\
&(ii)& m_0\approx m_1 \gg m_e,\quad m_i < m_e, \label{eq:conditions}\\
&(iii)& m_0 \gg m_e > m_1,\quad m_i < m_e\,,\nonumber
\label{eq:cases}
\end{eqnarray}
where $m_i$ is the mediator mass with $i=V,a,\phi$ and $m_0$ is again assumed to be greater than $m_1$ in all cases.
Note that $(i)$ and $(ii)$ represent the upper-right region of the $(m_1, E_1)$ parameter plane with respect to $(m_e, m_e)$ in figure~\ref{fig:m0-m1}, while $(iii)$ does the upper-left region.
\medskip
\noindent ($a$) \underline{Vector mediator}:
We first consider the VF case (i.e., fermionic BDM), displaying example unit-normalized recoil energy spectra (solid lines) in figure~\ref{fig:Espec} with our benchmark parameter choices shown in the legend.
\begin{figure}[t]
\centering
\includegraphics[width=11cm]{recoil.png}
\caption{Unit-normalized electron recoil energy spectra (solid lines) in the VF case for three sets of mass values as shown in the legend.
BDM and the mediator are a Dirac fermion and a massive vector.
The dashed lines are the corresponding plots with detector resolution ($\sigma_{\rm res}=0.45$~keV) and efficiency reported in Ref.~\cite{Aprile:2020yad} and Ref.~\cite{Aprile:2020tmw}, respectively.}
\label{fig:Espec}
\end{figure}
In the first benchmark point (red), the BDM $\chi_1$ has a speed of $v_1=0.06c$, hence lies in the 68\% C.L.-favored region of Ref.~\cite{Kannike:2020agf} as also supported by the typical recoil energy of $\mathcal{O}({\rm keV})$.
Furthermore, since $m_1,m_V \gg m_e$ and $E_1 \approx m_1$, the spectral shape is almost flat over the allowed range in this limit:
\begin{equation}
\frac{d\sigma_{1e}}{dE_r}\approx \frac{(g_\chi^Vg_e^V)^2 m_e m_1^2}{2\pi p_1^2m_V^4}\,, \label{eq:approx1}
\end{equation}
from which we find the total cross-section\footnote{Our expression has mass dependence different from the finding in Ref.~\cite{Fornal:2020npv}.
Ours is proportional to $m_e^2$ (vs. $m_e m_1$ in Ref.~\cite{Fornal:2020npv}), resulting in smaller estimates of cross-section.} to be
\begin{equation}
\sigma_{1e}\approx \frac{(g_\chi^Vg_e^V)^2m_e^2}{\pi m_V^4}\,. \label{eq:xsform}
\end{equation}
This flat distribution can be distorted to a rising-and-falling shape by detector smearing and efficiency, as shown by the red dashed curve.
For the second benchmark point (green), we choose a mediator $V$ lighter than electron.
Unlike the previous case, the expected recoil energy spectrum is rapidly falling off:
\begin{equation}
\frac{d\sigma_{1e}}{dE_r} \approx \frac{(g_\chi^Vg_e^V)^2m_e m_1^2}{2\pi p_1^2(2m_eE_r+m_V^2)^2}\,, \label{eq:approx2}
\end{equation}
for which the total cross-section is dominated by the region of $E_r \to 0$.
The reason is because the differential cross-section in electron recoil momentum is peaking toward small $p_e(\ll m_e)$ due to the $t$-channel exchange of $V$ and this feature is more prominent for $m_V \ll m_e$~\cite{Kim:2020ipj}.
Once detector effects are included, events are expected to populate most densely around $2-3$~keV (see the green dashed curve).
However, a caveat to keep in mind is that too small $m_V$ values would lead most of events to lie below 2~keV since $d\sigma_{1e}/dE_r$ goes like $1/E_r^2$.
Our numerical study suggests that $m_V \gtrsim 5$~keV would be favored by the data for the chosen $(m_0,m_1)$ pair.
This observation motivates the third benchmark point (blue) where the BDM even lighter than electron acquires a significant boost factor.
An approximation similar to Eq.~\eqref{eq:approx2} goes through with $m_1^2$ replaced by $E_1^2$ since $E_1 \gg m_1$.
As also shown in figure~\ref{fig:Espec}, the differential spectrum is not much different from that of the second benchmark point, except a long tail beyond 7~keV which may not be appreciable at this earlier stage.
Moreover, the spectrum with detector effects (blue dashed) is quite similar to the second benchmark point.
This demonstrates that unlike the claim in Ref.~\cite{Kannike:2020agf} the favored region can be extended further below $\sim 0.1$~MeV and/or further beyond $v_1=0.3c$, as long as $m_V$ is smaller than $m_e$.
However, the preferred range of $m_V$ is more restricted than that in the second benchmark point.
Our numerical study shows that $m_V \gtrsim 50$~keV would result in more than half of events lying beyond 7~keV, so that $5 \lesssim m_V \lesssim 50$~keV would be favored for the chosen $(m_0,m_1)$ pair.
The cross-section $\sigma_{1e}$ also determines the mean free path $\bar{\ell}_1$ in the earth, which is given by $\sim 1/(\langle n_e \rangle \sigma_{1e})$ with $\langle n_e \rangle$ being the mean electron number density along the $\chi_1$ propagation line.
Here we assume that $\chi_1$ has negligible interactions with nuclei.
If $g_e^V$ is too large (with $g_1^V$ set to be $\mathcal{O}(1)$), $\chi_1$ may scatter multiple times inside the earth before reaching the XENON1T detector located $\sim 1,600$~m underground, resulting in a substantial loss of energy that $\chi_1$ initially carries out.
The situation becomes worse if $\chi_1$ comes from the opposite side of the earth.
As shown in Eq.~\eqref{eq:Nsig}, $\mathcal{F}_1$ and $\sigma_{1e}$ are complementary to each other for a fixed $N_{\rm sig}$, i.e., a small $\mathcal{F}_1$ would be compensated by a large $\sigma_{1e}$ at the expense of multiple scattering of $\chi_1$.
This scenario was explored in Ref.~\cite{Fornal:2020npv}.
In our study, we rather focus on the opposite case where $\sigma_{1e}$ is small (hence no worries about the issue of too many scatterings) but sub-GeV (and smaller) $\chi_0$ allows a large flux of $\chi_1$.
\begin{figure}
\centering
\includegraphics[width=11cm]{fit-v2.png}
\caption{Sample energy spectra for the same benchmark mass spectra and particle spins (i.e., the VF case) as in figure~\ref{fig:Espec}.
We assume $g_\chi^V=1$ and galactic BDM for which the flux is given by Eq.~\eqref{eq:flux}.
The values of $\sigma_{1e}$ and $g_e^V$ associated with these fits are shown in the legend. The quoted $\sigma_{1e}$ are roughly consistent with the value of $\sigma_{1e}^{100}$ at $E_1=10$~MeV.
The background model (with negligible tritium contributions) claimed by the XENON Collaboration and the data points are given by the gray line and the black dots, respectively.}
\label{fig:fit}
\end{figure}
In figure~\ref{fig:fit}, we now show sample energy distributions for the three benchmark mass spectra taken in figure~\ref{fig:Espec}, assuming $g_\chi^V=1$ and galactic BDM whose flux is given by Eq.~\eqref{eq:flux}.
The values of $\sigma_{1e}$ and $g_e^V$ associated with these fits are shown in the legend.
The black dots and the gray line are the data points and the background model (with negligible tritium contributions) are imported from Ref.~\cite{Aprile:2020tmw}.
A few comments should be made for the quoted $\sigma_{1e}$ and $g_e^V$ values.
First, the required $\sigma_{1e}$ is of order $10^{-35}-10^{-34}~{\rm cm}^2$ resulting in more than ten thousand km ($\sim$ the diameter of the earth) of mean free path, i.e., at most a handful of $\chi_1$ scattering would arise inside the earth before reaching the XENON1T detector. See also the reference lines for $\sigma_{1e}^{100}$ and $\bar{\ell}_1$ in figure~\ref{fig:m0-m1}.
Second, there are mild differences among the quoted $\sigma_{1e}$ values although the BDM flux is fixed for all benchmark points.
As discussed earlier, the nominal scattering cross-section to explain the excess can be different due to the detector effects.
As suggested by figure~\ref{fig:Espec}, the green and blue curves are more affected by the detector efficiency since more events are expected to populate toward the lower energy regime.
Therefore, these two points typically demand a nominal BDM scattering cross-section greater than that for the other one.
Third, some of the reported $g_e^V$ values might be in tension with existing limits, depending on the underlying model details.
We will revisit this potential issue in the next section.
Finally, we briefly discuss how the variation in the dark matter spin affects the conclusions that we have made so far for the VF case.
We see that $\overline{|\mathcal{A}|}^2$ for the VS case is approximated to $16m_e^2E_1^2$ just like the VF case, and therefore expect similar spectral behaviors.
We find that the actual distributions look very similar to the corresponding ones with $\chi_1$ for the same mass choices, holding similar conclusions.
\medskip
\noindent ($b$) \underline{Pseudo-scalar mediator}:
We perform similar analyses for the three regions of mass space discussed in the previous section.
For fermionic dark matter $\chi_1$ (i.e., the PF case), we find that
\begin{equation}
\hspace*{-0.1cm} \frac{d\sigma_{1e}}{dE_r}\approx \left\{
\begin{array}{ll}
\dfrac{(g_\chi^a g_e^a)^2m_e E_r^2}{8\pi p_1^2 m_a^4} & \hbox{ for }(i) \\ [1.5em]
\dfrac{(g_\chi^a g_e^a)^2m_e E_r^2}{8\pi p_1^2 (2m_e E_r+m_a^2)^2} & \hbox{ for }(ii)\hbox{ and }(iii),
\end{array}\right.
\end{equation}
and the corresponding energy spectra with the same benchmark mass spectra as in figure~\ref{fig:Espec} are shown in the left panel of figure~\ref{fig:Espec2}.
Unlike the vector mediator case, the differential cross-section rises in increasing $E_r$ due to the $E_r^2$ dependence in the numerators.
For ($i$) the recoil spectrum increases up to $E_r^{\max}$, whereas for ($ii$) it gradually saturates due to the competition with the $E_r$ dependence in the denominator.
All these expected behaviors are clearly shown by the solid red and the solid green curves in the left panel of figure~\ref{fig:Espec2}.
Interestingly enough, the differential cross-section for ($ii$) becomes constant in the limit of $m_a\to 0$, and the $m_a$ dependence gets negligible.
Therefore, if a small $m_a$ is preferred by the data, it may be challenging to determine $m_a$.
For region ($iii$), exactly the same spectral behavior as in region ($ii$) is expected.
However, $E_r^{\max}$ approaches 9.75~MeV so that events accompanying keV-scale energy are very unlikely to arise.
Indeed, the blue curve clings to the $x$ axis.
\begin{figure}[t]
\centering
\includegraphics[width=0.496\linewidth]{recoil-alp.png}
\includegraphics[width=0.496\linewidth]{fit-alp-v2.png}
\caption{[Left] The corresponding unit-normalized plots with the same benchmark mass spectra as in figure~\ref{fig:Espec} but with fermionic BDM and pseudo-scalar mediator (i.e., the PF case).
The solid and the dashed lines are the spectra without and with detector effects, respectively.
For the ($iii$) region, the spectrum is rising very slowly toward $E_r^{\max}\approx 9.75$~MeV so that events with keV-scale recoil energy are very unlikely to arise and the corresponding blue curve appears invisible.
[Right] Sample energy spectra for the first two benchmark mass spectra.
We assume $g_\chi^a=1$ and galactic BDM for which the flux is given by Eq.~\eqref{eq:flux}.
The values of $\sigma_{1e}$ and $g_e^a$ associated with these fits are shown in the legend.}
\label{fig:Espec2}
\end{figure}
This rising feature of the recoil energy spectra implies that less events are affected by the XENON1T detector efficiency unlike the ($ii$) and ($iii$) regions with a vector mediator.
In other words, nominal cross-sections differ not much from the corresponding fiducial cross-sections.
On the other hand, the total cross-section is much smaller than that of the vector mediator scenario for the same mass spectra and the same coupling strengths, because $E_r^2$ dependence (i.e., $\sim1-10~{\rm keV}^2$) is much smaller than $E_1^2$ dependence (see the discussions near Eqs.~\eqref{eq:approx1} and \eqref{eq:approx2}).
This implies that in order to obtain a required cross-section for a given BDM flux, a significantly larger coupling strength should be needed, compared to the corresponding value for the vector mediator.
The right panel of figure~\ref{fig:Espec2} shows sample energy spectra for the first two benchmark mass spectra with $g_\chi^a=1$ and galactic BDM, and clearly advocates all these expectations. The quoted $\sigma_{1e}$ are slightly smaller than the $\sigma_{1e}$ in figure~\ref{fig:fit}.
We also find that the required values of $g_e^a$ are larger than $g_e^V$ in figure~\ref{fig:fit} by roughly four orders of magnitude.
They may be strongly disfavored by the existing limits.
We again revisit this issue in the next section.
When it comes to the case with scalar BDM (i.e., the PS case), we see that $E_r$ dependence in the numerator is linear so that the rising feature becomes mitigated.
In particular, for regions ($ii$) and $(iii$) the recoil energy distributions can be described by a rising-and-falling shape, so it is possible to find ranges of parameter space to explain the XENON1T excess.
We do not pursue an investigation to identify such parameter space here, reserving it for future work.
\medskip
\noindent ($c$) \underline{Scalar mediator}:
Given the discussions thus far, we are now equipped with enough intuitions to understand the scalar mediator case qualitatively.
In the SF case, $\overline{|\mathcal{A}|}^2$ behaves like $\sim m_e^2m_1^2$ for the ($i$) and ($ii$) regions, so the argument for the ($i$) and ($ii$) regions of the vector mediator scenario essentially gets through modulo numerical prefactors.
By contrast, the linear $E_r$ dependence can survive for the ($iii$) region, i.e., $\overline{|\mathcal{A}|}^2 \propto 2m_1^2+m_eE_r$, and as a result, the recoil energy spectrum can be of rising-and-falling shape like the ($ii$) and ($iii$) regions of the PS case.
In the SS case, $\overline{|\mathcal{A}|}^2\propto m_e^2m_1^2 = {\rm const.}$, so the overall expectations can be referred to those in the VF case except the fact that the scattering cross-sections are much smaller than those in the VF case for a given set of mass values and coupling strengths.
\section{Discussions}
In this section, we discuss implications of our findings: fit parameter consistency with existing limits and scattering of BDM on xenon nuclei.
As mentioned before, the quoted parameter values to explain the XENON1T excess may be in tension with existing bounds.
Identifying $V$ as a dark photon and considering the first benchmark point in figure~\ref{fig:fit}, we find that the $(m_V, g_e^V)$ pair is safe from the existing bounds.
In terms of the standard kinetic mixing parameter $\epsilon$, $g_e^V=2.4\times 10^{-4}$ is translated to $\epsilon=7.9\times10^{-4}$ which is not yet excluded by the latest limits~\cite{Banerjee:2019hmi}.
However, the parameter values for the other two benchmark points are strongly constrained by the limits from various astrophysical searches.
The same tension arises for the second benchmark point in the right panel of figure~\ref{fig:Espec2} with $a$ identified as, say axion-like particle.
Indeed, it was argued that there are ways to circumvent those astrophysical bounds that would rule out such dark photons and axion-like particles.
The main idea is that if the coupling constant and the mass parameter have effective dependence upon environmental conditions of astrophysical objects such as temperature and matter density, which are very different in the XENON1T experiment, the limits can be relaxed by several orders of magnitude~\cite{Jaeckel:2006xm,Ahlers:2006iz,Jaeckel:2010ni,An:2013yfc}.
There are several works to discuss relevant mechanisms in the context of specific particle physics models, e.g., Refs.~\cite{Khoury:2003aq, Masso:2005ym, Masso:2006gc, Mohapatra:2006pv, Dupays:2006dp, Brax:2007ak, Kim:2007wj}, for which concise summaries are referred to Refs.~\cite{Bonivento:2019sri, Dent:2019ueq}.
Furthermore, Ref.~\cite{An:2013yfc} pointed out that the energy loss process inside the stellar medium could be quenched because of absorption for large values of coupling.
Therefore, a careful check is needed to see if these parameter points are disfavored by the astrophysical bounds.
Finally, in regard to the $(m_a, g_e^a)$ values for the first benchmark point in the right panel of figure~\ref{fig:Espec2}, it seems that there are no existing searches that are sensitive to this parameter point to the best of our knowledge.
However, due to a relatively large size of coupling we expect that existing or near-future laboratory-based experiments such as accelerator experiments can test this parameter point.
Moving onto the second issue, one may ask whether BDM would scatter off a xenon nucleus and whether this dark matter interpretation would be contradictory to the null signal observation in the nuclear recoil channel at the XENON1T detector.
A possible solution is to assume that the mediator is ``baryo-phobic'' or ``electro-philic''.
Aside from model dynamics, we can check this issue using kinematics.
The maximum kinetic energy of a recoiling xenon nucleus $E_{r,{\rm Xe}}^{\max}$ is simply given by Eq.~\eqref{eq:maxE} with $m_e$ replaced by $m_{\rm Xe}$ and with $s$ approximated to $m_{\rm Xe}^2$.
For the first two benchmark mass points $p_1\approx 630$~keV gives $E_{r,{\rm Xe}}^{\max}\approx 6\times 10^{-3}$~keV, whereas for the last one $p_1=10$~MeV results in $E_{r,{\rm Xe}}^{\max}\approx 1.6$~keV.
Therefore, XENON1T is not sensitive enough to the dark matter signals from the three benchmark points in the nucleus scattering channel.
However, if $E_1$ increases, XENON1T starts to be sensitive to the signals belonging to region ($iii$) in the nucleus scattering channel, allowing for complementarity between the electron and nucleus recoil channels.
\begin{figure}[t!]
\centering
\includegraphics[width=11cm]{Fig_5_updated.pdf}
\caption{
Sample energy spectra including the ionization factor for the same benchmark mass spectra and particle spins (VF case) as in figure~\ref{fig:Espec}. Three dashed curves are the same as those appearing in figure~\ref{fig:fit}, which already incorporate the detector resolution and the detector efficiency. The solid curves are the corresponding one further including effects of the ionization factor by the electrons in three outer shells.
}
\label{fig:ionization}
\end{figure}
Finally, we would like to comment on the effects of the ionization form factor. In general, the form factors fall steeply with the momentum recoil, and therefore the ionization form factor strongly biases the scattering towards low-momentum recoil.
In addition, the form factor does not necessarily fall monotonically and thus could modify the recoil energy spectrum~\cite{Bunge:1993jsz, Kopp:2009et, Essig:2011nj, Lee:2015qva, Roberts:2016xfw, Catena:2019gfa, AKKMPS}.
The ionization factor can be calculated by using the Roothaan-Hatree-Fock wave function for the initial state electron \cite{Bunge:1993jsz} and applying the plane wave approximation for the final state electron.
We have followed the procedure described in Ref.~\cite{Cao:2020bwd,Kopp:2009et} to compute the ionization form factor for the interaction between BDM and the electrons in a xenon atom. We consider three outermost orbitals ($5p$, $5s$, and $4d$), with respective binding energies $\sim$12, 26 and 76 eV, which are known to be the dominant contribution \cite{Essig:2011nj,Lee:2015qva}.
As a cross-check, we have reproduced relevant results such as the ionization form factor from each shell and the differential recoil spectra for some physics examples as in Ref. \cite{Essig:2011nj,Lee:2015qva}.
We have also compared our approach against more sophisticated method where the final electron state is described by a positive energy continuum solution of the Schr\"odinger equation with a hydrogen potential~\cite{Roberts:2016xfw, Roberts:2019chv, Catena:2019gfa}.
We find that the plane wave approximation provides a reasonably good approximation for the low-momentum transfer as noted in Ref.~\cite{Roberts:2019chv, Catena:2019gfa}.
In figure~\ref{fig:ionization} we show the energy spectra including the ionization factor for the same benchmark mass spectra and particle spins (VF case) as in figure~\ref{fig:Espec}.
Three dashed curves represent the energy spectra in figure~\ref{fig:fit}, which already incorporate the detector resolution and the detector efficiency, while the solid curves take into account effects of the ionization factor by considering 18 electrons in three outermost orbitals.
As the detector efficiency and resolution affect the shape of the energy spectra, the ionization form factor also gives additional distortion.
Nevertheless, the effects of the ionization factor in the shape of energy spectra are mild and the main features remain very similar.
We find that the spectra for other particle spins (PF) also remain very similar to those in figure~\ref{fig:Espec2}.\footnote{More detailed analyses on the ionization form factor for the scattering of fast-moving dark matter and bound electrons are reserved for our future work~\cite{AKKMPS}.}
\section{Conclusions}
\begin{table}[t]
\centering
\begin{tabular}{c|c c c}
\hline \hline
& Region ($i$) & Region ($ii$) & Region ($iii$)\\
\hline
$\gamma_{\rm BDM}$ & $\approx 1$ & $\approx 1$ & $\gg1$ \\
\hline
VF & \cmark (flat) & \comark (falling) & \comark (falling) \\
VS & \cmark (flat) & \comark (falling) & \comark (falling) \\
PF & \cmark (rising) & \cmark (rising) & \xmark (--) \\
PS & \cmark (rising) & \comark (rising-and-falling) & \comark (rising-and-falling) \\
SF & \cmark (flat) & \comark (falling) & \comark (rising-and-falling) \\
SS & \cmark (flat) & \comark (falling) & \comark (falling) \\
\hline \hline
\end{tabular}
\caption{A summary of our case studies.
Conditions of each region are defined in Eq.~\eqref{eq:conditions}.
$\gamma_{\rm BDM}$ denotes the Lorentz boost factor of BDM.
\cmark and \comark ~indicate that one can find mass spectra to reproduce the XENON1T excess and satisfy the conditions of the associated regions, while for entries marked with \comark~a certain range of mediator mass may not reproduce the XENON1T excess.
By contrast, \xmark~indicates that it is generally hard to find a mass spectrum to explain the excess.
The general shape of expected recoil energy spectra is described in the parentheses.}
\label{tab:summarytab}
\end{table}
The dark matter interpretation for the XENON1T anomaly is in favor of the existence of fast-moving or boosted dark matter component(s) in the present universe, which may require non-conventional dark matter dynamics.
We investigated various cases in which such dark matter of spin 1/2 and 0 interacts with electrons via the vector, pseudo-scalar, or scalar mediator in the context of the two-component boosted dark matter model as a concrete example.
Our findings are summarized in Table~\ref{tab:summarytab}.
We found that there exist a set of parameter choices to be compatible with existing bounds as well as to accommodate the anomaly.
In particular, the scales of mass and coupling parameters are sensitive to the mediator choice.
Our study further suggested that with appropriate choices of mediator and its mass, significantly boosted dark matter can be allowed on top of the moderately fast-moving dark matter.
Finally, we emphasize that the analysis method that we have proposed in this work is general, so we expect that it is readily applicable to the interpretation of observed data in other dark matter direct detection experiments.
\medskip
\noindent {\bf \emph{Note Added. - }} We confirm that our total cross-section formula in Eq.~\eqref{eq:xsform} agrees with the corresponding expression in the updated version of Ref.~\cite{Fornal:2020npv}.
\medskip
\noindent {\bf \emph{Acknowledgement. - }} HA and KK acknowledge support from the US DOE, Office of Science under contract DE-SC0019474.
DK acknowledges support from DOE Grant DE-FG02-13ER41976/DE-SC0009913/DE-SC0010813.
GM acknowledges support from DOE Grant Contract de-sc0012704.
JCP acknowledges support from the National Research Foundation of Korea (NRF-2019R1C1C1005073 and NRF-2018R1A4A1025334).
SS acknowledges support from the National Research Foundation of Korea (NRF-2020R1I1A3072747).
|
1,116,691,500,421 | arxiv | \section{Introduction}
The study of quantum mechanics in periodic structures is one of the central
topics in condensed matter physics since many decades \cite{Ashcroft}. The
behavior of electrons in a crystal or, more generally, interacting particles
in a periodic potential, even at the few-body level, puts forward a major
theoretical and numerical challenge for theorists. Therefore, oversimplified
models which are still able to capture certain qualitative features of the
original problem have been proposed. The so-called Hubbard model
\cite{Hubbard} for electrons in a metal and its bosonic counterpart
\cite{bosMI,Jaksch}, assume that the particles only populate a single energy band of
the periodic potential and that the effective interaction has a short range
(if not zero range) character.
Recent advances in the physics of ultracold atoms in optical lattices
\cite{OptLatRev} have opened a fascinating framework in which it is possible
to simulate with unprecedent accuracy some of the models traditionally used in
condensed matter physics. In particular, the transition from a superfluid to a Mott
insulator of bosons loaded in an optical lattice has been successfully
observed \cite{transition}. In another spectacular experiment \cite{Winkler}
, repulsively
bound pairs of atoms have been produced, and their main properties have been
measured. This experiment has stimulated renewed interest in the study of
few-body effects in discrete lattices
\cite{PiilMolmer,MVDP1,MVDP2,JinSong,weiss,MVDPAS,PiilMolmerdifferentmass}, which had
been almost only
studied by Mattis {\cite{RevMattis}} about twenty years ago.
Both two-body problems on a one-dimensional tight-binding lattice with on-site
\cite{Winkler,MVDP1,PiilMolmer} as well as nearest-neighbor \cite{MVDP2}
interactions can be solved exactly. Several conclusions about them can be obtained \cite{MVDP2}: (i) Bound states can be calculated via a certain
polynomial equation of different degree depending on the range of the
interaction. (ii) The scattering states, both symmetric (bosonic) or
antisymmetric (spin-polarized fermionic) are well described, asymptotically,
by a single phase shift which depends again on the range of the
potential. (iii) The ``low-energy'' properties of those systems, characterized
by the scattering lengths, can always be calculated and appear to be rather
simple expressions of the respective interaction potentials.
Therefore, the next relevant question to ask is if there is a general pattern
followed by the solutions and properties of the two-body problem when the
interactions are of arbitrary but finite range. In this article we deal with
this question and find that, indeed, there is a well defined pattern for the
bound states, and that all scattering properties can be calculated very
efficiently. For the particular, yet very important case of ``low-energy''
scattering
we show how to obtain the scattering lengths very accurately even
without knowing the phase shift in general. We treat both
identical and distinguishable particles which can, of course, have different
tunneling rates, thanks to a generalization of the center of mass separation
ansatz for the two-body problem. To illustrate our results, we apply them to a
model dipolar potential with a cutoff at a certain, long enough range.
\section{General separation of the two-body problem}
We consider two particles, labeled A and B, in general having different tunneling
rates \cite{footnote1} $J_A$ and $J_B$, and interacting via a symmetric two-body
potential $V(z)=V(-z)$. The reduction of the two-body to a one-body problem
was first carried out in \cite{PiilMolmerdifferentmass,Martikainen}.
The two-body one dimensional discrete Schr\"odinger
operator $H$ for the two-body system, acting on a two-body wave function $u$ in
$\ell^2(\mathbb{Z})\otimes
\ell^2(\mathbb{Z})$ is, in first-quantized form,
\begin{align}
(Hu)(n_A,n_B)&= \nonumber\\
&-J_A\left[u(n_A+1,n_B) + u(n_A-1,n_B)\right]\nonumber\\
&- J_B\left[u(n_A,n_B+1) + u(n_A,n_B-1)\right] \nonumber\\
&+ V(|n_A-n_B|)u(n_A,n_B),
\end{align}
where $n_A$ and $n_B$ are the (integer) lattice positions of particle $A$ and
$B$, respectively. For simplicity, we set the lattice spacing $s\equiv 1$, so
that distances, lengths and quasi-momenta are dimensionless.
For the moment, we do not allow $V$ to become infinitely large, that is, $|V(|n|)|<\infty$ for all $n\in
\mathbb{Z}$; this condition can be relaxed by allowing $|V(0)|\to \infty$ and then
applying the Bose-Fermi mapping theorem (BFMT) \cite{Girardeau} once the
problem is reduced to a one-body equation. Moreover, we assume that $V$ is an arbitrary finite
range potential of range $\rho\in \mathbb{Z}$, that is, $V(|n|>\rho)=0$ with at least $V(\rho)\neq
0$.\\
In order to solve this problem exactly we need to transform the Hamiltonian $H$ to a
single particle operator. For this purpose, consider the ansatz
\begin{equation}
u(n_A,n_B)=u_K(z) e^{-i\beta_K z+iKR},\label{generalseparation}
\end{equation}
where $R=(n_A+n_B)/2$ and $z=n_A-n_B$ are, respectively, the center of mass
and relative coordinates; $K$ is the total quasi-momentum and
\begin{equation}
\tan{\beta_K}=\frac{J_A-J_B}{J_A+J_B}\tan{(K/2)}.\label{tanbetaK}
\end{equation}
Note that when $J_A=J_B\equiv J$, the ansatz
(\ref{generalseparation}) reduces to the well known case of identical
particles \cite{Winkler,PiilMolmer,MVDP1}.
By inserting the choice (\ref{generalseparation}) in the Schr\"odinger
equation $Hu=Eu$ we arrive at the desired
single-particle Hamiltonian for each value of the total quasi-momentum $K$
\begin{equation}
(\tilde{H}u_K)(z)=-|J^{(K)}|[u_K(z+1)+u_K(z-1)]+V(|z|)u_K(z),\label{ham}
\end{equation}
where the so-called collective tunneling rate \cite{PiilMolmerdifferentmass} has the form
\begin{equation}
|J^{(K)}|=\sqrt{J_A^2+J_B^2+2J_AJ_B\cos{K}}.
\end{equation}
Note that the reduced Hamiltonian of Eq. (\ref{ham}) is equivalent to the
findings in \cite{PiilMolmerdifferentmass,Martikainen}.
At this point it is convenient to introduce an adimensional Hamiltonian by dividing
it by the collective tunneling, which is equivalent to setting
$J^{(K)}\equiv 1$ (energies become dimensionless) in Eq. (\ref{ham}), and rename $u_K\equiv u$ for
simplicity. We will assume this in the subsequent discussions.
\section{Bound states}\label{boundgeneral}
We pursue the exact solution for the bound states of any two-body system on
the lattice with finite range interactions. Before doing so, we need to define
what is actually meant by bound state, mathematically, for the convenience of
the reader.
We define a bound state of $\tilde{H}$, Eq. (\ref{ham}), as any square-summable solution $u(z)$ of the
discrete time-independent Schr\"odinger equation $\tilde{H}u=Eu$ with its associated
eigenvalue $E$ lying
outside the essential spectrum of $\tilde{H}$,
$\sigma_{\mathrm{ess}}=[-2,2]$. Recall that ``outside the essential spectrum'' can
actually mean above \cite{Winkler,PSAF,PiilMolmer,MVDP1} and not only below
the continuum.
It is
already known that for any finite range potential $V$ there exists at least
one symmetric bound state \cite{Damanik}; it is also known that the maximum
number of symmetric (antisymmetric) bound states of $\tilde{H}$ is $\rho+1$
($\rho$) \cite{Teschl}. Now we show rigorously {\it how} to calculate all these
bound states exactly. The formulation of this result is as follows:\\
{\it Theorem.\\
Let $\tilde{H}$ be the Hamiltonian (\ref{ham}) with $V$ a range-$\rho$ ($<\infty$)
potential. Then all bound states $u(z)$ of $\tilde{H}$ have the decay property
$u(z)\propto\alpha^{|z|-\rho}$ for $|z|\ge \rho$, $0<|\alpha|<1$; the energies of the bound states are
given by $E=-\alpha-1/\alpha$. If $u(z)$ is symmetric then $\alpha$ is a root of a polynomial of
degree $2\rho+1$ if $\rho\ge 1$ and, if $\rho=0$, its degree is $2$; if $u(z)$ is
antisymmetric and $\rho>0$ then $\alpha$ is the root of a polynomial of degree
$2\rho-1$.}
{\it Proof}. Applying the exponential ansatz for $u(z)$ with $|z|\ge
\rho$ yields immediately
\begin{equation}
E=-\alpha-1/\alpha\equiv f(\alpha).
\end{equation}
Since
$f((-1,0)\cup (0,1)) = (-\infty,-2)\cup (2,\infty)$ and $f$ is injective in
$(-1,0)\cup(0,1)$, we have that the exponential ansatz is the only possible form
for the bound states outside the range of $V$.\\
To see that $\alpha$ is a root of
a polynomial one shows by induction, for $\rho\ge 2$, that if $u(z)$ is exponentially decaying,
then $\alpha^n u(\rho-n)=Q^{(n)}_{2n-1}(\alpha)$ and $\alpha^{n-1}
u(\rho-n-1)=Q^{(n-1)}_{2n-3}(\alpha)$, where $Q^{(m)}_k$ are polynomials of
degree $k$. For symmetric solutions the polynomial equation is then obtained
by setting $u(1)=u(-1)$ and, for
antisymmetric solutions, by setting $u(0)=0$, which proves our
statement. For $\rho=0$ and $\rho=1$ the result can be proved by explicitly
obtaining the polynomial equation \cite{MVDP1,MVDP2}.
The theorem presented here implies that for any finite range potential one has to solve
a polynomial equation whose degree grows slowly with increasing $\rho$. The way
of obtaining such polynomials is, as can be observed from the proof, inductive: we start by setting $u(\rho)=1$
and proceed to calculate $u(\pm 1)$ and $u(0)$ by recurrence and solve the
respective symmetry constrains $u(1)=u(-1)$ or $u(0)=0$. Certainly if $\rho$
gets too large it becomes inconvenient to get such polynomials for a general
potential $V$, and in this case we should obtain the coefficients
of the polynomial for the given particular potential.
Consider now the specific choice of the potential
\begin{equation}
V(z)=\left\{
\begin{array}{rl}
-\frac{1}{|z|^3} & \text{if } 0<|z|\le 10 \\
0 & \text{if } |z|>10 \\
-9.7313 & \text{if } |z|=0, \label{Vz3}
\end{array} \right.
\end{equation}
which corresponds to a dipole-dipole interaction with a cutoff at a finite but
long range, and where the divergence of the potential at $z=0$ has been
substituted by a finite value. Note that such dipolar interactions, with a
tunable on-site interaction $V(0)$, can be realized with dipolar atoms or
molecules in optical lattices \cite{Trefzger1,Trefzger2,Menotti}.
We have
calculated both polynomials $P(\alpha)$ for the symmetric and
antisymmetric bound states numerically, with their roots characterizing the
bound states. The results are shown in Fig. \ref{fig:FIG2-general}. For symmetric bound
states the polynomial has only one root in $(-1,1)$, and
therefore only one bound state \cite{footnote2}. The polynomial has a root at $\alpha=1$, which
means that it has low-energy resonance. We will discuss these resonances in Section \ref{scatlengthgeneral}. The polynomial for antisymmetric bound states has also
one and only one root in $(-1,1)$, in agreement with the discrete Bargmann's
bound \cite{HundertmarkSimon}.
\begin{figure}[t]
\includegraphics[width=0.38\textwidth]{FIG2-general.eps}
\caption{(Color online). Polynomials for symmetric (blue solid line) and antisymmetric (red
dashed-dotted line) bound states with the interaction (\ref{Vz3}), whose roots $\alpha$ characterize the bound states
with energy $E=-\alpha-1/\alpha$. The horizontal line corresponds to zero ordinate.}
\label{fig:FIG2-general}
\end{figure}
\section{Scattering states}
After having introduced the first main result of this paper, which deals exclusively with
bound states, it is natural to ask about the
exact scattering properties of the system. For finite range potentials, the scattering states of the Hamiltonian $\tilde{H}$ are
asymptotically plane waves, that is, for $|z|\ge \rho$ we have
\begin{align}
u_S(z)&\propto \cos (k|z|+\delta_S),\label{scatteringsym}\\
u_A(z)&\propto \operatorname{sgn}(z) \cos(k|z|+\delta_A),\label{scatteringant}
\end{align}
where $S$ and $A$ denote, respectively, symmetric and antisymmetric
solutions. Their associated eigenenergies are given by the well known
tight-binding energy dispersion relation \cite{Ashcroft}
\begin{equation}
E=-2\cos(k).
\end{equation}
However, a general result concerning
the phase shifts $\delta_S$ and $\delta_A$ does not seem feasible, and it is quite cumbersome to obtain them in closed
form for long enough ranges. One can, however, calculate the phase shifts (and
from them the exact solution at all $z$) numerically by recurrence. To this
end, we set $u_S(\rho+1)=\cos(k(\rho+1)+\delta_S)$ and
$u_S(\rho)=\cos(k\rho+\delta_S)$ and analogously for antisymmetric solutions,
from equations (\ref{scatteringsym}) and (\ref{scatteringant}). Then we calculate
$u_S(-1)$ and $u_S(1)$ for symmetric solutions with the help of the
Schr\"odinger equation (\ref{ham}) and solve $u_S(-1)=u_S(1)$. In the case of
antisymmetric solutions, the relevant equation is $u_A(0)=0$. We have done so
for the example potential of Eq. (\ref{Vz3}), as is plotted in
Fig. \ref{fig:FIG3-general}. There, we clearly observe that the main
differences between both phase shifts occur at low quasi-momenta where the
symmetric solution is resonant (see Fig. \ref{fig:FIG2-general}); looking at slightly higher quasi-momenta already
shows good agreement between both phase shifts. This means that far from $k=0$
fermionization appears rapidly: the large on-site interaction $V(0)$ in (\ref{Vz3}) acts as a
hard-core at high energies, for which the resonance plays no role (it is located
at the bottom of the continuum), and therefore the symmetric phase shifts are
close to the antisymmetric (``fermionic''). At low quasi-momenta, the resonance
obviously dominates the asymptotic behavior of the symmetric scattering states.
In the insets of Fig. \ref{fig:FIG3-general}, we plot the
comparison of the phase shifts for the potential in Eq. (\ref{Vz3}) and a
model range-1 potential, whose analytic solution is known \cite{MVDP2}. The model potential $W$ is chosen so
as to be consistent with Bargmann's bound \cite{HundertmarkSimon}, and to be resonant for
the lowest-energy symmetric solution. We obtain \cite{MVDP2}
\begin{align}
W(1)&=-\sum_{z=1}^{10}z^{-3}=-1.19753\nonumber\\
W(0)&=-12.125.\label{modelpot}
\end{align}
The qualitative agreement between the results using $V$ or $W$ for the
symmetric eigenstates is manifest in Fig. \ref{fig:FIG3-general} and, as
expected, the differences are most noticeable in the high quasi-momentum
regime. For antisymmetric eigenstates the agreement is very good, even
quantitatively, until $|k|\simeq \pi/2$. The simplified potential (\ref{modelpot})
can thus be used as a good approximation for the interaction (\ref{Vz3}) in
the problem of many (spin-polarized) fermions or
hard-core bosons on a one dimensional lattice at low energies (around the
ground state) and low filling (typically much smaller than half the number of
lattice sites). Such a problem can then be solved exactly by means of the
Bethe ansatz \cite{BetheI}.
\begin{figure}[t]
\includegraphics[width=0.38\textwidth]{FIG3-general-fin.eps}
\caption{(Color online). The calculated phase shifts
($\tan(\delta)$ with $\delta=\delta_S$ or $\delta=\delta_A$)
for symmetric (blue circles) and antisymmetric (red squares) scattering
wave functions, Eqs. (\ref{scatteringsym}) and (\ref{scatteringant}), as functions of the
relative quasi-momentum $k$, for the potential (\ref{Vz3}). Left inset:
comparison of the symmetric phase shift (blue circles) with the one obtained
with a model range-1 potential (red triangles), Eq. (\ref{modelpot}). Right
inset: antisymmetric phase shift (red squares) compared to the result with the
model range-1 potential (blue triangles). The axes of the insets have the same
meaning as those of the main figure.}
\label{fig:FIG3-general}
\end{figure}
\section{Scattering lengths and zero-energy resonances}\label{scatlengthgeneral}
\subsection{Low-energy scattering}
The ``low-energy'' ($k\to 0,\pi$) scattering properties of the
two-body system can be understood via a simple, yet exact, calculation of the
scattering lengths. Indeed, the solution of the
time-independent Schr\"odinger equation when $k\to 0$ ($k\to \pi$) has an energy
$E=-2$ ($E=+2$), and has the asymptotic ($|z|\ge \rho$) behavior
\begin{align}
u_S(z) &= (\mp 1)^z \frac{|z|-a_S^{\pm}}{\rho-a_S^{\pm}}\\
u_A(z) &= \operatorname{sgn}(z) (\mp 1)^z \frac{|z|-a_A^{\pm}}{\rho-a_A^{\pm}} ,
\end{align}
where $a_i^{-}$ ($a_i^{+}$) is the scattering length at $k\to 0$ ($k \to
\pi$), $i=S,A$. It
must be noted that, in the case of the lattice, there are {\it four} different
scattering lengths, two for ``bosons'' (symmetric solutions) and two for
spin-polarized ``fermions'' (antisymmetric solutions). In
order to calculate the scattering lengths we proceed as follows : using
the recurrence relation from $z=\rho$ by setting $u_i(\rho+1)=(\mp 1)^{\rho+1}
[1+1/(\rho-a^{\pm})]$,
$u_i(\rho)=(\mp 1)^{\rho}$ and $E\equiv E_{\pm}=\pm 2$, the scattering lengths for the symmetric states are obtained
by solving the equation
$(V(0)-E_{\pm})u_S(0)-2u_S(1)=0$ (see proof of the theorem), while for
the antisymmetric states the
equation to solve is $u_A(0)=0$. It is remarkable that the resulting
equations for the scattering lengths as functions of the potential can be cast
as linear in $a^{\pm}$, that
is, are of the form $s_0a^{\pm}+b_0=0$ with $s_0$ and $b_0$ real constants
which depend on $V(z)$. In fact, this is an alternative way of defining the
four lattice scattering lengths, totally equivalent to the definition
$a^{\pm} = -\lim _ {k \to \pi,0} \partial_k \delta$ \cite{PiilMolmerFeshbach}, with the advantage of not
needing to know the phase shift explicitly. It must be noted
at this point that, strictly speaking, scattering lengths are unique of one-dimensional lattices
since the radial symmetry is lost in dimensions $d>1$.
As an example, we have calculated $a_S^-$ for the dipolar potential with
a cutoff as $V(0<|z|\le \rho)=-1/|z|^3$, $V(|z|>\rho)=0$, and leaving $V(0)$
as a free parameter. The results are shown in Fig. \ref{fig:FIG1-general} for
a range $\rho=10$, where there is a resonance clearly marked at the point were
the scattering length diverges.
\begin{figure}[t]
\includegraphics[width=0.38\textwidth]{FIG1-general.eps}
\caption{(Color online). Scattering length $a_S^-$
(see text) as a function of the free parameter $V(0)$, with $V(0<|z|\le
10)=-1/|z|^3$ and $V(|z|>10)=0$.}
\label{fig:FIG1-general}
\end{figure}
The divergence of one of the scattering lengths can happen for different values of the total
quasi-momentum $K$ \cite{footnote3}. In the simplest case of a zero range interaction with
$V(0)\equiv U$, the system is known to have no ``zero-energy''
resonances \cite{MVDP1}. For longer ranges, already starting with $\rho=1$
\cite{MVDP2}, these resonances can occur. With the method outlined in this
work we are able to predict when, for a given range-$\rho$ potential with one or
more free parameters $\{V(z_1),V(z_2),\ldots,V(z_n)\}$, there is
such a resonance. To do so, one sets $u(z\ge \rho)=(\mp 1)^z$ and iterates recursively as
has been explained, and then solves the resulting equation for symmetric and
antisymmetric states, getting a relation among the free parameters of
the potential to have a resonance.\\
We consider again the example of Fig. \ref{fig:FIG1-general}. As we
have already noted, the system can admit one resonance at the bottom of the continuum for
the symmetric states. From Fig. \ref{fig:FIG1-general}, the approximate location
of the resonance can be inferred. However, once the scattering length starts
to diverge, an accurate location of the resonance grafically is very hard (if
not impossible), especially if the resonance is very sharp as a function of
$V(0)$. With our method, we are able to locate the resonance
very precisely, obtaining for the example we are dealing with a value of $V(0)
= -9.7313$ \cite{footnote4}. This is exactly the value chosen in the previous sections to
match this resonance.
\subsection{Near-resonant bound states}
It is well known that the binding energy and size of a near-resonant bound
state (NRBS) are closely related to the (large) scattering
length $a^{\pm}\lessgtr 0$. On a one dimensional
lattice, the relation between the binding energy $E_b$ and the scattering
length in the effective mass approximation is given by
\begin{equation}
|E_b| \approx \frac{\hbar^2}{2|m^{\pm}|(a^{\pm})^2},\label{Eb}
\end{equation}
where $m^{\pm} = \mp \hbar^2/2$ is the effective mass of the pair with an
obvious notation. Note that Eq. (\ref{Eb}) is also valid in the context of
Feshbach resonances \cite{PiilMolmerFeshbach}.
When the scattering length is large, the parameter $\alpha$ characterizing
a NRBS is close to $\pm 1$ (for $E=-2$ and $E=2$,
respectively). By writing $|\alpha|=1-|\xi|$, it is not difficult to show that
the size of the bound state, for $\rho \ge 1$, behaves as $\ell ^* \equiv \langle |z| \rangle \sim 1/(2|\xi|)$ for $|\xi| \ll 1$. Mathematically, one has
\begin{equation}
\lim_{\xi \to 0} |\xi| \ell^* = \lim_{\xi \to 0} |\xi| \frac{\bra{u} |z| \ket{u}}{\Vert u \Vert ^2} = \frac{1}{2}.
\end{equation}
Since for small $|\xi|$, $|E_b|\approx \xi^2$, we have that the
relation between the size of the NRBS and its associated scattering length is
given by
\begin{equation}
\ell ^* \approx \sqrt{\frac{|m^{\pm}|}{2\hbar^2}} |a^{\pm}| = \frac{|a^{\pm}|}{2},
\end{equation}
showing that the size of the NRBS grows linearly with the scattering length,
as we expected.
\subsection{The number of bound states}
The scattering lengths are very useful quantities in the sense that knowing their
precise values implies knowing the total number of bound states with energies lying below or above
the continuum. For this purpose, we use the discrete analog of Sturm
oscillation theory \cite{Teschl}. In simple terms, oscillation theory states
that the number of nodes \cite{footnote5} of the zero-energy ($E=E_{-}=-2$)
symmetric (antisymmetric) solution
$u$ of $\tilde{H}u=Eu$ in $\mathbb{Z}_+$ is exactly the number of
symmetric (antisymmetric) eigenstates
below $E_{-}=-2$. Since any state with energy below $E_{-}$ is a bound state, the
number of nodes of $u$ is the number of bound states below the continuum. To see
how many bound states there are with energies above the continuum, we make use of the
transformation $\hat{G}$ of Appendix \ref{app}, Eq. (\ref{trans}), or,
equivalently, count the number of missing nodes.
We apply oscillation theory now to our example with the potential of
Eq. (\ref{Vz3}) leaving again $V(0)$ as a free parameter. We calculate (not shown) the symmetric zero-energy solution
for a given scattering length $a_S^-$ with the methods introduced in this
section and see that for $V(0)<-9.7313$ there are two bound states with energies
below $E_{-}$ and for $V(0)\ge -9.7313$ there is exactly one bound state below the
continuum.
\section{A generalization}
More general Hamiltonians with exchange
operators appear when dealing with the problem of one ``free'' boson and a
bound pair \cite{MVDPAS}. In such
case, the effective particles are distinguishable, there is a hardcore on-site
interaction, an effective range-1 potential and a first order exchange term. We assume now
that we have the following general one-body Schr\"odinger operator
\begin{align}
(H_{\mathrm{ex}}u)(z)=&-[u(z+1)+u(z-1)]+V(|z|)u(z)\nonumber\\
&+\Omega(|z|)(\hat{P} u)(z),
\end{align}
where $V(0)$ can be finite or infinite and where $\hat{P}$ is the discrete parity
operator. We further assume that $\Omega(|z|)$ has a finite range $\rho_{\mathrm{ex}}$
with no on-site exchange, $\Omega(0)=0$, since it can be included in $V(0)$.
Obviously $[\hat{P},H_{\mathrm{ex}}]=0$, and therefore we can look for
symmetric and antisymmetric solutions. However, the Hamiltonian does not
commute with the exchange operator $\Omega \hat{P}$. With the parity as a good quantum number
it is straightforward to generalize the theorem of Section \ref{boundgeneral} to include exchange. To see
this, take $\rho_M\equiv \operatorname{max}(\rho,\rho_{\mathrm{ex}})$. If $u(z)$ is
symmetric, the exchange
shifts the potential to $V(|z|)+\Omega(|z|)$, while if $u(z)$ is antisymmetric
it shifts the potential to $V(|z|)-\Omega(|z|)$. Therefore, obtaining the
bound states of $H_{\mathrm{ex}}$ reduces again to a polynomial equation of degree
$2\rho_M\pm 1$, and all the results of our theorem apply by changing $\rho$ by
$\rho_M$ and $V$ by $V\pm \Omega$. However, it is no longer true that a
hardcore condition $|V(0)|\to \infty$ maps ``bosons''
onto ``fermions'' (symmetric onto antisymmetric solutions). Indeed, the non-trivial dependence of $H_{\mathrm{ex}}$ on the
parity of the eigenstates makes it possible to have states above as well as
below its continuum even if $V$ and $\Omega$ have both the same definite
sign. This is the fact that makes $H_{\mathrm{ex}}$ violate the hypotheses of
the BFMT \cite{Girardeau}, and
it explains the appearance of exotic three-body bound states on a 1D lattice
\cite{MVDPAS}.
\section{Conclusions}
In this paper, we have shown how the exact wave functions and energies
of any bound state
of two particles on a one-dimensional tight-binding lattice can be calculated
by solving a polynomial equation whose order increases slowly with increasing
range of the two-body interaction potential, which can also include
parity-dependent terms, such as effective particle exchange. We have
shown that the calculation of the exact scattering states is possible, and simple. We have also shown how the
zero-energy resonances associated with the entry or exit of a bound state can
be trivially and exactly located, and related the scattering lengths to the
number of bound states above or below the continuum by making reference to the
discrete version of Sturm theory.
There are, on the other hand, many open problems in the physics of few particles
on a lattice. At the two-body level, there is still no general prescription for the
calculation of bound and scattering states on two- and three-dimensional lattices with
arbitrary finite range interactions. For the
three-body case, the Efimov effect \cite{Efimov}, which was shown to
appear on a three-dimensional (3D) simple cubic lattice \cite{RevMattis}, has not
been quantitatively examined yet; the lattice Efimov effect should depend largely
on the total quasi-momentum of the system and, moreover, there
should appear new kinds of exotic three-body bound states in 3D with no analog
in continuous space, as has been shown to be the case in 1D
\cite{MVDPAS}. It would also be very interesting to explore few-body effects
in other lattice geometries.
\begin{acknowledgments}
Useful discussions with David Petrosyan, Luis Rico and especially Gerald Teschl
are gratefully acknowledged.
This work was supported by the EU network EMALI.
\end{acknowledgments}
|
1,116,691,500,422 | arxiv | \section{Introduction}
A quadratic extension of a cyclotomic field, which is non-abelian Galois
over the rational number field $\Bbb Q$, is called a quasi-cyclotomic field.
All quasi-cyclotomic fields are described explicitly in [8] followed the works
in [1] and [3]. They are generated by a canonical $\mathbb Z/2\mathbb Z$-basis. The minimal
quasi-cyclotomic field containing the quadratic roots of one element of the
basis is called a primary quasi-cyclotomic field.
L.Yin and C.Zhang [7] have studied the arithmetic of any quasi-cyclotomic field. In
this paper we determine all irreducible representations of primary quasi-cyclotomic
fields. The methods apply to determine the irreducible representations
of an arbitrary quasi-cyclotomic field. We also compute
the Artin $L$-functions for a class of quasi-cyclotomic fields.
First we recall the constructions of primary quasi-cyclotomic fields.
Let $S$ be the set consisting of $-1$ and all prime numbers. For $p\in S$,
we put $\bar p=4,8,p$ and set $p^*=-1, 2,(-1)^{\frac{p-1}2}p$ if $p=-1,2$
and an odd prime number, respectively. Let
$K=\Bbb Q(\zeta_{\bar pq})$ be the cyclotomic field of conductor $\bar pq$.
For a class $[a]\in\Bbb Q/\Bbb Z$, we set $\sin[a]=2\sin a\pi$ for $0<a<1$ and
$\sin[0]=1$. For prime numbers $p<q$, we define
\[
v_{pq} =
\prod_{i=0}^{\frac{p-1}2}\prod_{j=0}^{\frac{q-1}2}\frac{\sin[\frac{iq+j}{pq}]}
{\sin[\frac{jp+i}{pq}]}\quad \qquad(p>2)
\]
and
\[
v_{2q} =\frac{\sin[\frac 14]}{\sin[\frac 1{4q}]}\prod_{j=0}^{\frac{q-1}2}
\frac{\sin[\frac{j}{2q}]\cdot\sin[\frac{2j-1}{4q}]}
{\sin[\frac{4j+i}{4q}]\cdot\sin[\frac{j}{q}]\cdot\sin[\frac{2j-1}{2q}]}.
\]
For $p<q\in S$, we put
\begin{displaymath}
u_{pq}:= \left \{
\begin{array}{ll}
\sqrt{q^*} &\text{if }p=-1\\
v_{pq} &\text{if }p=2\text{ or } p \equiv q \equiv 1 \mathrm{mod} 4 \\
\sqrt{p}\cdot v_{pq} &\text{if } p \equiv 1, ~ q \equiv 3 \mathrm{mod} 4 \\
\sqrt{q}\cdot v_{pq} &\text{if } p \equiv 3, ~ q \equiv 1 \mathrm{mod} 4 \\
\sqrt{pq}\cdot v_{pq} &\text{if } p \equiv q \equiv 3 \mathrm{mod} 4.
\end{array}
\right.
\end{displaymath}
Let $\tilde K=K(\sqrt{u_{pq}})$. Then $\tilde K$ is the minimal one in all quasi-cyclotomic
fields which contain $\sqrt{u_{pq}}$. We call these fields $\tilde K$ primary quasi-cyclotomic
fields. Let $G=\text{Gal}(K/\Bbb Q)$ and $\tilde G=\text{Gal}(\tilde K/\Bbb Q)$. We always
denote by $\epsilon$ the unique non-trivial element of $\text{Gal}(\tilde K/K)$.
If $(p,q)=(-1,2)$, then the group $G$ is generated by two elements $\sigma_{-1}$ and
$\sigma_2$, where $\sigma_{-1}(\zeta_8)=\zeta_8^{-1}$ and $\sigma_2(\zeta_8)=\zeta_8^{5}$.
If $p>2$, then $G$ is generated by two elements $\sigma_p$ and $\sigma_q$,
where $\sigma_p(\zeta_{p})=\zeta_{p}^a,\sigma_p(\zeta_{q})=\zeta_{q}$
and $\sigma_q(\zeta_{p})=\zeta_{p},\sigma_q(\zeta_{q})=\zeta_{q}^b$,
with $a,b$ being generators of $(\Bbb Z/p\Bbb Z)^*$ and $(\Bbb Z/q\Bbb Z)^*$
respectively. If $p=2$, then $G$ is generated by three elements $\sigma_{-1},\sigma_2$
and $\sigma_q$, where $\sigma_{-1},\sigma_2$ act on $\zeta_8$ as above and on $\zeta_q$
trivially, and $\sigma_q$ acts on $\zeta_q$ as above and on $\zeta_8$ trivially.
Next we describe the group $\tilde G$ by generators and relations.
An element $\sigma\in G$ has two liftings in $\tilde G$. By [Sect.3, 7] the action of the
two liftings on $\sqrt{u_{pq}}$ has the form $\pm\alpha\sqrt{u_{pq}}$ or
$\pm\alpha\sqrt{u_{pq}}/\sqrt{-1}$ with $\alpha>0$. We fix the lifting $\tilde\sigma$
of $\sigma$ to be the one with the positive sign. Then the other lifting of $\sigma$ is
$\tilde\sigma\epsilon$. The group $\tilde G$ is generated by $\epsilon,\tilde\sigma_p$
and $\tilde\sigma_q$ (and $\tilde\sigma_{-1}$ if $p=2$). Clearly $\epsilon$ commutes
with the other generators. In addition, we have
$\tilde\sigma_p\tilde\sigma_q=\tilde\sigma_q\tilde\sigma_p\epsilon$
(and $\tilde\sigma_{-1}$ commutes with $\tilde\sigma_2$ and $\tilde\sigma_q$ if $p=2$).
For an element $g$ of a group, we denote by $|g|$ the order of $g$ in the group.
Let $\log_{-1}:\{\pm 1\}\rightarrow\Bbb Z/2\Bbb Z$ be the unique isomorphism.
For an odd prime number $p$ and an integer $a$ with $p\nmid a$, let $(\frac ap)$ be the quadratic
residue symbol. We also define $(\frac a2)=(\frac a{-1})=1$ for any $a$. Then we have, see [7, Th.3],
\[
|\tilde\sigma_p|=(1+\log_{-1}(\frac{q^*}p))|\sigma_p|\quad
\text{ and }\quad |\tilde\sigma_q|=(1+\log_{-1}(\frac{p^*}q))|\sigma_q|,
\]
with the exception that $\tilde\sigma_2=2|\sigma_2|$ when $(p,q)=(-1,2)$. If $p=2$, we
have furthermore $|\tilde\sigma_{-1}|=|\sigma_{-1}|$.
Thus we have determined the group $\tilde G$ by generators and relations.
\section{Abelian subgroup of index 2}
In this section we construct a subgroup of $\tilde G$ of index 2 and determine the
structure of the subgroup. We consider the following three cases separately:
Case A:\quad~ $|\wt{\sigma}_p| =|\sigma_p|~\text{ and }~|\wt{\sigma}_q| =|\sigma_q|$;
Case B:\quad~ $|\wt{\sigma}_p|=2|\sigma_p|,~~ |\wt{\sigma}_q| =|\sigma_q|~\text{ or }~
|\wt{\sigma}_p| =|\sigma_p|,~~|\wt{\sigma}_q| =2|\sigma_q|$;
Case C:\quad~ $|\wt{\sigma}_p| =2|\sigma_p|~\text{ and }~ |\wt{\sigma}_q| =2|\sigma_q|$.\\
All the three cases may happen. In fact, the case (A) happens if and only if
$(\frac{p^*}q)=(\frac{q^*}p)=1$; the case (B) happens if and only if
$(\frac{p^*}q)\ne(\frac{q^*}p)$ or $(p,q)=(-1,2)$; and the case (C) happens if and only if
$(\frac{p^*}q)=(\frac{q^*}p)=-1$.
In the case A, we define the subgroup $N$ of $\wt G$ to be
\[
N=\begin{cases}<\wt\sigma_{-1},\wt\sigma_2,\wt\sigma_q^2,\varepsilon> &~~\text{ if }~~p=2\\
<\wt\sigma_p,\wt{\sigma}^2_q,\varepsilon> &~~\text{ if }~~p\ne 2
\end{cases}\eqno(A2.1)
\]
It is easy to see that the subgroup $N$ is abelian of index 2 and is a direct sum of the cyclic
groups generated by the elements. Thus we have
\[
N\cong\begin{cases}
\mathbb Z/2\mathbb Z\oplus \mathbb Z/((q-1)/2)\mathbb Z\oplus\mathbb Z/2\mathbb Z &~~\text{ if }~~p=-1\\
\mathbb Z/2\mathbb Z\oplus\mathbb Z/2\mathbb Z\oplus\mathbb Z/((q-1)/2)\mathbb Z\oplus\mathbb Z/2\mathbb Z &~~\text{ if }~~p=2\\
\mathbb Z/(p-1)\mathbb Z\oplus\mathbb Z/((q-1)/2)\mathbb Z\oplus\mathbb Z/2\mathbb Z &~~\text{ if }~~p>2.
\end{cases}\eqno (A2.2)
\]
In the case B, we define the the subgroup $N$ of $\wt G$ to be
\[
N=\begin{cases}
<\wt\sigma_{-1},\wt{\sigma}_2,\wt{\sigma}^2_q> &~~\text{ if }~~p=2\\
<\wt{\sigma}_p,\wt{\sigma}^2_q> &~~\text{ if }~~p\ne 2\text{ and }
|\wt{\sigma}_q| =2|\sigma_q|\\
<\wt{\sigma}^2_p,\wt{\sigma}_q> &~~\text{ if }~~|\wt{\sigma}_p| =2|\sigma_p|.
\end{cases}\eqno(B2.1)
\]
Again $N$ is abelian and has index 2 in $\wt G$. In addition, we have
\[
N\cong\begin{cases} \mathbb Z/2\mathbb Z\oplus\mathbb Z/2\mathbb Z &~\text{ if }~(p,q)=(-1,2)\\
\mathbb Z/2\mathbb Z\oplus\mathbb Z/(q-1)\mathbb Z &~\text{ if }~p=-1,~ q>2\\
\mathbb Z/2\mathbb Z\oplus\mathbb Z/2\mathbb Z\oplus\mathbb Z/(q-1)\mathbb Z &~\text{ if }~p=2\\
\mathbb Z/(p-1)\mathbb Z\oplus\mathbb Z/(q-1)\mathbb Z &~\text{ if }~p>2.
\end{cases}\eqno(B2.2)
\]
In the case C, we must have $p,q$ are odd prime numbers. Let $v_2(p-1)$ denote the power
of 2 in $p-1$. We define the subgroup $N$ of $\wt G$ to be
\[
N=\begin{cases}
<\wt{\sigma}^2_p\,,\wt{\sigma}_q> &~~\text{ if }~~v_2(p-1)\le v_2(q-1)\\
<\wt{\sigma}_p\,,\wt{\sigma}^2_q> &~~\text{ if }~~v_2(p-1)> v_2(q-1).
\end{cases}\eqno(C2.1)
\]
Then $N$ is an abelian subgroup of $\wt G$. When $v_2(p-1)\le v_2(q-1)$, we have
\[
|N|=\frac{|\wt\sigma_p^2|\cdot|\wt\sigma_q|}{|<\wt\sigma_p^2>\cap<\wt\sigma_q>|}
=\frac{(p-1)\cdot 2(q-1)}{2}.
\]
So $[\wt G : N]=2$ and $N$ is a normal subgroup of $\wt G$. We have the same result
when $v_2(p-1)> v_2(q-1)$. The subgroup
$<\wt{\sigma}^2_p\,,\wt{\sigma}_q>$ is always an abelian subgroup of $\wt G$ of index 2.
But we can not get all irreducible representations from the inducement
of the representations of this subgroup when $v_2(p-1)> v_2(q-1)$. So we define $N$ in two cases.
Next we determine the structure of the subgroup $N$ in the case C. We consider the
case $v_2(p-1)\le v_2(q-1)$ in detail.
Let $d=\gcd(\frac{p-1}2,q-1)$, $s=(p-1)/2d$ and $t=(q-1)/d$. Choose
$u,v\in\mathbb Z$ such that $us+vt=1$. We have the relations
\[(\wt{\sigma}^2_p)^{p-1}=1,~~~
(\wt{\sigma}^2_p)^{\frac{p-1}2}=\varepsilon=\wt{\sigma}^{q-1}_q.
\]
Let $M$ be the free abelian group generated by two words $\alpha\,,\,\beta$. Let
\[
\alpha_1=(p-1)\alpha\;\;;\;\;\beta_1=\frac{p-1}2\alpha-(q-1)\beta\;;
\]
and let $M_1$ be the subgroup of $M$ generated by
$\alpha_1\,,\beta_1$. Then $M_1$ is the kernel of the homomorphism
\[
M\longrightarrow N;~~\alpha\mapsto\wt{\sigma}^2_p,~\beta\mapsto\wt{\sigma}_q\;\;.
\]
So we have $N\cong M/M_1$. Define the matrix
\[
A:=\matr{p-1}{\frac{p-1}2}{0}{1-q}\;.
\]
Then $(\alpha_1\,,\beta_1)=(\alpha\,,\beta)\cdot A$.
We determine the structure of $M_1$ by considering the standard form
of $A$. Define
\[
P:=\matr{u}{v}{-t}{s}\in\mathrm{SL}_2(\mathbb Z)\;;
\quad\;Q:=\matr{1}{2tv-1}{-1}{-2tv+2}\in\mathrm{SL}_2(\mathbb Z).
\]
Then
\[
B:=PAQ=\matr{d}{0}{0}{-2s(q-1)}
\]
is the standard form of $A$. Let
\[
(\tau\,,\mu)=(\alpha\,,\beta)P^{-1}\quad\text{and}\quad(\tau_1\,,\mu_1)=(\alpha_1\,,\beta_1)Q.
\]
Then $(\tau_1\,,\mu_1)=(\tau\,,\mu)B$, $M=\mathbb Z\tau\oplus\mathbb Z\mu$ and
$M_1=\mathbb Z d\tau\oplus \mathbb Z 2s(q-1)\mu$. We thus have
\[
N\cong M/M_1\cong \mathbb Z/d\mathbb Z\oplus\mathbb Z/2s(q-1)\mathbb Z\;.
\]
By abuse of notation, we also write
\[
(\tau\,,\mu)=(\wt{\sigma}^2_p\,,\wt{\sigma}_q)P^{-1}=
(\wt{\sigma}^{2s}_p\wt{\sigma}^{t}_q\,,\,\wt{\sigma}_p^{-2v}\wt{\sigma}^u_q)\;.
\]
Then $\tau\,,\mu$ are of order $d\,,2s(q-1)$ respectively, and $N$ is a direct sum of
$<\tau>$ and $<\mu>$. We have $\wt\sigma_p^2=\tau^u\mu^{-t}$ and $\wt\sigma_q=\tau^v\mu^s$.
When $v_2(p-1)>v_2(q-1)$, we get the structure of $N$ in the same way. So in the case (C)
we have
\[
N\cong\begin{cases}
\mathbb Z/d\mathbb Z\oplus\mathbb Z/2s(q-1)\mathbb Z\ &~~\text{ if }~~v_2(p-1)\le v_2(q-1)\\
\mathbb Z/d'\mathbb Z\oplus\mathbb Z/2s'(p-1)\mathbb Z\ &~~\text{ if }~~v_2(p-1)> v_2(q-1),
\end{cases}\eqno(C2.2)
\]
where $d=\gcd(\frac{p-1}2,q-1), s=(p-1)/2d$ and $d'=\gcd(p-1,\frac{q-1}2), s'=(q-1)/2d'$.
Now we summarize our results in the following
\begin{prop}\label{thm2p1}
The abelian subgroup $N$ of the group $\wt G$ of index 2 defined in (A2.1), (B2.1) and (C2.1)
has the structure described in (A2.2), (B2.2) and (C2.2) in the cases (A), (B) and (C), respectively.
In particular, every irreducible representation of $\wt G$ has dimension 1 or 2.
\end{prop}
\section{2-dimensional representations}
We determine all irreducible representations of $\wt G$ in this section. We will use
some basic facts from representation theory freely. For the details, see [6].
It is well-known that the 1-dimensional representations of $\wt G$ correspond bijectively to
those of the maximal abelian quotient $G$ of $\wt G$, which are Dirichlet characters. So we mainly
construct the 2-dimensional irreducible representations of $\wt G$. From the dimension formula of
all irreducible representations, we see that $\wt G$ has $|G|/4$ irreducible representations of
dimension 2, up to isomorphism. Let $N$ be the subgroup of $\wt G$ defined in last section.
Let $\wt G=N\cup\sigma N$ be a decomposition of cosets.
If $\rho:N\rightarrow\Bbb C^*$ is a representation of $N$, the induced representation $\wt\rho$
of $\rho$ is a representation of $\wt G$ of dimension $2$. The space of the representation
$\wt\rho$ is $V=\text{Ind}_N^{\wt G}(\Bbb C)=\Bbb C[\wt G]\otimes_{\Bbb C[N]}\Bbb C$ with basis
$e_1=1\otimes 1$ and $e_2=\sigma\otimes 1$. The group homomorphism
\[
\wt\rho : \wt G\longrightarrow \GL(V)\simeq\GL_2(\mathbb C)
\]
is given under the basis by
\begin{equation}\label{eq2p2}
\wt\rho(\wt\sigma)=\matr{\rho(\wt\sigma)}{\rho(\wt\sigma\sigma)}
{\rho(\sigma^{-1}\wt{\sigma})}{\rho(\sigma^{-1}\wt{\sigma}\sigma)},\quad\;\forall\;\wt\sigma\in\wt
G,
\end{equation}
where $\rho(\wt\sigma)=0$ if $\wt\sigma\notin N$. The representation $\wt\rho$ is irreducible
if and only if $\rho\not\cong \rho^{\tau}$ for every $\tau\in\wt G\setminus N$,
where $\rho^{\tau}$ is the conjugate representation of $\rho$ defined by
\[
\rho^{\tau}(x)=\rho(\tau^{-1}x\tau)\;,\;\;\forall\;x\in N\,.
\]
Since $N$ is abelian, we only need to check $\rho\not\cong \rho^{\sigma}$.
Now we begin to construct all 2-dimensional irreducible representations of $\wt G$.
As in last section,
we consider the three cases separately. In addition, we consider the case when $p$ and $q$
are odd prime numbers in details, and only state the results in the cases when $p=-1$ or $2$.
\noindent {\bf 3.1. Case A.} Assume $p>2$. We have in this case
$N=\langle\wt{\sigma}_p\,,\,\wt{\sigma}^2_q\,,\varepsilon\rangle$
and
\[
N\cong\mathbb Z/(p-1)\mathbb Z\oplus\mathbb Z/((q-1)/2)\mathbb Z\oplus\mathbb Z/2\mathbb Z\;.
\]
Every irreducible representation of $N$ can be written as
$\rho_{ijk}: N\longrightarrow\mathbb C^*$ with
\[
\rho_{ijk}(\wt{\sigma}_p)=\zeta^i_{p-1}\;;\;
\rho_{ijk}(\wt{\sigma}^2_q)=\zeta^{2j}_{q-1}\;;\;
\rho_{ijk}(\varepsilon)=(-1)^k\;.
\]
where $0\le i<p-1,~0\le j<\frac{q-1}2$ and $k=0,1$.
Since $\wt G=N\cup\wt\sigma_q N$ and $\rho^{\wt{\sigma}_q}_{ijk}(\wt{\sigma}_p)
=\rho_{ijk}(\varepsilon)\rho_{ijk}(\wt{\sigma}_p)=(-1)^k\rho_{ijk}(\wt{\sigma}_p)$, we have
\[
\rho^{\wt{\sigma}_q}_{ijk}\not\cong\rho_{ijk}\Longleftrightarrow
k=1.
\]
Write $\rho_{ij}=\rho_{ij1}$. The induced representation $\wt\rho_{ij}:
\wt{G}\longrightarrow \mathrm{GL}_2(\mathbb C)$ of $\rho_{ij}$ is given by
\[
\wt\rho_{ij}(\wt{\sigma}_p)=\matr{\zeta_{p-1}^i}{0}{0}{-\zeta_{p-1}^i},~
\wt\rho_{ij}(\wt{\sigma}_q)=\matr{0}{\zeta^{2j}_{{q-1}}}{1}{0},~
\wt\rho_{ij}(\varepsilon)=-I,\eqno(A3.1)
\]
where $I$ is the identity matrix of degree 2. Since
\[
\wt\rho_{ij}(\wt{\sigma}^2_p)=\matr{\zeta^{2i}_{p-1}}{0}{0}{\zeta^{2i}_{p-1}}\quad\text{and}\quad
\wt\rho_{ij}(\wt{\sigma}^2_q)=\matr{\zeta^{2j}_{{q-1}}}{0}{0}{\zeta^{2j}_{{q-1}}},
\]
we see that the representations $\wt\rho_{ij}$ with $0\le i<\frac{p-1}2,~0\le j<\frac{q-1}2$
are irreducible and are not isomorphic to each other, by considering the values of the characters
of these representations at $\wt\sigma_p^2$ and $\wt\sigma_q^2$. The number of these representations
is $\frac{p-1}{2}\cdot\frac{q-1}{2}=\frac{|G|}{4}$. So they are all the irreducible
representations of $\wt G$ of dimension 2.
Similarly, when $p=-1$, all irreducible representations of $\wt G$ of dimension 2 are
$\wt\rho_{j}$ with $0\le j<\frac{q-1}2$, where
\[
\wt\rho_j(\wt\sigma_{-1})=\matr{1}{0}{0}{-1},\quad
\wt\rho_{j}(\wt{\sigma}_q)=\matr{0}{\zeta^{2j}_{{q-1}}}{1}{0},\quad
\wt\rho(\varepsilon)=-I\eqno(A3.2)
\]
and when $p=2$, all irreducible representations of $\wt G$ of dimension 2 are
$\bar\rho_{ij}$ with $0\le i\le 1$ and $0\le j<\frac{q-1}2$, where $\bar\rho_{ij}(\varepsilon)=-I$
and
\[
\bar\rho_{ij}(\wt\sigma_{-1})=(-1)^iI,~\bar\rho_{ij}(\wt\sigma_{2})=\matr{1}{0}{0}{-1},~
\bar\rho_{ij}(\wt{\sigma}_q)=\matr{0}{\zeta^{2j}_{{q-1}}}{1}{0}.
\eqno (A3.3)\]
\noindent {\bf 3.2 Case B.} Assume $p>2$ and $|\wt\sigma_q|=2|\sigma_q|$. Then
$N=\langle\wt{\sigma}_p\,,\,\wt{\sigma}^2_q\rangle$, and
\[
N\cong\mathbb Z/(p-1)\mathbb Z\oplus\mathbb Z/(q-1)\mathbb Z.
\]
Any irreducible representation of $N$ has the form $\rho_{ij}: N\longrightarrow\mathbb C^*$, where
\[
\rho_{ij}(\wt{\sigma}_p)=\zeta^i_{p-1}\;,\;
\rho_{ij}(\wt{\sigma}^2_q)=\zeta^j_{q-1}\;,\;
\rho_{ij}(\varepsilon)=\rho_{ij}(\wt{\sigma}^2_q)^{\frac{q-1}2}=(-1)^j\;,
\]
and $0\le i<p-1,~0\le j<q-1$. It is easy to check that
\[
\rho^{\wt{\sigma}_q}_{ij}\not\cong\rho_{ij}\Longleftrightarrow
j\equiv 1\;\pmod{2}\;.
\]
The induced representation $\wt\rho_{ij}:
\wt{G}\longrightarrow \mathrm{GL}_2(\mathbb C)$ of $\rho_{ij}$ with odd $j$ is given by
\[
\wt\rho_{ij}(\wt{\sigma}_p)=\matr{\zeta^i_{p-1}}{0}{0}{-\zeta^i_{p-1}},~~
\wt\rho_{ij}(\wt{\sigma}_q)=\matr{0}{\zeta^j_{q-1}}{1}{0}.\eqno(B3.1)
\]
Since
\[
\wt\rho_{ij}(\wt{\sigma}^2_p)=\matr{\zeta^{2i}_{p-1}}{0}{0}{\zeta^{2i}_{p-1}}\quad\text{and}\quad
\wt\rho_{ij}(\wt{\sigma}^2_q)=\matr{\zeta^{j}_{q-1}}{0}{0}{\zeta^{j}_{q-1}},
\]
we see that the representations $\wt\rho_{ij}$ with $0\le i<\frac{p-1}2$ and $0\le j< q-1,~2\nmid j$
are irreducible and are not isomorphic to each other. The number of these representations is
$\frac{|G|}4$. So they are all the irreducible representations of $\wt G$ of dimension 2.
Similarly, when $(p,q)=(-1,2)$, there is only one irreducible representation $\wt\rho_0$ of dimension 2
defined by
\[
\wt\rho_0(\wt\sigma_{-1})=\matr{1}{0}{0}{-1},\quad\text{and}\quad
\wt\rho_0(\wt\sigma_{2})=\matr{0}{-1}{1}{0}.\eqno(B3.2)
\]
When $p=-1$ and $q>2$, all irreducible representations of dimension 2 are $\wt\rho_{j}$ with
$0\le j< q-1,~2\nmid j$, where $\wt\rho_j$ is defined by
\[
\wt\rho_j(\wt\sigma_{-1})=\matr{1}{0}{0}{-1}\quad\text{and}\quad
\wt\rho_{j}(\wt{\sigma}_q)=\matr{0}{\zeta^j_{q-1}}{1}{0}. \eqno(B3.3)
\]
When $p=2$, all irreducible representations of dimension 2 are $\bar\rho_{ij}$ with
$0\le i\le 1$ and $0\le j< q-1,~2\nmid j$, where $\bar\rho_{ij}$ is defined by
\[
\bar\rho_{ij}(\wt\sigma_{-1})=(-1)^iI,~
\bar\rho_{ij}(\wt\sigma_{2})=\matr{1}{0}{0}{-1},~
\bar\rho_{ij}(\wt{\sigma}_q)=\matr{0}{\zeta^j_{q-1}}{1}{0}.
\eqno(B3.4)
\]
When $|\wt\sigma_p|=2|\sigma_p|$, all irreducible representations of dimension 2 are $\hat\rho_{ij}$
with $0\le i< p-1,~2\nmid i$ and $0\le j<\frac{q-1}2$, where $\hat\rho_{ij}$ is defined by
\[
\hat\rho_{ij}(\wt{\sigma}_p)=\matr{0}{\zeta^i_{p-1}}{1}{0},~~
\hat\rho_{ij}(\wt{\sigma}_q)=\matr{\zeta^j_{q-1}}{0}{0}{-\zeta^j_{q-1}}.\eqno(B3.5)
\]
\noindent {\bf 3.3. Case C.} Assume $v_2(p-1)\le v_2(q-1)$. Let
\[
d=\gcd(\frac{p-1}{2}\,,q-1)\;,\;s=\frac{p-1}{2d}\,,t=\frac{q-1}d\;;\;\;us+vt=1
\]
as before. We must have that $t$ is even and $u$ is odd. Let
$\tau=\wt{\sigma}^{2s}_p\cdot\wt{\sigma}^{t}_q$
and $\mu=\wt{\sigma}^{-2v}_p\cdot\wt{\sigma}^u_q$.
Then
$N=\langle\wt{\sigma}^2_p\,,\,\wt{\sigma}_q\rangle=\langle\tau\,,\mu\rangle$
and
\[
N\cong\mathbb Z/d\mathbb Z\oplus\mathbb Z/2s(q-1)\mathbb Z.
\]
Any irreducible representation $\rho_{ij}: N\longrightarrow\mathbb C^*$ is of the form
\[
\rho_{ij}(\tau)=\zeta^i_d=\zeta^{2s(q-1)i}_{(p-1)(q-1)}\quad\text{and}\quad
\rho_{ij}(\mu)=\zeta^j_{2s(q-1)}=\zeta^{dj}_{(p-1)(q-1)}.
\]
From $\wt\sigma_p^2=\tau^u\mu^{-t}$ and $\wt\sigma_q=\tau^v\mu^s$, we have
\[
\rho_{ij}(\wt{\sigma}^2_p)=\zeta^{2sui-j}_{p-1}\;;\;
\rho_{ij}(\wt{\sigma}_q)=\zeta^{2tvi+j}_{2(q-1)}\;;\;
\rho_{ij}(\varepsilon)=\rho_{ij}(\wt{\sigma}^2_p)^{\frac{p-1}2}=(-1)^j\;.
\]
It is easy to show
\[
\rho^{\wt{\sigma}_p}_{ij}\not\cong\rho_{ij}\Longleftrightarrow
j\equiv 1\;\pmod{2}\;.
\]
The induced representation $\wt\rho_{ij}:\wt{G}\longrightarrow \mathrm{GL}_2(\mathbb C)$
of $\rho_{ij}$ with odd $j$ is given by
\[
\wt\rho_{ij}(\tau)=\matr{\zeta^i_d}{0}{0}{\zeta^i_d}\;;\quad\;
\wt\rho_{ij}(\mu)=\matr{\zeta^j_{2s(q-1)}}{0}{0}{-\zeta^j_{2s(q-1)}}.
\]
Here in the first equality we used the fact that $t$ is even, and in the second equality
we used the fact that $u$ is odd. Furthermore we have
\[
\wt\rho_{ij}(\wt{\sigma}_p)=\matr{0}{\zeta^{2sui-j}_{p-1}}{1}{0}\;;\;
\wt\rho_{ij}(\wt{\sigma}_q)=\matr{\zeta^{2tvi+j}_{2(q-1)}}{0}{0}{-\zeta^{2tvi+j}_{2(q-1)}}.
\eqno(C3.1)\]
By considering the values of the character of $\wt\rho_{ij}$ at $\tau$ and $\mu^2$, we see that
all the representations $\wt\rho_{ij}$ with $0\le i<d$ and $0\le j<s(q-1),~2\nmid j$ are
irreducible and are not isomorphic to each other. The number of these representations is
$d\cdot \frac{s(q-1)}{2}=\frac{|G|}{4}$. So they are all the irreducible representations
of $\wt G$ of dimension 2.
Similarly, if $v_2(p-1)> v_2(q-1)$, we let
\[
d'=\gcd(p-1\,,\frac{q-1}2)\;,\;s'=\frac{p-1}{d}\,,t'=\frac{q-1}{2d}\;;\;\;u's'+v't'=1.
\]
Then all the irreducible representations of $\wt G$ of dimension 2 are $\hat\rho_{ij}$ with
$0\le i<d'$ and $0\le j<t'(p-1),~2\nmid j$, where $\hat\rho_{ij}$ is defined by
\[
\hat\rho_{ij}(\wt{\sigma}_p)=\matr{\zeta^{2s'u'i+j}_{2(p-1)}}{0}{0}{-\zeta^{2s'u'i+j}_{2(p-1)}};~~
\hat\rho_{ij}(\wt{\sigma}_q)=\matr{0}{\zeta^{2t'v'i-j}_{q-1}}{1}{0}\; .
\eqno(C3.2)
\]
Let R$^2(\wt G)$ be the set of all irreducible representations, up to isomorphism, of
$\wt G$ of dimension 2. As a summary, we have proved the following.
\begin{thm} All 2-dimensional irreducible representations of $\wt G$ are induced from the
representations of $N$. In detail, we have
In the case (A)
\[
\mathrm{R}^2(\wt G)=\begin{cases}
\{\wt\rho_j\mid 0\le j<\frac{q-1}2\}&\mathrm{ if }~p=-1\\
\{\bar\rho_{ij}\mid i=0,1,~0\le j<\frac{q-1}2\} &\mathrm{ if }~p=2\\
\{\wt\rho_{ij}\mid 0\le i<\frac{p-1}2,~0\le j<\frac{q-1}2\} &\mathrm{ if }~p>2,
\end{cases}
\]
where $\wt\rho_j,~\bar\rho_{ij}$ and $\wt\rho_{ij}$ are defined in (A3.2), (A3.3) and
(A3.1) respectively.
In the case (B)
\[
\mathrm{R}^2(\wt G)=\begin{cases}
\{\wt\rho_0\}&\mathrm{ if }~(p,q)=(-1,2)\\
\{\wt\rho_j\mid 0\le j<q-1,~2\nmid j\}&\mathrm{ if }~p=-1,~q>2\\
\{\bar\rho_{ij}\mid i=0,1,~0\le j<q-1,~2\nmid j\} &\mathrm{ if }~p=2\\
\{\hat\rho_{ij}\mid 0\le i<p-1,~2\nmid i,~0\le j<\frac{q-1}2\} &\mathrm{ if }~|\wt\sigma_p|=2|\sigma_p|\\
\{\wt\rho_{ij}\mid 0\le i<\frac{p-1}2,~0\le j<q-1,~2\nmid j\} &\mathrm{ otherwise },
\end{cases}
\]
where $\wt\rho_0,~\wt\rho_j,~\bar\rho_{ij},~\hat\rho_{ij}$ and $\wt\rho_{ij}$ are defined in
(B3.2), (B3.3), (B3.4), (B3.5) and (B3.1) respectively.
In the case (C)
\[
\mathrm{R}^2(\wt G)=\begin{cases}
\{\wt\rho_{ij}\mid 0\le i<d,~0\le j<s(q-1),~2\nmid j\} &\mathrm{ if }~v_2(p-1)\le v_2(q-1),\\
\{\hat\rho_{ij}\mid 0\le i<d',~0\le j<t'(p-1),~2\nmid j\} &\mathrm{ otherwise},
\end{cases}
\]
where $\wt\rho_{ij}$ and $\hat\rho_{ij}$ are defined in (C3.1) and (C3.2) respectively.
\end{thm}
\section{The Frobenious maps}
This section is a preparation for the next section to compute the Artin $L$-functions of the
quasi-cyclotomic fields $\wt K$ when $p=-1$. For a prime number $\ell$ which is unramified
in $\wt K/K$, let $I_\ell$ (resp. $\wt I_\ell$) be the inertia group of $\ell$ in the extension
$K/\Bbb Q$ (resp. $\wt K/\Bbb Q$). Let ${\mathrm{Fr}}_\ell$ be the Frobenious automorphism of $\ell$
in $G/I_\ell$ and $\wt{\mathrm{Fr}}_\ell$ the Frobenious automorphism of $\ell$ in $\wt G/\wt I_\ell$
associated to some prime ideal over $\ell$. In this section we determine $\wt{\mathrm{Fr}}_\ell$ by
Fr$_\ell$ for $\ell=2$.
From now on, we always assume that $p=-1$, namely, $K=\Bbb Q(\zeta_{4q})$ and $\wt K=K(\sqrt[4]{q^*})$.
For a prime number $\ell$, we say that $\ell$ is ramified (resp. inertia, splitting)
in the relative quadratic extension $\tilde K/K$
if the prime ideals of $K$ over $\ell$ are ramified (resp. inertia, splitting) in $\tilde K$.
In [Sect.5, 7] we have determined the decomposition nature of odd prime numbers in $\wt K/K$. Now we determine
the decomposition nature of 2 in $\wt K/K$.
\begin{prop} If $q=2$, then 2 is ramified in $\wt K/K$. If $q$ is odd, then 2 is unramified in $\wt K/K$
if and only if $(\frac 2q)=1$, and in this case 2 splits in $\wt K/K$ if $q^*\equiv 1\mathrm{mod} 16$ and is inertia
in $\wt K/K$ if $q^*\equiv 1\mathrm{mod} 8$ but $q^*\not\equiv 1\mathrm{mod} 16$.
\end{prop}
\begin{proof} We first consider the case $q=2$. The unique prime ideal of $K$ over 2 is the principal
ideal generated by $\pi:=1-\zeta_8$. Since the ramification degree of 2 in $K/\Bbb Q$ is 4 and
$\sqrt 2=\pi(\pi+2\zeta_8)\zeta_8$, we have that 2 is ramified in $\wt K/K$ if and only if
$x^2\equiv\sqrt 2\mathrm{mod}\pi^{10}$ is not solvable in the ring $O_K$ of the integers of $K$ by [2], which
is equivalent to that $(1+\frac 2\pi\zeta_8)\zeta_8\mathrm{mod}\pi^{8}$ is not a square. Since $2=u\pi^4$ for some
unit $u$, we have
\[
(1+\frac 2\pi\zeta_8)\zeta_8\equiv\zeta_8\equiv (1-\pi)\mathrm{mod}\pi^3,
\]
namely $(1+\frac 2\pi\zeta_8)\zeta_8\mathrm{mod}\pi^3$ is not a square. So 2 is ramified in $\wt K/K$.
Now we assume that $q$ is odd. Let $\pi_2=1-\zeta_4$. Then 2 is unramified in $\wt K/K$ if and only
if $x^2\equiv\sqrt{q^*}\mathrm{mod}\pi_2^4$ is solvable in $O_K$. Furthermore, 2 splits in $\wt K/K$ if and
only if $x^2\equiv\sqrt{q^*}\mathrm{mod}\pi_2^5$ is solvable in $O_K$. By Gauss sum we have
\[
\sqrt{q^*}=\sum_{a=1}^{q-1}(\frac aq)\zeta_q^a=1+2\sum_{(\frac aq)=1}\zeta_q^a.
\]
Let $\alpha=\sum_{(\frac aq)=1}\zeta_q^a,~\beta=\sum_{(\frac aq)=1}\zeta_{2q}^a$, and
$\gamma=\sum_{(\frac aq)=1}\sum_{(\frac bq)=1,~a<b}\zeta_{2q}^{a+b}$,
where in the summations $a,b$ run over $1,2,\cdots,q-1$.
Then $\alpha=\beta^2-2\gamma$, from which and the equality $2=\pi_2^2-\pi_2^3$, we have
\[\begin{split}
\sqrt{q^*}&=1+2\beta^2-4\gamma=1+\pi_2^2\beta^2-\pi_2^3\beta^2-4\gamma\\
&\equiv(1+\pi_2\beta)^2-\pi_2^3(\beta+\beta^2)+\pi_2^4(\beta-\gamma)\\
&\equiv(1+\pi_2\beta)^2-\pi_2^3(\alpha+\beta)+\pi_2^4(\beta+\gamma)\mathrm{mod}\pi_2^5.
\end{split}\]
Since $\zeta_{2q}=-\zeta_q^{-\frac{q-1}2}=-\zeta_q^t$, where $t$ is the inverse of 2 in $(\mathbb Z/q\mathbb Z)^*$,
we see $\beta=\sum_{(\frac aq)=1}(-1)^a\zeta_{q}^{ta}\equiv\sum_{(\frac aq)=1}\zeta_{q}^{ta}\mathrm{mod} 2$.
So if $(\frac 2q)=1$ we have $\alpha\equiv\beta\mathrm{mod} 2$ and thus $2$ is unramified in $\wt K/K$,
and if $(\frac 2q)=-1$ we have $\alpha+\beta\equiv\sum_{a=1}^{q-1}\zeta_q^a=-1\mathrm{mod} 2$ and thus
$2$ is ramified in $\wt K/K$.
Now we assume $(\frac 2q)=1$. Then $\sqrt{q^*}\mathrm{mod}\pi_2^5$ is a square if and only if
$\pi_2\mid\beta+\gamma$. We consider $2(\beta+\gamma)$. Since $\alpha\equiv\beta\mathrm{mod} 2$, we have
\[
2(\beta+\gamma)=2\beta+\beta^2-\alpha\equiv \alpha(\alpha+1)\mathrm{mod} 4.
\]
From $\sqrt{q^*}=1+2\alpha$, we see $\alpha(\alpha+1)=\frac{q^*-1}4$. Since $8\mid q^*-1$ under
the assumption $(\frac 2q)=1$, we have $\beta+\gamma\equiv\frac{q^*-1}8\mathrm{mod} 2$. So
$\pi_2\mid\beta+\gamma$ if and only if $\pi_2\mid\frac{q^*-1}8$, namely $2\mid\frac{q^*-1}8$.
We complete the proof.
\end{proof}
By the way, we have determined the ring $O_{\wt K}$ of the integers of $\wt K$. In fact, we have
\begin{cor}
Assume that $q$ is an odd prime number. Let $t_q=(q-1)/4$ if $q\equiv 1\mathrm{mod} 4$, and
$t_q=(q-3)/4$ if $q\equiv 3\mathrm{mod} 4$. Then
\[
O_{\wt K}=\begin{cases}
\Bbb Z\left[\zeta_{4q},\frac{\sqrt[4]{q^*}+1+\pi_2\beta}{\pi_2(\sin[\frac 1q])^{t_q}}\right]
&\mathrm{if}~ (\frac 2q)=-1\\
\Bbb Z\left[\zeta_{4q},\frac{\sqrt[4]{q^*}+1+\pi_2\beta}{2(\sin[\frac 1q])^{t_q}}\right]
&\mathrm{if}~ (\frac 2q)=1.
\end{cases}\]
\end{cor}
\begin{proof}
See [Th.2, 7].
\end{proof}
Now we assume that 2 is unramified in $\wt K/K$. Let Fr$_2\in G$ such
that Fr$_2(\zeta_4)=1$ and Fr$_2(\zeta_q)=\zeta_q^2$. It is a Frobenious element of 2 in
$G$ modulo $I_2$. We have Fr$_2=\sigma_2^{b_2}$ with $2\mid b_2$ for $(\frac 2q)=1$. Thus
$\wt{\text{Fr}}_2=\wt\sigma_2^{b_2}$ or $\wt{\text{Fr}}_2=\wt\sigma_2^{b_2}\varepsilon$.
We need to determine $\wt{\text{Fr}}_2$ completely. Since $(\frac 2q)=1$, we have
\[
\sqrt{q^*}\equiv(1+\pi_2\alpha)^2+\pi_2^4(\beta+\gamma)\mathrm{mod}\pi_2^5.
\]
Write $u=1+\pi_2\alpha$ for simplicity. Since $\sqrt{q^*}\equiv u^2\mathrm{mod}\pi_2^4$, we see
$\frac{\sqrt[4]{q^*}-u}2\in O_{\wt K}$. Let $\wp$ be the prime ideal of $\wt K$ over 2
associated to $\wt{\text{Fr}}_2$. By the definition, we have
\[
\wt{\text{Fr}}_2\left(\frac{\sqrt[4]{q^*}-u}2\right)\equiv\left(\frac{\sqrt[4]{q^*}-u}2\right)^2
\equiv(\beta+\gamma)+\frac{\sqrt[4]{q^*}-u}2\mathrm{mod}\wp.
\]
On the other hand, since $\wt\sigma_q^{b_2}(\sqrt[4]{q^*})=(-1)^{\frac{b_2}2}\sqrt[4]{q^*}$ and
$\wt\sigma_q^{b_2}(u)=u$ for $2\mid b_2$, we have
\[
\wt\sigma_q^{b_2}(\frac{\sqrt[4]{q^*}-u}2)=\frac{(-1)^{\frac{b_2}2}\sqrt[4]{q^*}-u}2
\]
and
\[
\wt\sigma_q^{b_2}\varepsilon(\frac{\sqrt[4]{q^*}-u}2)=\frac{(-1)^{\frac{b_2}2+1}\sqrt[4]{q^*}-u}2.
\]
So if $2\mid\frac{b_2}2$ we have $\wt{\text{Fr}}_2=\wt\sigma_2^{b_2}$ if and only if
$\pi_2\mid\beta+\gamma$ (namely 2 splits in $\wt K/K$), and if $2\nmid\frac{b_2}2$ we have
$\wt{\text{Fr}}_2=\wt\sigma_2^{b_2}$ if and only if $\pi_2\nmid\beta+\gamma$ (namely 2 is inertia
in $\wt K/K$). In the case $q\equiv 3\mathrm{mod} 4$, we can always assume that $2\nmid\frac{b_2}2$,
since if $4\mid b_2$, we may replace $b_2$ by $b_2+(q-1)$. In the case $q\equiv 1\mathrm{mod} 4$, we have
$2\mid\frac{b_2}2\Longleftrightarrow 2^{\frac{q-1}4}\equiv 1\mathrm{mod} q\Longleftrightarrow q$ has the
form $A^2+64B^2$ for $A,B\in\Bbb Z$, by the Exercise 28 in Chap.5 in [5]. So we get the following result
\begin{prop} Assume that 2 is unramified in $\wt K/K$. Let $\mathrm{Fr}_2=\sigma_2^{b_2}$. We have
$2\mid b_2$. If $q\equiv 3\mathrm{mod} 4$, we always assume $b_2\equiv 2\mathrm{mod} 4$. Let $P_0$ be the set of
the prime numbers of the form $A^2+64B^2$ with $A,B\in\Bbb Z$. Then we have
\[
\wt{\mathrm{Fr}}_2=\begin{cases}\wt\sigma_2^{b_2}&\mathrm{if}~~q\not\in P_0,16\nmid q^*-1,
~\mathrm{or}~q\in P_0,16\mid q^*-1\\
\wt\sigma_2^{b_2}\varepsilon &\mathrm{if}~~q\in P_0,16\nmid q^*-1,
~\mathrm{or}~q\not\in P_0,16\mid q^*-1.
\end{cases}
\]
\end{prop}
The following lemma is useful in next section.
\begin{lemma} We have $\varepsilon\in\wt I_\ell$ if and only if $\ell$ is ramified in
$\wt K/K$.
\end{lemma}
\begin{proof}
The canonical projection $\wt G\longrightarrow G\simeq\wt G/\langle\varepsilon\rangle$
induces a surjective homomorphism $\wt{I}_\ell\longrightarrow I_\ell$ which implies
the isomorphism $\wt{I}_\ell/<\varepsilon>\cap\wt{I}_\ell\cong I_\ell$. Thus
$\ell$ is ramified in $\wt K/K\Longleftrightarrow |\wt{I}_\ell|=2|I_\ell|
\Longleftrightarrow |\wt{I}_\ell\cap<\varepsilon>|=2\Longleftrightarrow\varepsilon\in\wt{I}_\ell$.
\end{proof}
\section{The Artin $L$-functions}
In this section we compute the Artin $L$-functions of the quasi-cyclotomic fields
$\wt K=\Bbb Q(\zeta_{4q},\sqrt[4]{q^*})$.
The $L$-functions associated to the 1-dimensional representations of $\wt G$ are the well-known
Dirichlet $L$-functions. So we mainly compute the $L$-functions associated to the 2-dimensional
irreducible representation of $\wt G$. Let $\varphi:\wt G\rightarrow\text{GL}(V)$ be a 2-dimensional
irreducible representations. The Artin $L$-function $L(\varphi,s)$ associated to $\varphi$ is
defined as the product of the local factors
\[
L(\varphi,s)=\prod_{\ell:\mathrm{prime}}L_\ell(\varphi,s),
\]
where the local factors are defined as
$L_\ell(\varphi,s)=\mathrm{det}(1-\varphi(\wt{\mathrm{Fr}}_\ell)\ell^{-s}|V^{\wt I_\ell})^{-1}$.
Now we begin to compute them. First we notice that if $\ell$ is ramified in $\wt K/K$, then
$V^{\wt I_\ell}=0$ and $L_\ell(\varphi,s)=1$, which is due to the facts that
$\varepsilon\in\wt I_\ell$ by Lem.4.4 and $\varphi(\varepsilon)=-I$ for any irreducible representation
$\varphi$ of $\wt G$ by Th.3.1.
\noindent {\bf 5.1 The case $q=2$.} By section 3, there is only one 2-dimensional representation
$\wt\rho_0$ in this case, which is defined as
\[
\wt\rho_0(\wt\sigma_{-1})=\matr{1}{0}{0}{-1},\quad\text{and}\quad
\wt\rho_0(\wt\sigma_{2})=\matr{0}{-1}{1}{0}.
\]
Since 2 is ramified in $\wt K/K$, we have $L_2(\wt\rho_0,s)=1$. Assume that $\ell$ is an odd prime
number.
If $\ell\equiv 7\mathrm{mod} 8$, then Fr$_\ell=\sigma_{-1}$ and thus $\wt{\text{Fr}}_\ell=\wt\sigma_{-1}$
or $\wt\sigma_{-1}\varepsilon$. In any case we have
\[
L_\ell(\wt\rho_0,s)=\mathrm{det}\left(I\pm\matr{1}{0}{0}{-1}\ell^{-s}\right)^{-1}
=(1-\ell^{-2s})^{-1}.
\]
If $\ell\equiv 5\mathrm{mod} 8$, then Fr$_\ell=\sigma_{2}$ and thus $\wt{\text{Fr}}_\ell=\wt\sigma_{2}$
or $\wt\sigma_{2}\varepsilon$. We have
\[
L_\ell(\wt\rho_0,s)=\mathrm{det}\left(I\pm\matr{0}{-1}{1}{0}\ell^{-s}\right)^{-1}
=(1+\ell^{-2s})^{-1}.
\]
If $\ell\equiv 3\mathrm{mod} 8$, then Fr$_\ell=\sigma_{-1}\sigma_{2}$ and thus
$\wt{\text{Fr}}_\ell=\wt\sigma_{-1}\wt\sigma_{2}$
or $\wt\sigma_{-1}\wt\sigma_{2}\varepsilon$. We have
\[
L_\ell(\wt\rho_0,s)=\mathrm{det}\left(I\pm\matr{1}{0}{0}{-1}\matr{0}{-1}{1}{0}\ell^{-s}\right)^{-1}
=(1-\ell^{-2s})^{-1}.
\]
If $\ell\equiv 1\mathrm{mod} 8$, then Fr$_\ell=1$ and thus $\wt{\text{Fr}}_\ell=1$ or $\varepsilon$.
In this case, we must determine $\wt{\text{Fr}}_\ell$ completely.
Since $\wt{\text{Fr}}_\ell(\sqrt[4]2)\equiv(\sqrt[4]2)^\ell\mathrm{mod}\wp$ for the prime ideal $\wp$
of $\wt K$ over $\ell$ associated to $\wt{\text{Fr}}_\ell$, we have $\wt{\text{Fr}}_\ell=1$
if $2^{\frac{\ell-1}4}\equiv 1\mathrm{mod}\ell$, and $\wt{\text{Fr}}_\ell=\varepsilon$ if
$2^{\frac{\ell-1}4}\equiv -1\mathrm{mod}\ell$. As in last section, we have that for $\ell\equiv 1\mathrm{mod} 8$,
$2^{\frac{\ell-1}4}\equiv 1\mathrm{mod}\ell$ if and only if $\ell\in P_0$. So we have
\[
L_\ell(\wt\rho_0,s)=\begin{cases}
(1-\ell^{-s})^{-2}\quad &\text{if }\ell\in P_0\\
(1+\ell^{-s})^{-2}\quad &\text{otherwise. }
\end{cases}
\]
We get the Artin $L$-function in the case $(p,q)=(-1,2)$ as follows.
\begin{equation}\label{eq2p3}\begin{split}
L(\wt\rho_0,s)=&\prod_{\ell\equiv 3\text{ or }7(8)}(1-\ell^{-2s})^{-1}\cdot
\prod_{\ell\equiv 5(8)}(1+\ell^{-2s})^{-1}\\
&\times\prod_{\ell\in P_0}(1-\ell^{-s})^{-2}\cdot
\prod_{\ell\equiv 1(8),~\ell\not\in P_0}(1+\ell^{-s})^{-2}.
\end{split}\end{equation}
\noindent {\bf 5.2 The case $q$ is odd.} All 2-dimensional irreducible representations
of $\wt G$ are $\wt\rho_j$ with $0\le j<q-1,2\mid j$ if $q\equiv 1\mathrm{mod} 4$, and
$0\le j<q-1,2\nmid j$ if $q\equiv 3\mathrm{mod} 4$, where $\wt\rho_j$ is defined by
\[
\wt\rho_j(\wt\sigma_{-1})=\matr{1}{0}{0}{-1},\quad\wt\rho_{j}(\wt{\sigma}_q)
=\matr{0}{\zeta^{j}_{q-1}}{1}{0}\quad\text{and}\quad\wt\rho_{j}(\varepsilon)=-I.
\]
We first determine the local factors $L_\ell(\wt\rho_j,s)$ for $\ell\neq 2,q$. For such
$\ell$, we have $V^{\wt I_\ell}=V$. Let Fr$_\ell=\sigma_{-1}^{a_\ell}\sigma_q^{b_\ell}$,
which is equivalent to
$\ell\equiv(-1)^{a_\ell}\mathrm{mod} 4$ and $\ell\equiv g^{b_\ell}\mathrm{mod} q$, where $g$ is the primitive
root $\mathrm{mod} q$ associated to $\sigma _q$. It is easy to compute that
\[
\wt\rho_j(\wt\sigma_q^{b_\ell})=\matr{0}{\zeta_{q-1}^{j}}{1}{0}^{b_\ell}=\begin{cases}
\zeta_{2(q-1)}^{jb_\ell}I&\quad\text{ if }2\mid b_\ell\\
\matr{0}{\zeta_{2(q-1)}^{j(b_\ell+1)}}{\zeta_{2(q-1)}^{j(b_\ell-1)}}{0}
&\quad\text{ if }2\nmid b_\ell.
\end{cases}
\]
Furthermore, we have
\[
\text{det}(I-\wt\rho_j(\wt\sigma_{-1}^{a_\ell}\wt\sigma_q^{b_\ell})\ell^{-s})=\begin{cases}
(1-\zeta_{2(q-1)}^{jb_\ell}\ell^{-s})^2 &\text{ if }a_\ell=0,~2\mid b_\ell\\
1-\zeta_{q-1}^{jb_\ell}\ell^{-2s} &\text{ if }a_\ell=0,~2\nmid b_\ell\\
&\text{ or }a_\ell=1,~2\mid b_\ell\\
1+\zeta_{q-1}^{jb_\ell}\ell^{-2s} &\text{ if }a_\ell=1,~2\nmid b_\ell
\end{cases}
\]
and
\[
\text{det}(I+\wt\rho_j(\wt\sigma_{-1}^{a_\ell}\wt\sigma_q^{b_\ell})\ell^{-s})=\begin{cases}
(1+\zeta_{2(q-1)}^{jb_\ell}\ell^{-s})^2 &\text{ if }a_\ell=0,~2\mid b_\ell\\
1-\zeta_{q-1}^{jb_\ell}\ell^{-2s} &\text{ if }a_\ell=0,~2\nmid b_\ell\\
&\text{ or }a_\ell=1,~2\mid b_\ell\\
1+\zeta_{q-1}^{jb_\ell}\ell^{-2s} &\text{ if }a_\ell=1,~2\nmid b_\ell.
\end{cases}
\]
So we get
\[
L_\ell(\wt\rho_j,s)=(1-\zeta_{q-1}^{jb_\ell}\ell^{-2s})^{-1}
\]
if $\ell\equiv 1\mathrm{mod} 4$ and $\ell\equiv g^{b_\ell}\mathrm{mod} q$ with $2\nmid b_\ell$, or if
$\ell\equiv 3\mathrm{mod} 4$ and $\ell\equiv g^{b_\ell}\mathrm{mod} q$ with $2\mid b_\ell$, and
\[
L_\ell(\wt\rho_j,s)=(1+\zeta_{q-1}^{jb_\ell}\ell^{-2s})^{-1}
\]
if $\ell\equiv 3\mathrm{mod} 4$ and $\ell\equiv g^{b_\ell}\mathrm{mod} q$ with $2\nmid b_\ell$.
To compute the local factors when $\ell\equiv 1\mathrm{mod} 4$ and $\ell\equiv g^{b_\ell}\mathrm{mod} q$
with $2\mid b_\ell$, we must determine $\wt{\text{Fr}}_\ell$ completely. Since $(\frac\ell q)=1$,
we have $(\frac q\ell)=1$ and $(\frac{q^*}\ell)=1$. Let $\alpha_\ell\in\mathbb Z$ such that
$\alpha_\ell^2\equiv q^*\mathrm{mod}\ell$. From
$\wt\sigma_q^{b_\ell}(\sqrt[4]{q^*})=(-1)^{\frac{b_\ell}2}\sqrt[4]{q^*}$, we see
$\wt{\text{Fr}}_\ell=\wt\sigma_q^{b_\ell}$ if $(\frac{\alpha_\ell}\ell)=(-1)^{\frac{b_\ell}2}$, and
$\wt{\text{Fr}}_\ell=\wt\sigma_q^{b_\ell}\varepsilon$ if $(\frac{\alpha_\ell}\ell)=(-1)^{\frac{b_\ell}2+1}$.
So when $\ell\equiv 1\mathrm{mod} 4$ and $\ell\equiv g^{b_\ell}\mathrm{mod} q$ with $2\mid b_\ell$, we have
\[
L_\ell(\wt\rho_j,s)=\begin{cases}
(1-\zeta_{2(q-1)}^{jb_\ell}\ell^{-s})^{-2}&\text{ if }(\frac{\alpha_\ell}\ell)=(-1)^{\frac{b_\ell}2}\\
(1+\zeta_{2(q-1)}^{jb_\ell}\ell^{-s})^{-2}&\text{ if }(\frac{\alpha_\ell}\ell)=(-1)^{\frac{b_\ell}2+1}.
\end{cases}
\]
Next we compute the local factors $L_2(\wt\rho_j,s)$ and $L_q(\wt\rho_j,s)$. When $(\frac 2q)=-1$,
we know from last section that 2 is ramified in $\wt K/K$. So $L_2(\wt\rho_j,s)=1$ in this case. Now
we assume $(\frac 2q)=1$. Since $I_2=<\sigma_{-1}>$ and 2 is unramifed in $\wt K/K$, we have
$\wt I_2=<\wt\sigma_{-1}>$ or $\wt I_2=<\wt\sigma_{-1}\varepsilon>$.
The matrixes $I+\wt\rho_j(\wt\sigma_{-1})$ and $I+\wt\rho_j(\wt\sigma_{-1}\varepsilon)$ have rank 1.
So $V^{\wt I_2}$ has dimension 1. Write Fr$_2=\sigma_2^{b_2}$ with $2\mid b_2$. As in last section,
we always assume $b_2\equiv 2\mathrm{mod} 4$ if $q\equiv 3\mathrm{mod} 4$. Recall that $P_0$ be the set of the prime
numbers of the form $A^2+64B^2$ with $A,B\in\Bbb Z$. Since
$\wt\rho_j(\wt{\sigma}_2^{b_2})=\zeta_{2(q-1)}^{jb_2}I$, by Prop.4.3 we have
\[
L_2(\wt\rho_j,s)=\begin{cases}1-\zeta_{2(q-1)}^{jb_2}2^{-s}&\mathrm{if}~~q\not\in P_0,16\nmid q^*-1,
~\mathrm{or}~q\in P_0,16\mid q^*-1\\
1+\zeta_{2(q-1)}^{jb_2}2^{-s} &\mathrm{if}~~q\in P_0,16\nmid q^*-1,
~\mathrm{or}~q\not\in P_0,16\mid q^*-1.
\end{cases}
\]
When $q\equiv 3\mathrm{mod} 4$, we know that $q$ is ramified in $\wt K/K$. So $L_q(\wt\rho_j,s)=1$ for
odd $j$ in this case. Assume $q\equiv 1\mathrm{mod} 4$. Since $I_q=<\sigma_q>$ and $q$ is unramifed in
$\wt K/K$, we have $\wt I_q=<\wt\sigma_q>$ or $\wt I_2=<\wt\sigma_q\varepsilon>$. Thus
$V^{\wt I_q}=0$ if $j\ne 0$, and $V^{\wt I_q}$ has dimension 1 if $j=0$.
The Frobenious map Fr$_q$ of $q$ in $G$ modulo $I_q$ is the identity map. So $\wt{\mathrm{Fr}}_q=1$ or
$\varepsilon$. In [Sect.5, 7] we have showed that $q$ splits in $\wt K/K$ if $q\equiv 1\mathrm{mod} 8$ and is inertia
if $q\equiv 5\mathrm{mod} 8$. So $\wt{\mathrm{Fr}}_2=1$ if $q\equiv 1\mathrm{mod} 8$ and $\wt{\mathrm{Fr}}_2=\varepsilon$
if $q\equiv 5\mathrm{mod} 8$. Thus we get
\[
L_q(\wt\rho_j,s)=\begin{cases}
1&\text{ if }j\ne 0\\
1-q^{-s}&\text{ if }j=0,~q\equiv 1\mathrm{mod} 8\\
1+q^{-s}&\text{ if }j=0,~q\equiv 5\mathrm{mod} 8.
\end{cases}\]
We have computed all the local factors. So we have
\begin{equation}\label{eq2p3}\begin{split}
L(\wt\rho_j,s)=&(1-u_q\zeta_{2(q-1)}^{jb_2}2^{-s})^{-1}(1-(-1)^{\frac{q-1}4}q^{-s})^{-n_j}\\
&\times\prod_{\ell\equiv 1,~2\nmid b_\ell\text{ or } \ell\equiv 3,~2\mid b_\ell}
(1-\zeta_{q-1}^{jb_\ell}\ell^{-2s})^{-1}\\
&\times\prod_{\ell\equiv 3,~2\nmid b_\ell}(1+\zeta_{q-1}^{jb_\ell}\ell^{-2s})^{-1}
\prod_{\ell\equiv 1,~2\mid b_\ell}(1-u_\ell\zeta_{2(q-1)}^{jb_\ell}\ell^{-s})^{-2},
\end{split}
\end{equation}
where $u_q=1$ if $q\not\in P_0,16\nmid q^*-1,
~\mathrm{or}~q\in P_0,16\mid q^*-1$ and $u_q=-1$ otherwise; $n_j=0$ if $j\ne 0$ and $n_0=1$;
and $u_\ell=(\frac{\alpha_\ell}\ell)(-1)^{\frac{b_\ell}2}$. Here in the products,
$"\equiv"$ means the congruence modulo 4.
\begin{thm} Except for the Dirichlet $L$-functions, all Artin $L$-functions of the
Galois extension $\wt K/\Bbb Q$ are explicitly given by (5.1) in the case $q=2$ and
by (5.2) in the case $q$ is odd, where in (5.2) $0\le j<q-1,~2\mid j$ if $q\equiv 1\mathrm{mod} 4$
and $0\le j<q-1,~2\nmid j$ if $q\equiv 3\mathrm{mod} 4$.
\end{thm}
\noindent {\bf 5.3 A formula.} Let $\zeta_{\wt K}(s)$ and $\zeta_K(s)$ be the Dedekind zeta functions of
$\wt K$ and $K$ respectively. By Artin's formula of the decomposition of Dedekind zeta functions,
we have
\[
\frac{\zeta_{\wt K}(s)}{\zeta_K(s)}=\prod_{\wt\rho_j}\prod_{\ell:\text{ prime}}
L_\ell(\wt\rho_j,s)^2,
\]
where $\wt\rho_j$ runs over all 2-dimensional irreducible representations of $\wt G$.
When $q=2$, there is only one 2-dimensional irreducible representation of $\wt G$. So the
square of (5.1) gives the formula. When $q$ is odd, by computing
$\prod_{\wt\rho_j}L_\ell(\wt\rho_j,s)$, we get the following
\begin{cor} For a prime number $\ell\ne q$, let $f_\ell=\frac{q-1}{\gcd(b_\ell,q-1)}$ be the order
of $\ell\mathrm{mod} q$ and let $g_\ell=\gcd(b_\ell,q-1)=\frac{q-1}{f_\ell}$. If $q\equiv 1\mathrm{mod} 4$, we have
\[\begin{split}
\frac{\zeta_{\wt K}(s)}{\zeta_K(s)}=&(1-u_q^{f_2}2^{-f_2s})^{-g_2}(1-(-1)^{\frac{q-1}4}q^{-s})^{-2}
\prod_{\ell\equiv 1,~2\nmid b_\ell\text{ or } \ell\equiv 3}(1-\ell^{-f_\ell s})^{-2g_\ell}\\
&\times\prod_{\ell\equiv 1,~2\mid b_\ell}(1-u_\ell^{f_\ell}\ell^{-f_\ell s})^{-2g_\ell},
\end{split}\]
and if $q\equiv 3\mathrm{mod} 4$, we have
\[\begin{split}
\frac{\zeta_{\wt K}(s)}{\zeta_K(s)}=&(1+u_q^{f_2}2^{-f_2s})^{-g_2}
\prod_{\ell\equiv 1,~2\nmid b_\ell}(1+\ell^{-f_\ell s})^{-2g_\ell}
\prod_{\ell\equiv 3}(1-\ell^{-2f_\ell s})^{-g_\ell}\\
&\times\prod_{\ell\equiv 1,~2\mid b_\ell}(1+u_\ell^{f_\ell}\ell^{-f_\ell s})^{-2g_\ell},
\end{split}\]
where $u_q$ and $u_\ell$ are as above.
\end{cor}
|
1,116,691,500,423 | arxiv | \section{Introduction}
The corrosion of a metal after the damage of its protective layer is a problem
of wide technological interest. The evolution of the
corrosion front is the result of a competition between localized dissolution
and passivation processes. The latter consists of the formation of a
passivation layer which reduces the corrosion rate and prevents a fast
propagation of the damage. However, the breakdown of the passivation layer
leads to the so-called pitting corrosion \cite{frankel}, with an increase in
the dissolution rate. It is generally accepted that the propagation of the pit
is preceded by a nucleation regime, and the transition between these regimes
takes place at the so-called pit initiation time or incubation time. This
characteristic time was experimentally observed for a long time
\cite{hoar,shibata}. In stainless steel, the process is often connected with
the presence of inclusions ($MnS$) from which the pits begin \cite{alkire}. In
weakly passivated materials, it often begins at surface defects and surface
inhomogeneities \cite{frankel}, with only a small fraction giving rise to
indefinitely developing pits. Some recent works determined relations between
the incubation times and physicochemical conditions in different processes of
technological interest (see. e. g. Refs.
\protect\cite{fukumoto,hassan,rehim,amin,zaky}).
There is also extensive literature covering the theoretical aspects of
pitting corrosion. In some cases, specific applications are considered, such as
stainless steel \cite{nagatani1,laycock1, laycock2}, while some works focus on
universal features of that process \cite{meakin1}. Among the models we may
also distinguish the ones based on stochastic approaches
\cite{nagatani1, meakin1, nagatani2, nagatani3, meakin2} from those based on
analytical
formulations of the corrosion problem \cite{laycock1, laycock2, digby,organ}.
These papers are devoted to the study of pit propagation, with the focus on the
pit shapes \cite{laycock1,meakin2,nagatani3} and their evolution or the
investigation of the interaction between pits \cite{organ}.
On the other hand, the transition between the nucleation regime and pit
propagation has attracted less attention. It motivated the recent study of a
stochastic model with mechanisms that may be involved in this transition
\cite{vautrin1,stafiej,passivity}.
These mechanisms are passivation/depassivation phenomena, generation of local pH
inhomogeneities by spatially separated cathodic and anodic reactions and
smoothing out of these inhomogeneities by diffusion. Simulation of the
corrosion process initiated by small damage to a protective surface showed
the existence of an incubation time separating a nucleation regime of slow
corrosion from a regime with a much higher growth rate of the corrosion front.
Qualitative features of the growing cavities in two-dimensional
simulations have already been addressed \cite{vautrin1,stafiej}. However,
a thorough analysis of the quantitative effects of different model parameters
is still lacking.
In this paper, we extend the study of this corrosion model by combining scaling
ideas and simulation results. Since the number of parameters of the original
model is large, our study focuses on their values in the most realistic ranges
for possible applications.
Thus, while keeping the model amenable for a combination of theoretical and
numerical work, we attempt to preserve the perspective of applications.
The damage is represented by two lattice sites initially exposed to the
environment,
which mimics local damage of a relatively cheap (not stainless steel) painted
material in its natural environment. This is certainly a situation of
practical interest.
The starting point of our scaling analysis is to estimate
the characteristic times of the physicochemical mechanisms involved in the
corrosion process. By a suitable matching of those characteristic
times, we estimate the incubation radius, which is defined as the effective
radius of the dissolved region at the incubation time. Such a
procedure resembles those recently used
to estimate crossover exponents in statistical growth models
\cite{rdcor,lam}. The incubation radius is related to model
parameters such as probabilities of spatially separated reactions (cathodic and
anodic), probabilities of acidity-enhanced depassivation and the diffusion
coefficients. Subsequently, the dependence of the average incubation
time on the model parameters is also analyzed.
The theoretical predictions are supported by simulation data for two dimensional
square lattices. This dimensionality is forced by computational limitations.
However, the physicochemical and geometrical arguments of the theoretical
analysis are independent of the underlying lattice structure. Thus we expect
that this independence extends to the results in a certain range of the model
parameters. This is important to justify the reliability of the model for a
mesoscopic description of corrosion phenomena.
The paper is organized as follows. In Sec. II we review the
statistical model and give a summary of the results of previous papers. In Sec.
III we present the scaling theory for the model, which predicts the average
radius of the dissolved cavity at the incubation time. In Sec. IV we compare
the theoretical predictions with simulation data. In Sec. V we discuss the
scaling properties of the incubation time. In Sec. VI we discuss the relations
of this model with other reaction-diffusion models and other models for pitting
corrosion. In Sec. VII we summarize our results and present our conclusions.
\section{The corrosion model}
\label {model}
\subsection{The electrochemical basis of the model}
\label{electrochemicalbasis}
In an acidic or neutral medium, the anodic dissolution of commonly used metallic
material such as steel, iron and aluminum, can be described by the reaction
\begin{equation}
Me + H_2O \to {MeOH}_{aq} + e^- + H^+ ,
\label{dissolacidic}
\end{equation}
where $Me$ represents the bulk metal material and ${MeOH}_{aq}$ represents
products which may be composed of hydroxides, oxides and water.
For example, the mechanisms of $Ni$ dissolution in acid phosphate solutions was
recently discussed in Ref. \protect\cite{munoz}. However, the precise
chemical nature of these products is not important for the model presented here,
but only the assumption that they are detached from the corroding material
and belong to its environment.
On the other hand, in a basic environment the surface is expected to
repassivate. In other words, the chemical species produced there are adherent to
the surface and compact enough to prevent further dissolution. We represent it
by
the reaction
\begin{equation}
Me + {OH}^- \to {MeOH}_{ads} + e^- ,
\label{dissolbasic}
\end{equation}
where ${MeOH}_{ads}$ refers to the adherent species forming the passive layer at
the metal surface.
In both cases the pH of the solution at the locus of the reaction
decreases.
In acidic deoxygenated media the associated cathodic reaction is
\begin{equation}
H^+ + e^- \to \frac{1}{2} H_ 2 ,
\label{cathodicacidic}
\end{equation}
while in a basic environment we have
\begin{equation}
H_2O + e^- \to \frac{1}{2} H_ 2 + {OH}^- .
\label{cathodicbasic}
\end{equation}
As expected, these reactions increase the local pH.
If anodic and cathodic reactions occur next to each other there
is a mutual compensation of their effect on pH and eventual neutralization.
Thus the
pairs of reactions (\ref{dissolacidic})-(\ref{cathodicacidic}) (in neutral or
acidic medium) or (\ref{dissolbasic})-(\ref{cathodicbasic}) (in basic medium)
do not alter pH of the solution at a significant lengthscale.
Consequently, they can be simply combined as:
(\ref{dissolacidic}-\ref{cathodicacidic})
\begin{equation}
Me + H_2O \to {MeOH}_{aq} + \frac{1}{2}H_2
\label{reacacidic}
\end{equation}
if the surrounding solution is acidic, or as
(\ref{dissolbasic}-\ref{cathodicbasic})
\begin{equation}
Me + H_2O \to {MeOH}_{ads} + \frac{1}{2}H_2
\label{reacbasic}
\end{equation}
if the surrounding solution is basic.
Reaction (\ref{reacbasic}), in which the oxidation of the metal is followed by
the deposition of a passive species, is much more frequent than the
reaction (\ref{reacacidic}), in which the product of the corrosion immediately
leaves the surface. In a neutral medium, the former also tends to be dominant.
However, the spreading of the electric signal in the metal is considered as
instantaneous when compared to the other processes. Therefore
oxidation (\ref{dissolacidic} or \ref{dissolbasic}) and reduction
(\ref{cathodicacidic}
or \ref{cathodicbasic}) may occur at distant points simultaneously. These
spatially separated
electrochemical (SSE) reactions change the local pH of the solution where they
occur
in favor of occurrence of the reactions of the same type at the location. On
the other
hand, pH inhomogeneities (local excess of $H^+$ over ${OH}^-$) may be
suppressed by diffusion, which tends to reimpose uniformity and brings about
neutralization.
Finally, it is also important to recall that the dissolution of the adsorbed
species (${MeOH}_{ads}$ in the above reactions) is very slow in a neutral
medium, and even more difficult in a basic one. For instance, simulation of
corrosion on steel by $CO_2$ at a high pH \cite{nesic} shows low diffusion
rates,
with the formation of very protective films.
However, the dissolution is significantly
enhanced in acidic medium, following the reaction
\begin{equation}
{MeOH}_{ads} + H^+ \to {Me}^+ + H_2O .
\label{dissolaggressive}
\end{equation}
As the anions ${Me}^+$ leave the surface, the metal is again exposed to the
corrosion agents.
\subsection{The statistical model}
\label{statisticalmodel}
Here we rephrase the model of Ref. \protect\cite{vautrin1}, which
amounts to describing the above processes at a mesoscopic level. Metal and
solution are represented in
a lattice, whose sites may assume six different states. These states represent
the presence of significant amounts of the most important chemical species,
while the chemical reactions are represented by stochastic rules for the changes
of those states after each time step.
The sites representing the bulk metal (unexposed to the corroding solution) are
labeled M and called metal or M sites. The so-called R (reactive) sites
represent the
"bare" metal exposed to the corrosive environment. We consider the passivation
layer at these points permeable enough to allow for anodic dissolution of the
metal. In contrast, the passivated
regions, labeled P sites, represent sites covered with a passive
layer which is compact enough to prevent their anodic dissolution. The other
lattice sites, labeled E, A and B, denote the neutral
environment, the acidic and basic regions, respectively (compared to previous
work on this model \cite{vautrin1,stafiej,passivity}, label C is here replaced
by B in
order to make a clearer
association with the basic character of the solution and emphasize the pH
role).
The evolution of the corrosion process amounts to
transformations of the interfacial sites (R and P) into solution sites (E, A or
B), followed by the conversion into R sites of those M sites that are put in
contact with the solution. This accounts for the displacement of
the interface. In the following, we will use the label S to denote a surface
site which can be either R or P.
The possible changes of site labels certainly depend on the pH of the
surrounding solution. Thus, for a given S site, our pH related scale will be
represented by the algebraic excess of A
sites over B sites among the nearest neighbors. We denote it by $N_{exc}$, so
that the pH decreases as $N_{exc}$ increases.
During the simulation of the process, SSE reactions, local vicinity reactions
and depassivation reactions are performed in this order. For each type of
reaction, the lists of R and/or P sites are looked up in random order, and the
decision of each one to undergo that reaction is taken with a prescribed
probability. These reactions are
followed by a certain number of random steps of A and B particles in the
solution and possible annihilation of their pairs. This series of events
takes place in one time unit.
The first set of transformations of the lattice sites account for the SSE
reactions. In one time unit, each R site may undergo an SSE reaction with a
probability $p_{SSE}$. In neutral or acidic medium ($N_{exc}\geq 0$), reaction
(\ref{dissolacidic}) is represented by
\begin{equation}
R\to A ,
\label{reacRA}
\end{equation}
and in basic medium ($N_{exc}<0$), reaction (\ref{dissolbasic}) is represented
by
\begin{equation}
R+B(nn) \to P+E(nn)
\label{reacRBPE}
\end{equation}
(here $(nn)$ refers to a B site which is neareast neighbor of
the R site).
Any of the above anodic reactions is possible only if there is another surface
site (R or P) which can mediate the associated cathodic process. This
corresponds to the
electrochemical reactions (\ref{cathodicacidic}-\ref{cathodicbasic}), and is
represented in the model by
\begin{equation}
S + A(nn) \to S + E(nn) ,
\label{reacSASE}
\end{equation}
or
\begin{equation}
S+E(nn) \to S+B(nn) .
\label{reacSESB}
\end{equation}
If none of the S sites has a nearest neighbor A or E, then the cathodic reaction
is
impossible, and consequently, the anodic one does not occur.
Subsequently, the possibility of local reactions
is examined in a random sweep of the lists of R and P sites. The
covering of the metal by the passive layer in neutral or basic medium is
represented by
\begin{equation}
R\to P .
\label{reacRP}
\end{equation}
We assume that it occurs with probability $1$ in basic medium ($N_{exc}<0$) and
probability $p_{cor1}$ in neutral medium ($N_{exc}=0$). On the other hand, in
acidic medium ($N_{exc}>0$), R sites may be immediately dissolved, which
corresponds to the reaction
\begin{equation}
R\to E .
\label{reacRE}
\end{equation}
In order to account for the effects of increasing acidity, this process is
assumed to occur with probability $p_{cor2}N_{exc}$, where $p_{cor2}$ is
constant. Here we assume that $p_{cor2}\ll 1$, following the idea that R sites
are preferably dissolved in an anodic reaction.
The last set of possible transformations in a time unit account for
depassivation. It is assumed that this process is not possible in a basic medium
($N_{exc}<0$) and rather slow in a neutral medium, so that the transformation
\begin{equation}
P\to E
\label{reacPE}
\end{equation}
occurs with a small probability $p_{oxi}$ when $N_{exc}=0$. On the other hand,
in order to represent the effect of aggressive anions (reaction
\ref{dissolaggressive}) in acidic medium ($N_{exc}>0$), we assume that
(\ref{reacPE}) occurs with probability ${p'}_{oxi}N_{exc}$, where ${p'}_{oxi}$
is constant, but not very small. In previous works on the model
\cite{vautrin1,stafiej}, a fixed value ${p'}_{oxi}=1/4$ was
used. However, this process plays an essential role in the crossover from slow
to rapid corrosion, thus possible variations of its rate will be considered
here.
Finally, to represent diffusion in the solution, the random walk of particles
A and B is considered. During one time unit,
each A and B particle tries to perform $N_{diff}$ steps to nearest neighbor
sites. If the step takes that particle to a site with neutral solution (E), then
the particles exchange their positions:
$A_1+E_2\to E_1+A_2$ or $B_1+E_2\to E_1+B_2$, where indexes $1$ and $2$ refer to
neighboring lattice sites. If the step takes an A particle to a
site with a B particle or vice versa, then they are annihilated and both are
replaced by E particles: $A_1+B_2\to E_1+E_2$. This represents the
reestablishment of neutrality by diffusion of the local excess of $H^+$ and
${OH}^-$ and by mutual irreversible neutralization.
In all other cases A and B particles remain at their position.
It is important to recall that the probabilities defined above, as well as
$N_{diff}$, can be viewed as the time rates for the corresponding processes. If
the lattice parameter is $a$, then the diffusion coefficient of the A and B
particles is $a^2N_{diff}/2d$, where $d$ is the system dimensionality.
To simplify and reduce the number of parameters we take equal diffusion
coefficients
for both species.
\subsection{Initial conditions and values of the model parameters}
\label{parameters}
Although this model can be investigated with several different initial
conditions, one of the most interesting cases is the study of the corrosion
process taking place after a small damage of a protective layer.
This is the case when a painted metal surface is locally damaged, which puts the
metal in contact with an aggressive environment. It frequently occurs in
relatively cheap (not stainless steel) painted materials in their natural
environment, thus it is of great practical interest to investigate corrosion in
such conditions.
In the model, such damage can be represented by the passivation of two sites
of the top layer of a metallic matrix, while it is assumed that all sites
around the metal are inert, i. e. they cannot be dissolved. This initial
condition is illustrated in Fig. 1a and will be explored by a scaling theory
and numerical methods in the next sections.
The most rapid process in this corrosion problem is expected to be that for
passivation of reactive regions in a neutral or basic medium (reaction
\ref{reacRP}). For this reason, the unit rate is associated to this event in
the basic medium, and $p_{cor1}\lesssim 1$ is adopted. On the other hand, the
passivation in acidic medium is difficult, which justifies the choice
$p_{cor2}\ll 1$. In the simulations of this paper, we will consider
$p_{cor1}=0.9$, and $p_{cor2}=0.02$ or $p_{cor2}=0.005$.
The slowest process is expected to be the dissolution of passivated regions in
a neutral medium. Since it occurs with probability $p_{oxi}$ per unit time, the
characteristic time of this process is $\tau \equiv 1/p_{oxi}\gg 1$. However,
dissolution is enhanced in acidic medium, which justifies the use of
${p'}_{oxi} \gg p_{oxi}$.
It is also reasonable to assume that $p_{SSE}\ll 1$, since this is the fraction
of reactions which
generate significant pH inhomogeneities. However, this is the mechanism
responsible for the onset of high corrosion rates after an incubation time,
which suggests that $p_{SSE}\ll p_{oxi}$. Otherwise, the effective rate of metal
dissolution would not be affected by the pH inhomogeneities.
Finally, diffusion rates of particles A and B are also of order $1$ or larger,
which accounts for rapid cancellation of pH fluctuations in the solution.
\subsection{Summary of previous results on the model}
\label{previous}
The above model without the SSE reactions was formerly considered in Ref.
\protect\cite{vautrin}, where the evolution of the corrosion front was analyzed
via simulations and a mean-field approach. The first study of the corrosion
model with SSE reactions was presented in Ref. \protect\cite{vautrin1}, where
the formation of domains of A and B particles in a highly irregular dissolved
region was observed, as well as the existence of a nucleation regime before the
rapid pit propagation.
The first study which focused the nucleation regime was presented in Ref.
\protect\cite{stafiej}, where the distribution of incubation times was obtained
from simulation. It was shown that, at short times, the size of
the dissolved region slowly increased because the rates of dissolution of P
particles were very small and the repassivation of reactive sites was very
frequent. The products of anodic and cathodic reactions create inhomogeneities
in the local pH of the solution, represented by A and B particles, but these
particles mutually neutralize in an $ A+B \rightarrow 0$ reaction. These
features characterize the nucleation regime.
However, anodic and cathodic reactions may occur at two distant points, which
makes the neutralization slow, while the regions enriched in anions are noxious
for repassivation, promoting further dissolution of the metal. Consequently,
after a certain time the size of the cavity and the number of A and B particles
rapidly increase. This corresponds to the onset of pitting corrosion. The
distribution of incubation times characterizing the transition from the
nucleation regime to pitting corrosion was obtained in Ref.
\protect\cite{stafiej} for two different values of the model parameters. It was
shown that the average incubation time decreased as $p_{oxi}$ increased, but no
the study of the relations between incubation time and the other model
parameters was presented there.
\section{Scaling theory}
\label{scaling}
Starting from the configuration in Fig. 1a, one of the initial P sites is
dissolved after a characteristic time of order $\tau/2$ (reaction
\ref{reacPE}). It leads to the appearance of reactive sites at its
neighborhood, as illustrated in Fig. 1b for the case of depassivation of the P
site at the left.
When $p_{SSE}=0$, the subsequent steps of the process are the passivation of
the reactive sites, which leads to the configuration shown in Fig. 1c.
Consequently, an additional time of order $\tau/3$ will
be necessary for dissolution of a region of the passive layer in Fig. 1c and the
onset of new reactive sites. After that, the new R sites will also be rapidly
passivated. Thus, for small $p_{oxi}$ (large $\tau$), the corrosion process is
always slow. A mean-field theory is able to predict its long-term behavior
\cite{vautrin}.
The case of nonzero but small $p_{SSE}$ is much more complex due to the
appearance of the pH inhomogeneities in the solution. Fig. 1d illustrates the
result of an SSE reaction after the configuration of Fig. 1b, followed by
diffusion which leads to the annihilation of the pair AB.
Simulation of the model \cite{stafiej} shows that, at short times, the size of
the dissolved region increases and the region acquires the approximate shape of
a semicircle centered at the point of the initial damage. After a certain
time, the rate of production of A and B particles remarkably increases,
compensating the annihilation effects. This is mainly a consequence of
the enhanced dissolution and the impossibility of passivation in acidic
regions. Consequently, the total rate of dissolution of the metal becomes much
larger and the cavity develops an irregular shape. The time evolution of the
area of the dissolved region (number of dissolved sites) obtained in simulation
is illustrated in Fig. 2.
In order to relate the geometrical features of the dissolved cavity with the
model parameters, we will consider that it has the semicircular shape of radius
$R$ shown in Fig. 3. In the following, we will estimate the characteristic
times for creation of new A and B particles in that cavity and the
characteristic time for the annihilation of one of those pairs. Matching these
time scales, we can predict conditions for the incubation to occur.
This reasoning follows the same lines which are successfully applied to study
the crossover between different growth kinetics in Ref. \protect\cite{rdcor}.
First we consider the mechanisms for depassivation followed by SSE reactions.
There are two possible paths for the dissolution of particles P and creation of
new reactive sites: dissolution in a neutral environment, with probability
$p_{oxi}$, and dissolution in acidic medium, i. e. in contact with A particles,
which typically occurs with probability ${p'}_{oxi}$ (contact with a
single A particle). In any case, the generation of a new particle A requires a
subsequent SSE reaction.
In the case of dissolution in a neutral environment, the average
time necessary for dissolution of a single P particle is $\tau/N_P$, where $N_P$
is the current number of P particles at the surface of the cavity.
This is illustrated in Fig. 4a. After
dissolution, one or two R particles are generated, and each one may undergo an
SSE reaction with probability $p_{SSE}$. Otherwise, these R particles are
rapidly passivated, and new SSE reactions will be possible only after another
depassivation event. These two possibilities are also illustrated in Fig. 4a.
Thus, the average time for creation of
a pair AB inside the cavity from this path is of order $\tau_{cre}\sim
\tau/N_P/p_{SSE}$. While $\tau$ must be interpreted as a time
interval, here $p_{SSE}$ must be viewed as a dimensionless probability which
indicates the fraction of depassivation events that are followed by SSE
reactions. Now, since $N_P$ is of the order of the number of sites at the
surface of the cavity, we have
\begin{equation}
N_P\sim \pi R/a
\label{np}
\end{equation}
for the cavity of radius $R$ (Fig. 3 - $a$ is the lattice parameter). This leads
to
\begin{equation}
\tau_{cre}\sim \frac{\tau a}{\pi R p_{SSE}} .
\label{taucre}
\end{equation}
In the case of dissolution in an acidic environment, we assume that there is
an A particle present in the cavity. The fraction of the time it spends in the
surface of the cavity is approximately the ratio between the number of perimeter
sites and the total number of sites in the cavity (Fig. 3):
$\left( \pi R/a\right) /\left( \pi R^2/2a^2\right) = 2a/R$. The
probability of dissolution of P in contact with a single A is ${p'}_{oxi}$, thus
the characteristic time for dissolution is $1/{p'}_{oxi}$ in this case. This
process is illustrated in Fig. 4b. Again, the subsequent processes may be SSE
reactions or passivation of the R particles (see Fig. 4b). Thus, the average
time for production of a new A particle is
\begin{equation}
{\tau '}_{cre}\sim \frac{1}{2a/R} \frac{1}{{p'}_{oxi}} \frac{1}{p_{SSE}} =
\frac{R}{2a{p'}_{oxi} p_{SSE}}
\label{taucre1}
\end{equation}
Again, in this case $p_{SSE}$ must be viewed as a dimensionless fraction of SSE
reactions.
Certainly the above estimates of $\tau_{cre}$ and ${\tau '}_{cre}$ contain
inexact numerical factors, but the dependence on the model parameters is
expected to be captured. However, it is interesting to stress that the
superestimation of ${\tau '}_{cre}$ not only comes from geometrical aspects but
also from diffusion. Indeed, the random movement of the A particle tends to be
slower in contact with the surface, where the number of empty neighbors is
smaller. This increases the average time of A in contact with P.
Now we estimate the characteristic time for annihilation of a pair of particles
A and B, which represents the smoothing out of local pH inhomogeneities and
leads
to a decrease of the dissolution rate. Such a pair is illustrated
in Fig. 3. In a first approximation we expect that the average time for
its annihilation, $\tau_{ann}$, is of the order of the number of sites inside
the cavity divided by the number of steps per time unit, $N_{diff}$. Thus
\begin{equation}
\tau_{ann}\sim \frac{\left( R^2/2a^2\right)}{N_{diff}} .
\label{tauann}
\end{equation}
Notice that, when the cavity is small, $\tau_{ann}\ll
{\tau '}_{cre}\ll \tau_{cre}$,
thus all pairs AB are rapidly annihilated after their production. It means that
the solution is neutral during most of the time. This conclusion is realistic
because, at large length scales, the solution is expected to always be neutral.
However, as $R$ grows, the number of pairs AB in the cavity increases, which
leads to a more rapid dissolution of the metal and slower smoothing out the of
pH
inhomogeneities. For sufficiently low probabilities $p_{oxi}$ of dissolution in
a
neutral medium, as discussed in Sec. \ref{parameters}, we expect that
$\tau_{cre}$ is always large compared to the other time scales. Thus,
a series of reactions appears when dissolution in acidic medium becomes
rapid enough to counterbalance the neutralizing effect of diffusion. At
this time, if a new pair AB is created, it will induce the creation
of other pairs and, consequently, a succession of SSE reactions. This is
clear from Fig. 4b, in which we observe the production of two A particles in a
small region of the lattice. Thus, at the incubation time, we expect that
$\tau_{ann}\sim {\tau '}_{cre}$. Using Eqs. (\ref{taucre1}) and (\ref{tauann}),
we obtain an average radius
\begin{equation}
R_I \sim \frac{N_{diff}}{\pi p_{SSE} {p'}_{oxi} } \qquad ,\qquad d=2 ,
\label{ri}
\end{equation}
where the index $I$ refer to incubation time and the spatial dimension $d$ is
emphasized.
This result was derived for the model in two dimensions because our simulations
were performed in these conditions. In a three-dimensional system, the
dependence of ${\tau '}_{cre}$ on $R$ does not change, since it is derived from
a boundary to volume ratio. However, $\tau_{ann}$ may increase as $R^3$ instead
of $R^2$ for a three-dimensional cavity, since that is the order of the number
of sites. Consequently, Eq. (\ref{ri}) is expected to be changed to
\begin{equation}
R_I \sim \sqrt{ \frac{N_{diff}}{p_{SSE} {p'}_{oxi}} } \qquad ,\qquad d=3 .
\label{ri3d}
\end{equation}
Thus, independently of the system dimensionality, we observe that $R_I$
decreases as
$p_{SSE}$ or ${p'}_{oxi}$ increase.
One of the interesting points of these results is their independence on
the lattice structure, which is important for comparisons between such a
mesoscopic model and experimental results. In the following, we show that
simulation data are in a good agreement with the predictions in two dimensions
in the cases where the sizes of the cavities and incubation times are large
enough for such a continuous description to apply. This adds more confidence to
the extension of this theoretical analysis to three-dimensional systems.
\section{Simulation results}
\label{simulation}
Our simulations are performed in a square lattice of vertical and
horizontal sizes $L=1000$, with the central sites of the top layer (largest $y$)
initially
labeled as P and the rest labeled as M (Fig. 1a). The process is interrupted
before or when the corrosion front reaches the sides or bottom of the box.
Typically one hundred independent runs have been performed for each set of the
model parameters.
Following previous experience with simulation of this model, the incubation time
is defined as that in which the number of particles A (or B) is 20. Although
our scaling theory predicted the time necessary for the creation of the second
A particle, it is observed that within a very narrow time interval the number
of A and B particles rapidly increases from values near 2
or 3 to some tenths. Thus,
defining the incubation time in the presence of a smaller numbers of particles
A, such as 10 or 15, leaded to negligible changes in the final estimates of
incubation times and radii of the cavities.
The first test of the relation (\ref{ri}) addresses the dependence of $R_I$ on
$p_{SSE}$. Simulations for $0.01\leq p_{SSE}\leq 0.1$ are performed,
considering fixed values of the other parameters: $p_{oxi}={10}^{-3}$,
${p'}_{oxi}=0.25$, $p_{cor1}=0.9$, $p_{cor2}=0.02$, $N_{diff}=1$. In Fig. 5 we
show a log-log plot of $R_I$ versus $p_{SSE}$ with a linear fit of slope
$-1.05$. This is in a good agreement with Eq. (\ref{ri}) for
constant ${p'}_{oxi}$ and $N_{diff}$, which suggests $R_I\sim 1/p_{SSE}$. This
is certainly the most important test to be performed with this model because it
is much more difficult to estimate a relative probability of spatially
separated reactions in an experiment than to measure diffusion coefficients or
dissolution rates.
We also test the predicted dependence of $R_I$ on $N_{diff}$ by simulations
with the same parameters above, except that we fix $p_{SSE}=0.1$ and
$N_{diff}$ is varied between $1$ and $10$. In Fig. 6 we show a log-log plot of
$R_I$
versus $N_{diff}$ with a linear fit of the data points with larger $N_{diff}$.
That fit gives $R_I\sim {N_{diff}}^{0.7}$, which is a slightly slower dependence
than the linear one predicted by Eq. (\ref{ri}). However, it illustrates the
rapid increase of $R_I$ with the diffusion coefficient, and from Fig. 6 it is
clear that the effective exponent in the relation between $R_I$ and $N_{diff}$
tends to increase as $N_{diff}$ increases \cite{effectiveexp}. The deviation
from the linear relation may be attributed, among other factors, to the
assumption of free random walk properties for A and B particles in our scaling
picture.
We also confirm that $R_I$ rapidly increases as ${p'}_{oxi}$ decreases,
particularly when the latter is small, as suggested by Eq. (\ref{ri}). We
performed simulations with $p_{oxi}={10}^{-4}$,
${p}_{SSE}=0.01$, $p_{cor1}=0.9$, $p_{cor2}=0.02$, $N_{diff}=1$, and various
${p'}_{oxi}$ between $0.25$ and $0.025$. In Fig. 7 we show a log-log plot of
$R_I$ versus ${p'}_{oxi}$. For small ${p'}_{oxi}$ the incubation radius $R_I$
rapidly increases with decreasing ${p'}_{oxi}$, although the exact dependence
predicted in Eq. (\ref{ri}) is not observed. However, when ${p'}_{oxi}$ is very
small, the average time of creation of new A particles in acidic media (${\tau
'}_{cre}$) becomes very large and possibly comparable to the time for
creation in neutral
media ($\tau_{cre}$), even if $p_{oxi}$ is small. Consequently, a competition
between these mechanisms may appear and rule out the above scaling picture.
Finally, we also test a possible dependence of $R_I$ on $p_{oxi}$. Simulations
are performed with ${p'}_{oxi}=0.25$, $p_{cor1}=0.9$, $p_{cor2}=0.02$,
$N_{diff}=1$ and $p_{SSE}=0.1$, while $p_{oxi}$ varies between ${10}^{-2}$ and
${10}^{-4}$. The estimates of $R_I$ range between $9.67\pm 1.50$
and $9.87\pm 1.48$, i. e. they fluctuate $2\%$ while $p_{oxi}$ varies two
orders of magnitude. This supports our prediction that $R_I$ does not depend on
this rate. On the other hand, as discussed below, the average incubation time
strongly depends on that quantity.
Another interesting point of our scaling analysis is the possibility of
obtaining reliable estimates of the order of magnitude of the radius of the
dissolved region at the incubation time. For instance, simulations with
$p_{oxi}={10}^{-3}$, ${p'}_{oxi}=0.25$, $p_{cor1}=0.9$, $p_{cor2}=0.02$,
$N_{diff}=1$ and $p_{SSE}=0.01$ gave $R_I/a=53\pm 8$, while Eq. (\ref{ri}) gives
$R_I=127$.
\section{Incubation time}
Here we derive relations for the average incubation time by extending the
analysis that leads to the scaling theory of Sec. \ref{scaling}. Again, we
focus on the ranges of realistic values of the model parameters presented in
Sec. \ref{parameters}.
The initial configuration of the system, shown in Fig. 1a, evolves to that shown
in Fig. 1b after an average time $\tau/2$. The latter has a large probability
[${\left( 1-p_{SSE}\right)}^2$] of being passivated, which
gives rise to the configuration with $3$ particles P of Fig. 1c. Otherwise, with
a small probability of order $2p_{SSE}$, one of the R particles undergoes an SSE
reaction, increasing the size of the dissolved region, as shown in Fig. 1d.
However,
due to the diffusion of A and B and the high probability of passivation of new
R sites, there is a large probability that the first configuration of Fig. 1d
evolves to passivated configurations with two or three E particles and four or
five P particles. For instance, for $p_{sse}=0.1$, we estimate
that nearly $80\%$ of the initial configurations will evolve to the passivated
state of Fig. 1c, nearly $10\%$ will evolve to passivated states with four P
and two E particles (last configuration of Fig. 1d with R replaced by P), and
nearly $10\%$ will evolve to passivated states with five P and three E
particles. The probability that A and B particles of the configuration in Fig.
1d survive and their number eventually increases is negligible.
Notice that the time intervals for diffusion of A and B particles and for the
SSE reactions before passivation are both very small compared to
$\tau$. Thus, the new passivated configurations described above are obtained
after a time which is also approximately $\tau/2$, corresponding to the first
dissolution event (process from Fig. 1a to Fig. 1b).
After repassivation of the surface, one of the P particles will be dissolved
after a characteristic time of order $\tau/N_P$. From the above discussion, $N_P
=3$ is the most probable value for small $p_{SSE}$ (Fig. 1b), although $N_P=4$
and $N_P=5$ have non-negligible probabilities of occurrence. Thus, new
configurations with reactive sites will appear after the total time
$\tau/2+\tau/N_P\left( 2\right)$, where ${N_P\left( 2\right)}$ is an average
number of P particles before the second dissolution event
(in neutral media). From the above discussion we see that ${N_P\left( 2\right)}$
is slightly
larger than 3 for small $p_{SSE}$ (${N_P\left( 2\right)}=3$ for $p_{SSE}=0$).
Subsequently, passivation of reactive sites, diffusion and SSE reactions
take place, but in all cases the time intervals are much smaller than
$\tau$. New passive configurations will be generated, with larger $N_P$ and
consequently, characteristic times $\tau/N_P$ to be depassivated (dissolution
of one P). Thus, the average incubation time is expected to have the general
form
\begin{equation}
\langle \tau_{INC} \rangle \approx \frac{\tau}{2} + \frac{\tau}
{N_P\left( 2\right)} +
\frac{\tau}{N_P\left( 3\right)} + \dots + \frac{\tau}{{N_P}^{INC}} .
\label{tauinc}
\end{equation}
Here, $N_P\left( i\right)$ is the average number of P particles in the
passivated
states just before the $i^{th}$ dissolution event, and ${N_P}^{INC}$ is
the number of P particles at the time in which the rapid dissolution begins, i.
e. at the incubation time. From Eq. (\ref{np}), we have ${N_P}^{INC}\sim \pi
R_I/a$, thus $R_I$ will determine the term where the series in Eq.
(\ref{tauinc}) will be truncated.
This series resembles a harmonic series which is truncated at a certain term.
Indeed, for very small $p_{SSE}$, the subsequent terms are expected to be close
to consecutive integer values. Moreover, for $p_{SSE}=0$ the harmonic series is
recovered and $\langle \tau_{INC} \rangle$ diverges, which is consistent with
the fact that there is no incubation process without the SSE reactions
\cite{vautrin}.
One of the interesting features of the expansion in Eq. (\ref{tauinc}) is that
it allows for the separation of the effects of the main chemical reactions and
the
effects of dissolution in neutral environment. While the former are important
to determine $R_I$ (Eq. \ref{ri}) and, consequently, the number of terms in the
expansion, the latter determine the value of the characteristic time $\tau$. On
the other hand, the sensitivity of $\langle \tau_{INC} \rangle$ on variations
of $R_I$ is not high because the main contributions to the sum in Eq.
(\ref{tauinc}) come from the terms with small $N_P$. Thus, significant
variations in the incubation time are only expected from variations in the
probability of dissolution in neutral medium, $p_{oxi}=1/\tau$. This is a
somewhat expected feature because $\tau$ is the largest characteristic time of
the relevant events in this system.
The above analysis is fully supported by our simulation data. In Fig. 8 we show
a log-log plot of $\langle \tau_{INC} \rangle$ versus $\tau\equiv 1/p_{oxi}$,
obtained in simulations with fixed values of the other parameters:
${p'}_{oxi}=0.25$, $p_{cor1}=0.9$, $p_{cor2}=0.02$, $N_{diff}=1$, $p_{SSE}=0.1$.
The linear fit of the data, shown in Fig. 8, has a slope $0.96$, which is very
close to the value $1$ expected from Eq.
(\ref{tauinc}) with constant denominators in all terms.
The above results imply that $\langle \tau_{INC} \rangle$ decreases as
$p_{oxi}$ increases.
Simulations of the model with different values of ${p'}_{oxi}$, keeping the
other parameters fixed, also show a decrease of $\langle \tau_{INC} \rangle$ as
${p'}_{oxi}$ increases. On the other hand, both oxidation probabilities are
expected to be enhanced by increasing the concentration of aggressive anions in
solution. This parallels the experimental observation of a decrease in the
incubation time as the concentration of those anions increases
\cite{hassan,rehim,amin,zaky}, which shows the reliability of our model.
In Fig. 9 we show
$\langle \tau_{INC} \rangle$ versus $p_{SSE}$ obtained with the same set of
parameters of the data as in Fig. 5. The observed dependence qualitatively
agrees with Eq.(\ref{tauinc}): as $p_{SSE}$ increases, the radius $R_I$
decreases
and, consequently, the number of terms in the series of Eq. (\ref{tauinc}) also
decreases. However, due to the particular structure of Eq.
(\ref{tauinc}), it
is difficult to predict a simple relation between $\langle \tau_{INC} \rangle$
and $p_{SSE}$.
From Eq. (\ref{tauinc}) we are also able to provide the correct order of
magnitude
of the average incubation time. For instance, considering the simulation with
$p_{oxi}={10}^{-3}$,
${p'}_{oxi}=0.25$, $p_{cor1}=0.9$, $p_{cor2}=0.02$, $N_{diff}=1$ and
$p_{SSE}=0.01$, we obtained $R_I/a\approx 53$ and $\langle \tau_{INC} \rangle
\approx 1.2\times {10}^4$. On the other hand, using Eq. (\ref{np}), that value
of $R_I$ leads to ${N_P}^{INC}\approx 166$. Since $p_{SSE}$ is small, we may
approximate the series in Eq. (\ref{tauinc}) by a harmonic series truncated at
this value of ${N_P}^{INC}$, so that the theoretical estimate of incubation time
is $4.7\times {10}^{3}$, i. e. of the same order of magnitude of the simulation
value.
\section{Relations to other corrosion models and reaction-diffusion systems}
\label{relations}
In our model, the crossover from slow corrosion to a rapid corrosion process
takes place when regions with large number of A and B particles are produced in
the solution. Indeed, previous simulation work on the model has already
shown the presence of spatially separated domains of A and B particles
in the solution \cite{vautrin1} after the incubation time, and this separation
slows down the annihilation of A and B pairs.
This phenomenon resembles the segregation effects
observed in reaction-diffusion systems of the type $A+B\to 0$ in confined
media \cite{wilczek,oz,anacker,lindenberg,lin,kopelman,reigada}, such as
systems with tubular geometries and fractal lattices. In those systems, there
is no injection of new particles and the initial distribution of reactants is
random. However, after a certain time, large domains of A and B particles are
found, and the annihilation process is possible only at the frontiers of
those domains. Consequently, the concentration of A and B particles slowly decay
when compared to the reactions of the type $A+A\to 0$
\cite{anacker,lindenberg}. As far as we know, this phenomenon was not
experimentally observed yet, although the depletion zones of a related model
($A+B\to B$ reactions) were already observed in photobleaching of fluorescein
dye by a focused laser beam \cite{trap1,trap2}.
However, the segregation is not expected for the reaction
$A+B\to 0$ in three dimensions, and in two dimensions it is expected to be
marginal \cite{lindenberg}. This is the case in our model. The production of A
and B particles takes place at the surface of the cavity, and this surface may
play the same role of the rigid boundaries of confined media (restricting the
diffusion of A and B particles) during a short times after their creation.
However, the main mechanism contributing to segregation in our model is the
preferential production of new A particles by dissolution in acidic media, i. e.
production of new A particles close to the other A.
Despite the differences in the mechanisms leading to segregation between our
model and the reaction-diffusion systems of the type $A+B\to 0$, it is clear
that in both cases the mixing of A and B domains is slow due to diffusion
restrictions.
It is also important to notice that our model has significant differences from
those of Refs. \protect\cite{meakin1,meakin2}, which also aim at representing
universal features of corrosion processes. They also consider the interplay
between dissolution and passivation, with the former being limited by diffusion
of
an aggressive species in the solution. The concentration of the aggressive
species depends only on the properties of the solution in contact with the
metal. However, in our model the particles responsible for enhancing the
dissolution (A and B, or pH inhomogeneities) are themselves dissolution
products. This feature implies that the transition from the nucleation stage
to pitting corrosion is a consequence of an auto-catalytic production of those
particles. Instead, the models of Refs. \protect\cite{meakin1,meakin2} and
related stochastic models of pitting corrosion
\cite{nagatani1,nagatani2,nagatani3} were suitable to represent features of the
regime of pit propagation.
\section{Discussion and conclusion}
We studied a model for corrosion and passivation of a metallic surface after a
small damage to its protective layer, in which an initial regime of slow
corrosion crosses over at the incubation time to a regime of rapid corrosion.
The dramatic increase of the corrosion rate is related to the presence of acidic
regions in the solution and the autocatalytic enhancement of pH inhomogeneities
due to spatially separated anodic and cathodic reactions. Our scaling analysis
of the model is based on the matching of the characteristic
times of creation and neutralization of pH inhomogeneities in the solution and
leads to an estimate of the average radius $R_I$ of the dissolved region
at the incubation time. That radius decreases with the rate of spatially
separated reactions and the rate of dissolution in acidic media, and it
increases with the diffusion coefficient of particles A and B in the
solution, which tells us how fast the suppression of pH inhomogeneities takes
place.
The average incubation time is written as the sum of a series of
characteristic times for the slow dissolution in neutral media. It
has a complex dependence on $R_I$ and linearly increases with the rate of
dissolution in neutral media. These results are confirmed by
numerical simulation in two-dimensional lattices, but the extension of
the theory to three dimensions is also discussed. Relations to other
reaction-diffusion systems with segregation and other corrosion models are
discussed. Since the relative values of
the model parameters are expected to provide a realistic description of real
corrosion processes, we believe that this work may be useful for the analysis
of experimental work in this field.
|
1,116,691,500,424 | arxiv | \section{Introduction}
The idea behind the Minkowski curvature flow treated in \cite{mink} is to consider the plane $\mathbb{R}^2$ with a different norm and study the motion of curves evolving by the minkowskian curvature in this new context. But, since we are working with (almost) arbitrary norms in $\mathbb{R}^2$ is quite natural to ask what happens if the norms change along the motion. There is a (maybe naive) physical application for this: one can imagine a motion in an ambient where the "resistence" varies with time. We will do our generalization as follows: consider $\mathbb{R}^2\times [0,T)$ "sliced" in the following way: each plane $\mathbb{R}^2\times\{t\}$ is identified with the Minkowski plane $\mathbb{R}^2$ endowed with a norm given by a $\mathcal{P}_t$-unit ball with boundary parameterized as usual by $p_t(\theta)$. The idea is to consider each curve of the flow lying in one of these planes and evolving according to its geometry. We could also think about this as a motion in a 2-dimensional fibration over an interval. In this paper we study some results concerning the existence and convergence of the flow under certain conditions to the family of norms. In section two we define the flow, derive some evolution formulas and give some conditions on the family of norms that guarantee desired properties for the flow. In section 3 we study a particular case for what we can ensure existence of the solution until the (usual) area converges to zero. To finish the paper we give, using a geometric argument, a estimate for a uniform blow up time for the solutions of a family of parabolic PDE's. \\
\section{The time-dependent minkowskian curvature flow}
Let $a:S^1\times[0,T)\rightarrow \mathbb{R}$ a strictly positive smooth function which is $\pi$-periodic on $\theta$ for every fixed $t$. For a function $f(\theta,t)$ we will denote the spatial derivatives by $f'$ and the time derivative by $\dot{f}$. Suppose that we have $a(\theta,t)+a''(\theta,t) > 0$ everywhere. We can define a family of $\mathcal{P}$-unit balls on $\mathbb{R}^2$ defining $\mathcal{P}_t$ to be the region enclosed by the curve: \\
\[p_t(\theta) = p(\theta,t) = a(\theta,t)e_r + a'(\theta,t)e_\theta, \] \\
where $e_r = (\cos\theta, \sin\theta)$ and $e_\theta = (-\sin\theta,\cos\theta)$, as usual. For each $t$ we have also the dual ball $\mathcal{Q}_t$ to $\mathcal{P}_t$. For every $t$ the following holds: \\
\[ q(\theta,t) = \frac{p'(\theta,t)}{[p(\theta,t),p'(\theta,t)]}, \ \ \mathrm{and} \] \\
\[ p(\theta,t) = - \frac{q'(\theta,t)}{[q(\theta,t),q'(\theta,t)]} \] \\
We define the time-dependent minkowskian curvature flow (that, for sake of convenience, will be called "generalized flow", in constrast with the "usual flow" studied in \cite{mink}) associated to the family of norms given by $\mathcal{P}_t$ as an application $F:S^1\times[0,T_0)\rightarrow\mathbb{R}^2$, $T_0 \leq T$, which satisfies \\
\[ \frac{\partial F}{\partial u}(u,t) = v(u,t).q(\theta(u,t),t); \ \ \mathrm{and} \] \\
\[ \frac{\partial F}{\partial t}(u,t) = -k(u,t).p(\theta(u,t),t), \]\\
where $k(u,t)$ is the minkowskian curvature of the curve $u \mapsto F_t(u)$ calculated with respect to the $\mathcal{P}_t$ norm, and $\theta(u,t) + \pi/2$ is the angle between $\partial F/\partial u$ and the $x$-axis. Thus, we can write: \\
\[ \frac{\partial F}{\partial\theta}(u,t) = \lambda(u,t).q(\theta(u,t),t) \] \\
and then, the curvature is given by \\
\[ k(u,t) = \frac{[p(\theta(u,t),t),p'(\theta(u,t),t)]}{\lambda(u,t)} \] \\
Notice that we have $\lambda d\theta = vdu$. We derive now some evolution formulas. \\
\begin{lemma} We have, for every $(u,t)\in S^1\times[0,T)$: \\
\[\displaystyle\frac{\partial v}{\partial t} = -k^2v + \frac{\partial\log(a)}{\partial t}v\] \\
\end{lemma}
\begin{proof}
First, we compute \\
\[ \frac{\partial}{\partial t}\left(\frac{\partial F}{\partial u}\right) = \frac{\partial v}{\partial t}q + v\left(\frac{\partial q}{\partial\theta}\frac{\partial\theta}{\partial t} + \frac{\partial q}{\partial t}\right) \] \\ \\
Is easy to check that $\displaystyle\frac{\partial q}{\partial t} = -\frac{\partial\log(a)}{\partial t}q$. Since $\partial q/\partial\theta$ points on the $p$ direction, writing $\displaystyle\frac{\partial}{\partial t}\left(\frac{\partial F}{\partial u}\right)$ in the basis $\{p,q\}$ the coefficient of $q$ is $\displaystyle\frac{\partial v}{\partial t}-\frac{\partial\log(a)}{\partial t}v$. Let us now compute \\
\[ \frac{\partial}{\partial u}\left(\frac{\partial F}{\partial t}\right) = -\frac{\partial k}{\partial u}p - k\frac{\partial p}{\partial\theta}\frac{\partial\theta}{\partial u} = -\frac{\partial k}{\partial u}p -k^2vq \] \\
And then, since $u$ and $t$ are independent parameters we have the desired equality. \\
\end{proof}
Notice that we can still work with a $\mathcal{Q}_t$-arclength parameter $s_t$, but only for a fixed time. As a corollary of the previous lemma we have an evolution formula to the $\mathcal{Q}_t$-arclength of the curves as follows: \\
\[ \frac{\partial L_{\mathcal{Q}_{t}}}{\partial t} = \int_0^{2\pi}\frac{\partial v}{\partial t}\ du = -\int_0^{2\pi}k^2v \ du + \int_0^{2\pi}\frac{\partial\log(a)}{\partial t}v \ du, \] \\
\begin{lemma} \label{qlenght} If there exists $M \in \mathbb{R}$ such that $\dot{a}(\theta,t) \leq Ma(\theta,t)$ in $S^1\times[0,T)$, then the $\mathcal{Q}_t$-lenght of the curves is bounded along the motion.\\
\end{lemma}
\begin{proof} The hypotesis gives $\displaystyle\frac{\partial\log(a)}{\partial t} \leq M$ in $S^1\times [0,T)$. Then, we have \\
\[ \frac{\partial L_{\mathcal{Q}_t}}{\partial t} \leq \int_0^{2\pi}\frac{\partial\log(a)}{\partial t}v \ du \leq ML_{\mathcal{Q}_t}. \] \\
So, Gronwall's inequality yields $L_{\mathcal{Q}_t} \leq e^{MT}L_{\mathcal{Q}_0}$. \\
\end{proof}
We can also derive an evolution formula for the area $A(t)$ enclosed by the curves: \\
\begin{eqnarray*} \frac{\partial A}{\partial t} = \frac{\partial}{\partial t}\left(\frac{1}{2}\int_0^{2\pi}\left[F(u,t),\frac{\partial F}{\partial u}(u,t)\right]\ du\right) = \\ \\ = \frac{1}{2}\int_0^{2\pi}\left[\frac{\partial F}{\partial t}(u,t),\frac{\partial F}{\partial u}(u,t)\right]\ du + \frac{1}{2}\int_0^{2\pi}\left[F(u,t),\frac{\partial^2 F}{\partial t\partial u}(u,t)\right]\ du = \\ \\ = \int_0^{2\pi}-kv \ du = -\int_0^{L_{\mathcal{Q}_t}}k \ ds_t = -2A(\mathcal{P}_t)\end{eqnarray*} \\
Now, is easy to see that the evolution of the isoperimetric ratio is given by \\
\[ \frac{\partial}{\partial t}\left(\frac{L^2_{\mathcal{Q}_t}}{A(t)}\right) = -\frac{2L_{\mathcal{Q}_t}}{A(t)}\left(\int_0^{L_{\mathcal{Q}_t}}k^2 ds_t - A(\mathcal{P}_t)\frac{L_{\mathcal{Q}_t}}{A(t)}\right) + \frac{2L_{\mathcal{Q}_t}}{A(t)}\int_0^{L_{\mathcal{Q}_t}}\frac{\partial\log(a)}{\partial t} \ ds_t \] \\
Here we observe that the isoperimetric ratio may be an increasing function. Moreover, the evolution of the isoperimetric ratio depends on the choice of the considered family of norms. \\ \\
Let us now turn our attention to the existence of such a generalized flow. For a fixed $t$ we know that a $2\pi$-periodic, positive and $C^1$ function $k:[0,2\pi]\rightarrow \mathbb{R}$ is the $t$-minkowskian curvature of a simple, closed, strictly convex and $C^2$ curve if and only if the equalities \\
\begin{eqnarray} \int_0^{2\pi}\frac{a(\theta,t)+a''(\theta,t)}{k(\theta)}\sin(\theta) \ d\theta = \int_0^{2\pi}\frac{a(\theta,t)+a''(\theta,t)}{k(\theta)}\cos(\theta) \ d\theta = 0 \end{eqnarray} \\
hold. With this in mind we claim that we don't need too strong hypotesis to ensure short-time existence for the generalized flow. This is justified in the next theorem.
\begin{teo} Consider a function $k:S^1\times [0,T_0)\rightarrow \mathbb{R}$, $T_0 \leq T$, such that $k\in C^{2+\alpha,1+\alpha}(S^1\times[0,T_0-\epsilon])$ for all $\epsilon > 0$, satisfying the evolution equation: \\
\begin{eqnarray} \frac{\partial k}{\partial t} = \frac{a}{a+a''}k^2k'' + \frac{2a'}{a+a''}k^2k' + k^3 + \frac{\partial\log(a+a'')}{\partial t}k \end{eqnarray} \\
with initial condition $k(\theta,0) = \varphi(\theta)$, where $\varphi$ is a strictly positive $C^{1+\alpha}$ function satistying (1). Assume that $t \mapsto a(\theta,t) + a''(\theta,t)$ is nondecreasing for each $\theta \in S^1$. Then, using this function (whose short term existence is guaranteed by the standard theory on parabolic equations) one can build the family of curves on parameter t: \\
\begin{eqnarray*} F(\theta,t) = \left( -\int_0^{\theta}\frac{a(\sigma,t)+a''(\sigma,t)}{k(\sigma,t)}\sin\sigma \ d\sigma - \int_0^ta(0,s)k(0,s) \ ds,\right. \\
\left.\int_0^{\theta}\frac{a(\sigma,t)+a''(\sigma,t)}{k(\sigma,t)}\cos\sigma \ d\sigma - \int_0^ta(0,s)k'(0,s)+a'(0,s)k(0,s) \ ds \right) \\ \end{eqnarray*}
for which the following holds: \\
\begin{description}
\item[(a)] for each fixed $t$ the map $t \mapsto F(\theta,t)$ is a simple, closed and strictly convex curve parameterized as usual whose $t$-minkowskian curvature is given by $\theta \mapsto k(\theta,t)$ \\
\item[(b)] $\displaystyle\frac{\partial F}{\partial\theta}(\theta,t) = -k(\theta,t).p(\theta,t) - a(\theta,t)^2k'(\theta,t).q(\theta,t)$ \\
\end{description}
\end{teo}
\begin{proof}
The proofs of \textbf{(b)} and of \\
\[\int_0^{2\pi}\frac{a(\sigma,t)+a''(\sigma,t)}{k(\sigma,t)}\sin\sigma \ d\sigma = \int_0^{2\pi}\frac{a(\sigma,t)+a''(\sigma,t)}{k(\sigma,t)}\cos\sigma \ d\sigma = 0 \] \\
for every $t$ are straightfoward calculations just like in the usual minkowskian curvature flow. Thus, we only need to prove that $k$ is strictly positive in $S^1\times [0,T_0)$. But the hypotesis on $a + a''$ guarantees that we have \\
\[ \frac{\partial\log(a+a'')}{\partial t} \geq 0 \ \ \mathrm{in} \ S^1\times [0,T_0) \] \\
and then we can repeat the proof of the usual case to prove that $k_{\mathrm{MIN}}(t)$ is bounded by below by $k_{\mathrm{MIN}}(0)$. This concludes the proof. \\
\end{proof}
From the evolution formulas we can see that the area $A(t)$ enclosed by the curve at time $t$ is a decreasing function. But the decay ratio depends on $A(\mathcal{P}_t)$, and then we cannot readily say that the area converges to $0$ even for infinite time. Furthermore, we don't even can tell if the $\mathcal{Q}_t$-lengths have an upper bound. But, working on a certain class of $\mathcal{P}_t$ families we can maybe give good answers to these questions. We work with a particular case in the next section. \\
\section{Homothetic family of $\mathcal{P}$-balls}
Consider a Minkowski norm on $\mathbb{R}^2$ given by the set $\mathcal{P} = \mathcal{P}_0$, whose boundary is parameterized by \\
\[ p_0(\theta) = a_0(\theta)e_r + a_0'(\theta)e_{\theta} \] \\
where $a_0$ is a $C^{\infty}$ and $\pi$-periodic function. Let $f:[0,\infty) \rightarrow \mathbb{R}$ be a $C^{\infty}$, positive and nondecreasing function such that $f(0) = 1$. Then, we can define a family of Minkowski norms putting \\
\[ a(\theta,t) = f(t)a_0(\theta) \] \\
and, naturally, $p(\theta,t) = a(\theta,t)e_r + a'(\theta,t)e_{\theta} = f(t)p_0(\theta)$. \\
\noindent\textit{Remark.} Even if this looks like a simple reparametrization at the time we point out that, here, we are calculating the curvatures in a different way at each time. This interpretation considers the flow evolving with respect to the geometry of each space. \\
The hypotesis that $f$ is nondecreasing guarantees short term existence of the generalized flow associated to this family of norms, taking by initial conditon a closed, strictly convex and smooth curve. In this particular case the evolution equation becames \\
\[ \frac{\partial k}{\partial t} = \frac{a_0}{a_0+a_0''}k^2k'' + \frac{2a_0'}{a_0+a_0''}k^2k' + k^3 + \frac{\dot{f}(t)}{f(t)}k, \] \\ We claim that in this case, if the solution continues until the area enclosed by the curves goes to 0, then the area converges to 0 in finite time. In fact, notice first that \\
\[ A(\mathcal{P}_t) = \int_0^{2\pi}[p(\theta,t),p'(\theta,t)] \ d\theta = f(t)^2\int_0^{2\pi}[p_0(\theta),p_0'(\theta)]\ d\theta = f(t)^2A(\mathcal{P}_0) \] \\
Now, the evolution formula for the area guarantees that \\
\[ A(t) = A(0) - 2\int_0^tA(\mathcal{P}_s) \ ds = A(0) - 2\int_0^tf(s)^2A(\mathcal{P}_0)\ ds \leq A(0) - 2tA(\mathcal{P}_0), \] \\
since $f(0) = 1$ and $f$ is nondecreasing. Then, $A(t)$ converges to zero for some time $T_1 \leq A(0)/A(\mathcal{P}_0)$. \\
Our next claim is that we have an upper bound for the $\mathcal{Q}_t$-length of the curves in finite time. In fact, for finite time we can take an upper bound for $\displaystyle\frac{\dot{f}(t)}{f(t)}$ and use Lemma \ref{qlenght}. This proves that the $\mathcal{Q}_t$-lenght doesn't blow up along the motion. \\
We will now prove that in this case we also have solution until the area converges to 0. The strategy is basically the same that in the usual case treated in \cite{mink}. We consider a solution $k$ of (2) defined on $S^1\times [0,T)$ such that the area enclosed by the associated curves remains bounded away from zero (i.e., we have $T \leq T_1$) and prove that $k$ and its derivatives are bounded in $S^1\times [0,T)$. Finally, we use Ascoli-Arzela's theorem to extend $k$ past $T$. The difference here is that we have to deal with bounds to $f$ and its derivatives. This shouldn't be a problem since we are working with a finite time interval $[0,T) \subseteq [0,T_1]$, and then compacity and the smoothness of $f$ guarantee the needed bounds. \\
Recall that in the minkowski plane we still can define the median curvature of a curve parameterized by the usual $\theta$ as the supremum of the values $x$ for which we have $k(\theta) > x$ in some interval of length $\pi$. We have the estimate \\
\[ k^* \leq C\frac{L_{\mathcal{Q}}}{A} \] \\
for a constant $C$ given by \\
\[ C = \left(\max_{\theta\in [0,2\pi]}|q(\theta)|\right)^2\max_{\theta\in [0,2\pi]}[p(\theta),p'(\theta)] \] \\
And then, in our generalized case we have \\
\[ k^*(t) \leq C(t)\frac{L_{\mathcal{Q}_t}}{A(t)}, \] \\
with \\
\[C(t) = \left(\max_{\theta\in [0,2\pi]}\left| \frac{1}{a(\theta,t)}\right|\right)^2\max_{\theta \in [0,2\pi]}\left[p(\theta,t),p'(\theta,t)\right] \] \\
The point here is that there is an uniform upper bound for the median curvature along the motion if the area is bounded by below by a constant greater then zero. We already know that we have an upper bound for $L_{\mathcal{Q}_t}$. We also can (in such a natural way!) produce an uniform upper bound for $C(t)$ just rewriting \\
\begin{eqnarray*} C(t) = \frac{1}{f(t)^2}\left(\max_{\theta\in [0,2\pi]}\frac{1}{a_0(\theta)}\right)^2f(t)^2\max_{\theta \in [0,2\pi]}[p_0(\theta),p_0'(\theta)] = \\ \\ = \left(\max_{\theta\in [0,2\pi]}\frac{1}{a_0(\theta)}\right)^2\max_{\theta \in [0,2\pi]}[p_0(\theta),p_0'(\theta)] \end{eqnarray*} \\
This is summarized as follows \\
\begin{lemma} If $A(t)$ is bounded away from zero on $[0,T)$, then there is an uniform upper bound for $k^*(t)$ in $[0,T)$. \\
\end{lemma}
By consequence we have the following proposition \\
\begin{prop}If $k^*(t)$ is bounded in $[0,T)$, then the function\\
\[ t \mapsto \displaystyle\int_0^{2\pi}(a_0(\theta)+a_0''(\theta))a_0(\theta)\log\left(\frac{k(\theta,t)}{f(t)}\right)\ d\theta \] \\
is also bounded in $[0,T)$. \\
\end{prop}
\begin{proof} A straightfoward calculation gives us \\
\begin{eqnarray*} \frac{d}{dt}\left(\int_0^{2\pi}(a_0(\theta)+a_0''(\theta))a_0(\theta)\log\left(\frac{k(\theta,t)}{f(t)}\right)\ d\theta\right) = \\ \\ = \int_0^{2\pi}(a_0(\theta)k(\theta,t))^2 - \left((a_0(\theta)k(\theta,t))'\right)^2d\theta \end{eqnarray*} \\
From now on we use basically the same strategy used in the usual case. Fix any $t \in [0,T)$ and let $A = \left\{\theta \in [0,2\pi] \mid k(\theta,t) > k^*(t)\right\}$. We have the estimate on A: \\
\[ \int_0^{2\pi}(a_0k)^2-\left((a_0k)'\right)^2d\theta \leq 2k^*(t)\int_0^{2\pi}a_0(a_0+a_0'')k \ d\theta - k^*(t)^2\int_A(a_0')^2+2a_0a_0''+a_0^2d\theta \] \\
Here we must take some care with the first integral on the right side. First, notice that \\
\begin{eqnarray*} \int_0^{2\pi}a_0(a_0+a_0'')k \ d\theta = \frac{1}{f(t)^2}\int_0^{2\pi}[p(\theta,t),p'(\theta,t)]k\ d\theta = \frac{1}{f(t)^2}\int_0^{2\pi}k^2\lambda \ d\theta = \\ \\ \frac{1}{f(t)^2} \int_0^{2\pi}k^2v \ du \end{eqnarray*} \\
This is not the derivative of $L_{\mathcal{Q}_t}$ with changed sign as in the usual case, but this won't be a problem. Once we have \\
\[ \int_0^{2\pi}k^2v \ du = -\frac{\partial L_{\mathcal{Q}_t}}{\partial t} + \frac{\dot{f}(t)}{f(t)}\int_0^{2\pi}v \ du = -\frac{\partial L_{\mathcal{Q}_t}}{\partial t} + \frac{\dot{f}(t)}{f(t)}L_{\mathcal{Q}_t} \] \\
we can use uniform bounds (remember $f$ is bounded by below by 1) for $f$, $\dot{f}$ and $L_{\mathcal{Q}_t}$ to write \\
\[ \frac{1}{f(t)^2}\int_0^{2\pi}k^2v \ du \leq -C_0\frac{\partial L_{\mathcal{Q}_t}}{\partial t} + C_1 \] \\
for constants $C_0,C_1>0$ which don't depend on $t$. This yields \\
\[ \int_0^{2\pi}(a_0k)^2-\left((a_0k)'\right)^2d\theta \leq -2k^*(t)C_0\frac{\partial L_{\mathcal{Q}_t}}{\partial t} + 2k^*(t)C_1 + 2\pi k^*(t)C_2, \] \\
where $C_2 = \displaystyle\max_{\theta\in [0,2\pi]}\left|(a_0')^2+2a_0a_0'' + a_0^2\right|$. Now, integrating and using, again, a bound for $L_{\mathcal{Q}_t}$ yields the desired estimate for finite time, which is what we want. \\
\end{proof}
\begin{coro} In the same conditions of the proposition the function $t \mapsto \displaystyle\int_0^{2\pi}a_0(a_0+a_0'')\log(k) \ d\theta$ is bounded in $[0,T)$. \\
\end{coro}
\begin{proof} Let $M$ be an upper bound given for $\displaystyle\int_0^{2\pi}a_0(a_0+a_0'')\log\left(\frac{k}{f}\right)\ d\theta$. Then, \\
\[ \int_0^{2\pi}a_0(a_0+a_0'')\log(k) \ d\theta \leq M + \log(f(t))\int_0^{2\pi}a_0(a_0+a_0'') \ d\theta, \] \\
and then we have the desired since $\log(f(t))$ is bounded for finite time. Notice also that we have an obvious lower bound (not necessarily positive): \\
\[ \int_0^{2\pi}a_0(a_0+a_0'')\log\left(k_{\mathrm{MIN}}(0)\right) \ d\theta \leq \int_ 0^{2\pi}a_0(a_0+a_0'')\log(k) \ d\theta \]
for every $t\in [0,T)$.
\end{proof}
\begin{prop} There exists a constant $N \geq 0$ such that \\
\[ \int_0^{2\pi}\left((a_0(\theta)k(\theta,t))'\right)^2d\theta \leq \int_0^{2\pi}\left(a_0(\theta)k(\theta,t)\right)^2d\theta + N \]
for every $T \in [0,T)$.
\end{prop}
\begin{proof} We will first consider the function $g:[0,T) \rightarrow \mathbb{R}$ given by : \\
\[ g(t) = \int_0^{2\pi}\left(a_0(\theta)k(\theta,t)\right)^2 - \left((a_0(\theta)k(\theta,t))'\right)^2 + 2a_0(\theta)\left(a_0(\theta)+a_0''(\theta)\right)\log(k(\theta,t))\frac{\dot{f}(t)}{f(t)} \ d\theta \] \\
After some calculations we have that its derivative is given by \\
\[ \frac{dg}{dt} = \int_0^{2\pi}\frac{2a_0(a_0+a_0'')}{k^2}\left(\frac{\partial k}{\partial t}\right)^2 + 2a_0(a_0+a_0'')\log(k)\frac{\partial^2\log(f)}{\partial t^2} \ d\theta \] \\
and then, \\
\begin{eqnarray*} \frac{dg}{dt} \geq 2\int_0^{2\pi}a_0(a_0+a_0'')\log(k)\frac{\partial^2\log(f)}{\partial t}^2\ d\theta \geq \\ \\ \geq 2\frac{\partial^2\log(f)}{\partial t^2}\int_0^{2\pi}a_0(a_0+a_0'')\log\left(k_{\mathrm{MIN}}(0)\right) \ d\theta \geq C \end{eqnarray*}
for some constant $C$ that we cannot take positive because we may have $k_{\mathrm{MIN}}(0) < 1$. In particular we can assume $C <0$. Notice that we used, again, uniform bounds for $f$ and its derivatives in finite time. By integration we have, for each $t \in [0,T)$ \\
\[ g(t) \geq g(0) + Ct \geq g(0) + CT, \] \\
since $C < 0$. Then, substituting $g$ by its expression and rearranging the terms we have for every $t \in [0,T)$, \\
\begin{eqnarray*} \int_0^{2\pi}\left((a_0k)'\right)^2d\theta \leq -g(0) - CT + \int_0^{2\pi}(a_0k)^2d\theta + 2\frac{\dot{f}(t)}{f(t)}\int_0^{2\pi}a_0(a_0+a_0'')\log k \ d\theta \leq \\ \\ \leq \int_0^{2\pi}(a_0k)^2d\theta -g(0) - CT + M, \end{eqnarray*}
where $M$ is a time-independent constant given by bounds on $f$ and $f'$ and by the previous corollary. Taking $N$ to be any positive number greater than $-g(0) - CT + M$ yields the desired. \\
\end{proof}
\begin{lemma} If $\displaystyle\int_0^{2\pi}a_0(a_0+a_0'')\log\left(k\right) \ d\theta$ is bounded in $[0,T)$, then for any $\delta > 0$ there exists a constant $C$ such that if $k(\theta,t) > C$ in an interval $J$ (varying the parameter $\theta$) then we have necessarily $|J| \leq \delta$. \\
\end{lemma}
\begin{proof} The proof is identical to the proof in the usual case. \\
\end{proof}
\begin{prop} If $\displaystyle\int_0^{2\pi}a_0(a_0+a_0'')\log\left(k\right) \ d\theta$ is bounded in $[0,T)$, then $k(\theta,t)$ has an upper bound in $S^1\times [0,T)$. \\
\end{prop}
\begin{proof} The proof is, again, identical to the proof in the usual case. \\
\end{proof}
Combining these lemmas and propositions yields \\
\begin{teo}If the area $A(t)$ enclosed by the curves associated to the $t$-minkowskian curvature function $k$ admits a strictly positive lower bound on $[0,T)$, then $k$ is uniformly bounded in $S^1\times [0,T)$. \\
\end{teo}
Let us prove now that the first spatial derivative of $k$ is also bounded provided $k$ is bounded. \\
\begin{prop} If $k$ is bounded in $S^1\times [0,T)$, then $k'$ is also bounded in $S^1\times [0,T)$. \\
\end{prop}
\begin{proof} As in the usual case, consider the function $u:S^1\times [0,T) \rightarrow \mathbb{R}$ given by $u = k'e^{ct+h(\theta)}$, where $h(\theta) = \log\left(a_0(\theta)^2\right)$ and $c$ is to be choosen later. After some calculations we have that $u$ is a solution of the second-order linear parabolic equation: \\
\[ \frac{\partial u}{\partial t} = \left(c + 3k^2 + \frac{\dot{f}(t)}{f(t)}\right)u - k^2\frac{2a_0'}{a_0+a_0''}\frac{\partial u}{\partial\theta} + \frac{\partial}{\partial\theta}\left(k^2\frac{a_0}{a_0+a_0''}\frac{\partial u}{\partial\theta}\right) \] \\
Now, using bounds for $k$, $f$ and $\dot{f}$ one can choose $c$ such that the coeficient of $u$ is nonpositive. Then, using the maximum principle we have that $u$ is bounded in $S^1\times [0,T)$, and then $k'$ is also bounded since we are working with finite time. \\
\end{proof}
To show that the higher derivatives of $k$ are also bounded in $S^1\times [0,T)$ we repeat the proof of the usual case adding some terms given by the new term on the evolution equation. The bounds for $f$ and its derivatives will provide the necessary bounds for this new terms. Thus we have, as in the usual case: \\
\begin{teo}The solution to the generalized minkowskian curvature flow associated to the family of norms given by $a(\theta,t)=f(t)a_0(\theta)$ where $a_0$ is a positive, $\pi$-periodic and $C^{\infty}$ function, and $f$ is a positive, nondecreasing, $C^{\infty}$ function with $f(0) = 1$ continues until the area enclosed by the curves converges to zero. \\
\end{teo}
Since the solution cannot continues after the area goes to zero we have an corollary concerning the blow up of a family of PDE's.\\
\begin{coro} Fix a function $g:S^1\rightarrow \mathbb{R}$ which is smooth and $\pi$-periodic, and a smooth and strictly positive function $u_0:S^1 \rightarrow\mathbb{R}$ such that \\
\[ \int_0^{2\pi}\frac{g(\theta)+g''(\theta)}{u_0(\theta)}\sin(\theta)d\theta = \int_0^{2\pi}\frac{g(\theta)+g''(\theta)}{u_0(\theta)}\cos(\theta)d\theta = 0. \] \\
Let $\mathcal{F}$ be the family of the positive, nondecreasing and smooth functions $f:[0,\infty) \rightarrow \mathbb{R}$ with $f(0) = 1$, and consider the associated family of PDE's: \\
\[ \left\{ \begin{array}{lll} u_t = \displaystyle\frac{g}{g+g''}u^2u_{\theta\theta}+\frac{2g'}{g+g''}u^2u_{\theta}+u^3+\frac{\dot{f}}{f}u \\ \\ u(0) = u_0 \\ \end{array}\right. \] \\
Then, we must have an uniform upper bound for the blow up time of the solutions to these PDE's that only depends on $g$ and $u_0$. In other words, there exists a time $T = T(u_0,g)$ such that picking any solution $u$ for some of these PDE's we must have $u$ or some of its derivatives blowing up for some $t \leq T$. The time $T$ is explicitly given by the ratio $T = A/2B$ where $A$ is the area of the curve whose minkowskian curvature with respect to the norm given by the curve $a:\theta \mapsto g(\theta)e_r + g'(\theta)e_\theta$ is $u_0$; and $B$ is the area of the curve $a$. \\
\end{coro}
Even though this might be obvious for someone who has familiarity with nonlinear parabolic PDE's, we think the interest here is that we arrived at this result using, essencially, geometric methods. \\
|
1,116,691,500,425 | arxiv | \section{Introduction}
In \cite{Manu:2020tty}, we studied aspects of entanglement and quantum
extremal surfaces (QES) in various families of holographic spacetimes
exhibiting cosmological singularities. This is inspired by the
exciting discoveries made recently on the black hole information
paradox
\cite{Penington:2019npb,Almheiri:2019psf,Almheiri:2019hni,Penington:2019kki,Almheiri:2019qdq},
unravelled via the study of entanglement, quantum extremal surfaces
and islands: by now there is a large body of literature on various
aspects of these issues, reviewed in {\it e.g.}\
\cite{Almheiri:2020cfm,Raju:2020smc,Chen:2021lnq,Kibe:2021gtw}. Quantum
extremal surfaces are extrema of the generalized entropy
\cite{Faulkner:2013ana,Engelhardt:2014gca} obtained by incorporating
the bulk entanglement entropy of matter alongwith the classical area
of the entangling RT/HRT surface
\cite{Ryu:2006bv}-\cite{Rangamani:2016dms}. These lead to various new
insights on black holes. Explicit calculation is possible in effective
2-dimensional models where the bulk entanglement entropy can be
studied through 2-dim CFT techniques.
It is interesting to ask if quantum extremal surfaces might be used to
probe cosmological, Big-Crunch or -Bang, singularities. While the
vicinity of the singularity is expected to be rife with severe
stringy/quantum gravity effects, one might hope to gain some insight
into how these extremal surfaces probe such singularities. Some
interesting recent work on QES and cosmologies
appears in \cite{Chen:2020tes,Hartman:2020khs} and also
{\it e.g.}\ \cite{Krishnan:2020fer}-\cite{Aguilar-Gutierrez:2021bns}.
The investigations in \cite{Manu:2020tty} pertained to various
Big-Crunch singularities, in particular the isotropic $AdS$ Kasner
spacetime. These spacetimes have no horizons and no significant
entropy, so they are somewhat unlike black hole horizons. Further, we
are considering closed universes with no entanglement with
``elsewhere'' ({\it e.g.}\ other universes). Part of the goal here is to
gain some understanding of how quantum extremal surfaces probe such
spacetime singularities in closed universes with no horizons and no
entanglement with regions external to these universes. The
time-dependence implies that the classical extremal RT/HRT surface
dips into the bulk radial and as well as time directions. Explicitly
analysing the extremization equations in the semiclassical region far
from the singularity can be carried out in detail: we find the surface
bends in the direction away from the singularity. In the 2-dim
cosmologies \cite{Bhattacharya:2020qil} obtained by dimensional
reduction of these and other singularities, quantum extremal surfaces
can be studied by extremizing the generalized entropy, with the bulk
matter taken to be in the ground state (which is reasonable in the
semiclassical region far from the singularity). The resulting
extremization shows the quantum extremal surfaces to always be driven
to the semiclassical region far from the singularity. In
sec.~\ref{sec:rev}, we review the analysis in \cite{Manu:2020tty}. The
2-dim dilaton gravity theories in these cases are somewhat more
complicated than Jackiw-Teitelboim gravity and are not ``near JT'' in
essential ways. The cosmological solutions here are sourced by an
extra scalar which descends from the scalar in the higher dimensional
theory. These theories capture a subset of the observables of the
higher dimensional theory and so are best regarded as models of
``effective holography'' \cite{Narayan:2020pyj}, UV-incomplete in
totality but adequate for capturing various aspects including
entanglement. Since the quantum extremal surfaces are driven to the
semiclassical region far from the singularity, the approximation of
using the 2-dimensional theory is consistent and the other higher
dimensional modes do not make any significant contribution.
In this paper, we continue our investigations there and develop them
further: wherever possible we look for quantum extremal surfaces
spacelike-separated from the observer location. We first do a careful
study of QES focussing on $AdS$ Kasner singularities
(sec.~\ref{sec:AdSKreg}), by introducing a spatial regulator. This
enables relating the locations in time of the observer on the
holographic boundary and the QES with the bulk matter central charge
and the regulator. In the semiclassical region, this shows that the
quantum extremal surface lags behind the observer location (in the
direction away from the singularity). A potential island-like region,
upon analysing in detail near the island boundary, turns out to be
inconsistent. We then extend this to more general singularities
admitting a holographic interpretation, which exhibit similar
behaviour. In sec.~\ref{sec:null}, we study certain families of null
Kasner Big-Crunch singularities: these exhibit a certain
``holomorphy'' due to special properties of null backgrounds. Further
they are also distinct in the behaviour of the QES, which now can
reach the singularity (although the generalized entropy continues to
be singular). We then discuss aspects of 2-dimensional effective
theories involving dimensional reduction of other cosmologies in
sec.~\ref{sec:dSFRW}, including de Sitter space (Poincare slicing) and
FRW cosmologies under certain conditions. Sec.~\ref{sec:Disc} contains
some conclusions. Some details appear in two Appendices.
\section{Review: Big-Crunches \& quantum extremal surfaces}\label{sec:rev}
There is a long history of studying cosmological singularities in
string theory and holography: see \cite{Bhattacharya:2020qil} for a
partial list of references in this regard, and
{\it e.g.}\ \cite{Craps:2006yb,Burgess:2011fa} for reviews of cosmological
singularities in string theory.
In \cite{Manu:2020tty}, various families of cosmological spacetimes
with spacelike Big-Crunch singularities were considered: the higher
dimensional space and its reduction ansatz are of the form
\begin{equation}\label{redux+Weyl}
ds^2_D = g^{(2)}_{\mu\nu} dx^\mu dx^\nu + \phi^{2\over d_i} d\sigma_{d_i}^2\ ;
\qquad\quad g_{\mu\nu}=\phi^{{d_i-1\over d_i}} g^{(2)}_{\mu\nu}\ ,
\qquad D=d_i+2\ .
\end{equation}
$d_i$ is the dimension of the transverse space.
The Weyl transformation from $g^{(2)}_{\mu\nu}$ to the 2-dim metric
$g_{\mu\nu}$ ensures that the dilaton kinetic energy vanishes and
the action is
\begin{equation}\label{actionXPsiU}
S= {1\over 16\pi G_2} \int d^2x\sqrt{-g}\, \Big(\phi\mathcal{R}
- U(\phi,\Psi) -\frac{1}{2} \phi (\partial\Psi)^2 \Big)\ ,
\end{equation}
The dilaton potential $U(\phi,\Psi)$ potentially couples the dilaton
$\phi$ to $\Psi$.\ Certain aspects of generic dilaton gravity theories
of this kind (and these 2-dim cosmological backgrounds), dimensional
reduction and holography were discussed in \cite{Narayan:2020pyj} (see
also \cite{Grumiller:2021cwg}): these theories are more complicated
than JT gravity and are not ``near JT''. they capture a subset of the
observables of the higher dimensional theory and so are best regarded
as UV-incomplete models of ``effective holography''. There is
nontrivial dynamics in the theory (\ref{actionXPsiU}) driven by the
extra scalar $\Psi$ which descends from the scalar in the higher
dimensional theory. In particular there are nontrivial cosmological
singularity solutions here, which were analysed in
\cite{Bhattacharya:2020qil}. See Appendix~\ref{App:2dgES} for some
details. The power-law scaling ansatze for the 2-dim fields and the
corresponding higher dimensional spacetimes are
\begin{equation}\label{phie^fPsi-ansatz}
\phi=t^kr^m,\quad e^f=t^ar^b,\quad e^\Psi=t^\alpha r^\beta \quad\rightarrow\quad
ds_D^2 = {e^f\over \phi^{(d_i-1)/d_i}}\big(-dt^2+dr^2\big)+\phi^{2/d_i}dx_i^2\ .
\end{equation}
The universality (\ref{univSing}) implies that $k=1$.
Note that $r=0$ is the asymptotic (holographic) boundary. The
equations of motion (\ref{2dimseom-EMD1-Psi}) then lead to algebraic
relations between the various exponents above, which can then be
solved for, leading to nontrivial families of cosmological solutions
\cite{Bhattacharya:2020qil}.
A prototypical example is $AdS$ Kasner and its reduction to 2-dimensions,
\begin{eqnarray}\label{AdSDK-2d}
&& U=2\Lambda\phi^{1/d_i}\,,\quad \Lambda=-{1\over 2}\,d_i(d_i+1)\ ,\qquad
p={1\over d_i}\ , \quad
\alpha = \sqrt{{2(d_i-1)\over d_i}} , \nonumber\\
&& ds^2 = {R^2\over r^2} (-dt^2 + dr^2) + {t^{2p}\,R^2\over r^2} dx_i^2\,,
\qquad e^\Psi = t^\alpha\,,\qquad d_ip^2=1-{1\over 2}\alpha^2\ , \nonumber\\ [1mm]
\rightarrow\ \ && \phi={t\,R^{d_i}\over r^{d_i}}\,,\qquad
ds^2={t^{(d_i-1)/d_i}\,R^{d_i+1}\over r^{d_i+1}}(-dt^2+dr^2)\,,\qquad
e^\Psi=t^{\sqrt{2(d_i-1)/d_i}}\ .\quad
\end{eqnarray}
$R$ is the $AdS$ length scale. We are suppressing an implicit Kasner
scale $t_K$: {\it e.g.}\ $t^{2p}\rightarrow (t/t_K)^{2p}$. We will reinstate this as
required.
The higher dimensional spacetimes and their dual field theories were
in fact studied long back in \cite{Das:2006dz}-\cite{Awad:2008jf} as
certain kinds of time-dependent deformations of $AdS/CFT$ with the
hope of gaining insights via gauge/gravity duality into cosmological
(Big-Bang or -Crunch) singularities: some aspects of these were
reviewed in \cite{Bhattacharya:2020qil}. See also
\cite{Engelhardt:2014mea}-\cite{Engelhardt:2016kqb} for further
investigations on some of these. While the bulk spacetime develops a
cosmological Big-Crunch (or -Bang) singularity and breaks down, the
holographic dual field theory (in the $AdS_5$ case), living on a space
that itself crunches, is subject to a severe time-dependent gauge
coupling $g_{YM}^2=e^\Psi$ and may be hoped to provide insight into
the dual dynamics. In this case the scalar $\Psi$ controls the
gauge/string coupling. Generically it was found by analysing at weak
coupling that the gauge theory response also ends up appearing
singular \cite{Awad:2008jf}: however null singularities appear
better-behaved admitting weakly coupled CFT descriptions in certain
variables \cite{Das:2006pw}\ (a string worldsheet analysis in related
null Kasner singularities appears in \cite{Madhu:2009jh}). There is a
large family of such backgrounds exhibiting cosmological singularities
found long back: in these the deformations of the metric and string
dilaton $\Psi$ are constrained, suggesting that the dual CFT state is
likely nontrivial, with nontrivial non-generic initial conditions
required to create Big-Crunch singularities which are perhaps
qualitatively different from black holes (note that generic severe
time-dependent deformations on the vacuum state are expected to
thermalize on long timescales, dual to black hole formation in the
bulk). Some of these backgrounds have the technical feature of spatial
isotropy which allows studying these backgrounds from a possibly
simpler perspective, by carrying out a dimensional reduction on the
spatial directions with the ansatz (\ref{redux+Weyl}). This enables
the 2-dim dilaton gravity perspective (\ref{actionXPsiU}) formulated
in \cite{Bhattacharya:2020qil}, and also helps uncover new cosmologies
of the form (\ref{phie^fPsi-ansatz}) including ones with
nonrelativistic (hyperscaling violating Lifshitz) asymptotics,
reviewed briefly in Appendix~\ref{App:2dgES}.
\begin{figure}[h]
\hspace{1pc}
\includegraphics[width=15pc]{adsbbRT.pdf}
\hspace{2pc}
\begin{minipage}[b]{19pc}
\caption{{ \label{cosRT1}
\footnotesize{
Cartoon of extremal surfaces in $AdS$ Kasner spacetime,
anchored on a boundary time slice $t_0$ (extended as the grey
horizontal plane in the bulk). The extremal surface (red) bends
away from the singularity at $t=0$ (dotted line), {\it i.e.}\ $t_*>t_0$,\
with $(t_*,r_*)$ the turning point. \newline\newline
}}}
\end{minipage}
\end{figure}
Extremal surfaces can be studied as codim-2 surface probes of these
cosmological spacetimes. This is reliable if the surface is anchored on
a boundary subregion in the semiclassical region far from the singularity
where stringy or quantum gravity effects are not large. This can be
analysed in great detail as in \cite{Manu:2020tty}. The time-dependence
of the cosmology implies that the RT/HRT surface dips into the time
direction also, besides the radial (holographic) direction.
The resulting picture (focussing on strip-shaped subregions consistent
with the symmetries here) is as in Figure~\ref{cosRT1}.
The surface is parametrized as $(t(r),x(r))$ stretching in all $x_i$
directions except $x\in \{x_i\}$ which represents the width (size)
direction of the strip with $\Delta x=l$. The anchoring time slice is
$t(0)=t_0$: some details appear in Appendix~\ref{App:2dgES}.
The extremization of the time function $t(r)$ is more complicated due
to the time-dependence and gives a second order nonlinear differential
equation for $t(r)$. In the semiclassical region, we expect the
time-dependence to be mild so that ${dt\over dr}\equiv t'\ll 1$. This
leads to a slightly simpler, but still nonlinear, equation, which
however can be shown to admit power-series solutions,
\begin{equation}\label{t(r)Expn}
t(r) = t_0 + \sum_n c_n r^n\,, \quad c_n\sim {1\over t_0^\#}
\qquad\Rightarrow\quad t_*>t_0\ ,
\end{equation}
which can then be shown to satisfy $t_*\equiv t(r_*)>t_0$, in other
words, the surface bends in the direction away from the singularity.
This is straightforward to see (although involved) in the regime of
small subregion width (where $A={t_*\over r_*^2}\gtrsim {1\over t_0^2}$).
The analysis is a little more delicate in the IR limit where the
subregion becomes the whole space (and $A\rightarrow 0$): here we find
\begin{equation}
r_*\rightarrow\infty\,,\quad t_0\rightarrow\infty\,,\quad {t_0\over r_*}\lesssim 1;
\qquad t_*\gtrsim t_0\ .
\end{equation}
Thus the RT/HRT surface is driven to the region far from the singularity
(see also \cite{Engelhardt:2013jda} for similar observations in a
different context): in the IR limit this effectively means infinitely
far from the singularity since $t_*\gtrsim t_0\rightarrow\infty$.
Another notable example with similar behaviour is \cite{Hartman:2013qma},
where the Hartman-Maldacena surfaces exhibit a limiting surface in the
black hole interior.
\medskip
\noindent \underline{{\bf Quantum extremal surfaces}}
Now we briefly review the discussion of quantum extremal surfaces in
\cite{Manu:2020tty}.
Quantum extremal surfaces are extrema of the generalized entropy
$S_{gen} = S_{cl}+S_{bulk}$, the leading classical term being the area
of the extremal surface while the second term is the entropy of the
bulk matter in the region enclosed by the extremal surface and the
boundary. In 2-dim theories, the bulk entropy can be calculated by
using 2-dim CFT techniques. For instance, if the bulk matter is
approximated by a CFT in a curved space and is taken to be in the
ground state, then the bulk entropy can be obtained by a
generalization of the Calabrese-Cardy replica formulation
\cite{Calabrese:2004eu,Calabrese:2009qy} for a single spacetime
interval $\Delta^2$, giving (see Appendix~\ref{App:EE2dCFT})
\begin{equation}\label{Sgen0}
S_{gen} = {\phi\over 4G} +
{c\over 12}\log \big(\Delta^2 e^f|_{(t,r)} e^f|_{(t_0,r_0)}\big) ;\qquad
1\ll c\ll {1\over G}\ .
\end{equation}
The last condition arises from requiring that the bulk matter entropy
is non-negligible but not so large as to destabilize the leading
classical area contribution.
The 2-dim space is of the form $ds^2=e^f\eta_{\mu\nu}dx^\mu dx^\nu$ and
the Weyl factors above arise from the conformal transformation of
the twist operator 2-point function in the replica formulation, the
twist operators located at the endpoints of the interval in question
(between the boundary and the extremal surface).
As a simple time-independent example consider the 2-dim dilaton-gravity
background obtained from the dimensional reduction (\ref{redux+Weyl}) of
$AdS_{d_i+2}$ with metric in the Poincare slicing\
$ds^2_{AdS_{d_i+2}}={R^2\over r^2} (-dt^2+dr^2)+{R^2\over r^2} dx_i^2$\
($R$ is the $AdS$ scale). Some aspects of such generic 2-dim
dilaton gravity theories have been discussed in \cite{Narayan:2020pyj}.
This 2-dim background, the corresponding generalized entropy (\ref{Sgen0})
and its extremization give
\begin{eqnarray}\label{SgenAdSDred}
\phi = {R^{d_i}\over r^{d_i}}\ ,\ \ &&
ds^2 = {R^{d_i+1}\over r^{d_i+1}} (-dt^2+dr^2)\ ,\\
S_{gen} = {\phi_r\over 4G}\,{R^{d_i}\over r^{d_i}}
+ {c\over 12} \log \left( {r^2/\epsilon_{UV}^2\over (r/R)^{d_i+1}} \right)\ \
&\Rightarrow &\ \
\partial_rS_{gen} = -{d_i\phi_rR^{d_i}\over 4G\, r^{d_i+1}}
- {c\over 6}\,\Big({d_i-1\over 2}\Big)\,{1\over r} = 0\ . \nonumber
\end{eqnarray}
(We have written $S_{bulk}$ using the rules of boundary CFT since the
effective space is the half-line with one end of the interval at the
boundary $r=0$: see Appendix~\ref{App:EE2dCFT}. A useful resource for
QES calculations in time-independent cases is \cite{MahajanTalk}.)\
We see that both terms are of the same sign since $c>0$ and $d_i>1$.
Thus the solution is\ $r\equiv r_*\rightarrow\infty$ for the location of the QES:
this leads to the entire Poincare wedge which is the expected answer
(also in the higher dimensional point $AdS_D$ when the subsystem
becomes the whole space).
Thus in this case, there are no islands, {\it i.e.}\ regions disconnected from the
boundary defined {\it e.g.}\ by a finite location of the quantum extremal
surface\footnote{\label{IslAMM}
In {\it e.g.}\ \cite{Almheiri:2019yqk}, a flat non-gravitating (bath)
region was appended beyond the boundary $r=0$ of an $AdS_2$ region,
giving the generalized entropy\
$S_{gen}\sim {\phi_r\over 4G}{1\over r} + {c\over 6}\log ((r+r')^2\,{1\over r})$.
The interval in question has endpoints $r\in AdS_2$ and $r'$ in the
flat space region beyond the boundary: the warp factor at the $r'$ end
does not contribute since it is trivial in that flat region.
Both $r, r'>0$ in this parametrization: the space is not a half-line
now. Setting $r'\sim 0$ for simplicity and extremizing gives\
$-{\phi_r\over 4G}{1\over r^2} + {c\over 6}{1\over r} = 0$\,: the
competition between the two terms leads to a finite value
$r_*\sim {\phi_r\over Gc}$ for the QES location, {\it i.e.}\ an island.}.
One way to understand this is in terms of the violation of the
Bekenstein bound, as discussed in \cite{Hartman:2020khs}: if the
classical dilatonic term is overpowered by the subleading bulk
entropy contribution, we may expect islands. To see this, note that
(\ref{SgenAdSDred}) can be recast as
\begin{equation}\label{noIsl}
S_{gen} = {\phi\over 4G} + {c\over 12}\,{d_i-1\over d_i}\,\log\phi\ ,
\end{equation}
with a relative plus sign in the two contributions, retaining
only terms relevant for extremization. As long as $\phi$ is not too
small, the bulk entropy term scaling as $\log\phi$ is subdominant to
the classical area term scaling as $\phi$. If we entangle the bulk
matter with ``elsewhere'' then $S_{bulk}$ could increase possibly
leading to nontrivial competition with the classical area term and
thereby islands, as is the case in \cite{Almheiri:2019yqk}
and in various cases in \cite{Hartman:2020khs}.
Now we will study quantum extremal surfaces in the 2-dim cosmological
backgrounds reviewed earlier. We focus first on the 2-dim cosmology
obtained by reduction of the $AdS_D$ Kasner spacetime (\ref{AdSDK-2d}),
restricting attention to the observer at the holographic boundary at
$r=0$.
We carry out the extremization in the reliable semiclassical region far
from the singularity at $t=0$: the observer is at $(t_0,0)$.
Assuming for simplicity that the QES lies on the same time slice as
the observer {\it i.e.}\ $t=t_0$, equivalently that the QES is maximally
spacelike separated from the observer, it turns out that (\ref{Sgen0})
can be recast as
\begin{eqnarray}\label{SgenAdSKas-t=t0}
t=t_0:\qquad\quad
S_{gen} = {\phi\over 4G} + {c\over 6}\,{d_i-1\over d_i} \log\phi\ ,&& \quad
\phi={t\over r^{d_i}}\ ,\nonumber \\ [2mm]
\partial_rS_{gen} \sim
-\left( {\phi_r\over 4G}\,{d_i\,t\over r^{d_i+1}} + {c\over 12} {d_i-1\over r}
\right) = 0 ,&&
\partial_tS_{gen} \sim
{\phi_r\over 4G}\,{1\over r^{d_i}} + {c\over 12}\,{d_i-1\over d_i\,t} = 0\ .\ \
\end{eqnarray}
Since $c>0$ and $d_i>1$, both contributions in both derivative
expressions appear with the same sign. Note also that in this entire
discussion, we are on one side (the past) of the singularity at $t=0$,
so the range of the time variable is\ $t\equiv |t|\geq 0$.\ Then it is
clear that the only QES solution $(t_*,r_*)$ to extremization is
\cite{Manu:2020tty}
\begin{equation}\label{AdSKas-t*r*}
t\sim t_0\ ,\qquad r\equiv r_*\rightarrow\infty\ ,\qquad t\equiv t_*\rightarrow\infty\ ;
\qquad t_* \lesssim r_*\ ,
\end{equation}
{\it i.e.}\ the quantum extremal surface is driven to the semiclassical
region, infinitely far from the singularity at $t=0$. A more general
analysis vindicates this. Further, in this semiclassical region the
dilaton is not too small so there are no islands here since the
Bekenstein bound is not violated, as in (\ref{noIsl}).
\section{$AdS$ Kasner, quantum extremal surfaces, regulated}\label{sec:AdSKreg}
In what follows we will study various 2-dim backgrounds given by the
dilaton $\phi$ and the 2-dim metric $e^f$ and analyse quantum extremal
surfaces obtained from the extremization of the generalized entropy
(\ref{Sgen0}): in general these are of the form
\begin{eqnarray}\label{Sgen1}
&& \qquad\qquad
S_{gen}={\phi\over 4G}+{c\over 12}\log(\Delta^2\,e^f|_{(t,r)})\ ,
\qquad\qquad \Delta^2=r^2-(t-t_0)^2\,,\nonumber\\ [1mm]
&& {\partial_r\phi\over 4G} + {c\over 6}\,{r\over\Delta^2}
+ {c\over 12}\partial_rf = 0\,,
\qquad
{\partial_t\phi\over 4G} - {c\over 6}\,{t-t_0\over\Delta^2} +
{c\over 12} \partial_tf = 0\ ,
\end{eqnarray}
where we have retained only terms relevant for extremization.
These are all spaces with a holographic boundary so we are using
the corresponding expression for the generalized entropy in
Appendix~\ref{App:EE2dCFT}.
We would like to understand the dependence of the quantum extremal
surface ($t_*,r_*)$ on the observer location $(t_0,r_0)\equiv (t_0,0)$\,:
we will focus on the observer at the holographic boundary $r=0$.\
Here we study the $AdS$ Kasner case: we will put back the $AdS$ scale
$R$ and the Kasner scale $t_K$ in (\ref{AdSDK-2d}) so the lengthscales
are manifest. Then the dilaton and 2-dim metric become
\begin{equation}\label{AdSK-RtK}
\phi={t/t_K\,\over (r/R)^{d_i}}\,,\qquad
ds^2={(t/t_K)^{(d_i-1)/d_i}\over (r/R)^{d_i+1}}(-dt^2+dr^2)\ .
\end{equation}
Towards understanding quantum extremal surfaces, let us study
(\ref{Sgen1}) with the scales put in explicitly as in (\ref{AdSK-RtK}).
If we assume $t=t_0$, we obtain (\ref{SgenAdSKas-t=t0}),
(\ref{AdSKas-t*r*}), which are structurally similar to the $AdS$ case
(\ref{SgenAdSDred}). We will instead attempt solving for $t$ as a
function of $t_0$. Then the extremization equations are
(introducing $\phi_r$ as bookkeeping for now)
\begin{equation}\label{ExtSgen-r0t0}
{c\over 6}\,{r\over \Delta^2} =
{\phi_r\over 4G}\,{d_i\,t/t_K\over \,r^{d_i+1}/R^{d_i}}
+ {c\over 12}\,{d_i+1\over r}\,,\qquad\quad
{c\over 6}\,{t-t_0\over \Delta^2} =
{\phi_r\over 4G}\,{1/t_K\over \,r^{d_i}/R^{d_i}}
+ {c\over 12}\,{d_i-1\over d_i\,t}\ .
\end{equation}
Note that each term now has dimensions of inverse length manifestly.
In the parametrization of these cosmologies (\ref{AdSDK-2d}), the
singularity is at $t=0$: regarding this as a Big-Crunch, we take
the time coordinate $t$ to represent $|t|$ so that $t>0$ in our
entire discussion.
We require that the QES is spacelike-separated from the observer,
consistent with the interpretation of these extremal surfaces as
holographic entanglement. This implies
\begin{equation}\label{t>t0}
\qquad \Delta^2 > 0\quad\Rightarrow\quad t_*>t_0\ ,\qquad\qquad\qquad
[\Delta^2=r^2-(\Delta t)^2]
\end{equation}
from the $t$-equation in (\ref{ExtSgen-r0t0}). This means that the
QES always lags behind the observer, in the direction away from the
singularity ($t=0$).
Let us now look in more detail at QES solutions near the semiclassical
solution (\ref{AdSKas-t*r*}), where $\Delta t\sim 0$ and
$r, t\rightarrow\infty$. Let us first rewrite the $r$-extremization equation
in (\ref{ExtSgen-r0t0}) as
\begin{equation}\label{rExt-reg}
{3\phi_r\over Gc}{d_i\,t/t_K\over \,r^{d_i+1}/R^{d_i}}
+ \Big({d_i+1\over r} - {2r\over\Delta^2}\Big)
= {3\phi_r\over Gc}{d_i\,t/t_K\over \,r^{d_i+1}/R^{d_i}}
+ {d_i+1\over r} \Big( {{d_i-1\over d_i+1} r^2 - (\Delta t)^2
\over r^2-(\Delta t)^2} \Big) = 0
\end{equation}
As long as $\Delta t$ is small, {\it i.e.}\ $\Delta^2\sim r^2$, the second
term is positive: thus both terms are positive, the only solution
to this being $r\equiv r_*\rightarrow\infty$. This is very similar to the
time-independent $AdS$ case in (\ref{SgenAdSDred}), giving the
entire Poincare wedge as the entanglement wedge: there are no islands.
Analysing the $t$-extremization equation is rendered tricky with
$r_*\rightarrow\infty$ strictly. Towards obtaining insight into the $t_0$
dependence of $t_*$, let us regulate as $r_*=R_c\sim\infty$ with some
large but finite spatial cutoff $R_c$ that represents the boundary
of the entanglement wedge. Then the $t$-equation in (\ref{ExtSgen-r0t0})
becomes
\begin{equation}\label{tExt-reg}
\qquad{\Delta t\over R_c^2-(\Delta t)^2} =
{1\over 2K_c}
+ {d_i-1\over 2 d_i\,t}\ ,
\qquad\qquad
{1\over K_c} = {3\phi_r\over Gc} {1/t_K\over \,R_c^{d_i}/R^{d_i}}\ .
\end{equation}
This expression is manifestly satisfied semiclassically as in
(\ref{AdSKas-t*r*}). Taking these regulated equations as containing
finite terms we can solve for $t_*$\,: with $\Delta t\ll R_c$, we
obtain the approximate regulated expression
\begin{equation}\label{tExt-reg2}
\qquad {\Delta t\over R_c^2} \sim
{1\over 2K_c}
+ {d_i-1\over 2 d_i\,t_0}\ ,
\qquad\quad \Delta t = t_*-t_0\ ,
\end{equation}
where we have approximated $\Delta^2\sim R_c^2$ and set $t\sim t_0$ in
the last expression (with $t_0$ large, as in (\ref{AdSKas-t*r*}))\
(there is some similarity with the semiclassical expansion (\ref{t(r)Expn})).
We see that the QES (\ref{tExt-reg2}) lags behind the observer, in
the direction away from the singularity.
We now see that as $t_0$ decreases, $\Delta t$ increases, {\it i.e.}\ the
lag of the QES is increasing: see the top part of Figure~\ref{cosQES3}
for a heuristic depiction (the lag is exaggerated!).
\begin{figure}[h]
\hspace{2pc}
\includegraphics[width=11pc]{figcosQES3.pdf}
\hspace{3pc}
\begin{minipage}[b]{21pc}
\caption{{ \label{cosQES3}
\footnotesize{Cartoon of the 2-dim $AdS$ Kasner geometry
(singularity at $t=0$), the holographic boundary at $r=0$ and
the QES at $(t_*,r_*)$, with a time-independent $AdS$ space
appended for $t>t_K$. The boundary observer $(t_0,0)$ moves in
time from the time-independent region to the $AdS$ Kasner region.
The QES lags behind in time, {\it i.e.}\ $t_*>t_0$, when $t_0$ is in
the Kasner region.
\newline \newline
}}}
\end{minipage}
\end{figure}
The on-shell generalized entropy (\ref{Sgen1}) in the semiclassical
regime where $\Delta^2\sim R_c^2$ becomes
\begin{equation}\label{SgenOS-semicl}
S_{gen}^{o.s.} \sim {\phi_r\over 4G}\,{t_*/t_K\over (R_c/R)^{d_i}} +
{c\over 12} \log \left( {R_c^2\over \epsilon_{UV}^2}\,
{(t_*/t_K)^{(d_i-1)/d_i}\over (R_c/R)^{d_i+1}} \right)\ ,
\end{equation}
with $t_*$ in (\ref{tExt-reg2}).
Since $t_*\gtrsim t_0$ and $R_c$ is large, $S_{gen}^{o.s.}$ is not
dramatically different structurally from the $AdS$ value
(\ref{SgenAdSDred}), without the $t_*/t_K$ factors. In more detail, we see that the on-shell
$AdS$ expression (\ref{SgenAdSDred}) with $r_*=R_c$ and $\phi|_{r_*}=\phi_*$
becomes\
$S^{o.s.}={\phi_*\over 4G} + {c\over 12}\log\big({R^2\over\epsilon_{UV}^2}
({\phi_*\over\phi_r})^{^{(d_i-1)/d_i}}\big)$ so the log
vanishes when its argument becomes $O(1)$, {\it i.e.}\ when $\phi_*$ is
sufficiently small. At this point, $S^{o.s.}\sim {\phi_*\over 4G}\sim 0$,
in accord with the physical expectation that the $AdS$ ground state
has zero entropy. In this sense the spatial regulator $R_c$ has
physical meaning as the effective physical boundary of the
entanglement wedge, where $\phi_*$ becomes small enough to be comparable
with $({\epsilon_{UV}\over R})^{^{\#}}$. Note that we can recast $S^{o.s.}$
as (\ref{noIsl}) exactly setting\
${1\over\phi_r^{(d_i-1)/d_i}} {R^2\over\epsilon_{UV}^2}\sim 1$ thus fixing
$\phi_r$, which can possibly be regarded as renormalizing
${\phi_r\over G}\equiv {1\over G_r}$\ (and rendering $S_{gen}$ finite).
The above expression (\ref{SgenOS-semicl}) is similar when the
$t_*/t_K$ factors are $O(1)$ so the above arguments apply, and the
overall entropy is not appreciable.
As a further check, note that this QES solution vindicates the maximin
property\footnote{In the semiclassical regime, the
second derivatives\ \ $\partial_t^2S_{gen}|_*\sim
-{c\over 12}\,{d_i-1\over d_i\,t_*^2} - {c\over 6}\,{1\over\Delta_*^2}
- {c\over 3}\,{(t_*-t_0)^2\over \Delta_*^4} < 0$\ \ and \ $\partial_r^2S_{gen}|_*\sim {\phi_r\over 4G}\,{d_i\,(d_i+1)\,t_*\,R^d_i\over t_k\, R_c^{d_i+2}} + {c\over 12}\,{d_i+1\over R_c^2} + {c\over 6}\,{1\over \Delta_*^2}\,{(1-{2R_c^2\over \Delta_*^2})} >0$\ confirm time-maximization and spatial minimization, with
the regulator $R_c$ finite.}.
Naively it appears that ${\Delta t\over R_c^2} \sim {1\over t_0}$ shows
a growth as $t_0$ decreases. Rewriting (\ref{tExt-reg}) and solving
as a quadratic, taking $\Delta t > 0$, gives
\begin{equation}
{\Delta t\over R_c} = {1\over R_c} \left( \sqrt{ {1\over
({1\over K_c} + {d_i-1\over d_i\,t} )^2} + R_c^2}\ -\
{1\over {1\over K_c} + {d_i-1\over d_i\,t}} \right) \ ,
\end{equation}
showing a slow growth in $\Delta t$ as $t$ decreases, for fixed
regulator $R_c$. Extrapolating and setting $t_0=0$ shows that $t=0$
is not a solution (this can also be seen in (\ref{ExtSgen-r0t0})).
Our analysis is best regarded as valid in the semiclassical regime,
far from the singularity, approximating bulk matter to be in the
ground state. However perhaps the qualitative feature of the quantum
extremal surfaces and the associated entanglement wedge excluding the
near singularity region (depicted schematically in the top, $AdS$
Kasner, part of Figure~\ref{cosQES3}) will remain as a reliable result
even with better near singularity bulk entropy models.
We note that the $S_{gen}^{o.s.}$ (\ref{SgenOS-semicl}) decreases with
time evolution towards the singularity (this is reminiscent of
{\it e.g.}\ \cite{Barbon:2015ria,Caputa:2021pad} revealing low complexity in
such singularities). Recasting this semiclassical value in the form
(\ref{noIsl}), we note that as long as $\phi$ is not too small, the
bulk entropy term is subleading to the area term. Thus the Bekenstein
bound is not violated and there are no spatially disconnected islands
of the kind noted in black holes. In some qualitative sense, it is
tempting to regard the excluded near singularity region as a timelike
separated island-like region: it would be interesting to understand
this better.
\subsection{Searching for islands}\label{sec:AdSK-islands}
Looking now at (\ref{rExt-reg}), we see that for
\begin{equation}\label{islandRegime}
{d_i-1\over d_i+1} r^2 < (\Delta t)^2 < r^2\ ,
\end{equation}
a spacelike-separated island appears to emerge. Unlike the semiclassical
region with $\Delta t\ll r$ (where both terms are the same sign), the
numerator in the term in brackets in (\ref{rExt-reg}) now changes sign
indicating a large but finite $r\sim ({\phi_r\over Gc})^\#$ solution
leading to a disconnected region: there is some structural similarity
to the discussion in \cite{Almheiri:2019yqk} (see Footnote~\ref{IslAMM}).
Towards exploring this in detail, first, note that the $\partial_r$-equation
in (\ref{ExtSgen-r0t0}) can be rewritten as
\begin{equation}\label{delr-Delta^2}
\qquad
\Delta^2 = {2 r^2\over d_i+1}\, {1\over 1+{d_i\over d_i+1} {t\over K}}\ ,
\qquad\quad {1\over K} = {3\phi_r\over Gc} {1/t_K\over \,r^{d_i}/R^{d_i}}\ ,
\end{equation}
so
\begin{equation}\label{Deltat/r-3}
\Delta^2=r^2-(\Delta t)^2\quad\Rightarrow\quad
{\Delta t\over r} = \sqrt{ {{d_i-1\over d_i+1}\
+\ {d_i\over d_i+1} {t\over K} \over
1\ +\ {d_i\over d_i+1} {t\over K} }}\ .
\end{equation}
The potential island arises at large finite $r$ in (\ref{rExt-reg}) when
\begin{equation}\label{islandBndry}
(\Delta t)^2\gtrsim {d_i-1\over d_i+1} r^2
\end{equation}
so that $\Delta t$ is not small but in fact scales as $r$ which is
large.
Expanding (\ref{Deltat/r-3}) in the vicinity of (\ref{islandBndry}) gives
\begin{equation}\label{Deltat/r-3-Expn}
{\Delta t\over r^2} \sim \sqrt{{d_i-1\over d_i+1}}\,{1\over r}
\left( 1 + {d_i\over d_i^2-1}\,{t\over K} + \ldots \right) .
\end{equation}
Now the $\partial_t$-equation (\ref{ExtSgen-r0t0}) as an exact quadratic
can be solved to obtain (choosing $\Delta t>0$)
\begin{equation}\label{Dtr-tEx-quadr}
{\Delta t\over r} = \sqrt{ {{d_i^2\over (d_i-1)^2}\,{t^2\over r^2}\over
({d_i\over d_i-1}{t\over K} + 1 )^2} + 1}\ -\
{{d_i\over d_i-1}\,{t\over r}\over {d_i\over d_i-1}{t\over K} + 1}\ ,
\end{equation}
with $K$ defined in (\ref{delr-Delta^2}).\ For a nontrivial island-like
solution, this expression for ${\Delta t\over r}$ must match that in
(\ref{Deltat/r-3}) in the vicinity of the island boundary
(\ref{islandBndry}). With ${t\over K}\sim \epsilon$ being small, we
expand and obtain at leading order
\begin{equation}\label{Isl-leadingterm0}
\sqrt{1+x^2} - x = \sqrt{{d_i-1\over d_i+1}}\quad\xrightarrow{solving}\quad
x\equiv {d_i\over d_i-1}\,{t\over r} = {1\over\sqrt{d_i^2-1}}
\end{equation}
This gives
\begin{equation}\label{Isl-leadingterm}
\Delta t \gtrsim \sqrt{{d_i-1\over d_i+1}}\ r \sim d_i\, t\ .
\end{equation}
The last condition $\Delta t \gtrsim d_i\,t$ is clearly impossible
with $\Delta t=t-t_0$ for any $d_i>1$.
In addition, using the leading term matching condition
(\ref{Isl-leadingterm}) and expanding (\ref{Dtr-tEx-quadr}) about
the potential island boundary (\ref{islandBndry}) shows that the
first subleading term in ${t\over K}$ is\
${1\over d_i}\sqrt{{d_i+1\over d_i-1}} {t\over K}$\ which does not
match the first subleading term in (\ref{Deltat/r-3-Expn}).
We have been looking for an island-like solution in the vicinity of
the potential island boundary (\ref{islandBndry}) emerging
continuously from the semiclassical region where $r_*\rightarrow\infty$ as
discussed after (\ref{rExt-reg}). So we require a simultaneous
solution to the extremization equations (\ref{ExtSgen-r0t0}) recast as
(\ref{Deltat/r-3}) and (\ref{Dtr-tEx-quadr}), just inside the island
region. Then at the very least the leading and first subleading terms
in the expansions of (\ref{Deltat/r-3}) and (\ref{Dtr-tEx-quadr}) near
(\ref{islandBndry}) must agree, which is not the case. Thus this
potential island solution is inconsistent.
One could ask if there are nontrivial islands further away, towards
the singularity (although they may not be physically reliable). In
this regard, we can write $\Delta t=t-t_0$ and expand out the $r$- and
$t$-extremization equations (\ref{rExt-reg}), (\ref{tExt-reg})\,: this
leads to two cubic equations in $t$. However, taking them as
simultaneously true (and {\it e.g.}\ eliminating the $t^3$ term), it appears
that there are no consistent finite $r, t$ solutions to these, {\it i.e.}\ no
islands.
\subsection{Appending a time-independent far region}
Let us now consider appending the $AdS$ Kasner space with a
time-independent $AdS$ region far from the singularity, joined
at the Kasner scale $t=t_K$. See Figure~\ref{cosQES3}. So we have
$AdS$ Kasner for $t<t_K$ and the time-independent $AdS$ space for
$t>t_K$, {\it i.e.}\
\begin{eqnarray}\label{AdSK-RtK-reg}
&& \phi={t/t_K\,\over (r/R)^{d_i}}\,,\qquad
ds^2={(t/t_K)^{(d_i-1)/d_i}\over (r/R)^{d_i+1}}(-dt^2+dr^2)
\qquad\quad [t<t_K]\ , \nonumber\\
&& \phi={1\over (r/R)^{d_i}}\,,\qquad\ \ \
ds^2={1\over (r/R)^{d_i+1}}(-dt^2+dr^2) \qquad\qquad [t>t_K]\ .
\end{eqnarray}
The spaces are joined continuously at $t=t_K$ but the joining is not
smooth.
Now the extremization equations must be analysed separately as the
observer at $t_0$ moves through each region. The generalized entropy
and its extremization (\ref{Sgen1}) applied to the background profiles
(\ref{AdSK-RtK-reg}) in both regions give
\begin{eqnarray}
t_0> t_K: &&
{c\over 6}\,{r\over \Delta^2} =
{\phi_r\over 4G}\,{d_i\over \,r^{d_i+1}/R^{d_i}}
+ {c\over 12}\,{d_i+1\over r}\,,\qquad
{c\over 6}\,{t-t_0\over \Delta^2} = 0\ ; \\
t_0< t_K: && {c\over 6}\,{r\over \Delta^2} =
{\phi_r\over 4G}\,{d_i\,t/t_K\over \,r^{d_i+1}/R^{d_i}}
+ {c\over 12}\,{d_i+1\over r}\,,\qquad
{c\over 6}\,{t-t_0\over \Delta^2} =
{\phi_r\over 4G}\,{1/t_K\over \,r^{d_i}/R^{d_i}}
+ {c\over 12}\,{d_i-1\over d_i\,t}\ . \nonumber
\end{eqnarray}
In the time-independent region $t>t_K$ we see it is physically reasonable
to set $t_*=t_0$, {\it i.e.}\ the QES lies on the same time slice as the observer.
This follows from time-translation invariance in that region at least
for $t_0\gg t_K$ (far from the junction at $t_K$). Since the joining
slice $t_K$ is in the semiclassical region far from the singularity,
it is adequate to use (\ref{tExt-reg2}) with the regulator to study
the time evolution of the QES in the Kasner region. The lagging (or
repulsive) feature of the QES thus begins once the observer transits
into the Kasner region (the sharp joining at $t_k$ implies that the
lag does not evolve smoothly).
To see this in more detail, consider the time $t_0=t_K-\delta t_0$ when
the observer is just entering the Kasner region: then we expect that
the quantum extremal surface is just a little away from the observer
time slice $t_0$. To quantify this, let us compare $\delta t_*$ in
(\ref{tExt-reg2}) with $\delta t_0$\ (and $K_c$ defined in
(\ref{tExt-reg})): we have
\begin{equation}
\delta t_0=t_K-t_0>0\ ;\qquad
{\delta t_*\over R_c^2} = {t_*-t_0\over R_c^2} \sim
{1\over 2K_c} + {d_i-1\over 2d_i\,t_K} \Big( 1 + {\delta t_0\over t_K} \Big) ,
\end{equation}
so that for small $\delta t_0$ {\it i.e.}\ $t_0\sim t_K$, the quantum extremal
surface ends up being pushed to the time-independent region ($t_*>t_K$).
Of course as the observer moves in time further, the QES enters the
Kasner region as well. To see this further, let us compare the QES
location with the Kasner scale: with $t_0\lesssim t_K$, we
have
\begin{equation}
t_*\lesssim t_K \qquad\Rightarrow\qquad
{t_K-t_0\over R_c^2} \gtrsim {1\over 2K_c} + {d_i-1\over 2d_i\,t_0}
\end{equation}
In other words, the quantum extremal surface is within the
Kasner region if the observer is sufficiently further within.
The cross-over of the QES to the Kasner region occurs when $t_*\sim t_K$,
{\it i.e.}\ when the above inequality is saturated (giving
${t_0-t_K\over R_c^2} \sim - {1\over 2K_c} - {d_i-1\over 2d_i t_K}$).
The model (\ref{AdSK-RtK-reg}) is just meant as a simple toy model for
gaining some insight into the evolution of the quantum extremal
surface as the observer transits from the time-independent far region
into the time-dependent $AdS$ Kasner region towards the
singularity. The existence of the time-independent far region suggests
that one can prepare the initial state as the ground state via a
Euclidean continuation. Putting this on firmer footing is however more
tricky. There is a discontinuity at the $t=t_K$ slice perhaps
reflecting the fact that the Kasner time-dependence does not switch
off at $t_K$: this might imply additional concerns in smooth time
evolution into the Kasner region (without any external energy-momentum
inflow). More detailed analysis of this requires detailed
understanding of the junction conditions for joining up the spacetimes
at $t_K$. Perhaps rather than a sharp time slice at $t_K$, it would be
more physical to find a thickened spacetime region interpolating
smoothly between the time-independent far region and the Kasner
region: then the QES lag is likely to evolve smoothly. We will leave
these questions for the future.
\subsection{More general 2-dim cosmologies, QES, regulated}
In the previous subsections, we studied $AdS$-Kasner cosmologies and
their 2-dim reflections obtained by dimensional reduction
(\ref{AdSDK-2d}), and quantum extremal surfaces. Now we will extend
this to more general 2-dim cosmologies (\ref{phie^fPsi-ansatz}).
We have the 2-dim dilaton and metric fields of the form
\begin{equation}\label{genCos}
\phi=t r^m\,,\quad e^f=t^ar^b\,,\qquad a>0,\ \ \ m<0,\ \ \ b<0\ .
\end{equation}
Note that we have taken the time exponent of the dilaton in accord
with the universality (\ref{univSing}) of the near singularity region
found in \cite{Bhattacharya:2020qil}. We take $a>0$ to simulate a
Big-Crunch singularity at $t=0$. Further we assume $m, b<0$ in accord
with the intuition that the dilaton and the 2-dim metric grow towards
the holographic boundary at $r=0$.
The generalized entropy (\ref{Sgen1}) and its extremization with $r, t$, give
\begin{equation}\label{ExtSgen-r0t0-gen}
{c\over 6} {r\over\Delta^2}
= {\phi_r\over 4G} {|m| t\over r^{|m|+1}} + {c\over 12} {|b|\over r}\ ,
\qquad
{c\over 6} {t-t_0\over\Delta^2}
= {\phi_r\over 4G} {1\over r^{|m|}} + {c\over 12} {a\over t}\ ,\quad
\end{equation}
analogous to (\ref{ExtSgen-r0t0}), except that we have suppressed
length scales analogous to $R,\, t_K$ here. Firstly, requiring the
spacelike condition $\Delta^2>0$ implies $t_*>t_0$, analogous to
(\ref{t>t0}): this means the QES lags behind the observer, in the
direction away from the singularity at $t=0$.
As noted already in \cite{Manu:2020tty}, it is clear that the QES
solution to these extremization equations is again of the form
(\ref{AdSKas-t*r*}), {\it i.e.}\ $r_*\rightarrow\infty,\ t_*\sim t_0\rightarrow\infty$ with
$t_*\lesssim r_*$. In the vicinity of the semiclassical region,
analogous to the $AdS$ Kasner case (\ref{rExt-reg}) we can recast
the $r$-equation as
\begin{equation}\label{rExt-reg-gen}
{3\phi_r\over Gc} {|m| t\over r^{|m|+1}}
+ \Big({|b|\over r} - {2r\over\Delta^2}\Big) =
{3\phi_r\over Gc} {|m| t\over r^{|m|+1}}
+ {|b|\over r} \Big({{|b|-2\over |b|}\, r^2 - (\Delta t)^2
\over r^2-(\Delta t)^2} \Big) = 0\ .
\end{equation}
As in that case, with $\Delta t$ small, {\it i.e.}\ $\Delta^2\sim r^2$,
both terms are positive and the only solution to this is $r_*\rightarrow\infty$,
giving the entire Poincare wedge as the entanglement wedge: there are
no islands.\
Now, the $t$-equation becomes
\begin{equation}\label{tExt-reg-gen}
{\Delta t\over R_c^2-(\Delta t)^2} =
{3\phi_r\over 2Gc} {1/t_K\over \,R_c^{d_i}/R^{d_i}} + {d_i-1\over 2 d_i\,t}\ ,
\end{equation}
analogous to (\ref{tExt-reg}).
As before, we are regulating the QES solution as $r_*=R_c\sim\infty$
with some large but finite spatial cutoff $R_c$ representing the
boundary of the entanglement wedge.
Taking these regulated equations as containing finite terms we can
solve for $t_*$\,, obtaining an approximate regulated expression
analogous to (\ref{tExt-reg2}) after setting $\Delta^2\sim R_c^2$
and $t\sim t_0$. The resulting semiclassical picture is similar to
the discussion in the $AdS$ Kasner case, with the QES lag increasing
as $t_0$ decreases.
Now let us look for island-like solutions in these more general
holographic cosmologies, analogous to Sec.~\ref{sec:AdSK-islands}.
The corresponding island boundary here, analogous to (\ref{islandBndry}), is
\begin{equation}\label{islandBndry-gen}
(\Delta t)^2\gtrsim {|b|-2\over |b|} r^2\ .
\end{equation}
Analogous to (\ref{Deltat/r-3}) and (\ref{Dtr-tEx-quadr}) in the
$AdS$ Kasner case, we obtain, respectively,
\begin{equation}\label{Deltat/r-3-gen}
\Delta^2=r^2-(\Delta t)^2\quad\Rightarrow\quad
{\Delta t\over r} = \sqrt{ {{|b|-2\over |b|}\
+\ {|m|\over |b|} {t\over K} \over
1\ +\ {|m|\over |b|} {t\over K} }}\ ,
\qquad {1\over K} = {3\phi_r\over Gc} {1\over r^{|m|}}\ ,
\end{equation}
rearranging (\ref{rExt-reg-gen}), and
\begin{equation}\label{Dtr-tEx-quadr-gen}
{\Delta t\over r} = \sqrt{ {{1\over a^2}\,{t^2\over r^2}\over
({1\over a}{t\over K} + 1 )^2} + 1}\ -\
{{1\over a}\,{t\over r}\over {1\over a}{t\over K} + 1}\ ,
\end{equation}
from the $\partial_t$-equation in (\ref{ExtSgen-r0t0-gen}) regarded as
a quadratic, choosing $\Delta t>0$.
For a nontrivial island-like solution emerging in the vicinity of
(\ref{islandBndry-gen}), these two expressions for ${\Delta t\over r}$
must match: expanding, the leading order terms give
\begin{equation}\label{Isl-leadingterm-gen}
x\equiv {1\over a}\,{t\over r}\,:\quad
\sqrt{1+x^2} - x = \sqrt{{|b|-2\over |b|}}\quad\xrightarrow{solving}\quad
{t\over r} = {a\over\sqrt{|b| (|b|-2)}}\ ,
\end{equation}
while matching the first subleading terms requires
\begin{equation}
{1\over a\sqrt{|b| (|b|-2)}} \Big( 1 - {1\over \sqrt{|b| (|b|-2) + 1}} \Big)
\,{t\over K} = {|m|/|b|\over \sqrt{|b| (|b|-2)}}\,{t\over K}
\end{equation}
{\it i.e.}\
\begin{equation}\label{Isl-subleadingterm-gen}
{a |m|\over |b|} = 1 - {1\over \sqrt{|b| (|b|-2) + 1}}
\end{equation}
For the $AdS$ Kasner values $a={d_i-1\over d_i}\,,\
m=-d_i\,,\ b=-(d_i+1)$, these agree with the conditions obtained in
Sec.~\ref{sec:AdSK-islands}, which were not consistent as we saw.
The condition (\ref{Isl-leadingterm-gen}) gives
$\Delta t = {|b|-2\over a} t$\,: this is impossible in the $AdS$
Kasner case (\ref{Isl-leadingterm}) as we saw.
For the hyperscaling violating cosmologies (\ref{hvL-2d}), this
condition can again be shown to be impossible to satisfy ($a$ takes its
maximum value for $\gamma=0$). The hyperscaling violating Lifshitz
cosmologies in \cite{Bhattacharya:2020qil} require $a=|b|-2\,,\ m=-1$\
(reviewed very briefly after (\ref{hvL-2d})). This gives\
$\Delta t = {|b|-2\over a} t = t$\,,\ which is satisfied for $t_0=0$,
but this is the location of the singularity which is unreliable\
(the condition (\ref{Isl-subleadingterm-gen}) becomes\
${2\over b}={1\over \sqrt{|b| (|b|-2) + 1}}$\ giving $b=-2, a=0$).
Thus overall, these more general holographic cosmologies appear
qualitatively similar to the $AdS$ Kasner case.
The conditions (\ref{genCos}) on the exponents are motivated by the
more general investigations on 2-dimensional cosmologies in
\cite{Bhattacharya:2020qil}. These investigations employ fairly
general and minimal assumptions on the effective action governing such
cosmological spacetimes: the resulting space of cosmologies is quite
rich, including ones with nonrelativistic ({\it e.g.}\ hyperscaling violating
Lifshitz) asymptotics and boundary conditions, and they all satisfy
the conditions (\ref{genCos}). However it would be interesting to
explore the space of such cosmologies, possibly enlarging them
(including those that do not admit reduction to 2-dimensions), towards
understanding the behaviour of quantum extremal surfaces with regard
to the Big-Crunch (-Bang) singularities they may exhibit.
\section{Null cosmologies and quantum extremal surfaces}\label{sec:null}
We consider cosmological spacetimes with null time-dependence in this
section: there are parallels with the discussions in
\cite{Das:2006dz,Das:2006pw,Madhu:2009jh}, as well as
{\it e.g.}\ \cite{Horowitz:1989bv,Craps:2005wd,Chu:2006pa,Lin:2006ie,Craps:2008bv}.
If we further require that the higher dimensional spacetime admits
dimensional reduction (\ref{redux+Weyl}) to 2-dimensions, this reduces
to a restricted family of 2-dimensional backgrounds of the form
\begin{equation}\label{nullBgnd}
ds^2=-dx^+dx^-\,,\qquad \phi=\phi(x^+)\,,\qquad \Psi=\Psi(x^+)\ ,
\qquad x^\pm = t\pm r\ .
\end{equation}
The 2-dim metric can always be coordinate-transformed to be flat if
we only have $x^+$-dependence in $\phi, e^f$ in the reduction ansatz
(\ref{redux+Weyl}), leading to the above. The upstairs spacetime
(\ref{phie^fPsi-ansatz}) then is
\begin{equation}\label{nullUpstairs}
ds^2 = -\phi^{-(d_i-1)/d_i} dx^+dx^- + \phi^{2/d_i} dy_i^2\ ,
\qquad x_i=\{r,y_i\}\ .
\end{equation}
This comprises various higher dimensional backgrounds with null
singularities {\it e.g.}\
\begin{equation}\label{nullUpstairs2}
ds^2=(x^+)^a (-dx^+dx^-) + (x^+)^bdy_i^2
\end{equation}
which however are somewhat special, given the restriction to
the 2-dimensional reduction ansatz (\ref{redux+Weyl}): thus it also
does not include the null holographic $AdS$ cosmologies in
\cite{Das:2006dz,Chu:2006pa,Lin:2006ie,Das:2006pw} which are of the form\
$ds^2={R^2\over r^2}[e^{f(x^+)}(-dx^+dx^-+dx_i^2)+dr^2]$\,. There are
qualitative parallels however. The exponents $a,b$ in
(\ref{nullUpstairs2}) are related by the Einstein equations. These
are a bit similar to the null Kasner backgrounds considered in
\cite{Madhu:2009jh}, except that the 2-dim restriction implies that
$e^f\equiv (x^+)^a$ can be absorbed by redefining the null time
variable $x^+\rightarrow X^+=\int e^f dx^+$. In writing the 2-dim backgrounds
(\ref{nullBgnd}) we have effectively redefined the lightcone variables
$x^\pm$ in this manner. These backgrounds are likely supersymmetric.
Now the equations of motion (\ref{2dimseom-EMD0-Psi}) simplify
tremendously since there is only null-time dependence in the
background ansatze (\ref{nullBgnd}): for instance all nontrivial
contractions of the form $g^{\mu\nu}\partial_\mu\Psi\partial_\nu\Psi\sim
g^{+-}\partial_+\Psi\partial_-\Psi$ vanish since there is no $x^-$-dependence.
We also have ${\cal R}=0$ since the 2-dim space is flat. Thus the
equations of motion give
\begin{eqnarray}\label{EOMnull}
(++): && -\partial_+^2\phi - {\phi\over 2} (\partial_+\Psi)^2 = 0\ ;\qquad \\
(\phi):\ \ \ {\partial U\over \partial\phi} = {\cal R}
- {1\over 2}(\partial\Psi)^2 = 0\ ; &&\quad
(\Psi):\ \ \ {\partial U\over\partial\Psi} = \partial_\mu(\phi g^{\mu\nu}\partial_\nu\Psi) = 0\ .
\nonumber
\end{eqnarray}
These imply that the dilaton potential is trivial and give a single
nontrivial condition from the $(++)$ equation relating $\phi, \Psi$.
We want to consider a Big-Crunch singularity arising at $x^+=0$ as a
future null singularity, so we take $x^+<0$ in our entire discussion
below. Then
\begin{eqnarray}\label{null2d}
&& \phi=(-x^+)^k\,,\quad \Psi=\Psi(x^+)\quad \Rightarrow\quad
(\partial_+\Psi)^2 = -2 {\partial_+^2\phi\over\phi} = - {2k(k-1)\over (x^+)^2}\ ,
\nonumber\\
&& \Rightarrow\quad 0<k\leq 1\,,\qquad \phi=(-x^+)^k\,,\qquad
e^\Psi = (-x^+)^{\pm\sqrt{2k(1-k)}}\ .
\end{eqnarray}
While $k>0$ gives vanishing dilaton as $x^+\rightarrow 0$, the exponent of
$e^\Psi$ could have either sign. The single $\phi,\Psi$-relation allows
extrapolating $\phi, \Psi$ above to asymptotically constant functions
{\it i.e.}\ flat space.
This 2-dim background implies the upstairs background
(\ref{nullUpstairs}) with $\phi$ as above: this is of the form
(\ref{nullUpstairs2}) with $a=-{k(d_i-1)\over d_i}$ and $b={2k\over d_i}$\,.
These have\
$R^i{_{+i+}}={k(1-k)\over d_i\,(x^+)^2}$ so tidal forces diverge\
(all curvature invariants vanish due to the null nature of the
backgrounds). To see this in more detail, consider a null geodesic
congruence propagating along $x^+$ with cross-section along some
$y^i$-direction: the geodesic equation then gives
\begin{equation}
{dx^+\over d\lambda^2} + \Gamma^+_{++}\Big({dx^+\over d\lambda}\Big)^2 = 0
\quad\rightarrow\quad \lambda={(x^+)^{a+1}\over a+1}\ ,
\end{equation}
where $\Gamma^+_{++}={a\over x^+}$\ is the only nonvanishing
$\Gamma^+_{ij}$ component. Solving this leads to the affine parameter
above and the tangent vector becomes
$\xi=\partial_\lambda=({dx^+\over d\lambda})\partial_+$ so $\xi^+=(x^+)^a$. The
relative acceleration of neighbouring geodesics then is\
$a^M=R^M{_{CDB}}\xi^C\xi^Dn^B$ with $n=n^B\partial_B$ the unit normalized
cross-sectional separation vector. Then it can be seen that
$a^i=R^i{_{+i+}}(\xi^+)^2n^i$ so $|a^i|^2$ diverges for all $0<k<1$
leading to diverging tidal forces, somewhat similar to the
corresponding discussion in \cite{Madhu:2009jh}. For $k=1$
the spacetimes (\ref{nullUpstairs}) have all Riemann components
vanishing: these can be recast as $ds^2=-dX^+dx^-+(X^+)^2dy_i^2$ which
can be shown to be flat space in null Milne coordinates (redefining
$Y_i=X^+y_i\,,\ y^-=x^-+y_i^2X^+$).
Now we analyze quantum extremal surfaces. These cosmologies have no
holographic boundary: introducing a bookkeeping $\phi_r$, the
generalized entropy (Appendix~\ref{App:EE2dCFT}) is
\begin{equation}\label{SgenNull}
S_{gen} = {\phi_r\over 4G} (-x^+)^k + {c\over 6}\log(-\Delta x^+ \Delta x^-)\ ,
\end{equation}
where $\Delta x^\pm=x^\pm-x_0^\pm$ characterizes the spacetime interval
between the observer O and the QES (see Figure~\ref{nullQES}).
Strictly speaking, there is a null Kasner scale $t_N$ here appearing
as $\phi=({-x^+\over t_N})^k$ so $\phi$ is dimensionless: however
since the 2-dim metric is flat in these variables, $t_N$ can be
absorbed into the definition of $\phi_r$ above: so we will suppress
this (unlike the spacelike cases in sec.~\ref{sec:AdSKreg} earlier).\
The extremization with respect to $x^-$ and $x^+$ gives
\begin{equation}\label{SgenExtNull}
\partial_-S_{gen} = {c\over 6} {-\Delta x^+\over -\Delta x^+ \Delta x^-} = 0\,,
\qquad
\partial_+S_{gen} = -{\phi_r\over 4G}\, {k\over (-x^+)^{1-k}} +
{c\over 6} {\partial_+\Delta^2\over \Delta^2} = 0\, .
\end{equation}
With $0<k<1$, the classical extremization ($c=0$) gives $x^+\rightarrow\infty$\,:
in full, we have
\begin{equation}\label{nullQESext0}
\Delta^2=-\Delta x^+ \Delta x^->0\,,\quad
\Delta x^-=x^--x_0^-\rightarrow-\infty\,,\quad
-{1\over (-x^+)^{1-k}} + {2Gc\over 3\phi_rk}\,{1\over x^+-x^+_0} = 0\ ,
\end{equation}
so
\begin{equation}\label{nullQESext}
\Delta x^+ > 0, \quad
x_*^+=x^+_0 + {2Gc\over 3k\phi_r} (-x_*^+)^{1-k} > x^+_0\ ;
\qquad \Delta x^-<0,\quad x_*^-\rightarrow X_c^-\sim-\infty\ .
\end{equation}
\begin{figure}[h]
\hspace{2pc}
\includegraphics[width=11pc]{nullQES.pdf}
\hspace{3pc}
\begin{minipage}[b]{20pc}
\caption{{ \label{nullQES}
\footnotesize{Cartoon of the 2-dim geometry with the
null singularity at $x^+=0$, the worldline $(x^+_0,x^-_0)$
of a timelike observer (vertical trajectory, representing
for simplicity a fixed spatial location), and the quantum
extremal surface at $(x^+_*,x^-_*)$.
As can be seen, the QES is spacelike separated from the
observer ($\Delta^2>0$) if $\Delta x^+>0$ and $\Delta x^-\sim-\infty$,\
and lies towards the singularity in terms of $x^+$-slices.
The entanglement wedge defined by the QES is shown as the blue
wedge.
\newline \newline
}}}
\end{minipage}
\end{figure}
This is best visualized as in Figure~\ref{nullQES}\,: we describe this
further below.
From (\ref{Sgen0}), we have $Gc\ll 1$ so that $x^+\sim x^+_0$ upto
small corrections (with $k\neq 0$). Thus employing perturbation
theory in $Gc$, we obtain
\begin{equation}\label{nullQESx+}
x_*^+ \sim x^+_0 + {2Gc\over 3k\phi_r} (-x^+_0)^{1-k}\ ,
\end{equation}
{\it i.e.}\ the QES is almost on the same null-time ($x^+$) slice as the
observer, but just a little \emph{towards} the null singularity\
(using absolute values gives\
$|x^+| - |x^+_0| \sim -{2Gc\over 3k\phi_r} |x^+_0|^{1-k}$). The
location of the QES as being towards the singularity rather than away
as in the spacelike cases may look surprising at first sight. However
from Figure~\ref{nullQES}, drawing constant $x^+$ and $x^-$ slices, it
is clear that the location of the QES with $\Delta x^+>0$ and $\Delta
x^-\rightarrow-\infty$ is geometrically reasonable and expected if the QES and
the observer are to be spacelike separated\ ($\Delta x^+<0$ gives
timelike separation between the QES and the observer). In terms of
the $(t,r)$-coordinates (\ref{nullBgnd}), Figure~\ref{nullQES}
can be taken to depict the region with $x^+=t+r<0$ in the
$(t,r)$-plane, with the singularity locus being $t+r=0$ and the
timelike observer worldline having some fixed $r_0$ with $t_0<0$.
The description in Figure~\ref{nullQES} continues to hold as long
as the observer remains timelike: it also holds if the observer is
moving along a null trajectory along $x^+$ with fixed $x^-$.
As a further check, we see that this extremization exhibits
time-maximization with null time ($x^+$): we have, using (\ref{nullQESext}),
\begin{equation}
\partial_+^2S_{gen} = -k(1-k){\phi_r\over 4G} (-x^+)^{k-2}
- {c\over 6} {1\over (x^+-x_0^+)^2}\quad\rightarrow\quad \partial_+^2S_{gen}|_*<0\ .
\end{equation}
Note however that\ $\partial_-^2S_{gen}=-{c\over 6}{1\over(\Delta x^-)^2}\rightarrow 0^-$
from (\ref{SgenExtNull}). This should not be surprising: the
2-dim space here is flat and the absence of the bulk gravitational field
makes it quite different from $AdS$-like spaces\ ({\it e.g.}\ an expression
like $S\sim \log r$ gives $\partial_r^2S\sim -{1\over r^2}\rightarrow 0^-$).
As examples of (\ref{SgenNull}), we see that for a nearly smooth space
{\it e.g.}\ with $k=\epsilon\ll 1$, (\ref{nullQESext}) gives\
$x_*^+ \sim (1-{2Gc\over 3\epsilon\phi_r}) x^+_0$\,.\
The case $k={2\over 3}$ gives the cubic
\begin{equation}\label{nullQESk=2/3}
x^+_*=-t^3\,:\qquad t^3 + {Gc\over \phi_r}\,t - |x_0^+| = 0\ ,
\end{equation}
which can be shown to have one real root which satisfies $\Delta x^+>0$
and agrees with (\ref{nullQESx+}) in perturbation theory in $Gc$.
For generic $k$ values, recasting using $x^+=-y^{{1\over 1-k}}$\,,
it can be seen numerically that there is one real root satisfying
$\Delta x^+>0$. Along these lines, for values such as $k={1\over 2}$
we choose the positive root of the resulting quadratic in continuity
with neighbouring $k$ values, which then again gives $\Delta x^+>0$.
Note that these null cosmological singularities are somewhat different
from the spacelike ones: for instance the extremization
(\ref{nullQESext}) shows that the singularity locus $x^+=0$ is in fact
an allowed QES solution when $x_0^+=0$. The behaviour near $x^+=0$ can
be seen explicitly in examples including (\ref{nullQESk=2/3}), {\it e.g.}\
numerically. Thus these null singularities appear to not be excluded
from the entanglement wedge of the observer.\
However the on-shell generalized entropy (\ref{SgenNull}) continues
to be singular generically in the vicinity of the singularity:
(\ref{nullQESext}) gives
\begin{equation}
S_{gen}^{o.s.} = {\phi_r\over 4G} (-x_*^+)^k +
{c\over 6} \log \left({2Gc\over 3k\phi_r}\,
{(-x_*^+)^{1-k}\, |X_c^-|\over\epsilon_{UV}^2}\right)\ .
\end{equation}
Thus although formal extrapolation to the singularity appears possible,
the above implies that the QES (\ref{nullQESext}) is only reliable in
the semiclassical regime with large $x^+_*$ and $Gc\ll 1$\ (where the
Bekenstein bound does not appear violated). Also since $S_{gen}^{o.s.}$
appears singular, further subleading contributions beyond the bulk
entropy term presumably also must be considered.
It was observed in \cite{Madhu:2009jh} that strings become highly
excited in the vicinity of a null Kasner Big-Crunch singularity\ (see
also \cite{Horowitz:1989bv,Craps:2008bv}). It is likely that this will
be true for (\ref{nullUpstairs2}) as well. In this regard, note that
the backgrounds (\ref{nullBgnd}) necessarily require the extra scalar
$e^\Psi$ to be nontrivial: interpreting this as the string coupling
$g_s=e^\Psi$ and choosing the negative sign exponent for $e^\Psi$ in
(\ref{null2d}) suggests large string interactions in the vicinity of
the singularity $x^+=0$. It is conceivable however that in some
appropriate double-scaling limit\ $x_*^+\rightarrow 0,\ X_c^-\rightarrow-\infty$,
with\ ${2Gc\over 3k\phi_r}\, {(-x_*^+)^{1-k}\, |X_c^-|\over\epsilon_{UV}}$\
held fixed, the generalized entropy can be rendered nonsingular. It
would be nice to explore this more carefully, perhaps dovetailing with
the positive sign exponent for $e^\Psi$ in (\ref{null2d}) and
suppressed string interactions.
It is interesting to note that there is an entire function-worth of
nontrivial null backgrounds in (\ref{nullBgnd}), as (\ref{EOMnull})
shows. This is a special feature of 2-dim spacetimes that have a
``holomorphic'' structure, as is the case here with solely
$x^+$-dependence: for instance the backgrounds (\ref{nullUpstairs2})
can be recast by redefining the null-time variable to give
(\ref{nullBgnd}), so that the 2-dim metric is flat in these
$x^\pm$-coordinates\footnote{Instead of these ``flat'' variables, had
we taken the background to be
\begin{equation}
e^f=(X^+)^\alpha\,,\ \ \phi=(X^+)^K\,,\quad\rightarrow\quad
(\partial_+\Psi)^2 = {2K (\alpha-K+1)\over (X^+)^2}\ \ \rightarrow\ \
0<K\leq \alpha+1\ .\nonumber
\end{equation}
In other words, the exponent $k$ earlier is related as $k={K\over\alpha+1}$\,.
Now the generalized entropy contains the metric factor $e^{f/2}|_*$,
thus appearing singular.
}.
Spacelike cosmological singularities generically do not exhibit any
such ``holomorphy'' and cannot generically be recast in flat
coordinates and the metric factor $e^f$ lingers. This holomorphy
shows up in the extremization equations (\ref{SgenExtNull}),
(\ref{nullQESext}), where the $x^\pm$ sectors decouple (in contrast
with {\it e.g.}\ (\ref{ExtSgen-r0t0}) in the AdS Kasner case, and more
generally (\ref{Sgen1})). In fact, considering generic 2-dim
backgrounds (\ref{nullBgnd}), extremizing the generalized entropy
gives
\begin{equation}
\partial_+S_{gen} = 0 \quad \rightarrow\quad
\partial_+\phi + {2Gc\over 3\phi_r} {1\over x^+-x_0^+} = 0\quad\rightarrow\quad
x^+ - x_0^+ = -{2Gc\over 3\phi_r} {1\over \partial_+\phi}\ ,
\end{equation}
again exhibiting this holomorphicity.
From the logic in Figure~\ref{nullQES} with $\Delta^2>0$ and\
$\Delta x^+>0$,\ $\Delta x^-\rightarrow-\infty$, this implies that the
quantum extremal surface must lie in the direction of decreasing
dilaton, {\it i.e.}\ $\partial_+\phi<0$. This is consistent with our earlier
discussion since the dilaton Crunches towards decreasing $x^+$.
\section{Other cosmologies and QES}\label{sec:dSFRW}
In this section, we will study other cosmological backgrounds, in
particular de Sitter space in the Poincare slicing and FRW universes
under certain conditions. One might take these to have natural
asymptotics at future or past timelike infinity, and are thus quite
different from the previous discussions on $AdS$-like or null
cosmologies where the asymptotics are at spatial or null infinity. As
we will see (and has been noted previously), the extremal surface
structure is rather different in these cases below: in some sense we
are simply extending our previous investigations in some formal way to
the cosmologies below, with the hope that better understanding will
emerge over time.
\subsection{de Sitter, Poincare}\label{sec:dSqes}
de Sitter space $dS_{d_i+1}$ in the Poincare slicing and its 2-dim
reduction are
\begin{equation}
ds^2 = {R^2\over\tau^2}(-d\tau^2+dx^2+dy_i^2)\quad\rightarrow\quad
\phi={R^{d_i}\over(-\tau)^{d_i}}\,,\ \ \
ds^2 = {R^{d_i+1}\over(-\tau)^{d_i+1}}(-d\tau^2+dx^2)\ .
\end{equation}
We are parametrizing the upper Poincare patch with the future boundary
$I^+$ at $\tau=0$ and the past horizon at $\tau\rightarrow-\infty$, and
$-\infty<\tau<0$ generically so the minus signs are explicitly retained.
As $\tau$ increases to the future, the dilaton grows. There is a
singularity at $\tau\rightarrow-\infty$ in the effective 2-dim space: the
space is conformally $dS_2$\ (there are some parallels with
the discussions of $AdS_D$ reductions in \cite{Narayan:2020pyj}).
In this inflationary patch, we take the observer to be in the ground
state, so the bulk entropy is given by the ground state expression.
Then the generalized entropy for a bulk observer on a static worldline
at say $(x_0,\tau_0)$ is (see Appendix~\ref{App:EE2dCFT})
\begin{equation}\label{SgendS}
S_{gen} = {\phi_r\over 4G}\,{R^{d_i}\over(-\tau)^{d_i}} +
{c\over 6}\log \left(\Delta^2\, {R^{(d_i+1)/2}\over(-\tau)^{(d_i+1)/2}}
\right)\ ,
\end{equation}
retaining only terms relevant for extremization.
Then extremization gives
\begin{equation}\label{SgendSext}
{c\over 3}\,{\Delta x\over\Delta^2} = 0\,,\qquad
- \left( -{d_i\phi_r\over 4G}\,{R^{d_i}\over(-\tau)^{d_i+1}}
- {c\over 12}\,{d_i+1\over(-\tau)} \right)
- {c\over 3}\,{\tau-\tau_0\over \Delta^2} = 0\ .
\end{equation}
One solution to this is
\begin{equation}\label{dStimelikeQES}
\Delta x = 0\,,\ \ \Delta^2=-(\tau-\tau_0)^2\,;\ \qquad
{d_i\phi_r\over 4G}\,{R^{d_i}\over(-\tau)^{d_i+1}}
+ {c\over 12}\,{d_i+1\over(-\tau)}
+ {c\over 3}\,{1\over\tau-\tau_0} = 0\ .
\end{equation}
For a late time observer with $\tau_0\sim 0$, we have
\begin{equation}
\Delta x = 0\,,\quad
{d_i\phi_r\over 4G}\,{R^{d_i}\over(-\tau)^{d_i+1}}
+ {c\over 12}\,{3-d_i\over\tau} = 0
\ \ \ \rightarrow\ \ \
\tau_*=-R\left({d_i\over 3-d_i}\,{3\phi_r\over Gc}\right)^{1/d_i}
\end{equation}
Note that we are looking for a solution with $\tau<0$ as per our
parametrization: so for $d_i\geq 3$ ({\it i.e.}\ $dS_5$ and higher) the
only real QES solution is $\tau\rightarrow-\infty$.
For $d_i=1$ this matches the result in \cite{Chen:2020tes}\
((\ref{SgendS}) matches eq.6.7 there).
Most notably, the above QES solution is timelike-separated from the
observer: so $\Delta^2<0$\ (unlike {\it e.g.}\ (\ref{t>t0}),
(\ref{nullQESext0})) and the generalized entropy (\ref{SgendS}) has
an imaginary part
from $\log(-1)$. In the spirit of our earlier discussions, it is
interesting to look for quantum extremal surfaces that are
spacelike-separated from the observer: towards this, note that there
is a distinct family of solutions which by construction are
spacelike-separated, along the lines of the discussions in the
cosmologies earlier with a regulator. Then (\ref{SgendSext}) gives
\begin{equation}\label{dSspacelikeQES}
\Delta^2\sim R_c^2\ ,\qquad
{d_i\phi_r\over 4G}\,{R^{d_i}\over(-\tau)^{d_i+1}}
+ {c\over 12}\,{d_i+1\over(-\tau)}
\sim {c\over 3}\,{\tau-\tau_0\over R_c^2}\ ,
\end{equation}
regulating the QES as before with a spatial cutoff $R_c$.
First, note that if we remove the regulator so $R_c\rightarrow\infty$, then
we obtain (with $t\equiv-\tau>0$)
\begin{equation}
{d_i\phi_r\over 4G}\,{R^{d_i}\over t^{d_i+1}}
+ {c\over 12}\,{d_i+1\over t} = 0\ .
\end{equation}
Both terms have the same sign so the only real QES is at $\tau\rightarrow-\infty$.
This is spacelike separated only if the observer is also localized
at sufficiently early times.
With a finite spatial regulator $R_c$, we see that in general
$\tau>\tau_0$, {\it i.e.}\ the QES lies on time slices later than the observer.
As a first approximation, note that in the classical limit $c\rightarrow 0$,
the solution is $\tau\rightarrow-\infty$\,: this is the location where the
dilaton is minimized. For early times also, the solution
is similar, {\it i.e.}\
\begin{equation}\label{dSrealQES}
|\tau_0|\gg R:\qquad \tau\rightarrow-\infty\,,\ \ \tau\sim\tau_0\ ,
\end{equation}
{\it i.e.}\ the QES is in the far past when the observer is also in the
far past. This can be seen to exhibit time-maximization.
Let us analyze (\ref{dSspacelikeQES}) for 4-dim de Sitter upstairs
($d_i=2$): then
\begin{equation}\label{dS4spqes}
\Delta^2\sim R_c^2\ ,\qquad
{\phi_r\over 2G}\,{R^{2}\over(-\tau)^{3}} + {c\over 12}\,{3\over(-\tau)}
\sim {c\over 6}\,{\tau-\tau_0\over R_c^2}\ .
\end{equation}
With $t\equiv -\tau$, we can rewrite this as
\begin{equation}
\Delta^2\sim R_c^2\ ,\qquad
{6\phi_r\over Gc}\,R^{2}R_c^2 + 3R_c^2\,t^2 \sim 2t^3 (t_0-t)\quad\rightarrow\quad
t^4-t_0t^3+{3R_c^2\over 2}\,t^2+{3\phi_r\over Gc}\,R^2R_c^2 \sim 0\ .
\end{equation}
Clearly as $t_0\rightarrow 0$, there is no real QES solution since all terms are
positive (there appears to be a critical $t_0$ where the real QES
(\ref{dSrealQES}) stops existing). $dS_{d_i+1}$ can be seen to exhibit
similar behaviour.
Overall, in some essential sense, the physical interpretation of the
generalized entropy in these cases is not transparent, for instance as
holographic entanglement in the dual boundary theory, along the lines
of the $AdS$ cases\ (even from a bulk point of view alone, the
timelike separation is unconventional compared with the usual
formulations of entanglement on a spatial slice). However our
discussion appears to corroborate previous work on classical de Sitter
extremal surfaces. Taking the future boundary as a natural anchor in
$dS$, there are either complex extremal surfaces
\cite{Narayan:2015vda,Sato:2015tta,Miyaji:2015yva} or future-past
(timelike) extremal surfaces
\cite{Narayan:2017xca,Narayan:2020nsc}. The latter future-past
surfaces perhaps suggest some new ``temporal entanglement'' between
$I^\pm$: taking the area of such surfaces to be real is effectively
removing an overall $i$-factor which would arise from rotating a
spatial extremal surface to a timelike one (this is also vindicated by
the complex generalized entropy (\ref{SgendS}), (\ref{dStimelikeQES}),
generalizing \cite{Chen:2020tes} for $dS_2$). Overall perhaps this
suggests new interpretations towards entanglement in de Sitter space
based on the future boundary and $dS/CFT$
\cite{Strominger:2001pn,Witten:2001kn,Maldacena:2002vr}. The $dS/CFT$
dictionary $\Psi_{dS}=Z_{CFT}$ suggests that boundary entanglement is
not bulk entanglement (quite unlike $Z_{bulk}=Z_{CFT}$ in $AdS$). Bulk
observables require $|\Psi_{dS}|^2$ suggesting two copies of the dual
CFT: this is reflected in the future-past extremal surfaces
\cite{Narayan:2017xca,Narayan:2020nsc} alluded to above. Other recent
perspectives on extremal surfaces anchored on the de Sitter horizon
include {\it e.g.}\ \cite{Shaghoulian:2021cef}.
\subsection{FRW cosmologies, 2-dim gravity and QES}
Consider FRW cosmologies with flat spatial sections sourced by a
scalar field $\Psi$\ (general reviews include
{\it e.g.}\ \cite{Trodden:2004st,Baumann:2009ds}): we choose one of the
spatial directions to be noncompact and perform dimensional reduction
on the others to obtain a 2-dim background
\begin{equation}
ds^2 = -dt^2 + a(t)^2 dx_i^2 \quad\rightarrow\quad
\phi=a^{d_i}\,,\ \ \ ds^2 = a^{d_i+1} (-d\tau^2+dx^2)\ ,
\end{equation}
as a solution to (\ref{actionXPsiU}), (\ref{2dimseom-EMD0-Psi}),
(\ref{2dimseom-EMD1-Psi}).
The energy-momentum conservation equation gives\
$dE+pdV=d(\rho\,a^{d_i+1})+pd(a^{d_i+1})=0$, {\it i.e.}\
${\dot\rho}+(d_i+1)H(\rho+p)=0$.
This along with the Friedmann equation and the equation of state
$p=w\rho$ gives FRW cosmologies with
\begin{equation}
p=w\rho\,,\quad a \sim t^k\,,\ \ k={2\over (1+d_i)(1+w)}
\qquad\
\Big[\rho={1\over 2}{\dot\Psi}^2 - V\,,\ \ p={1\over 2}{\dot\Psi}^2 + V\Big]
\end{equation}
Now using conformal time $\tau$ gives
\begin{equation}\label{ataunu}
d\tau = {dt\over a(t)}\ \ \rightarrow\ \ \tau\sim t^{1-k}\quad\rightarrow\quad
a(\tau) \sim \Big({\tau\over\tau_F}\Big)^{{k\over 1-k}}\equiv
\Big({\tau\over\tau_F}\Big)^\nu\ ,
\end{equation}
introducing the FRW scale $\tau_F$ so the scale factor
becomes dimensionless: $\tau_F$ controls the strength of
time-dependence in these backgrounds, analogous to $t_K$ in
(\ref{AdSK-RtK}).
Note that the above FRW description is slightly different from
focussing on the vicinity of the singularity as in
\cite{Bhattacharya:2020qil}: taking dominant time derivatives implies
${\dot\Psi}^2\gg V$ so $p\sim \rho$, {\it i.e.}\ $w\sim 1$, giving
$\nu={1\over d_i}$ so $\phi=a^{d_i}\sim \tau$ in agreement with the
universality (\ref{univSing}).
More generally the physical bounds on the equation of state parameter
$w$ translate to corresponding regimes for $\nu$:
\begin{equation}\label{nu-w}
\nu = {2\over (1+d_i)(1+w) - 2}\ ;\qquad\ \ -1\leq w\leq 1\ \ \Rightarrow\ \
\nu>{1\over d_i}\ \ {\rm or}\ \ \nu\leq -1\ .
\end{equation}
Now we analyse quantum extremal surfaces here. In general the bulk
matter entropy corresponds to some excited state, such as the thermal
state. A variety of such studies for FRW cosmologies including
entanglement with auxiliary universes appears in \cite{Hartman:2020khs},
revealing islands in various cases. Our discussion here will be
limited to simply extending the earlier de Sitter QES solutions to
certain FRW cases, which correspond to matter in the ground state
(as may arise for pressureless matter with $w=0$).
The generalized entropy for an observer in such a background is
(see Appendix~\ref{App:EE2dCFT})
\begin{equation}\label{SgenFRW}
S_{gen} = {a^{d_i}\over 4G} + {c\over 6} \log
\left(\Delta^2\, a^{(d_i+1)/2}|_{(\tau,r)}\right)\ ,\qquad
\Delta^2 = (\Delta x)^2-(\tau-\tau_0)^2\ ,
\end{equation}
where we are using conformal time $\tau$ in the 2-dim theory.
Now extremization gives
\begin{eqnarray}\label{SgenFRWext}
\partial_xS_{gen} = {c\over 6}{\partial_x\Delta^2\over\Delta^2} = 0\ ,\ \ &&\ \
\partial_\tau S_{gen} = {d_ia^{d_i-1}\,\partial_\tau a\over 4G}
+ {c\over 12} {(d_i+1)\partial_\tau a\over a}
+ {c\over 6} {\partial_\tau\Delta^2\over\Delta^2} = 0\ ,\nonumber\\ [1mm]
\longrightarrow\qquad {c\over 3}\,{\Delta x\over\Delta^2} = 0\ ,\ \ &&\ \
{d_i\nu\over 4G} {\tau^{\nu d_i-1}\over\tau_F^{\nu d_i}}
+ {c\over 12} {(d_i+1)\nu\over\tau}
- {c\over 3} {\tau-\tau_0\over\Delta^2} = 0\ ,
\end{eqnarray}
using (\ref{ataunu}), (\ref{nu-w}).
First considering timelike separated QES, we have
\begin{equation}\label{FRWtimelikeQES}
\Delta x = 0\ ,\ \ \Delta^2=-(\tau-\tau_0)^2\,;
\qquad
{d_i\nu\over 4G} {\tau^{\nu d_i-1}\over\tau_F^{\nu d_i}}
+ {c\over 12} {(d_i+1)\nu\over\tau}
+ {c\over 3} {1\over\tau-\tau_0} = 0\ ,
\end{equation}
analogous to (\ref{dStimelikeQES}) in the de Sitter case $\nu=-1$:
for $\nu<-1$ the nature of these timelike QES is similar.
For $\nu d_i>1$, taking the first term to be dominant over the second
gives\
\begin{equation}
\nu>{1\over d_i}\,:\qquad\quad
{d_i\nu\over 4G} {\tau^{\nu d_i-1}\over\tau_F^{\nu d_i}}
+ {c\over 3} {1\over\tau-\tau_0} \sim 0\ \qquad\quad [\tau\gtrsim\tau_F]\ ,
\end{equation}
which is structurally similar to (\ref{nullQESext0}), with
corresponding QES solutions (with $\tau_*-\tau_0<0$), valid for $\tau$
large compared to $\tau_F$. Since these are timelike-separated, the
on-shell generalized entropy acquires an imaginary part from $\log
(-1)$ in $\Delta^2<0$, similar to (\ref{dStimelikeQES}).
Alternatively, looking for spacelike separated QES along the lines of
(\ref{dSspacelikeQES}) gives
\begin{equation}
\Delta^2\sim R_c^2\ , \qquad
{d_i\nu\over 4G} {\tau^{\nu d_i-1}\over\tau_F^{\nu d_i}}
+ {c\over 12} {(d_i+1)\nu\over\tau} \sim {c\over 3} {\tau-\tau_0\over R_c^2}\ .
\end{equation}
We are looking in the region of slow time evolution {\it i.e.}\ large
$\tau\gtrsim\tau_F$\ (far from the singularity at $\tau=0$), towards
understanding the evolution of the QES with the observer time $\tau_0$.
Then for any $\nu>0$, we have $\tau^{\nu-1}>\tau^{-1}$ so we can
approximate the time extremization equation as
\begin{equation}
{d_i\nu\over 4G} {\tau^{\nu d_i-1}\over\tau_F^{\nu d_i}}
\sim {c\over 3} {\tau-\tau_0\over R_c^2}
\quad\rightarrow\quad
\tau^{\nu d_i-1} \sim
{4Gc\over 3d_i\nu}\,{\tau_F^{\nu d_i}\over R_c^2}\ (\tau-\tau_0).
\end{equation}
This equation while tricky in general does have solutions at least
for specific families of $\nu$. For instance pressureless dust has
$w=0$ so using (\ref{nu-w}) we have
\begin{eqnarray}
w=0\ \ {\it i.e.}\ \ \nu={2\over d_i-1} &\xrightarrow{d_i=2} &
\tau^3\sim {Gc\,\tau_F^4\over 3 R_c^2}\ (\tau-\tau_0)\ ,\nonumber\\
&\xrightarrow{d_i=3} &
\tau^2\sim {4Gc\,\tau_F^3\over 9 R_c^2}\ (\tau-\tau_0)\ ,
\end{eqnarray}
both of which admit real solutions as long as
${Gc\tau_F^{\nu d_i}\over R_c^2}$ lies in appropriate regimes with
regard to $\tau_0$. For instance the $d_i=3$ case requires
${4Gc\,\tau_F^3\over 9 R_c^2}>4\tau_0$ for reality. Since the spatial
regulator $R_c\rightarrow\infty$ strictly speaking, it is clear that these
solutions only make sense in a limit where we take $c$ small and
$R_c$ large holding the above condition fixed: so the existence of
these spacelike-separated QES solutions is not generic.
For generic scalar configurations, it is more appropriate
to consider bulk entropy contributions that are not those pertaining
to the ground state: then\ $S_{gen} = {a^{d_i}\over 4G} + S_b$\ gives\
${d_i\,\partial_\tau a\over 4G} + \partial_\tau S_b = 0$.
Discussions of this sort have previously appeared in {\it e.g.}\
\cite{Hartman:2020khs}. When $S_b$ overpowers the classical area term,
the Bekenstein bound is violated and islands can arise if further
conditions hold. For $S_b$ representing bulk matter in some mixed
state, one might imagine some auxiliary purifying universe ``elsewhere''
which could then lead to islands. We will not discuss this further here.
\section{Discussion}\label{sec:Disc}
We have discussed quantum extremal surfaces in various cosmological
spacetimes with Big-Crunch singularities, developing further the
investigations in \cite{Manu:2020tty}. The generalized entropy here is
studied in 2-dimensional cosmologies obtainable in part from
dimensional reduction of higher dimensional cosmologies: the bulk
matter is taken to be in the ground state, which is reasonable in the
semiclassical region far from the singularity. First we focussed on
the isotropic $AdS$ Kasner spacetime and its reduction to
2-dimensions: the quantum extremal surfaces in \cite{Manu:2020tty}
were found to be driven to the semiclassical region infinitely far
from the Big-Crunch singularities present in these backgrounds (the
classical RT/HRT surfaces for finite subregion size bend in the
direction away from the singularity, Figure~\ref{cosRT1}). Analysing
further, the spatial extremization equation (\ref{rExt-reg}) shows
that in the semiclassical region, the QES location leads to the entire
Poincare wedge, with no island-like regions. Introducing a spatial
regulator in the time extremization equation (\ref{tExt-reg}) enables
understanding the dependence of the QES on the observer's location in
time. This shows that the QES lags behind the observer location, in
the direction away from the singularity, as in
Figure~\ref{cosQES3}. The lag can be seen to increase slowly as the
observer evolves towards the singularity: extrapolating shows that the
singularity $t=0$ is not a solution to the extremization equations.
Thus the entanglement wedge appears to exclude the near singularity
region. Removing the regulator recovers the results in
\cite{Manu:2020tty}. The spatial extremization equation
(\ref{rExt-reg}) shows an island-like region emerging for
(\ref{islandBndry}). However analysing carefully the extremization
equations recast as (\ref{Deltat/r-3}), (\ref{Dtr-tEx-quadr}), in the
vicinity of this island boundary reveals that the potential
island-like solution is in fact inconsistent. Appending a
time-independent far region joined with the $AdS$ Kasner region at the
time slice $t=t_K$ as in (\ref{AdSK-RtK-reg}) gives further insight on
the QES behaviour. This QES analysis in the $AdS$ Kasner case extends
to more general singularities admitting a holographic interpretation,
with similar QES behaviour (\ref{rExt-reg-gen}), (\ref{tExt-reg-gen}),
in the semiclassical region, and inconsistencies near a potential
island boundary (\ref{islandBndry-gen}). These cosmologies include
nonrelativistic asymptotics: the assumptions on the exponents
(\ref{genCos}) are fairly general.
In sec.~\ref{sec:null}, we studied certain families of null Big-Crunch
singularities, which exhibit a certain ``holomorphy'' due to special
properties of null backgrounds. These are distinct in the behaviour of
the quantum extremal surface Figure~\ref{nullQES}, which can now reach
the singularity: however the on-shell generalized entropy continues to
be singular so the vicinity of singularity is not reliable. In all
these cases, the QES is manifestly spacelike-separated from the
observer ({\it e.g.}\ (\ref{t>t0}), (\ref{nullQESext0})), consistent with its
interpretation as holographic entanglement. We then discuss aspects of
2-dimensional effective theories involving dimensional reduction of
other cosmologies including de Sitter space (Poincare slicing) and FRW
cosmologies. In these cases, there are families of QES solutions which
are timelike-separated from the observer (\ref{dStimelikeQES}),
(\ref{FRWtimelikeQES}) \ (the $dS$ case here is in part a
generalization of some results in \cite{Chen:2020tes} for the $dS_2$
case): correspondingly the generalized entropy acquires an imaginary
part. We also find real spacelike-separated QES solutions in the
presence of finite spatial regulators (\ref{dSspacelikeQES}). In de
Sitter, these real solutions cease to exist for the late-time
observer. Overall this perhaps corroborates earlier studies of
classical extremal surfaces anchored at the future boundary
\cite{Narayan:2015vda,Sato:2015tta,Miyaji:2015yva,
Narayan:2017xca,Narayan:2020nsc}: see the discussion at the end
of sec.~\ref{sec:dSqes}.
Our investigations here have been on using quantum extremal surfaces
to gain some insights on cosmological spacetimes containing Big-Crunch
singularities: all these admit the form of a 2-dimensional cosmology
and thus exclude more general cosmologies that do not admit a
reduction to 2-dimensions. Most of our discussions pertain to bulk
matter in the ground state, which is reasonable far from the
singularities in the cosmologies we have discussed. Overall the
cosmologies we have considered are closed universes with no horizons,
no appreciable entropy and no additional non-gravitating bath regions:
in such cases islands are not generic\ (there are parallels with some
discussions in \cite{Geng:2021hlu}). This is consistent with previous
studies of closed universes with no entanglement with ``elsewhere'',
{\it i.e.}\ regions external to the universes in question which might act as
purifiers for mixed states. This is consistent with the Bekenstein
bound not being violated, {\it i.e.}\ the bulk entropy does not overpower the
classical area in the generalized entropy. Our discussion of de
Sitter space pertains only to the Poincare slicing: see
{\it e.g.}\ \cite{Hartman:2020khs,Sybesma:2020fxg} for other discussions of
de Sitter. The FRW discussions also must be extended to cases with
bulk matter in excited states far from the ground state: in
this case islands will appear, corresponding to violations of the
Bekenstein bound.
Perhaps the most interesting question pertains to studying more
interesting models for bulk matter in the near-singularity spacetime
region where the matter might be expected to get highly excited.
Presumably incorporating analogs of more ``stringy'' or quantum
entanglement will give more insights into how the near singularity
region is accessible via entanglement (with the null singularities
perhaps more tractable).
At a more broad brush level, in some essential ways, cosmological
singularities in holography are perhaps qualitatively different from
black holes. They appear to require nontrivial non-generic initial
conditions: generic time-dependent deformations of the CFT vacuum are
expected to thermalize on long timescales, leading to black hole
formation in the bulk rather than a Big-Crunch. This appears
consistent with our finding that {\it e.g.}\ the $AdS$ Kasner and other
holographic cosmological singularities are inaccessible via
entanglement with conventional ground state bulk matter: perhaps this
corroborates the expectation of non-generic holographic dual
states\ (see discussion after (\ref{AdSDK-2d}) and also other related
studies {\it e.g.}\ \cite{Barbon:2015ria,Caputa:2021pad} of such
singularities and complexity). It would be interesting to gain more
insights into the role of holographic entanglement, quantum extremal
surfaces and islands in cosmology more broadly.
\vspace{10mm}
{\footnotesize \noindent {\bf Acknowledgements:}\ \ It is a pleasure
to thank Dileep Jatkar and A. Manu for comments on a draft. We also
thank Debangshu Mukherjee for some discussions on the FRW cases.
This work is partially supported by a grant to CMI from the Infosys
Foundation.}
\vspace{5mm}
|
1,116,691,500,426 | arxiv | \section{Introduction}
The Standard Model (SM) accounts for almost all experimental high
energy physics data; however, the observation of neutrino oscillations
requires that the SM be extended to include nonzero neutrino masses.
While there are many ways to expand the SM to account for
neutrino oscillations, we attempt to do so with the following goals.
First, the neutrino mass scale is significantly lower than the mass
scales of the other fermions, so we would like the model to account
for this without the addition of many tiny parameters. Second, lepton
number violation has not yet been observed, so we would like the model
to give rise to Dirac neutrino masses, with Majorana masses forbidden.
Third, we would like the model to be testable at the CERN Large Hadron
Collider (LHC).
Most neutrino mass models give rise to Majorana masses for the SM
neutrinos, with many predicting TeV-scale new physics accessible at
the LHC. In contrast, only a few models for Dirac neutrinos have been
proposed. These typically involve a second Higgs doublet with very
small vacuum expectation value (vev) that couples only to the
left-handed lepton doublets and the right-handed neutrinos, resulting in
neutrino masses of the same order as the very small vev. The original
SM-like Higgs doublet couples to all of the quarks and charged leptons
in the usual way. Such a Yukawa coupling structure can be obtained by
imposing a global $Z_2$ symmetry, as proposed in the models of
Refs.~\cite{Ma:2000cc,Gabriel:2006ns}; however, this does not by
itself forbid neutrino Majorana mass terms, which must instead be
eliminated by imposing an additional lepton number symmetry. The
required Yukawa coupling structure can also be obtained by imposing a
global U(1) symmetry; this idea was first proposed in
Ref.~\cite{Fayet:1974fj} as a way of ensuring the (then-assumed)
masslessness of the neutrinos in the presence of right-handed neutrino
states, and has the virtue of forbidding Majorana mass terms by
itself.
In order to generate neutrino masses, the global symmetry used to
ensure the desired Yukawa structure has to be broken. Spontaneous
breaking leads to a very light scalar which can cause problems with
standard big-bang nucleosynthesis~\cite{Gabriel:2006ns}, as well as
having significant effects on the phenomenology of the new Higgs
particles~\cite{Gabriel:2008es}. By instead breaking a global U(1)
symmetry explicitly, the model proposed by us in
Ref.~\cite{Davidson:2009ha} generates Dirac neutrino masses while
avoiding very light scalars.\footnote{A similar mechanism was used to
explain the top-bottom quark mass hierarchy in
Ref.~\cite{Hashimoto:2004xp}.} A supersymmetric version of this
model was studied in Ref.~\cite{Marshall:2009bk}, which found
spectacular multi-lepton signals from cascade decays of the
supersymmetric partners of the new Higgs bosons and right-handed
neutrinos at the LHC.
In this paper we study the LHC detection prospects of the
non-supersymmetric model of Ref.~\cite{Davidson:2009ha}. This model
expands the SM by adding a second Higgs doublet $\Phi_2$ with the same
electroweak quantum numbers as the SM Higgs doublet $\Phi_1$, as well
as adding three gauge-singlet right-handed Weyl spinors $\nu_{R_i}$
which will become the right-handed components of the three Dirac
neutrinos. The model imposes a global U(1) symmetry under which the
second Higgs doublet and the right-handed neutrinos have charge $+1$,
while all the SM fields have charge zero. This allows Yukawa
couplings of the second Higgs doublet only to the right-handed
neutrinos and the SM lepton doublet, and forbids Majorana masses for
the right-handed neutrinos. It also tightly constrains the form of
the Higgs potential. Breaking the U(1) symmetry explicitly using a
term $m_{12}^2 \Phi_1^\dagger \Phi_2 + {\rm h.c.}$ in the Higgs
potential yields a vev $v_2$ for the second Higgs doublet and
consequently gives the neutrinos Dirac masses proportional to $v_2$.
By requiring that $v_2 \sim \mathcal{O}({\rm eV})$, the Dirac neutrino
masses are made suitably small without requiring tiny Yukawa couplings.
The characteristic feature of the model is that the couplings of the charged
scalar pair $H^{\pm}$ and two neutral scalars $H^0$ and $A^0$ from the
second Higgs doublet to leptons and neutrinos are
controlled by the neutrino masses and mixing angles. In this paper we
take advantage of the distinctive decay of the charged Higgs boson
$H^+$ into charged leptons and neutrinos. We focus on electroweak
pair production of $H^+H^-$ at the LHC followed by decays to $\ell
\ell^\prime p_T^{\rm miss}$, where $\ell \ell^\prime$ can be any
combination of opposite-sign $e$, $\mu$, and $\tau$ leptons and
$p_T^{\rm miss}$ denotes missing transverse momentum (carried away by
the neutrinos). Because $\tau$ leptons are more difficult to
reconstruct experimentally, we concentrate on the final states with
$\ell \ell^{\prime} = e^+ e^-$, $\mu^+ \mu^-$, and $e^\pm \mu ^\mp$.
The major backgrounds are diboson production ($W^+W^-$, $ZZ$, and $Z
\gamma$) and top quark pair production with both tops decaying
leptonically.
To determine whether the $H^+H^-$ signal will be detectable at the
LHC, we generated signal and background events using MadGraph/MadEvent
version 4~\cite{Alwall:2007st} assuming 14~TeV $pp$ center-of-mass
energy. We present results both at parton level, and after
hadronization with PYTHIA~\cite{PYTHIA} and fast detector simulation
with PGS~\cite{PGS}. With appropriate cuts, we find that a 5$\sigma$
discovery can be achieved with luminosity in the range 8--75~fb$^{-1}$
for $M_{H^+} = 100$~GeV, depending on
the neutrino mixing parameters. For $M_{H^+} = 300$~GeV a 5$\sigma$
discovery can be made with luminosity in the range 24--460~fb$^{-1}$.
The higher luminosity requirements
occur when the neutrino parameters are such that $H^+$ decays mostly
to $\tau\nu$, leading to final states not considered in our analysis.
We find that the kinematic variable $M_{T2}$ is very effective at
separating the signal from the $t \bar t$ and $WW$ backgrounds for
charged Higgs masses above the $W$ mass, and also provides sensitivity
to the charged Higgs mass.\footnote{While we have not made a detailed study of charged Higgs detection
prospects at 7~TeV $pp$ centre-of-mass energy, we note that the cross
section for the most dangerous $WW$ background is about 2.5 times
smaller at 7~TeV. However, the signal cross section is also about 2.5
(4.5) times smaller at this energy for $M_{H^+} = 100$ (300)~GeV.
Furthermore, the LHC is anticipated to collect only about 1~fb$^{-1}$
of integrated luminosity at 7~TeV. We thus expect detection or even
exclusion of the process considered here to be unfeasible in the
current 7~TeV LHC run.}
This paper is organized as follows. In the next section we review the
model and present the charged Higgs decay branching ratios. In
Sec.~\ref{sec:SB} we describe the signal and background processes, our
event generation procedure and selection cuts, and the resulting
signal significance. In Sec.~\ref{sec:conclusions} we summarize our
conclusions.
\section{The model}
As outlined in the introduction, we start with the field content of
the SM and add to it a new scalar SU(2)$_L$ doublet $\Phi_2$ (the SM
Higgs is denoted $\Phi_1$) and three right-handed gauge singlets
$\nu_{R_i}$ (these are the right-handed neutrinos). We impose a U(1)
symmetry under which $\Phi_2$ and the three $\nu_{R_i}$ have charge
+1 and all the other fields are uncharged, which leads to the
Yukawa coupling structure~\cite{Davidson:2009ha}
\begin{equation}
\mathcal{L}_{\rm Yuk} = - y^d_{ij} \bar d_{R_i} \Phi_1^{\dagger} Q_{L_j}
- y^u_{ij} \bar u_{R_i} \tilde \Phi_1^{\dagger} Q_{L_j}
- y^{\ell}_{ij} \bar e_{R_i} \Phi_1^{\dagger} L_{L_j}
- y^{\nu}_{ij} \bar \nu_{R_i} \tilde \Phi_2^{\dagger} L_{L_j}
+ {\rm h.c.}
\label{eq:smyuk}
\end{equation}
Here $\tilde \Phi_i \equiv i \sigma_2 \Phi_i^*$ is the conjugate Higgs
doublet and $y^f_{ij}$ are the 3$\times$3 Yukawa matrices for fermion
species $f$.
The Higgs doublets can be written explicitly as
\begin{eqnarray}
\Phi_i=\left( \begin{array} {c} \phi^+_i \\
(v_i + \phi^{0,r}_i + i \phi^{0,i}_i)/\sqrt{2} \end{array} \right),
\end{eqnarray}
where $v_1$ will be generated by the usual spontaneous symmetry
breaking mechanism of the SM and $v_2$ will be generated by the
explicit breaking of the global U(1), described below. Inserting
these expressions for $\Phi_i$ into Eq.~(\ref{eq:smyuk}), we obtain
the fermion masses and couplings to scalars. In particular, the
fourth term in Eq.~(\ref{eq:smyuk}) gives rise to the neutrino mass
matrix and interactions:
\begin{eqnarray}
\mathcal{L}_{\rm Yuk}
&\supset& - \frac{y^{\nu}_{ij} v_2}{\sqrt 2} \bar \nu_{R_i} \nu_{L_j}
- \frac{y^{\nu}_{ij}}{\sqrt 2} \phi_2^{0,r} \bar \nu_{R_i} \nu_{L_j}
- i \frac{y^{\nu}_{ij}}{\sqrt 2} \phi_2^{0,i} \bar \nu_{R_i} \nu_{L_j}
+ y^{\nu}_{ij} \phi_2^+ \bar \nu_{R_i} \ell_{L_j} + {\rm h.c.}
\label{eq:nuyuk}
\end{eqnarray}
After diagonalizing the mass matrix in the first term, the neutrino
mass eigenvalues are given by $m_{\nu_i} = y^{\nu}_i v_2/\sqrt{2}$,
where $y^{\nu}_i$ are the eigenvalues of $y^{\nu}_{ij}$. In this way,
the small masses of the three neutrinos can be traced to the small
value of $v_2$.
We obtain the vevs of the scalar doublets from the Higgs potential as
follows. The most general gauge-invariant scalar potential for two
Higgs doublets is (see, e.g., Ref.~\cite{HHG}),
\begin{eqnarray}
V &=& m_{11}^2 \Phi_1^{\dagger} \Phi_1 + m_{22}^2 \Phi_2^{\dagger} \Phi_2
- \left[ m_{12}^2 \Phi_1^{\dagger} \Phi_2 + {\rm h.c.} \right]
\nonumber \\
&& + \frac{1}{2} \lambda_1 \left( \Phi_1^{\dagger} \Phi_1 \right)^2
+ \frac{1}{2} \lambda_2 \left( \Phi_2^{\dagger} \Phi_2 \right)^2
+ \lambda_3 \left( \Phi_1^{\dagger} \Phi_1 \right)
\left( \Phi_2^{\dagger} \Phi_2 \right)
+ \lambda_4 \left( \Phi_1^{\dagger} \Phi_2 \right)
\left( \Phi_2^{\dagger} \Phi_1 \right) \nonumber \\
&& + \left\{ \frac{1}{2} \lambda_5 \left( \Phi_1^{\dagger} \Phi_2 \right)^2
+ \left[ \lambda_6 \Phi_1^{\dagger} \Phi_1
+ \lambda_7 \Phi_2^{\dagger} \Phi_2 \right] \Phi_1^{\dagger} \Phi_2
+ {\rm h.c.} \right\}.
\end{eqnarray}
Imposing the global U(1) symmetry eliminates $m_{12}^2$, $\lambda_5$,
$\lambda_6$, and $\lambda_7$. The global U(1) symmetry is broken
explicitly by reintroducing a small value for $m_{12}^2$. This leaves
the Higgs potential~\cite{Davidson:2009ha},\footnote{Note that
using a $Z_2$ symmetry instead of the global U(1) would allow a
nonzero $\lambda_5$ term.}
\begin{eqnarray}
V &=& m_{11}^2 \Phi_1^{\dagger} \Phi_1 + m_{22}^2 \Phi_2^{\dagger} \Phi_2
- \left[ m_{12}^2 \Phi_1^{\dagger} \Phi_2 + {\rm h.c.} \right]
\nonumber \\
&& + \frac{1}{2} \lambda_1 \left( \Phi_1^{\dagger} \Phi_1 \right)^2
+ \frac{1}{2} \lambda_2 \left( \Phi_2^{\dagger} \Phi_2 \right)^2
+ \lambda_3 \left( \Phi_1^{\dagger} \Phi_1 \right)
\left( \Phi_2^{\dagger} \Phi_2 \right)
+ \lambda_4 \left( \Phi_1^{\dagger} \Phi_2 \right)
\left( \Phi_2^{\dagger} \Phi_1 \right).
\end{eqnarray}
Stability of the potential at large field values requires
$\lambda_1,\lambda_2>0$, $\lambda_3 > -\sqrt{\lambda_1 \lambda_2}$,
and $\lambda_4 > -\sqrt{\lambda_1 \lambda_2} - \lambda_3$. We want
$v_1$ to arise through the usual spontaneous symmetry breaking
mechanism, which is achieved when $m_{11}^2<0$. We do not want the
global U(1) to also be broken spontaneously, as that will create a
very light pseudo-Nambu-Goldstone boson, which is incompatible with
standard big-bang nucleosynthesis; thus we require that the curvature
of the potential in the $v_2$ direction at zero $\Phi_2$ field value
be positive, i.e., $m_{22}^2 + (\lambda_3+\lambda_4) v_1^2/2 > 0$.
To find the values of the vevs in terms of the parameters of the Higgs
potential, we apply the minimization conditions,
\begin{eqnarray}
\left. \frac{\partial V}{\partial |\Phi_1|}\right|_{\rm min}
&=& m_{11}^2 v_1 - m_{12}^2 v_2
+ \frac{1}{2} \lambda_1 v_1^3
+ \frac{1}{2} ( \lambda_3 + \lambda_4 ) v_1 v_2^2 = 0
\nonumber \\
\left. \frac{\partial V}{\partial |\Phi_2|}\right|_{\rm min}
&=& m_{22}^2 v_2 - m_{12}^2 v_1
+ \frac{1}{2} \lambda_2 v_2^3
+ \frac{1}{2} ( \lambda_3 + \lambda_4 ) v_1^2 v_2 = 0.
\end{eqnarray}
Since we will require $m_{12}^2 \ll v_1^2$, we can ignore $m_{12}^2$ and
$v_2$ when finding the value of $v_1$. This yields
\begin{equation}
v_1^2 = \frac{-2 m_{11}^2}{\lambda_1}.
\end{equation}
For $v_2$, we need to consider $m_{12}^2$, although again we may
ignore higher order terms in $m_{12}^2/v_1^2$; this yields
\begin{equation}
v_2 = \frac{m_{12}^2 v_1}
{m_{22}^2 + \frac{1}{2} (\lambda_3 + \lambda_4) v_1^2}.
\label{eq:v2}
\end{equation}
We will choose parameters so that $v_1 \simeq 246$~GeV and $v_2
\sim$~eV. This requires $m_{12}^2 \sim ({\rm MeV})^2$. We note that
because $m_{12}^2$ is the only source of breaking of the global U(1)
symmetry, its size is technically natural; i.e., radiative corrections
to $m_{12}^2$ are proportional to $m_{12}^2$ itself and are only
logarithmically sensitive to the high-scale
cut-off~\cite{Davidson:2009ha}.
The mass eigenstates of the charged and CP-odd neutral scalars are given by
\begin{eqnarray}
G^+ &=& \phi_1^+ \sin\beta + \phi_2^+ \cos\beta \simeq \phi_1^+
\nonumber \\
H^+ &=& \phi_1^+ \cos\beta - \phi_2^+ \sin\beta \simeq - \phi_2^+
\nonumber \\
G^0 &=& \phi_1^{0,i} \sin\beta + \phi_2^{0,i} \cos\beta \simeq \phi_1^{0,i}
\nonumber \\
A^0 &=& \phi_1^{0,i} \cos\beta - \phi_2^{0,i} \sin\beta
\simeq - \phi_2^{0,i},
\end{eqnarray}
where we define $\tan\beta \equiv v_1/v_2 \sim 10^{11}$. $G^+$ and
$G^0$ are the Goldstone bosons, which do not appear as physical
particles in the unitarity gauge. $H^+$ and $A^0$ are the physical
charged and CP-odd neutral Higgs states and are almost entirely
contained in $\Phi_2$. Neglecting contributions of order $m_{12}^2$
and $v_2^2$, the masses of $H^+$ and $A^0$ are~\cite{Davidson:2009ha}
\begin{eqnarray}
M_{H^+}^2 &=& m_{22}^2 + \frac{1}{2} \lambda_3 v_1^2
\nonumber \\
M_A^2 &=& m_{22}^2 + \frac{1}{2} (\lambda_3 + \lambda_4) v_1^2
= M_{H^+}^2 + \frac{1}{2} \lambda_4 v_1^2.
\end{eqnarray}
The mass matrix for the CP-even neutral states is almost diagonal,
yielding only very tiny mixing of order $v_2/v_1$. Ignoring the
mixing, the eigenstates are $h^0 \simeq \phi^{0,r}_1$ (SM-like) and
$H^0 \sim \phi^{0,r}_2$, with masses~\cite{Davidson:2009ha}
\begin{eqnarray}
M_h^2 &=& m_{11}^2 + \frac{3}{2} \lambda_1 v_1^2
= \lambda_1 v_1^2 \nonumber \\
M_H^2 &=& m_{22}^2 + \frac{1}{2} (\lambda_3 + \lambda_4) v_1^2
= M_A^2.
\end{eqnarray}
After diagonalizing the neutrino mass matrix, Eq.~(\ref{eq:nuyuk})
yields the following couplings to the new physical Higgs states:
\begin{equation}
\mathcal{L}_{\rm Yuk} \supset -\frac{m_{\nu_i}}{v_2} H^0 \bar \nu_i \nu_i
+ i \frac{m_{\nu_i}}{v_2} A^0 \bar \nu_i \gamma_5 \nu_i
- \frac{\sqrt{2} m_{\nu_i}}{v_2} [U_{\ell i}^* H^+ \bar \nu_i P_L e_{\ell}
+ {\rm h.c.}],
\end{equation}
where $U_{\ell i}$ is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
matrix, defined according to $\nu_{\ell} = \sum_i U_{\ell i} \nu_i$,
where $\nu_{\ell}$ are the neutrino flavor eigenstates.
The PMNS matrix can be parameterized in terms of three mixing angles
$\theta_{ij}$ (with $ij = 12$, 23, and 13) and a phase $\delta$ according
to (see, e.g., Ref.~\cite{Fogli:2005cq}),
\begin{equation}
U_{\ell i} = \left( \begin{array}{ccc}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-i \delta} \\
-s_{12} c_{23} - c_{12} s_{23} s_{13} e^{i \delta}
& c_{12} c_{23} - s_{12} s_{23} s_{13} e^{i \delta}
& s_{23} c_{13} \\
s_{12} s_{23} - c_{12} c_{23} s_{13} e^{i \delta}
& -c_{12} s_{23} - s_{12} c_{23} s_{13} e^{i \delta}
& c_{23} c_{13} \end{array} \right),
\end{equation}
where $c_{ij} \equiv \cos \theta_{ij}$ and $s_{ij} \equiv \sin
\theta_{ij}$. The 2$\sigma$ experimentally-allowed ranges for the
three mixing angles and the neutrino mass-squared differences are
given in Table~\ref{tab:nuparams}. The phase $\delta$ and the mass of
the lightest neutrino are undetermined, although tritium beta decay
experiments set an upper limit on the neutrino masses of about
2~eV~\cite{Amsler:2008zzb}.
\begin{table}
\begin{tabular}{cc}
\hline \hline
Parameter & Value \\
\hline
$\sin^2 \theta_{12}$ & $0.314(1^{+0.18}_{-0.15})$ \\
$\sin^2 \theta_{23}$ & $0.44(1^{+0.41}_{-0.22})$ \\
$\sin^2 \theta_{13}$ & $0.9^{+2.3}_{-0.9} \times 10^{-2}$ \\
\hline
$\Delta m^2 \equiv m^2_{\nu_2} - m^2_{\nu_1}$
& $7.92(1 \pm 0.09) \times 10^{-5}$~eV$^{2}$ \\
$\Delta M^2 \equiv m^2_{\nu_3} - \frac{1}{2}(m^2_{\nu_1} + m^2_{\nu_2})$
& $\pm 2.4(1^{+0.21}_{-0.26}) \times 10^{-3}$~eV$^{2}$ \\
\hline \hline
\end{tabular}
\caption{Current values of the neutrino mixing parameters and
mass-squared differences, from the global fit to neutrino
oscillation data performed in Ref.~\cite{Fogli:2005cq}.
Uncertainties quoted are the 2$\sigma$ ranges. Note that the
constraint on $\sin\theta_{13}$ is only an upper bound, and that the
sign of $\Delta M^2$ is not yet known.}
\label{tab:nuparams}
\end{table}
Since the decays of $H^0$ and $A^0$ to two neutrinos will be invisible
to a collider detector, the decay of most interest is $H^+ \rightarrow
\ell^+ \nu$. The charged Higgs can decay into all nine combinations
of $\ell_i \nu_j$; summing over neutrino mass eigenstates, the partial
width to a particular charged lepton $\ell$ is~\cite{Davidson:2009ha}
\begin{equation}
\Gamma \left(H^+ \to \ell^+ \nu \right)
= \frac{M_{H^+} \langle m^2_{\nu} \rangle_{\ell}}{8\pi v_2^2},
\end{equation}
where we define the expectation value of the neutrino mass-squared in
a flavor eigenstate by~\cite{Fukuyama:2008sz}
\begin{equation}
\langle m^2_{\nu} \rangle_{\ell} = \sum_i m^2_{\nu_{i}} |U_{\ell i}|^2.
\end{equation}
In what follows we work under the assumption that $M_{H^0, A^0} >
M_{H^+}$, i.e., $\lambda_4 > 0$, so that the decays $H^+ \rightarrow
W^+ H^0$, $W^+ A^0$ will be kinematically forbidden. The branching
ratios of the charged Higgs are then completely determined by the
neutrino masses and mixing:
\begin{equation}
{\rm BR}(H^+ \to \ell^+ \nu)
= \frac{\langle m^2_{\nu} \rangle_{\ell}}
{\sum_{\ell} \langle m^2_{\nu} \rangle_{\ell}}
= \frac{\langle m^2_{\nu} \rangle_{\ell}}
{\sum_i m^2_{\nu_i}},
\end{equation}
where we used the unitarity of the PMNS matrix to simplify the
denominator.
The sign of the larger neutrino mass splitting $\Delta M^2$ is unknown
(see Table~\ref{tab:nuparams}). The situation in which $\Delta M^2$
is positive, so that $\nu_3$ is the heaviest neutrino, is called the
normal neutrino mass hierarchy, while the situation in which $\Delta
M^2$ is negative, so that $\nu_1$ and $\nu_2$ are heavier, is called
the inverted hierarchy. We compute the charged Higgs branching
fractions as a function of the lightest neutrino mass for both
hierarchies, scanning over the 2$\sigma$ allowed ranges of the
neutrino parameters as given in Table~\ref{tab:nuparams}. Results are
shown in Fig.~\ref{fig:brs}.\footnote{We disagree with the charged
Higgs branching fractions to leptons presented in
Ref.~\cite{Gabriel:2008es} for the $Z_2$ model of
Ref.~\cite{Gabriel:2006ns}; these decays should have the same
relative branching fractions as in our model.} The large
spread in the branching ratios to $\mu \nu$ and $\tau \nu$ for
lightest neutrino masses below about 0.06~eV is due to the current
experimental uncertainty in $\sin^2\theta_{23}$, which controls the
relative amount of $\nu_{\mu}$ and $\nu_{\tau}$ in the isolated mass
eigenstate $\nu_3$.
\begin{figure}
\resizebox{1\textwidth}{!}{
{\rotatebox{270}{\includegraphics{hcbr_nh.eps}}}
{\rotatebox{270}{\includegraphics{hcbr_ih.eps}}}}
\caption{Charged Higgs decay branching fractions to $e\nu$, $\mu\nu$,
and $\tau\nu$ as a function of the lightest neutrino mass.}
\label{fig:brs}
\end{figure}
Limits on the model parameters were discussed in
Ref.~\cite{Davidson:2009ha}. The most significant for our purposes is
from searches for leptons plus missing energy at the CERN Large
Electron-Positron Collider, which put a lower bound on the charged
Higgs mass of 65--85 GeV, depending on the mass of the lightest
neutrino. Big-bang nucleosynthesis also puts an upper bound on the
neutrino Yukawa couplings of
\begin{equation}
y^{\nu}_i \equiv \frac{\sqrt{2} m_{\nu_i}}{v_2} \lesssim
\frac{1}{30}\left[\frac{M_{H^+}}{100 \ {\rm GeV}}\right]
\left[\frac{1/\sqrt{2}}{|U_{\ell i}|} \right].
\end{equation}
\section{Signal and background at the LHC}
\label{sec:SB}
In most other two-Higgs-doublet models, the charged Higgs decay rate
to a particular charged lepton is proportional to the square of the
charged lepton mass (see, e.g., Ref.~\cite{HHG}). Such a charged
Higgs therefore decays predominantly to $\tau \nu$, with decays to
$\mu\nu$, $e \nu$ below 1\%. In our neutrino-mass model, however, the
charged Higgs decay rate to a particular charged lepton is instead
proportional to the square of the mass of the corresponding neutrino
flavor eigenstate. As a result, the branching fraction to $e \nu$
and/or $\mu \nu$ will always be sizable. In particular, in the normal
hierarchy BR($H^+ \to \mu \nu) \simeq 1/2$, in the inverted hierarchy
BR($H^+ \to e \nu) \simeq 1/2$ and BR($H^+ \to \mu \nu) \simeq 1/4$,
and for a degenerate neutrino spectrum BR($H^+ \to e \nu) \simeq$
BR($H^+ \to \mu \nu) \simeq 1/3$, as shown in Fig.~\ref{fig:brs}.
Considering the high detection efficiency and lower fake rates of $e$
and $\mu$ compared to $\tau$, we study $H^+H^-$ pair production at the
LHC mediated by a photon or $Z$, followed by decays to $e$ or $\mu$
with missing transverse momentum. We consider two scenarios, $M_{H^+}
= 100$ and 300~GeV, and present results as a function of the lightest
neutrino mass for both the normal and inverted hierarchy.
The process of interest is $pp \rightarrow H^+ H^- \rightarrow \ell
\ell^{\prime} \bar \nu_\ell \nu_{\ell^{\prime}}$, with $\ell
\ell^{\prime} = e^+ e^-$, $\mu^+ \mu^-$, and $e^{\pm} \mu^{\mp}$. The
relevant backgrounds are $pp\rightarrow VV \rightarrow
\ell \ell^{\prime} \bar \nu \nu$ with $VV = W^+W^-$, $ZZ$, or $Z \gamma$
and the neutrinos of any type, and $pp \rightarrow t \bar t \rightarrow
\ell \ell^{\prime} \bar \nu_{\ell} \nu_{\ell^{\prime}} b \bar b$. In
spite of the presence of the extra $b$ jets that can be vetoed, the $t
\bar t$ process is important because of its exceptionally high cross
section at the LHC.
\subsection{Event generation}
\label{sec:evtgen}
We simulated the signal and background processes with the parton-level
Monte Carlo MadGraph/MadEvent version~4~\cite{Alwall:2007st}. We
present both a parton-level analysis and an analysis including
showering, hadronization, and a fast detector simulation using a
PYTHIA-PGS package designed to be used with MadEvent. PYTHIA (version
6.4.20)~\cite{PYTHIA} generates initial- and final-state radiation and
hadronizes the final-state quarks and gluons, while PGS (Pretty Good
Simulation of High Energy Collisions, version 4)~\cite{PGS} is a basic
detector simulator---we used the default settings for ATLAS. For the
signal process we generated 10,000 unweighted events in each of the
$e^+e^-$, $\mu^+ \mu^-$, and $\mu^+ e^-$ final states. For both the
$VV$ and $t \bar t$ backgrounds we generated 100,000 unweighted events
in each of the three leptonic final states. We incorporated the $\mu^- e^+$ final state by
doubling the $\mu^+ e^-$ cross sections. For the backgrounds we used
the default SM branching fractions from MadGraph/MadEvent, given in
Table~\ref{tab:WZbrs}.
\begin{table}
\begin{tabular}{cc}
\hline \hline
Process & Branching fraction \\
\hline
$W^+ \rightarrow \ell^+\nu_\ell$ ($\ell = e$ or $\mu$) & 0.1068 \\
$Z \rightarrow \ell^+ \ell^-$ ($\ell = e$ or $\mu$) & 0.0336 \\
$Z \rightarrow \nu \bar \nu$ (all 3 neutrinos) & 0.2000 \\
$t \rightarrow W^+b$ & 1.0000 \\
\hline \hline
\end{tabular}
\caption{Default SM branching fractions used in
MadGraph/MadEvent~\cite{Alwall:2007st}.}
\label{tab:WZbrs}
\end{table}
Although MadGraph/MadEvent is a tree-level event generator, we
partially incorporated next-to-leading order (NLO) QCD corrections.
We did this for two reasons. First, QCD corrections have a
significant effect on the signal and background (especially $t \bar
t$) cross sections, as well as significantly reducing the QCD scale
uncertainty, so that using NLO cross sections lets us obtain more
reliable results. Second, for the $M_{H^+}=100$~GeV simulation we will apply a jet veto to reduce the $t
\bar t$ background, which will also affect the signal and $VV$
background once initial-state radiation is included. While this could
be simulated by running the no-jet events through PYTHIA, a
parton-level simulation of the $H^+H^-j$ and $VVj$ processes provides
a more accurate description of jet kinematics. Because these one-jet
processes make up part of the NLO QCD cross section for the
corresponding no-jet processes, we must incorporate the NLO cross
sections for consistency, as follows.
In the absence of a full NLO Monte Carlo, NLO QCD corrections are
usually incorporated by multiplying the leading-order (LO) cross
section---and the cross section corresponding to each simulated event
both before and after cuts---by a $k$-factor equal to the ratio of the
NLO cross section to the tree-level cross section. In our case,
however, our jet veto will affect LO events (which have no jet) and
NLO events (which can have a final-state jet) differently. We deal
with this by simulating $pp \to H^+H^- j$ and $pp \to VVj$ with the
same decay final states as considered in the no-jet processes. For
simplicity we generate the same number of events with an additional
jet at the parton level as were generated for the no-jet processes.
Because the $t \bar t$ background already contains two jets at leading
order, we do not separately generate events with additional jets for
this background. To avoid the collinear and infrared singularities,
we apply a minimum $p_T$ cut of 10~GeV on the jet at the
event-generation level.
The square of the NLO matrix element can be expressed up to order
$\alpha_s$ as
\begin{equation}
\mathcal{|M|}_{\rm NLO}^2
= | \mathcal{M}_{\rm LO} + \mathcal{M}_{\rm 1 \, loop} |^2
+ \mathcal{|M|}_{\rm 1 \, jet}^2.
\end{equation}
We used MadGraph/MadEvent to calculate the cross sections
corresponding to $\mathcal{M}_{\rm LO}$ and $\mathcal{M}_{\rm 1 \,
jet}$. We computed the NLO cross-section for $pp \to H^+H^-$ at the
LHC using the public FORTRAN code {\tt
PROSPINO}~\cite{Beenakker:1999xh,Alves:2005kr} with CTEQ6 parton
densities~\cite{Pumplin:2002vw}, with the renormalization and
factorization scales set equal to $M_{H^+}$. We took the NLO cross
sections for the SM $W^+W^-$ and $ZZ$ background processes from
Ref.~\cite{Campbell:1999ah}. This paper quotes results using both the
MRS98 and CTEQ5 parton densities, with results differing by $\sim$6\%;
since we use CTEQ6 for the tree-level MadGraph/MadEvent calculation,
we take the results using the CTEQ5 parton densities for consistency.
For events with $e^{\pm} \mu^{\mp}$ in the final state, only the cross
section for $W^+W^-$ is relevant; for events with $\mu^+\mu^-$ or
$e^+e^-$ in the final state, both the $W^+W^-$ and $ZZ$ processes
contribute and we add the cross sections at both LO and NLO. We took
the $t \bar t$ cross section from Ref.~\cite{Bonciani:1998vc}, which
includes both NLO and next-to-leading logarithmic corrections.
The remaining scale uncertainty is about $\pm$5\% when the
factorization and renormalization scales are varied between $m_t/2$
and $2m_t$. The relevant cross sections are given in
Table~\ref{tab:xsecs}.
\begin{table}
\begin{tabular}{ccc}
\hline \hline
Process & Cross section & Source \\
\hline
$pp \to H^+H^-$ ($M_{H^+} = 100$~GeV) & 295 fb
& {\tt PROSPINO}~\cite{Beenakker:1999xh,Alves:2005kr} \\
$pp \to H^+H^-$ ($M_{H^+} = 300$~GeV) & 5.32 fb
& {\tt PROSPINO}~\cite{Beenakker:1999xh,Alves:2005kr} \\
$pp \to W^+W^-$ & 127.8 pb & Ref.~\cite{Campbell:1999ah} \\
$pp \to ZZ$ & 17.2 pb & Ref.~\cite{Campbell:1999ah} \\
$pp \to t \bar t$ & 833 pb & Ref.~\cite{Bonciani:1998vc} \\
\hline \hline
\end{tabular}
\caption{NLO cross sections for signal and background processes (before
decays) at the LHC (14~TeV). The $t \bar t$ cross section also
includes a resummation of next-to-leading logarithmic corrections.}
\label{tab:xsecs}
\end{table}
We find that with our generator-level jet
$p_T$ cut on $\sigma_{\rm 1 \, jet}$, $\sigma_{\rm NLO} < \sigma_{\rm
LO} + \sigma_{\rm 1 \, jet}$, so the one-loop matrix element must
interfere destructively with the LO matrix element. Thus the
generated cross section from the LO process must be scaled down in
order to incorporate the effects of the one-loop correction.
For the parton-level simulation, the relevant scale factor is
determined by solving for $k$ in the equation,
\begin{equation}
\sigma_{\rm NLO} = k \, \sigma_{\rm LO} + \sigma_{\rm 1 \, jet},
\label{eq:k}
\end{equation}
before cuts are applied, and then using this equation with the same
value of $k$ to calculate the surviving $\sigma_{\rm NLO}$ after the
cuts are applied to the LO and one-jet MadGraph/MadEvent simulated
results.
For the PYTHIA-PGS simulation, the simulated events have extra jets
produced by PYTHIA and ``measured'' jet $p_T$ smeared by PGS. To
avoid double-counting, we use the following equation with two
constants:
\begin{equation}
\sigma_{\rm NLO} = m \, \sigma_{\rm LO}^{\rm cut}
+ n \, \sigma_{\rm 1 \, jet}^{\rm cut},
\label{eq:mn}
\end{equation}
where $\sigma_{\rm LO}^{\rm cut}$ and $\sigma_{\rm 1 \, jet}^{\rm
cut}$ are the cross sections identified by PGS as having no jets and
at least one jet, respectively, with $p_T > 10$ GeV, out of the
combined LO and one-jet generated samples. The constants $m$ and $n$
are determined by $m \, \sigma_{\rm LO}^{\rm cut} = k \, \sigma_{\rm
LO}$ and $n \, \sigma_{\rm 1 \, jet}^{\rm cut} = \sigma_{\rm 1 \,
jet}$ using $k$ from Eq.~(\ref{eq:k}). Equation~(\ref{eq:mn}) with
the same values of $m$ and $n$ is then used after cuts to calculate
the surviving $\sigma_{\rm NLO}$.
\subsection{Cuts}
We apply four cuts to reduce the background, summarized in
Table~\ref{tab:cuts}. The first cut checks for the presence of two
opposite-sign leptons each with $p_T > 20$~GeV and missing transverse
momentum of at least 30~GeV. For the parton-level simulation, we also
apply acceptance cuts on the pseudorapidity of both leptons, $|\eta| <
3.0$ for electrons and $|\eta| < 2.4$ for muons. Second, for the
$e^+e^-$ and $\mu^+\mu^-$ final states we veto events for which the
dilepton invariant mass falls between 80 and 100~GeV, in order to
eliminate background from $Z (\to \ell \ell) + p_T^{\rm miss}$. This
will also eliminate the majority of any background from $Z + jets$
with fake $p_T^{\rm miss}$, which we did not simulate. The third cut
vetoes events containing a jet with $p_T > 30$~GeV; for the
parton-level simulation, we require that this jet falls in the
rapidity range $|\eta| < 5.0$. This eliminates more than 97\% of the
$t \bar t$ backgound, but also reduces the signal by about a factor of
two. We find that this cut is useful for $M_{H^+} = 100$~GeV. For
$M_{H^+} = 300$~GeV the signal cross section is considerably smaller
and the signal events will be better separated from background in our
final cut variable, so that we obtain better sensitivity without the
jet veto.
\begin{table}
\begin{tabular}{lp{13cm}}
\hline \hline
Cut name & Explanation \\
\hline
Basic cuts & Present are a lepton and antilepton, each with
$p_T^{\ell} > 20$~GeV, and missing transverse momentum $p_T^{\rm miss}
> 30$~GeV. For the parton level results, we also apply lepton
acceptance cuts of $|\eta_e|<3.0$ and $|\eta_\mu|<2.4$.\\
$Z$ pole veto \ \ & To eliminate events that include $Z \rightarrow
\ell^+\ell^-$, we veto events in which the invariant mass of $e^+e^-$
or $\mu^+\mu^-$ is between 80 and 100~GeV (not applied to the
$e^{\pm}\mu^{\mp} p_T^{\rm miss}$ final state).\\
Jet veto & Designed to reduce $t \bar t$ background, any event with a
jet with $p_T^{\rm jet} > 30$~GeV was rejected. For the parton level
results, this veto is only applied when $|\eta_{\rm jet}| < 5.0$.
(Applied only for $M_{H^+} = 100$~GeV.)\\
$M_{T2}$ cut & To reduce the $W^+W^-$ and $t \bar t$ backgrounds, we
make use of the larger mass of $H^+$ compared to the intermediate $W$
bosons in both backgrounds by cutting on $M_{T2}$ (defined in
Eq.~(\ref{eq:mT2})). For $M_{H^+} = 100$~GeV we require $M_W < M_{T2}
< 100$~GeV and for $M_{H^+} = 300$~GeV we require 150~GeV~$< M_{T2} <
300$~GeV.\\
\hline \hline
\end{tabular}
\caption{Summary of cuts.}
\label{tab:cuts}
\end{table}
The final cut is on the variable $M_{T2}$, defined as~\cite{Lester:1999tx}
\begin{equation}
M_{T2}^2 = \begin{array}{c} \rm min \\
q_T^{\rm miss(1)} + q_T^{\rm miss(2)} = p_T^{\rm miss} \end{array}
\left[ {\rm max} \left\{ m_T^2 \left( p_T^{\ell(1)}, q_T^{\rm miss(1)}
\right) ,
m_T^2 \left(p_T^{\ell(2)}, q_T^{\rm miss(2)} \right) \right\} \right],
\label{eq:mT2}
\end{equation}
where $m_T^2$ is the square of the transverse mass (ignoring the charged
lepton and neutrino masses),
\begin{equation}
m_T^2 \left( p_T^{\ell}, q_T^{\rm miss} \right)
= 2 \left( |\vec p_T^{\, \ell}| |\vec q_T^{\, \rm miss}|
- \vec p_T^{\, \ell} \cdot \vec q_T^{\, \rm miss} \right).
\end{equation}
In other words, $M_{T2}$ is determined by making a guess for the
transverse momenta of the two neutrinos (constrained by the measured
total missing transverse momentum) and computing the transverse masses
of the two $\ell \nu$ systems; the guess is then varied until the
larger of the two reconstructed transverse masses is minimized.
For equal-mass intermediate particles each decaying to $\ell \nu$, the
$M_{T2}$ distribution has an upper endpoint at the mass of the
intermediate particle. Thus by cutting out events with $M_{T2} <
M_W$, all the $W^+W^-$ background should be eliminated (the endpoint
is in fact smeared out by the finite $W$ width and momentum resolution
of the detector). Since the leptons and missing transverse momentum
in the $t \bar t$ process also come from decays of on-shell $W^+W^-$,
this background should be eliminated as well. There is also a
small contribution to the $VV$ background from nonresonant processes
that can have $M_{T2} > M_W$. Since all signal events will have
$M_{T2} < M_{H^+}$, we also cut out events with $M_{T2} > M_{H^+}$ in
an effort to reduce the background from these nonresonant $VV$
processes. For $M_{H^+}=300$~GeV, we find that raising the minimum
cut on $M_{T2}$ to 150~GeV reduces the tail of the nonresonant $VV$
events without reducing the signal too much.
In Fig.~\ref{fig:hist} we show the $M_{T2}$ distributions for signal
and background processes in the $e^+e^- p_T^{\rm miss}$ channel for
$M_{H^+}=100$ and 300~GeV after the other cuts have been applied, for
the PYTHIA-PGS simulation. Note that the $t \bar t$ background
distribution has a maximum $M_{T2}$ value a little above the $W$ mass,
so that it can be eliminated with a high enough cut on $M_{T2}$, as we
do for the case of $M_{H^+} = 300$~GeV. (The higher $M_{T2}$ endpoint
for $t \bar t$ in the right-hand plot in Fig.~\ref{fig:hist} is due to
the absence of the jet veto, resulting in much higher $t \bar t$
statistics.) The $VV$ background also falls off dramatically around
$M_{T2} \sim M_W$; however, due to nonresonant diagrams without
on-shell intermediate $W$ pairs, this background extends to much
higher values of $M_{T2}$. With our simulation statistics, a single
$\bar t t$ event corresponds to a cross section of about 0.1~fb, while
a single $VV$ event corresponds to a cross section of about 0.01~fb.
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics{hist100.eps}
\includegraphics{hist300.eps}}
\caption{$M_{T2}$ distributions after other cuts have been applied for
the $e^+ e^- p_T^{\rm miss}$ final state, with $M_{H^+}=100$~GeV
(left, with jet veto) and 300~GeV (right, no jet veto). For the
signal we take BR($H^+ \to e^+ \nu) = 1/3$, which occurs for a
degenerate neutrino spectrum. The $M_{T2}$ cut window is shown by
the vertical lines.}
\label{fig:hist}
\end{figure}
\subsection{Results}
The efficiency of each cut on $\sigma_{\rm NLO}$ for the $e^+ e^-
p_T^{\rm miss}$ final state is displayed in Tables~\ref{eehh},
\ref{eebg100} and~\ref{eebg300}. Cut efficiencies for $\mu^+
\mu^-p_T^{\rm miss}$ are displayed in Tables~\ref{mmhh}, \ref{mmbg100}
and~\ref{mmbg300}, and for $e^\pm \mu^\mp p_T^{\rm miss}$ in
Tables~\ref{emhh}, \ref{embg100} and~\ref{embg300}. We give
efficiencies for both the parton-level simulation and the simulation
including showering, hadronization, and fast detector simulation using
the PYTHIA-PGS package. All results incorporate NLO corrections as
described in Sec.~\ref{sec:evtgen}.
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$M_{H^+}=100$~GeV}&\multicolumn{2}{c}{$M_{H^+}=300$~GeV}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.62644&0.47860&0.92961&0.72105\\
$Z$ pole veto&0.90754&0.90165&0.97732&0.97724\\
Jet veto&0.68433&0.60717&--&--\\
$M_{T2}$ cut&0.17075&0.15542&0.47317&0.45945\\
\hline
Cumulative&0.06643&0.04072&0.42988&0.32375\\
\hline \hline
\end{tabular}
\caption{Cut efficiencies for the signal process $pp \rightarrow e^+
e^- p_T^{\rm miss}$ via $H^+ H^-$. The efficiency of each cut is
defined as the cross section that passed the cut divided by the
cross section that passed the previous cut. The cumulative efficiency
is the cross section that passed all the cuts divided by the original
cross section. The jet veto is not applied for $M_{H^+} = 300$~GeV.
The $M_{T2}$ cut is $M_W < M_{T2} < 100$~GeV for
$M_{H^+}=100$~GeV, and 150~GeV $ < M_{T2} < 300$~GeV for $M_{H^+}=300$~GeV.}
\label{eehh}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$VV$ Background}&\multicolumn{2}{c}{$t \bar t$ Background}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.42708&0.33912&0.58407&0.40612\\
$Z$ pole veto&0.74727&0.73255&0.86236&0.85501\\
Jet veto&0.63306&0.67299&0.01318&0.02856\\
$M_{W^+} < M_{T2} < 100$ GeV&0.01401&0.01147&0.01205&0.02716\\
\hline
Cumulative&0.00283&0.00192&0.00008&0.00027\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for background for
$pp \rightarrow e^+ e^- p_T^{\rm miss}$, with cuts for $M_{H^+}=100$~GeV.}
\label{eebg100}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$VV$ Background}&\multicolumn{2}{c}{$t \bar t$ Background}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.42708&0.33912&0.58407&0.40612\\
$Z$ pole veto&0.74727&0.73255&0.86236&0.85501\\
150 GeV $ < M_{T2} < 300$ GeV&0.00260&0.00196&0.00000&0.00000\\
\hline
Cumulative&0.00083&0.00049&0.00000&0.00000\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for background for
$pp \rightarrow e^+ e^- p_T^{\rm miss}$, with cuts for $M_{H^+}=300$~GeV.}
\label{eebg300}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$M_{H^+}=100$ GeV}&\multicolumn{2}{c}{$M_{H^+}=300$ GeV}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.51713&0.43680&0.84810&0.69845\\
$Z$ pole veto&0.90869&0.90075&0.97756&0.97696\\
Jet veto&0.68310&0.57831&--&--\\
$M_{T2}$ cut&0.16875&0.17320&0.47802&0.46847\\
\hline
Cumulative&0.05417&0.03941&0.39632&0.31966\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for the signal process
$pp \rightarrow \mu^+ \mu^- p_T^{\rm miss}$ via $H^+ H^-$.}
\label{mmhh}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$VV$ Background}&\multicolumn{2}{c}{$t \bar t$ Background}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.32959&0.28226&0.52593&0.39048\\
$Z$ pole veto&0.73839&0.73098&0.86021&0.85489\\
Jet veto&0.62703&0.63204&0.01346&0.02282\\
$M_{W^+} < M_{T2} < 100$ GeV&0.01324&0.01554&0.00657&0.03675\\
\hline
Cumulative&0.00202&0.00203&0.00004&0.00028\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for background for
$pp \rightarrow \mu^+ \mu^- p_T^{\rm miss}$, with cuts for $M_{H^+}=100$~GeV.}
\label{mmbg100}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$VV$ Background}&\multicolumn{2}{c}{$t \bar t$ Background}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.32959&0.28226&0.52593&0.39048\\
$Z$ pole veto&0.73839&0.73098&0.86021&0.85489\\
150 GeV $ < M_{T2} < 300$ GeV&0.00288&0.00239&0.00000&0.00000\\
\hline
Cumulative&0.00070&0.00049&0.00000&0.00000\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for background for
$pp \rightarrow \mu^+ \mu^- p_T^{\rm miss}$, with cuts for $M_{H^+}=300$~GeV.}
\label{mmbg300}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$M_{H^+}=100$ GeV}&\multicolumn{2}{c}{$M_{H^+}=300$ GeV}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.56131&0.45743&0.88249&0.70832\\
Jet veto&0.68783&0.59528&--&--\\
$M_{T2}$ cut&0.16857&0.16121&0.47427&0.46373\\
\hline
Cumulative&0.06508&0.04390&0.41854&0.32847\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for the signal process
$pp \rightarrow e^\pm \mu^\mp p_T^{\rm miss}$ via $H^+ H^-$.}
\label{emhh}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$VV$ Background}&\multicolumn{2}{c}{$t \bar t$ Background}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.35835&0.30423&0.55297&0.39556\\
Jet veto&0.65590&0.68572&0.01255&0.02592\\
$M_{W^+} < M_{T2} < 100$ GeV&0.00860&0.01207&0.01585&0.03018\\
\hline
Cumulative&0.00202&0.00252&0.00011&0.00031\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for background for
$pp \rightarrow e^\pm \mu^\mp p_T^{\rm miss}$, with cuts for
$M_{H^+}=100$~GeV.}
\label{embg100}
\end{table}
\begin{table}
\begin{tabular}{lcccc}
\hline \hline
&\multicolumn{2}{c}{$VV$ Background}&\multicolumn{2}{c}{$t \bar t$ Background}\\
Cuts&Parton&PYTHIA/PGS&Parton&PYTHIA/PGS \\
\hline
Basic cuts&0.35835&0.30423&0.55297&0.39556\\
150 GeV $ < M_{T2} < 300$ GeV&0.00057&0.00049&0.00000&0.00000\\
\hline
Cumulative&0.00021&0.00015&0.00000&0.00000\\
\hline \hline
\end{tabular}
\caption{As in Table~\ref{eehh} but for background for
$pp \rightarrow e^\pm \mu^\mp p_T^{\rm miss}$, with cuts for
$M_{H^+}=300$~GeV.}
\label{embg300}
\end{table}
Consider for example the PYTHIA-PGS results in the $e^+e^- p_T^{\rm
miss}$ final state, and assume a degenerate neutrino spectrum so
that BR($H^+ \to e^+ \nu) = 1/3$. In this case, for $M_{H^+} =
100$~GeV, the cuts reduce the charged Higgs signal cross section in
this channel from 32.8~fb to 1.34~fb, while reducing the $VV$
background from 1570~fb to 3.01~fb and the $t \bar t$ background from
9500~fb to 2.57~fb. The ratio of signal to background cross sections
(S/B) is then 0.24. For $M_{H^+} = 300$~GeV, S/B is comparable.
These are displayed for all channels for a degenerate neutrino
spectrum in Table~\ref{SB}. In all cases S/B is at least 0.22,
comfortably larger than the QCD and parton density uncertainties on
the $VV$ and $t \bar t$ backgrounds; the overall cross sections of
these backgrounds can also be normalized experimentally using $M_{T2}$
regions below $M_W$.
\begin{table}
\begin{tabular}{cccc}
\hline \hline
$M_{H^+}$&Channel& S/B & \ Luminosity for 5$\sigma$ \ \\
\hline
& \ $e^+e^-p_T^{\rm miss}$ \ & 0.24 & 78 fb$^{-1}$ \\
100~GeV&$\mu^+\mu^-p_T^{\rm miss}$& 0.22 & 88 fb$^{-1}$ \\
&$e^\pm \mu^\mp p_T^{\rm miss}$& 0.22 & 40 fb$^{-1}$ \\
\hline
&$e^+e^-p_T^{\rm miss}$& 0.25 & 526 fb$^{-1}$ \\
300~GeV&$\mu^+\mu^-p_T^{\rm miss}$& 0.25 & 540 fb$^{-1}$ \\
&$e^\pm \mu^\mp p_T^{\rm miss}$& 0.89 & 73 fb$^{-1}$ \\
\hline \hline
\end{tabular}
\caption{Signal over background (S/B) and luminosity required for a
5$\sigma$ discovery in a single channel for the three signal
processes studied, for $M_{H^+} = 100$ and 300~GeV, assuming a
degenerate neutrino spectrum so that BR($H^+ \to e^+ \nu) = {\rm
BR}(H^+ \to \mu^+ \nu) = 1/3$.}
\label{SB}
\end{table}
For $M_{H^+} = 100$~GeV, the background after cuts depends sensitively
on the shape of the background $M_{T2}$ distribution just above $M_W$.
This is controlled by the $W$ width and the detector resolution for
lepton momenta and missing $p_T$; its shape should not suffer from QCD
or parton-density uncertainties. For $M_{H^+} = 300$~GeV, the shape
and normalization of the nonresonant tail of the $VV$ background is
especially important. This background is mostly Drell-Yan with an
additional on-shell $W$ boson radiated from one of the final-state
leptons; the QCD corrections to such processes are well understood.
Given enough statistics, the shape of this background could also be
normalized using the $M_{T2}$ region above $M_{H^+}$. Note also that
the nonresonant tail of the $VV$ background is significantly smaller
for the $e^{\pm} \mu^{\mp}$ final state than for the $e^+e^-$ and
$\mu^+\mu^-$ final states, leading to a much higher signal purity in
this final state for $M_{H^+}=300$~GeV as shown in the last line of
Table~\ref{SB} (for the lower charged Higgs mass this effect is
swamped by the resonant-$W$ contribution).
The integrated luminosity required for a 5$\sigma$ discovery of
$H^+H^-$ is displayed in Fig.~\ref{sig100} for $M_{H^+} = 100$~GeV and
Fig.~\ref{sig300} for $M_{H^+} = 300$~GeV, for each channel separately
and for all three channels combined. We use the PYTHIA-PGS results
and compute only the statistical significance. For the normal
hierarchy with $M_{H^+} = 100$ (300)~GeV, we find 5$\sigma$ discovery
statistics with a minimum of 9 (56)~fb$^{-1}$. For the inverted
hierarchy, the minimum is 8 (24)~fb$^{-1}$. For the case of
degenerate neutrino masses, 20 (57)~fb$^{-1}$ is needed. For
degenerate neutrino masses, the luminosity needed for a 5$\sigma$
discovery in each channel separately is given in Table~\ref{SB}.
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics{nh100c.eps}
\includegraphics{ih100c.eps}}
\caption{Luminosity required at the LHC (14~TeV) for a 5$\sigma$
discovery if $M_{H^+} = 100$~GeV, for the normal hierarchy (NH,
left) and inverted hierarchy (IH, right). The lines for each
channel bound the range of required luminosities obtained by
scanning over the 2$\sigma$ allowed ranges of the parameters of the
neutrino mixing matrix and mass-squared differences.}
\label{sig100}
\end{figure}
\begin{figure}
\resizebox{\textwidth}{!}{
\includegraphics{nh300c.eps}
\includegraphics{ih300c.eps}}
\caption{As in Fig.~\ref{sig100} but for $M_{H^+} = 300$~GeV.}
\label{sig300}
\end{figure}
\section{Discussion and conclusions}
\label{sec:conclusions}
The two-Higgs-doublet model for Dirac neutrino masses studied here
provides distinctive leptonic signatures at the LHC due to the
characteristic decay pattern of the charged Higgs boson, controlled by
the neutrino masses and mixing. We have shown that a simple set of
cuts allows discovery of charged Higgs pairs with decays to
$\ell \ell^{(\prime)} p_T^{\rm miss}$ with relatively modest
integrated luminosity. In particular we found that a cut on the
kinematic variable $M_{T2}$ provides very effective suppression of $W$
pair and $t \bar t$ backgrounds for charged Higgs masses sufficiently
above the $W$ mass.
In the inverted neutrino mass hierarchy, the large branching fractions
of the charged Higgs to $e \nu$ and $\mu \nu$ guarantees a 5$\sigma$
discovery for any allowed neutrino mass and mixing parameter values
with only 20 (57) fb$^{-1}$ for $M_{H^+} = 100$ (300)~GeV. The
discovery potential remains remarkably good at $M_{H^+} = 300$~GeV
despite the rapidly falling charged Higgs pair production cross
section because of the increasing separation of the signal $M_{T2}$
distribution from the background.
In the normal neutrino mass hierarchy, the large uncertainty on the
neutrino mixing angle $\theta_{23}$ leads to parameter regions in
which the charged Higgs decays predominantly to $\tau\nu$, with a
branching fraction to light leptons below 40\%, resulting in poor
discovery sensitivity in the light lepton channels studied in this
paper. Away from these parameter regions, the discovery prospects are
only slightly worse than in the inverted hierarchy.
As more stringent experimental limits are placed on the neutrino
parameters from neutrino oscillation experiments and direct searches
for the kinematic neutrino mass in beta decay, the predictions for the
charged Higgs branching ratios in this model will tighten. For
example, one goal of the currently-running T2K long-baseline neutrino
oscillation experiment in Japan is to improve the measurement accuracy
of $\sin^2(2 \theta_{23})$ by an order of
magnitude~\cite{Zito:2008zza}, which would reduce the 2$\sigma$ spread
in the charged Higgs branching ratios to $\mu\nu$ and $\tau\nu$ at low
lightest-neutrino mass from the current $\pm$30\% to about $\pm$10\%.
Sensitivity to the neutrino mass hierarchy relies on detection of a
nonzero $\theta_{13}$, a major goal of T2K and the longer-baseline
U.S.-based experiment NO$\nu$A currently under
construction~\cite{NOvATDR}. The ratios of the signal rates in the
three channels considered here would allow the normal, inverted, and
degenerate neutrino spectra to be differentiated, providing a key test
of the connection of the model to the neutrino sector.
Measurement of the charged Higgs branching fractions will also provide
some sensitivity to the mass of the lightest neutrino. For a lightest
neutrino mass between about 0.01 and 0.1~eV, the charged Higgs
branching ratios vary dramatically with the value of the lightest
neutrino mass (Fig.~\ref{fig:brs}); once the measurement of
$\theta_{23}$ from neutrino oscillations has improved, measurement of
the ratio of the $e \nu$ and $\mu\nu$ modes will provide sensitivity
to the lightest neutrino mass in this range. This is nicely
complementary to the prospects for direct kinematic neutrino mass
determination from the tritium beta decay experiment KATRIN, which is
designed to be sensitive down to neutrino masses of about
0.2~eV~\cite{Valerius:2005aw}---i.e., at the lower end of the
degenerate part of the spectrum---and is scheduled to begin
commissioning in 2012~\cite{KatrinTalk}. We note that, because the
neutrinos in this model are Dirac particles, neutrinoless double beta
decay experiments will have no signal and will thus not be sensitive
to the neutrino mass scale.
The mass of the charged Higgs is also accessible at the LHC through
the signal event kinematics. In particular, the signal $M_{T2}$
distribution is flat up to an endpoint at the charged Higgs mass, as
shown in Fig.~\ref{fig:hist}. A fit to this distribution on top of
the background should provide a measurement of the charged Higgs mass.
This would allow a valuable cross-check of the charged Higgs pair
production cross section together with the visible branching fractions
as predicted by the neutrino parameters. The pair production cross
section is sensitive to the isospin of the charged Higgs through its
coupling to the $Z$ boson, allowing the two-doublet nature of the
model to be established~\cite{Davidson:2009ha}.
We finally comment on the applicability of our results to two other
neutrino mass models that contain a charged Higgs boson. First, the
$Z_2$ model of Ref.~\cite{Gabriel:2006ns} contains a charged Higgs
with partial widths to leptons and LHC production cross section
identical to those in our model. The charged Higgs in the $Z_2$ model
differs from ours in that it can also decay to $W^+ \sigma$, where the
neutral scalar $\sigma$ is extremely light due to the spontaneous
breaking of the $Z_2$ symmetry. This competing mode dominates unless
$H^+$ is not much heavier than $M_W$ and the neutrino Yukawa couplings
are $\mathcal{O}(1)$~\cite{Gabriel:2008es} (this parameter region is
forbidden by standard big-bang nucleosynthesis, but the $Z_2$ model
already requires nonstandard cosmology due to the very light scalar
$\sigma$). For this parameter range, then, our results should carry
over directly. For smaller Yukawa couplings or a heavier $H^+$, the
decays to leptons used in our analysis are suppressed, resulting in a
smaller signal on top of the same background.
Second, neutrino masses of Majorana type can be generated by the
so-called Type II seesaw mechanism~\cite{type2seesaw}, in which an
SU(2)-triplet Higgs field $X \equiv (\chi^{++}, \chi^+, \chi^0)^T$
with very small vev is coupled to a pair of SM lepton doublets. LHC
phenomenology for this Higgs-triplet model was studied in
Ref.~\cite{Perez:2008ha}, which considered signatures from
$\chi^{++}\chi^{--}$ and $\chi^{++}\chi^-$ (and the conjugate process)
at the LHC. While the decay branching fractions of $\chi^+$ in this
model are identical to those of the charged Higgs in our Higgs-doublet
model, the LHC production cross section for $\chi^+ \chi^-$ in the
triplet model is about 2.7 times smaller than for $H^+H^-$ in the
doublet model~\cite{Davidson:2009ha}, due to the different isospin of
$\chi^+$ which modifies its coupling to the $Z$ boson. The signals
studied here would thus have a S/B of less than 10\% for most
channels, potentially leading to problems with background systematics.
For sufficiently high charged Higgs mass, though, the $\mu^\pm e^\mp$
channel would still have a decent S/B (33\% for $M_{H^+}=300$~GeV and
a degenerate neutrino spectrum); the reduced cross section in the
triplet model would however require an integrated luminosity close to
300 fb$^{-1}$ for discovery. In any case, searches for the
doubly-charged scalar would yield an earlier discovery of the triplet
model.
\begin{acknowledgments}
This work was supported by the Natural Sciences and Engineering
Research Council of Canada.
\end{acknowledgments}
|
1,116,691,500,427 | arxiv | \section{Introduction}
\IEEEPARstart{R}{ecent} years have witnessed the rapid evolution of communication technologies and fast proliferation of Internet of Things (IoT) devices with growing capabilities of data gathering, analyzing and knowledge utilization assisted by wireless networks~\cite{1,2,3,4}, which also enable a wide range of advanced applications, e.g., smart factory, intelligent transportation, augmented reality (AR)/virtual reality (VR) games, etc. However, constrained computing resources and capabilities of IoT devices pose great challenges in handling ever-growing computation-intensive and time-sensitive application data~\cite{5}. Besides, limited power and battery supply~\cite{6} of smart devices may further prevent smooth on-board application processing. Such challenges urgently call for cost-effective, reliable and efficient computing resource provisioning techniques over connected IoT systems to secure necessary resources for computation-intensive applications.
\vfill
\subsection{Motivations}
\begin{table*}[t!]
\caption{Comparison among different resource trading modes}
\label{table_example}
\centering
\setlength{\tabcolsep}{7mm}{
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Resource trading mode} & \textbf{Spot trading} & \textbf{Conventional booking} & \textbf{Overbooking}\\ \hline
Data basis & Current statistics & Historical statistics & Historical statistics \\ \hline
Decision-making overhead ($d$) &\multicolumn{3}{|c|}{~$d\left(\text{Overbooking}\right)$~$\leq $~$d\left(\text{Conventional booking}\right)$~$<$~$d\left(\text{Spot trading}\right)$}\\ \hline
Time efficiency ($t$) &\multicolumn{3}{|c|}{$t\left(\text{Overbooking}\right)$~$>$~$t\left(\text{Conventional booking}\right)$~$>$~$t\left(\text{Spot trading}\right)$ }\\ \hline
Resource utilization rate ($r$) &\multicolumn{3}{|c|}{$r\left(\text{Conventional booking}\right)$~$\leq$~$r\left(\text{Overbooking}\right)$~$\leq$~$r\left(\text{Spot trading}\right)$}\\ \hline
\end{tabular}}
\end{table*}
\noindent
This article studies a novel overbooking-promoted resource provisioning paradigm under hybrid device-edge-cloud network architecture~\cite{7,8,9}, in supporting effective and reliable computing services. The following key questions have been identified, which represent our major motivations.
\textit{(i). Why computing resources should be booked in advance?} Mutually beneficial resource trading mechanisms are the foundation of distributed computing resource sharing due to the selfishness of every participant. Consequently, incentives for resource sharing plays a critical role in facilitating consensual and reliable resource provisioning among multiple parties. For instance, an end-user can offload a certain amount of application data to edge server for execution by getting access to a nearby access point (AP, e.g, base stations, etc.), via paying for the obtained resources and computing services. At the same time, the edge server could be charged for purchasing resources from a remote cloud server.
In securing necessary distributive resources, conventional onsite spot trading presents a widely adopted paradigm that enables resource selling/buying among resource owners and requesters according to the current system conditions (e.g., resource supply/demand and wireless channel qualities between end-users and APs at present), which, however, can cause significant performance degradation, e.g., overhead can be incurred for discussing/negotiating the final trading agreement~\cite{3,10,4} which thus can lead to unsatisfying time/resource efficiency. Take online auction as an example, computing resources that have been put aside for unsuccessful trading during the decision-making procedure may cause resource underutilization. Moreover, spot trading participants are generally risking failures to access the required resources, e.g., only a finite number of winners can finally obtain limited resources during an onsite auction, while losers receive nothing even though they have spent both time and energy on bidding/waiting/negotiating during the auction procedure. This case can further result in unsatisfying trading experience~\cite{4,11}.
Since the big data generated by massive IoT devices often expects real-time processing, the above-mentioned shortcomings prompt the authors to investigate efficient resource trading mechanisms. To this end, \textit{resource booking} is considered in this article which facilitates a \textit{forward trading} manner (namely, presale), where a resource owner and a requester can reach an agreement for future practical trading in advance~\cite{3,10}, via signing a forward contract associated with contract terms, such as reasonable resource price, the amount of trading resources, and default clause if either party breaks the contract, etc. The benefit of the pre-signed trading contracts is that participants will no longer have to spend extra time/energy on onsite decision-making, which can thus improve time efficiency. An example of timeline comparison is depicted by Fig. 1, where in forward trading, the actual service can be directly delivered without any negotiation thanks to pre-signed contracts.
\begin{figure}[h!t]
\centerline{\includegraphics[width=1\linewidth]{fig1}}
\caption{Timeline comparison associated with spot trading and forward trading.}
\label{fig1}
\end{figure}
\textit{(ii). Why resource overbooking is critical in dynamic networks?} Conventional resource booking mechanism that allows the equal amount of reserved (booked) and available resources for requesters is generally ineffective in handling networks with dynamic resource demands. This is provoked by factors such as uncertain mobility and willingness of requester, varying wireless communication conditions, etc. For example, "\textit{no shows}" of requesters are common in real-life networks, where smart devices that lose connections with the edge server, or have run out of power, will not participate in a trading and thus prevent the originally confirmed utilization of booked resources as stipulated in pre-signed contracts. This case can further incur the underutilization of dynamic resources.
To achieve better resource usage, \textit{overbooking}~\cite{12,13} has been introduced as a common practice in many fields (e.g., airlines and hotels, spectrum reservation, etc.), encouraging the promissory reserved resources in excess of actual resource supply owing to dynamic demands. For example, airlines routinely overbook flight tickets by ensuring the maximum number of travelers since some of them may be absent from the planned trip; otherwise, each flight usually takes off with roughly 15\% seats empty, which further incurs nonnegligible economic losses\cite{11, 12}.
Similarly, encouraging proper overbooking rate in computing resource trading market can greatly support substantial resource utilization and profit advantages, via analyzing historical statistics associated with the uncertain resource demand and supply (e.g., uncertain user’s participation, varying channel quality).
\textit{(iii). Why resource overbooking in hybrid device-edge-cloud networks is challenging?}
Integrating cloud and edge into a hybrid computing system represents a viable solution~\cite{8,9} to overcome the possible resource shortage of the edge server, where the remote cloud server
plays the role of a powerful backup resource supply center, in supporting more end-users and applications while attracting better profits. Although device-edge-cloud network architecture efficiently unifies distributed heterogeneous resources for service provisioning, additional challenges would be incurred. For example, since resources should be overbooked across three parties: end-users, edge server, and cloud server, the amount of resources that the edge decides to purchase from the cloud server relies heavily on the dynamic resource demand of end-users, where an inappropriate overbooking rate can result in performance degradation for computing service delivery. Besides, the cloud server generally has to serve other requesters, where the uncertain resource supply can leave impacts on the cloud server's willingness to sell resources to the edge server.
Thus, dynamic resource supply/demand, and the individual rationality associated with different parties in hybrid device-edge-cloud networks greatly call for designing feasible resource overbooking rate. For example, the overbooking rate should be beneficial to different parties, which considers both the overbooking procedure among end-users and edge, as well as that between edge and cloud. The above discussions represent the most significant motivations. Specifically, Table 1 shows the conclusive differences among different resource trading methods on critical evaluation indicators.
\begin{figure*}[t!]
\centerline{\includegraphics[width=1\linewidth]{fig2}}
\caption{Framework of the proposed OATF under hybrid device-edge-cloud network architecture, where this article mainly relies on investigating the resource trading among Cloud server, Edge 1, and end-users associated with Edge 1.}
\label{fig2}
\end{figure*}
\subsection{Contributions}
\noindent
Most existing works mainly focus on spot trading \cite{8,15} or conventional booking\cite{3}, which are facing challenges in handling dynamic and unpredictable nature of resource trading market. This article proposes a novel overbooking-promoted trading mechanism for computing resources named ``\textit{Overbook in Advance, Trade in Future}'' (OATF), under device-edge-cloud network architecture, which contains three layers: user layer, an edge layer, and cloud layer (namely, three different parties). This article aims to investigate comprehensive insights on how the proposed OATF mechanism can facilitate efficient resource provisioning associated with end-users, an edge server, and a cloud server. Major contributions are summarized below:
\noindent
$\bullet$ OATF, a novel overbooking-enabled forward trading mechanism for computing service, is proposed under device-edge-cloud network architecture. Various uncertainties are considered to capture the random and unpredictable nature of resource trading market. Specifically, the edge server can overbook resources to multiple end-users while purchasing backup resources from the cloud server, by signing forward contracts in advance, via analyzing historical trading statistics. The overall framework and relevant key issues, e.g., contract term determination and risk evaluation, are analyzed in detail.
\noindent
$\bullet$ A case study is investigated to describe how the proposed OATF mechanism can be implemented in practice. For which, a multi-objective optimization (MOO) problem is formulated, while a two-way multilateral negotiation scheme is designed that facilitates the trading among different parties.
\noindent
$\bullet$ Comprehensive experimental results demonstrate that the proposed OATF mechanism achieves commendable benefits for participants from three different parties, while outperforming benchmark methods on critical evaluation factors, e.g., application completion time, undesired trading failure, time-efficiency, and resource utilization.
\section{Overbook in Advance, Trade in Future}
\subsection{Overview}
\noindent
The hybrid device-edge-cloud network architecture contains three key layers: user layer, edge layer, and cloud layer, as illustrated in Fig. 2.
\noindent
\textbf{User Layer} mainly includes smart devices (e.g., smartphone, smart vehicle, drone, etc.) with intelligent applications, which, however, are facing difficulties to process application data locally, owing to limited on-board computing/storage resources and capabilities. Fortunately, this framework allows end-users to purchase resources and computing services from edge server (or cloud server) by offloading a certain amount of application data via getting access to nearby APs. Notably, APs are connected to edge servers through fiber-optic links \cite{14}.
\noindent
\textbf{Edge Layer} is generally composed of several edge servers close to end-users, which can offer computing services under cost-effective and responsive manner. However, the limited computing/storage resource supply of a single edge server brings challenges to meet the ever-growing resource demands, mainly incurred by the big data generated on countless IoT devices and the wide range of innovative mobile applications. Thus, an edge server may have to purchase more resources from the remote cloud server especially during peak hours. Specifically, edge servers are connected to the cloud server via fiber-optic links\cite{14}.
\begin{figure*}[t!]
\centering
\centerline{\includegraphics[width=0.99\linewidth]{fig3}}
\caption{Timelines and trading examples upon comparing different resource trading modes.}
\end{figure*}
\noindent
\textbf{Cloud layer} considers a remote cloud server distant from end-users as a relatively powerful data/computing center, which provides highly precise computing service for mobile applications. However, due to the potential excessive transmission delay and burdens on wireless links as well as cloud server, direct communications among end-users and cloud server are generally not recommended. Instead, the cloud server is seen as an effective backup resource supply center that lends resources to edge servers, and thus helping with attracting more end-users and revenue.
Resource trading considered in this article mainly investigates the interactions among the cloud server, one edge server (e.g., Edge 1 in Fig. 2), and the relevant end-users (e.g., end-users of Edge 1 in Fig. 2). Note that a resource market under mobile wireless communication networks is always dynamic, inherent uncertainties should thus be carefully concerned from two key perspectives: resource demand/supply, and network condition. The uncertainty associated with resource demand mainly refers to the fluctuating number of applications, as well as "show/no show" cases of end-users. Considering a resource trading at time $t$, end-users may carry different number of heterogeneous applications, which directly impacts the amount of required resources. Besides, end-users may not always participate in a trading, e.g., a smart device outside of computing server's communication coverage or has run out of power will be absent from time $t$ and thus not using the booked resources (namely, ``no show''). Uncertain resource supply generally depends on the cloud server, since it may have to offer services to many other customers. For instance, in Fig. 2, the amount of cloud resources provisioned to Edge 2 and Edge 3 can directly affect the available resources for Edge 1. Then, the uncertain network condition is mainly reflected by varying wireless channel qualities among edge server (namely, APs) and end-users, incurred by factors such as users' mobility and transmission power, etc. Apparently, a poor channel quality poses significant impact on application execution performance, e.g., a large data transmission delay.
By analyzing historical statistics associated with the above-mentioned uncertainties, the proposed OATF mechanism encourages two forward trading contract types (see Fig. 2): \textit{Type 1} indicates the forward contract among end-users and the edge server; and \textit{Type 2} represents the forward contract between edge server and the cloud server. Particularly, every practical resource trading is performed among participants depending on pre-determined forward contracts without further onsite negotiation. Specifically, aiming to achieve substantial utilization and profit advantages under dynamic resource supply/demand, a certain overbooking rate is encouraged, which allows the amount of booked resources stipulated in forward contracts to exceed the available resource supply. For example, the total promissory reserved resources for end-users $r^{User}$ can be larger than the available resource supply $r^{Edge}+r^{Backup}$, where $r^{Edge}$ and $r^{Backup}$ denote the edge server's local resources, and the available backup resources borrowed from cloud server, respectively.
\subsection{Significant Issues}
Timeline associated with the proposed OATF can generally be divided into two phases: before practical trading, and during practical trading, where significant issues of the former phase are analyzed below.
\noindent
$\bullet$\textbf{Contract term design (rights and obligations)}: By signing a forward contract with the edge server, each end-user can enjoy the following rights during practical trading: \textit{(i)} a certain amount of reserved resources; \textit{(ii)} reasonable price for trading resources; and \textit{(iii)} a compensation from the edge server if the end-user fails to acquire the promissory resources owing to insufficient resource supply. Besides, each contractual end-user also has to follow the obligation by paying a certain penalty to edge server when it is absent from a trading (namely, breaks the contract).
Apparently, the above-mentioned \textit{(i)}-\textit{(iii)} are obligations of the edge server, while the penalty paid from end-users stands for its right in contract type 1. It is noteworthy that since each contract involves one specific user, the user selection problem can be figured out accordingly. Besides, edge server can also purchase computing services from the cloud server to meet the growing resource demands of end-users, by enjoying a certain amount of reserved cloud resources, proper trading price, and a compensation when the cloud server breaks the contract since it also has to serve other customers (namely, edge's rights, cloud's obligations associated with contract type 2). Similarly, edge server has to pay a penalty for not buying the confirmed cloud resources (namely, edge's obligation, cloud's right associated with contract type 2). For example, the edge's local resources may be sufficient to cover resource demands when few end-users participate in a trading. More importantly, our proposed OATF greatly supports fairness, since the pre-determined prices will not be impacted by the uncertainties in resource trading market. Apparently, inappropriate rights and obligations associated with different forward contracts can definitely bring performance degradations, e.g., large resource price/penalty may lead to negative utilities to end-users. Thus, designing feasible contract terms represents a significant problem.
\begin{figure*}[t!]
\centering
\centerline{\includegraphics[width=1\linewidth]{fig4}}
\caption{Procedure of the proposed negotiation scheme for contract design.}
\end{figure*}
\noindent
$\bullet$ \textbf{Overbooking rate design}: Overbooking rate refers to the proportion of promissory reserved resources that exceeds the available resource supply, e.g., $\frac{r^{User}-\left(r^{Edge}+r^{Backup}\right)}{r^{Edge}+r^{Backup}}$. Infeasible overbooking rate may incur two major problems: \textit{(i)} a large value of overbooking rate, namely, overmuch promissory reserved resources, can prevent some users from enjoying computing service owing to limited resource supply, and thus results in poor trading experience; and \textit{(ii)} a small value of overbooking rate, namely, deficient resources available for booking, can lead to underutilization of dynamic resources and further bring economic losses to computing servers. Besides, additional challenges can be incurred since overbooking is considered across three layers, where the dynamic resource demands from end-users can definitely impact the amount of trading resources of both two contracts. Consequently, overbooking rate should be well designed by comprehensively analyzing historical statistics of various uncertainties.
\noindent
$\bullet$ \textbf{Risk management}: Uncertainties can generally bring risks, mainly in forms of \textit{(i)} participants' utilities, and \textit{(ii)} resource usage. The former indicates that participants are at risk of obtaining undesired utilities during each trading. For example, a contractual end-user who is suffering from a poor wireless channel quality and a high pre-determined resource price may receive negative utility during a practical trading, due to excessive data transmission delay and large payment. The edge server may get unsatisfying utility for paying high penalty to cloud server when lots of end-users are absent from a trading. Besides, since the cloud server generally offers services to multiple customers (e.g., Edges 2-3 in Fig. 2), a large amount of reserved resources for the concerned edge server (Edge 1) can directly reduce the resource supply and thus the relevant revenue. The later risk is mainly caused by overbooking, where a contractual end-user may still fail to access the required resources. Although he gets compensation, this case can definitely lead to poor trading experience. Additionally, the edge server is risking inadequate resource usage rate due to possible "no shows". Therefore, risks should be properly managed and controlled.
During each practical trading (the later phase), the following key issues should be carefully considered.
\noindent
$\bullet$ \textbf{Compensated user selection}: Limited resource supply can incur the case where end-users who have signed contracts but finally fail to acquire resources, and should process applications locally. Since different end-users may have heterogeneous applications (e.g., data size, etc.) and requirements (e.g., tolerant completion time, etc.), a proper selection strategy should be concerned for choosing appropriate compensated end-users (if any), during each practical trading. Common methods can refer to first-come-first-serve (FCFS), random selection (e.g., users are randomly be compensated), greedy-based selection (e.g., users with the worst channel qualities will be compensated), etc.
\noindent
$\bullet$ \textbf{Application transfer decision}:
Note that different computing servers can provide heterogeneous resources and services. Thus, which application(s) of which end-user(s) could be transferred to the cloud server, or stay on the edge server, presents another noteworthy problem, since cloud can offer rather powerful computing capability and may directly impact the performance of applications. Factors such as the tolerant delay of mobile application, preference of end-user, distance and channel qualities between users and APs have to be taken into consideration.
Fig. 3 shows the timeline and trading examples related to the proposed OATF mechanism, in comparison with conventional booking method (e.g., equal-booking-related trading~\cite{4}) and spot trading. Apparently, forward contracts are pre-signed among participants in Fig. 3(a) and Fig. 3(b), where contractual users will no longer spend extra time and energy on decision-making during each practical trading. Specifically, OATF in Fig. 3(a) encourages a certain overbooking rate calculated by $(10-8)/8=25\%$ in case of possible "no shows". Fig. 3(b) depicts the conventional booking method where the overall resources booked to end-users can not exceed resource supply. For example, in Trading 2, Fig. 3(b), where user 2 is absent, the resource utilization is calculated by $80\%$; while that of our proposed OATF achieves $100\%$ (see Fig. 3(a)) which thus can better deal with dynamic resource demands. As a comparison, Fig. 3(c) illustrates the onsite spot trading mode, where the actual data offloading and service delivery procedure can only start after an onsite trading agreement has been reached. This case can definitely lead to extra latency and energy costs incurred by onsite decision-making. Besides, undesired trading failures might be incurred. For example, in Trading 3 (Fig. 3(c)), although users 1 and 4 have spent a certain amount of time negotiating the trading agreement with the edge server, they finally fail to obtain the required service due to resource shortage.
\section{Case Study}
This section investigates a case study associated with OATF upon considering three key parties: multiple end-users, an edge server with $r^{Edge}$ resources, and a cloud server with $r^{Cloud}$ resources. Namely, edge server and cloud server can theoretically process a maximum of $r^{Edge}+r^{Cloud}$ applications in parallel during a trading, for analytical simplicity.
\subsection{Basic Modeling}
Considering multiple independent identically distributed (i.i.d) end-users $\mathbb{U}=\left\{u_x\left|x\in\right.\left\{1,2,...,|\mathbb{U}|\right\}\right\}$ which are supposed to have same computing capability (e.g., similar smartphone types with the same processors), number of applications $n$, etc., for analytical simplicity. Consequently, terms of contract $\mathbb{M}^{Edge}$ offered by the edge server can thus be the same among different users, which is also general in real-life networks~\cite{4}. Specifically, each user $u_x$ may encounter two uncertain factors: $\alpha_x$ and $\gamma_x$, where random variable $\alpha_x$ defines the attendance
and absence
of $u_x$ during a trading, that obeys a Bernoulli distribution represented by $\alpha_x\sim\text{B}\left\{\left(1,0\right),\left(a,1-a\right)\right\}$. Besides, random variable $\gamma_x$ describes the changing channel quality of the link between $u_x$ and the nearby AP, which follows a uniform distribution denoted by $\gamma_x\sim \text{U}\left(y_1,y_2\right)$, where a small value of $\gamma_x$ can lead to excessive transmission latency of application data. Specifically, contract $\mathbb{M}^{Edge}$ offered by edge server is denoted as a tetrad,
where the utility $\mathcal{U}^{u_x}$ of an end-user $u_x$ who has signed contract $\mathbb{M}^{Edge}$ considers the following factors: \textit{(i)} the saved time and energy as benefited from the amount of reserved resources $r^{user}$; \textit{(ii)} payment to edge server $p^{UtoE}$ for required resources and service; \textit{(iii)} penalty to edge server $q^{UtoE}$ for possible absence; and \textit{(iv)} possible compensation $c^{EtoU}$ obtained from edge server.
Let $\mathbb{M}^{Cloud}$ indicate the forward contract between cloud server and edge server. Relying on both $\mathbb{M}^{Edge}$ and $\mathbb{M}^{Cloud}$, the utility of edge server $\mathcal{U}^{Edge}$ concerns four parts: \textit{(i)}. revenue obtained from end-users; \textit{(ii)}. compensation that edge should pay to the end-users who fail to acquire resources; \textit{(iii)}. payment $p^{EtoC}$ for the predetermined amount of cloud resources $r^{Backup}$; and \textit{(iv)}. penalty $q^{EtoC}$ if not purchasing cloud resources.
This case study considers an interesting assumption that the cloud server will offer services to the studied edge server as the highest priority (namely, the cloud server will not break $\mathbb{M}^{Cloud}$). Accordingly, the uncertain resource demand from other customers obeys a discrete uniform distribution denoted by $\beta\sim\text{U}\left(0,1,...,r^{Cloud}\right)$. Since the pre-determined contract $\mathbb{M}^{Cloud}$ has set aside $r^{Backup}$ resources for the studied edge server, some of these requesters may have to wait for resource release during a trading when cloud server is fully occupied.
Consequently, utility $\mathcal{U}^{Cloud}$ of the cloud server is defined via considering four key parts: \textit{(i)} revenue obtained from other resource requesters; \textit{(ii)} partial refund for requesters who have to wait for available resources; \textit{(iii)} income $p^{EtoC}$ obtained from the edge server; and \textit{(iv)} possible penalty $q^{EtoC}$ paid from the edge server.
\begin{figure*}[h!t]
\centering
\subfigure[]{\includegraphics[width=.33\linewidth]{fig5a}}\hfill
\subfigure[]{\includegraphics[width=.33\linewidth]{fig5b}}\hfill
\subfigure[]{\includegraphics[width=.327\linewidth]{fig5c}}
\caption{Performance evaluation from long-term perspective via simulating 5000 trading.}
\end{figure*}
\subsection{Analysis of Key Issues}
Analysis of key issues mentioned in the previous section based on the above models are discussed hereafter. First, before practical trading, the design of contracts, as well as overbooking rate is formulated as a multi-objective optimization (MOO) problem ($r^{User}=|\mathbb{U}|\times r^{user}$), aiming to maximize the expected utilities of end-users $\text{E}\left[\sum_{x=1}^{x=|\mathbb{U}|}\mathcal{U}^{u_x}\right]$, edge server $\text{E}\left[\mathcal{U}^{Edge}\right]$, and cloud server $\text{E}\left[\mathcal{U}^{Cloud}\right]$, while meeting tolerable risks as major constraints.
Specifically, each end-user $u_x$ considers two key risks: \textit{(i)} the risk of receiving a negative utility, which is defined as the probability that $\mathcal{U}^{u_x}$ is less than or equal to 0; and \textit{(ii)} the risk of failing to acquire resources due to overbooking, which is calculated by the probability that conditions $\alpha_{x^\prime}=1$ and $r^{user}\sum_{x\neq x^\prime}\alpha_x>r^{Edge}+r^{Backup}-r^{User}$ are both met. Namely, the overall resources offered by the edge and cloud server fail to afford the demand of end-users, where some users thus have to process their applications locally. Similarly, the edge server is facing two major risks: \textit{(i)} the risk of obtaining an unsatisfying utility as defined by the probability that $\mathcal{U}^{Edge}$ is less than its expectation $\text{E}\left[\mathcal{U}^{Edge}\right]$; and \textit{(ii)} the risk of resource underutilization, which is reflected by the probability that resource usage stays below a certain rate (mainly caused by an improper overbooking rate). In addition, cloud server is undergoing the risk represented by the probability that the value of $\mathcal{U}^{Cloud}$ is smaller than or equal to $\text{E}\left[\mathcal{U}^{Cloud}\right]$. Apparently, all the above-mentioned risks should be well controlled within a certain range (e.g., each probability should not exceed a threshold. e.g., 30\%).
The proposed MOO problem faces difficulties to be solved directly by traditional algorithms such as the weighted sum method and $\epsilon$-constrained method, since it involves non-convex objective functions and complicated probabilistic constraints.
To this end, a two-way multilateral negotiation scheme is designed that alternatively optimizes expected utilities,
while meeting acceptable risks. Specifically, "two-way" indicates that the edge server has to communicate with both end-users and the cloud server. Fig. 3 shows a diagram of how the proposed negotiation scheme is implemented among different parties, to reach the final agreement on forward contracts. As illustrated by Fig. 4, the edge server first starts a quotation process (step 1), while end-users can determine the acceptable range of $r^{user}$ (step 2) under tolerable risks under given the current price and default clause associated with contract $\mathbb{M}^{Edge}$. Then, the cloud server starts its quotation process, where the edge server decides the acceptable range of $r^{Backup}$, according to the price and default clause (step 4) related to both contracts
, under tolerable risks. Based on which, the cloud server determines feasible values of $r^{Backup}$ (step 5). When the three parties reach a consensus (step 6), end-users select the optimal pair of $r^{user}$ and $r^{Backup}$ that can maximize their expected utilities, which are regarded as candidate contract term, together with the corresponding prices and penalties (step 7). After the quotations of both edge and cloud servers are completed, the edge server chooses the final contract terms for both forward contracts $\mathbb{M}^{Edge}$ and $\mathbb{M}^{Cloud}$, to maximize its expected utility. Apparently, the edge server should understand both end-users and could server, which calls for two-way communication.
In particular, the corresponding computational complexity relies the total number of quotations of edge and cloud servers, which can be denoted by $O(Number~of~quotations)$. Notably, the overhead of OATF will only be incurred during contract negotiation (e.g. 10:00am in Fig. 1), while responsive and cost-effective computing services can be delivered directly during each practical trading, thanks to pre-signed contracts.
Then, each practical trading will proceed among participants on the basis of pre-determined forward contracts. Since end-users are i.i.d, FCFS mechanism is utilized for both compensated user selection and application transfer decision in this case study, which is also fair in real-world trading market.
\subsection{Results and Evaluation}
Simulation results associated with the proposed case study are carefully analyzed mainly from the long-term perspective, comparing with representative baseline methods listed below.
\noindent
$\bullet$ \textbf{Conventional resource booking method (CBooking)} that doesn't allow the amount of resources reserved for end-users $r^{User}$ to exceed the available resource supply $r^{Edge}+r^{Backup}$.
\noindent
$\bullet$ \textbf{Spot trading method} where each practical trading is performed relying on the current market/network conditions (e.g., the current value of $\alpha_x$, $\gamma_x$ and $\beta$). Specifically, both uniform and differential pricing rules~\cite{4}~are considered, as abbreviated to ``SpotT\_UP'' and ``SpotT\_DP'', respectively.
Major parameters are set as follows: $a=76\%$, $y_1=100$, $y_2=400$, $r^{Edge}=197$, $r^{Cloud}=600$, $\mathbb{U}=137$, $e^t=550\text{mW}$ ($e^t$ indicates the transmission power, while $e^t\gamma_x$ represents the received SNR of AP which is roughly within [17dB, 23dB]), $n=5$. Data size of each application is set to be 1Mb, while the corresponding required computing resources can be calculated as $600\times1024\text{bit}\times{1024}^2 \text{CPU cycles}$. Bandwidth of the channel between each end-user and the nearby AP (connected to edge server) is set by 6MHz. Besides, risks are controlled within interval $[20\%,40\%]$.
Since onsite decision-making relies heavily on the end-to-end (E2E) delay of wireless communication channels, a random variable $\tau_x$ (ms) is applied for each end-user in this simulation to describe E2E delay of the wireless channel between end-user $u_x$ and the nearby AP, which follows a uniform distribution denoted by $\tau_x\sim \text{U}\left(2,15\right)$~\cite{3}.
Evaluations upon considering the average value (per trading) of different indicators are shown in Fig. 5, via simulating 5000 trading. In Fig. 5(a), the proposed OATF mechanism outperforms Spot\_UP and Spot\_DP on the average value of participants' utilities, since onsite users have to spend extra time/energy on negotiating a trading agreement with the edge server which directly reduce the usable time for actual computing service delivery and thus time efficiency (also see Fig. 5(b)). Besides, onsite edge server may suffer from insufficient resource supply (e.g., cloud resources are occupied by other customers) and off-peak period (e.g., large number of users are absent from a trading).Besides, although overbooking can better handle dynamic resource demands, it also faces risks. This has led to similar performance of OATF in comparison with CBooking in Fig. 5(a), mainly caused by risks due to the random and unpredictable nature.
Namely, the studied OATF relies on a risk and opportunity coexisting resource market, which is closer to real-world networks. For example, in Fig. 5(a), the cloud server’s utility of CBooking slightly outperforms that of OATF mainly because that overbooking may cause larger refunds paid to other customers of the cloud server (e.g., parameter $\beta$).
Fortunately, the proposed OATF mechanism can get far better average resource usage rate than CBooking as illustrated by Fig. 5(b), since a feasible overbooking rate is encouraged that greatly supports dynamic resource demands.
Fig. 5(c) depicts the average trading failures and the relevant rate, where all the trading in CBooking are successful owing to that the amount of promissory reserved resources for end-users in forward contracts equals to resource supply. In addition, the proposed OATF mechanism enables roughly 99.1\% and 99.3\% performance improvement on the number of failed users, and failure rate, rather than Spot\_UP and Spot\_DP, respectively, due to that onsite end-users may undergo insufficient resources and more severe competition, according to the current market/network conditions. For example, an end-user $u_x$ with large $\gamma_x$ can afford a higher resource price than $u_{x^\prime}$ with a small $\gamma_{x^\prime}$, which may lead to a failure although $u_{x^\prime}$ has spent time on negotiating with the edge server.
\section{Conclusion and Future Direction}
\noindent
Big data generated by countless IoT devices highly requests real-time and cost-effective application processing, which calls for additional requirements on resource provisioning techniques in dynamic networks. This article investigates a novel overbooking-enabled forward trading mechanism called OATF, under device-edge-cloud network architecture with various uncertainties. OATF relies on pre-determined forward contracts negotiated among end-users and the edge server, as well as between the edge server and cloud server in advance, which will be fulfilled accordingly during each future practical trading. Specifically, a certain overbooking rate is encouraged that greatly supports the substantial resource utilization under dynamic resource demand. Framework and key issues associated with the proposed OATF mechanism are carefully analyzed, based on which, an interesting case study is investigated via specific mathematical models. Comprehensive simulation results illustrate that the proposed OATF mechanism can achieve commendable performance on various evaluation indicators such as time efficiency and resource usage, in comparison with conventional trading methods.
Several interesting research directions are worthy of consideration. For example, smart and alterable rights/obligations rather than fixed contract terms are noteworthy to study via adopting machine learning approaches, to better capture the dynamics and unpredictable nature of resource trading market. Moreover, multiple edge servers can be considered, where competition and cooperation among them stand for another future direction.
\section*{Acknowledgement}
\noindent
This work was supported in part by the Fundamental Research Funds for the Central Universities under No. 20720220104, the Discovery Program of Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant RGPIN2018-06254, and the Canada Research Chair Program.
\section{Introduction}
\IEEEPARstart{R}{ecent} years have witnessed the rapid evolution of communication technologies and fast proliferation of Internet of Things (IoT) devices with growing capabilities of data gathering, analyzing and knowledge utilization assisted by wireless networks~\cite{1,2,3,4}, which also enable a wide range of advanced applications, e.g., smart factory, intelligent transportation, augmented reality (AR)/virtual reality (VR) games, etc. However, constrained computing resources and capabilities of IoT devices pose great challenges in handling ever-growing computation-intensive and time-sensitive application data~\cite{5}. Besides, limited power and battery supply~\cite{6} of smart devices may further prevent smooth on-board application processing. Such challenges urgently call for cost-effective, reliable and efficient computing resource provisioning techniques over connected IoT systems to secure necessary resources for computation-intensive applications.
\vfill
\subsection{Motivations}
\begin{table*}[t!]
\caption{Comparison among different resource trading modes}
\label{table_example}
\centering
\setlength{\tabcolsep}{7mm}{
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Resource trading mode} & \textbf{Spot trading} & \textbf{Conventional booking} & \textbf{Overbooking}\\ \hline
Data basis & Current statistics & Historical statistics & Historical statistics \\ \hline
Decision-making overhead ($d$) &\multicolumn{3}{|c|}{~$d\left(\text{Overbooking}\right)$~$\leq $~$d\left(\text{Conventional booking}\right)$~$<$~$d\left(\text{Spot trading}\right)$}\\ \hline
Time efficiency ($t$) &\multicolumn{3}{|c|}{$t\left(\text{Overbooking}\right)$~$>$~$t\left(\text{Conventional booking}\right)$~$>$~$t\left(\text{Spot trading}\right)$ }\\ \hline
Resource utilization rate ($r$) &\multicolumn{3}{|c|}{$r\left(\text{Conventional booking}\right)$~$\leq$~$r\left(\text{Overbooking}\right)$~$\leq$~$r\left(\text{Spot trading}\right)$}\\ \hline
\end{tabular}}
\end{table*}
\noindent
This article studies a novel overbooking-promoted resource provisioning paradigm under hybrid device-edge-cloud network architecture~\cite{7,8,9}, in supporting effective and reliable computing services. The following key questions have been identified, which represent our major motivations.
\textit{(i). Why computing resources should be booked in advance?} Mutually beneficial resource trading mechanisms are the foundation of distributed computing resource sharing due to the selfishness of every participant. Consequently, incentives for resource sharing plays a critical role in facilitating consensual and reliable resource provisioning among multiple parties. For instance, an end-user can offload a certain amount of application data to edge server for execution by getting access to a nearby access point (AP, e.g, base stations, etc.), via paying for the obtained resources and computing services. At the same time, the edge server could be charged for purchasing resources from a remote cloud server.
In securing necessary distributive resources, conventional onsite spot trading presents a widely adopted paradigm that enables resource selling/buying among resource owners and requesters according to the current system conditions (e.g., resource supply/demand and wireless channel qualities between end-users and APs at present), which, however, can cause significant performance degradation, e.g., overhead can be incurred for discussing/negotiating the final trading agreement~\cite{3,10,4} which thus can lead to unsatisfying time/resource efficiency. Take online auction as an example, computing resources that have been put aside for unsuccessful trading during the decision-making procedure may cause resource underutilization. Moreover, spot trading participants are generally risking failures to access the required resources, e.g., only a finite number of winners can finally obtain limited resources during an onsite auction, while losers receive nothing even though they have spent both time and energy on bidding/waiting/negotiating during the auction procedure. This case can further result in unsatisfying trading experience~\cite{4,11}.
Since the big data generated by massive IoT devices often expects real-time processing, the above-mentioned shortcomings prompt the authors to investigate efficient resource trading mechanisms. To this end, \textit{resource booking} is considered in this article which facilitates a \textit{forward trading} manner (namely, presale), where a resource owner and a requester can reach an agreement for future practical trading in advance~\cite{3,10}, via signing a forward contract associated with contract terms, such as reasonable resource price, the amount of trading resources, and default clause if either party breaks the contract, etc. The benefit of the pre-signed trading contracts is that participants will no longer have to spend extra time/energy on onsite decision-making, which can thus improve time efficiency. An example of timeline comparison is depicted by Fig. 1, where in forward trading, the actual service can be directly delivered without any negotiation thanks to pre-signed contracts.
\begin{figure}[h!t]
\centerline{\includegraphics[width=1\linewidth]{fig1}}
\caption{Timeline comparison associated with spot trading and forward trading.}
\label{fig1}
\end{figure}
\textit{(ii). Why resource overbooking is critical in dynamic networks?} Conventional resource booking mechanism that allows the equal amount of reserved (booked) and available resources for requesters is generally ineffective in handling networks with dynamic resource demands. This is provoked by factors such as uncertain mobility and willingness of requester, varying wireless communication conditions, etc. For example, "\textit{no shows}" of requesters are common in real-life networks, where smart devices that lose connections with the edge server, or have run out of power, will not participate in a trading and thus prevent the originally confirmed utilization of booked resources as stipulated in pre-signed contracts. This case can further incur the underutilization of dynamic resources.
To achieve better resource usage, \textit{overbooking}~\cite{12,13} has been introduced as a common practice in many fields (e.g., airlines and hotels, spectrum reservation, etc.), encouraging the promissory reserved resources in excess of actual resource supply owing to dynamic demands. For example, airlines routinely overbook flight tickets by ensuring the maximum number of travelers since some of them may be absent from the planned trip; otherwise, each flight usually takes off with roughly 15\% seats empty, which further incurs nonnegligible economic losses\cite{11, 12}.
Similarly, encouraging proper overbooking rate in computing resource trading market can greatly support substantial resource utilization and profit advantages, via analyzing historical statistics associated with the uncertain resource demand and supply (e.g., uncertain user’s participation, varying channel quality).
\textit{(iii). Why resource overbooking in hybrid device-edge-cloud networks is challenging?}
Integrating cloud and edge into a hybrid computing system represents a viable solution~\cite{8,9} to overcome the possible resource shortage of the edge server, where the remote cloud server
plays the role of a powerful backup resource supply center, in supporting more end-users and applications while attracting better profits. Although device-edge-cloud network architecture efficiently unifies distributed heterogeneous resources for service provisioning, additional challenges would be incurred. For example, since resources should be overbooked across three parties: end-users, edge server, and cloud server, the amount of resources that the edge decides to purchase from the cloud server relies heavily on the dynamic resource demand of end-users, where an inappropriate overbooking rate can result in performance degradation for computing service delivery. Besides, the cloud server generally has to serve other requesters, where the uncertain resource supply can leave impacts on the cloud server's willingness to sell resources to the edge server.
Thus, dynamic resource supply/demand, and the individual rationality associated with different parties in hybrid device-edge-cloud networks greatly call for designing feasible resource overbooking rate. For example, the overbooking rate should be beneficial to different parties, which considers both the overbooking procedure among end-users and edge, as well as that between edge and cloud. The above discussions represent the most significant motivations. Specifically, Table 1 shows the conclusive differences among different resource trading methods on critical evaluation indicators.
\begin{figure*}[t!]
\centerline{\includegraphics[width=1\linewidth]{fig2}}
\caption{Framework of the proposed OATF under hybrid device-edge-cloud network architecture, where this article mainly relies on investigating the resource trading among Cloud server, Edge 1, and end-users associated with Edge 1.}
\label{fig2}
\end{figure*}
\subsection{Contributions}
\noindent
Most existing works mainly focus on spot trading \cite{8,15} or conventional booking\cite{3}, which are facing challenges in handling dynamic and unpredictable nature of resource trading market. This article proposes a novel overbooking-promoted trading mechanism for computing resources named ``\textit{Overbook in Advance, Trade in Future}'' (OATF), under device-edge-cloud network architecture, which contains three layers: user layer, an edge layer, and cloud layer (namely, three different parties). This article aims to investigate comprehensive insights on how the proposed OATF mechanism can facilitate efficient resource provisioning associated with end-users, an edge server, and a cloud server. Major contributions are summarized below:
\noindent
$\bullet$ OATF, a novel overbooking-enabled forward trading mechanism for computing service, is proposed under device-edge-cloud network architecture. Various uncertainties are considered to capture the random and unpredictable nature of resource trading market. Specifically, the edge server can overbook resources to multiple end-users while purchasing backup resources from the cloud server, by signing forward contracts in advance, via analyzing historical trading statistics. The overall framework and relevant key issues, e.g., contract term determination and risk evaluation, are analyzed in detail.
\noindent
$\bullet$ A case study is investigated to describe how the proposed OATF mechanism can be implemented in practice. For which, a multi-objective optimization (MOO) problem is formulated, while a two-way multilateral negotiation scheme is designed that facilitates the trading among different parties.
\noindent
$\bullet$ Comprehensive experimental results demonstrate that the proposed OATF mechanism achieves commendable benefits for participants from three different parties, while outperforming benchmark methods on critical evaluation factors, e.g., application completion time, undesired trading failure, time-efficiency, and resource utilization.
\section{Overbook in Advance, Trade in Future}
\subsection{Overview}
\noindent
The hybrid device-edge-cloud network architecture contains three key layers: user layer, edge layer, and cloud layer, as illustrated in Fig. 2.
\noindent
\textbf{User Layer} mainly includes smart devices (e.g., smartphone, smart vehicle, drone, etc.) with intelligent applications, which, however, are facing difficulties to process application data locally, owing to limited on-board computing/storage resources and capabilities. Fortunately, this framework allows end-users to purchase resources and computing services from edge server (or cloud server) by offloading a certain amount of application data via getting access to nearby APs. Notably, APs are connected to edge servers through fiber-optic links \cite{14}.
\noindent
\textbf{Edge Layer} is generally composed of several edge servers close to end-users, which can offer computing services under cost-effective and responsive manner. However, the limited computing/storage resource supply of a single edge server brings challenges to meet the ever-growing resource demands, mainly incurred by the big data generated on countless IoT devices and the wide range of innovative mobile applications. Thus, an edge server may have to purchase more resources from the remote cloud server especially during peak hours. Specifically, edge servers are connected to the cloud server via fiber-optic links\cite{14}.
\begin{figure*}[t!]
\centering
\centerline{\includegraphics[width=0.99\linewidth]{fig3}}
\caption{Timelines and trading examples upon comparing different resource trading modes.}
\end{figure*}
\noindent
\textbf{Cloud layer} considers a remote cloud server distant from end-users as a relatively powerful data/computing center, which provides highly precise computing service for mobile applications. However, due to the potential excessive transmission delay and burdens on wireless links as well as cloud server, direct communications among end-users and cloud server are generally not recommended. Instead, the cloud server is seen as an effective backup resource supply center that lends resources to edge servers, and thus helping with attracting more end-users and revenue.
Resource trading considered in this article mainly investigates the interactions among the cloud server, one edge server (e.g., Edge 1 in Fig. 2), and the relevant end-users (e.g., end-users of Edge 1 in Fig. 2). Note that a resource market under mobile wireless communication networks is always dynamic, inherent uncertainties should thus be carefully concerned from two key perspectives: resource demand/supply, and network condition. The uncertainty associated with resource demand mainly refers to the fluctuating number of applications, as well as "show/no show" cases of end-users. Considering a resource trading at time $t$, end-users may carry different number of heterogeneous applications, which directly impacts the amount of required resources. Besides, end-users may not always participate in a trading, e.g., a smart device outside of computing server's communication coverage or has run out of power will be absent from time $t$ and thus not using the booked resources (namely, ``no show''). Uncertain resource supply generally depends on the cloud server, since it may have to offer services to many other customers. For instance, in Fig. 2, the amount of cloud resources provisioned to Edge 2 and Edge 3 can directly affect the available resources for Edge 1. Then, the uncertain network condition is mainly reflected by varying wireless channel qualities among edge server (namely, APs) and end-users, incurred by factors such as users' mobility and transmission power, etc. Apparently, a poor channel quality poses significant impact on application execution performance, e.g., a large data transmission delay.
By analyzing historical statistics associated with the above-mentioned uncertainties, the proposed OATF mechanism encourages two forward trading contract types (see Fig. 2): \textit{Type 1} indicates the forward contract among end-users and the edge server; and \textit{Type 2} represents the forward contract between edge server and the cloud server. Particularly, every practical resource trading is performed among participants depending on pre-determined forward contracts without further onsite negotiation. Specifically, aiming to achieve substantial utilization and profit advantages under dynamic resource supply/demand, a certain overbooking rate is encouraged, which allows the amount of booked resources stipulated in forward contracts to exceed the available resource supply. For example, the total promissory reserved resources for end-users $r^{User}$ can be larger than the available resource supply $r^{Edge}+r^{Backup}$, where $r^{Edge}$ and $r^{Backup}$ denote the edge server's local resources, and the available backup resources borrowed from cloud server, respectively.
\subsection{Significant Issues}
Timeline associated with the proposed OATF can generally be divided into two phases: before practical trading, and during practical trading, where significant issues of the former phase are analyzed below.
\noindent
$\bullet$\textbf{Contract term design (rights and obligations)}: By signing a forward contract with the edge server, each end-user can enjoy the following rights during practical trading: \textit{(i)} a certain amount of reserved resources; \textit{(ii)} reasonable price for trading resources; and \textit{(iii)} a compensation from the edge server if the end-user fails to acquire the promissory resources owing to insufficient resource supply. Besides, each contractual end-user also has to follow the obligation by paying a certain penalty to edge server when it is absent from a trading (namely, breaks the contract).
Apparently, the above-mentioned \textit{(i)}-\textit{(iii)} are obligations of the edge server, while the penalty paid from end-users stands for its right in contract type 1. It is noteworthy that since each contract involves one specific user, the user selection problem can be figured out accordingly. Besides, edge server can also purchase computing services from the cloud server to meet the growing resource demands of end-users, by enjoying a certain amount of reserved cloud resources, proper trading price, and a compensation when the cloud server breaks the contract since it also has to serve other customers (namely, edge's rights, cloud's obligations associated with contract type 2). Similarly, edge server has to pay a penalty for not buying the confirmed cloud resources (namely, edge's obligation, cloud's right associated with contract type 2). For example, the edge's local resources may be sufficient to cover resource demands when few end-users participate in a trading. More importantly, our proposed OATF greatly supports fairness, since the pre-determined prices will not be impacted by the uncertainties in resource trading market. Apparently, inappropriate rights and obligations associated with different forward contracts can definitely bring performance degradations, e.g., large resource price/penalty may lead to negative utilities to end-users. Thus, designing feasible contract terms represents a significant problem.
\begin{figure*}[t!]
\centering
\centerline{\includegraphics[width=1\linewidth]{fig4}}
\caption{Procedure of the proposed negotiation scheme for contract design.}
\end{figure*}
\noindent
$\bullet$ \textbf{Overbooking rate design}: Overbooking rate refers to the proportion of promissory reserved resources that exceeds the available resource supply, e.g., $\frac{r^{User}-\left(r^{Edge}+r^{Backup}\right)}{r^{Edge}+r^{Backup}}$. Infeasible overbooking rate may incur two major problems: \textit{(i)} a large value of overbooking rate, namely, overmuch promissory reserved resources, can prevent some users from enjoying computing service owing to limited resource supply, and thus results in poor trading experience; and \textit{(ii)} a small value of overbooking rate, namely, deficient resources available for booking, can lead to underutilization of dynamic resources and further bring economic losses to computing servers. Besides, additional challenges can be incurred since overbooking is considered across three layers, where the dynamic resource demands from end-users can definitely impact the amount of trading resources of both two contracts. Consequently, overbooking rate should be well designed by comprehensively analyzing historical statistics of various uncertainties.
\noindent
$\bullet$ \textbf{Risk management}: Uncertainties can generally bring risks, mainly in forms of \textit{(i)} participants' utilities, and \textit{(ii)} resource usage. The former indicates that participants are at risk of obtaining undesired utilities during each trading. For example, a contractual end-user who is suffering from a poor wireless channel quality and a high pre-determined resource price may receive negative utility during a practical trading, due to excessive data transmission delay and large payment. The edge server may get unsatisfying utility for paying high penalty to cloud server when lots of end-users are absent from a trading. Besides, since the cloud server generally offers services to multiple customers (e.g., Edges 2-3 in Fig. 2), a large amount of reserved resources for the concerned edge server (Edge 1) can directly reduce the resource supply and thus the relevant revenue. The later risk is mainly caused by overbooking, where a contractual end-user may still fail to access the required resources. Although he gets compensation, this case can definitely lead to poor trading experience. Additionally, the edge server is risking inadequate resource usage rate due to possible "no shows". Therefore, risks should be properly managed and controlled.
During each practical trading (the later phase), the following key issues should be carefully considered.
\noindent
$\bullet$ \textbf{Compensated user selection}: Limited resource supply can incur the case where end-users who have signed contracts but finally fail to acquire resources, and should process applications locally. Since different end-users may have heterogeneous applications (e.g., data size, etc.) and requirements (e.g., tolerant completion time, etc.), a proper selection strategy should be concerned for choosing appropriate compensated end-users (if any), during each practical trading. Common methods can refer to first-come-first-serve (FCFS), random selection (e.g., users are randomly be compensated), greedy-based selection (e.g., users with the worst channel qualities will be compensated), etc.
\noindent
$\bullet$ \textbf{Application transfer decision}:
Note that different computing servers can provide heterogeneous resources and services. Thus, which application(s) of which end-user(s) could be transferred to the cloud server, or stay on the edge server, presents another noteworthy problem, since cloud can offer rather powerful computing capability and may directly impact the performance of applications. Factors such as the tolerant delay of mobile application, preference of end-user, distance and channel qualities between users and APs have to be taken into consideration.
Fig. 3 shows the timeline and trading examples related to the proposed OATF mechanism, in comparison with conventional booking method (e.g., equal-booking-related trading~\cite{4}) and spot trading. Apparently, forward contracts are pre-signed among participants in Fig. 3(a) and Fig. 3(b), where contractual users will no longer spend extra time and energy on decision-making during each practical trading. Specifically, OATF in Fig. 3(a) encourages a certain overbooking rate calculated by $(10-8)/8=25\%$ in case of possible "no shows". Fig. 3(b) depicts the conventional booking method where the overall resources booked to end-users can not exceed resource supply. For example, in Trading 2, Fig. 3(b), where user 2 is absent, the resource utilization is calculated by $80\%$; while that of our proposed OATF achieves $100\%$ (see Fig. 3(a)) which thus can better deal with dynamic resource demands. As a comparison, Fig. 3(c) illustrates the onsite spot trading mode, where the actual data offloading and service delivery procedure can only start after an onsite trading agreement has been reached. This case can definitely lead to extra latency and energy costs incurred by onsite decision-making. Besides, undesired trading failures might be incurred. For example, in Trading 3 (Fig. 3(c)), although users 1 and 4 have spent a certain amount of time negotiating the trading agreement with the edge server, they finally fail to obtain the required service due to resource shortage.
\section{Case Study}
This section investigates a case study associated with OATF upon considering three key parties: multiple end-users, an edge server with $r^{Edge}$ resources, and a cloud server with $r^{Cloud}$ resources. Namely, edge server and cloud server can theoretically process a maximum of $r^{Edge}+r^{Cloud}$ applications in parallel during a trading, for analytical simplicity.
\subsection{Basic Modeling}
Considering multiple independent identically distributed (i.i.d) end-users $\mathbb{U}=\left\{u_x\left|x\in\right.\left\{1,2,...,|\mathbb{U}|\right\}\right\}$ which are supposed to have same computing capability (e.g., similar smartphone types with the same processors), number of applications $n$, etc., for analytical simplicity. Consequently, terms of contract $\mathbb{M}^{Edge}$ offered by the edge server can thus be the same among different users, which is also general in real-life networks~\cite{4}. Specifically, each user $u_x$ may encounter two uncertain factors: $\alpha_x$ and $\gamma_x$, where random variable $\alpha_x$ defines the attendance
and absence
of $u_x$ during a trading, that obeys a Bernoulli distribution represented by $\alpha_x\sim\text{B}\left\{\left(1,0\right),\left(a,1-a\right)\right\}$. Besides, random variable $\gamma_x$ describes the changing channel quality of the link between $u_x$ and the nearby AP, which follows a uniform distribution denoted by $\gamma_x\sim \text{U}\left(y_1,y_2\right)$, where a small value of $\gamma_x$ can lead to excessive transmission latency of application data. Specifically, contract $\mathbb{M}^{Edge}$ offered by edge server is denoted as a tetrad,
where the utility $\mathcal{U}^{u_x}$ of an end-user $u_x$ who has signed contract $\mathbb{M}^{Edge}$ considers the following factors: \textit{(i)} the saved time and energy as benefited from the amount of reserved resources $r^{user}$; \textit{(ii)} payment to edge server $p^{UtoE}$ for required resources and service; \textit{(iii)} penalty to edge server $q^{UtoE}$ for possible absence; and \textit{(iv)} possible compensation $c^{EtoU}$ obtained from edge server.
Let $\mathbb{M}^{Cloud}$ indicate the forward contract between cloud server and edge server. Relying on both $\mathbb{M}^{Edge}$ and $\mathbb{M}^{Cloud}$, the utility of edge server $\mathcal{U}^{Edge}$ concerns four parts: \textit{(i)}. revenue obtained from end-users; \textit{(ii)}. compensation that edge should pay to the end-users who fail to acquire resources; \textit{(iii)}. payment $p^{EtoC}$ for the predetermined amount of cloud resources $r^{Backup}$; and \textit{(iv)}. penalty $q^{EtoC}$ if not purchasing cloud resources.
This case study considers an interesting assumption that the cloud server will offer services to the studied edge server as the highest priority (namely, the cloud server will not break $\mathbb{M}^{Cloud}$). Accordingly, the uncertain resource demand from other customers obeys a discrete uniform distribution denoted by $\beta\sim\text{U}\left(0,1,...,r^{Cloud}\right)$. Since the pre-determined contract $\mathbb{M}^{Cloud}$ has set aside $r^{Backup}$ resources for the studied edge server, some of these requesters may have to wait for resource release during a trading when cloud server is fully occupied.
Consequently, utility $\mathcal{U}^{Cloud}$ of the cloud server is defined via considering four key parts: \textit{(i)} revenue obtained from other resource requesters; \textit{(ii)} partial refund for requesters who have to wait for available resources; \textit{(iii)} income $p^{EtoC}$ obtained from the edge server; and \textit{(iv)} possible penalty $q^{EtoC}$ paid from the edge server.
\begin{figure*}[h!t]
\centering
\subfigure[]{\includegraphics[width=.33\linewidth]{fig5a}}\hfill
\subfigure[]{\includegraphics[width=.33\linewidth]{fig5b}}\hfill
\subfigure[]{\includegraphics[width=.327\linewidth]{fig5c}}
\caption{Performance evaluation from long-term perspective via simulating 5000 trading.}
\end{figure*}
\subsection{Analysis of Key Issues}
Analysis of key issues mentioned in the previous section based on the above models are discussed hereafter. First, before practical trading, the design of contracts, as well as overbooking rate is formulated as a multi-objective optimization (MOO) problem ($r^{User}=|\mathbb{U}|\times r^{user}$), aiming to maximize the expected utilities of end-users $\text{E}\left[\sum_{x=1}^{x=|\mathbb{U}|}\mathcal{U}^{u_x}\right]$, edge server $\text{E}\left[\mathcal{U}^{Edge}\right]$, and cloud server $\text{E}\left[\mathcal{U}^{Cloud}\right]$, while meeting tolerable risks as major constraints.
Specifically, each end-user $u_x$ considers two key risks: \textit{(i)} the risk of receiving a negative utility, which is defined as the probability that $\mathcal{U}^{u_x}$ is less than or equal to 0; and \textit{(ii)} the risk of failing to acquire resources due to overbooking, which is calculated by the probability that conditions $\alpha_{x^\prime}=1$ and $r^{user}\sum_{x\neq x^\prime}\alpha_x>r^{Edge}+r^{Backup}-r^{User}$ are both met. Namely, the overall resources offered by the edge and cloud server fail to afford the demand of end-users, where some users thus have to process their applications locally. Similarly, the edge server is facing two major risks: \textit{(i)} the risk of obtaining an unsatisfying utility as defined by the probability that $\mathcal{U}^{Edge}$ is less than its expectation $\text{E}\left[\mathcal{U}^{Edge}\right]$; and \textit{(ii)} the risk of resource underutilization, which is reflected by the probability that resource usage stays below a certain rate (mainly caused by an improper overbooking rate). In addition, cloud server is undergoing the risk represented by the probability that the value of $\mathcal{U}^{Cloud}$ is smaller than or equal to $\text{E}\left[\mathcal{U}^{Cloud}\right]$. Apparently, all the above-mentioned risks should be well controlled within a certain range (e.g., each probability should not exceed a threshold. e.g., 30\%).
The proposed MOO problem faces difficulties to be solved directly by traditional algorithms such as the weighted sum method and $\epsilon$-constrained method, since it involves non-convex objective functions and complicated probabilistic constraints.
To this end, a two-way multilateral negotiation scheme is designed that alternatively optimizes expected utilities,
while meeting acceptable risks. Specifically, "two-way" indicates that the edge server has to communicate with both end-users and the cloud server. Fig. 3 shows a diagram of how the proposed negotiation scheme is implemented among different parties, to reach the final agreement on forward contracts. As illustrated by Fig. 4, the edge server first starts a quotation process (step 1), while end-users can determine the acceptable range of $r^{user}$ (step 2) under tolerable risks under given the current price and default clause associated with contract $\mathbb{M}^{Edge}$. Then, the cloud server starts its quotation process, where the edge server decides the acceptable range of $r^{Backup}$, according to the price and default clause (step 4) related to both contracts
, under tolerable risks. Based on which, the cloud server determines feasible values of $r^{Backup}$ (step 5). When the three parties reach a consensus (step 6), end-users select the optimal pair of $r^{user}$ and $r^{Backup}$ that can maximize their expected utilities, which are regarded as candidate contract term, together with the corresponding prices and penalties (step 7). After the quotations of both edge and cloud servers are completed, the edge server chooses the final contract terms for both forward contracts $\mathbb{M}^{Edge}$ and $\mathbb{M}^{Cloud}$, to maximize its expected utility. Apparently, the edge server should understand both end-users and could server, which calls for two-way communication.
In particular, the corresponding computational complexity relies the total number of quotations of edge and cloud servers, which can be denoted by $O(Number~of~quotations)$. Notably, the overhead of OATF will only be incurred during contract negotiation (e.g. 10:00am in Fig. 1), while responsive and cost-effective computing services can be delivered directly during each practical trading, thanks to pre-signed contracts.
Then, each practical trading will proceed among participants on the basis of pre-determined forward contracts. Since end-users are i.i.d, FCFS mechanism is utilized for both compensated user selection and application transfer decision in this case study, which is also fair in real-world trading market.
\subsection{Results and Evaluation}
Simulation results associated with the proposed case study are carefully analyzed mainly from the long-term perspective, comparing with representative baseline methods listed below.
\noindent
$\bullet$ \textbf{Conventional resource booking method (CBooking)} that doesn't allow the amount of resources reserved for end-users $r^{User}$ to exceed the available resource supply $r^{Edge}+r^{Backup}$.
\noindent
$\bullet$ \textbf{Spot trading method} where each practical trading is performed relying on the current market/network conditions (e.g., the current value of $\alpha_x$, $\gamma_x$ and $\beta$). Specifically, both uniform and differential pricing rules~\cite{4}~are considered, as abbreviated to ``SpotT\_UP'' and ``SpotT\_DP'', respectively.
Major parameters are set as follows: $a=76\%$, $y_1=100$, $y_2=400$, $r^{Edge}=197$, $r^{Cloud}=600$, $\mathbb{U}=137$, $e^t=550\text{mW}$ ($e^t$ indicates the transmission power, while $e^t\gamma_x$ represents the received SNR of AP which is roughly within [17dB, 23dB]), $n=5$. Data size of each application is set to be 1Mb, while the corresponding required computing resources can be calculated as $600\times1024\text{bit}\times{1024}^2 \text{CPU cycles}$. Bandwidth of the channel between each end-user and the nearby AP (connected to edge server) is set by 6MHz. Besides, risks are controlled within interval $[20\%,40\%]$.
Since onsite decision-making relies heavily on the end-to-end (E2E) delay of wireless communication channels, a random variable $\tau_x$ (ms) is applied for each end-user in this simulation to describe E2E delay of the wireless channel between end-user $u_x$ and the nearby AP, which follows a uniform distribution denoted by $\tau_x\sim \text{U}\left(2,15\right)$~\cite{3}.
Evaluations upon considering the average value (per trading) of different indicators are shown in Fig. 5, via simulating 5000 trading. In Fig. 5(a), the proposed OATF mechanism outperforms Spot\_UP and Spot\_DP on the average value of participants' utilities, since onsite users have to spend extra time/energy on negotiating a trading agreement with the edge server which directly reduce the usable time for actual computing service delivery and thus time efficiency (also see Fig. 5(b)). Besides, onsite edge server may suffer from insufficient resource supply (e.g., cloud resources are occupied by other customers) and off-peak period (e.g., large number of users are absent from a trading).Besides, although overbooking can better handle dynamic resource demands, it also faces risks. This has led to similar performance of OATF in comparison with CBooking in Fig. 5(a), mainly caused by risks due to the random and unpredictable nature.
Namely, the studied OATF relies on a risk and opportunity coexisting resource market, which is closer to real-world networks. For example, in Fig. 5(a), the cloud server’s utility of CBooking slightly outperforms that of OATF mainly because that overbooking may cause larger refunds paid to other customers of the cloud server (e.g., parameter $\beta$).
Fortunately, the proposed OATF mechanism can get far better average resource usage rate than CBooking as illustrated by Fig. 5(b), since a feasible overbooking rate is encouraged that greatly supports dynamic resource demands.
Fig. 5(c) depicts the average trading failures and the relevant rate, where all the trading in CBooking are successful owing to that the amount of promissory reserved resources for end-users in forward contracts equals to resource supply. In addition, the proposed OATF mechanism enables roughly 99.1\% and 99.3\% performance improvement on the number of failed users, and failure rate, rather than Spot\_UP and Spot\_DP, respectively, due to that onsite end-users may undergo insufficient resources and more severe competition, according to the current market/network conditions. For example, an end-user $u_x$ with large $\gamma_x$ can afford a higher resource price than $u_{x^\prime}$ with a small $\gamma_{x^\prime}$, which may lead to a failure although $u_{x^\prime}$ has spent time on negotiating with the edge server.
\section{Conclusion and Future Direction}
\noindent
Big data generated by countless IoT devices highly requests real-time and cost-effective application processing, which calls for additional requirements on resource provisioning techniques in dynamic networks. This article investigates a novel overbooking-enabled forward trading mechanism called OATF, under device-edge-cloud network architecture with various uncertainties. OATF relies on pre-determined forward contracts negotiated among end-users and the edge server, as well as between the edge server and cloud server in advance, which will be fulfilled accordingly during each future practical trading. Specifically, a certain overbooking rate is encouraged that greatly supports the substantial resource utilization under dynamic resource demand. Framework and key issues associated with the proposed OATF mechanism are carefully analyzed, based on which, an interesting case study is investigated via specific mathematical models. Comprehensive simulation results illustrate that the proposed OATF mechanism can achieve commendable performance on various evaluation indicators such as time efficiency and resource usage, in comparison with conventional trading methods.
Several interesting research directions are worthy of consideration. For example, smart and alterable rights/obligations rather than fixed contract terms are noteworthy to study via adopting machine learning approaches, to better capture the dynamics and unpredictable nature of resource trading market. Moreover, multiple edge servers can be considered, where competition and cooperation among them stand for another future direction.
\section*{Acknowledgement}
\noindent
This work was supported in part by the Fundamental Research Funds for the Central Universities under No. 20720220104, the Discovery Program of Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant RGPIN2018-06254, and the Canada Research Chair Program.
|
1,116,691,500,428 | arxiv | \section{Introduction}
\label{sec_intro}
In four-dimensional General Relativity, a fundamental connection between geometric optics and the algebraic structure of the Weyl tensor is provided by the Goldberg-Sachs (GS) theorem \cite{GolSac62,NP}, which states that an Einstein spacetime is algebraically special if, and only if, it contains a shearfree geodetic null congruence (cf.~\cite{Stephanibook,penrosebook2} for related results and generalizations). This theorem plays an important role in the construction of algebraically special exact solutions of the Einstein equations \cite{Stephanibook}, as the remarkable discovery of the Kerr metric shows \cite{Kerr63}.
In recent years, the growing interest in higher dimensional gravity has motivated the study of extensions of the above concepts to $n>4$ dimensions. An algebraic classification of the Weyl tensor based on the notion of Weyl Aligned Null Directions (WANDs) has been put forward in \cite{Coleyetal04} (see also \cite{OrtPraPra12rev} for a recent review). Furthermore, a higher dimensional version of Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) formalisms have been presented in \cite{Pravdaetal04,Coleyetal04vsi,OrtPraPra07} and \cite{Durkeeetal10}, respectively. However, simple examples reveal that neither the ``geodesic part'' nor the ``shearfree part'' of the GS theorem extend in an obvious way to higher dimensions (see \cite{Ortaggioetal12} for a brief summary, along with the original references \cite{MyePer86,FroSto03,Pravdaetal04,OrtPraPra07,PraPraOrt07,GodRea09,Durkee09}).
The proper formulation of the geodesic part of the higher dimensional GS theorem has been proven in \cite{DurRea09} (see also \cite{PraPraOrt07,Durkee09}): in particular, {\em an Einstein spacetime admits a multiple WAND if, and only if, it admits a geodesic multiple WAND} -- hence one can restrict to geodesic multiple WANDs {(mWANDs)} with no loss of generality. Very recent work \cite{Ortaggioetal12} has analyzed the shearfree part in five dimensions by proving necessary conditions on the form of the optical matrix $\rhob$ (defined below in \eqref{expmatrix}) following from the existence of an mWAND.\footnote{See \cite{Taghavi-Chabert11} for a different formulation of the ``shearfree'' condition and \cite{Ortaggioetal12} for a comparison between the two approaches.} Contrary to the 4d case, the conditions obtained in \cite{Ortaggioetal12} are, in general, not sufficient. In fact, it seems that in higher dimensions conditions that are both necessary and sufficient in general do not exist.
In the present contribution we focus on $n$-dimensional Einstein (including Ricci-flat) spacetimes of type~II (type D being understood as a special subcase thereof) in more than five dimensions and work out the corresponding necessary conditions on $\rhob$ { {\em under the assumption that the mWAND is twistfree} (i.e., $\rhob$ is symmetric, {equivalent to $A_{ij}=0$, cf.~\eqref{S_A} below)}}.\footnote{This automatically guarantees that $\bl$ is geodesic. Except for a few remarks, we assume {that $\rhob$ is non-zero}, since we are interested in studying its possible non-trivial forms.}
Since the types N and III were already studied in \cite{Pravdaetal04}, we will simply connect the results of \cite{Pravdaetal04} to our analysis when appropriate. As it turns out, in $n>5$ dimensions the constraints on $\rhob$ are in general not as strong as those for $n=5$ \cite{Ortaggioetal12}. Nevertheless, combining our results for a type II spacetime with the results for type III and type N \cite{Pravdaetal04} gives
\begin{theorem}[Eigenvalue structure of $\rhob$ for $n\ge 6$ and $A_{ij}=0$]
\label{prop_GSHD}
In an algebraically special Einstein spacetime of dimension $n\ge 6$ that is not conformally flat, the $($symmetric$)$ optical matrix {$\rhob$} of a non-twisting multiple WAND has at least {\em one double eigenvalue}. In the following special cases stronger conditions hold and the most general permitted eigenvalue structures are
\begin{enumerate}[(i)]
\item if $\Phi^A_{ij} \ne 0$: $\{a,a,0,\ldots , 0\}$ {\rm\cite{Ortaggioetal12}},
\label{PhiA}
\item if $\det \rhob\not= 0$, $\Phi_{ij}\not=0$: $\{a,a,\ldots , a\}$ (Robinson-Trautman, $\Phi_{ij}\propto \delta_{ij}$, type D(bd)),
\label{RT}
\item if $\Phi_{ij}=0$ (type II(abd)): $\{a,a,b,b,c_1,\ldots , c_{n-6}\}$, \label{Phi0}
\item
if the type is N or III: $\{a,a,0,\ldots , 0\}$ {\rm\cite{Pravdaetal04}}.
\label{NIII}
\end{enumerate}
\end{theorem}
(The Weyl components $\Phi^A_{ij}$ and $\Phi_{ij}$ are defined below in \eqref{bw0}.) The proof of (\ref{RT}) and (\ref{Phi0}) is a new result and will be given in sections~\ref{sec_nondeg} and \ref{sec_nontwist}, respectively.
Note that in the above cases (\ref{PhiA})--(\ref{NIII}) the matrix $\rhob$ possesses at least {\em two} double eigenvalues. In particular, in six dimensions all possible cases can be listed explicitly:
\begin{corol}[Eigenvalue structure of $\rhob$ for $n=6$ and $A_{ij}=0$]
\label{corol6d}
In a 6d algebraically special Einstein spacetime that is not confomally flat, the $($symmetric$)$ optical matrix $\rhob$ of a non-twisting multiple WAND can have only one of the following eigenvalue structures (where $a,b $ might coincide, or vanish): (i) $\{a,a,b,b \}$; (ii) $\{a,a,b,0 \}$; (iii) $\{a,b,0,0 \}$. If the spacetime is type III or N then the structure is $\{a,a,0,0\}$.
\end{corol}
Note that we do not claim that all classes compatible with theorem~\ref{prop_GSHD} or corollary \ref{corol6d} are non-empty.
The structure of the paper is the following. In section~\ref{sec_genII} we briefly summarize results for general (possibly twisting) type II spacetimes in arbitrary dimension (already contained in \cite{Ortaggioetal12}), in particular the so called ``optical constraint''. These will be useful in the rest of the paper, where, however, we limit ourselves to the non-twisting case.
In section~\ref{sec_nontw_gen} we set up the study of general non-twisting type II spacetimes. It turns out that it is convenient to study the case of non-degenerate and degenerate $\rhob$ separately. This is done in sections~\ref{sec_nondeg} and \ref{sec_nontwist}, where we derive certain necessary conditions for $\rhob$ being compatible with a non-twisting mWAND,
in particular proving thus theorem~\ref{prop_GSHD}. Using various explicit examples we show in section~\ref{sec_counter} that the necessary conditions obtained in sections~\ref{sec_nondeg} and \ref{sec_nontwist} are in general not sufficient. In section~\ref{sec_totgeod} we elucidate the geometrical implications of theorem~\ref{prop_GSHD} in terms of existence of integrable null distributions with
totally geodesic integral null two-surfaces. In section~\ref{sec_6D} we study the six-dimensional case (corollary \ref{corol6d}) and provide various explicit examples of such metrics.
Some related results are given in the appendices. {{In appendix~\ref{app_alg_eq} we present new algebraic constraints for general type II Einstein spacetimes. Appendix~\ref{app_type_N_III} contains a new proof of the canonical form of the optical matrix $\rhob$ for type III Einstein spacetimes in the case of a non-twisting mWAND $\bl$ (the original, longer proof was given in \cite{Pravdaetal04}). In appendix~\ref{app_shearfree} we summarize various equivalent formulations of the ``geodesic and shearfree'' condition in four dimensions, and we discuss some extensions of those to higher dimensions (including optical structure, optical constraint, integrability of certain null distributions). Appendix~\ref{app_OS5d} contains the proof (mostly relying on results of \cite{Ortaggioetal12,Wylleman_priv}) of the existence of an optical structure in all algebraically special spacetimes in five dimensions. In appendix~\ref{app_shearfreetwist} we present examples of twisting but shearfree mWANDs (in six dimensions). Finally, in appendix~\ref{app_violating} examples of Einstein spacetimes of type D with non-degenerate mWANDs violating the optical constraint are given.}}
\paragraph{Notation}
Throughout the paper we use higher dimensional GHP formalism developed in \cite{Durkeeetal10}. We employ a null frame
\begin{equation}
\{\lb \equiv \eb_{(0)}=\eb^{(1)},
\bn \equiv \eb_{(1)} = \eb^{(0)},
\mmb{i}\equiv\eb_{(i)} = \eb^{(i)} \}
\end{equation}
with indices $i,j,k,\ldots$ running from $2$ to $n-1$. The null vector fields $\lb$ and $\bn$ and the orthonormal spacelike vector fields $\mmb{i}$ obey $\lb \cdot \bn =1$, $\mmb{i} \cdot \mmb{j} = \delta_{ij}$ and $\lb \cdot \mmb{i} = 0 = \bn \cdot \mmb{i}.$
The optical matrix $\rhob$ is defined by
\be
\rho_{ij} = m_{(i)}^a m_{(j)}^b \nabla_b \ell_a , \label{expmatrix}
\ee
and its trace gives the expansion scalar (up to normalization)
\be
\rho \equiv \rho_{ii}.
\ee
{Its (anti-)symmetric parts are, respectively,
\be
S_{ij}\equiv \rho_{(ij)} , \qquad A_{ij}\equiv \rho_{[ij]} .
\label{S_A}
\ee
We also define the rank of $\rhob$ as}
\be
m\equiv\mbox{rank}(\rhob).
\ee
Other Ricci rotation coefficients used in the paper are \cite{Pravdaetal04,OrtPraPra07,Durkeeetal10}
\be
L_{1i} = n^a m_{(i)}^b \nabla_b \ell_a , \quad \tau_i\equiv L_{i1} = m_{(i)}^a n^b \nabla_b \ell_a ,\quad \M{i}{j}{0}=m_{(j)}^a\ell^b \nabla_b m_{(i)a} , \quad \M{i}{j}{k}=m_{(j)}^am_{(k)}^b \nabla_b m_{(i)a} .
\label{L1i_M}
\ee
Boost weight (b.w.) zero components of the Weyl tensor
\be
\Phi_{ijkl} = C_{ijkl}, \ \ \ \Phi_{ij} = C_{0i1j},\ \ \ 2\Phia_{ij} = C_{01ij}, \ \ \ \Phi = C_{0101}
\label{bw0}
\ee
are subject to
\bea
&& \Phi_{ijkl} = \Phi_{[ij][kl]} = \Phi_{klij}, \qquad \Phi_{i[jkl]}=0, \label{Phicycl} \\
&& \Phis_{ij} \equiv \Phi_{(ij)} = -\pul\Phi_{ikjk}, \qquad \Phia_{ij} \equiv \Phi_{[ij]}, \qquad \Phi=\Phi_{ii}, \label{PhiS}
\eea
and thus they are determined by the $\frac{1}{12}(n-1)(n-2)^2(n-3)$ components of $\Phi_{ijkl}$ and the $\pul (n-2)(n-3)$ components of $\Phia_{ij}$. In a type II spacetime, at least one of the latter two quantities must be non-vanishing when $\bl$ is an mWAND (otherwise the spacetime would be type III or more special).
Recall that the algebraic subtypes II(a), II(b), II(c) and II(d) are defined by the vanishing of $\Phi$, the symmetric traceless part of $\Phi_{ij}$, the ``Weyl part'' of $\Phi_{ijkl}$, and $\Phia_{ij}$, respectively (see \cite{OrtPraPra12rev}).
There is no summation in case one or both repeated indices is/are in round brackets, unless it is said otherwise. E.g., there is no summation in $\Phi_{i(j)k(j)}$ and there is summation in $\Phi_{ijkj}$ and $\sum_j \Phi_{i(j)k(j)}$.
{In what follows we shall employ the Sachs equation \cite{Pravdaetal04,OrtPraPra07,Durkeeetal10}
\begin{eqnarray}
\tho \rho_{ij} &=& - \rho_{ik} \rho_{kj} \label{Sachs},
\end{eqnarray}
and the following Bianchi identities (eqs.~(A10), (A11) of \cite{Durkeeetal10})
\begin{eqnarray}
\tho \Phi_{ij} &=& -(\Phi_{ik} + 2\Phia_{ik} + \Phi \delta_{ik}) \rho_{kj}, \label{A2}\label{Bi2}\\[3mm]
-\tho \Phi_{ijkl} &=& 4\Phia_{ij} \rho_{[kl]} - 2\Phi_{[k|i} \rho_{j|l]} + 2\Phi_{[k|j} \rho_{i|l]}
+ 2\Phi_{ij[k|m} \rho_{m|l]} . \label{A4}\label{Bi3}
\end{eqnarray}
In a parallelly propagated frame, the GHP directional derivative $\tho$ reduces to the NP derivative along the $\bl$ direction, $D=\ell^a\nabla_a$.
For an affinely parametrized $\bl$ with an affine parameter $r$ this is simply $D=\partial_r$.}
There are two possible approaches to studying consequences of the Bianchi and Ricci equations. The first approach consists in applying the derivative operator $\tho$ on certain algebraic equations and in deriving new algebraic constraints, e.g. eq.~\eqref{thornB8}, using \eqref{Sachs}--\eqref{Bi3} (cf. also appendix~\ref{app_alg_eq} and \cite{Ortaggioetal12} (section~3.1 and appendix A therein)). In this paper we mainly use the second approach, which consists in solving the Sachs and Bianchi differential equations \eqref{Sachs}--\eqref{Bi3} and then in analyzing the compatibility of their solutions with the algebraic equations \eqref{B4}--\eqref{thornB8}.
\section{Results for general type II}
\label{sec_genII}
This section is devoted to summarizing some useful results that hold for all type II Einstein spacetimes, without assuming that $\rhob$ is non-twisting. In particular we present the algebraic restrictions on the Weyl components of b.w. zero that follow from the Bianchi identities, to be used in the next sections. More general new results are given in appendix~\ref{app_alg_eq} for future reference.
For genuine type II spacetimes the (unique) mWAND is necessarily geodesic, while for type D spacetimes there always exists a geodesic mWAND \cite{DurRea09}. In the frame used below we thus take $\bl$ to be a geodesic mWAND, without loss of generality.
\subsection{Algebraic constraints}
For type II Einstein spacetimes, the Weyl tensor must obey the algebraic constraints (A.12) and
(B8) of \cite{Durkeeetal10} (already discussed in \cite{PraPraOrt07,Durkee09}) and $\tho$(B8) (derived in \cite{Ortaggioetal12}), namely
\bea
2\Phia_{[jk|}\rho_{i|l]} -2\Phi_{i[j}\rho_{kl]} + \Phi_{im [jk|}\rho_{m|l]} = 0, \label{B4} \\
\Phi_{kj} \rho_{ij} - \Phi_{jk} \rho_{ij} + \Phi_{ij} \rho_{kj} - \Phi_{ji} \rho_{jk}
+ 2 \Phi_{ij} \rho_{jk} - \Phi_{ik} \rho + \Phi \rho_{ik} + \Phi_{ijkl} \rho_{jl} = 0 \label{B8} , \\
\left( 2 \Phi_{kj} - \Phi_{jk} \right) \rho_{il} \rho_{jl} + \left( 2 \Phi_{ij} - \Phi_{ji} \right) \rho_{jl} \rho_{kl} - \Phis_{ik} \rho_{jl} \rho_{jl} + \Phi \rho_{il} \rho_{kl} + \Phi_{ijkl} \rho_{js} \rho_{ls} =0. \label{thornB8}
\eea
Eq. \eqref{B8} is traceless and its
symmetric and antisymmetric parts read, respectively,
\bea
\left( 2 \Phi_{kj} - \Phi_{jk} \right) S_{ij} + \left( 2 \Phi_{ij} - \Phi_{ji} \right) S_{jk} - \Phis_{ik} \rho + \Phi S_{ik} + \Phi_{ijkl} S_{jl} =0, \label{B8sym}\\
\Phi_{jk} A_{ji} + \Phi_{ji} A_{kj} + \Phi_{ij} \rho_{jk} -\Phi_{kj} \rho_{ji} + \Phia_{k i} \rho + \Phi A_{ik} + \Phi_{ijkl} A_{jl} = 0 . \label{B8asym}
\eea
The reason for investigating algebraic conditions such as \eqref{B4}--\eqref{B8asym} (or those of \cite{Pravdaetal04} in the case of type III/N) is that they will in general constrain the possible form of $\rhob$. This way one obtains the standard Goldberg-Sachs theorem in four dimensions and one can arrive at similar conclusions also in higher dimensions, at least with additional assumptions (e.g. on the Weyl type \cite{Pravdaetal04}, on the number of dimensions \cite{Ortaggioetal12},
on the form of the line-element \cite{OrtPraPra09,MalPra11}, on the asymptotic behaviour \cite{OrtPraPra09b}, and/or on optical properties of $\bl$, as we shall discuss in the next sections).
\subsection{The optical constraint}
\label{subsec_OC}
It was observed in \cite{Ortaggioetal12} that the above conditions on the Weyl tensor appear to be less stringent when $\rhob$ satisfies
the {\em optical constraint} \cite{OrtPraPra09}
\be
\rho_{ik} \rho_{jk} \propto \rho_{(ij)} . \label{OC2}
\ee
In particular, when this holds eq.~\eqref{thornB8} is not an extra restriction.
{This thus suggests that the branch of type II solutions {whose mWAND} obeys \eqref{OC2} corresponds to the case of a ``generic'' Weyl tensor \cite{Ortaggioetal12}.}
It has been proven that \eqref{OC2} indeed holds for {the Kerr-Schild vector} of all (generalized) Kerr-Schild spacetimes \cite{OrtPraPra09,MalPra11} (including Myers-Perry black holes \cite{MyePer86}), for non-degenerate geodesic double WANDs in asymptotically flat type II spacetimes \cite{OrtPraPra09b} (see eq.~(14) therein) {and for the unique double WAND of all genuine type II Einstein spacetimes in five dimensions (see footnote~\ref{foot_typeD} for the type D case) \cite{Ortaggioetal12,Wylleman_priv}.}
Nevertheless, {double WANDs} violating the optical constraint also exist, as shown by explicit examples constructed below in section~\ref{subsubsec_examples_distinct} for $n\ge 6$, in appendix~\ref{app_violating} for $n\ge 7$, and in section~6.3 of \cite{Ortaggioetal12} for $n=5$.\footnote{We observe that all such examples are of type D. In fact, in five dimensions {\em all} type D Einstein spacetimes admitting a geodesic mWAND violating the optical constraint are known \cite{Ortaggioetal12} (and coincide with the class of Einstein spacetimes admitting a non-geodesic mWAND \cite{DurRea09}). Such ``exceptional'' null directions are necessarily twisting in 5d (but not in higher dimensions).
However, in all such 5d type D spacetimes there always also exists a pair of non-aligned (non-twisting) mWANDs that {\em do} obey the optical constraint \cite{Ortaggioetal12}. On the other hand, this is generically not true when $n>5$ (see appendix \ref{subsubsec_ex2} for a ten-dimensional example admitting exactly two mWANDs, both violating the optical constraint).\label{foot_typeD}}
Although the above analysis applies only to type D and genuine type II spacetimes (eqs.~(\ref{B4})--(\ref{thornB8}) become trivial identities for more special types), it is worth remarking that the optical constraint is also obeyed by the mWAND of all type N Einstein spacetimes \cite{Pravdaetal04,Durkeeetal10}. {The mWAND $\bl$ of type III Einstein spacetimes also obeys the optical constraint provided either \cite{Pravdaetal04}: (i) the spacetime is five-dimensional; (ii) the Weyl tensor satisfies a genericity condition (see~\cite{Pravdaetal04}); (iii) $\bl$ is {\it non-twisting} (see also appendix \ref{app_type_N_III} for a simpler proof in case (iii))}.
The optical constraint implies that $(\Id- \frac{2}{\alpha} \rhob)$ is an orthogonal matrix \cite{Ortaggioetal12} and that consequently $[\rhob, \rhob^T]$ vanishes \cite{OrtPraPra09,Ortaggioetal12}. The optical matrix $\rhob$ is therefore a {\em normal} matrix and can thus be put, using spins, into a convenient block-diagonal form {(see \cite{OrtPraPra09,OrtPraPra10} for extended related discussions)}, i.e.,
\bea
\rhob = \alpha{\rm diag}\left(1, \dots 1,
\frac{1}{1+ \alpha^2 b_1^2}\left[\begin {array}{cc} 1 & -\alpha b_1 \\
\alpha b_1 & 1 \label{canformL} \\
\end {array}
\right]
, \dots,
\frac{1}{1+ \alpha^2 b_\nu^2} \left[\begin {array}{cc} 1 & -\alpha b_\nu \\
\alpha b_\nu & 1 \\ \end {array}
\right]
, 0, \dots ,0
\right).
\eea
The block-diagonal form \eqref{canformL} {is useful for practical purposes because it} allows for determining the $r$-dependence of $\rhob$ by integrating the Sachs equation \eqref{Sachs} {(in a parallelly transported frame)} \cite{OrtPraPra10}. {\em In the non-twisting case it reduces to a sequence of 1s followed by a sequence of 0s} (up to an overall factor). Note that the symmetric part of each two-block is proportional to a two-dimensional identity matrix, i.e. it is ``shear-free''. In four dimensions the optical constraint implies that either there is a single such block (in which case $\rho_{ij}$ is shearfree) or that $\rho_{ij}$ is symmetric with exactly one non-vanishing eigenvalue. However, the Goldberg-Sachs theorem shows that the latter case cannot occur. Therefore in 4d the optical constraint is a necessary condition for $\bl$ to be a repeated principal null direction but it is not sufficient \cite{Ortaggioetal12}.
On the other hand, in higher dimensions a vector field $\bl$ obeying the optical constraint is in general {\em shearing}, as in the case of Myers-Perry black holes \cite{MyePer86} (see also \cite{FroSto03,PraPraOrt07} for a discussion of their optical properties). Together with other results \cite{Pravdaetal04,PodOrt06,OrtPraPra07}, this has made clear that the shearfree condition is in general ``too restrictive'' in higher dimensions, as opposed to the four-dimensional case. In particular, it was observed in \cite{OrtPraPra07} that in {\em odd} dimensions a twisting geodesic mWAND is necessarily shearing. By contrast, twisting geodesic mWANDs with zero shear are permitted in {\em even} dimensions and they have necessarily $\det(\rhob)\neq 0$ (as can be easily seen in a frame adapted to $A_{ij}$, using the fact that $S_{ij}\propto\delta_{ij}$). An explicit example in six dimensions is discussed in appendix \ref{app_shearfreetwist}.
In the non-twisting case, the existence of shearfree spacetimes has been already known for some time in all dimensions -- they are either \RT \cite{PodOrt06} or Kundt \cite{PodZof09} solutions, according to the presence/absence of expansion.
\subsection{Possible generalizations of the geodesic{\&}shearfree property}
In arbitrary dimensions various geometric conditions can be considered which are different from the standard geodesic{\&}shearfree condition (considered above) and which, however, become all equivalent (except for the optical constraint, cf.~\cite{Ortaggioetal12}) in the special case $n=4$. Further such conditions are discussed in appendix~\ref{app_shearfree}.
One could thus conceive that various formulations of the Goldberg-Sachs theorem are in principle possible when $n>4$. Some of these formulations have been studied in \cite{Taghavi-Chabert11,Taghavi-Chabert11b,Ortaggioetal12} (see also \cite{OrtPraPra09}) but none of these gives necessary and sufficient conditions. In the rest of this paper we will discuss necessary conditions determined by the presence of a non-twisting mWAND in an $n$-dimensional Einstein spacetime and we will present several explicit examples. A possible interpretation of these results in terms of the geometric conditions of appendix~\ref{app_shearfree} will be discussed in sections~\ref{sec_totgeod} (for $n\ge6$) and \ref{subsec_integrability} (for $n=6$).
\section{Non-twisting $\bl$: general properties}
\label{sec_nontw_gen}
Here we study the optics of a hypersurface orthogonal mWAND $\bl$ in a type II Einstein spacetime. This mWAND is automatically geodetic. Thus we have
\be
\kappa_i=0=A_{ij} ,
\ee
{so that $\rho_{ij}=S_{ij}$.} Since the algebraic equations \eqref{B4}--\eqref{B8asym} are trivial for Kundt spacetimes, in what follows we assume $\rhob\not= ${\bf 0}.
\subsection{Case $\Phia_{ij} \neq 0$}
\label{sec_PhiA}
This case has been already analyzed in \cite{Ortaggioetal12}, {arriving at} point (i) of theorem~\ref{prop_GSHD} of section~\ref{sec_intro}. As already observed there, in this case $\rhob$ satisfies the optical constraint and it is necessarily degenerate ($m=2$). Moreover it is shearing, except for $\rhob=0$ {(which is necessarily the case for $n=4$)}.
It remains to consider the case when $\Phi^A_{ij}=0$.
\subsection{Case $\Phia_{ij}=0$}
This case defines the subtype II(d) {in the notation of \cite{Coleyetal04}}. For $\Phia_{ij}=0$, eq.~\eqref{B8} reduces to (recall $A_{ij}=0$ here)
\be
-\rho \Phi_{ik} + \Phi \rho_{ik} + 2\Phi_{ij} \rho_{jk} + \Phi_{ijkl} \rho_{jl}=0 , \label{eqn:algI}
\ee
and eq. \eqref{B8asym} to
\be
[\rhob,\Phib^S]=0.\label{comm_rho_Phi}
\ee
Note that taking into account (\ref{comm_rho_Phi}), eq. (\ref{eqn:algI}) is symmetric and corresponds to eq.~\eqref{B8sym}.
Similarly, eq. \eqref{thornB8} yields
\be
-\Phi_{ik} \rho_{jl} \rho_{jl} + \Phi (\rho^2)_{ik} + 2 \Phi_{ij} (\rho^2)_{jk} + \Phi_{ijkl} (\rho^2)_{jl} = 0\label{eqn:rhoC}.
\ee
Thanks to \eqref{comm_rho_Phi}, one can choose a {basis where both $\rho_{ij}$ and $\Phi_{ij}$ are diagonal}, $\rho_{ij}=\diag(\rho_2,\rho_3,\dots)$,
$\Phi_{ij}=\diag(\Phi_2,\Phi_3,\dots)$, { therefore $\Phi_{ijkj}=0$ for $i\neq k$}.
In this frame, {the off-diagonal components of the algebraic constraint~(\ref{eqn:algI}) are}
\be
\sum_j\rho_{(j)}\Phi_{i(j)k(j)}=0 \qquad (\mbox{for } k\neq i) .
\label{nondiag_constr}
\ee
The diagonal part of~(\ref{eqn:algI}) can be expressed as ${\cal L}_{ij} \tilde \rho_j =0$,
where $\tilde \rho_{i}$ is the vector $\tilde \rho_{i}=(\rho_2, \rho_3 \dots)$
and the linear operator ${\cal L}_{ij}$ is given by
\[
{\cal L}_{ij} = W_{ij} + \Phi \delta_{ij} + 2 \diag (\Phi_{2},\Phi_{3},\dots \Phi_{n}) - \left[
\begin{array}{cccc}
\Phi_{2} & \Phi_{2} & \dots & \Phi_{2} \\
\Phi_{3} & \Phi_{3} & \dots & \Phi_{3} \\
\vdots & \vdots & \dots & \vdots\\
\Phi_{n} & \Phi_{n} & \dots & \Phi_{n}
\end{array}
\right]
\]
or, equivalently, ${\cal L}_{ij} = W_{ij} +( \Phi + 2\Phi_{(i)} ) \delta_{ij} -\Phi_{(i)} $, where {we have defined the symmetric and traceless matrix}
\be
W_{ij} \equiv C_{(i)(j)(i)(j)} \qquad {\mathrm{\ \ (no\ summation)}}.\label{def_W}
\ee
{Note that $W_{(i)(i)}=0$.} From the first of \eqref{PhiS}, it follows
\be
\sum_j W_{ij} = - 2 \Phi_i ,
\label{W_phi}
\ee
{which will be used in certain calculations throughout the paper.} (From \eqref{W_phi}
it follows that $W_{ij}=0\Rightarrow {\cal L}_{ij}=0$.)
Thus the sum of the rows of ${\cal L}_{ij}$ vanishes and so they are linearly dependent and
\be
\det {\cal {\bm{L}}} =0. \label{detL}
\ee
Therefore zero is an eigenvalue of ${\cal L}$ and (the diagonal part of) (\ref{eqn:algI}) admits non-trivial solutions.
Note that (\ref{eqn:rhoC}) has the form ${\cal L}_{ij} {\tilde \rho}^2_j =0$, where components of the $(n-2)$ dimensional vector ${\tilde \rho}^2$ are squares of components of ${\tilde \rho}$. The characteristic polynomial of ${\cal L}$ is
\be
\det({\cal L}-\lambda I)=\lambda^{n-2} + k_{n-3} \lambda^{n-3} + \dots + k_1 \lambda + k_0 ,
\ee
with $k_0=0$ due to (\ref{detL}).
Now zero is a multiple eigenvalue of ${\cal L}$ iff $k_1=0$. Let us observe that for a generic form of a type II Weyl tensor $k_1$ is non-vanishing {and thus zero is a single eigenvalue
of ${\cal L}$}.\footnote{For example in five dimensions $k_1 = 12 \left(\phi_2 \phi_4+ \phi_3 \phi_4 + \phi_2 \phi_3 \right)$. {If we consider, for instance, five-dimensional} \RT spacetimes ({which coincide with the Schwarzschild solution plus a possible cosmological constant}), characterized by $\rho_{ij} \propto \delta_{ij}$, equation \eqref{eqn:algI} implies $\Phi_{ij} \propto \delta_{ij}$, which clearly leads to $k_1 \not=0$.}
If this is the case then ${\tilde \rho}^2$ is proportional to ${\tilde \rho}$ and therefore
\be
\rho_{ij}= \alpha \diag(1,1, \dots 1,0,\dots 0).
\ee
Recalling also point (i) of theorem~\ref{prop_GSHD},
we can conclude with
\begin{prop}
For non-twisting type II Einstein spacetimes with $k_1 \not=0$, all non-vanishing eigenvalues of the optical matrix $\rho_{ij}$ coincide.
\label{prop_nontwist}
\end{prop}
If $\Phia_{ij} \not= 0$, the stronger result of point (i) of theorem~\ref{prop_GSHD} holds.
As we will see in sections~\ref{subsec_alldistinct} and \ref{subsec_permitted_6D} (table~\ref{tab_6D}),
there do exist non-twisting type II Einstein spacetimes with distinct non-vanishing eigenvalues of $\rhob$.
According to Proposition~\ref{prop_nontwist}, for these spacetimes $k_1$ vanishes. In particular $\cal L$, and therefore also $k_1$, vanishes for spacetimes with $W_{ij}=0$. We will study special cases with various parts of the Weyl tensor vanishing in the following sections.
\section{Non-twisting $\bl$: non-degenerate $\rhob$ ($m=n-2$)}
\label{sec_nondeg}
First, let us consider the case of a non-degenerate $\rhob$ (i.e., $\det\rhob\not= 0$ -- this is relevant, e.g., for asymptotically flat algebraically special spacetimes \cite{OrtPraPra09b}). When $\det\rhob\not= 0$ one necessarily has (see point (i) of theorem~\ref{prop_GSHD})
\be
\Phia_{ij}=0 ,
\ee
so that the type is II(d). As noticed above we can use a frame in which both $\rho_{ij}$ and $\Phi_{ij}$ are diagonal, which is moreover compatible with parallel transport \cite{PraPra08,OrtPraPra10}. Thus the eigenvalues of the optical matrix are \cite{PraPra08}
\be
\rho_i=\frac{1}{r-b_i} .
\label{rho_i}
\ee
{In order to discuss the possible structures of $\rhob$ it is convenient to discuss separately various cases in which different parts of the Weyl tensor are (non-)zero.}
\subsection{Cases $\Phi_{ij}\neq0$ and $\Phi_{ij}=0\neq W_{ij}$}
\label{subsec_nondeg_Phi}
From Bianchi equation~\eqref{Bi2} one has
\beqn
D\left(\frac{\Phi_{(i)}}{\rho_{(i)}}\right)=-\Phi .
\label{nondeg_DPhi_i}
\eeqn
The case when all $\rho_i$ coincide (and are non-zero) is the known \RT case, for which also all $\Phi_i$ must coincide {\cite{PraPra08,PodOrt06}.}
{Thus} let us consider here the case when at least two eigenvalues $\rho_i$ are different.
Since the r.h.s. {of~(\ref{nondeg_DPhi_i})} is the same for any $i$, we obtain that either all $\Phi_i=0$, or
\be
\Phi_{i}=\rho_i(A+\Phi^0_{(i)}) ,
\label{Phi_i}
\ee
where $A=A(r)$ is a function that must satisfy the equation $DA+A\rho=-\Phi^0_{j}\rho_j$ (since $\Phi=A\rho+\Phi^0_{j}\rho_j$) and, from now on, quantities with a superscript $^0$ do not depend on $r$. Using the intermediate substitution $A(r)=Y(r)\prod_k\rho_k$, one arrives at the solution
\be
A=\left(\prod_k\frac{1}{r-b_k}\right)\left[-\sum_j\Phi^0_{j}\int\d r\prod_{l\neq j}(r-b_l)+A^0\right] .\label{A_nondeg}
\ee
For subsequent discussions it is useful to rewrite \eqref{A_nondeg} using partial fraction decomposition, i.e.,
\be
A=\frac{P_{n-2}}{\prod_k {(r-b_k)}}=a^0+\frac{P_{n-3}}{\prod_k {(r-b_k)}}=a^0+
\sum_{K=1}^{\alpha_{max}} \sum_{\forall i, \alpha_i\geq K}\frac{c^{(K)}_{(i)}}{(r-b_i)^K} ,\label{A_decomposed}
\ee
where $P_k$ denotes a polynomial of order $k$ in $r$, $c^{(K)}_{(i)}$ {do not depend on $r$} and $\alpha_i$ denotes the multiplicity of $b_i$, with $\alpha_{max}$ being the maximal multiplicity. {In particular, the term $a^0$ can be determined by looking at} the leading term of \eqref{A_nondeg} in the limit $r\to\infty$, i.e.,
\be
A=-\frac{1}{n-2}\sum_k\Phi^0_k+O(r^{-1}) ,
\ee
so that
\be
a^0=-\frac{1}{n-2}\sum_k\Phi^0_k.\label{am}
\ee
In the case $\Phi^0_j=0$, $a^0$ vanishes and one has simply $A=A^0/\prod_k {(r-b_k)}$.
Next, the equation for $W_{ij}$ (see \eqref{Bi3}) can be written as (but recall $W_{(i)(i)}=0$)
\beqn
& & D\left(\frac{W_{ij}}{\rho_{(i)}-\rho_{(j)}}\right)=\frac{\Phi_{i}\rho_j+\Phi_{j}\rho_i}{\rho_{(i)}-\rho_{(j)}} \qquad (\rho_{j}\neq\rho_{i}) , \\
& & D\left(\frac{W_{ij}}{\rho_{(i)}^2}\right)=\frac{\Phi_{i}+\Phi_{j}}{\rho_{(i)}} \qquad (\rho_{j}=\rho_{i}) .
\eeqn
Using the above expression for $\Phi_i$, in both cases one can write the solution as
\be
W_{ij}=\frac{1}{(r-b_{(i)})(r-b_{(j)})}\left[2\int\d r A+(\Phi^0_{i}+\Phi^0_{j})r+ W^0_{ij}\right] .\label{Wij_nondeg}
\ee
Now, imposing $-2\Phi_i=\sum_jW_{ij}$ {(recall~(\ref{W_phi}))} we obtain the constraint
\be
-2(A+\Phi^0_{i})=\sum_{j\neq i}\frac{1}{r-b_{(j)}}\left[2\int\d r A+(\Phi^0_{i}+\Phi^0_{j})r+ W^0_{ij}\right] .
\label{constrW}
\ee
{For $r\to\infty$, using \eqref{A_decomposed} and \eqref{am} at
the leading order (\ref{constrW}) implies that all $\Phi^0_{i}$ coincide, i.e., }
\be
\Phi^0_{i}=\frac{1}{n-2}\sum_k\Phi^0_k\equiv{f^0}.
\label{Phi_i0}
\ee
$A$ thus becomes
\be
A=-f^0+A^0\prod_k\frac{1}{r-b_k} ,
\label{A_2}
\ee
where we have used $\sum_j\prod_{l\neq j}(r-b_l)=D[\prod_k(r-b_k)]$, so that
\be
\Phi_{i}=\frac{A^0}{r-b_{i}}\prod_k\frac{1}{r-b_k} , \qquad W_{ij}=\frac{1}{(r-b_{(i)})(r-b_{(j)})}\left[2A^0r\prod_k\frac{1}{r-b_k}+ W^0_{ij}\right] .
\label{nondeg_Weyl_simplif}
\ee
The constraint~(\ref{constrW}) can thus be written as
\be
-2A^0\prod_k\frac{1}{r-b_k}=\sum_{j\neq i}\frac{1}{r-b_{(j)}}\left[2A^0\int\d r\prod_k\frac{1}{r-b_k}+W^0_{ij}\right] \qquad (i=2,\ldots,n-1) .
\label{constrW2}
\ee
{It is now useful to discuss separately various subcases with different multiplicity of eigenvalues.}
\subsubsection{All the $b_i$ coincide: shearfree congruences (Robinson-Trautman spacetimes)}
In the \RT case all the $b_i$ {of (\ref{rho_i})} coincide {(and can be set to zero by shifting $r$, if desired)} and {from~(\ref{constrW2})} we simply obtain $\sum_{j}W^0_{ij}=0$.
On the other hand, let us assume in the following that not all $b_i$ coincide, i.e., at least two of these are distinct, say $b_2\neq b_3$.
\subsubsection{Shearing case with all $b_i$ distinct: not permitted}
First, if all $b_i$ are distinct, {i.e., $\alpha_{max}=1$ in \eqref{A_decomposed}}, we can compute explicitly the required integral using partial fraction decomposition
\be
\int\d r\prod_k\frac{1}{r-b_k}=\sum_k\frac{\ln(r-b_k)}{\prod_{l\neq k}(b_k-b_l)} \qquad \mbox{($b_k$ all distinct)}.
\ee
It is thus clear that the singularity structure of the l.h.s. and the r.h.s. of (\ref{constrW2}) (for any $i$) cannot be the same unless $A^0=0$. Therefore, {from~(\ref{constrW2})} also $\sum_{j}(r-b_{(j)})^{-1}W^0_{ij}=0$. However, this condition implies that $b_i$ cannot be all distinct, so that this case is in fact not permitted (unless $W^0_{ij}=0$, so that $W_{ij}=0$ and therefore also $\Phi_{ij}=0$, contrary to our assumptions here -- however, we will see in sections~\ref{subsec_nondeg_Phi3} and \ref{subsec_nondeg_Phi4} below that the case with all distinct $b_i$ is ruled out also for $W^0_{ij}=0$).
\subsubsection{Shearing case with at least one $b_i$ repeated}
\label{subsec_onerep}
Let us now consider the case when at least one $b_i$ is repeated, say $b_2$, with multiplicity $1<\alpha_2<n-2$, and there exists $b_3\neq b_2$ (this is not a restriction since we are excluding here the \RT case, which corresponds to $\alpha_2=n-2$).
Using partial fraction decomposition {similarly as in}~\eqref{A_decomposed}, the left and right hand sides {of (\ref{constrW2}) take} the form (for $i=3$)
\bea
\mbox{lhs}&=&
-2A^0\left[\frac{p_{\alpha_2}}{(r-b_2)^{\alpha_2}}+(\mbox{terms with lower order poles at } b_2 \mbox{ and with no poles at }b_2)\right],\nonumber\\
\mbox{rhs}&=&
-2A^0\frac{p_{\alpha_2}\alpha_2}{(\alpha_2-1)(r-b_2)^{\alpha_2}} +(\mbox{terms with lower order poles at } b_2 \mbox{ and with no poles at }b_2) , \nonumber
\eea
respectively, where $p_{\alpha_2}\not= 0$. Comparing the highest order terms in $1/(r-b_2)$ we thus get
\be
A^0=0, \qquad \sum_{j}\frac{W^0_{ij}}{r-b_{(j)}}=0 .
\ee
In particular, it follows from (\ref{nondeg_Weyl_simplif}) that
\be
\Phi_i=0 ,
\ee
so that the Weyl type is II(abd), and \eqref{Wij_nondeg} thus gives
\be
W_{ij}=\frac{W^0_{ij}}{(r-b_i)(r-b_j)} ,
\label{Wij}
\ee
with $W^0_{ij}=W^0_{ji}$ and $W^0_{(i)(i)}=0$.
Now, because of (\ref{Wij}) the condition $\sum_jW_{ij}=0$ constraints the possible multiplicities of $b_i$ (i.e., of the eigenvalues of $\rho_{ij}$). First, if $b_i$ are all distinct, we immediately get $W_{ij}^0=0$, {as already discussed above}. Similarly, recalling $W_{ij}^0=W_{ji}^0$ it is easy to see that for $W_{ij}^0\neq0$ the structures $\{a,a,c_1,\ldots,c_{n-4}\}$ and
$\{a,a,a,c_1,\ldots,c_{n-5}\}$ are also forbidden.\footnote{To see that
$\{a,a,a,c_2,\ldots,c_{n-5}\}$ cannot occur, assume $b_2=b_3=b_4$ and all remaining eigenvalues are single. Then $\sum_jW_{ij}=0$ with (\ref{Wij}) gives
$W^0_{i2}+W^0_{i3}+W^0_{i4}=0$ and $W^0_{i\mu}=0$ ($\mu\neq 2,3,4$). Since $W^0_{ij}=W^0_{ji}$, this implies that the only possible non-zero components are $W_{23}$, $W_{24}$ and $W_{34}$, with the conditions $W_{23}^0+W_{24}^0=0$, $W_{32}^0+W_{34}^0=0$ and $W_{42}^0+W_{43}^0=0$. However, this system admits only the solution $W_{23}^0=W_{24}^0=W_{34}^0=0$, so that $W_{ij}=0$.\label{foot_311}}
However, the structure
$\{a,a,b,b,c_1,\ldots,c_{n-6}\}$ is compatible with such constraints (take, e.g., $b_2=b_3$, $b_4=b_5$ and $W_{34}^0=-W_{35}^0=W_{25}^0=-W_{24}^0\neq0$, {and the remaining $W_{ij}$ equal to zero}).
To summarize, we have seen above that in type II Einstein spacetimes with a non-degenerate non-twisting mWAND $\bl$, one has $\Phia_{ij}=0$ and
\begin{enumerate}
\item if $\Phi_{ij}\neq0$, then $\bl$ must be shearfree and the corresponding spacetimes belong to the \RT class. {This includes, in particular, the result of \cite{OrtPraPra09b} for asymptotically flat type II vacuum spacetimes (restricted to non-twisting case)}\footnote{Ref.~\cite{OrtPraPra09b} used an expansion method. Instead of the condition $\Phi_{ij}\neq 0$, a condition on the asymptotic fall-off behaviour of the Weyl tensor was assumed there. In the present notation, that amounts to taking $W_{ij}^0=\Phi^{\{3\}}_{ijkl}=\Phi^{\{4\}}_{ijkl}=0$, which requires that $\Phi_{ij}\neq 0$ (otherwise the type would be III). The assumptions of \cite{OrtPraPra09b} in the non-twisting case were thus stronger than those used here (note indeed that \RT spacetimes with $W_{ij}^0\neq 0$ do exist \cite{PodOrt06}). {(The symbols $\Phi^{\{3\}}_{ijkl}$ and $\Phi^{\{4\}}_{ijkl}$ are defined in sections~\ref{subsec_nondeg_Phi3} and \ref{subsec_nondeg_Phi4} below.)}}
\item if $\Phi_{ij}=0$ (type II(abd)) the structure of eigenvalues of $\rho_{ij}$ can be more generic, however for $\Phi_{ij}=0\neq W_{ij}$ it must have the multiplicities $\{a,a,b,b,c_1,\ldots,c_{n-6}\}$ (or more special).
\end{enumerate}
Let us now discuss the remaining cases (in which, without loss of generality, we could assume $\Phi_{ij}=0=W_{ij}$ -- however this will actually not be used to prove the following results).
\subsection{Case $\Phi^{\{3\}}_{ijkl}\neq0$}
\label{subsec_nondeg_Phi3}
{Here we assume that a component of $\Phi_{ijkl}$ with precisely three distinct values of $i,j,k,l$ is non-vanishing,
i.e., for some $i\not=k$, we have $\Phi_{i(j)k(j)} \not=0$.} This case is possible {only for $n\ge6$} because we need at least three possible distinct values for $i,j,\ldots$ and because of the tracefree condition $\Phi_{ijik}=0$ with $k\neq j$ (however, {for $n=4,5$ necessarily $\Phi_{ij}\neq0$} and from the above discussion one already knows that \RT is the only possibility).
Eq.~\eqref{B4} gives
\be
\Phi_{(i)(j)(i)(k)}(\rho_{(j)}-\rho_{(k)})=0 \qquad (k\neq j) .\label{B4offdiag}
\ee
Without loss of generality, we can assume $\Phi_{2324}\neq0$, so that necessarily $b_3=b_4$.
Now, the off-diagonal ($i\not=k$) components of \eqref{Bi3} read
\be
-D \Phi_{i(j)k(j)} = \Phi_{i(j)k(j)} \rho_{(j)} + \Phi_{i(j)k(j)} \rho_{(k)}.
\ee
Taking into account $\rho_k = (r-b_k)^{-1}$, we arrive at
\be
\Phi_{i(j)k(j)} = \frac{\Phi^{0}_{i(j)k(j)}}{(r-b_{(j)})(r-b_{(k)})} \qquad (i \not= k). \label{Phi_ijkj}
\ee
Then,
the condition $\Phi_{ijkj}=0$, $k\not=i$, implies that at least another component $\Phi_{(i)3(i)4}$ must be non-zero, say $\Phi_{5354}\neq0$, and that $b_5=b_2$. The eigenvalue structure must therefore again be
$\{a,a,b,b,c_1,\ldots,c_{n-6}\}$ (or more special).
\subsection{Case $\Phi^{\{4\}}_{ijkl}\neq0$}
\label{subsec_nondeg_Phi4}
Here we assume that there exists a non-vanishing component of $\Phi_{ijkl}$ with all values of $i, j, k, l$ being distinct. This case is possible only for $n\ge6$.
Similarly as in the previous case, eq.~\eqref{Bi3} implies
\be
\Phi_{ijkl} = \frac{\Phi^{0}_{ijkl}}{(r-b_{(k)})(r-b_{(l)})}, \qquad {\mathrm{\ (for \ }} i,j,k,l {\mathrm{\ all \ distinct}}).
\ee
However, since $\Phi_{ijkl}=\Phi_{klij}$ we obtain $b_i=b_k$ and $b_j=b_l$ (or $b_i=b_l$ and $b_j=b_k$) and the eigenvalue structure is again
$\{a,a,b,b,c_1,\ldots,c_{n-6}\}$ (or more special).
\subsection{Summary}
No further cases are possible, since $\Phi^{\{4\}}_{ijkl}=\Phi^{\{3\}}_{ijkl}=W_{ij}=0$ implies the type III.
To summarize, {in sections~\ref{subsec_nondeg_Phi}--\ref{subsec_nondeg_Phi4}} we have thus shown that
\begin{prop}
\label{prop_non_deg2}
For type II Einstein spacetimes, {the existence of} a non-twisting, non-degenerate double WAND implies that the algebraic type is necessarily II(d), i.e. $\Phia_{ij}=0$, or more special, and
\begin{enumerate}
\item if $\Phi_{ij}\neq 0$ the spacetime is shearfree (Robinson-Trautman), i.e. the eigenvalue structure of $\rho_{ij}$ is
$\{a,\ldots,a\}$ {(with $a\neq0$)}. This is {the only possibility when} $n=4,5$.\footnote{Since type II(abd) coincides with type III in those dimensions.}
One has $\Phi_{ij}=A^0\delta_{ij}/(r-b_0)^{n-1}$, so that the type is D(bd)
(one can set $b_0=0$ by a shift of the affine parameter $r$).
\item if $\Phi_{ij}=0$ {(type II(abd))} the structure is
$\{a,a,b,b,c_1,\ldots,c_{n-6}\}$ (or more special; $a,b,c_\alpha\neq0$). In particular, in six dimensions this means
$\{a,a,b,b\}$ (see also section~\ref{sec_6D}).
\end{enumerate}
\end{prop}
This is nothing but a more detailed version of points (\ref{RT}) and (in the subcase $\det\rhob\neq0$) (\ref{Phi0}) of theorem~\ref{prop_GSHD} of section~\ref{sec_intro}, which are thus proven (the proof of (\ref{Phi0}) will be completed in section~\ref{sec_nontwist} for any value of rank($\rhob$), cf. remark~\ref{rem_Phi0}).
In particular, (for $n>5$) there are always at least two double eigenvalues.
In this sense, we will see that the situation is different in the degenerate case. {Note that spacetimes of point 1. (all explicitly known \cite{PodOrt06}) obey the optical constraint, whereas those of point 2. do not, in general. Examples of the latter in $n\ge7$ dimensions will be provided in appendix~\ref{app_violating}.}
\section{Non-twisting $\bl$: degenerate $\rhob$ ($0<m<n-2$) }
\label{sec_nontwist}
Let $m$ denote the rank of $\rho_{ij}$. The value $m=n-2$ corresponds to the previously considered non-degenerate case, while $m=0$ defines Kundt. Therefore here we shall restrict to $0<m<n-2$ .
We need to define two types of indices to distinguish between non-vanishing and vanishing eigenvalues, namely
\beqn
& & \rho_p=\frac{1}{r-b_p} \qquad (p,q,o,t=2,\ldots, m+1) , \\
& & \rho_z=0 \qquad (z,v,w,y=m+2,\ldots, n-1) . \label{rho_z_0}
\eeqn
Recall that (point (i) of theorem~\ref{prop_GSHD})
in the non-twisting case with $\Phi^A_{ij}\neq0$ one has $m=0,2$ and $\rho_{ij}=\mbox{diag}(a ,a ,0,\ldots,0)$ for any $n>4$, {so that this case does not require further investigation}.
In all remaining cases we thus have
\be
\Phi^A_{ij}=0 \qquad (m\neq 0,2).
\ee
{In the following we will give the $r$-dependence of the Weyl components, which is then used to constraint the possible forms of $\rhob$. In particular, we shall explore under what conditions all the eigenvalues of $\rhob$ {can} be distinct (which is not permitted in the non-degenerate case). We shall also discuss some special cases and construct explicit examples.}
\subsection{$\Phi_{ij}$ and $W_{ij}$ components}
\label{subsec_Phi_W_deg}
Proceeding similarly as in section~\ref{sec_nondeg} but {(thanks to (\ref{rho_z_0}))} with the additional equation $D\Phi_z=0$, one finds
\beqn
& & \Phi_{p}=\rho_p(A+\Phi^0_{(p)}) , \qquad \Phi_{z}=\Phi_{z}^0 , \\
& & W_{pq}=\frac{1}{(r-b_{(p)})(r-b_{(q)})}\left[2\int\d r A+(\Phi^0_{p}+\Phi^0_{q})r+ W^0_{pq}\right] , \\
& & W_{pz}=\frac{1}{r-b_{(p)}}(\Phi^0_zr+W_{pz}^0) , \qquad W_{zv}=W_{zv}^0 ,
\eeqn
where
\be
A=\left(\prod_o\frac{1}{r-b_o}\right)\left[-\sum_p\Phi^0_{p}\int\d r\prod_{q\neq p}(r-b_q)-\sum_z\Phi^0_z\int\d r\prod_{q}(r-b_q)+A^0\right] .
\label{A_degen}
\ee
Now, imposing $-2\Phi_z=\sum_pW_{zp}+\sum_vW_{zv}$ we obtain the constraints
\be
\sum_vW_{zv}^0=-(m+2)\Phi^0_z , \qquad \sum_p\frac{\Phi^0_zb_p+W^0_{pz}}{r-b_p}=0 .
\label{constr_Phi_z}
\ee
Next, imposing $-2\Phi_q=\sum_pW_{qp}+\sum_zW_{qz}$ gives
\be
-2(A+\Phi^0_{q})=\sum_{p\neq q}\frac{1}{r-b_{(p)}}\left[2\int\d r A+(\Phi^0_{q}+\Phi^0_{p})r+ W^0_{pq}\right]+\sum_z(\Phi^0_zr+W^0_{qz}) .
\label{constrW_deg}
\ee
By comparing the leading terms of the l.h.s. and of the r.h.s. for $r\to\infty$ one obtains
\be
m\Phi^0_{q}+\frac{1}{m+1}b_q\sum_z\Phi^0_z+\sum_zW^0_{qz}=\sum_p\Phi^0_{p}-\frac{1}{m+1}\sum_pb_p\sum_z\Phi^0_z .
\label{constrW_deg_leading}
\ee
\subsection{$\Phi^{\{3\}}_{ijkl}$ and $\Phi^{\{4\}}_{ijkl}$ components}
\label{subsec_Phi3_4_deg}
To complete the description of the Weyl tensor, we now give the general form of the $\Phi^{\{3\}}_{ijkl}$ and $\Phi^{\{4\}}_{ijkl}$ components (recall that these are non-zero only for $n\ge6$). In particular, we also discuss constraints following from the assumption that all non-vanishing eigenvalues of $\rho_{ij}$ are distinct (useful for later analysis).
\subsubsection{$\Phi^{\{3\}}_{ijkl}$ components}
\label{subsubPhi3deg}
Eq.~\eqref{B4} gives
\be
\Phi_{(o)(p)(o)(q)}(\rho_{(q)}-\rho_{(p)})=0=\Phi_{(z)(p)(z)(q)}(\rho_{(q)}-\rho_{(p)}) , \qquad \Phi_{(q)p(q)z}=0=\Phi_{(v)p(v)z} , \quad (p\neq q, \ v\neq z) .
\label{A12_deg_Phi3}
\ee
Non-vanishing components must satisfy the tracefree conditions (recall that $\Phi_{ij}$ is diagonal in the frame we are using)
\be
\Phi_{poqo}+\Phi_{pzqz}=0 , \qquad \Phi_{zpvp}+\Phi_{zwvw}=0 \qquad (p\neq q, \ v\neq z) .
\label{trace_deg_Phi3}
\ee
The $r$-dependence, following from~\eqref{Bi3}, is
\beqn
& & \Phi_{(p)z(p)v}=\frac{\Phi_{(p)z(p)v}^0}{r-b_{(p)}} \quad (v\neq z), \qquad \Phi_{(z)v(z)w}={\Phi_{(z)v(z)w}^0} \quad (v\neq w) , \nonumber \\
& & \Phi_{(p)q(p)o}=\frac{\Phi_{(p)q(p)(o)}^0}{(r-b_{(p)})(r-b_{(o)})} , \quad (q\neq o) \qquad \Phi_{(z)p(z)q}=\frac{\Phi_{(z)p(z)(q)}^0}{r-b_{(q)}} \quad (p\neq q) .
\label{r_deg_Phi3}
\eeqn
From~(\ref{trace_deg_Phi3}), the quantities in~(\ref{r_deg_Phi3}) must satisfy the constraints
\beqn
& & \Phi_{pzqz}^0=0 , \qquad \sum_o\frac{\Phi^0_{p(o)q(o)}}{r-b_{(o)}}=0 \qquad (p\neq q) , \label{r_deg_Phi3_constr_0}\\
& & \Phi^0_{zwvw}=0 , \qquad \sum_p\frac{\Phi_{z(p)v(p)}^0}{r-b_{(p)}}=0 \qquad (v\neq z) . \label{r_deg_Phi3_constr}
\eeqn
Note that, by the symmetries of the Weyl tensor, {from} the {third} equation of~(\ref{r_deg_Phi3}) it follows that
$\Phi_{(p)q(p)o}\neq0\Rightarrow b_q=b_o$ (for $q\neq o$), in agreement with~(\ref{A12_deg_Phi3}); using also the {second} equation of~\eqref{r_deg_Phi3_constr_0} we get
\be
\Phi^{\{3\}}_{(p)q(p)o}\neq0 \Rightarrow \rhob=\diag(a,a,b,b,c_1\dots) \qquad (a,b\neq0).
\label{Phi3_new}
\ee
Further, {the fourth} equation of~(\ref{r_deg_Phi3}) gives $\Phi_{(z)p(z)q}\neq0\Rightarrow b_q=b_p$ {(for $q\neq p$)}, i.e., the structure is $\{a,a,c_1\dots\}$ with $a\neq0$, in agreement with~(\ref{A12_deg_Phi3}). For $\Phi_{z(p)v(p)}\neq0$, {the second equation of}~\eqref{r_deg_Phi3_constr} also implies $\{a,a,c_1\dots\}$ with $a\neq0$.
We are interested, in particular, in determining what are the necessary conditions in order to have all the non-vanishing eigenvalues distinct. From the {above} observations it follows that one necessarily has $\Phi_{(p)q(p)o}=\Phi_{(z)p(z)q}=\Phi_{z(p)v(p)}=0$, therefore the only non-zero $\Phi^{\{3\}}_{ijkl}$ components can be the $\Phi_{z(w)v(w)}$, however with the constraint $\Phi^0_{zwvw}=0$ (eq.~(\ref{r_deg_Phi3_constr})).
It is easy to see that for this to be satisfied in a non-trivial way, indices $z,v\ldots$ must run at least over four values, i.e. there must be at least four zero eigenvalues of $\rho_{ij}$ (excluding Kundt, the spacetime must thus be at least seven-dimensional). In other words:
\begin{itemize}
\item all non-zero $\rho_{ij}$ are distinct and $m>n-6$ $\Rightarrow$ $\Phi^{\{3\}}_{ijkl}=0$.
\end{itemize}
\subsubsection{$\Phi^{\{4\}}_{ijkl}$ components}
\label{subsubPhi4deg}
From~\eqref{Bi3} and the symmetries of the Weyl tensor one finds
\beqn
& & \Phi_{pqot}=\frac{\Phi_{pq(o)(t)}^0}{(r-b_{(o)})(r-b_{(t)})} , \qquad \Phi_{pzqw}=\frac{\Phi_{pz(q)w}^0}{r-b_{(q)}} , \qquad \Phi_{zvwy}=\Phi_{zvwy}^0 , \\
& & \Phi_{pqoz}=0 , \qquad \Phi_{pqzw}=0, \qquad \Phi_{pzvw}=0 .
\label{r_deg_Phi4}
\eeqn
Similarly as in the case of $\Phi^{\{3\}}_{ijkl}$, we observe that
{if} $\Phi_{pqot}\neq0$ then $b_t=b_q$ and $b_o=b_p$ (or $b_t=b_p$ and $b_o=b_q$), i.e.,
\be
\Phi^{\{4\}}_{pqot}\neq0 \Rightarrow \rhob=\diag(a,a,b,b,c_1\dots) \qquad (a,b\neq0).
\label{Phi4_new}
\ee
{Additionally,} {if} $\Phi_{pzqw}\neq0$ then
$b_q=b_p$, {so that the structure is} { $\{a,a,c_1\dots\}$ with $a\neq0$}.
Having all the non-vanishing eigenvalues distinct thus requires $\Phi_{pqot}=0=\Phi_{pzqw}$ and the only non-zero components can be $\Phi_{zvwy}$. Therefore we conclude again that indices $z,v\ldots$ must run at least over four values unless $\Phi^{\{4\}}_{ijkl}=0$ (and, again, excluding Kundt, the spacetime must thus be at least seven-dimensional), i.e.,
\begin{itemize}
\item all non-zero $\rho_{ij}$ are distinct and $m>n-6$ $\Rightarrow$ $\Phi^{\{4\}}_{ijkl}=0$.
\end{itemize}
\subsection{Case $\Phi_{ij}=0$ (type II(abd))}
\label{subsec_Phi=0}
This case (non-trivial only for $n>5$) is obtained by setting
\be
A=-\Phi^0_p=K^0 , \qquad \Phi^0_z=0
\ee
{in the results obtained in~\ref{subsec_Phi_W_deg}.}
For the $W_{ij}$ components one thus has
\be
W_{pq}=\frac{W^0_{pq}}{(r-b_{(p)})(r-b_{(q)})} , \qquad W_{pz}=\frac{W_{(p)z}^0}{r-b_{(p)}} , \qquad W_{zv}=W_{zv}^0 ,\label{Phi=0_Wpq}
\ee
with
\be
\sum_vW_{zv}^0=0 , \qquad \sum_p\frac{W^0_{(p)z}}{r-b_{(p)}}=0 , \qquad \sum_zW^0_{qz}=0 , \qquad \sum_{p}\frac{W^0_{(p)q}}{r-b_{(p)}}=0 .
\label{Phi=0_constr}
\ee
In view of these constraints (and of the properties of $W_{ij}$) we can briefly comment on some special cases:
\begin{itemize}
\item $\Phi_{ij}=0$, $m=1$ $\Rightarrow$ $W_{pq}=0=W_{pz}$,
\item $\Phi_{ij}=0$, $m=2$ or $m=3$ $\Rightarrow$ $W_{pq}=0$,
\item $\Phi_{ij}=0$, $m=n-3$ $\Rightarrow$ $W_{zv}=0=W_{pz}$,
\item $\Phi_{ij}=0$, $m=n-4$ or $m=n-5$ $\Rightarrow$ $W_{zv}=0$.
\end{itemize}
For special values of $n$ and $m$ some of these can hold simultaneously, thus leading to $W_{ij}=0$. Recalling the trivial implications $m=0\Rightarrow W_{pq}=0=W_{pz}$ and $m=n-2\Rightarrow W_{zv}=0=W_{pz}$ (valid also for $\Phi_{ij}\neq 0$), we have in particular
\begin{itemize}
\item for $n=4$ and $m=0,1,2$, $\Phi_{ij}=0$ $\Rightarrow$ $W_{ij}=0$,
\item for $n=5$ and $m=0,1,2,3$, $\Phi_{ij}=0$ $\Rightarrow$ $W_{ij}=0$,
\item for $n=6$ and $m=1,3$, $\Phi_{ij}=0$ $\Rightarrow$ $W_{ij}=0$.
\end{itemize}
While the first two implications simply reproduce the known result that $\Phi_{ij}=0\Leftrightarrow W_{ij}=0$ for $n=4,5$ (for any permitted $m$), the last remark will be useful for later purposes.
Using \eqref{Phi=0_Wpq}, \eqref{Phi=0_constr} and a reasoning similar to that of section \ref{subsec_onerep} (in the paragraph after eq.~\eqref{Wij}, including footnote~\ref{foot_311}) one can also show that
\be
W_{pq}\neq0 \Rightarrow \rhob=\diag(a,a,b,b,c_1\dots) \qquad (a,b\neq0),
\label{W_new}
\ee
which will be useful in the following.
\subsubsection{No repeated non-zero eigenvalues}
\label{subsubsec_Phi=0_norepeated}
In case all the $b_p$ are distinct, {equations~(\ref{Phi=0_constr})} imply
\be
W^0_{pz}=0=W^0_{pq} ,
\ee
so that the only non-zero components of $W_{ij}$ are the $W_{zv}=W_{zv}^0$, with $\sum_vW_{zv}^0=0$. This implies that, in order to have $W_{ij}\neq0$, one needs that indices $z,v,\ldots$ run at least over four values, i.e., the cases $m=n-3$, $m=n-4$ and $m=n-5$ imply $W_{ij}=0$. Thus we have proven: $\Phi_{ij}=0$, all non-zero $\rho_{ij}$ are distinct and $m>n-6$ $\Rightarrow$ $W_{ij}=0$. {Together with the results of subsection~\ref{subsec_Phi3_4_deg}, we see that if such assumptions hold all boost weight zero components must actually vanish, so that we can conclude}
\begin{prop}
\label{prop_IIabd}
For type II(abd) ($\Phi_{ij}=0$) non-twisting Einstein spacetimes with (degenerate) $\rhob$ of rank $n-6<m<n-2$, the eigenvalue structure of $\rhob$ is
$\{a,a, c_1\ldots, 0,\ldots\}$ (with $a\neq0$) or more special {(i.e., at least two non-zero eigenvalues of $\rhob$ coincide)}.
\label{prop_Phi=0}
\end{prop}
The condition on $m$ means that either $m=n-3$ or $m=n-4$ or $m=n-5$. We will see {in section~\ref{subsec_n-3}} below that in the case $m=n-3$ the assumption $\Phi_{ij}=0$ can in fact be dropped {(since $\Phi_{ij}\neq0$ is not possible if all non-zero eigenvalues are distinct). For $m=n-4$ the eigenvalue structure is thus $\{a,a,0,0,c_1,\ldots,c_{n-6}\}$ and for $m=n-5$ it is $\{a,a,0,0,0,c_1,\ldots,c_{n-7}\}$ (while for any smaller $m$ there are, of course, $n-2-m\ge 4$ vanishing eigenvalues).}
Recall also that for $m=n-2$ (non-degenerate case) we had a similar {result, cf.~proposition~\ref{prop_non_deg2}.}
\subsection{The special case $m=n-3$}
\label{subsec_n-3}
For $n=4$ this case is not possible since it gives $m=1$ (see section~\ref{subsec_m=1}).
{In higher dimensions, the simplest spacetimes of this class are obtained by taking a Brinkmann warp {\cite{Brinkmann25} (see also \cite{OrtPraPra11})} of a type II Einstein spacetime possessing a non-degenerate mWAND with a single (spacelike) extra-dimension (cf. also \cite{OrtPraPra11}). They include, e.g., {static} black strings and} this case may thus be of special interest.
Since there is only one vanishing eigenvalue of $\rho_{ij}$, i.e., $z$ can take only one value, we have $W_{zv}=0$ and, from~(\ref{constr_Phi_z}),
\be
\Phi^0_z=0 , \qquad \sum_p\frac{W^0_{pz}}{r-b_p}=0 .
\label{constr_m=n-3}
\ee
Therefore (\ref{constrW_deg_leading}) reduces to
\be
(n-3)\Phi^0_{q}+W^0_{qz}=\sum_p\Phi^0_{p} .
\ee
From the results of subsection~\ref{subsec_Phi3_4_deg}, when $m=n-3$ the only possible non-zero components of $\Phi^{\{3\}}_{ijkl}$ can be $\Phi_{(p)q(p)o}$, with $q\neq o$ (in particular, $\Phi_{pzqz}=\Phi_{p(z)q(z)}=0$, see the first of \eqref{r_deg_Phi3_constr_0}), which satisfy
\be
\sum_o\frac{\Phi^0_{p(o)q(o)}}{r-b_{(o)}}=0 \qquad (p\neq q) ,
\ee
while the only possible non-zero components of $\Phi^{\{4\}}_{ijkl}$ are $\Phi_{pqot}$ ($\Phi_{pqoz}=0$, see the first of \eqref{r_deg_Phi4}) .
From sections \ref{subsubPhi3deg} and \ref{subsubPhi4deg} (see \eqref{Phi3_new}, \eqref{Phi4_new})
it thus follows that if $\Phi^{\{3\}}_{ijkl}\neq0$ ($\Leftrightarrow\Phi^{\{3\}}_{(p)q(p)o}\neq0$ here) or $\Phi^{\{4\}}_{ijkl}\neq0$ ($\Leftrightarrow\Phi^{\{4\}}_{pqot}\neq0$ here) then the eigenvalue structure is $\{a,a,b,b,c_1\dots\}$, where $a,b\neq0$. Moreover, it is easy to see that when $m=n-3$ both $\Phi^{\{3\}}_{ijkl}$ and $\Phi^{\{4\}}_{ijkl}$ can be non-zero only for $n\ge7$. This is of interest only in six dimensions ($\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$ identically for $n=4,5$) and can be summarized as
\begin{itemize}
\item $n=6$, $m=3$ $\Rightarrow$ $\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$.
\end{itemize}
If we additionally assume $\Phi_{ij}=0$ (then also $W_{zv}=0=W_{pz}$, see comments after \eqref{Phi=0_constr}), then some of the components $W_{pq}$, $\Phi^{\{3\}}_{p(o)q(o)}$ or $\Phi^{\{4\}}_{pqot}$ have to be non-vanishing (otherwise the spacetime would be of type III). Then using also results of section~\ref{subsec_Phi=0} {(eq.~\eqref{W_new})} it follows that type II spacetimes {with $m=n-3$ and $\Phi_{ij}=0$} {must have} the eigenvalue structure $\{a,a,b,b,c_1\dots c_{n-7},0\}$.
\subsubsection{No repeated non-zero eigenvalues}
\label{subsubsec_m=n-3_no_rep}
If all $b_p$ are distinct from the second of~(\ref{constr_m=n-3}) we obtain
\be
W^0_{pz}=0 ,
\ee
so that $W_{pz}=0$ and all the $\Phi^0_{q}$ coincide, namely,
\be
\Phi^0_{q}=\frac{1}{(n-3)}\sum_p\Phi^0_{p}\equiv f^0 .
\ee
Therefore
\be
A=-f^0+A^0\prod_p\frac{1}{r-b_p} ,
\ee
and one can basically proceed as in the non-degenerate case. Comparing the structures of the poles of the l.h.s. and r.h.s. of (\ref{constrW_deg}) one again arrives at
\be
A^0=0, \qquad W^0_{pq}=0 .
\ee
Combining this with the previous results we conclude $\Phi_q=0$, $\Phi_z=0$, $W_{qz}=0$, $W_{qp}=0$, $W_{zv}=0$, i.e.
\be
\Phi_{ij}=0=W_{ij} .
\ee
Since here $m=n-3>n-6$, we can use the results of subsection~\ref{subsec_Phi3_4_deg} to conclude that also $\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$, so that all b.w. 0 components of the Weyl tensor vanish. Thus we have proven: {\it all non-zero $\rho_{ij}$ are distinct and $m=n-3$ $\Rightarrow$ all b.w. 0 components vanish.}
In other words, {\it in the case $m=n-3$, the eigenvalue structure of $\rhob$ must be $\{a,a,c_1, \dots ,c_{n-5}, 0\}$
(or more special)}. If, additionally, $\Phi_{ij}=0$, there is in fact another repeated non-zero eigenvalue \cite{Wylleman_priv}\footnote{We thank Lode Wylleman for pointing this out.} {(see the comment just before section \ref{subsubsec_m=n-3_no_rep}), so that}
Proposition~\ref{prop_Phi=0} can accordingly be reformulated as (see also \cite{Wylleman_priv})
\begin{prop}
\label{prop_n-3_text}
In a type II Einstein spacetime of dimension $n>4$ with a non-twisting multiple WAND of rank $m=n-3$, the eigenvalue structure of $\rhob$ is $\{a,a,c_1,\ldots c_{n-5},0\}$ where $a,c_\alpha \ne 0$ (and necessarily $\Phi^A_{ij}=0$ if $n>5$). If, additionally, $\Phi_{ij}=0$, the structure becomes $\{a,a,b,b,c_1,\ldots c_{n-7},0\}$ where $a,b,c_\alpha \ne 0$.
\end{prop}
\begin{rem}
\label{rem_Phi0}
{Together with propositions~\ref{prop_non_deg2} and \ref{prop_Phi=0}, this shows that, for any possible value of $m$ ($0<m\le n-2$), the mWAND of type II(abd) (i.e., $\Phi_{ij}=0$) Einstein spacetimes has $\rhob$ of the form $\{a,a,b,b,c_1,\ldots c_{n-6}\}$ (where $a\ne 0$ if $m=n-5$ or $m=n-4$ and $a,b\ne 0$ if $m=n-3$ or $m=n-2$ ).} In particular, point~(\ref{Phi0}) of theorem~\ref{prop_GSHD} is thus now also proven.
\end{rem}
\begin{rem}
\label{rem_5d}
In five dimensions (for which always $\Phi_{ij}\neq0$), recalling also the results for the non-degenerate case {(proposition~\ref{prop_non_deg2})}, this enables us to list all the admitted eigenvalues structures, {for $m=3,2,1,0$, respectively}: $\rho_{ij}=\mbox{diag}(a,a,a)$,
$\rho_{ij}=\mbox{diag}(a,a,0)$,
$\rho_{ij}=\mbox{diag}(a,0,0)$,
or $\rho_{ij}=0$ (Kundt). See \cite{Ortaggioetal12} for more details and examples.
\end{rem}
\begin{rem}
\label{rem_6d}
In six dimensions, we conclude that the case $m=3$ requires $\rho_{ij}=\mbox{diag}(a,a,b,0)$. However, as shown above in this case one has $\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$ and additionally the implication $\Phi_{ij}=0$ $\Rightarrow$ $W_{ij}=0$ holds (see subsection~\ref{subsec_Phi=0}). Therefore one necessarily has $\Phi_{ij}\neq0$ when $n=6$ and $m=3$ (or all Weyl boost zero components would vanish). {See section~\ref{sec_6D} for more details.}
\end{rem}
\subsection{The special case $m=1$}
\label{subsec_m=1}
From a viewpoint complementary to that of the previous subsection, it is also interesting to analyze the case when only one eigenvalue is non-zero. Let us associate the index $p=2$ to this eigenvalue, while $z,v=3,\ldots,n-1$. Using the general equations presented above, one obtains (cf. also~\cite{PraPra08})
\be
\Phi_z=\Phi^0_z , \qquad \Phi_2=-\frac{1}{2}\sum_z\Phi^0_z, \qquad W_{2z}=\Phi^0_z , \qquad \sum_vW^0_{zv}=-3\Phi^0_z ,
\ee
while clearly here $W_{pq}=0$ (note that for $n=4$ one gets
$\Phi^0_z=0$
and $\Phi_{ij}=0=W_{ij}$, so this case does not occur). Therefore all components of $\Phi_{ij}$ and $W_{ij}$ are $r$-independent. Note that $\Phi_{ij}=0\Leftrightarrow \Phi^0_z=0$. When this happens the only non-zero components of $W_{ij}$ can be the $W^0_{zv}$, but with the constraint $\sum_vW^0_{zv}=0$. Therefore it is easy to see that $W_{ij}$ can be non-zero only if $z,v$ can take at least four values, i.e., $z,v=3,4,5,6,\ldots$, so that $n\ge 7$. To summarize \begin{itemize}
\item $n<7$, $m=1$, $\Phi_{ij}=0$ $\Rightarrow$ $W_{ij}=0$ .
\end{itemize}
This can be understood as a special instance of the results of subsection~\ref{subsec_Phi=0}.
This result is of interest in six dimensions, since one always has $\Phi_{ij}=0\Rightarrow W_{ij}=0$ in four and five dimensions. Namely, for $n=6$ the case $\{a ,0,0,0\}$ is forbidden if $\Phi_{ij}=0\neq W_{ij}$.
From the results of subsection~\ref{subsec_Phi3_4_deg}, when $m=1$ the only possible non-zero components of $\Phi^{\{3\}}_{ijkl}$ can be $\Phi_{(z)v(z)w}$ (with $v\neq w$), which satisfy
\be
\Phi^0_{zvzw}=0 \qquad (v\neq w) ,
\ee
while the only possible non-zero components of $\Phi^{\{4\}}_{ijkl}$ are $\Phi_{zvwy}$. It is easy to see that, since $m=1$, both sets of components vanish identically unless $n\ge7$, so that
\begin{itemize}
\item $n=6$, $m=1$ $\Rightarrow$ $\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$.
\end{itemize}
\begin{rem}
\label{rem_6d_m=1}
Since for $n=6$, $m=1$ one has $\Phi_{ij}=0$ $\Rightarrow$ $W_{ij}=0$ (see above), we conclude that in six dimensions spacetimes with $m=1$ require $\Phi_{ij}\neq 0$.
\end{rem}
\subsubsection{Examples}
\label{subsubsec_m=1}
For any $n\ge5$, simple examples of spacetimes with $m=1$ {(and thus $\Phi^A_{ij}=0$)} are given by:
\begin{enumerate}
\item dS$_3\times$S$_{n-3}$ (or AdS$_3\times$H$_{n-3}$): $n\ge5$, $\Phi_{ij}\neq 0$, $\Lambda\neq 0$, \label{dSxS_m=1}
\item as above but with a Brinkmann warp: $n\ge6$, $\Phi_{ij}\neq 0$, also $\Lambda=0$ is possible, \label{Brink_m=1}
\item Minkowski$_3\times$(Ricci-flat)$_{n-3}$: $n\ge7$, $\Phi_{ij}=0$, $\Lambda=0$. \label{MinkxRc_m=1}
\end{enumerate}
In the first case, if {we restrict to} $n\ge7$ the second factor space can be in fact {\em any} Einstein space (with Ricci scalar given by $2(n-3)K$). The metric of the first and third case can thus be given in a unified way as
\be
\d s^2 =\left(1+\frac{K}2ur\right)^{-2}(2\d u\d r+r^2\d x^2)+\d\Sigma^2 ,
\ee
where $\d\Sigma^2$ is the metric of an $(n-3)$-dimensional Einstein space with Ricci scalar $R_\Sigma=2(n-3)K$ and $K$ is a constant related to the $n$-dimensional Ricci scalar by $R=2nK$. A geodesic twistfree mWAND $\bl$ (degenerate but expanding along $x$) is
\be
{\ell}_a\d x^a=\d u .
\ee
Note that here $r$ is an affine parameter along $\bl$ only in the case $K=0$. {Such product spaces are of type D \cite{PraPraOrt07} and in fact any null vector field tangent to the three-dimensional Lorentzian factor is an mWAND \cite{GodRea09,PraPraOrt07}.}
Using these results metrics for the second case can also be constructed straightforwardly. {These will be also of type D \cite{OrtPraPra11}.}
\subsection{Case when all non-zero eigenvalues are distinct}
\label{subsec_alldistinct}
By combining the results of subsections~\ref{subsec_Phi=0} and \ref{subsec_Phi3_4_deg} we have
\begin{itemize}
\item all non-zero $\rho_{ij}$ are distinct, $m=n-4$ or $m=n-5$ and $\Phi_{ij}=0$ $\Rightarrow$ all b.w. 0 components vanish,
\item all non-zero $\rho_{ij}$ are distinct and $m=n-3$ $\Rightarrow$ all b.w. 0 components vanish.\footnote{By adding the results of \cite{Pravdaetal04} we can in fact conclude that in both cases the whole Weyl tensor must vanish.}
\end{itemize}
Therefore type II Einstein spacetimes for which all non-zero $\rho_{ij}$ are distinct require that at least one of the following holds ({obviously $m>0$ or we would simply have Kundt}):
\begin{enumerate}
\item $1\le m\le n-6$ (i.e., at least four eigenvalues of $\rho_{ij}$ vanish),
\item $\Phi_{ij}\neq 0$ and $1\le m\le n-4$ (i.e., at least two eigenvalues of $\rho_{ij}$ vanish).
\end{enumerate}
The first case can occur only for $n\ge7$: for $n=7$ one can have only $\rho_{ij}=\mbox{diag}(a ,0,0,0,0)$, for $n=8$ either $\rho_{ij}=\mbox{diag}(a ,0,0,0,0,0)$ or $\rho_{ij}=\mbox{diag}(a ,b ,0,0,0,0)$, etc.. The second case can occur for $n\ge5$: for $n=5$ one has simply $\rho_{ij}=\mbox{diag}(a ,0,0)$, for $n=6$ either $\rho_{ij}=\mbox{diag}(a ,0,0,0)$ or $\rho_{ij}=\mbox{diag}(a ,b ,0,0)$, for $n=7$ either $\rho_{ij}=\mbox{diag}(a ,0,0,0,0)$ or $\rho_{ij}=\mbox{diag}(a ,b ,0,0,0)$ or $\rho_{ij}=\mbox{diag}(a ,b ,c ,0,0,0)$, etc.. These results are summarized in table~\ref{tab_distinct_eigenval}. For such solutions {\em the optical constraint is clearly violated if $m>1$}. The {previously discussed} case $m=1$ has been included for completeness but it is of course ``trivial'' in this context since there is a single non-zero eigenvalue.
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|c|l|c|}
\hline Case & $n$ & Possible $m$ & Examples \\\hline
$\Phi_{ij}\not=0$ & 5 & 1 & \\
& 6 & 1,2 & \\
& 7 & 1,2,3 & \\
& 8 & 1,2,3,4 & \\
& \vdots & & \\
& $n$ & 1,2,3\ldots, $n-4$ & (A)dS$_{m+2}\times$S$_{n-2-m}$(H$_{n-2-m}$) \\\hline
$\Phi_{ij} =0$ & 7 & 1 & \\
& 8 & 1,2 & \\
& \vdots & & \\
& $n$ & 1,2,3,\ldots, $n-6$ & Mink$_{m+2}\times$(Ricci-flat)$_{n-2-m}$ \\
\hline
\end{tabular}
\caption{Non-twisting Einstein spacetimes for which there are no repeated non-zero eigenvalues of $\rho_{ij}$: permitted values of the rank $m$ of $\rho_{ij}$ in various dimensions {(necessarily $1\le m \le n-4$ thanks to propositions~\ref{prop_non_deg2} and \ref{prop_n-3_text})}. {As discussed in the text, different examples can also be generated using Brinkmann's warp, or by replacing the S$_{n-2-m}$(H$_{n-2-m}$) factor by a more general Einstein space. {Note that here solutions with $m>1$ violate the optical constraint.\label{tab_distinct_eigenval}}
}}
\end{center}
\end{table}
\subsubsection{Examples}
\label{subsubsec_examples_distinct}
The case $n=4$ is not possible. Examples with $m=1$ have been already presented {in subsection~\ref{subsec_m=1} and will not be discussed again here}. For $m\ge2$ the $n=5$ case is also forbidden {(cf. Remark~\ref{rem_5d} and \cite{Ortaggioetal12})}.
For $n=6$ the only possibility is $m=2$ (with $\Phi_{ij}\neq0$) and one can construct explicit Einstein spacetimes by taking dS$_4\times$S$^2$ or AdS$_4\times$H$^2$. These can then be extended to any dimension $n\ge7$ by a simple Brinkmann warp (and if one starts from dS$_4\times$S$^2$ the cosmological constant of the resulting spacetime can be arbitrary, cf.~\cite{OrtPraPra11}), so to have explicit examples of the type $\rho_{ij}=\mbox{diag}(a ,b ,0,\ldots,0)$ (i.e., with $m=2$) for any $n>5$.
However, according to the previous comments (see also table~\ref{tab_distinct_eigenval}), for $n=7$ one can have $m=2,3$, for $n=8$ it is possible $m=2,3,4$ and so on: in general, and for any $n>5$, $2\le m\le n-4$ explicit metrics can be constructed similarly as in six dimensions by taking dS$_{m+2}\times$S$_{n-2-m}$ or AdS$_{m+2}\times$H$_{n-2-m}$ (where the two factor spaces must have {Ricci scalars} given by $(m+2)(m+1)K$ and $(n-2-m)(m+1)K$, respectively), and appropriate hypersurface orthogonal null congruences living in the (A)dS$_{m+2}$ factor {(explicit examples are given below)}. These have necessarily $\Phi_{ij}\neq0$ and $\Lambda\neq0$. Using Brinkmann's warp one can also generate Ricci-flat solutions (again with $\Phi_{ij}\neq0$) for which, however, the stronger restriction $n>6$, $2\le m<n-4$ holds (at least if this method is used). Products of Ricci-flat spaces are possible if each factor is at least four-dimensional (therefore there are always at least four vanishing eigenvalues) and the Lorentzian factor is flat and fall into the $\Phi_{ij}=0$ class, with $\Lambda=0$ (these metrics belong to the type D \pp waves mentioned in \cite{OrtPraPra09}).
Summary:
\begin{enumerate}
\item dS$_{m+2}\times$S$_{n-2-m}$ (or AdS$_{m+2}\times$H$_{n-2-m}$)\footnote{Note that if $n-2-m\ge4$ the second factor space can be in fact any Einstein space with Ricci scalar given by $(n-2-m)(m+1)K$.}: $n>5$, $2\le m\le n-4$, $\Phi_{ij}\neq 0$, $\Lambda\neq 0$, \label{dSxS}
\item as above + Brinkmann, also $\Lambda=0$: $n>6$, $2\le m\le n-5$, $\Phi_{ij}\neq 0$, \label{Brink}
\item Minkowski$_{m+2}\times$(Ricci-flat)$_{n-2-m}$: $n>7$, $2\le m\le n-6$, $\Phi_{ij}=0$, $\Lambda=0$. \label{MinkxRc}
\end{enumerate}
Note that all the above spacetimes {also} belong to the Kundt class, although w.r.t. a null congruence different from the one which is of interest for our discussion (which is expanding and shearing). {In fact they all admit $\infty^m$ mWANDs:} similarly as in section~\ref{subsubsec_m=1}, they are of type D (with $\Phi^A_{ij}=0$) and any null vector field tangent to the Lorentzian factor is an mWAND -- so one can always also find an mWAND with an optical matrix having all non-zero eigenvalues identical. {However, this mWAND ``degeneracy'' is not the generic situation} (see appendix~\ref{app_violating} for an example of a class of type D spacetimes which admit (only) two double WANDs, both violating the optical constraint).
Explicitly, for cases~\ref{dSxS} and \ref{MinkxRc} one can take, e.g., the metric
\be
\d s^2 =\Omega^2(2\d u\d r+P_q^2\d x_q^2)+\d\Sigma^2 ,
\label{ex_distinct}
\ee
where $q=2,\dots,m+1$, $\d\Sigma^2$ is the metric of an $(n-2-m)$-dimensional Einstein space with Ricci scalar $R_\Sigma=(n-2-m)(m+1)K$ and
\be
P_q=r-b_q, \qquad \Omega^{-1}=1+\frac{K}{4}\left[2r\left(u-\frac{1}{2}P_qx_q^2\right)+P_q^2x_q^2\right ] ,
\ee
{in which $b_q$ and $K$ are constants.}
The first factor space in~(\ref{ex_distinct}) is a space of constant curvature with Ricci scalar given by $(m+2)(m+1)K$, so that the Ricci scalar of the full spacetime is
\be
R=n(m+1)K .
\ee
The geodesic, twistfree mWAND $\bl$ and the remaining frame vectors are given by
\be
{\ell}_a\d x^a=\d u , \qquad n_a\d x^a=\Omega^2\d r, \qquad m_{(q)a}\d x^a=\Omega P_{(q)}\d x_{(q)} ,
\ee
while vectors $m_{(v)a}$ will depend on the specific form of $\d\Sigma^2$. The optical matrix $\rho_{ij}$ is given by
\be
\rho_{pq}=\delta_{(p)q}\frac{(\Omega P_{(p)})_{,r}}{\Omega^{3}P_{(p)}} , \qquad \rho_{pv}=0=\rho_{vz} .
\ee
In the case $K=0$ (i.e., $\Lambda=0$) one thus has $\rho_{pq}=\delta_{(p)q}/(r-b_{(p)})$. Note, however, that $r$ is {\em not} an affine parameter when $K\neq 0$ (one has $\bl=\Omega^{-2}\pa_r$).
For case~\ref{Brink} one can use the method illustrated in detail in \cite{OrtPraPra11}.
\section{Counterexamples}
\label{sec_counter}
In \cite{Ortaggioetal12}, a counterexample to the converse of the five-dimensional ``shear-free'' part of the GS theorem was presented, thus demonstrating explicitly that the condition that the optical matrix $\rhob$ admits a canonical form compatible with a geodetic mWAND is not sufficient for the null geodetic being an mWAND. Similarly, also our results above give conditions that are (necessary but) not sufficient. In order to demonstrate that, here we present a few ``counterexamples'', i.e., certain Einstein spacetimes that admit a non-twisting null (thus geodesic) vector field $\bl$ with $\rhob$ taking one of the permitted ``canonical forms'' (cf. theorem~\ref{prop_GSHD} and propositions~\ref{prop_non_deg2}, \ref{prop_IIabd} and \ref{prop_n-3_text}) and yet with $\bl$ not being an mWAND. Note that such counterexamples will necessarily be shearing: a twistfree shearfree null vector field is automatically an mWAND \cite{PodOrt06,OrtPraPra07}.
\subsection{Non-degenerate $\rhob=\{a,a,b,b,c_1, \ldots, c_{n-6} \}$}
In the non-degenerate case a ``counterexample'' to the canonical form $\{a,a,b,b,c_1, \ldots, c_{n-6} \}$ (with $a,b,c_\alpha \ne 0$ and $a\neq b$ -- see points (ii) and (iii) of theorem~\ref{prop_GSHD} and proposition~\ref{prop_non_deg2}) is given by the following six-dimensional Ricci-flat spacetime, which belongs to a class of metrics considered by Robinson (as described in \cite{Trautman02b}),
\be
\d s^2=2\d u\d r+2\cosh^2r\d w\d\bar w+2\sin^2r\d\zeta\d\bar\zeta .
\label{robinson}
\ee
Here the hypersuface orthogonal null vector field $\ell_a\d x^a=\d u$ is {\em not} an mWAND (not even a single one) and yet the corresponding
$\rhob$ has the eigenvalue structure $\{a,a,b,b\}$ (with $a\neq0\neq b$, $a\neq b$). Associated to $\bl$ there is also an optical structure \cite{Trautman02b}.
Note, however, that spacetime~\eqref{robinson} is a type N \pp wave (albeit in {so-called Rosen coordinates, see, e.g., metric~(39) of \cite{Brinkmann25}}), with a covariantly constant mWAND given by $n_a\d x^a=\d r$.
\subsection{Case $n-6<m<n-2$}
Direct products of~\eqref{robinson} with flat extra dimensions clearly provide non-WANDs with eigenvalues $\{a,a,b,b,0,\ldots,0 \}$. In particular, for $n=7$ one obtains $\{a,a,b,b,0\}$, which is thus a counterexample to the $\rhob$-form of Proposition~\ref{prop_n-3_text}. For $n=8,9$ this is also a counterexample to the $\rhob$-form of Proposition~\ref{prop_IIabd}. Another counterexample to Proposition~\ref{prop_n-3_text} can be obtained for $n=6$ by taking the direct product of a vacuum black ring \cite{EmpRea02prl} with a flat dimension: in the coordinates of \cite{ElvEmp03}, the hypersurface orthogonal null vector field (also considered in 5d in \cite{Taghavi-Chabert11} for different purposes)
\be
\ell_a\d x^a = \frac{\sqrt{F(y)}}{G(y)}\d y+\d\psi ,
\ee
corresponds to $\rhob$ of the form $\{a,a,b,0\}$ and is not a WAND.
Alternatively, the same form $\{a,a,b,0\}$ can be obtained by taking the direct product of a 5d vacuum static KK bubble \cite{GodRea09} with a flat dimension (with a non-WAND $\ell_a\d x^a=\d t+V(r)^{-1/2}\d r$ -- this 6d spacetime is of type D but the mWANDs are different from $\bl$ \cite{OrtPraPra11}).
\subsection{Case when all non-zero eigenvalues are distinct}
Examples of spacetimes with a non-twisting, non-WAND $\bl$ having all non-zero eigenvalues distinct can be constructed as a direct product of a 4d \pp wave with flat space, i.e.,
\be
\d s^2=2\d u\d r+\cosh^2r\d x^2+\sin^2r\d y^2+\d z_i\d z^i .
\label{4d_pp}
\ee
Here, $\ell_a\d x^a=\d u$ has a $\rhob$ with eigenvalue structure $\{a,b,0,\ldots,0\}$ (with $a\neq0\neq b$, $a\neq b$). This is the form of $\rhob$ discussed in section~\ref{subsec_alldistinct} (see also table~\ref{tab_6D} for the $n=6$ case). Similarly as the spacetime~\eqref{robinson}, also \eqref{4d_pp} is a type N \pp wave with mWAND $n_a\d x^a=\d r$.
\subsection{Case $m=1$}
Examples of non-twisting null congruences with $m=1$ (the case discussed in section~\ref{subsec_m=1}) that are not multiple WANDs can be obtained from the (Ricci-flat) Newman-Tamburino lift considered in \cite{Ortaggioetal12}, i.e.,
\be
{\rm d}s^2 = r^2 {\rm d} x^2 + x^2 {\rm d} y^2 - \frac{4 r}{x } {\rm d} u {\rm d} x - 2 {\rm d} u {\rm d} r + x^{-2} \left[c + \ln (r^2 x^4)\right] {\rm d} u^2+\d z_i\d z^i , \label{NTmetric}
\ee
where $c$ is a constant. In this case $\ell_a\d x^a = {\rm d} u$ is a {\em single} WAND with optical matrix of the form $\{a,0,\ldots,0\}$ (with $a\neq 0$).
\section{Existence of totally geodesic null two-surfaces}
\label{sec_totgeod}
Theorem~\ref{prop_GSHD} comprises the main results proven in the previous sections. In particular, it implies that there always exists at least a pair of repeated (possibly vanishing) eigenvalues and that (recall that $n\ge 6$) there are in fact at least two such pairs when, e.g., $\det\rhob\neq0$ (see also proposition~\ref{prop_non_deg2}) or $\Phi_{ij}=0$ (see also propositions~\ref{prop_IIabd} and \ref{prop_n-3_text}). The following proposition (see also the definitions in appendix~\ref{app_shearfree}, in particular eq.~\eqref{D}) elucidates the geometrical meaning of this fact (and provides a connection with the standard 4d version of the Goldberg-Sachs theorem, cf. appendix~\ref{app_GS_4D}).
\begin{prop}
In an algebraically special Einstein spacetime of dimension $n\ge 6$ with a non-twisting {mWAND} $\bl$, using a parallelly transported eigenframe of $\rhob$
\begin{enumerate}
\item if the eigenvalue structure of $\rhob$ is $\{a,a,c_1,\ldots, c_{n-5},0 \}$ with $c_\alpha\neq a\neq 0$ then the totally null distribution ${\cal D}_{23}$ is integrable with totally geodesic integral surfaces,
\item if the eigenvalue structure of $\rhob$ is $\{0,0,c_1,\ldots, c_{n-4}\}$ with $c_\alpha\neq 0$ then the totally null distribution ${\cal D}_{23}$ is integrable with totally geodesic integral surfaces,
\item if the eigenvalue structure of $\rhob$ is $\{a,a,b,b,c_1, \ldots, c_{n-6} \}$ with $b\neq a$ and $a\neq c_\alpha\neq b$ then the totally null distributions ${\cal D}_{23}$ and ${\cal D}_{45}$ are integrable with totally geodesic integral surfaces (either $a$ or $b$ can be zero).
\end{enumerate}
\label{prop_integrab}
\end{prop}
\begin{proof}
The proof is similar in all cases and relies on the use of the Ricci identity (11k,\cite{OrtPraPra07}) (equivalent to the NP equation (A4,\cite{Durkeeetal10})), which for algebraically special Einstein spacetimes reduces to
\be
\delta_{[j|} \rho_{i|k]} = L_{1[j|} \rho_{i|k]}+ \tau_i \rho_{[jk]}+\rho_{il}\M{l}{[j}{k]}
+ \rho_{l[j|}\M{l}{i|}{k]} , \label{11k}
\ee
where $\delta_i=m_{(i)}^a\nabla_a$ (recall also the definitions in \eqref{L1i_M}). Since here $\bl$ is twistfree we have $\rho_{[jk]}=0$ in the above equation. Assume the form of $\rhob$ specified in 1., i.e., $\rho_{22}=\rho_{33}=a\neq 0$ (and all remaining eigenvalues different from $a$) and consider \eqref{11k} for $i,j,k=23\hat k/32\hat k$ (with $\hat k\neq 2,3$). This gives $\M{2}{\hat k}{3}=0=\M{3}{\hat k}{2}$. Then, \eqref{11k} with $i,j,k=22\hat k/33\hat k$ gives $\M{2}{\hat k}{2}=\M{3}{\hat k}{3}$, which completes the proof of point 1. (see eqs.~\eqref{2totgeod_real}). The proof of point 2. is identical. The proof of point 3. is analogous, after extending the reasoning to the index pair (45); in this case, however, either $a$ or $b$ can be zero, provided all $c_\alpha\neq 0$.
\end{proof}
\begin{rem}
In the case there are further pairs of repeated eigenvalues, the corresponding two-spaces will also define integrable totally geodesic distributions, provided these pairs are not repeated. In particular in even dimensions, if there are $(n-2)/2$ (non-repeated) pairs of equal eigenvalues there will be $(n-2)/2$ such totally geodesic two-surfaces.
\end{rem}
\begin{rem}
Point 1. of the above proposition includes, in particular, all non-twisting but expanding type III/N Einstein spacetimes (for which $c_{\alpha}=0$ \cite{Pravdaetal04}) and type II non-twisting non-Kundt spacetimes with $\Phi^A_{ij} \ne 0$ (point \eqref{PhiA} of theorem~\ref{prop_GSHD}).
Point 3. is relevant to the case $\det\rhob\neq0$ with $\Phi_{ij}=0$ (proposition~\ref{prop_non_deg2} and point \eqref{Phi0} of theorem~\ref{prop_GSHD}), which can only be of type II(abd).
\end{rem}
\begin{rem}
The proposition does not explicitly include the $n=5$ case. However, from Proposition~4 of \cite{Ortaggioetal12} one can easily obtain the corresponding results (also in the presence of twist). In particular, in the non-twisting case ${\cal D}_{23}$ is totally geodesic when the eigenvalue structure of $\rhob$ is either $\{a,a,0\}$ or $\{0,0,a\}$ (with $a\neq 0$ in both cases). The same is true in the case $\{a,a,a\}\neq0$, although this does not follow from (11k,\cite{OrtPraPra07}) (instead, one can easily adapt the argument of footnote~10 of \cite{Ortaggioetal12}).
\end{rem}
\begin{rem}
Recall that in 4d the Goldberg-Sachs theorem implies that $\bl$ is geodesic and the eigenvalue structure of $\rho_{(ij)}$ is $\{a,a\}$, which is {\em equivalent} to ${\cal D}_{23}$ being integrable (and thus automatically totally geodesic, in 4d, see appendix~\ref{app_GS_4D}).
\end{rem}
\section{Type II spacetimes with a non-twisting multiple WAND in six dimensions}
\label{sec_6D}
As already mentioned, five-dimensional algebraically special Einstein spacetimes have been studied in detail in \cite{Ortaggioetal12} and, for $n=5$, the results obtained above are also contained in that reference. The next lower dimension to consider is thus $n=6$ and in this section our general results are specialized to this case. This is of interest also because of a qualitative difference between $n<6$ and $n\ge 6$ dimensions: for $n<6$ type II spacetime necessarily have $\Phi_{ij}\neq 0$ (since type II coincides with type II(c) {for $n=4,5$}), while this is not so in higher dimensions.
\subsection{Permitted forms of $\rhob$}
\label{subsec_permitted_6D}
One has to consider the various cases $m=4,3,2,1$ ($m=0$ is Kundt).
First, recall that for $m\neq 2$ point \eqref{PhiA} of theorem~\ref{prop_GSHD} implies that $\Phi^A_{ij}=0$. Let us thus first discuss this case.
Now, for $m=4$ proposition~\ref{prop_non_deg2} tells us that if $\Phi_{ij}\neq 0$ we are in the \RT class, i.e. $\rhob=\mbox{diag}(a,a,a,a)\neq 0$ (in which case $\Phi_{ij}\propto \delta_{ij}$), otherwise we have $\rhob=\mbox{diag}(a,a,b,b)$ (with $a,b\neq 0$).
For $m=3$ we learn from Proposition~\ref{prop_n-3_text} that $\rhob=\mbox{diag}(a,a,b,0)$, while from section~\ref{subsec_n-3} we know that {$\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$ and $\Phi_{ij}\neq 0$ (Remark~\ref{rem_6d}).} Substituting the form of $\rhob$ into (\ref{eqn:algI})--\eqref{eqn:rhoC} {{and}} (\ref{eqn:algII}) further reveals that $\Phi_{ij}=\diag(\alpha,-\alpha,0,0)\neq 0$.
For $m=1$ one obviously has $\rhob=\mbox{diag}(a,0,0,0)\neq 0$. From section~\ref{subsec_m=1} we know that { $\Phi^{\{3\}}_{ijkl}=0=\Phi^{\{4\}}_{ijkl}$ and $\Phi_{ij}\neq 0$ (Remark~\ref{rem_6d_m=1}).}
Finally, for $m=2$ we can have two possibilities. If $\Phi^A_{ij}\neq0$ point \eqref{RT} of theorem~\ref{prop_GSHD}
shows that $\rhob=\mbox{diag}(a,a,0,0)\neq 0$. On the other hand, if $\Phi^A_{ij}=0=\Phi^S_{ij}$ we still get $\rhob=\mbox{diag}(a,a,0,0)\neq 0$ from Proposition~\ref{prop_IIabd}, whereas if $\Phi^A_{ij}=0\neq\Phi^S_{ij}$ the more general form $\rhob=\mbox{diag}(a,b,0,0)$ is permitted (with $a\neq 0\neq b$).
We have checked in the various permitted cases that the constraints (\ref{eqn:algI})--\eqref{eqn:rhoC} and \eqref{alg_no_Phi4}--\eqref{eqn:algII_N}
allow for nontrivial type II Weyl tensor. {These results are summarized in table~\ref{tab_6D}, together with a few examples. Examples of spacetimes with distinct non-zero eigenvalues follow from the general discussion of section~\ref{subsubsec_examples_distinct}, which need not be repeated here.}
\begin{table}[t]
\begin{center}
\begin{tabular}{|c|l|c|c|c|l|c|}
\hline Case & $m$ & Possible form of $\rhob$ & O.C. & O.S. & I.T.G. & Examples \\\hline
$\Phi_{ij} \not=0$ & 4 & $\diag(a,a,a,a)$ & $\surd$ & $\surd *$ & ${\cal D}_{23}*$, ${\cal D}_{45}*$ & RT \\
$ $ & 3 & $\diag(a,a,b,0)$ & if $b=a$ & X & ${\cal D}_{23}$ (if $b\neq a$) & ?($\dag$) \\
$ $ & 2 \ ($\Phi^A_{ij}=0$) & $\diag(a,b,0,0)$ & if $b=a$ & & ${\cal D}_{45}$ (${\cal D}_{23}$ if $b=a$) & {dS$_4\times$S$^2$} \\
$ $ & 2 \ ($\Phi^A_{ij}\neq 0$) & $\diag(a,a,0, 0)$ & $\surd$ & $\surd$ & ${\cal D}_{23}$, ${\cal D}_{45}$ & ? \\
$ $ & 1 & $\diag(a,0,0,0)$ & $\surd$ & X & & {dS$_3\times$S$^3$} \\
$ $ & 0 & $\diag(0,0,0,0)$ & $\surd$ & & & Kundt \\\hline
$\Phi_{ij} =0 $ & 4 & $\diag(a,a,b,b)$ & if $b=a$ & $\surd$ & ${\cal D}_{23}$, ${\cal D}_{45}$ (if $b\neq a$)& ?($\dag$) \\
$ $ & 3 & X & & & & X \\
$ $ & 2 & $\diag(a,a,0,0)$ & $\surd$ & $\surd$ & ${\cal D}_{23}$, ${\cal D}_{45}$ & ? \\
$ $ & 1 & X & & & & X \\
$ $ & 0 & $\diag(0,0,0,0)$ & $\surd$ & & & Kundt \\
\hline
\end{tabular}
\caption{Permitted forms of the optical matrix associated with a non-twisting mWAND in a six-dimensional type II spacetime in the two cases $\Phi_{ij} \not=0$ and $\Phi_{ij}=0$ (subtype II(abd)). Recall that $\Phi^A_{ij}\neq 0$ is possible only for $m=2$. A question mark indicates subcases that are in principle permitted but for which no examples are known (so we do not claim that these classes are non-empty), while X corresponds to forbidden subcases.
Although we are not aware of examples with $m=3$ and $\Phi_{ij} \not=0(=\Phi^A_{ij})$ having the most general structure, special solutions with $a=b$ (e.g., a static black string) can be obtained by warping once a type II 5d Einstein metric. {Similarly, generic $m=4$ solutions with $\Phi_{ij}=0$ are not known, but in the special case $a=b$ they} exist within the \RT family \cite{PodOrt06} (the ``$\mu=0$'' subcase). {These special examples are denoted by a $\dag$ in the table.} In the O.C. column we have indicated if/under what conditions $\rhob$ satisfies the optical constraint. In the O.S. column whether ${\cal D}=\mbox{Span}\{\mbb{2}+i\mbb{3},\mbb{4}+i\mbb{5},\bl\}$ defines an optical structure and in the I.T.G. column we listed integrable complex two-dimensional distribution that admit totally geodesic integral surfaces (cf. section~\ref{subsec_integrability} and appendix~\ref{app_shearfree}). The symbol $*$ means that the corresponding statement is true in a special subcase specified in the text (see section~\ref{subsec_integrability}).
\label{tab_6D}}
\end{center}
\end{table}
\subsection{Additional examples}
In addition to the examples mentioned in table~\ref{tab_6D} and in section~\ref{subsubsec_examples_distinct}, here we give two more solutions with $m=1$ in six dimensions (and, necessarily, $\Phi_{ij}\neq0$).
\subsubsection{Example with $\Lambda\neq0$}
We can use, e.g., AdS$_4\times$H$^2$ in the form
\be
\d s^2=\frac{3\beta^2}{z^2}(-\d t^2+\d\rho^2+\rho^2\d\phi^2+\d z^2)+\frac{\beta^2}{x^2}(\d x^2+\d y^2) .
\ee
For the null congruence
\be
\ell_a\d x^a=\d t+\d\rho ,
\ee
with the spacelike frame vectors $\bbm_{(2)}=z\rho^{-1}/(\sqrt{3}\beta)\pa_\phi$, $\bbm_{(3)}=z/(\sqrt{3}\beta)\pa_z$, $\bbm_{(4)}=x\beta^{-1}\pa_x$, $\bbm_{(5)}=x\beta^{-1}\pa_y$, one finds $\rho_{ij}=\mbox{diag}(a,0,0,0)$, where $a=z^2/(3\beta^2\rho)$.
\subsubsection{Example with $\Lambda$ arbitrary}
One can alternatively construct an example with {$\rho_{ij}=\mbox{diag}(a,0,0,0)$} by starting from a null geodesic congruence in dS$_3\times$S$^2$ and adding an extra dimension by Brinkmann's warp. For example,
\be
\d s^2=f(z)\left[-V(r)\d t^2+\frac{1}{V(r)}\d r^2+r^2\d\chi^2+\beta^2(\d\theta^2+\sin^2\theta\d\phi^2)\right]+\frac{\d z^2}{f(z)},
\ee
with the null congruence
\be
\ell_a\d x^a=\d t+V^{-1}\d r ,
\ee
where
\be
V(r)=1-\frac{r^2}{2\beta^2}, \qquad f(z)=-\lambda z^2+2dz+b , \qquad d^2=\frac{1}{4\beta^2}-\lambda b.
\ee
Using the spacelike frame vectors $\bbm_{(2)}=f^{-1/2}r^{-1}\pa_\chi$, $\bbm_{(3)}=f^{-1/2}\beta^{-1}\pa_\theta$, $\bbm_{(4)}=f^{-1/2}\beta^{-1}\sin^{-1}\theta\pa_\phi$, $\bbm_{(5)}=f^{1/2}\pa_z$, one finds $\rho_{ij}=\mbox{diag}(a,0,0,0)$, where $a=\frac{1}{rf}$.
It follows from \cite{PraPraOrt07,OrtPraPra11} that $\Phi_{ij}\neq0$, {as it should be}. In general $\Lambda$ can have any sign but, in particular, we can set $\Lambda=0$ by choosing the parameter $\lambda=0$ in $f(z)$.
\subsection{Optical constraint, optical structures and totally geodesic complex two-dimensional null surfaces}
\label{subsec_integrability}
First, in the non-twisting case the optical constraint \eqref{canformL} is satisfied iff the optical matrix $\rhob$ possesses only one (possibly repeated) non-zero eigenvalue plus, possibly, some vanishing eigenvalues. This explains column O.C. (``optical constraint'') of table~\ref{tab_6D}.
Next, when $\bl$ is non-twisting one can always take an eigenframe of $\rhob$ to be parallelly transported \cite{OrtPraPra10,PraPra08} and thus in particular set $\M{i}{j}{0}=0$. It is then obvious that to each pair of repeated eigenvalues of $\rhob$ there corresponds an integrable two-dimensional totally null distribution, defined by two unit vectors of the corresponding eigenspace and by $\bl$ (cf. eq.~\eqref{D23_6d}).
Further conditions on the Ricci rotation coefficients can be obtained by considering the Ricci identity (11k,\cite{OrtPraPra07}), i.e., eq.~\eqref{11k}, similarly as in section~\ref{sec_totgeod}.
Consequences of this equation enable one to prove the statements in columns O.S. (``optical structure'') and I.T.G. (``integrable totally geodesic'' complex two-dimensional subspace) of table~\ref{tab_6D}. Since this is rather straightforward, let us just exemplify it to show that when $\rho_{ij}=\diag(a,a,b,b)$ (with $b\neq a$) the maximally totally null distribution ${\cal D}=\mbox{Span}\{\mbb{2}+i\mbb{3},\mbb{4}+i\mbb{5},\bl\}$ defines an optical structure (cf. eqs.~\eqref{6D_OS}). Namely, from \eqref{11k} with $i,j,k=224$ and $i,j,k=334$ one gets $\M{4}{2}{2}=\M{4}{3}{3}$, while $i,j,k=225$ and $i,j,k=335$ give $\M{5}{2}{2}=\M{5}{3}{3}$. Similarly, by taking $i,j,k=442/552$ and $i,j,k=443/553$ one obtains, respectively, $\M{2}{4}{4}=\M{2}{5}{5}$ and $\M{3}{4}{4}=\M{3}{5}{5}$. Next, with $i,j,k=324,234,325,235$ one finds $\M{3}{4}{2}=\M{2}{4}{3}=\M{3}{5}{2}=\M{2}{5}{3}=0$, while $i,j,k=524,425,534,435$ give $\M{5}{2}{4}=\M{4}{2}{5}=\M{5}{3}{4}=\M{4}{3}{5}=0$. This implies that eqs.~\eqref{6D_OS} are satisfied, as we wanted to prove.
Let us finally remark that in the Robinson-Trautman case, corresponding to the first row of table~\ref{tab_6D}, eq.~\eqref{11k} reduces to $L_{1i}=0$
(cf. also the results of \cite{PodOrt06,PraPra08}) and thus does not constraint the $\M{i}{j}{k}$. However, in the special subcase (cf.~\cite{PodOrt06}) in which the transverse space is of constant curvature one can show by inspection that there exists a natural parallelly transported frame such that ${\cal D}$ defines an optical structure and ${\cal D}_{23}$, ${\cal D}_{45}$ correspond to totally geodesic null two-surfaces.\footnote{This is a straightforward extension of the discussion in the 5d case, see footnote~10 of \cite{Ortaggioetal12}.} {{In fact the real $\mbb{i}$ can be paired arbitrarily in this case, giving rise also to other optical structures and totally geodesic null two-surfaces, and since }} there is a continuous freedom in spins there exist {{actually}} infinitely many optical structures.
This partly answers a question raised in remark~5.3 of \cite{Taghavi-Chabert11b}, by showing that also in even dimensions a spacetime may admit more than $2^{[n/2]}$ optical structures (noting also that in this example replacing $\bl\leftrightarrow\bn$ gives other optical structures).
\subsection{Type III/N spacetimes}
By the results of \cite{Pravdaetal04}, in this case there are only two admitted structures of $\rhob$: either $\rhob=0$ (Kundt spacetimes) or $\rhob=\diag(a,a,0,0)\neq0$. Since eq.~\eqref{11k} does not contain any Weyl components of zero boost weight, the above discussion still applies and implies that in the case $\rhob=\diag(a,a,0,0)\neq0$ both ${\cal D}_{23}$ and ${\cal D}_{45}$ correspond to totally geodesic null two-surfaces and that ${\cal D}$ defines an optical structure -- explicit examples can be constructed as explained in \cite{OrtPraPra10}.
\section*{Acknowledgments}
We are grateful to Lode Wylleman for useful discussions. The authors acknowledge support from research plan {RVO: 67985840} and research grant no P203/10/0749.
\renewcommand{\thesection}{\Alph{section}}
\setcounter{section}{0}
\renewcommand{\theequation}{{\thesection}\arabic{equation}}
\section{Algebraic constraints for type II spacetimes}
\setcounter{equation}{0}
\label{app_alg_eq}
\subsection{{General mWAND}}
For completeness and future reference, let us give here some additional algebraic constraints following from the Bianchi and Ricci identities for type II Einstein spacetimes in arbitrary dimensions.
Note that rhs of eq. \eqref{A4} is not symmetric under $ij\leftrightarrow kl$ and thus \eqref{A4}$_{ijkl}-$\eqref{A4}$_{klij}$ leads to an algebraic equation
\be
2(\Phia_{ij} A_{kl}-\Phia_{kl} A_{ij}) +\Phi_{k[j}\rho_{i]l}-\Phi_{l[j}\rho_{i]k}-\Phi_{j[k}\rho_{l]i}+\Phi_{i[k}\rho_{l]j}
+\Phi_{ij[k|s}\rho_{s|l]}-\Phi_{kl[i|s}\rho_{s|j]}=0 , \label{alg_no_Phi4}
\ee
and its trace gives the antisymmetric part of \eqref{B8}.
By multiplying by $\rho_{lj}$ and using \eqref{B8}--\eqref{B8asym} and symmetries of the Weyl tensor, after lengthy calculations one arrives at an equation with no $\Phi_{ijkl}$ terms
\be
\rho \Phi_{j[i|}\rho_{j|k]}+\Phi_{jl}\rho_{jl}A_{ki}+\rho_{jl}\rho_{jl}\Phia_{ik}
+\Phi_{jl}\left[ 3\rho_{[j|i}\rho_{|l]k} +\rho_{[k|l}\rho_{j|i]} \right]
+2\rho_{jl}\left[ \Phi_{j[i|}A_{l|k]}+2\Phia_{l[i|}\rho_{j|k]}\right]=0.
\label{constr_twist}
\ee
\subsection{{Non-twisting mWAND with $\Phia_{ij}=0$}}
As already mentioned, by differentiating algebraic constraints and using the Bianchi and Ricci identities one can arrive at further algebraic constraints.
For example, in the {\em non-twisting} case {with $\Phia_{ij}=0$},\footnote{{{Recall that in the non-twisting case with $\Phia_{ij}\neq0$ the form of $\rhob$ is fully determined by point \eqref{PhiA} of theorem~\ref{prop_GSHD}, cf.~\cite{Ortaggioetal12}.}}}
by differentiating~\eqref{eqn:algI}
we arrive at \eqref{eqn:rhoC} {(equivalent to setting $A_{ij}=0=\Phia_{ij}$ in \eqref{thornB8}), which can be rewritten as}
\be
- [\rhob^2] \Phi_{ik} + \Phi (\rhob^2)_{ik} + 2 \Phi_{ij} (\rhob^2)_{jk} + {\Phi}_{ijkl} {(\rhob^2)}_{jl} =0 \qquad (A_{ij}=0{=\Phia_{ij}}),
\label{eqn:algII_0}
\ee
where $[\rhob^2]$ denotes the trace of {$\rhob^2$}, etc. The trace of \eqref{eqn:algII_0} vanishes and multiplying \eqref{eqn:algII_0} by $\rho_{ik}$ gives
\be
[\rhob^2][\Phib \cdot \rhob]-[\rhob][\Phib \cdot\rhob^2]=0 \qquad {(A_{ij}=0{=\Phia_{ij}})} .
\label{trace}
\ee
Differentiation {of \eqref{eqn:algII_0}} leads to
\bea
& & -3 \left[ - [\rhob^3] \Phi_{ik} + \Phi (\rhob^3)_{ik} + 2 \Phi_{ij} (\rhob^3)_{jk} + {\Phi}_{ijkl} {(\rhob^3)}_{jl} \right] \nonumber\\
& & {}+ \left( \Phi [\rhob^2] + [\Phib \cdot\rhob^2]
\right) \rho_{ik}
- \left(\Phi \rho + [\Phib \cdot \rhob]
\right) (\rhob^2)_{ik}=0 \qquad (A_{ij}=0{=\Phia_{ij}})
\label{eqn:algII} ,
\eea
with trace {of \eqref{eqn:algII}} vanishing due to \eqref{trace}.
For $\Phi_{ij}=0$, the $(N-1)$th derivative of \eqref{B8} reads
\be
{\Phi}_{ijkl} {(\rhob^N)}_{jl} =0, \quad N =1,2,\dots \qquad (A_{ij}=0{=\Phi_{ij}}).
\label{eqn:algII_N}
\ee
\section{{Non-twisting type III Einstein spacetimes in arbitrary dimension}}
\setcounter{equation}{0}
\label{app_type_N_III}
For non-twisting Einstein spacetimes of type N, $\rho_{ij}=\diag({a},{a},0,\dots,0)$ \cite{Pravdaetal04,Durkeeetal10}, where ${a}$ {vanishes} in the case of Kundt spacetimes. Let us present here a new derivation of the form of the optical matrix for a non-twisting type III Einstein spacetime in arbitrary
dimension. This is considerably simpler than the original derivation of \cite{Pravdaetal04}.
For non-twisting type III, eq. (59) of \cite{Pravdaetal04} reduces to {(in the notation of \cite{Durkeeetal10})}
\be
2 \Psi'_{lij} \rho_{lk} = \rho \Psi'_{kij}. \label{typeIIInon-twist}
\ee
In the Kundt case ($\rhob=0$) this becomes an identity, while for non-twisting non-Kundt Einstein spacetimes (to which we restrict from now on) necessarily $\rho\neq0$ (cf. Proposition~1 of \cite{OrtPraPra07}) and \eqref{typeIIInon-twist} gives non-trivial information.
Following \cite{OrtPraPra10} we choose a parallelly propagated frame with diagonal form of $\rho_{ij}=\diag(\rho_2,\rho_3,\dots)$.
Without loss of generality we can assume that $\Psi'_{2ij} \not=0$ for some values of $i,\ j$. Eq.~\eqref{typeIIInon-twist} then implies
\be
\rho_2 = \frac{\rho}{2}{\neq0} \label{typeIIIrho}.
\ee
Using the $r$-dependence of the non-vanishing eigenvalues of $\rho_{ij}$ (see eq.~\eqref{rho_i}), this gives
\be
\frac{2}{r-b_2} = \sum_p \frac{1}{r-b_p},
\ee
where $p$ corresponds to all non-vanishing eigenvalues of $\rho_{ij}$.
Apart from the Kundt case with $\rho_{ij}=0$, this equation can be satisfied only for $\rho_{ij}$ of rank 2, with ${a\equiv}\rho_2=\rho_3$.
Thus {\em the optical matrix for the non-twisting type III Einstein spacetime is either vanishing (Kundt) or
$\rho_{ij}=\diag({a},{a},0,\dots,0)$}, exactly as in the type N case.
\section{On the geodetic and shearfree condition in four and higher dimensions}
\setcounter{equation}{0}
\label{app_shearfree}
\subsection{Goldberg-Sachs in four dimensions}
\label{app_GS_4D}
The four-dimensional Goldberg-Sachs theorem is usually expressed as a statement about a null congruence being geodesic ($\kappa=0$) and shearfree ($\sigma=0$). As well-known, this property can be interpreted as different but equivalent geometrical statements, which we now briefly review. This will later be useful for discussing the higher dimensional case.
\subsubsection{Integrability of two-spaces}
\label{subsubsec_integr_d4}
By considering the commutator $[\delta,D]$ (see, e.g., p.~77 of \cite{Stephanibook})
one immediately sees that $\bl$ is geodesic and shearfree if and only if the distribution
\be
{\cal D}=\mbox{Span}\{\bbm,\bl\}
\ee
is integrable (cf.~\cite{penrosebook2}), i.e. $[{\cal D},{\cal D}]\subset{\cal D}$.
A similar conclusion holds of course for $\bar{\cal D}$. The geodesic{\&}shearfree property can thus be seen as a statement about the null bivector $\bl\wedge\bbm$ instead of one about the vector field $\bl$. While the vector field $\bl$ defines a privileged real null direction lying in ${\cal D}\cap\bar{\cal D}$, $\bbm$ is not fixed uniquely (null rotations with $\bl$ fixed, spins and boost can be performed, under which $\bl\wedge\bbm$ is fixed, up to a rescaling, and $\kappa=0=\sigma$ is preserved). The significance of this viewpoint appears clearly when considering complex extensions of the Goldberg-Sachs theorem \cite{PlebHac75}.
\subsubsection{Conformal structure of the screen space}
Given a generic null vector field $\bl$, its flow preserves the conformal structure of the ``screen space'' $L^\bot/L$ when \cite{RobTra83} (see also \cite{NurTra02,Trautman02a,Trautman02b})
\be
{\cal L}_\l g=\rho g+\l\otimes\xi+\xi\otimes\l \quad \Leftrightarrow \quad \kappa=0=\sigma .
\label{Lie_4}
\ee
In four dimensions only, this is equivalent to requiring that the complex structure of the screen space is preserved by the flow of $\bl$ \cite{RobTra83,NurTra02,Trautman02a,MasTag08}.
This condition can also be derived by imposing Maxwell equations on a null field (two-form) \cite{robinsonnull} (cf. also, e.g., \cite{Stephanibook,penrosebook2,NurTra02,Trautman02a,Trautman02b}).
\subsection{Conformal structure of the screen space: the standard geodesic{\&}shearfree condition}
\label{subsec_screenHD}
Ref.~\cite{RobTra83} (see also \cite{Trautman02b}) proposed to extend (\ref{Lie_4}) to higher dimensions as a definition of the geodesic{\&}shearfree condition. As it turns out,
\be
{\cal L}_\l g=\rho g+\l\otimes\xi+\xi\otimes\l \quad \Leftrightarrow \quad \kappa_{i}=0=S_{ij}-\frac{S_{kk}}{n-2}\delta_{ij} ,
\label{Lie}
\ee
so that indeed $\bl$ is geodesic and shearfree in the standard sense. Some comments on algebraically special Einstein spacetimes admitting such a congruence have been already given in section~\ref{subsec_OC} and appendix~\ref{app_shearfreetwist}.
Note that for $n>4$ eq.~(\ref{Lie}) does {\em not} follow from Maxwell's equations for a null field (intended again as a two-form), while geodeticity {($\kappa_{i}=0$)} does, and shear is non-zero for expanding solutions \cite{Ortaggio07,Durkeeetal10}.
\subsection{Existence of an optical structure}
\label{subsubsec_optical}
\subsubsection{Complex notation}
For later discussions it will be useful to use a complex basis. Namely, in {\em even} dimensions one can take the frame $(\bl,\bn,\bmu_A,\bar\bmu_A)$ and in {\em odd} dimensions the frame $(\bl,\bn,\bmu_A,\bar\bmu_A,\bx)$. Except for the unit spacelike vector $\bx$, all the vectors are null, the complex $\bmu_A$ are defined by
\be
\bmu_2=\frac{1}{\sqrt{2}}(\mbb{2}+i\mbb{3}) , \qquad \bmu_4=\frac{1}{\sqrt{2}}(\mbb{4}+i\mbb{5}) , \qquad \ldots ,
\label{mu}
\ee
and ${\bar\bmu_A}$ by their complex conjugates, where $A=2\mu$, $\mu=1,\ldots,(n-2-\epsilon)/2$, with $\epsilon=0,1$ for even and odd dimensions, respectively (in 4d this would be the standard NP frame). The metric becomes
\be
g=\bl\otimes \bn+\bn\otimes\bl+\bmu_A\otimes\bar\bmu_A+\bar\bmu_A\otimes\bmu_A+\epsilon \bx\otimes\bx .
\label{complex_g}
\ee
Then one can define the complex counterpart of the Ricci rotation coefficients. These are defined as an obvious extension of the real coefficients, e.g.,
\be
\cL_{AB}=\mu^a_A\mu^b_B\nabla_b l_{a} , \qquad \cL_{A\bar B}=\mu^a_A\bar\mu^b_B \nabla_b l_{a} , \qquad \cM{A}{C}{B}=\mu^a_C\mu^b_{{B}}\nabla_b \mu_{A a},
\ee
and other coefficients (and their complex-conjugates) are defined similarly.
\subsubsection{Optical structure}
\label{subsubsec_OS}
Ref.~\cite{HugMas88} studied the consequences of Maxwell's equations for a null field defined as an $n/2$-form in $n$ even dimensions. From these, they arrived at a generalization of the geodesic{\&}shearfree condition different from the $\kappa=0=\sigma$ condition discussed in section~\ref{subsec_screenHD} (see also \cite{NurTra02,Trautman02a,Trautman02b,MasTag08}). Recently, this has been extended also to odd dimensions \cite{Taghavi-Chabert11}. Namely, consider the totally null $(n-\epsilon)/2$-dimensional distribution (recall that $A$ in $\mu_A$ can be only even)
\be
{\cal D}=\mbox{Span}\{\bl,\bmu_2,\ldots,\bmu_{(n-2-\epsilon)}\} .
\label{D_gen}
\ee
If ${\cal D}$ and ${\cal D}^\bot$ are integrable, i.e.,
\be
[{\cal D},{\cal D}]\subset{\cal D} , \qquad [{\cal D}^\bot,{\cal D}^\bot]\subset{\cal D}^\bot ,
\label{Robinson}
\ee
${\cal D}$ is said to define an ``{\em optical structure}'' \cite{Taghavi-Chabert11} (note that ${\cal D}^\bot={\cal D}$ in even dimensions). In 4d, eq.~\eqref{Robinson} reduces to the standard conditions $\kappa=0=\sigma$ (as discussed in section~\ref{subsubsec_integr_d4}), which indeed corresponds to the conditions coming from the Mariot-Robinson theorem \cite{Stephanibook,penrosebook2} for a Maxwell null two-form. In higher dimensions, using the complex counterpart of the commutators given in \cite{Coleyetal04vsi}, one finds that (\ref{Robinson}) is equivalent to
\be
\cL_{A0}=\cL_{x0}=\cL_{AB}=\cL_{Ax}=\cL_{xA}=\cM{A}{B}{0}=\cM{A}{x}{0}=\cM{A}{B}{C}=\cM{A}{B}{x}=\cM{x}{A}{B}=0 .
\label{optical_complex}
\ee
(In fact one first obtains conditions such as $\cM{A}{B}{C}=\cM{C}{B}{A}$, but $\cM{A}{B}{C}=-\cM{B}{A}{C}$, etc.) The equations above represent the conditions obtained when $n$ is odd, however, if $n$ is even one can simply drop all equations containing $x$ (this will be understood from now on).
Note, in particular, that the vector $\bl$ is {\em geodesic} ($\cL_{A0}=\cL_{x0}=0$), but generally is not required to be shearfree (except when $n=4$).
In even dimensions it has also been observed that (\ref{Robinson}) means that the complex structure of the screen space is preserved \cite{NurTra02,Trautman02a,MasTag08}. We have shown in \cite{Ortaggioetal12} that in 5d a very large class of algebraically special Einstein spacetimes possesses an optical structure. In appendix \ref{app_OS5d} we extend the results of \cite{Ortaggioetal12} by showing that, in fact, all algebraically special 5d Einstein spacetimes possess (at least) one optical structure.
\paragraph{Optical structure in six dimension} Let us rewrite the conditions \eqref{optical_complex} in real notation in the special case $n=6$, which is useful for the discussion in the main text (see \cite{Ortaggioetal12} for the case $n=5$). One readily gets
\beqn
& & \kappa_{i}=0 \quad (i=2,\ldots,5) , \nonumber \\
& & \rho_{22}=\rho_{33}, \quad \rho_{23}=-\rho_{32}, \quad \rho_{44}=\rho_{55}, \quad \rho_{45}=-\rho_{54}, \nonumber \\
& & \rho_{24}=\rho_{35}, \quad \rho_{42}=\rho_{53}, \quad \rho_{34}=-\rho_{25}, \quad \rho_{43}=-\rho_{52}, \nonumber \\
& & \M{2}{4}{0}=\M{3}{5}{0}, \quad \M{3}{4}{0}=-\M{2}{5}{0}, \qquad\qquad\qquad\qquad\qquad\qquad\qquad (n=6) \label{6D_OS} \\
& & \M{2}{4}{2}-\M{3}{4}{3}=\M{2}{5}{3}+\M{3}{5}{2}, \quad -\M{2}{5}{2}+\M{3}{5}{3}=\M{2}{4}{3}+\M{3}{4}{2}, \nonumber \\
& & \M{2}{4}{4}-\M{2}{5}{5}=\M{3}{4}{5}+\M{3}{5}{4}, \quad \M{2}{4}{5}+\M{2}{5}{4}=-\M{3}{4}{4}+\M{3}{5}{5} . \nonumber
\eeqn
In particular, if $\bl$ is twistfree then $\rhob$ has two pairs of repeated eigenvalues.
\subsection{Optical constraint}
Based on results for Kerr-Schild spacetimes, Ref.~\cite{OrtPraPra09} put forward yet another possible generalization of the shearfree condition for mWANDs in higher dimensions. This is the so called ``optical constraint'', already discussed in section~\ref{subsec_OC}, which involves only the null direction $\bl$ (as opposed to the optical structure discussed above). The Lorentz transformation freedom of null rotations preserving $\bl$, boosts and spins is thus retained in this case (indeed spins can be used to arrive at the canonical form~\eqref{canformL}), see also \cite{OrtPraPra10} for related comments. We have shown in \cite{Ortaggioetal12} that in 5d a very large class of algebraically special Einstein spacetimes admits an mWAND obeying the optical constraint, {{and Ref.~\cite{Wylleman_priv} extended our result to prove that in fact {\em all} algebraically special Einstein spacetimes admit such an mWAND} (see also \cite{OrtPraPra12rev})}.
Note that in 4d the optical constraint is a necessary condition for $\bl$ to be a repeated principal null direction but is not {sufficient} \cite{Ortaggioetal12}.
\subsection{Integrability of a (complex) two-dimensional totally null distribution}
\subsubsection{${\cal D}_{23}$ integrable}
As a further generalization of the geodesic{\&}shearfree condition one can consider the integrability of the complex two-dimensional totally null distribution
\be
{\cal D}_{23}=\mbox{Span}\{\bmu_{2},\bl\} .
\label{D}
\ee
It is easy to show that ${\cal D}_{23}$ is integrable if and only if
\be
\cL_{20}=0, \qquad \cL_{B2}=\cM{2}{B}{0}, \qquad \cL_{\bar B2}=\cM{2}{\bar B}{0} \quad (B\neq 2) , \qquad \cL_{x2}=\cM{2}{x}{0} .
\label{2integr}
\ee
In 4d this again reduces to the standard $\kappa=0=\sigma$ condition.
In particular, in six dimensions \eqref{2integr} can be rewritten in real notation as
\beqn
& & \kappa_{2}=0=\kappa_{3} , \qquad \rho_{22}=\rho_{33}, \quad \rho_{23}=-\rho_{32}, \nonumber \label{D23_6d} \\
& & \rho_{42}=\M{2}{4}{0}, \quad \rho_{43}=\M{3}{4}{0}, \quad \rho_{52}=\M{2}{5}{0}, \quad \rho_{53}=\M{3}{5}{0} \qquad\qquad (n=6) .
\eeqn
\subsubsection{${\cal D}_{23}$ integrable with totally geodesic integral surfaces}
We can strengthen the above conditions by further requiring the integral surfaces of ${\cal D}_{23}$ to be {\em totally geodesic}. The corresponding equations read
\beqn
& & \cL_{B0}=0=\cL_{x0}, \qquad \cL_{B2}=0=\cL_{x2}, \qquad \cL_{\bar B2}=0 \quad (B\neq 2) , \nonumber \\
& & \cM{2}{B}{0}=0=\cM{2}{x}{0}, \qquad \cM{2}{B}{2}=0=\cM{2}{x}{2}, \qquad \cM{2}{\bar B}{0}=0 \quad (B\neq 2) , \qquad \cM{2}{\bar B}{2}=0 \quad (B\neq 2) .
\label{2totgeod}
\eeqn
Now $\bl$ is necessarily geodesic ($\cL_{B0}=0=\cL_{x0}$). In 4d \eqref{2totgeod} is equivalent to \eqref{2integr} because $B=2$ is the only possibility (and there are no $x$-components). In real notation \eqref{2totgeod} can be rewritten as (where we define $\hat k\neq 2,3$)
\beqn
& & \kappa_{i}=0 , \qquad \rho_{22}=\rho_{33}, \quad \rho_{23}=-\rho_{32}, \qquad \rho_{\hat k2}=0=\rho_{\hat k3} \nonumber \\
& & \M{2}{\hat k}{0}=0=\M{3}{\hat k}{0} , \label{2totgeod_real} \\
& & \M{2}{\hat k}{2}=\M{3}{\hat k}{3}, \quad \M{2}{\hat k}{3}=-\M{3}{\hat k}{2} \nonumber .
\eeqn
\section{Optical structures in five dimensions}
\label{app_OS5d}
Proposition~4 of \cite{Ortaggioetal12} gives a set of sufficient conditions for a five-dimensional Einstein spacetime of type II or more special to possess an {\em optical structure} \cite{Taghavi-Chabert11}. In particular, it shows that, except possibly for Kundt spacetimes (and for a special subclass ``(iii)'' of genuine type II, later proven not to exist \cite{Wylleman_priv}), all algebraically special Einstein spacetimes admit an optical structure in five dimensions. It is the purpose of this appendix to show that this in fact holds also for Kundt spacetimes. (Obviously, if a complex optical structure is integrable its complex conjugate is integrable too and this will be understood in the following.) Combining this with the proof of \cite{Wylleman_priv} that the special subclass (iii) of genuine type II is empty, we arrive at
\begin{prop}
\label{prop_integrab_5D}
In a five-dimensional Einstein spacetime admitting a multiple WAND~$\lb$ there always exists an optical structure. In the case of type D spacetimes there exist in fact (at least) two optical structures.
\end{prop}
\begin{proof}
Let us show that Einstein spacetimes of the Kundt class always possess an optical structure in five dimensions. In other words, we need to show that in such spacetimes there always exists a null frame $\{\bl,\bn,\mbb{2},\mbb{3},\mbb{4}\}$ such that
the totally null distribution
\be
{\cal D}=\mbox{Span}\{\mbb{2}+i\mbb{3},\bl\} ,
\ee
and its orthogonal complement
\be
{\cal D}^\bot=\mbox{Span}\{\mbb{2}+i\mbb{3},\mbb{4},\bl\} ,
\ee
are both integrable. This is equivalent to \cite{Ortaggioetal12} (see also appendix~\ref{app_shearfree}, and
{\eqref{L1i_M}} for the definition of $\M{a}{b}{c}$)
\beqn
& & \kappa_{i}=0, \qquad \rho_{33}=\rho_{22}, \qquad \rho_{32}=-\rho_{23},\qquad \rho_{24}=0=\rho_{34}, \qquad \rho_{42}=0=\rho_{43}, \label{Dorth_integr_k_rho}
\\ & & \M{2}{4}{0}=0=\M{3}{4}{0} , \qquad \M{2}{4}{2}=\M{3}{4}{3}, \qquad \M{2}{4}{3}=-\M{3}{4}{2} .
\label{Dorth_integr_M}
\eeqn
Kundt spacetimes admit a metric in the form \cite{Coleyetal03,ColHerPel06,PodZof09}
\be
\d s^2 =2\d u\left[\d r+H(u,r,x)\d u+W_\alpha(u,r,x)\d x^\alpha\right]+ g_{\alpha\beta}(u,x) \d x^\alpha\d x^\beta , \label{Kundt_gen}
\ee
where $\alpha,\beta=2,3,4$ in five dimensions. Here $\bl=\partial_r$ is a geodesic, twistfree, shearfree, non-expanding mWAND, so that \eqref{Dorth_integr_k_rho} is automatically satisfied. Now, define a null frame with $\ell_a\d x^a=\d u$, $n_a\d x^a=\d r+H\d u+W_\alpha\d x^\alpha$ and the spacelike vectors $\mbb{i} $ living in the three-dimensional transverse (Euclidean) space spanned by the $x^\alpha$ (their components will be, in particular, independent of $r$). Then one immediately finds
\be
\M{i}{j}{0}=0,
\ee
so that the first of \eqref{Dorth_integr_M} is satisfied in this frame.
Next, using the Christoffel symbols given in \cite{PodZof09} one can check that
\be
m_{(i)\a;\beta}=m_{(i)\a||\beta} ,
\ee
where the covariant derivatives on the r.h.s. is taken w.r.t. the transverse metric $g_{\a\beta}$.
It follows that
\be
{\M{i}{j}{k}} ={\Mt{i}{j}{k}} ,
\ee
where the connection coefficients on the r.h.s. are those computed w.r.t. the transverse metric $g_{\a\beta}$ (and the o.n. frame vectors of the transverse space are simply the ``projections'' $\tilde m_{(i)\a}$ of the $m_{(i)\a}$ vectors).
One then finds
\be
[\mbtt{2}+i\mbtt{3},\mbtt{4}]=(\Mt{4}{2}{4}+i\Mt{4}{3}{4})\mbtt{4} +[-\Mt{2}{4}{2}+i(\Mt{2}{3}{4}-\Mt{2}{4}{3})]\mbtt{2}+[\Mt{3}{2}{4}-\Mt{3}{4}{2}-i\Mt{3}{4}{3}]\mbtt{3} ,
\ee
so that $\mbox{Span}\{\mbtt{2}+i\mbtt{3},\mbtt{4}\}$ is integrable iff
\be
\Mt{3}{4}{3}=\Mt{2}{4}{2} , \qquad \Mt{3}{4}{2}=-\Mt{2}{4}{3} .
\label{integr_234}
\ee
Proving the integrability of ${\cal D}$ and ${\cal D}^\bot$ is thus now reduced to proving the integrability of $\mbox{Span}\{\mbtt{2}+i\mbtt{3},\mbtt{4}\}$. The transverse frame is arbitrary, so we just need to show that there exists at least one frame satisfying this integrability property.
If we define the complex null vector field
\be
\bmu=\frac{1}{\sqrt{2}}(\mbtt{2}+i\mbtt{3}) ,
\label{def_mu}
\ee
using \eqref{integr_234} the required integrability condition $[\bmu,\mbtt{4}]=\alpha\bmu+\beta\mbtt{4}$ reads
\be
\mu_{a||b}m_{(4)}^a\mu^b=0 .
\ee
Thus by choosing a complex null geodesics $\bmu$ in the transverse three-space we automatically obtain the integrability of the corresponding distributions ${\cal D}$ and ${\cal D}^\bot$ and {(together with the result of \cite{Wylleman_priv})} our proof is complete.
\end{proof}
\section{Shearfree twisting spacetimes (even dimensions)}
\label{app_shearfreetwist}
Twisting geodesic mWANDs with zero shear are forbidden in odd dimensions \cite{OrtPraPra07} but they are permitted in {\em even} dimensions and they have necessarily $\det(\rhob)\neq 0$ (as can be easily seen in a frame adapted to $A_{ij}$, using the fact that $S_{ij}\propto\delta_{ij}$). Here we present an explicit example in six dimensions. {To our knowledge, this is the first such example that has been identified.}
First, consider the six-dimensional Ricci flat Taub-NUT metric \cite{ManSte04}
\beqn
\d s^2= & & -F(r)(\d t-2n_1\cos\theta_1\d\phi_1-2n_2\cos\theta_2\d\phi_2)^2+\frac{\d r^2}{F(r)} \nonumber \\
& & {}+(r^2+n_1^2)(\d\theta_1^2+\sin\theta_1^2\d\phi_1^2)+(r^2+n_2^2)(\d\theta_2^2+\sin\theta_2^2\d\phi_2^2) ,
\label{MasSte6D}
\eeqn
where
\be
F(r)=\frac{r^4/3+(n_1^2+n_2^2)r^2-2mr-n_1^2n_2^2}{(r^2+n_1^2)(r^2+n_2^2)} .
\ee
We observe that this is a spacetime of type D. A geodetic mWAND is given by
\be
\ell_a\d x^a=\d t+F(r)^{-1}\d r-2n_1\cos\theta_1\d\phi_1-2n_2\cos\theta_2\d\phi_2 ,
\ee
while a second one can simply be obtained by reflecting $\bl$ as $t\to-t$, $\phi_1\to-\phi_1$, $\phi_2\to-\phi_2$ \cite{PraPraOrt07}. Using the frame vectors
\beqn
& & m_{(2)a}\d x^a=\sqrt{r^2+n_1^2}\d\theta_1, \qquad m_{(3)a}\d x^a=\sqrt{r^2+n_1^2}\sin\theta_1\d\phi_1, \nonumber \\
& & m_{(4)a}\d x^a=\sqrt{r^2+n_2^2}\d\theta_2, \qquad m_{(5)a}\d x^a=\sqrt{r^2+n_2^2}\sin\theta_2\d\phi_2,
\eeqn
one finds
\be
\rhob = \left(
\begin{array}{cccc}
\displaystyle \frac{r}{r^2+n_1^2} & \displaystyle -\frac{n_1}{r^2+n_1^2} & 0 & 0\\
\displaystyle \frac{n_1}{r^2+n_1^2} & \displaystyle \frac{r}{r^2+n_1^2} & 0 & 0 \\
0 & 0 & \displaystyle \frac{r}{r^2+n_2^2} & \displaystyle -\frac{n_2}{r^2+n_2^2}\\
0 & 0 & \displaystyle \frac{n_2}{r^2+n_2^2} & \displaystyle \frac{r}{r^2+n_2^2}
\end{array}
\right) .
\ee
One can easily check that $\rhob$ obeys the optical constraint~(\ref{OC2}). {Moreover, we also observe that in the spacetime~\eqref{MasSte6D} the maximally totally null distribution ${\cal D}=\mbox{Span}\{\mbb{2}+i\mbb{3},\mbb{4}+i\mbb{4},\bl\}$
defines an {\em optical structure} (concept introduced and discussed in \cite{HugMas88,NurTra02,Trautman02a,Trautman02b,Taghavi-Chabert11,Taghavi-Chabert11b}), and both the totally null distributions ${\cal D}_{23}=\mbox{Span}\{\mbb{2}+i\mbb{3},\bl\}$ and ${\cal D}_{45}=\mbox{Span}\{\mbb{4}+i\mbb{5},\bl\}$ are integrable, with totally geodesic integral surfaces (see appendix~\ref{app_shearfree} for the corresponding definitions and conditions).}
The null vector field $\bl$ is generically shearing ($S_{22}=S_{33}\neq S_{44}=S_{55}$), however in the special case $n_1=n_2$ (corresponding to the solutions of \cite{AwaCha02}) it becomes {\em shearfree} (while still being expanding and twisting). The class of shearfree twisting spacetimes is thus non-empty in higher dimensions.
\section{{Non-degenerate, non-twisting geodesic mWANDs violating the optical constraint ($n\ge 7$)}}
\setcounter{equation}{0}
\label{app_violating}
In the main text we have seen examples of Einstein spacetimes admitting a non-twisting mWAND violating the optical constraint, see e.g. table~\ref{tab_distinct_eigenval}. However, all of them have a degenerate $\rhob$ (i.e., $m<n-2$). In this appendix we provide also some examples with $m=n-2$, relevant to theorem~\ref{prop_GSHD} (point \eqref{Phi0}) and proposition~\ref{prop_non_deg2} (point 2.).
\subsection{Metric}
Using the theory of conformal Einstein spaces \cite{Brinkmann25} (reviewed, e.g., in \cite{petrov}), by taking a double Brinkmann warp one can construct the following Einstein space (satisfying $R_{ab}=2\Lambda g_{ab}/(n-2)$):
\be
\d s^2=\lambda r^2\d u^2+2\d u\d r+(\lambda ur-1)^2\d\sigma_\lambda^2+r^2\d\Sigma_0^2 ,
\label{general_violating}
\ee
where
\be
\lambda=2\frac{\Lambda}{(n-1)(n-2)} ,
\ee
$\d\sigma_\lambda^2$ is a Riemannian Einstein space of dimension $n_\sigma$ with Ricci scalar $R_\sigma=n_\sigma(n_\sigma-1)\lambda$ and $\d\Sigma_0^2$ a Ricci-flat Riemannian space of dimension $n_\Sigma$. In order for $\d s^2$ to have a non-zero Weyl tensor, the metrics $\d\sigma_\lambda^2$ or $\d\Sigma_0^2$ cannot be both of constant curvature, so that either $n_\sigma\ge 4$ or $n_\Sigma\ge 4$ (or both). The complete spacetime has thus dimension $n=2+n_\sigma+n_\Sigma\ge 7$ (unless it is of constant curvature, which is of no interest to us).
Let us define the vector field $\bl$ with covariant and contravariant components
\be
\ell_a\d x^a=\d u , \qquad \ell^a\pa_a=\pa_ r ,
\label{l_ex}
\ee
which is obviously null and hypersurface-orthogonal, thus automatically geodesic and twistfree, and $r$ is an affine parameter along it. This can be accompanied by another null vector $\bn$ (also hypersurface-orthogonal)
\be
n_a\d x^a=\d r+\frac{\lambda}{2}r^2\d u ,
\label{n_ex}
\ee
which satisfies the normalization condition $\bl\cdot\bn=1$.
\subsection{Weyl tensor}
From the results of \cite{Brinkmann25} together with those on direct product spacetimes \cite{PraPraOrt07} it readily follows that
\be
C_{uabc}=0=C_{rabc} .
\label{C_ur_ex}
\ee
Let us consider a set of $n-2$ spacelike o.n. vectors $\bbm_{(i)}$ orthogonal to $\bl$ and $\bn$, composed of a subset of vectors denoted by $i=A, B, \ldots$ living in the subspace of $\d\sigma_\lambda^2$ and a subset denoted by $i=I,J, \ldots$ living in the subspace of $\d\Sigma_0^2$. Then the only non-zero Weyl frame components are given by $C_{ABCD}$ and $C_{IJKL}$, and their $r$-dependence is (recall the first definition in \eqref{bw0})
\be
\Phi_{ABCD}=\frac{1}{(\lambda ur-1)^2}C^0_{ABCD} , \qquad \Phi_{IJKL}=\frac{1}{r^2}C^0_{IJKL} ,
\label{C_ABIJ_ex}
\ee
where quantities with superscript $^0$ do not depend on $r$ (these are in fact the Weyl components of the respective ``subspaces'').
In particular, one has
\be
\WD{ij} =0 .
\ee
Note also that one has generically $W_{ij}\neq 0$ (recall definition \eqref{def_W} and constraint~\eqref{W_phi}).
\subsection{Multiple WAND(s) and optics}
From the above results it follows that both $\bl$ and $\bn$ are {double} WANDs and the spacetime is thus of type~D. In fact a bit more than that, since
\be
C_{abcd}\ell^d=0=C_{abcd}n^d \label{Wandcond}
\ee
(cf. \cite{PraPra05,Ortaggio09} for the meaning of such conditions in terms of the Weyl type) and the type is D(abd).\footnote{The construction {of metric \eqref{general_violating}} is carried out more naturally in a different coordinate system (not adapted to $\bl$) in which $\bl$ and $\bn$ are related by a coordinate transformation (``time-reflection'') that leaves the metric invariant: therefore they share the same geometric properties.}
The optical matrix $\rhob$ of $\bl$ is obviously symmetric, it is moreover diagonal and non-degenerate with components
\be
\rho_{AB}=\frac{\lambda u}{\lambda ur-1}\delta_{AB} , \qquad \rho_{IJ}=\frac{1}{r}\delta_{IJ} .
\ee
There are thus two eigenvalue-blocks of dimension $n_\sigma$ and $n_\Sigma$, or $[n_\sigma,n_\Sigma]$. For instance, for $n=7$ we can construct explicit solutions with eigenvalues $\{a,a,a,a,b\}$, for $n=8$ we can have
$\{a,a,a,a,a,b\}$ and $\{a,a,a,a,b,b\}$ and so on (in all cases $a\neq0\neq b$, $a\neq b$). These provide examples for of the shearing spacetimes of theorem~\ref{prop_GSHD}
and proposition~\ref{prop_non_deg2}, although an eigenvalue structure {more general than the one of metric \eqref{general_violating} can in principle exist}. {In particular, the optical constraint is clearly violated (cf. eq.~\eqref{canformL}).} Note also that we cannot construct a six-dimensional example with this method. By setting the cosmological constant to zero in the above metrics ($\lambda=0$) one obtains Ricci-flat spacetimes -- however, these are direct products and $\rhob$ becomes degenerate, which is of no interest to the present discussion.
By taking direct products one can trivially generate many similar examples with a degenerate $\bl$. These will have $\WDS{ij} \neq 0$, however still with $\WDA{ij} =0$ (see \cite{PraPraOrt07}), in agreement with point (i) of theorem~\ref{prop_GSHD}.
\subsubsection{On possible additional (m)WANDs}
\label{subsubsec_additionalW}
One might wonder whether spacetimes \eqref{general_violating} admit other multiple (double) WANDs (different from $\bl$ and $\bn$, eqs.~\eqref{l_ex} and \eqref{n_ex}) and in particular whether those can obey the optical constraint. Take a generic null vector $\bk=\alpha\pa_u+\beta\pa_r+\gamma^A\bbm_{(A)}+\gamma^I\bbm_{(I)}$ (where the $\bbm_{(A)}$ [$\bbm_{(I)}$] have components only in the subspace of $\d\sigma_\lambda^2$ [$\d\Sigma_0^2$]), i.e., $\alpha^2\lambda r^2+2\alpha\beta+\delta_{AB}\gamma^A\gamma^B+\delta_{IJ}\gamma^I\gamma^J=0$. Since for $\gamma^A=0=\gamma^I$ $\bk$ gives the directions of $\bl$ and $\bn$ (respectively for $\alpha=0$ and $\beta=-\frac{1}{2}\alpha\lambda r^2$), we now assume that $\gamma^A$ and $\gamma^I$ are not both zero (so that $\alpha\neq0$). Using \eqref{C_ur_ex}, \eqref{C_ABIJ_ex} and the Bel-Debever criteria \cite{Ortaggio09} it follows that: (i) $\bk$ is a WAND $\Leftrightarrow C^0_{ABCD}\gamma^A\gamma^C=0=C^0_{IJKL}\gamma^I\gamma^K$
(so that, using also \eqref{C_ur_ex}, one gets $C_{abcd}k^bk^d=0$); (ii) $\bk$ is an mWAND $\Leftrightarrow C^0_{ABCD}\gamma^A=0=C^0_{IJKL}\gamma^I$ (in which case, using also \eqref{C_ur_ex}, one gets $C_{abcd}k^d=0$).
These conditions on the Weyl tensors of $\d\sigma_\lambda^2$ and $\d\Sigma_0^2$ are generically not satisfied (see an example in section~\ref{subsubsec_ex2}), so in general spacetimes \eqref{general_violating} do not admit any other WANDs (not even single) apart from $\bl$ and $\bn$. However, it is also clear that in special cases additional mWANDs may exist. For example, when either $\d\sigma_\lambda^2$ or $\d\Sigma_0^2$ are conformally flat (which is necessarily the case if $n_\sigma<4$ or $n_\Sigma<4$) spacetimes \eqref{general_violating} admit a continuous infinity of mWANDs (generically non-geodesic but some can be geodesic, see an example in section~\ref{subsubsec_ex1}).
\subsection{An explicit example with additional mWANDs}
\label{subsubsec_ex1}
For the sake of definiteness, one can for instance construct an explicit solution for $n=7$ by taking (\ref{general_violating}) with
\beqn
& & \d\sigma_\lambda^2=V(\rho)\d\tau^2+V^{-1}(\rho)\d\rho^2+\rho^2(\d\theta^2+\sin\theta^2\d\phi^2) , \qquad V(\rho)=1-\frac{\mu}{\rho}-\lambda \rho^2 , \nonumber \\
& & \d\Sigma_0^2=\d z^2 , \label{NOCmetric}
\eeqn
and a coordinate range such that $V(\rho)>0$.
Taking $\bl=\pa_r$ and the orthonormal vectors
\beqn
& & \bbm_{(2)}=A^{-1}V^{-1/2}(\rho)\pa_\tau , \qquad \bbm_{(3)}=A^{-1}V^{1/2}(\rho)\pa_\rho, \qquad \bbm_{(4)}=A^{-1}\rho^{-1}\pa_\theta, \nonumber \\
& & \bbm_{(5)}=A^{-1}\rho^{-1}\sin\theta^{-1}\pa_\phi, \qquad \bbm_{(6)}=r^{-1}\pa_z , \qquad A=\lambda ur-1 ,
\eeqn
one finds
\be
\rho_{22}=\rho_{33}=\rho_{44}=\rho_{55}=\frac{\lambda u}{\lambda ur-1} , \qquad \rho_{66}=\frac{1}{r} .
\ee
The only non-zero Weyl frame components read (recall definition \eqref{def_W})
\be
W_{23}=W_{45}=\frac{1}{(\lambda ur-1)^2}\frac{\mu}{\rho^3} , \qquad W_{24}=W_{25}=W_{34}=W_{35}=-\frac{1}{(\lambda ur-1)^2}\frac{\mu}{2\rho^3} .
\ee
However, note that since $\d\Sigma_0^2=\d z^2$ is (conformally) flat, it follows from section~\ref{subsubsec_additionalW} that metric \eqref{general_violating} with \eqref{NOCmetric} admits also other mWANDs of the form
$\bk=\frac{1}{r^2}[\pa_u-\frac{1}{2}r^2(\lambda+\gamma^2)\pa_r+\gamma\pa_z]$, where $\gamma$ is an arbitrary function.
It turns out that if $\gamma_{,r}=\gamma_{,u}=\gamma_{,z}=0$ the mWAND $\bk$ is geodesic, in which case it becomes twistfree iff $\gamma$ is a constant and it obeys the optical constraint iff $\gamma^2=|\lambda|$ (so that it is also twistfree), i.e.,
\be
\bk=\frac{1}{r^2}\left[\partial_u-\frac{1}{2}(\lambda+|\lambda|)r^2\pa_r\pm\sqrt{|\lambda|}\pa_z\right] .
\ee
In this case the corresponding optical matrix is of the form $\{a,a,a,a,a\}$ for $\lambda>0$ and $\{a,a,a,a,0\}$ for $\lambda<0$ (in both cases $a\neq0$).
\subsection{An explicit example without additional mWANDs}
\label{subsubsec_ex2}
{It follows from section~\ref{subsubsec_additionalW} that in order for metric \eqref{general_violating} to admit only two double WANDs we need $n_\sigma\ge4$ and $n_\Sigma\ge4$, i.e., at least 10 spacetime dimensions. If we now take $\d\sigma_\lambda^2$ as in \eqref{NOCmetric} but} $\d\Sigma_0^2$ corresponding to the Riemannian version of 4d Schwarzschild, {by looking at the conditions of section~\ref{subsubsec_additionalW} it is easy to see that indeed} the {\em only} mWANDS are the $\bl$ and $\bn$ discussed above (eqs.~\eqref{l_ex} and \eqref{n_ex}). Such metric thus constitutes an example of an Einstein spacetime with {\em all double WANDs violating the optical constraint}. Recall also that such mWANDs \eqref{l_ex} and \eqref{n_ex} are geodesic and non-degenerate.
|
1,116,691,500,429 | arxiv |
\section{Introduction}\label{sec:intro}
Memristors are used for memory applications~\cite{Niu2010, Ho2011, Mohammad2013, Baghel2015, Radakovits2019}, where even storage of multiple bits per device is feasible~\cite{zangeneh_design_of_1t1r_reram,kim2010cnna,Taherinejad2015ems,Taherinejad2016cce}. In addition, memristors have become increasingly popular for neural network and learning applications~\cite{Pershin2012ieee,Thomas2013applied}, by exploiting their analog, synapses-like nature. Another application of memristors is implementing digital (in-memory) logic~\cite{borghetti_memristive_stateful_logic, talati_logic_design_magic, CRS_proposal}, such as \gls{imply}, for various computations~\cite{Gupta2018, Papandroulidakis2017, Rohani2017, Taherinejad2019newcas, guckert2018system, Radakovits2020tcasi}. At the moment these applications -- more often than not -- are
not verified by physical implementation and experimental data~\cite{CAS_physRealization}. This imbalance leads to many problems when actual physical implementation is desired. While material sciences have certainly progressed in this field~\cite{waser_redox_reram_physics,menzel_switching_kinetics_ecm,menzel_reram_physics}, the circuit-level interface to higher abstraction levels is not yet ready to provide a reliable base for proposed applications~\cite{CAS_physRealization}. Some of the fundamental problems, that need to be considered at design time, are {inter-device variability} and {cyclic variability}. In larger structures, usually implemented within crossbar arrays, sneak paths and wire resistance are an even bigger issue~\cite{cassuto_sneak_path_constraints,sneak_paths_closed_form}. While the two latter have received an acceptable level of attention from the community, the two former have been less explored and addressed by the community. We hope that this work encourages and provides pointers to the community to move in that direction.\par
Here, we aim to provide a better insight into the operation of a single memristive \gls{imply} logic gate by considering device variations. Similar works on that topic already exist, such as~\cite{kvatinsky_device_variations,xie_robustness_of_memristor_logic,chen_imply_ron_not_reached}, which mainly focus on other types of memristor-based logic. The most relevant work to ours is~\cite{chen_imply_ron_not_reached}, where the focus is set on the design of the \gls{imply} circuit itself, and an alternative operation ``\gls{nimp}'' is proposed to mitigate certain problems. In contrast, our work explores regular \gls{imply} in more detail, particularly regarding the effect of device variations on the \gls{imply} operation, and leaves the \gls{nimp} approach for future works. We note that there is a variety of memristors based on different physical effects~\cite{waser_redox_reram_physics}. From the perspective of this work, the internal mechanism is to a large extent inconsequential. Hence, we use the general term, memristor, to refer to \emph{resistive switching elements} or \glspl{reram} and, when needed, specify what may make the internal mechanisms important.
The main contribution of this work to the field of memristor-based logic, is an in-depth mathematical analysis of memristive \gls{imply} regarding its constraints due to device variation. Plausibility of the proposed constraints is verified via simulations using a popular model. \par
The rest of this paper is organized as follows: \Cref{sec:imply_logic}, particularly \Cref{subsec:gate_and_constr}, reviews memristive \gls{imply} logic and shows its limitations. An introduction to the crossbar architecture is given in \Cref{sec:xbar_fundamental} and the device model is described in \Cref{sec:model}.
In \Cref{sec:math}, we formulate new constraints for the \gls{imply} gate, before comparing them to the single gate simulation results in \Cref{sec:sim_single}. The results of the crossbar simulations are presented in \Cref{sec:sim_xbar} and compared against the single gate simulation and constraints. We conclude the paper in \Cref{sec:conclusion}.
\section{Material implication (IMPLY)}\label{sec:imply_logic}
The truth table of \gls{imply}, and its four different cases, are shown in~\Cref{tab:truth_table_imply}. It takes two input states $p$ and $q$ and outputs $q'$. Not every type of memristor is suitable for material implication. The devices have to exhibit \emph{voltage threshold behavior}. Moreover, all devices used for an operation shall have the same parameter values (resistance range, threshold voltages, switching speed).
\begin{table}
\centering
\caption{Truth table of material implication and its four cases}
\label{tab:truth_table_imply}
\begin{tabular}{c|cc|c}
\hline\hline
Cases & $p$ & $q$ & $q'$\\
\hline
{Case} 1 &0 & 0 & 1 \\
{Case} 2 &0 & 1 & 1 \\
{Case} 3 &1 & 0 & 0 \\
{Case} 4 &1 & 1 & 1 \\
\hline\hline
\end{tabular}
\end{table}
\subsection{Gate structure and constraints}\label{subsec:gate_and_constr}
Two memristors and a resistor are necessary for a single memristive \gls{imply} gate. \Cref{fig:imply_gate} shows such a gate, with abstracted drivers and sense circuitry. Each memristor can be \textit{set} (forced to \gls{lrs}) by applying a voltage $|\ensuremath{V_\mathrm{set}}|>|\ensuremath{v_\mathrm{on}}|$ with appropriate (in our case \textit{negative}) polarity; and can be \textit{reset} (forced to \gls{hrs}) by applying $|\ensuremath{V_\mathrm{reset}}|>|\ensuremath{v_\mathrm{off}}|$ with an opposite polarity\footnote{This is true for bipolar switching mechanisms. \gls{pc} based devices, for example, may use the same voltage polarity for set and reset.}. If a memristor is set, it represents logic state `1'; if it is reset, it represents logic state `0'~\cite{borghetti_memristive_stateful_logic}. During initialization, these voltage amplitudes are applied to each device, while the other is kept floating. For the actual logic operation both devices are driven at the same time: $\ensuremath{V_\mathrm{cond}}$ is applied to node $R$ and $\ensuremath{V_\mathrm{set}}$ to node $T$ of \Cref{fig:imply_gate}.
For a correct operation
\begin{align}
|\ensuremath{V_\mathrm{set}}| > |\ensuremath{v_\mathrm{on}}| \label{eqn:vset_relation}\\
|\ensuremath{V_\mathrm{set}} - \ensuremath{V_\mathrm{cond}}| < |\ensuremath{v_\mathrm{on}}| \label{eqn:vcond_relation1}\\
|\ensuremath{V_\mathrm{reset}}| > |\ensuremath{v_\mathrm{off}}| \label{eqn:vclear_relation}
\end{align}
must hold. Moreover, the circuit designer needs to select a valid value for $\ensuremath{R_\mathrm{G}}$, as described in~\cite{kvatinsky_imply_logic_design}.\par
\begin{figure}[!b]
\centering
\subfigure[]{
\begin{circuitikz}
\draw(0,0) to[memristor,i=$i$] (2,0);
\draw[-stealth](.2,.5)node[above]{$+$} -- (1,.5)node[above]{$v$} -- (1.8,.5)node[above]{$-$};
\path(0,0) -- (0,-1.5);
\end{circuitikz}
\label{fig:symbol_polarity}
}
\subfigure[]{
\tikzset{varr/.style={-stealth}}
\begin{circuitikz}[scale=.6,transform shape]
\draw (.7,-1.5) to[short] (.7,-1)
(3.3,-1.5) to[short] (3.3,-1)
(3.3,-3.5) -- (-1,-3.5) to[R,l_=$R_G$,font=\Large] (-1,-5) node[ground]{};
\draw(.7,-3.5) to[Mr,*-o,l^=$P$,font=\Large] (.7,-1.5) node[left]{$R$}
(3.3,-3.5) to[Mr,-o,l^=$Q$,font=\Large](3.3,-1.5) node[left]{$T$};
\draw(-.5,-1) rectangle (4.5,0)
(2,-.5) node[font=\Large] {Driver \& Sensing};
\draw[varr] (1.2,-1.75)node[right]{$+$} --(1.2,-2.5) node[right]{\Large$V_P$}-- (1.2,-3.25)node[right]{$-$};
\draw[varr] (3.8,-1.75)node[right]{$+$} --(3.8,-2.5) node[right]{\Large$V_Q$}-- (3.8,-3.25)node[right]{$-$};
\draw[varr] (-.5,-3.7)node[right]{$+$} --(-.5,-4.5) node[right]{\Large$V_G$}-- (-.5,-5.3)node[right]{$-$};
\end{circuitikz}
\label{fig:imply_gate}}
\caption{\subref{fig:symbol_polarity} Memristor symbol and defined voltage polarity used in this work. \subref{fig:imply_gate} A single IMPLY gate.}
\label{fig:symbol_and_imply_gate}
\end{figure}
It is important to note that only in Case~1 of \Cref{tab:truth_table_imply} the output memristor $Q$ is actually changing its state. However, during this process the voltage across each device changes too. It is valid to ask if this has an effect on the result of the operation, and the answer is yes. Using \gls{kcl}, Chen et al.~\cite{chen_imply_ron_not_reached} showed that there are two possible final steady states of the operation:
\begin{enumerate}
\item The normalized state variable $s$ reaches the upper boundary of $1$ ($R_Q=\ensuremath{R_\mathrm{on}}$) before the voltage across $Q$ falls below the threshold $\ensuremath{v_\mathrm{on}}$. The final steady state is $R_Q=\ensuremath{R_\mathrm{on}}$.
\item $\ensuremath{V_Q}$ falls below the threshold $\ensuremath{v_\mathrm{on}}$ before $s$ reaches $1$. In this case the steady state resistance can be expressed as\footnote{Note that in our convention $\ensuremath{v_\mathrm{on}}<0$.}:
\begin{align}
R_\mathrm{min} &= \frac{-\ensuremath{v_\mathrm{on}}\,\ensuremath{R_\mathrm{G}}\,\ensuremath{R_\mathrm{off}}}{(\ensuremath{R_\mathrm{G}}+\ensuremath{R_\mathrm{off}}) (\ensuremath{V_\mathrm{set}}+\ensuremath{v_\mathrm{on}}) - \ensuremath{R_\mathrm{G}} \ensuremath{V_\mathrm{cond}}}\label{eqn:Rmin}
\end{align}
\end{enumerate}
An important point to mention is that this calculation is based on the premise, that the driving voltages of $P$ and $Q$ are chosen such that there is \emph{no state drift in $P$} during operation.
\subsection{Crossbar principles}\label{sec:xbar_fundamental}
Crossbar architectures are a natural candidate for memristor-based logic, as high integration density can be reached.
In so called 1R (or 1M) crossbars, a memristor device is fabricated at each intersection of bit- and word-lines, which act as the access medium for the cell. 1R crossbars are very difficult to handle~\cite{wan2010edl,li2018imw,li2018nature}, even if parasitics are not considered. Many works have been carried out to study effects~\cite{cassuto_sneak_path_constraints,chen_crossbar_array_model,shin_data_dependent_statistical_model_analysis,shin_data_dependend_model}, or solve them~\cite{chen_crossbar_array_model,sneak_paths_closed_form, CRS_proposal}, but thus far 1T1R has been the preferred implementation~\cite{wan2010edl,li2018imw,li2018nature}. 1T1R (or 1T1M) crossbars,consist of a transistor and a memristor in each cell. The transistor in each cell cost extra area but they prevent the cells from switching state when a cell is not part of an operation (not selected). One possible, readout scheme is provided by~\cite{sneak_paths_closed_form}, which we use in this work. The chosen readout scheme~\cite{sneak_paths_closed_form} provides a closed-form solution.
Moreover, it introduces very little additional complexity, which enables this work to remain focused on issues regarding \gls{imply} itself. In \Cref{sec:sim_xbar} we compare crossbar simulation results against single gate results and outline the differences.
\section{Device model}\label{sec:model}
There are a range of different simulation models for memristors~\cite{kvatinsky_team,kvatinsky_vteam,strachan_memristor_model,jiang_stanford_model}. For the simulations presented in this paper, the TU Wien LTSpice implementation~\cite{VTEAM_Spice, CAS_physRealization} of VTEAM~\cite{kvatinsky_vteam} was used. An overview of the model is given in \Cref{eqn:w,eqn:s,eqn:dwdt,eqn:tu_ui,eqn:s_ui}, with a memristor polarity as shown in \Cref{fig:symbol_polarity}. \par
In VTEAM, $w$ acts as the state variable and represents a length between the extrema $\ensuremath{w_\mathrm{on}}$ and $\ensuremath{w_\mathrm{off}}$ ($w\in[\ensuremath{w_\mathrm{off}},\ensuremath{w_\mathrm{on}}]$).
Here, we define the relation of these state variable boundaries
\begin{align}
\ensuremath{w_\mathrm{on}} &> \ensuremath{w_\mathrm{off}}\label{eqn:w}
\end{align}
and define the normalized state variable ($s\in[0,1]$):
\begin{align}
s(w)= w' &= \frac{w-\ensuremath{w_\mathrm{off}}}{\ensuremath{w_\mathrm{on}}-\ensuremath{w_\mathrm{off}}}\label{eqn:s}
\end{align}
These definitions may be changed, as long as the model equations are updated, too.
The rate of change of the state variable, $w$, is defined by
\begin{align}
\frac{\mathrm{d} w}{\mathrm{d} t} &= \begin{cases}
\ensuremath{k_\mathrm{off}} \left(\frac{v}{\ensuremath{v_\mathrm{off}}}-1\right)^{\ensuremath{\alpha_\mathrm{off}}} \ensuremath{f_\mathrm{off}}(w)\ \ &0<\ensuremath{v_\mathrm{off}}<v\\
0 & \ensuremath{v_\mathrm{on}} < v < \ensuremath{v_\mathrm{off}}\\
\ensuremath{k_\mathrm{on}} \left(\frac{v}{\ensuremath{v_\mathrm{on}}}-1\right)^{\ensuremath{\alpha_\mathrm{on}}} \ensuremath{f_\mathrm{on}}(w) & v<\ensuremath{v_\mathrm{on}}<0\\
\end{cases}\label{eqn:dwdt}
\end{align}
which is the essential building block of the model~\cite{kvatinsky_vteam}. In this equation $\ensuremath{k_\mathrm{on}},~\ensuremath{k_\mathrm{off}},~\ensuremath{v_\mathrm{on}},~\ensuremath{v_\mathrm{off}},~\ensuremath{\alpha_\mathrm{on}}$ and $\ensuremath{\alpha_\mathrm{off}}$ represent fitting parameters, while $\ensuremath{f_\mathrm{on}}(w)$ as well as $\ensuremath{f_\mathrm{off}}(w)$ are window functions that limit $\mathrm{d} w/\mathrm{d} t$. \par
$I/V$-characteristics and window functions are not defined in the model and thus can be freely chosen. We chose a linear current/voltage dependency:
\begin{align}
R(w)=\ensuremath{R_\mathrm{off}}+\left(\ensuremath{R_\mathrm{on}}-\ensuremath{R_\mathrm{off}}\right)\cdot s(w)
\label{eqn:tu_ui}
\end{align}
By rearranging the equation we can further express $s(R)$ for any (measured) $R$:
\begin{align}
s(R) &= \frac{R-\ensuremath{R_\mathrm{off}}}{\ensuremath{R_\mathrm{on}} - \ensuremath{R_\mathrm{off}}}\label{eqn:s_ui}
\end{align}
The same expressions as for the Simmon's Tunnel Barrier model in~\cite{kvatinsky_team} were chosen as window functions. In addition, $w$ is bounded and thus cannot exceed $\ensuremath{w_\mathrm{on}}$ or $\ensuremath{w_\mathrm{off}}$. \par
The studies presented in this paper are kept as general as possible, however, simulations need model parameters. Rather than introducing arbitrary parameter values, we experimentally fitted~\cite{Taherinejad2019newcas} our VTEAM model to Knowm BS-AF-W~\cite{knowm_datasheet} memristors we had at the time.
Parameters shown in \Cref{tab:parameters}, represent a best effort fitting we conducted previously~\cite{Taherinejad2019newcas}.
\section{Formulating Constraints}\label{sec:math}
This section marks the beginning of our new contributions. In this section, we mathematically extract device variability constraints which govern and limit operations of \gls{imply}. At first, we define the notation: Each parameter involved in the analysis is written as $\xi_{i,M}$, where $\xi\in\{R,v,k\}$, $M \in \{P, Q\}$ and $i \in \{\mathrm{off}, \mathrm{on}\}$.
For example, the off-resistance of memristor $P$ in this notation would be denoted as $\ensuremath{R_{\mathrm{off},P}}$. \par
Logic thresholds determine the logic state of a device. They are defined separately for input (I) and output (O), as well as logic `1' (H) and `0' (L). Indices are used to denote the respective logic thresholds, e.g. $\Rtext{IL}$ is the input threshold for logic `0'.
\subsection{Static behavior}\label{subsec:stat_math}
Each case in the truth table (\Cref{tab:truth_table_imply}) imposes constraints onto the voltage $\ensuremath{V_Q}$ across memristor $Q$, as certain \textit{switching conditions} must be met. They can be analyzed via \gls{kcl} and represent a \textit{static} view of the circuit. The constraints can be used to find limits for $\ensuremath{R_{\mathrm{on},P}}$, $\ensuremath{R_{\mathrm{off},P}}$, $\ensuremath{v_{\mathrm{on},Q}}$ and $\ensuremath{v_{\mathrm{off},Q}}$. They do not provide limits for $\ensuremath{R_{\mathrm{on},Q}}$ or $\ensuremath{R_{\mathrm{off},Q}}$, as $R_Q$ in this context is the target output resistance state that $Q$ must reach during \gls{imply}. Therefore, $R_Q$ is later set according to the chosen output logic threshold: $R_Q\leq R_\mathrm{OH}$ or $R_Q\geq R_\mathrm{OL}$.Applying \gls{kcl} in \Cref{fig:imply_gate} gives us the voltage across $Q$ as
\begin{align}
\ensuremath{V_Q} &= \frac{\ensuremath{R_Q} (\ensuremath{R_P}+\ensuremath{R_\mathrm{G}})\ensuremath{V_\mathrm{set}}-\ensuremath{R_Q}\ensuremath{R_\mathrm{G}}\ensuremath{V_\mathrm{cond}}}{\Rp\RG+\Rp\Rq+\Rq\RG}\label{eqn:VQ}.
\end{align}
First we solve \Cref{eqn:VQ} with a generalized threshold voltage, $v$, and the solution is specialized for each case afterwards.
The first switching condition is:
\begin{align}
\ensuremath{V_Q}&> v.
\label{eqn:VQ_constr}
\end{align}
Plugging \Cref{eqn:VQ} into \Cref{eqn:VQ_constr} and isolating $\ensuremath{R_P}$ leads to
\begin{align}
\ensuremath{R_P}\cdot b&> a\label{eqn:Rp_intermediate},
\end{align}
where
\begin{align}
a= \ensuremath{R_Q}\ensuremath{R_\mathrm{G}} (v&+\ensuremath{V_\mathrm{cond}}-\ensuremath{V_\mathrm{set}})\label{eqn:a},\\
b= \ensuremath{R_Q}\ensuremath{V_\mathrm{set}}-&v(\ensuremath{R_\mathrm{G}}+\ensuremath{R_Q})\label{eqn:b}.
\end{align}
At this point the relation in \ref{eqn:Rp_intermediate}, is divided by $b$. Therefore, depending on the value of $b$, we have
\begin{align}
\begin{cases}
\ensuremath{R_P}>\frac{a}{b}\ &\text{if}\ \ \ b > 0\\
\ensuremath{R_P}<\frac{a}{b}\ &\text{if}\ \ \ b < 0\\
\ensuremath{R_P}\rightarrow\pm\infty\ &\text{if}\ \ \ b=0
\end{cases}\label{eqn:Rp_case}
\end{align}
Next, the switching condition
\begin{align}
\ensuremath{V_Q}&< v
\end{align}
is examined. Following the same steps as before, we have
\begin{align}
\begin{cases}
\ensuremath{R_P}<\frac{a}{b}\ &\text{if}\ \ \ b > 0\\
\ensuremath{R_P}>\frac{a}{b}\ &\text{if}\ \ \ b < 0\\
\ensuremath{R_P}\rightarrow\pm\infty\ &\text{if}\ \ \ b=0
\end{cases}\label{eqn:Rp_case2}
\end{align}
Since the third case ($b=0$) in \Cref{eqn:Rp_case,eqn:Rp_case2}
yields\footnote{That is, as long as $|a|$ is neither zero, nor $\infty$.} $\pm\infty$, it is of no interest for the rest of the analysis.
The first two cases in \Cref{eqn:Rp_case,eqn:Rp_case2} both provide limits for $v$ and $\ensuremath{R_P}$, respectively. \par
Here, the resulting equations (constraints) are specialized for each of the four cases of the truth table using the respective switching conditions. {$\ensuremath{R_Q}$ is set to the associated output logic threshold ($\Rtext{OH}$ or $\Rtext{OL}$).} Only the first case ($b>0$) of \Cref{eqn:Rp_case,eqn:Rp_case2} is considered, since the second case ($b<0$) only provides negative limits, and $\ensuremath{R_P} >0$.
For every case of the truth table, according to our notations, $\ensuremath{v_\mathrm{on}}<0$ and $\ensuremath{v_\mathrm{off}}>0$. Hence, we have
\begin{description}
\item[Case 1] $\ensuremath{V_Q} > -\ensuremath{v_{\mathrm{on},Q}}$
\begin{align}
\ensuremath{v_{\mathrm{on},Q}}&> -\ensuremath{V_\mathrm{set}}\frac{\Rtext{OH}}{\ensuremath{R_\mathrm{G}}+\Rtext{OH}}\label{eqn:vonq_case1}\\
\ensuremath{R_{\mathrm{off},P}} &> \frac{\Rtext{OH}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{on},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OH}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{on},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OH})}\label{eqn:Roffp_case1}
\end{align}
\item[Case 3:] $\ensuremath{V_Q} < -\ensuremath{v_{\mathrm{on},Q}}$
\begin{align}
\ensuremath{v_{\mathrm{on},Q}}&> -\ensuremath{V_\mathrm{set}}\frac{\Rtext{OL}}{\ensuremath{R_\mathrm{G}}+\Rtext{OL}}\label{eqn:vonq_case3}\\
\ensuremath{R_{\mathrm{on},P}} &< \frac{\Rtext{OL}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{on},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OL}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{on},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OL})}\label{eqn:Ronp_case3}
\end{align}
\item[Case 2/Case 4:] $\ensuremath{V_Q} > -\ensuremath{v_{\mathrm{off},Q}}$
\begin{align}
\ensuremath{v_{\mathrm{off},Q}}&> -\ensuremath{V_\mathrm{set}}\frac{\Rtext{OH}}{\ensuremath{R_\mathrm{G}}+\Rtext{OH}}\label{eqn:voffq_case24}\\
\ensuremath{R_{\mathrm{off},P}} &> \frac{\Rtext{OH}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{off},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OH}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{off},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OH})}\label{eqn:Roffp_case2}\\
\ensuremath{R_{\mathrm{on},P}} &> \frac{\Rtext{OH}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{off},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OH}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{off},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OH})}\label{eqn:Ronp_case4}
\end{align}
\end{description}
\Cref{eqn:vonq_case1,eqn:vonq_case3,eqn:voffq_case24} directly result from $b>0$, whereas \Cref{eqn:Roffp_case1,eqn:Ronp_case3,eqn:Roffp_case2,eqn:Ronp_case4} are the respective relations derived from $R_P>a/b$ in \Cref{eqn:Rp_case} and $R_P<a/b$ in \Cref{eqn:Rp_case2}.\par
Some additional static constraints are given by the choice of logic thresholds. That is,
\begin{align}
\ensuremath{R_{\mathrm{off},P}} > R_\mathrm{IL}\label{eqn:Roffp_greater_RIL}\\
\ensuremath{R_{\mathrm{off},Q}} > R_\mathrm{IL}\label{eqn:Roffq_greater_RIL}\\
\ensuremath{R_{\mathrm{on},P}} < R_\mathrm{IH}\label{eqn:Ronp_smaller_RIH}\\
\ensuremath{R_{\mathrm{on},Q}} < R_\mathrm{IH}\label{eqn:Ronq_smaller_RIH}.
\end{align}
Similar to standard logic families, input and output thresholds may differ. From the point-of-view of these constraints, only input thresholds need to be considered, as they determine whether or not the device states fed to the operation are valid in the first place.
\subsection{Dynamic behavior}\label{subsec:dyn_math}
With respect to the static analysis, the chosen timestep of operation can introduce much stricter constraints. For exact solutions, one would have to solve the differential state equation of the chosen model (in our case \Cref{eqn:dwdt} from VTEAM). This is not a trivial task and might not even be possible for all models. Thus, in this section we derive a lower boundary for $\ensuremath{v_{\mathrm{on},Q}}$, but not the infimum, which cannot be exceeded by the exact (or numeric) solution. That way we take into account the state change (dynamic behavior) of memristors during the operation, using an acceptable estimation. We note that in doing such an analysis, the chosen model is assumed to be accurate. However, in practice no existing model represents all the reality and physics involved. \par
\begin{figure}
\centering
\subfigure[]{
\begin{tikzpicture}[font=\footnotesize]
\draw[-stealth] (0,0) -- (2.5,0) node[right]{$t$};
\draw[-stealth] (0,0) -- (0,3) node[right]{$\ensuremath{V_Q}$};
\draw[] (.7,.1) --(.7,-.1)
(1.7,.1) --(1.7,-.1)
(.1,2.4) -- (-.1,2.4) node[left]{$V_{Q\mathrm{i}}$}
(.1,1.6) -- (-.1,1.6) node[left]{$V_{Q\mathrm{f}}$};
\draw[stealth-stealth] (.7,-.2) -- (1.2,-.2) node[below] {$\Delta T$} -- (1.7,-.2);
\draw[thick, red!70!yellow] (.7,2.4) -- (1.7,2.4);
\fill[red!70!yellow] (1.7,2.4)circle(.05);
\draw[dashed] (.7,0) -- (.7,2.4)
(1.7,0) --(1.7,1.6)
(0,2.4) -- (.7,2.4)
(0,1.6)--(1.7,1.6);
\fill[black] (.7,2.4) circle(.05)
(1.7,1.6) circle(.05);
\draw (.7,2.4) to[out=-45, in=160] (.9,2.1)
(1.7,1.6) to[out=160, in=-30] (1.5,1.8);
\draw[dotted] (.9,2.1)--(1.5,1.8);
\end{tikzpicture}
\label{fig:vq_during_imply}}
\subfigure[]{
\begin{tikzpicture}[font=\footnotesize]
\draw[-stealth] (0,0) -- (2.5,0) node[right]{$t$};
\draw[-stealth] (0,0) -- (0,2.5) node[right]{$\Delta w_Q$};
\draw[] (.7,.1) --(.7,-.1)
(1.7,.1) --(1.7,-.1);
\draw[stealth-stealth] (.7,-.2) -- (1.2,-.2) node[below] {$\Delta T$} -- (1.7,-.2);
\draw[thick, red!70!yellow] (.7,0) -- (1.7,1.6);
\fill[red!70!yellow] (1.7,1.6)circle(.05);
\fill[black] (.7,0) circle(.05)
(1.7,1) circle(.05);
\draw (.7,0) to[out=55, in=-130] (1,.4)
(1.7,1) to[out=-170, in=40] (1.5,.9);
\draw[dotted] (1,.4)--(1.5,.9);
\end{tikzpicture}
\label{fig:dw_during_imply}}
\caption{A symbolic voltage-time curve for $\ensuremath{V_Q}$ \subref{fig:vq_during_imply} an induced state change $\Delta w_Q$ \subref{fig:dw_during_imply} during a single \gls{imply} operation of duration $\Delta T$. The estimations for the formulation of constraints are drawn in orange, the symbolic representations of the actual curves in black.}
\label{fig:changes_during_imply}
\end{figure}
The main idea of our estimation is to look at how $\ensuremath{V_Q}$ changes over time in Case 1 of the truth table, while assuming negligible state drift in $P$. As $\ensuremath{R_Q}$ changes from \gls{hrs} to \gls{lrs}, $\ensuremath{V_Q}$ decreases. Thus, the initial voltage $V_{Q\mathrm{i}}$ is the highest occurring value of $\ensuremath{V_Q}$ during that timestep, while the final voltage $V_{Q\mathrm{f}}$ is the lowest -- symbolically shown in \Cref{fig:vq_during_imply}. If the device characteristics are such that the highest $\ensuremath{V_Q}$ corresponds to the maximum value of $\mathrm{d} w/ \mathrm{d} t$ -- in our case true due to \Cref{eqn:dwdt} -- a hard limit can be expressed. \gls{kcl} can be used to describe the initial voltage
\begin{align}
V_{Q\mathrm{i}} &= \frac{\ensuremath{R_{\mathrm{off},Q}}(\ensuremath{R_{\mathrm{off},P}}+\Rg)\ensuremath{V_\mathrm{set}}-\ensuremath{R_{\mathrm{off},Q}} \Rg \ensuremath{V_\mathrm{cond}}}{\ensuremath{R_{\mathrm{off},P}} \Rg+ \ensuremath{R_{\mathrm{off},P}} \ensuremath{R_{\mathrm{off},Q}} + \ensuremath{R_{\mathrm{off},Q}} \Rg}.
\label{eqn:VQi}
\end{align}
Plugging $V_{Q\mathrm{i}}$ into \Cref{eqn:dwdt} gives the inital rate of state change\footnote{$\ensuremath{f_\mathrm{on}}$ is missing in \Cref{eqn:dwqdt_initial} because
$\ensuremath{f_\mathrm{on}}\approx 1$ for $w_Q < \ensuremath{w_\mathrm{on}}$}:
\begin{align}
\frac{\mathrm{d} w_Q}{\mathrm{d} t}\bigg|_{\text{initial}} &= \frac{\Delta w_Q}{\Delta T}= \ensuremath{k_{\mathrm{on},Q}} \left(\frac{-V_{Q\mathrm{i}}}{\ensuremath{v_{\mathrm{on},Q}}}-1\right)^\alpha\label{eqn:dwqdt_initial}
\end{align}
Now we set the actual $\mathrm{d} w_Q/\mathrm{d} t$ equal to the initial rate for the whole timestep $\Delta T$. Through this simplification a maximum $\Delta w_Q$ for the given timestep $\Delta T$ can be found, which cannot be exceeded:
\begin{align}
\Delta w_Q &= \ensuremath{k_{\mathrm{on},Q}} \left(\frac{-V_{Q\mathrm{i}}}{\ensuremath{v_{\mathrm{on},Q}}}-1\right)^\alpha \Delta T\label{eqn:Dwq_max}
\end{align}
This is because the estimation provides a better overall situation towards the correct operation result, when compared to the actual situation. That is, as we see in \Cref{fig:dw_during_imply}, the estimated $\Delta w_Q$ is always larger than the actual value.
To obtain a correct result after the IMPLY operation, $\ensuremath{R_Q}$ must at least reach the logic threshold $R_\mathrm{OH}$. Otherwise the result would not be interpreted as logic `1'. Via \Cref{eqn:s_ui} we can find $s(R_\mathrm{OH})$. In combination with \Cref{eqn:s}, the necessary $\Delta w_\mathrm{min}$ can be expressed as
\begin{align}
\Delta w_\mathrm{min}&= \frac{R_\mathrm{OH}-\ensuremath{R_{\mathrm{off},Q}}}{\ensuremath{R_{\mathrm{on},Q}}-\ensuremath{R_{\mathrm{off},Q}}}(\ensuremath{w_\mathrm{on}}-\ensuremath{w_\mathrm{off}}) + \ensuremath{w_\mathrm{off}},
\label{eqn:wmin}
\end{align}
and
\begin{align}
\Delta w_Q &\geq \Delta w_\mathrm{min}\label{eqn:Dwq_greater_Dwmin}
\end{align}
shall be true. Plugging the previous terms into \Cref{eqn:Dwq_greater_Dwmin} gives
\begin{align}
\ensuremath{v_{\mathrm{on},Q}}&\geq\frac{-V_{Q\mathrm{i}}}{\sqrt[\leftroot{-1}\uproot{2}\scriptstyle\alpha]{\frac{\Delta w_\mathrm{min}}{\ensuremath{k_{\mathrm{on},Q}}\Delta T}} +1}, \label{eqn:vonq_wmin}
\end{align}
which is the newly found constraint.
As this relation contains multiple parameters of $P$ and $Q$ apart from $\ensuremath{v_{\mathrm{on},Q}}$, it provides boundaries for all of them. For example, a certain $\ensuremath{v_{\mathrm{on},Q}}$ restricts $\ensuremath{R_{\mathrm{off},P}}$ to a specific range, and in turn a certain $\ensuremath{R_{\mathrm{off},P}}$ restricts $\ensuremath{v_{\mathrm{on},Q}}$ to a specific range. \par
Considering \Cref{eqn:vonq_wmin}, a question is whether the
same estimation could be used to find an upper limit for $\ensuremath{v_{\mathrm{on},Q}}$. Such an analysis, however, is not meaningful for memristor $Q$. Both, a minimum value of $\mathrm{d} w/ \mathrm{d} t$ and a maximum value $\Delta w_\mathrm{max}$, must be specified. The later does not exist for $Q$ since a high $w_Q$ (ideally $\ensuremath{w_\mathrm{on}}$) is desired in Case~1.\par
There is, however, a $\Delta w_\mathrm{max}$ for memristor $P$, as a change of $\ensuremath{R_P}$ is generally not desired. By definition the logic state of $P$ remains unchanged if $\ensuremath{R_P}>R_\mathrm{IL}$. Only if this is true, it can be used as an input for an operation. As $\ensuremath{R_P}$ drifts away from $\ensuremath{R_{\mathrm{off},P}}$ (ideal \gls{hrs}), $\ensuremath{V_P}$ decreases. Thus, the minimum value of $\mathrm{d} w/\mathrm{d} t$ is the \textit{final value} at the end of the operation, in contrast to $V_{Q\mathrm{i}}$ being the initial value. The final value $V_{P\mathrm{f}}$ cannot be expressed easily, as $R_{P\mathrm{f}}$ and $R_{Q\mathrm{f}}$ are unknown. Hence, another simplification must be made: We evaluate $V_{P\mathrm{f}}$ using $\ensuremath{R_P}=\ensuremath{R_{\mathrm{off},P}}$, as if there was no state drift in the first place:
\begin{align}
V_{P\mathrm{f},j} &= \frac{\ensuremath{R_{\mathrm{off},P}}(R_{Q,j}+\Rg)\ensuremath{V_\mathrm{cond}}-\ensuremath{R_{\mathrm{off},P}}\Rg\ensuremath{V_\mathrm{set}}}{\ensuremath{R_{\mathrm{off},P}}\Rg+\ensuremath{R_{\mathrm{off},P}} R_{Q,j}+R_{Q,j}\Rg}\label{eqn:vpf}
\end{align}
Due to this very rough estimation, we expect the constraint to represent a fairly weak boundary. Therefore,
\begin{align}
R_{Q,1}&=R_{\mathrm{min},Q}\label{eqn:Rq1}\\
R_{Q,2}&=\frac{\ensuremath{R_{\mathrm{off},Q}}+R_{\mathrm{min},Q}}{2}\label{eqn:Rq2}\\
R_{Q,3}&=\sqrt{\ensuremath{R_{\mathrm{off},Q}}\cdot R_{\mathrm{min},Q}}\label{eqn:Rq3}
\end{align}
are used to evaluate $V_{P\mathrm{f}}$ in \Cref{eqn:vpf} and obtain a range within which the circuit is less likely to fail. The first value, \Cref{eqn:Rq1}, is the theoretical minimum for $\ensuremath{R_{\mathrm{on},Q}}$, which is the ideal $R_{Q\mathrm{f}}$. However, the actual $R_{Q,\mathrm{f}}$ can take on any value between $\ensuremath{R_{\mathrm{off},Q}}$ and $R_{\mathrm{min},Q}$. Hence, in a second estimation, \Cref{eqn:Rq2}, we assume that $R_{Q\mathrm{f}}$ is the arithmetic mean of $\ensuremath{R_{\mathrm{off},Q}}$ and $R_{\mathrm{min},Q}$. In other words, the final state is halfway between the initial state and the ideal endstate ($R_{\mathrm{min},Q}$). However, if $V_{P\mathrm{f}}$ is plotted over $R_Q$ on a linear scale, it reveals that the dependence is non-linear. Thus, $R_{Q,2}$ might not be the best estimation either. The dependence is, nevertheless, approximately linear on a semi-logarithmic scale; so our third estimation, \Cref{eqn:Rq3}, assumes $R_{Q\mathrm{f}}$ to be the geometric mean of $\ensuremath{R_{\mathrm{off},Q}}$ and $R_{\mathrm{min},Q}$. If the timestep of \gls{imply} operation is limited, we do not expect $R_{Q,1}$ to provide an appropriate estimation, since this is the overall optimum scenario. $R_{Q,2}$ and $R_{Q,3}$ might both be of value to the circuit designer, because they represent a non-ideal scenario, chosen based on design parameters. \par
Following the same steps as before, we can formulate the constraint for $\ensuremath{v_{\mathrm{on},P}}$:
\begin{align}
\Delta w_P &= \ensuremath{k_{\mathrm{on},P}} \left(\frac{-V_{P\mathrm{f},j}}{\ensuremath{v_{\mathrm{on},P}}}-1\right)^\alpha \Delta T\\
\Delta w_P &\leq \Delta w_\mathrm{max}\\
\ensuremath{v_{\mathrm{on},P}}&\leq\frac{-V_{P\mathrm{f},j}}{\sqrt[\leftroot{-1}\uproot{2}\scriptstyle\alpha]{\frac{\Delta w_\mathrm{max}}{\ensuremath{k_{\mathrm{on},P}}\Delta T}} +1}\label{eqn:vonp_wmax}
\end{align}
\Cref{tab:constraints_summary} provides a summary of relevant constraints on memristor parameters, that were derived in this section. Most of these relations depend on multiple parameters of both memristors. Thus, the permissible value range of one parameter is impacted by the values of other parameters, and vice versa. Once the value of a parameter is determined (either decided by the designer or given by the technology) respective equations in \Cref{tab:constraints_summary} determine the tolerable range of variation in others. This bidirectional view enables us to define an operating area, which, in turn, allows us to predict how the circuit will react to variations in the concerned parameters.
\begin{table}[t]
\scriptsize
\centering
\caption{Summary of related constraints on parameters of $P$ and $Q$.}
\def1.2{1.2}
\begin{tabular}{c|c}
\hline\hline
\multirow{2}{1cm}{Constraint} & Constrained \\
& parameters\\
\hline
$\ensuremath{v_{\mathrm{on},Q}}>f(\ensuremath{R_Q}\equiv\Rtext{OH})$ & \multirow{2}{.6cm}{$\ensuremath{v_{\mathrm{on},Q}}$} \\
\Cref{eqn:vonq_case1}&\\
\hline
$\ensuremath{R_{\mathrm{off},P}}>f(\ensuremath{v_{\mathrm{on},Q}},\ensuremath{R_Q}\equiv\Rtext{OH})$ & \multirow{2}{1.4cm}{$\ensuremath{R_{\mathrm{off},P}}, \ensuremath{v_{\mathrm{on},Q}}$} \\
\Cref{eqn:Roffp_case1}&\\
\hline
$\ensuremath{v_{\mathrm{on},Q}}>f(\ensuremath{R_Q}\equiv\Rtext{OL})$ & \multirow{2}{.6cm}{$\ensuremath{v_{\mathrm{on},Q}}$} \\
\Cref{eqn:vonq_case3}&\\
\hline
$\ensuremath{R_{\mathrm{on},P}}<f(\ensuremath{v_{\mathrm{on},Q}},\ensuremath{R_Q}\equiv\Rtext{OL})$ & \multirow{2}{1.4cm}{$\ensuremath{R_{\mathrm{on},P}}, \ensuremath{v_{\mathrm{off},Q}}$} \\
\Cref{eqn:Ronp_case3}&\\
\hline
$\ensuremath{v_{\mathrm{on},Q}}>f(\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{on},Q}},\ensuremath{R_{\mathrm{off},Q}},\ensuremath{k_{\mathrm{on},Q}})$ & $\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{on},Q}},\ensuremath{R_{\mathrm{off},Q}},$ \\
\Cref{eqn:vonq_wmin}& $\ensuremath{v_{\mathrm{on},Q}},\ensuremath{k_{\mathrm{on},Q}}$ \\
\hline
$\ensuremath{v_{\mathrm{on},P}}<f(\ensuremath{R_{\mathrm{on},P}},\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{off},Q}},\ensuremath{k_{\mathrm{on},P}})$ & $\ensuremath{R_{\mathrm{on},P}},\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{off},Q}},$ \\
\Cref{eqn:vonp_wmax}& $\ensuremath{v_{\mathrm{on},P}},\ensuremath{k_{\mathrm{on},P}}$ \\
\hline
$\ensuremath{R_{\mathrm{on},P}}<\Rtext{IH}$, \Cref{eqn:Ronp_smaller_RIH} &$\ensuremath{R_{\mathrm{on},P}}$\\
$\ensuremath{R_{\mathrm{off},P}}>\Rtext{IL}$, \Cref{eqn:Roffp_greater_RIL}& $\ensuremath{R_{\mathrm{off},P}}$ \\
$\ensuremath{R_{\mathrm{on},Q}}<\Rtext{IH}$, \Cref{eqn:Ronq_smaller_RIH} & $\ensuremath{R_{\mathrm{on},Q}}$\\
$\ensuremath{R_{\mathrm{off},Q}}>\Rtext{IL}$, \Cref{eqn:Roffq_greater_RIL} & $\ensuremath{R_{\mathrm{off},Q}}$\\
\hline\hline
\end{tabular}
\label{tab:constraints_summary}
\end{table}
\section{Simulation -- Single Gate}\label{sec:sim_single}
\subsection{Circuit design}\label{subsec:sim_circuit}
The simulated circuit corresponds to the circuit shown in \Cref{fig:imply_gate}, with the addition that $\ensuremath{R_\mathrm{G}}$ can be shorted by a parallel switch. The driver circuits are ideal voltage sources with serial switches for High-Z mode. Each switch is modeled with an on-resistance of $1\,\mathrm{n\Omega}$ and an off-resistance of $1\,\mathrm{G\Omega}$. Given the memristor properties, especially $\ensuremath{R_\mathrm{on}}$ and $\ensuremath{R_\mathrm{off}}$, five circuit-level parameters have to be determined. These are $\ensuremath{R_\mathrm{G}}, \ensuremath{V_\mathrm{set}}, \ensuremath{V_\mathrm{cond}}, \ensuremath{V_\mathrm{reset}}$ and $\ensuremath{V_\mathrm{read}}$. Choosing $\ensuremath{V_\mathrm{reset}}$ is somewhat straightforward, as it is only used for initialization and not the \gls{imply} operation per s\'{e}. \par
For this work, this voltage was set to $\ensuremath{V_\mathrm{reset}}=-1\,\mathrm{V}$.
Next, $\ensuremath{V_\mathrm{set}}$ and $\ensuremath{V_\mathrm{cond}}$ are determined. We define $\ensuremath{V_\mathrm{set}}=1\,\mathrm{V}, \ensuremath{V_\mathrm{cond}}=0.9\,\mathrm{V}$, based on the memristor's properties and~\Cref{eqn:vset_relation,eqn:vcond_relation1}. With the voltages set, the constraints on $\ensuremath{R_\mathrm{G}}$~\cite{kvatinsky_imply_logic_design} can be evaluated, which leads to: $5.000\,\mathrm{k\Omega} < \ensuremath{R_\mathrm{G}} < 230.769\,\mathrm{k\Omega}$. $\ensuremath{R_\mathrm{G}}=40\,\mathrm{k\Omega}$ was chosen as the value of this resistor, which is lower than the geometric mean ($100\,\mathrm{k\Omega}$) proposed by~\cite{kvatinsky_imply_principles}. A summary of model and circuit parameters is shown in~\Cref{tab:parameters}, where the former are based on experimental results from previous works~\cite{Taherinejad2019newcas,semiparallel_imply_adder}.
\begin{table}[t]
\centering
\caption{Nominal values of model and circuit parameters.}
\label{tab:parameters}
\scriptsize
\begin{tabular}{l c c c c c c}
\hline\hline
Parameter &$\ensuremath{v_\mathrm{on}}$&$\ensuremath{v_\mathrm{off}}$&$\ensuremath{R_\mathrm{on}}$&$\ensuremath{R_\mathrm{off}}$&\multicolumn{2}{c}{$\ensuremath{k_\mathrm{on}}$}\\
Value &$-0.7\,\mathrm{V}$&$10\,\mathrm{mV}$&$10\,\mathrm{k\Omega}$&$1\,\mathrm{M\Omega}$&\multicolumn{2}{c}{$1\,\mathrm{cm/s}$} \\
\hline
Parameter&$\ensuremath{\alpha_\mathrm{on}}$&$\ensuremath{\alpha_\mathrm{off}}$&$\ensuremath{w_\mathrm{on}}$&$\ensuremath{w_\mathrm{off}}$&\multicolumn{2}{c}{$\ensuremath{k_\mathrm{off}}$}\\
Value&3&3&$3\,\mathrm{nm}$&$0\,\mathrm{nm}$&\multicolumn{2}{c}{$-0.5\,\mathrm{nm/s}$}\\
\hline
Parameter& $\ensuremath{a_\mathrm{on}}$ &$\ensuremath{a_\mathrm{off}}$&$w_\mathrm{c}$\\
Value& $3\,\mathrm{nm}$&$0\,\mathrm{nm}$&$0.1\,\mathrm{nm}$\\
\hline
Parameter & $\ensuremath{V_\mathrm{set}}$&$\ensuremath{V_\mathrm{cond}}$&$\ensuremath{V_\mathrm{reset}}$&$\ensuremath{V_\mathrm{read}}$&$\ensuremath{R_\mathrm{G}}$&$T$\\
Value& $1.0\,\mathrm{V}$&$0.9\,\mathrm{V}$& $-1.0\,\mathrm{V}$&$0.1\,\mathrm{V}$&$40\,\mathrm{k\Omega}$&$15\,\mathrm{\mu s}$\\
\hline\hline
\end{tabular}
\end{table}
\Cref{eqn:Rmin,eqn:s_ui} are evaluated in order to get the operation constraints imposed by the circuit. Namely, $R_{\mathrm{min},Q} = 101.449\,\mathrm{k\Omega}$ and $s_{\mathrm{min},Q} = 0.908$. We can see that, in Case 1 and assuming no state drift in $P$, the output memristor $Q$ can never reach a state higher than $s_{\mathrm{min},Q}$ or, equivalently, can never have a resistance lower than $R_{\mathrm{min},Q}$.
\subsection{Methodology \& Setup}\label{subsec:method}
\begin{figure}[t]
\centering
\tikzset{%
Hstyle/.style={fill,RoyalBlue},%
Lstyle/.style={fill,LimeGreen},%
lbl/.style={left,font=\sffamily\bfseries\color{black}},%
logic/.style={left,font=\sffamily\bfseries\color{black}\footnotesize},%
thresh/.style={logic,right,align=center},
Indef/.style={fill,gray!60},%
Ondef/.style={fill,gray}
}
\def.8{.8}
\begin{tikzpicture}[scale=.6, transform shape]
\node[lbl] at (0,5.55) {Scheme:};
\begin{scope}[shift={(0,0)}]
\node[lbl] at (.8,5.5) {1/2};
\fill[Hstyle] (0,2.5) rectangle (.8,5);
\fill[Lstyle] (0,0) rectangle (.8,2.5);
\path[] (0,0) node[lbl]{0}
(0,.5*5) node[lbl]{0.5}
(.8,.5*5) node[thresh] {\footnotesize s$_\mathbf{\mathsf{OH}}$,\,s$_\mathbf{\mathsf{OL}}$\\ s$_\mathbf{\mathsf{IH}}$,\,s$_\mathbf{\mathsf{IL}}$}
(0,5) node[lbl]{1};
\end{scope}
\begin{scope}[shift={(3.5,0)}]
\node[lbl] at (.8,5.5) {1/3};
\fill[Hstyle] (0,.667*5) rectangle (.8,5);
\fill[Lstyle] (0,0) rectangle (.8,.333*5);
\fill[Indef] (0,.333*5) rectangle (.8,.667*5);
\path[] (0,0) node[lbl]{0}
(0,.333*5) node[lbl]{0.333}
(.8,.333*5) node[thresh] {s$_\mathbf{\mathsf{OL}}$,\,s$_\mathbf{\mathsf{IL}}$}
(0,.667*5) node[lbl]{0.667}
(.8,.667*5) node[thresh] {s$_\mathbf{\mathsf{OH}}$,\,s$_\mathbf{\mathsf{IH}}$}
(0,5) node[lbl]{1};
\end{scope}
\begin{scope}[shift={(7,0)}]
\node[lbl] at (.8+.2,5.5) {TTL};
\fill[Hstyle] (0,2.4) rectangle (.8,5);
\fill[Lstyle] (0,0) rectangle (.8,.4);
\fill[Ondef] (0,.4) rectangle (.8,2.4);
\fill[Indef] (0,.8) rectangle (.8,2.0);
\path[] (0,0) node[lbl]{0}
(0,.4) node[lbl]{0.08}
(.8,.4) node[thresh] {s$_\mathbf{\mathsf{OL}}$}
(0,.8) node[lbl]{0.16}
(.8,.8) node[thresh] {s$_\mathbf{\mathsf{IL}}$}
(0,2) node[lbl]{0.40}
(.8,2) node[thresh] {s$_\mathbf{\mathsf{IH}}$}
(0,2.4) node[lbl]{0.48}
(.8,2.4) node[thresh] {s$_\mathbf{\mathsf{OH}}$}
(0,5) node[lbl]{1};
\end{scope}
\begin{scope}[shift={(-1.5,-1)}]
\fill[Hstyle] (0,0) rectangle (.5,.5);
\fill[Lstyle] (2.5,0) rectangle (3,.5);
\fill[Ondef] (5,0) rectangle (5.5,.5);
\fill[Indef] (8,0) rectangle (8.5,.5);
\node[thresh] at (.5,.2) {Logic `1'};
\node[thresh] at (3,.2) {Logic `0'};
\node[thresh] at (5.5,.2) {Undefined\\\scriptsize Output};
\node[thresh] at (8.5,.2) {Undefined\\\scriptsize Input \& Output};
\end{scope}
\end{tikzpicture}
\caption{Different logic thresholds used in this paper.}
\label{tab:logic_thresholds}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=.25\textwidth]{img/corner_analysis_legend_special.pdf}
\caption{Four squares show the state of each variable in a simulation set and the outline color (green or red) shows the simulation result (correct or failed, respectively).}
\label{fig:legend_results_variation_von_voff}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=.85\textwidth]{img/corner_analysis_multiplot_special_regular.pdf}
\caption{Results summary for different degrees of variation in $\ensuremath{v_\mathrm{on}}, \ensuremath{v_\mathrm{off}}$ of $P$ and $Q$. The $1/3$ logic thresholds scheme was used here.}
\label{fig:full_results_variation_von_voff}
\end{figure*}
Proper \gls{imply} operation results -- with respect to the output logic thresholds -- are used to determine reliability. Correct operation is ensured when state changes within the memristors are occurring (switching conditions met) and are fast enough to exceed the given logic thresholds. We apply three different logic threshold schemes (shown in \Cref{tab:logic_thresholds}) to evaluate the operation results in relation to the chosen logic threshold. Each scheme defines separate, normalized input ($s_\mathrm{IH}$, $s_\mathrm{IL}$) and output ($s_\mathrm{OH}$, $s_\mathrm{OL}$) thresholds, as in conventional digital logic. Whereas the ``1/2'' and ``1/3'' scheme were chosen arbitrarily, the ``TTL'' scheme is derived from standard TTL ($V_\mathrm{CC}=5\,\mathrm{V}$)~\cite{TI_ttl_thresholds}. This is done by normalizing the threshold voltages $V_\mathrm{IH}$, $V_\mathrm{IL}$, $V_\mathrm{OH}$ and $V_\mathrm{OL}$ to $V_\mathrm{CC}$ -- e.g. $s_\mathrm{IH}=V_\mathrm{IH}/V_\mathrm{CC}$. The range between high and low thresholds, $[s_{\mathrm{IL}},s_{\mathrm{IH}}]$ and $[s_{\mathrm{OL}},s_{\mathrm{OH}}]$, is forbidden; in other words, the logic values and states in those ranges are considered undefined.
Reasons for failures are not separately determined in our setup. Hence, failures during initialization, which lead to erroneous operation results, are counted as regular failures and are not distinguished from errors during the operation itself. Further, our simulation setup utilizes constant timesteps, so actual switching time are not explicitly measured. \par
To obtain a nominal timebase for the \gls{imply} gate, a transient analog simulation of the memristor model was conducted. Examining the resulting waveform of the normalized state $s$ after the simulation showed that it takes $15\,\mathrm{\mu s}$ to switch from $1\%$ to $99\%$ of the state boundaries. Thus, the timestep of circuit operation is set to $T=15\,\mathrm{\mu s}$. Every action (initialization, \gls{imply} operation, readout) is executed using this fixed timestep. \par
Analog transient simulations were conducted in LTSpice, making use of this setup. Two memristor parameters per device ($\ensuremath{R_\mathrm{on}}, \ensuremath{R_\mathrm{off}}$ or $\ensuremath{v_\mathrm{on}}, \ensuremath{v_\mathrm{off}}$ or $\ensuremath{k_\mathrm{on}}, \ensuremath{k_\mathrm{off}}$) were varied simulatenously within the ranges reported in measurements~\cite{CAS_physRealization} and relative to the nominal state with a maximum deviation of $\pm 50\%$.
\subsection{Result Presentation Method}\label{subsec:postprocess}
To display the numerous results, we have come up with a presentation method of our own, which we introduce here. \par
Each parameter set is represented by a group of four squares. The left two squares, as displayed in \Cref{fig:legend_results_variation_von_voff}, show parameter values of memristor $P$, and the right two show that of memristor $Q$. The filling of each square represents the state of the corresponding parameter: empty means minimum, half-filled nominal and fully filled
maximum. \Cref{fig:legend_results_variation_von_voff} shows this concept and provides an example, too. The outline color of the squares shows whether the simulation result for a set of parameter variation ($\Delta$) was correct (highlighted by green) or incorrect (highlighted by red). In general, any combination of four parameters can be varied concurrently and displayed this way. However, our approach was to use three parameter sets: $\{\ensuremath{R_{\mathrm{on},P}}, \ensuremath{R_{\mathrm{off},P}}, \ensuremath{R_{\mathrm{on},Q}},\ensuremath{R_{\mathrm{off},Q}}\}$, $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{off},P}},\ensuremath{v_{\mathrm{on},Q}},\ensuremath{v_{\mathrm{off},Q}}\}$ and $\{\ensuremath{k_{\mathrm{on},P}},\ensuremath{k_{\mathrm{off},P}},\ensuremath{k_{\mathrm{on},Q}},\ensuremath{k_{\mathrm{off},Q}}\}$, as explained in \Cref{subsec:method}. \Cref{fig:full_results_variation_von_voff} shows a {complete set of simulations} for the parameters $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{off},P}},\ensuremath{v_{\mathrm{on},Q}},\ensuremath{v_{\mathrm{off},Q}}\}$. These resulting sets are then used to quickly identify those parameters that are common between different failed runs. For example, \Cref{fig:full_results_variation_von_voff} shows that the \gls{imply} operation produces no correct output if either, $\ensuremath{v_{\mathrm{on},Q}}$ or $\ensuremath{v_{\mathrm{on},P}}$, is at its maximum value for variations greater than or equal to $10\%$.
\subsection{Results analysis}\label{subsec:sim_analysis}
Combining the math provided in \Cref{sec:math} and the simulation results obtained in \Cref{subsec:postprocess} into joint graphs gives us \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m2_R,fig:R_P_limit_m2_k,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}. First, we take a closer look at \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m2_R,fig:R_P_limit_m2_k}, because they represent the most relatable logic threshold scheme, derived from traditional TTL thresholds. \Cref{fig:R_P_limit_m0_v,fig:R_P_limit_m1_v} show the same equations as in \Cref{fig:R_P_limit_m2_v}, plotted for the 1/2 and 1/3 threshold scheme, respectively. The other two graphs for these logic threshold schemes are omitted as they lead to the same conclusions as \Cref{fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}. Furthermore, the threshold voltages turned out to be the most critical parameters, so special attention is given to their results.
\subsubsection{Graph structure}\label{subsubsec:graph_structure}
Here, we explain how these graphs are composed.
Parameters $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{off},P}}$ of memristor $P$ are always shown on the y-axis since $R_P$ is crucial for the outcome of the operation.
We can also see that from the fact that $\ensuremath{R_{\mathrm{off},P}}$ or $\ensuremath{R_{\mathrm{on},P}}$ are present in all of the constraints described in \Cref{sec:math}. Different parameters are used in each graph for the x-axes. \par
Colored curves and areas are used to show constraints and important ranges:
\newif\ifgrayscale\grayscalefalse
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\definecolor{SandPurple}{HTML}{B57DA5}
\grayscalefalse
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\ifgrayscale
\colorlet{RonAreaColor}{black!10}
\colorlet{RonHatchColor}{black}
\colorlet{RoffVqAreaColor}{black!50}
\colorlet{RoffHatchColor}{black}
\colorlet{RoffVpAreaColor}{black!30}
\colorlet{OkColor}{white}
\colorlet{FailColor}{black}
\colorlet{UnsureColor}{black!50}
\colorlet{OtherErrorColor}{black!90}
\colorlet{StaticRColor}{black!80}
\colorlet{StaticvColor}{black!30}
\colorlet{StaticRonColor}{black!80}
\colorlet{StaticvRonColor}{black!30
\colorlet{DynVqColor}{black!60}
\colorlet{DynVpRoughColor}{black!40}
\colorlet{DynVpArMeanColor}{black!20}
\colorlet{DynVpGeoMeanColor}{black!30}
\colorlet{ThreshColor}{black}
\colorlet{ZerovColor}{white}
\colorlet{RoffAreaColor}{black!50}
\colorlet{DynRoffqColor}{black!60}
\colorlet{DynRonqColor}{black!40}
\colorlet{ThreshpColor}{black}
\colorlet{ThreshqColor}{black!30}
\colorlet{RoffKqAreaColor}{black!50}
\colorlet{RoffKpAreaColor}{black!30}
\colorlet{DynKqColor}{black!60}
\colorlet{DynKpRoughColor}{black!40}
\colorlet{DynKpArMeanColor}{black!20}
\colorlet{DynKpGeoMeanColor}{black!30}
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\else
\colorlet{RonAreaColor}{SandPurple!60}
\colorlet{RonHatchColor}{SandPurple}
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\colorlet{RoffHatchColor}{SandBlue!45
\colorlet{RoffVpAreaColor}{white
\colorlet{OkColor}{green!70}
\colorlet{FailColor}{red!80}
\colorlet{UnsureColor}{Orange!50!Yellow}
\colorlet{OtherErrorColor}{FailColor}
\colorlet{StaticRColor}{ForestGreen}
\colorlet{StaticvColor}{Orange!90!black
\colorlet{StaticRonColor}{RedViolet}
\colorlet{StaticvRonColor}{Black!70
\colorlet{DynVqColor}{blue!50!SandBlue}
\colorlet{DynVpRoughColor}{Sepia}
\colorlet{DynVpArMeanColor}{lime}
\colorlet{DynVpGeoMeanColor}{OliveGreen}
\colorlet{ThreshColor}{Cyan}
\colorlet{ZerovColor}{RoyalBlue}
\colorlet{RoffAreaColor}{SandBlue!60}
\colorlet{DynRoffqColor}{StaticvColor}
\colorlet{DynRonqColor}{blue!50!SandBlue}
\colorlet{ThreshpColor}{Cyan}
\colorlet{ThreshqColor}{LimeGreen}
\colorlet{RoffKqAreaColor}{white
\colorlet{RoffKpAreaColor}{white
\colorlet{DynKqColor}{blue!50!SandBlue}
\colorlet{DynKpRoughColor}{Sepia}
\colorlet{DynKpArMeanColor}{lime}
\colorlet{DynKpGeoMeanColor}{OliveGreen}
\defnone{none}
\fi
\def\itlen{5mm}
\def\itsep{1.5mm}
\def\itdum{-.8ex}
\begin{itemize}
\item[%
\tikz{\draw[black,thin,dashed] (0mm,0mm)--(\itlen,0mm);%
\path (0mm,\itdum)--(\itlen,\itdum);}]%
Black, dashed lines indicate nominal parameter values
\item[%
\tikz{\draw[ThreshColor, thick] (0mm,0mm)--(\itlen,0mm);%
\draw[ThreshColor,thick,dashed](0mm,-\itsep)--(\itlen,-\itsep);}]%
Light blue lines show input logic thresholds $R_\mathrm{IH}$ (solid) and $R_\mathrm{IL}$ (dashed)
for memristor $P$.
\item[%
\tikz{\draw[StaticRonColor, thick] (0mm,0mm)--(\itlen,0mm);%
\draw[DynVqColor, thick] (0mm,-.5*\itsep)--(.5*\itlen,-.5*\itsep);%
\draw[DynVqColor, thick,dotted] (.5*\itlen,-.5*\itsep)--(\itlen,-.5*\itsep);%
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\draw[DynVpArMeanColor, thick,dotted] (.5*\itlen,-\itsep)--(\itlen,-\itsep);}]%
Colored curves show the constraints from \Cref{sec:math}. Dotted parts indicate invalid plotting ranges, which do not correspond to any real value in physical devices.
\item[%
\tikz{\draw[StaticRColor, thick,-latex] (0mm,.5*\itsep)--(\itlen,.5*\itsep);%
\draw[DynVqColor, thick,latex-] (0mm,-.5*\itsep)--(\itlen,-.5*\itsep);}]%
Arrows indicate how the constraints restrict the operating area of a parameter, i.e., which side of the curve is acceptable due to the given constraint.
\item[%
\tikz{\fill[RoffHatchColor] (0mm,0mm) rectangle (\itlen,-\itsep);}]%
Blue areas show valid ranges of $\ensuremath{R_{\mathrm{off},P}}$ and the respective parameters on the x-axes. For example, in \Cref{fig:R_P_limit_m2_v}, this area represents valid ranges of $\ensuremath{R_{\mathrm{off},P}}$ versus $\ensuremath{v_{\mathrm{on},Q}}$, $\ensuremath{v_{\mathrm{off},Q}}$, $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{off},P}}$. Note that for $\ensuremath{v_{\mathrm{on},P}}$ our recommended range was used to limit the valid area, as the three different curves are only weak constraints.
\item[%
\tikz{\fill[RonAreaColor] (0mm,0mm) rectangle (\itlen,-\itsep);}]%
Purple areas show valid ranges of $\ensuremath{R_{\mathrm{on},P}}$ and the respective parameters on the x-axes. For example, in \Cref{fig:R_P_limit_m2_k}, this area represents valid ranges of $\ensuremath{R_{\mathrm{on},P}}$ versus $\ensuremath{k_{\mathrm{on},Q}}$, $\ensuremath{k_{\mathrm{off},Q}}$, $\ensuremath{k_{\mathrm{on},P}}$ and $\ensuremath{k_{\mathrm{off},P}}$. Note that restrictions on x-axis parameters are inherited from the $\ensuremath{R_{\mathrm{off},P}}$ operating area.
\item[%
\tikz{\draw[FailColor,line width=1mm] (0mm,0mm)--(.33*\itlen,0mm);%
\draw[UnsureColor,line width=1mm](.33*\itlen,0mm)--(.66*\itlen,0mm);%
\draw[OkColor,line width=1mm] (.66*\itlen,0mm)--(\itlen,0mm);%
\path(0mm,\itdum)--(\itlen,\itdum);}]%
The bars at each side of the graphs overlay our simulation results. Red sections show incorrect \gls{imply} results, green sections show correct results and orange sections are used for ranges in between, which are not explicitly covered by the simulations.
\end{itemize}
\subsubsection{Variation in voltage threshold}\label{par:v_var}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m2_v.pdf}
\caption{Analytical constraints and logic thresholds for the TTL scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},v_P, v_Q\}$. }
\label{fig:R_P_limit_m2_v}
\end{figure}
\Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v} depict voltage thresholds $\ensuremath{v_{\mathrm{on},P}}$, $\ensuremath{v_{\mathrm{off},P}}$, $\ensuremath{v_{\mathrm{on},Q}}$ and $\ensuremath{v_{\mathrm{off},Q}}$ of memristor $P$ and $Q$, as well as resistance parameters $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{off},P}}$ of $P$ using different logic threshold schemes (\Cref{tab:logic_thresholds}). For the analysis we concentrate on the TTL scheme, \Cref{fig:R_P_limit_m2_v}. \par
The logic thresholds ($R_\mathrm{IH}$, $R_\mathrm{IL}$) divide the plot into two parts: The bottom part concerning $\ensuremath{R_{\mathrm{on},P}}$ and the top part concerning $\ensuremath{R_{\mathrm{off},P}}$. Adding the static constraints, \Cref{eqn:vonq_case1,eqn:vonq_case3,eqn:Roffp_case1,eqn:Ronp_case3}, on top of the logic thresholds decreases the valid range of $\ensuremath{R_{\mathrm{off},P}}$, $\ensuremath{v_{\mathrm{on},Q}}$ and in particular $\ensuremath{R_{\mathrm{on},P}}$. The latter is evident from the purple area in \Cref{fig:R_P_limit_m2_v}, which is smaller than the plotted range. However, regarding $\ensuremath{R_{\mathrm{off},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$, the dynamic constraint, \Cref{eqn:vonq_wmin}, is even stricter than the static constraint. \par
There are no static constraints for $\ensuremath{v_{\mathrm{on},P}}$. A rough dynamic estimation is provided by \Cref{eqn:vonp_wmax}, which depends on $V_{P\mathrm{f}}$. As discussed in \Cref{subsec:dyn_math}, \Cref{eqn:vonp_wmax} is evaluated three times, using $R_{Q,1}$, $R_{Q,2}$ and $R_{Q,3}$, respectively. The three curves are drawn in brown, dark green and light green. No constraint for $\ensuremath{v_\mathrm{off}}$ has been found (\Cref{sec:math}). Hence, the valid ranges of $R_P$ over $\{\ensuremath{v_{\mathrm{off},P}}, \ensuremath{v_{\mathrm{off},Q}}\}$ are only limited by logic thresholds, \Cref{eqn:Roffp_greater_RIL,eqn:Ronp_smaller_RIH}. As a consequence of the above constraints, the valid range for each parameter is decreased and thus the advisable operating area remains as shown by the colored areas.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m0_v.pdf}
\caption{Analytical constraints and logic thresholds for the 1/2 scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},v_P, v_Q\}$. }
\label{fig:R_P_limit_m0_v}
\end{figure}
Simulation results for variation in $\ensuremath{v_{\mathrm{on},Q}}$ show very good agreement with the mathematical analysis, especially the dynamic estimation in \Cref{eqn:vonq_wmin}, which depends on \Cref{eqn:VQi,eqn:wmin}. At $+10\%$ variation of $\ensuremath{v_{\mathrm{on},Q}}$ and nominal $\ensuremath{R_{\mathrm{off},P}}$, the simulation fails (indicated by the thin red line), as the analysis predicted. \Cref{fig:R_P_limit_m2_v} shows very clearly that this failure is not accurately predicted by the static constraints from \Cref{subsec:stat_math} alone. Hence, the dynamic estimation (\Cref{subsec:dyn_math}) is vital. Variation in $\ensuremath{v_{\mathrm{on},P}}$ strengthens this point further, since different methods of estimating the dynamic behavior leads to important changes regarding the agreement of the simulations and the derived analytical constraints. On the upper end of the $\ensuremath{v_{\mathrm{on},P}}$ range, \Cref{eqn:vonp_wmax} ({evaluated using $R_Q=R_{Q,3}$ for $V_{P\mathrm{f}}$, \Cref{eqn:vpf}}) provides good congruence with our simulations, whereas \Cref{eqn:vonp_wmax} (evaluated using $R_Q=R_{Q,2}$ for $V_{P\mathrm{f}}$, \Cref{eqn:vpf}) represents a more conservative estimation. In contrast, evaluating \Cref{eqn:vonp_wmax} using the theoretical minimum $R_Q=R_{Q,1}=R_{\mathrm{min},Q}$ in \Cref{eqn:vpf}, does not yield a good estimation. On the lower end of the $\ensuremath{v_{\mathrm{on},P}}$ range, simulation results indicate some failures for $\ensuremath{v_{\mathrm{on},P}}\leq -0.84\,\mathrm{V}$ ($+20\%$). This behavior cannot be explained by any of the constraints from \Cref{sec:math}.
According to the simulation results (\Cref{subsec:postprocess}, \Cref{fig:full_results_variation_von_voff}), these specific failures only occur when $\ensuremath{v_{\mathrm{on},Q}}\geq -0.7\,\mathrm{V} (\pm 0\%)$, which leads us to believe that the reason for failure is the $20\%$ mismatch between $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$. Regarding both, $\ensuremath{v_{\mathrm{off},P}}$ and $\ensuremath{v_{\mathrm{off},Q}}$, there are almost no failures as expected, except for a (minor) failure during initialization for $\ensuremath{v_{\mathrm{off},Q}}$ at $+50\%$.\par
In terms of $R_P$ variation, the simulation results suggest that $\ensuremath{R_{\mathrm{off},P}}$ can lie within the uncertain range between logic thresholds while the \gls{imply} operation still outputs correct results. This stands to reason since the thresholds are artificial limits not governed by the circuit behavior. Further, $\ensuremath{R_{\mathrm{on},P}}$ is fine up to the lowest simulated value of $\ensuremath{R_{\mathrm{off},P}}$, because at that point $\ensuremath{R_{\mathrm{off},P}}>\ensuremath{R_{\mathrm{on},P}}$ changes to $\ensuremath{R_{\mathrm{off},P}}<\ensuremath{R_{\mathrm{on},P}}$, and hence the operation fails. \par
Combining all the simulation results and their respective analytical constraints, we can identify the areas in which the circuit is most likely to operate correctly. These are the areas highlighted in \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}. \Cref{eqn:vonq_wmin,eqn:vonp_wmax} and their respective dependencies, \Cref{eqn:VQi,eqn:wmin,eqn:vpf} (evaluated using $R_Q=R_{Q,3}$), are recommended for estimating the valid ranges of $\ensuremath{R_{\mathrm{off},P}}$ versus $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{on},Q}}\}$; whereas the static constraints {\Cref{eqn:vonq_case1,eqn:vonq_case3,eqn:Roffp_case1,eqn:Ronp_case3}} are sufficient for $\ensuremath{R_{\mathrm{on},P}}$ versus $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{on},Q}}\}$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m1_v.pdf}
\caption{Analytical constraints and logic thresholds for the 1/3 scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},v_P, v_Q\}$.}
\label{fig:R_P_limit_m1_v}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{R_P_limits_m2_R.pdf}
\caption{Analytical constraints and logic thresholds for the TTL scheme plotted over a range of memristor parameters $\{\ensuremath{R_P}, \ensuremath{R_Q}\}$.}
\label{fig:R_P_limit_m2_R}
\end{figure}
\subsubsection{Variation in resistance limits}\label{par:r_var}
There are no static constraints limiting $\ensuremath{R_{\mathrm{on},Q}}$ or $\ensuremath{R_{\mathrm{off},Q}}$. Therefore, only logic thresholds and the dynamic estimation of \Cref{eqn:vonq_wmin} can be applied. The latter depends on \Cref{eqn:VQi,eqn:wmin} and is evaluated in two ways: First, varying $\ensuremath{R_{\mathrm{on},Q}}$, but not $\ensuremath{R_{\mathrm{off},Q}}$; and second varying $\ensuremath{R_{\mathrm{off},Q}}$, but not $\ensuremath{R_{\mathrm{on},Q}}$. It is interesting to see that -- for any of the three schemes of \Cref{tab:logic_thresholds} -- the logic thresholds limit the operating areas (blue and purple) much more than the actual analytical constraints. Simulation results for $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{off},P}}$ are identical to \Cref{fig:R_P_limit_m2_v}, however, $\ensuremath{R_{\mathrm{off},Q}}$ cannot reach as low as $\ensuremath{R_{\mathrm{off},P}}$ without causing a failure. This is solely due to the chosen logic thresholds, as an \gls{imply} output of $\ensuremath{R_{\mathrm{off},Q}}<R_\mathrm{OL}$ is considered as failure. \par
Overall, resistance variation does not seem to hold as much potential for failures as variation in threshold voltage(s) does. \Cref{eqn:vonq_wmin} and its dependencies, \Cref{eqn:VQi,eqn:wmin}, can be used to identify valid parameter ranges, but -- based on our simulation results -- it is most likely not necessary. This is true for all three logic threshold schemes listed in \Cref{tab:logic_thresholds}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m2_k.pdf}
\caption{Analytical constraints and logic thresholds for the TTL scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},k_P,k_Q\}$. }
\label{fig:R_P_limit_m2_k}
\end{figure}
\subsubsection{Variation in switching speed}\label{par:k_var}
The dynamic constraint in \Cref{eqn:vonq_wmin} can be used to extract limits of $\ensuremath{k_{\mathrm{on},Q}}$, while \Cref{eqn:vonp_wmax} provides the basis for the analysis of $\ensuremath{k_{\mathrm{on},P}}$. \Cref{fig:R_P_limit_m2_k} shows the plotted equations and logic thresholds. \Cref{eqn:vonp_wmax} (evaluated using \Cref{eqn:vpf}, where $R_Q=R_{Q,1}$) is omitted, as well as all constraints containing $\ensuremath{k_{\mathrm{off},P}}$ and $\ensuremath{k_{\mathrm{off},Q}}$, since they are far outside of the plotted range.
The graph in \Cref{fig:R_P_limit_m2_k} shows that $\ensuremath{k_{\mathrm{on},P}}$ is hardly restricted by any constraint. Only at relatively high values, greater than $+50\%$ variation, \Cref{eqn:vonp_wmax} (evaluated using \Cref{eqn:vpf} with $R_Q=R_{Q,2}$) comes into effect, but cannot be compared to simulation results, as our simulated range ends at $+50\%$, in compliance with our methodology (\Cref{subsec:method}). In contrast, \Cref{eqn:vonq_wmin} provides a reasonable constraint for $\ensuremath{k_{\mathrm{on},Q}}$. Nonetheless, our simulated range only reaches down to $-50\%$ and thus results cannot be compared to the constraint. The other two logic threshold schemes show similar behavior. As before, the colored areas indicate the merged, predicted functional range of both, $\ensuremath{k_{\mathrm{on},P}}$ and $\ensuremath{k_{\mathrm{on},Q}}$.
\par
In conclusion, switching speed $k$ of both memristors can vary at least by $\pm50\%$ without performance issues, according to our simulation. Analytical constraints suggest that there is a lower boundary for $\ensuremath{k_{\mathrm{on},Q}}$ at approximately $2\,\mathrm{mm/s}$ ($-80\%$).
\section{Simulation -- Crossbar}\label{sec:sim_xbar}
\subsection{Setup}\label{subsec:xbar_circuit}
Analogous to the single \gls{imply} gate simulation setup (\Cref{subsec:sim_circuit}), the circuit in \Cref{fig:imply_gate} is the basis for the crossbar simulation. A complete $128\times 128${} cell 1T1R crossbar circuit was used. The \gls{imply} gate is formed by two memristors arbitrarily located within the crossbar. Each memristor has its own access device, in our case an ideal switch, and is connected to adjacent cells via resistors that model the nanowire resistances. The ideal switch is modeled using an on-resistance of $1\,\mathrm{\mu\Omega}$ and an off-resistance of $100\,\mathrm{M\Omega}$. Line resistances were chosen to be $10\,\mathrm{\Omega}$ each, according to the worst case in~\cite{sneak_paths_closed_form}. \Cref{fig:xbar_cell} shows the structure of a single cell.
\begin{figure}
\centering
\begin{circuitikz}[font=\Large,scale=.6,transform shape]
\node[rotate=0] at(-.7,0) {\dots};
\node at (-1.9,0) {word $y$};
\node[rotate=90] at(2.5,3.3) {\dots};
\node at (2.5,3.8) {bit $x$};
\node[rotate=0] at (3.2,0) {\dots};
\node[rotate=90] at (2.5,-.7) {\dots};
\draw(2.5,2.8) to[short,-*] (2.5,2.5) to [R,l=$R_{xy/x(y+1)}$] (2.5,0) to[short] (2.5,-.3)
(-.3,0) to[short,-*] (0,0) to[R,l_=$R_{xy/(x+1)y}$] (2.5,0) to[short](2.8,0);
\draw (0,0) to[memristor,*-,l=$M_{xy}$] (1.5,1.5) to[short] (1.7,1.7) to[short] (1.9,2.15) (2,2) to [short](2.5,2.5);
\node[rotate=45] at (1.5,2.3) {$T_{xy}$};
\end{circuitikz}
\caption{Structure of a single cell within the 1T1R crossbar, including line resistances.}
\label{fig:xbar_cell}
\end{figure}
Circuit parameters of the \gls{imply} gate are identical to \Cref{subsec:sim_circuit}, \Cref{tab:parameters}. Bit-line drivers are attached at the top and bottom for symmetry. The readout strategy described in \Cref{sec:xbar_fundamental} was implemented. Analog transient simulations were conducted in Cadence Spectre. The method of parameter variation is the same as defined for the single gate in \Cref{subsec:method}, except that only relative parameter variations ($\pm 50\%$) were conducted for the crossbar.
\begin{figure}[b]
\centering
\begin{tikzpicture}
\begin{axis}[
ybar,
ymin=0,
grid=major,
xtick distance=0.25,
width=0.6\linewidth,
height=.35\linewidth,
xlabel=$s$,
ylabel={$n$ per bin},
font=\footnotesize,
]
\addplot +[hist={bins=100}] table [y index=0] {data/init_states.csv};
\end{axis}
\end{tikzpicture}\vspace{-3mm}
\caption{Histogram of initial (normalized) device states, $s$, within the $128\times 128${} crossbar, plotted using 100 bins.}
\label{fig:xbar_initial_histogram}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.85\textwidth]{img/corner_analysis_multiplot_special_xbar.pdf}
\caption{Crossbar results summary for different degrees of variation in $\ensuremath{v_\mathrm{on}}, \ensuremath{v_\mathrm{off}}$ of $P$ and $Q$. Logic thresholds for `1' and `0' were set according to the TTL threshold scheme (\Cref{tab:logic_thresholds}). Memristor $P$ was at position $\{0,0\}$, while $Q$ was at the center, $\{63,63\}$.}
\label{fig:xbar_full_results_variation_von_voff}
\end{figure*}
\subsection{Methodology}\label{subsec:xbar_method}
\gls{imply} gates can be formed by any two memristors in the crossbar. Both, the worst case scenario in terms of parasitic resistance between the two memristors forming a gate, and the worst case voltage drop, were considered. Hence, four separate simulations were conducted with $P$ and $Q$ at different $\{\text{bit},\text{word}\}$ positions.
\begin{enumerate}
\item Memristor $P$ at position $\{0,0\}$, $Q$ at position $\{127,127\}$
\item Memristor $P$ at position $\{127,127\}$, $Q$ at position $\{0,0\}$
\item Memristor $P$ at position $\{0,0\}$, $Q$ at position $\{63,63\}$
\item Memristor $P$ at position $\{63,63\}$, $Q$ at position $\{0,0\}$
\end{enumerate}
Instead of using idealized ($s=0$ or $s=1$) or manually fixed initial memristor states, each cell is assigned a different initial state during (automated) netlist generation. The states are generated via Octave and follow a Gaussian distribution which has been cut in half as shown in \Cref{fig:xbar_initial_histogram}. Although this approach requires a greater effort, it represents a more realistic scenario than ideal initial states.
\begin{figure*}[]
\centering
\defSingle gate{Single gate}
\defCrossbar{Crossbar}
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\begin{tikzpicture}[scale=.6,transform shape,font=\large]
\begin{scope}[shift={(0,0)}]
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}
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\end{scope}
\begin{scope}[shift={(0,2*\barsep)}]
\node[] at (-8,0) {Single gate};
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{k_{\mathrm{off},Q}}/1\,\mathrm{\frac{nm}{s}}$} -- (4,0)node[right]{$\ensuremath{k_{\mathrm{on},Q}}/1\,\mathrm{\frac{mm}{s}}$};
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\node[] at (-8,0) {Crossbar};
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\end{scope}
\begin{scope}[shift={(\groupxsep,0)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},P}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},P}}/1\,\mathrm{mV}$};
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\end{scope}
\begin{scope}[shift={(\groupxsep,\barsep)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},P}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},P}}/1\,\mathrm{mV}$};
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\begin{scope}[shift={(\groupxsep,2*\barsep)}]
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\begin{scope}[shift={(\groupxsep,3*\barsep)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},Q}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},Q}}/1\,\mathrm{mV}$};
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\end{scope}
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\node[] at (-8,0) {Crossbar};
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\begin{scope}[shift={(\groupxsep,\groupysep+2*\barsep)}]
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{on},Q}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2,-.08) rectangle (3,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
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\end{scope}
\begin{scope}[shift={(\groupxsep,\groupysep+3*\barsep)}]
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\end{scope}
\end{tikzpicture}
\caption{Comparison of single gate and (combined) crossbar simulation results. A range of $\pm 50\%$ around the nominal value is plotted for each parameter. The results are color-coded: Green for correct \gls{imply} output, red for false output and orange for ranges inbetween, that are not covered by the simulation.}
\label{fig:xbar_vs_single}
\end{figure*}
\subsection{Results analysis}\label{subsec:xbar_analysis}
In this section we compare the crossbar simulation results against the single gate results.
As before, to be efficient, results are represented using our technique introduced in \Cref{subsec:postprocess}. \Cref{fig:xbar_full_results_variation_von_voff} shows a complete set of crossbar simulations for the parameters $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{off},P}},\ensuremath{v_{\mathrm{on},Q}},\ensuremath{v_{\mathrm{off},Q}}\}$.
\Cref{fig:xbar_vs_single} depicts the combined, i.e. worst case, results of all crossbar simulation setups explained in \Cref{subsec:xbar_method}, and the results of the single gate simulation, where the TTL threshold scheme was applied. The bars and color coding are identical to \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m2_R,fig:R_P_limit_m2_k,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}, \Cref{subsec:sim_analysis}. While the conclusions from \Cref{subsec:sim_analysis} remain true, unless noted otherwise, here we highlight the differences.
\subsubsection{Variation in voltage threshold}\label{subsubsec:xbar_v_var}
Given that the circuit is in a crossbar architecture, an increased number of errors due to threshold voltage variation can be expected in the crossbar simulation, when compared to the single gate simulation. Surprisingly, however, it is not significantly worse. \par
There are three main differences: First, the initialization failure of $\ensuremath{v_{\mathrm{off},Q}}$ (initially shown in \Cref{fig:R_P_limit_m2_v}) does not arise in the crossbar simulation. However, there were initialization failures in the crossbar for $8\,\mathrm{mV}\leq\ensuremath{v_{\mathrm{off},P}}\leq 9\,\mathrm{mV}$ ($-20\%$ to $-10\%$). Having said that, as $\ensuremath{v_\mathrm{off}}$ is of minor interest to the \gls{imply} operation, this can neither be considered an improvement, nor a degradation compared to the single gate. Second, results indicate failures if both $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$ are above $-0.63\,\mathrm{V}\ (-10\%)$ at the same time.
Based on the single gate simulation results (\Cref{subsec:sim_analysis}) and our recommendation to use \Cref{eqn:vonp_wmax} -- in combination with $R_Q=R_{Q,3}$ in \Cref{eqn:vpf} -- for device variability evaluation, this failure is predictable. As for the exact reason of this error, we assume that it is due to the increased state drift in $P$, as $|\ensuremath{v_{\mathrm{on},P}}|$ is so low. In terms of operational range, the valid values for $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$ are drastically restricted to the nominal value $\ensuremath{v_\mathrm{on}}$, as shown in \Cref{fig:xbar_vs_single}. It is only then that correct operations can be guaranteed.
However, if $\ensuremath{v_{\mathrm{on},P}} < -0.63\,\mathrm{V}\ (-10\%)$ is ensured, a much greater range for $\ensuremath{v_{\mathrm{on},Q}}$ is admissible, similar to the case of the single gate in \Cref{subsec:sim_analysis}. Finally, the third difference is that the \gls{imply} operation fails for $\ensuremath{v_{\mathrm{on},P}}<-0.77\,\mathrm{V}\ (+10\%)$ while $\ensuremath{v_{\mathrm{on},Q}}=\ensuremath{v_\mathrm{on}}$, as compared to $+20\%$ in the single gate simulation. Thus, the tolerable mismatch between $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$ shrinks to $10\%$ within the crossbar. \par
Apart from these differences the results of both simulations are identical, although \Cref{fig:xbar_vs_single} might not reveal it at the first look. This means that the proposed constraints for $\ensuremath{v_\mathrm{on}}$ and $\ensuremath{v_\mathrm{off}}$ from \Cref{sec:math} can be applied to get a basic understanding of threshold voltage variability within crossbar architectures. \par
\subsubsection{Variation in resistance limits}\label{subsubsec:xbar_r_var}
Varying the resistance limits of the memristors within the crossbar reveals some interesting results, as we can see in \Cref{fig:xbar_vs_single}. While \gls{imply} operations in the single gate simulation fail for $\ensuremath{R_{\mathrm{off},Q}}\leq 800\,\mathrm{k\Omega}\ (-20\%)$, the crossbar simulation shows correct results down to $\ensuremath{R_{\mathrm{off},Q}}=700\,\mathrm{k\Omega}\ (-30\%)$. We believe that this is due to the readout strategy applied to the crossbar, since the measured $R_Q$ after executing Case~3 (\Cref{tab:truth_table_imply}) is almost $1\,\mathrm{M\Omega}$ in a majority of the $-30\%$ simulation runs. Failures start occuring below $\ensuremath{R_{\mathrm{off},Q}}\leq 600\,\mathrm{k\Omega}\ (-40\%)$. The range between $-30\%$ and $-40\%$ variation is not explicitly covered by our simulation steps. \par
Furthermore, false \gls{imply} results within the crossbar come about at the upper and lower end of our simulated $\ensuremath{R_{\mathrm{off},Q}}$ range, as well as at the upper and lower end of the simulated $\ensuremath{R_{\mathrm{off},P}}$ range. This is a combined effect, since those errors only occur if both, $\ensuremath{R_{\mathrm{off},P}}\leq 500\,\mathrm{k\Omega}\ (-50\%)$ and $\ensuremath{R_{\mathrm{off},Q}}\geq 1.5\,\mathrm{M\Omega}\ (+50\%)$, or vice versa, are present at the same time. Interpreting this scenario based on the 1/2 or TTL logic thresholds from \Cref{tab:logic_thresholds}, one can see that if either $\ensuremath{R_{\mathrm{off},P}}$ or $\ensuremath{R_{\mathrm{off},Q}}$ are below $500\,\mathrm{k\Omega}$, they are not interpreted as logic `0', but logic `1'. Thus, they do not fulfill Case~1 of the truth table, where $p=0$ and $q=0$ must be true. Applying the 1/3 logic threshold scheme, an off-resistance of $500\,\mathrm{k\Omega}$ yields an undefined logic state. Therefore, none of the cases in the truth table is fulfilled. Hence, such errors are predicted via logic thresholds alone and do not require further evaluation using the constraints defined in \Cref{sec:math}. \par
Lastly, we should remark that simulation results for $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{on},Q}}$ in the crossbar are identical to the the single gate simulation results.
\subsubsection{Variation in switching speed}\label{subsubsec:xbar_k_var}
Swichting speed variation does not pose a threat to single \gls{imply} gates, as deduced in \Cref{subsec:sim_analysis}. However, based on our simulation results (\Cref{fig:xbar_vs_single}), behavior within a crossbar is very different. For variations in $\ensuremath{k_{\mathrm{on},P}}$ and $\ensuremath{k_{\mathrm{on},Q}}$ larger than $\pm 20\%$, the \gls{imply} operation fails. Further analysis of those failures reveals that it is the mismatch between $P$ and $Q$ which causes most errors.
If either $\Delta\ensuremath{k_{\mathrm{on},P}}\leq -20\%$ while $\Delta\ensuremath{k_{\mathrm{on},Q}}\geq +20\%$, or $\Delta\ensuremath{k_{\mathrm{on},P}}\geq +20\%$ while $\Delta\ensuremath{k_{\mathrm{on},Q}}\leq -20\%$, the operation result is wrong. This mismatch cannot be predicted by our constraints. Further, the simulation indicates failures for variation in $\ensuremath{k_{\mathrm{off},P}}$ larger than $\pm 20\%$, as well as for $\Delta\ensuremath{k_{\mathrm{off},Q}}=\pm 40\%$. As $\ensuremath{k_{\mathrm{off},P}}$ and $\ensuremath{k_{\mathrm{off},Q}}$ are never relevant during \gls{imply}, we infer that these are initialization errors. They can, however, be resolved by using a different initialization scheme than the one we applied. For example, using an additional readout cycle to confirm written initial states. Such a scheme provides feedback to resolve initialization errors before \gls{imply} is executed.
\section{Conclusion}\label{sec:conclusion}
Device variability is one of the main challenges when implementing memristor-based logic. In this paper, we formulated novel constraints based on static switching conditions and state change dynamics. We note that the underlying causes of variation in device parameters are not differentiated by our methodology. Hence, environmental effects (such as temperature) causing parameter variation are taken into account by our constraints, just as process variations are. \gls{imply} operation results after a fixed timestep of execution were used as the metric to assess gate performance. In addition, different logic threshold schemes were considered. The derived constraints were put to the test in an extensive analysis for single gate and $128\times 128${} 1T1R crossbar and their simulation results were compared. An efficient simulation results presentation method was introduced and applied to find critical parameters. \par
As a result of our analysis, variability in threshold voltages, especially $\ensuremath{v_{\mathrm{on},Q}}$, was identified as a major root of concern regarding correct operations. We conclude that the most dominant reasons for failure are predictable by our theoretical analysis for both the single gate and the crossbar. Therefore, our analysis and recommendations can be used for designing a reliable \gls{imply} gate. More specifically, we suggest to choose design parameters away from the borders of the recommended areas. Ideally, this distance should be chosen such that the typical (or maximum) variations, do not lead to crossing the borders of recommended area. Nonetheless, accompanying studies or simulations should be conducted to understand the non-deterministic errors, especially regarding voltage threshold- and switching speed mismatch within the crossbar, as well as state drift phenomena. \par
Lastly, we note that our analysis can be used to decide whether a specific memristor technology and \gls{imply} logic are compatible. To that end, technology parameters need to be assessed based on the constraints for reliable \gls{imply} operations we extracted in this work. Further, considering technology-dependent parameter variation, an acceptable margin from the borders of the operating area must be ensured. Otherwise, chances for failures in \gls{imply} operations are increased. Hence, it would be better to use other technologies to implement the intended \gls{imply}-based circuits, or use other logics to implement the intended functionalities on the given technology.\par
\bibliographystyle{unsrt2authabbrvpp}
\section{Introduction}\label{sec:intro}
Memristors are used for memory applications~\cite{Niu2010, Ho2011, Mohammad2013, Baghel2015, Radakovits2019}, where even storage of multiple bits per device is feasible~\cite{zangeneh_design_of_1t1r_reram,kim2010cnna,Taherinejad2015ems,Taherinejad2016cce}. In addition, memristors have become increasingly popular for neural network and learning applications~\cite{Pershin2012ieee,Thomas2013applied}, by exploiting their analog, synapses-like nature. Another application of memristors is implementing digital (in-memory) logic~\cite{borghetti_memristive_stateful_logic, talati_logic_design_magic, CRS_proposal}, such as \gls{imply}, for various computations~\cite{Gupta2018, Papandroulidakis2017, Rohani2017, Taherinejad2019newcas, guckert2018system, Radakovits2020tcasi}. At the moment these applications -- more often than not -- are
not verified by physical implementation and experimental data~\cite{CAS_physRealization}. This imbalance leads to many problems when actual physical implementation is desired. While material sciences have certainly progressed in this field~\cite{waser_redox_reram_physics,menzel_switching_kinetics_ecm,menzel_reram_physics}, the circuit-level interface to higher abstraction levels is not yet ready to provide a reliable base for proposed applications~\cite{CAS_physRealization}. Some of the fundamental problems, that need to be considered at design time, are {inter-device variability} and {cyclic variability}. In larger structures, usually implemented within crossbar arrays, sneak paths and wire resistance are an even bigger issue~\cite{cassuto_sneak_path_constraints,sneak_paths_closed_form}. While the two latter have received an acceptable level of attention from the community, the two former have been less explored and addressed by the community. We hope that this work encourages and provides pointers to the community to move in that direction.\par
Here, we aim to provide a better insight into the operation of a single memristive \gls{imply} logic gate by considering device variations. Similar works on that topic already exist, such as~\cite{kvatinsky_device_variations,xie_robustness_of_memristor_logic,chen_imply_ron_not_reached}, which mainly focus on other types of memristor-based logic. The most relevant work to ours is~\cite{chen_imply_ron_not_reached}, where the focus is set on the design of the \gls{imply} circuit itself, and an alternative operation ``\gls{nimp}'' is proposed to mitigate certain problems. In contrast, our work explores regular \gls{imply} in more detail, particularly regarding the effect of device variations on the \gls{imply} operation, and leaves the \gls{nimp} approach for future works. We note that there is a variety of memristors based on different physical effects~\cite{waser_redox_reram_physics}. From the perspective of this work, the internal mechanism is to a large extent inconsequential. Hence, we use the general term, memristor, to refer to \emph{resistive switching elements} or \glspl{reram} and, when needed, specify what may make the internal mechanisms important.
The main contribution of this work to the field of memristor-based logic, is an in-depth mathematical analysis of memristive \gls{imply} regarding its constraints due to device variation. Plausibility of the proposed constraints is verified via simulations using a popular model. \par
The rest of this paper is organized as follows: \Cref{sec:imply_logic}, particularly \Cref{subsec:gate_and_constr}, reviews memristive \gls{imply} logic and shows its limitations. An introduction to the crossbar architecture is given in \Cref{sec:xbar_fundamental} and the device model is described in \Cref{sec:model}.
In \Cref{sec:math}, we formulate new constraints for the \gls{imply} gate, before comparing them to the single gate simulation results in \Cref{sec:sim_single}. The results of the crossbar simulations are presented in \Cref{sec:sim_xbar} and compared against the single gate simulation and constraints. We conclude the paper in \Cref{sec:conclusion}.
\section{Material implication (IMPLY)}\label{sec:imply_logic}
The truth table of \gls{imply}, and its four different cases, are shown in~\Cref{tab:truth_table_imply}. It takes two input states $p$ and $q$ and outputs $q'$. Not every type of memristor is suitable for material implication. The devices have to exhibit \emph{voltage threshold behavior}. Moreover, all devices used for an operation shall have the same parameter values (resistance range, threshold voltages, switching speed).
\begin{table}
\centering
\caption{Truth table of material implication and its four cases}
\label{tab:truth_table_imply}
\begin{tabular}{c|cc|c}
\hline\hline
Cases & $p$ & $q$ & $q'$\\
\hline
{Case} 1 &0 & 0 & 1 \\
{Case} 2 &0 & 1 & 1 \\
{Case} 3 &1 & 0 & 0 \\
{Case} 4 &1 & 1 & 1 \\
\hline\hline
\end{tabular}
\end{table}
\subsection{Gate structure and constraints}\label{subsec:gate_and_constr}
Two memristors and a resistor are necessary for a single memristive \gls{imply} gate. \Cref{fig:imply_gate} shows such a gate, with abstracted drivers and sense circuitry. Each memristor can be \textit{set} (forced to \gls{lrs}) by applying a voltage $|\ensuremath{V_\mathrm{set}}|>|\ensuremath{v_\mathrm{on}}|$ with appropriate (in our case \textit{negative}) polarity; and can be \textit{reset} (forced to \gls{hrs}) by applying $|\ensuremath{V_\mathrm{reset}}|>|\ensuremath{v_\mathrm{off}}|$ with an opposite polarity\footnote{This is true for bipolar switching mechanisms. \gls{pc} based devices, for example, may use the same voltage polarity for set and reset.}. If a memristor is set, it represents logic state `1'; if it is reset, it represents logic state `0'~\cite{borghetti_memristive_stateful_logic}. During initialization, these voltage amplitudes are applied to each device, while the other is kept floating. For the actual logic operation both devices are driven at the same time: $\ensuremath{V_\mathrm{cond}}$ is applied to node $R$ and $\ensuremath{V_\mathrm{set}}$ to node $T$ of \Cref{fig:imply_gate}.
For a correct operation
\begin{align}
|\ensuremath{V_\mathrm{set}}| > |\ensuremath{v_\mathrm{on}}| \label{eqn:vset_relation}\\
|\ensuremath{V_\mathrm{set}} - \ensuremath{V_\mathrm{cond}}| < |\ensuremath{v_\mathrm{on}}| \label{eqn:vcond_relation1}\\
|\ensuremath{V_\mathrm{reset}}| > |\ensuremath{v_\mathrm{off}}| \label{eqn:vclear_relation}
\end{align}
must hold. Moreover, the circuit designer needs to select a valid value for $\ensuremath{R_\mathrm{G}}$, as described in~\cite{kvatinsky_imply_logic_design}.\par
\begin{figure}[!b]
\centering
\subfigure[]{
\begin{circuitikz}
\draw(0,0) to[memristor,i=$i$] (2,0);
\draw[-stealth](.2,.5)node[above]{$+$} -- (1,.5)node[above]{$v$} -- (1.8,.5)node[above]{$-$};
\path(0,0) -- (0,-1.5);
\end{circuitikz}
\label{fig:symbol_polarity}
}
\subfigure[]{
\tikzset{varr/.style={-stealth}}
\begin{circuitikz}[scale=.6,transform shape]
\draw (.7,-1.5) to[short] (.7,-1)
(3.3,-1.5) to[short] (3.3,-1)
(3.3,-3.5) -- (-1,-3.5) to[R,l_=$R_G$,font=\Large] (-1,-5) node[ground]{};
\draw(.7,-3.5) to[Mr,*-o,l^=$P$,font=\Large] (.7,-1.5) node[left]{$R$}
(3.3,-3.5) to[Mr,-o,l^=$Q$,font=\Large](3.3,-1.5) node[left]{$T$};
\draw(-.5,-1) rectangle (4.5,0)
(2,-.5) node[font=\Large] {Driver \& Sensing};
\draw[varr] (1.2,-1.75)node[right]{$+$} --(1.2,-2.5) node[right]{\Large$V_P$}-- (1.2,-3.25)node[right]{$-$};
\draw[varr] (3.8,-1.75)node[right]{$+$} --(3.8,-2.5) node[right]{\Large$V_Q$}-- (3.8,-3.25)node[right]{$-$};
\draw[varr] (-.5,-3.7)node[right]{$+$} --(-.5,-4.5) node[right]{\Large$V_G$}-- (-.5,-5.3)node[right]{$-$};
\end{circuitikz}
\label{fig:imply_gate}}
\caption{\subref{fig:symbol_polarity} Memristor symbol and defined voltage polarity used in this work. \subref{fig:imply_gate} A single IMPLY gate.}
\label{fig:symbol_and_imply_gate}
\end{figure}
It is important to note that only in Case~1 of \Cref{tab:truth_table_imply} the output memristor $Q$ is actually changing its state. However, during this process the voltage across each device changes too. It is valid to ask if this has an effect on the result of the operation, and the answer is yes. Using \gls{kcl}, Chen et al.~\cite{chen_imply_ron_not_reached} showed that there are two possible final steady states of the operation:
\begin{enumerate}
\item The normalized state variable $s$ reaches the upper boundary of $1$ ($R_Q=\ensuremath{R_\mathrm{on}}$) before the voltage across $Q$ falls below the threshold $\ensuremath{v_\mathrm{on}}$. The final steady state is $R_Q=\ensuremath{R_\mathrm{on}}$.
\item $\ensuremath{V_Q}$ falls below the threshold $\ensuremath{v_\mathrm{on}}$ before $s$ reaches $1$. In this case the steady state resistance can be expressed as\footnote{Note that in our convention $\ensuremath{v_\mathrm{on}}<0$.}:
\begin{align}
R_\mathrm{min} &= \frac{-\ensuremath{v_\mathrm{on}}\,\ensuremath{R_\mathrm{G}}\,\ensuremath{R_\mathrm{off}}}{(\ensuremath{R_\mathrm{G}}+\ensuremath{R_\mathrm{off}}) (\ensuremath{V_\mathrm{set}}+\ensuremath{v_\mathrm{on}}) - \ensuremath{R_\mathrm{G}} \ensuremath{V_\mathrm{cond}}}\label{eqn:Rmin}
\end{align}
\end{enumerate}
An important point to mention is that this calculation is based on the premise, that the driving voltages of $P$ and $Q$ are chosen such that there is \emph{no state drift in $P$} during operation.
\subsection{Crossbar principles}\label{sec:xbar_fundamental}
Crossbar architectures are a natural candidate for memristor-based logic, as high integration density can be reached.
In so called 1R (or 1M) crossbars, a memristor device is fabricated at each intersection of bit- and word-lines, which act as the access medium for the cell. 1R crossbars are very difficult to handle~\cite{wan2010edl,li2018imw,li2018nature}, even if parasitics are not considered. Many works have been carried out to study effects~\cite{cassuto_sneak_path_constraints,chen_crossbar_array_model,shin_data_dependent_statistical_model_analysis,shin_data_dependend_model}, or solve them~\cite{chen_crossbar_array_model,sneak_paths_closed_form, CRS_proposal}, but thus far 1T1R has been the preferred implementation~\cite{wan2010edl,li2018imw,li2018nature}. 1T1R (or 1T1M) crossbars,consist of a transistor and a memristor in each cell. The transistor in each cell cost extra area but they prevent the cells from switching state when a cell is not part of an operation (not selected). One possible, readout scheme is provided by~\cite{sneak_paths_closed_form}, which we use in this work. The chosen readout scheme~\cite{sneak_paths_closed_form} provides a closed-form solution.
Moreover, it introduces very little additional complexity, which enables this work to remain focused on issues regarding \gls{imply} itself. In \Cref{sec:sim_xbar} we compare crossbar simulation results against single gate results and outline the differences.
\section{Device model}\label{sec:model}
There are a range of different simulation models for memristors~\cite{kvatinsky_team,kvatinsky_vteam,strachan_memristor_model,jiang_stanford_model}. For the simulations presented in this paper, the TU Wien LTSpice implementation~\cite{VTEAM_Spice, CAS_physRealization} of VTEAM~\cite{kvatinsky_vteam} was used. An overview of the model is given in \Cref{eqn:w,eqn:s,eqn:dwdt,eqn:tu_ui,eqn:s_ui}, with a memristor polarity as shown in \Cref{fig:symbol_polarity}. \par
In VTEAM, $w$ acts as the state variable and represents a length between the extrema $\ensuremath{w_\mathrm{on}}$ and $\ensuremath{w_\mathrm{off}}$ ($w\in[\ensuremath{w_\mathrm{off}},\ensuremath{w_\mathrm{on}}]$).
Here, we define the relation of these state variable boundaries
\begin{align}
\ensuremath{w_\mathrm{on}} &> \ensuremath{w_\mathrm{off}}\label{eqn:w}
\end{align}
and define the normalized state variable ($s\in[0,1]$):
\begin{align}
s(w)= w' &= \frac{w-\ensuremath{w_\mathrm{off}}}{\ensuremath{w_\mathrm{on}}-\ensuremath{w_\mathrm{off}}}\label{eqn:s}
\end{align}
These definitions may be changed, as long as the model equations are updated, too.
The rate of change of the state variable, $w$, is defined by
\begin{align}
\frac{\mathrm{d} w}{\mathrm{d} t} &= \begin{cases}
\ensuremath{k_\mathrm{off}} \left(\frac{v}{\ensuremath{v_\mathrm{off}}}-1\right)^{\ensuremath{\alpha_\mathrm{off}}} \ensuremath{f_\mathrm{off}}(w)\ \ &0<\ensuremath{v_\mathrm{off}}<v\\
0 & \ensuremath{v_\mathrm{on}} < v < \ensuremath{v_\mathrm{off}}\\
\ensuremath{k_\mathrm{on}} \left(\frac{v}{\ensuremath{v_\mathrm{on}}}-1\right)^{\ensuremath{\alpha_\mathrm{on}}} \ensuremath{f_\mathrm{on}}(w) & v<\ensuremath{v_\mathrm{on}}<0\\
\end{cases}\label{eqn:dwdt}
\end{align}
which is the essential building block of the model~\cite{kvatinsky_vteam}. In this equation $\ensuremath{k_\mathrm{on}},~\ensuremath{k_\mathrm{off}},~\ensuremath{v_\mathrm{on}},~\ensuremath{v_\mathrm{off}},~\ensuremath{\alpha_\mathrm{on}}$ and $\ensuremath{\alpha_\mathrm{off}}$ represent fitting parameters, while $\ensuremath{f_\mathrm{on}}(w)$ as well as $\ensuremath{f_\mathrm{off}}(w)$ are window functions that limit $\mathrm{d} w/\mathrm{d} t$. \par
$I/V$-characteristics and window functions are not defined in the model and thus can be freely chosen. We chose a linear current/voltage dependency:
\begin{align}
R(w)=\ensuremath{R_\mathrm{off}}+\left(\ensuremath{R_\mathrm{on}}-\ensuremath{R_\mathrm{off}}\right)\cdot s(w)
\label{eqn:tu_ui}
\end{align}
By rearranging the equation we can further express $s(R)$ for any (measured) $R$:
\begin{align}
s(R) &= \frac{R-\ensuremath{R_\mathrm{off}}}{\ensuremath{R_\mathrm{on}} - \ensuremath{R_\mathrm{off}}}\label{eqn:s_ui}
\end{align}
The same expressions as for the Simmon's Tunnel Barrier model in~\cite{kvatinsky_team} were chosen as window functions. In addition, $w$ is bounded and thus cannot exceed $\ensuremath{w_\mathrm{on}}$ or $\ensuremath{w_\mathrm{off}}$. \par
The studies presented in this paper are kept as general as possible, however, simulations need model parameters. Rather than introducing arbitrary parameter values, we experimentally fitted~\cite{Taherinejad2019newcas} our VTEAM model to Knowm BS-AF-W~\cite{knowm_datasheet} memristors we had at the time.
Parameters shown in \Cref{tab:parameters}, represent a best effort fitting we conducted previously~\cite{Taherinejad2019newcas}.
\section{Formulating Constraints}\label{sec:math}
This section marks the beginning of our new contributions. In this section, we mathematically extract device variability constraints which govern and limit operations of \gls{imply}. At first, we define the notation: Each parameter involved in the analysis is written as $\xi_{i,M}$, where $\xi\in\{R,v,k\}$, $M \in \{P, Q\}$ and $i \in \{\mathrm{off}, \mathrm{on}\}$.
For example, the off-resistance of memristor $P$ in this notation would be denoted as $\ensuremath{R_{\mathrm{off},P}}$. \par
Logic thresholds determine the logic state of a device. They are defined separately for input (I) and output (O), as well as logic `1' (H) and `0' (L). Indices are used to denote the respective logic thresholds, e.g. $\Rtext{IL}$ is the input threshold for logic `0'.
\subsection{Static behavior}\label{subsec:stat_math}
Each case in the truth table (\Cref{tab:truth_table_imply}) imposes constraints onto the voltage $\ensuremath{V_Q}$ across memristor $Q$, as certain \textit{switching conditions} must be met. They can be analyzed via \gls{kcl} and represent a \textit{static} view of the circuit. The constraints can be used to find limits for $\ensuremath{R_{\mathrm{on},P}}$, $\ensuremath{R_{\mathrm{off},P}}$, $\ensuremath{v_{\mathrm{on},Q}}$ and $\ensuremath{v_{\mathrm{off},Q}}$. They do not provide limits for $\ensuremath{R_{\mathrm{on},Q}}$ or $\ensuremath{R_{\mathrm{off},Q}}$, as $R_Q$ in this context is the target output resistance state that $Q$ must reach during \gls{imply}. Therefore, $R_Q$ is later set according to the chosen output logic threshold: $R_Q\leq R_\mathrm{OH}$ or $R_Q\geq R_\mathrm{OL}$.Applying \gls{kcl} in \Cref{fig:imply_gate} gives us the voltage across $Q$ as
\begin{align}
\ensuremath{V_Q} &= \frac{\ensuremath{R_Q} (\ensuremath{R_P}+\ensuremath{R_\mathrm{G}})\ensuremath{V_\mathrm{set}}-\ensuremath{R_Q}\ensuremath{R_\mathrm{G}}\ensuremath{V_\mathrm{cond}}}{\Rp\RG+\Rp\Rq+\Rq\RG}\label{eqn:VQ}.
\end{align}
First we solve \Cref{eqn:VQ} with a generalized threshold voltage, $v$, and the solution is specialized for each case afterwards.
The first switching condition is:
\begin{align}
\ensuremath{V_Q}&> v.
\label{eqn:VQ_constr}
\end{align}
Plugging \Cref{eqn:VQ} into \Cref{eqn:VQ_constr} and isolating $\ensuremath{R_P}$ leads to
\begin{align}
\ensuremath{R_P}\cdot b&> a\label{eqn:Rp_intermediate},
\end{align}
where
\begin{align}
a= \ensuremath{R_Q}\ensuremath{R_\mathrm{G}} (v&+\ensuremath{V_\mathrm{cond}}-\ensuremath{V_\mathrm{set}})\label{eqn:a},\\
b= \ensuremath{R_Q}\ensuremath{V_\mathrm{set}}-&v(\ensuremath{R_\mathrm{G}}+\ensuremath{R_Q})\label{eqn:b}.
\end{align}
At this point the relation in \ref{eqn:Rp_intermediate}, is divided by $b$. Therefore, depending on the value of $b$, we have
\begin{align}
\begin{cases}
\ensuremath{R_P}>\frac{a}{b}\ &\text{if}\ \ \ b > 0\\
\ensuremath{R_P}<\frac{a}{b}\ &\text{if}\ \ \ b < 0\\
\ensuremath{R_P}\rightarrow\pm\infty\ &\text{if}\ \ \ b=0
\end{cases}\label{eqn:Rp_case}
\end{align}
Next, the switching condition
\begin{align}
\ensuremath{V_Q}&< v
\end{align}
is examined. Following the same steps as before, we have
\begin{align}
\begin{cases}
\ensuremath{R_P}<\frac{a}{b}\ &\text{if}\ \ \ b > 0\\
\ensuremath{R_P}>\frac{a}{b}\ &\text{if}\ \ \ b < 0\\
\ensuremath{R_P}\rightarrow\pm\infty\ &\text{if}\ \ \ b=0
\end{cases}\label{eqn:Rp_case2}
\end{align}
Since the third case ($b=0$) in \Cref{eqn:Rp_case,eqn:Rp_case2}
yields\footnote{That is, as long as $|a|$ is neither zero, nor $\infty$.} $\pm\infty$, it is of no interest for the rest of the analysis.
The first two cases in \Cref{eqn:Rp_case,eqn:Rp_case2} both provide limits for $v$ and $\ensuremath{R_P}$, respectively. \par
Here, the resulting equations (constraints) are specialized for each of the four cases of the truth table using the respective switching conditions. {$\ensuremath{R_Q}$ is set to the associated output logic threshold ($\Rtext{OH}$ or $\Rtext{OL}$).} Only the first case ($b>0$) of \Cref{eqn:Rp_case,eqn:Rp_case2} is considered, since the second case ($b<0$) only provides negative limits, and $\ensuremath{R_P} >0$.
For every case of the truth table, according to our notations, $\ensuremath{v_\mathrm{on}}<0$ and $\ensuremath{v_\mathrm{off}}>0$. Hence, we have
\begin{description}
\item[Case 1] $\ensuremath{V_Q} > -\ensuremath{v_{\mathrm{on},Q}}$
\begin{align}
\ensuremath{v_{\mathrm{on},Q}}&> -\ensuremath{V_\mathrm{set}}\frac{\Rtext{OH}}{\ensuremath{R_\mathrm{G}}+\Rtext{OH}}\label{eqn:vonq_case1}\\
\ensuremath{R_{\mathrm{off},P}} &> \frac{\Rtext{OH}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{on},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OH}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{on},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OH})}\label{eqn:Roffp_case1}
\end{align}
\item[Case 3:] $\ensuremath{V_Q} < -\ensuremath{v_{\mathrm{on},Q}}$
\begin{align}
\ensuremath{v_{\mathrm{on},Q}}&> -\ensuremath{V_\mathrm{set}}\frac{\Rtext{OL}}{\ensuremath{R_\mathrm{G}}+\Rtext{OL}}\label{eqn:vonq_case3}\\
\ensuremath{R_{\mathrm{on},P}} &< \frac{\Rtext{OL}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{on},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OL}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{on},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OL})}\label{eqn:Ronp_case3}
\end{align}
\item[Case 2/Case 4:] $\ensuremath{V_Q} > -\ensuremath{v_{\mathrm{off},Q}}$
\begin{align}
\ensuremath{v_{\mathrm{off},Q}}&> -\ensuremath{V_\mathrm{set}}\frac{\Rtext{OH}}{\ensuremath{R_\mathrm{G}}+\Rtext{OH}}\label{eqn:voffq_case24}\\
\ensuremath{R_{\mathrm{off},P}} &> \frac{\Rtext{OH}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{off},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OH}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{off},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OH})}\label{eqn:Roffp_case2}\\
\ensuremath{R_{\mathrm{on},P}} &> \frac{\Rtext{OH}\ensuremath{R_\mathrm{G}} (\ensuremath{V_\mathrm{cond}}-\ensuremath{v_{\mathrm{off},Q}}-\ensuremath{V_\mathrm{set}})}{\Rtext{OH}\ensuremath{V_\mathrm{set}}+\ensuremath{v_{\mathrm{off},Q}} (\ensuremath{R_\mathrm{G}}+\Rtext{OH})}\label{eqn:Ronp_case4}
\end{align}
\end{description}
\Cref{eqn:vonq_case1,eqn:vonq_case3,eqn:voffq_case24} directly result from $b>0$, whereas \Cref{eqn:Roffp_case1,eqn:Ronp_case3,eqn:Roffp_case2,eqn:Ronp_case4} are the respective relations derived from $R_P>a/b$ in \Cref{eqn:Rp_case} and $R_P<a/b$ in \Cref{eqn:Rp_case2}.\par
Some additional static constraints are given by the choice of logic thresholds. That is,
\begin{align}
\ensuremath{R_{\mathrm{off},P}} > R_\mathrm{IL}\label{eqn:Roffp_greater_RIL}\\
\ensuremath{R_{\mathrm{off},Q}} > R_\mathrm{IL}\label{eqn:Roffq_greater_RIL}\\
\ensuremath{R_{\mathrm{on},P}} < R_\mathrm{IH}\label{eqn:Ronp_smaller_RIH}\\
\ensuremath{R_{\mathrm{on},Q}} < R_\mathrm{IH}\label{eqn:Ronq_smaller_RIH}.
\end{align}
Similar to standard logic families, input and output thresholds may differ. From the point-of-view of these constraints, only input thresholds need to be considered, as they determine whether or not the device states fed to the operation are valid in the first place.
\subsection{Dynamic behavior}\label{subsec:dyn_math}
With respect to the static analysis, the chosen timestep of operation can introduce much stricter constraints. For exact solutions, one would have to solve the differential state equation of the chosen model (in our case \Cref{eqn:dwdt} from VTEAM). This is not a trivial task and might not even be possible for all models. Thus, in this section we derive a lower boundary for $\ensuremath{v_{\mathrm{on},Q}}$, but not the infimum, which cannot be exceeded by the exact (or numeric) solution. That way we take into account the state change (dynamic behavior) of memristors during the operation, using an acceptable estimation. We note that in doing such an analysis, the chosen model is assumed to be accurate. However, in practice no existing model represents all the reality and physics involved. \par
\begin{figure}
\centering
\subfigure[]{
\begin{tikzpicture}[font=\footnotesize]
\draw[-stealth] (0,0) -- (2.5,0) node[right]{$t$};
\draw[-stealth] (0,0) -- (0,3) node[right]{$\ensuremath{V_Q}$};
\draw[] (.7,.1) --(.7,-.1)
(1.7,.1) --(1.7,-.1)
(.1,2.4) -- (-.1,2.4) node[left]{$V_{Q\mathrm{i}}$}
(.1,1.6) -- (-.1,1.6) node[left]{$V_{Q\mathrm{f}}$};
\draw[stealth-stealth] (.7,-.2) -- (1.2,-.2) node[below] {$\Delta T$} -- (1.7,-.2);
\draw[thick, red!70!yellow] (.7,2.4) -- (1.7,2.4);
\fill[red!70!yellow] (1.7,2.4)circle(.05);
\draw[dashed] (.7,0) -- (.7,2.4)
(1.7,0) --(1.7,1.6)
(0,2.4) -- (.7,2.4)
(0,1.6)--(1.7,1.6);
\fill[black] (.7,2.4) circle(.05)
(1.7,1.6) circle(.05);
\draw (.7,2.4) to[out=-45, in=160] (.9,2.1)
(1.7,1.6) to[out=160, in=-30] (1.5,1.8);
\draw[dotted] (.9,2.1)--(1.5,1.8);
\end{tikzpicture}
\label{fig:vq_during_imply}}
\subfigure[]{
\begin{tikzpicture}[font=\footnotesize]
\draw[-stealth] (0,0) -- (2.5,0) node[right]{$t$};
\draw[-stealth] (0,0) -- (0,2.5) node[right]{$\Delta w_Q$};
\draw[] (.7,.1) --(.7,-.1)
(1.7,.1) --(1.7,-.1);
\draw[stealth-stealth] (.7,-.2) -- (1.2,-.2) node[below] {$\Delta T$} -- (1.7,-.2);
\draw[thick, red!70!yellow] (.7,0) -- (1.7,1.6);
\fill[red!70!yellow] (1.7,1.6)circle(.05);
\fill[black] (.7,0) circle(.05)
(1.7,1) circle(.05);
\draw (.7,0) to[out=55, in=-130] (1,.4)
(1.7,1) to[out=-170, in=40] (1.5,.9);
\draw[dotted] (1,.4)--(1.5,.9);
\end{tikzpicture}
\label{fig:dw_during_imply}}
\caption{A symbolic voltage-time curve for $\ensuremath{V_Q}$ \subref{fig:vq_during_imply} an induced state change $\Delta w_Q$ \subref{fig:dw_during_imply} during a single \gls{imply} operation of duration $\Delta T$. The estimations for the formulation of constraints are drawn in orange, the symbolic representations of the actual curves in black.}
\label{fig:changes_during_imply}
\end{figure}
The main idea of our estimation is to look at how $\ensuremath{V_Q}$ changes over time in Case 1 of the truth table, while assuming negligible state drift in $P$. As $\ensuremath{R_Q}$ changes from \gls{hrs} to \gls{lrs}, $\ensuremath{V_Q}$ decreases. Thus, the initial voltage $V_{Q\mathrm{i}}$ is the highest occurring value of $\ensuremath{V_Q}$ during that timestep, while the final voltage $V_{Q\mathrm{f}}$ is the lowest -- symbolically shown in \Cref{fig:vq_during_imply}. If the device characteristics are such that the highest $\ensuremath{V_Q}$ corresponds to the maximum value of $\mathrm{d} w/ \mathrm{d} t$ -- in our case true due to \Cref{eqn:dwdt} -- a hard limit can be expressed. \gls{kcl} can be used to describe the initial voltage
\begin{align}
V_{Q\mathrm{i}} &= \frac{\ensuremath{R_{\mathrm{off},Q}}(\ensuremath{R_{\mathrm{off},P}}+\Rg)\ensuremath{V_\mathrm{set}}-\ensuremath{R_{\mathrm{off},Q}} \Rg \ensuremath{V_\mathrm{cond}}}{\ensuremath{R_{\mathrm{off},P}} \Rg+ \ensuremath{R_{\mathrm{off},P}} \ensuremath{R_{\mathrm{off},Q}} + \ensuremath{R_{\mathrm{off},Q}} \Rg}.
\label{eqn:VQi}
\end{align}
Plugging $V_{Q\mathrm{i}}$ into \Cref{eqn:dwdt} gives the inital rate of state change\footnote{$\ensuremath{f_\mathrm{on}}$ is missing in \Cref{eqn:dwqdt_initial} because
$\ensuremath{f_\mathrm{on}}\approx 1$ for $w_Q < \ensuremath{w_\mathrm{on}}$}:
\begin{align}
\frac{\mathrm{d} w_Q}{\mathrm{d} t}\bigg|_{\text{initial}} &= \frac{\Delta w_Q}{\Delta T}= \ensuremath{k_{\mathrm{on},Q}} \left(\frac{-V_{Q\mathrm{i}}}{\ensuremath{v_{\mathrm{on},Q}}}-1\right)^\alpha\label{eqn:dwqdt_initial}
\end{align}
Now we set the actual $\mathrm{d} w_Q/\mathrm{d} t$ equal to the initial rate for the whole timestep $\Delta T$. Through this simplification a maximum $\Delta w_Q$ for the given timestep $\Delta T$ can be found, which cannot be exceeded:
\begin{align}
\Delta w_Q &= \ensuremath{k_{\mathrm{on},Q}} \left(\frac{-V_{Q\mathrm{i}}}{\ensuremath{v_{\mathrm{on},Q}}}-1\right)^\alpha \Delta T\label{eqn:Dwq_max}
\end{align}
This is because the estimation provides a better overall situation towards the correct operation result, when compared to the actual situation. That is, as we see in \Cref{fig:dw_during_imply}, the estimated $\Delta w_Q$ is always larger than the actual value.
To obtain a correct result after the IMPLY operation, $\ensuremath{R_Q}$ must at least reach the logic threshold $R_\mathrm{OH}$. Otherwise the result would not be interpreted as logic `1'. Via \Cref{eqn:s_ui} we can find $s(R_\mathrm{OH})$. In combination with \Cref{eqn:s}, the necessary $\Delta w_\mathrm{min}$ can be expressed as
\begin{align}
\Delta w_\mathrm{min}&= \frac{R_\mathrm{OH}-\ensuremath{R_{\mathrm{off},Q}}}{\ensuremath{R_{\mathrm{on},Q}}-\ensuremath{R_{\mathrm{off},Q}}}(\ensuremath{w_\mathrm{on}}-\ensuremath{w_\mathrm{off}}) + \ensuremath{w_\mathrm{off}},
\label{eqn:wmin}
\end{align}
and
\begin{align}
\Delta w_Q &\geq \Delta w_\mathrm{min}\label{eqn:Dwq_greater_Dwmin}
\end{align}
shall be true. Plugging the previous terms into \Cref{eqn:Dwq_greater_Dwmin} gives
\begin{align}
\ensuremath{v_{\mathrm{on},Q}}&\geq\frac{-V_{Q\mathrm{i}}}{\sqrt[\leftroot{-1}\uproot{2}\scriptstyle\alpha]{\frac{\Delta w_\mathrm{min}}{\ensuremath{k_{\mathrm{on},Q}}\Delta T}} +1}, \label{eqn:vonq_wmin}
\end{align}
which is the newly found constraint.
As this relation contains multiple parameters of $P$ and $Q$ apart from $\ensuremath{v_{\mathrm{on},Q}}$, it provides boundaries for all of them. For example, a certain $\ensuremath{v_{\mathrm{on},Q}}$ restricts $\ensuremath{R_{\mathrm{off},P}}$ to a specific range, and in turn a certain $\ensuremath{R_{\mathrm{off},P}}$ restricts $\ensuremath{v_{\mathrm{on},Q}}$ to a specific range. \par
Considering \Cref{eqn:vonq_wmin}, a question is whether the
same estimation could be used to find an upper limit for $\ensuremath{v_{\mathrm{on},Q}}$. Such an analysis, however, is not meaningful for memristor $Q$. Both, a minimum value of $\mathrm{d} w/ \mathrm{d} t$ and a maximum value $\Delta w_\mathrm{max}$, must be specified. The later does not exist for $Q$ since a high $w_Q$ (ideally $\ensuremath{w_\mathrm{on}}$) is desired in Case~1.\par
There is, however, a $\Delta w_\mathrm{max}$ for memristor $P$, as a change of $\ensuremath{R_P}$ is generally not desired. By definition the logic state of $P$ remains unchanged if $\ensuremath{R_P}>R_\mathrm{IL}$. Only if this is true, it can be used as an input for an operation. As $\ensuremath{R_P}$ drifts away from $\ensuremath{R_{\mathrm{off},P}}$ (ideal \gls{hrs}), $\ensuremath{V_P}$ decreases. Thus, the minimum value of $\mathrm{d} w/\mathrm{d} t$ is the \textit{final value} at the end of the operation, in contrast to $V_{Q\mathrm{i}}$ being the initial value. The final value $V_{P\mathrm{f}}$ cannot be expressed easily, as $R_{P\mathrm{f}}$ and $R_{Q\mathrm{f}}$ are unknown. Hence, another simplification must be made: We evaluate $V_{P\mathrm{f}}$ using $\ensuremath{R_P}=\ensuremath{R_{\mathrm{off},P}}$, as if there was no state drift in the first place:
\begin{align}
V_{P\mathrm{f},j} &= \frac{\ensuremath{R_{\mathrm{off},P}}(R_{Q,j}+\Rg)\ensuremath{V_\mathrm{cond}}-\ensuremath{R_{\mathrm{off},P}}\Rg\ensuremath{V_\mathrm{set}}}{\ensuremath{R_{\mathrm{off},P}}\Rg+\ensuremath{R_{\mathrm{off},P}} R_{Q,j}+R_{Q,j}\Rg}\label{eqn:vpf}
\end{align}
Due to this very rough estimation, we expect the constraint to represent a fairly weak boundary. Therefore,
\begin{align}
R_{Q,1}&=R_{\mathrm{min},Q}\label{eqn:Rq1}\\
R_{Q,2}&=\frac{\ensuremath{R_{\mathrm{off},Q}}+R_{\mathrm{min},Q}}{2}\label{eqn:Rq2}\\
R_{Q,3}&=\sqrt{\ensuremath{R_{\mathrm{off},Q}}\cdot R_{\mathrm{min},Q}}\label{eqn:Rq3}
\end{align}
are used to evaluate $V_{P\mathrm{f}}$ in \Cref{eqn:vpf} and obtain a range within which the circuit is less likely to fail. The first value, \Cref{eqn:Rq1}, is the theoretical minimum for $\ensuremath{R_{\mathrm{on},Q}}$, which is the ideal $R_{Q\mathrm{f}}$. However, the actual $R_{Q,\mathrm{f}}$ can take on any value between $\ensuremath{R_{\mathrm{off},Q}}$ and $R_{\mathrm{min},Q}$. Hence, in a second estimation, \Cref{eqn:Rq2}, we assume that $R_{Q\mathrm{f}}$ is the arithmetic mean of $\ensuremath{R_{\mathrm{off},Q}}$ and $R_{\mathrm{min},Q}$. In other words, the final state is halfway between the initial state and the ideal endstate ($R_{\mathrm{min},Q}$). However, if $V_{P\mathrm{f}}$ is plotted over $R_Q$ on a linear scale, it reveals that the dependence is non-linear. Thus, $R_{Q,2}$ might not be the best estimation either. The dependence is, nevertheless, approximately linear on a semi-logarithmic scale; so our third estimation, \Cref{eqn:Rq3}, assumes $R_{Q\mathrm{f}}$ to be the geometric mean of $\ensuremath{R_{\mathrm{off},Q}}$ and $R_{\mathrm{min},Q}$. If the timestep of \gls{imply} operation is limited, we do not expect $R_{Q,1}$ to provide an appropriate estimation, since this is the overall optimum scenario. $R_{Q,2}$ and $R_{Q,3}$ might both be of value to the circuit designer, because they represent a non-ideal scenario, chosen based on design parameters. \par
Following the same steps as before, we can formulate the constraint for $\ensuremath{v_{\mathrm{on},P}}$:
\begin{align}
\Delta w_P &= \ensuremath{k_{\mathrm{on},P}} \left(\frac{-V_{P\mathrm{f},j}}{\ensuremath{v_{\mathrm{on},P}}}-1\right)^\alpha \Delta T\\
\Delta w_P &\leq \Delta w_\mathrm{max}\\
\ensuremath{v_{\mathrm{on},P}}&\leq\frac{-V_{P\mathrm{f},j}}{\sqrt[\leftroot{-1}\uproot{2}\scriptstyle\alpha]{\frac{\Delta w_\mathrm{max}}{\ensuremath{k_{\mathrm{on},P}}\Delta T}} +1}\label{eqn:vonp_wmax}
\end{align}
\Cref{tab:constraints_summary} provides a summary of relevant constraints on memristor parameters, that were derived in this section. Most of these relations depend on multiple parameters of both memristors. Thus, the permissible value range of one parameter is impacted by the values of other parameters, and vice versa. Once the value of a parameter is determined (either decided by the designer or given by the technology) respective equations in \Cref{tab:constraints_summary} determine the tolerable range of variation in others. This bidirectional view enables us to define an operating area, which, in turn, allows us to predict how the circuit will react to variations in the concerned parameters.
\begin{table}[t]
\scriptsize
\centering
\caption{Summary of related constraints on parameters of $P$ and $Q$.}
\def1.2{1.2}
\begin{tabular}{c|c}
\hline\hline
\multirow{2}{1cm}{Constraint} & Constrained \\
& parameters\\
\hline
$\ensuremath{v_{\mathrm{on},Q}}>f(\ensuremath{R_Q}\equiv\Rtext{OH})$ & \multirow{2}{.6cm}{$\ensuremath{v_{\mathrm{on},Q}}$} \\
\Cref{eqn:vonq_case1}&\\
\hline
$\ensuremath{R_{\mathrm{off},P}}>f(\ensuremath{v_{\mathrm{on},Q}},\ensuremath{R_Q}\equiv\Rtext{OH})$ & \multirow{2}{1.4cm}{$\ensuremath{R_{\mathrm{off},P}}, \ensuremath{v_{\mathrm{on},Q}}$} \\
\Cref{eqn:Roffp_case1}&\\
\hline
$\ensuremath{v_{\mathrm{on},Q}}>f(\ensuremath{R_Q}\equiv\Rtext{OL})$ & \multirow{2}{.6cm}{$\ensuremath{v_{\mathrm{on},Q}}$} \\
\Cref{eqn:vonq_case3}&\\
\hline
$\ensuremath{R_{\mathrm{on},P}}<f(\ensuremath{v_{\mathrm{on},Q}},\ensuremath{R_Q}\equiv\Rtext{OL})$ & \multirow{2}{1.4cm}{$\ensuremath{R_{\mathrm{on},P}}, \ensuremath{v_{\mathrm{off},Q}}$} \\
\Cref{eqn:Ronp_case3}&\\
\hline
$\ensuremath{v_{\mathrm{on},Q}}>f(\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{on},Q}},\ensuremath{R_{\mathrm{off},Q}},\ensuremath{k_{\mathrm{on},Q}})$ & $\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{on},Q}},\ensuremath{R_{\mathrm{off},Q}},$ \\
\Cref{eqn:vonq_wmin}& $\ensuremath{v_{\mathrm{on},Q}},\ensuremath{k_{\mathrm{on},Q}}$ \\
\hline
$\ensuremath{v_{\mathrm{on},P}}<f(\ensuremath{R_{\mathrm{on},P}},\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{off},Q}},\ensuremath{k_{\mathrm{on},P}})$ & $\ensuremath{R_{\mathrm{on},P}},\ensuremath{R_{\mathrm{off},P}},\ensuremath{R_{\mathrm{off},Q}},$ \\
\Cref{eqn:vonp_wmax}& $\ensuremath{v_{\mathrm{on},P}},\ensuremath{k_{\mathrm{on},P}}$ \\
\hline
$\ensuremath{R_{\mathrm{on},P}}<\Rtext{IH}$, \Cref{eqn:Ronp_smaller_RIH} &$\ensuremath{R_{\mathrm{on},P}}$\\
$\ensuremath{R_{\mathrm{off},P}}>\Rtext{IL}$, \Cref{eqn:Roffp_greater_RIL}& $\ensuremath{R_{\mathrm{off},P}}$ \\
$\ensuremath{R_{\mathrm{on},Q}}<\Rtext{IH}$, \Cref{eqn:Ronq_smaller_RIH} & $\ensuremath{R_{\mathrm{on},Q}}$\\
$\ensuremath{R_{\mathrm{off},Q}}>\Rtext{IL}$, \Cref{eqn:Roffq_greater_RIL} & $\ensuremath{R_{\mathrm{off},Q}}$\\
\hline\hline
\end{tabular}
\label{tab:constraints_summary}
\end{table}
\section{Simulation -- Single Gate}\label{sec:sim_single}
\subsection{Circuit design}\label{subsec:sim_circuit}
The simulated circuit corresponds to the circuit shown in \Cref{fig:imply_gate}, with the addition that $\ensuremath{R_\mathrm{G}}$ can be shorted by a parallel switch. The driver circuits are ideal voltage sources with serial switches for High-Z mode. Each switch is modeled with an on-resistance of $1\,\mathrm{n\Omega}$ and an off-resistance of $1\,\mathrm{G\Omega}$. Given the memristor properties, especially $\ensuremath{R_\mathrm{on}}$ and $\ensuremath{R_\mathrm{off}}$, five circuit-level parameters have to be determined. These are $\ensuremath{R_\mathrm{G}}, \ensuremath{V_\mathrm{set}}, \ensuremath{V_\mathrm{cond}}, \ensuremath{V_\mathrm{reset}}$ and $\ensuremath{V_\mathrm{read}}$. Choosing $\ensuremath{V_\mathrm{reset}}$ is somewhat straightforward, as it is only used for initialization and not the \gls{imply} operation per s\'{e}. \par
For this work, this voltage was set to $\ensuremath{V_\mathrm{reset}}=-1\,\mathrm{V}$.
Next, $\ensuremath{V_\mathrm{set}}$ and $\ensuremath{V_\mathrm{cond}}$ are determined. We define $\ensuremath{V_\mathrm{set}}=1\,\mathrm{V}, \ensuremath{V_\mathrm{cond}}=0.9\,\mathrm{V}$, based on the memristor's properties and~\Cref{eqn:vset_relation,eqn:vcond_relation1}. With the voltages set, the constraints on $\ensuremath{R_\mathrm{G}}$~\cite{kvatinsky_imply_logic_design} can be evaluated, which leads to: $5.000\,\mathrm{k\Omega} < \ensuremath{R_\mathrm{G}} < 230.769\,\mathrm{k\Omega}$. $\ensuremath{R_\mathrm{G}}=40\,\mathrm{k\Omega}$ was chosen as the value of this resistor, which is lower than the geometric mean ($100\,\mathrm{k\Omega}$) proposed by~\cite{kvatinsky_imply_principles}. A summary of model and circuit parameters is shown in~\Cref{tab:parameters}, where the former are based on experimental results from previous works~\cite{Taherinejad2019newcas,semiparallel_imply_adder}.
\begin{table}[t]
\centering
\caption{Nominal values of model and circuit parameters.}
\label{tab:parameters}
\scriptsize
\begin{tabular}{l c c c c c c}
\hline\hline
Parameter &$\ensuremath{v_\mathrm{on}}$&$\ensuremath{v_\mathrm{off}}$&$\ensuremath{R_\mathrm{on}}$&$\ensuremath{R_\mathrm{off}}$&\multicolumn{2}{c}{$\ensuremath{k_\mathrm{on}}$}\\
Value &$-0.7\,\mathrm{V}$&$10\,\mathrm{mV}$&$10\,\mathrm{k\Omega}$&$1\,\mathrm{M\Omega}$&\multicolumn{2}{c}{$1\,\mathrm{cm/s}$} \\
\hline
Parameter&$\ensuremath{\alpha_\mathrm{on}}$&$\ensuremath{\alpha_\mathrm{off}}$&$\ensuremath{w_\mathrm{on}}$&$\ensuremath{w_\mathrm{off}}$&\multicolumn{2}{c}{$\ensuremath{k_\mathrm{off}}$}\\
Value&3&3&$3\,\mathrm{nm}$&$0\,\mathrm{nm}$&\multicolumn{2}{c}{$-0.5\,\mathrm{nm/s}$}\\
\hline
Parameter& $\ensuremath{a_\mathrm{on}}$ &$\ensuremath{a_\mathrm{off}}$&$w_\mathrm{c}$\\
Value& $3\,\mathrm{nm}$&$0\,\mathrm{nm}$&$0.1\,\mathrm{nm}$\\
\hline
Parameter & $\ensuremath{V_\mathrm{set}}$&$\ensuremath{V_\mathrm{cond}}$&$\ensuremath{V_\mathrm{reset}}$&$\ensuremath{V_\mathrm{read}}$&$\ensuremath{R_\mathrm{G}}$&$T$\\
Value& $1.0\,\mathrm{V}$&$0.9\,\mathrm{V}$& $-1.0\,\mathrm{V}$&$0.1\,\mathrm{V}$&$40\,\mathrm{k\Omega}$&$15\,\mathrm{\mu s}$\\
\hline\hline
\end{tabular}
\end{table}
\Cref{eqn:Rmin,eqn:s_ui} are evaluated in order to get the operation constraints imposed by the circuit. Namely, $R_{\mathrm{min},Q} = 101.449\,\mathrm{k\Omega}$ and $s_{\mathrm{min},Q} = 0.908$. We can see that, in Case 1 and assuming no state drift in $P$, the output memristor $Q$ can never reach a state higher than $s_{\mathrm{min},Q}$ or, equivalently, can never have a resistance lower than $R_{\mathrm{min},Q}$.
\subsection{Methodology \& Setup}\label{subsec:method}
\begin{figure}[t]
\centering
\tikzset{%
Hstyle/.style={fill,RoyalBlue},%
Lstyle/.style={fill,LimeGreen},%
lbl/.style={left,font=\sffamily\bfseries\color{black}},%
logic/.style={left,font=\sffamily\bfseries\color{black}\footnotesize},%
thresh/.style={logic,right,align=center},
Indef/.style={fill,gray!60},%
Ondef/.style={fill,gray}
}
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\begin{tikzpicture}[scale=.6, transform shape]
\node[lbl] at (0,5.55) {Scheme:};
\begin{scope}[shift={(0,0)}]
\node[lbl] at (.8,5.5) {1/2};
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(0,.5*5) node[lbl]{0.5}
(.8,.5*5) node[thresh] {\footnotesize s$_\mathbf{\mathsf{OH}}$,\,s$_\mathbf{\mathsf{OL}}$\\ s$_\mathbf{\mathsf{IH}}$,\,s$_\mathbf{\mathsf{IL}}$}
(0,5) node[lbl]{1};
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(0,.4) node[lbl]{0.08}
(.8,.4) node[thresh] {s$_\mathbf{\mathsf{OL}}$}
(0,.8) node[lbl]{0.16}
(.8,.8) node[thresh] {s$_\mathbf{\mathsf{IL}}$}
(0,2) node[lbl]{0.40}
(.8,2) node[thresh] {s$_\mathbf{\mathsf{IH}}$}
(0,2.4) node[lbl]{0.48}
(.8,2.4) node[thresh] {s$_\mathbf{\mathsf{OH}}$}
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\fill[Hstyle] (0,0) rectangle (.5,.5);
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\node[thresh] at (8.5,.2) {Undefined\\\scriptsize Input \& Output};
\end{scope}
\end{tikzpicture}
\caption{Different logic thresholds used in this paper.}
\label{tab:logic_thresholds}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=.25\textwidth]{img/corner_analysis_legend_special.pdf}
\caption{Four squares show the state of each variable in a simulation set and the outline color (green or red) shows the simulation result (correct or failed, respectively).}
\label{fig:legend_results_variation_von_voff}
\end{figure}
\begin{figure*}[t]
\centering
\includegraphics[width=.85\textwidth]{img/corner_analysis_multiplot_special_regular.pdf}
\caption{Results summary for different degrees of variation in $\ensuremath{v_\mathrm{on}}, \ensuremath{v_\mathrm{off}}$ of $P$ and $Q$. The $1/3$ logic thresholds scheme was used here.}
\label{fig:full_results_variation_von_voff}
\end{figure*}
Proper \gls{imply} operation results -- with respect to the output logic thresholds -- are used to determine reliability. Correct operation is ensured when state changes within the memristors are occurring (switching conditions met) and are fast enough to exceed the given logic thresholds. We apply three different logic threshold schemes (shown in \Cref{tab:logic_thresholds}) to evaluate the operation results in relation to the chosen logic threshold. Each scheme defines separate, normalized input ($s_\mathrm{IH}$, $s_\mathrm{IL}$) and output ($s_\mathrm{OH}$, $s_\mathrm{OL}$) thresholds, as in conventional digital logic. Whereas the ``1/2'' and ``1/3'' scheme were chosen arbitrarily, the ``TTL'' scheme is derived from standard TTL ($V_\mathrm{CC}=5\,\mathrm{V}$)~\cite{TI_ttl_thresholds}. This is done by normalizing the threshold voltages $V_\mathrm{IH}$, $V_\mathrm{IL}$, $V_\mathrm{OH}$ and $V_\mathrm{OL}$ to $V_\mathrm{CC}$ -- e.g. $s_\mathrm{IH}=V_\mathrm{IH}/V_\mathrm{CC}$. The range between high and low thresholds, $[s_{\mathrm{IL}},s_{\mathrm{IH}}]$ and $[s_{\mathrm{OL}},s_{\mathrm{OH}}]$, is forbidden; in other words, the logic values and states in those ranges are considered undefined.
Reasons for failures are not separately determined in our setup. Hence, failures during initialization, which lead to erroneous operation results, are counted as regular failures and are not distinguished from errors during the operation itself. Further, our simulation setup utilizes constant timesteps, so actual switching time are not explicitly measured. \par
To obtain a nominal timebase for the \gls{imply} gate, a transient analog simulation of the memristor model was conducted. Examining the resulting waveform of the normalized state $s$ after the simulation showed that it takes $15\,\mathrm{\mu s}$ to switch from $1\%$ to $99\%$ of the state boundaries. Thus, the timestep of circuit operation is set to $T=15\,\mathrm{\mu s}$. Every action (initialization, \gls{imply} operation, readout) is executed using this fixed timestep. \par
Analog transient simulations were conducted in LTSpice, making use of this setup. Two memristor parameters per device ($\ensuremath{R_\mathrm{on}}, \ensuremath{R_\mathrm{off}}$ or $\ensuremath{v_\mathrm{on}}, \ensuremath{v_\mathrm{off}}$ or $\ensuremath{k_\mathrm{on}}, \ensuremath{k_\mathrm{off}}$) were varied simulatenously within the ranges reported in measurements~\cite{CAS_physRealization} and relative to the nominal state with a maximum deviation of $\pm 50\%$.
\subsection{Result Presentation Method}\label{subsec:postprocess}
To display the numerous results, we have come up with a presentation method of our own, which we introduce here. \par
Each parameter set is represented by a group of four squares. The left two squares, as displayed in \Cref{fig:legend_results_variation_von_voff}, show parameter values of memristor $P$, and the right two show that of memristor $Q$. The filling of each square represents the state of the corresponding parameter: empty means minimum, half-filled nominal and fully filled
maximum. \Cref{fig:legend_results_variation_von_voff} shows this concept and provides an example, too. The outline color of the squares shows whether the simulation result for a set of parameter variation ($\Delta$) was correct (highlighted by green) or incorrect (highlighted by red). In general, any combination of four parameters can be varied concurrently and displayed this way. However, our approach was to use three parameter sets: $\{\ensuremath{R_{\mathrm{on},P}}, \ensuremath{R_{\mathrm{off},P}}, \ensuremath{R_{\mathrm{on},Q}},\ensuremath{R_{\mathrm{off},Q}}\}$, $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{off},P}},\ensuremath{v_{\mathrm{on},Q}},\ensuremath{v_{\mathrm{off},Q}}\}$ and $\{\ensuremath{k_{\mathrm{on},P}},\ensuremath{k_{\mathrm{off},P}},\ensuremath{k_{\mathrm{on},Q}},\ensuremath{k_{\mathrm{off},Q}}\}$, as explained in \Cref{subsec:method}. \Cref{fig:full_results_variation_von_voff} shows a {complete set of simulations} for the parameters $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{off},P}},\ensuremath{v_{\mathrm{on},Q}},\ensuremath{v_{\mathrm{off},Q}}\}$. These resulting sets are then used to quickly identify those parameters that are common between different failed runs. For example, \Cref{fig:full_results_variation_von_voff} shows that the \gls{imply} operation produces no correct output if either, $\ensuremath{v_{\mathrm{on},Q}}$ or $\ensuremath{v_{\mathrm{on},P}}$, is at its maximum value for variations greater than or equal to $10\%$.
\subsection{Results analysis}\label{subsec:sim_analysis}
Combining the math provided in \Cref{sec:math} and the simulation results obtained in \Cref{subsec:postprocess} into joint graphs gives us \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m2_R,fig:R_P_limit_m2_k,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}. First, we take a closer look at \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m2_R,fig:R_P_limit_m2_k}, because they represent the most relatable logic threshold scheme, derived from traditional TTL thresholds. \Cref{fig:R_P_limit_m0_v,fig:R_P_limit_m1_v} show the same equations as in \Cref{fig:R_P_limit_m2_v}, plotted for the 1/2 and 1/3 threshold scheme, respectively. The other two graphs for these logic threshold schemes are omitted as they lead to the same conclusions as \Cref{fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}. Furthermore, the threshold voltages turned out to be the most critical parameters, so special attention is given to their results.
\subsubsection{Graph structure}\label{subsubsec:graph_structure}
Here, we explain how these graphs are composed.
Parameters $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{off},P}}$ of memristor $P$ are always shown on the y-axis since $R_P$ is crucial for the outcome of the operation.
We can also see that from the fact that $\ensuremath{R_{\mathrm{off},P}}$ or $\ensuremath{R_{\mathrm{on},P}}$ are present in all of the constraints described in \Cref{sec:math}. Different parameters are used in each graph for the x-axes. \par
Colored curves and areas are used to show constraints and important ranges:
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\colorlet{DynVpRoughColor}{black!40}
\colorlet{DynVpArMeanColor}{black!20}
\colorlet{DynVpGeoMeanColor}{black!30}
\colorlet{ThreshColor}{black}
\colorlet{ZerovColor}{white}
\colorlet{RoffAreaColor}{black!50}
\colorlet{DynRoffqColor}{black!60}
\colorlet{DynRonqColor}{black!40}
\colorlet{ThreshpColor}{black}
\colorlet{ThreshqColor}{black!30}
\colorlet{RoffKqAreaColor}{black!50}
\colorlet{RoffKpAreaColor}{black!30}
\colorlet{DynKqColor}{black!60}
\colorlet{DynKpRoughColor}{black!40}
\colorlet{DynKpArMeanColor}{black!20}
\colorlet{DynKpGeoMeanColor}{black!30}
\defnone{black}
\else
\colorlet{RonAreaColor}{SandPurple!60}
\colorlet{RonHatchColor}{SandPurple}
\colorlet{RoffVqAreaColor}{white
\colorlet{RoffHatchColor}{SandBlue!45
\colorlet{RoffVpAreaColor}{white
\colorlet{OkColor}{green!70}
\colorlet{FailColor}{red!80}
\colorlet{UnsureColor}{Orange!50!Yellow}
\colorlet{OtherErrorColor}{FailColor}
\colorlet{StaticRColor}{ForestGreen}
\colorlet{StaticvColor}{Orange!90!black
\colorlet{StaticRonColor}{RedViolet}
\colorlet{StaticvRonColor}{Black!70
\colorlet{DynVqColor}{blue!50!SandBlue}
\colorlet{DynVpRoughColor}{Sepia}
\colorlet{DynVpArMeanColor}{lime}
\colorlet{DynVpGeoMeanColor}{OliveGreen}
\colorlet{ThreshColor}{Cyan}
\colorlet{ZerovColor}{RoyalBlue}
\colorlet{RoffAreaColor}{SandBlue!60}
\colorlet{DynRoffqColor}{StaticvColor}
\colorlet{DynRonqColor}{blue!50!SandBlue}
\colorlet{ThreshpColor}{Cyan}
\colorlet{ThreshqColor}{LimeGreen}
\colorlet{RoffKqAreaColor}{white
\colorlet{RoffKpAreaColor}{white
\colorlet{DynKqColor}{blue!50!SandBlue}
\colorlet{DynKpRoughColor}{Sepia}
\colorlet{DynKpArMeanColor}{lime}
\colorlet{DynKpGeoMeanColor}{OliveGreen}
\defnone{none}
\fi
\def\itlen{5mm}
\def\itsep{1.5mm}
\def\itdum{-.8ex}
\begin{itemize}
\item[%
\tikz{\draw[black,thin,dashed] (0mm,0mm)--(\itlen,0mm);%
\path (0mm,\itdum)--(\itlen,\itdum);}]%
Black, dashed lines indicate nominal parameter values
\item[%
\tikz{\draw[ThreshColor, thick] (0mm,0mm)--(\itlen,0mm);%
\draw[ThreshColor,thick,dashed](0mm,-\itsep)--(\itlen,-\itsep);}]%
Light blue lines show input logic thresholds $R_\mathrm{IH}$ (solid) and $R_\mathrm{IL}$ (dashed)
for memristor $P$.
\item[%
\tikz{\draw[StaticRonColor, thick] (0mm,0mm)--(\itlen,0mm);%
\draw[DynVqColor, thick] (0mm,-.5*\itsep)--(.5*\itlen,-.5*\itsep);%
\draw[DynVqColor, thick,dotted] (.5*\itlen,-.5*\itsep)--(\itlen,-.5*\itsep);%
\draw[DynVpArMeanColor, thick] (0mm,-\itsep)--(.5*\itlen,-\itsep);%
\draw[DynVpArMeanColor, thick,dotted] (.5*\itlen,-\itsep)--(\itlen,-\itsep);}]%
Colored curves show the constraints from \Cref{sec:math}. Dotted parts indicate invalid plotting ranges, which do not correspond to any real value in physical devices.
\item[%
\tikz{\draw[StaticRColor, thick,-latex] (0mm,.5*\itsep)--(\itlen,.5*\itsep);%
\draw[DynVqColor, thick,latex-] (0mm,-.5*\itsep)--(\itlen,-.5*\itsep);}]%
Arrows indicate how the constraints restrict the operating area of a parameter, i.e., which side of the curve is acceptable due to the given constraint.
\item[%
\tikz{\fill[RoffHatchColor] (0mm,0mm) rectangle (\itlen,-\itsep);}]%
Blue areas show valid ranges of $\ensuremath{R_{\mathrm{off},P}}$ and the respective parameters on the x-axes. For example, in \Cref{fig:R_P_limit_m2_v}, this area represents valid ranges of $\ensuremath{R_{\mathrm{off},P}}$ versus $\ensuremath{v_{\mathrm{on},Q}}$, $\ensuremath{v_{\mathrm{off},Q}}$, $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{off},P}}$. Note that for $\ensuremath{v_{\mathrm{on},P}}$ our recommended range was used to limit the valid area, as the three different curves are only weak constraints.
\item[%
\tikz{\fill[RonAreaColor] (0mm,0mm) rectangle (\itlen,-\itsep);}]%
Purple areas show valid ranges of $\ensuremath{R_{\mathrm{on},P}}$ and the respective parameters on the x-axes. For example, in \Cref{fig:R_P_limit_m2_k}, this area represents valid ranges of $\ensuremath{R_{\mathrm{on},P}}$ versus $\ensuremath{k_{\mathrm{on},Q}}$, $\ensuremath{k_{\mathrm{off},Q}}$, $\ensuremath{k_{\mathrm{on},P}}$ and $\ensuremath{k_{\mathrm{off},P}}$. Note that restrictions on x-axis parameters are inherited from the $\ensuremath{R_{\mathrm{off},P}}$ operating area.
\item[%
\tikz{\draw[FailColor,line width=1mm] (0mm,0mm)--(.33*\itlen,0mm);%
\draw[UnsureColor,line width=1mm](.33*\itlen,0mm)--(.66*\itlen,0mm);%
\draw[OkColor,line width=1mm] (.66*\itlen,0mm)--(\itlen,0mm);%
\path(0mm,\itdum)--(\itlen,\itdum);}]%
The bars at each side of the graphs overlay our simulation results. Red sections show incorrect \gls{imply} results, green sections show correct results and orange sections are used for ranges in between, which are not explicitly covered by the simulations.
\end{itemize}
\subsubsection{Variation in voltage threshold}\label{par:v_var}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m2_v.pdf}
\caption{Analytical constraints and logic thresholds for the TTL scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},v_P, v_Q\}$. }
\label{fig:R_P_limit_m2_v}
\end{figure}
\Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v} depict voltage thresholds $\ensuremath{v_{\mathrm{on},P}}$, $\ensuremath{v_{\mathrm{off},P}}$, $\ensuremath{v_{\mathrm{on},Q}}$ and $\ensuremath{v_{\mathrm{off},Q}}$ of memristor $P$ and $Q$, as well as resistance parameters $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{off},P}}$ of $P$ using different logic threshold schemes (\Cref{tab:logic_thresholds}). For the analysis we concentrate on the TTL scheme, \Cref{fig:R_P_limit_m2_v}. \par
The logic thresholds ($R_\mathrm{IH}$, $R_\mathrm{IL}$) divide the plot into two parts: The bottom part concerning $\ensuremath{R_{\mathrm{on},P}}$ and the top part concerning $\ensuremath{R_{\mathrm{off},P}}$. Adding the static constraints, \Cref{eqn:vonq_case1,eqn:vonq_case3,eqn:Roffp_case1,eqn:Ronp_case3}, on top of the logic thresholds decreases the valid range of $\ensuremath{R_{\mathrm{off},P}}$, $\ensuremath{v_{\mathrm{on},Q}}$ and in particular $\ensuremath{R_{\mathrm{on},P}}$. The latter is evident from the purple area in \Cref{fig:R_P_limit_m2_v}, which is smaller than the plotted range. However, regarding $\ensuremath{R_{\mathrm{off},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$, the dynamic constraint, \Cref{eqn:vonq_wmin}, is even stricter than the static constraint. \par
There are no static constraints for $\ensuremath{v_{\mathrm{on},P}}$. A rough dynamic estimation is provided by \Cref{eqn:vonp_wmax}, which depends on $V_{P\mathrm{f}}$. As discussed in \Cref{subsec:dyn_math}, \Cref{eqn:vonp_wmax} is evaluated three times, using $R_{Q,1}$, $R_{Q,2}$ and $R_{Q,3}$, respectively. The three curves are drawn in brown, dark green and light green. No constraint for $\ensuremath{v_\mathrm{off}}$ has been found (\Cref{sec:math}). Hence, the valid ranges of $R_P$ over $\{\ensuremath{v_{\mathrm{off},P}}, \ensuremath{v_{\mathrm{off},Q}}\}$ are only limited by logic thresholds, \Cref{eqn:Roffp_greater_RIL,eqn:Ronp_smaller_RIH}. As a consequence of the above constraints, the valid range for each parameter is decreased and thus the advisable operating area remains as shown by the colored areas.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m0_v.pdf}
\caption{Analytical constraints and logic thresholds for the 1/2 scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},v_P, v_Q\}$. }
\label{fig:R_P_limit_m0_v}
\end{figure}
Simulation results for variation in $\ensuremath{v_{\mathrm{on},Q}}$ show very good agreement with the mathematical analysis, especially the dynamic estimation in \Cref{eqn:vonq_wmin}, which depends on \Cref{eqn:VQi,eqn:wmin}. At $+10\%$ variation of $\ensuremath{v_{\mathrm{on},Q}}$ and nominal $\ensuremath{R_{\mathrm{off},P}}$, the simulation fails (indicated by the thin red line), as the analysis predicted. \Cref{fig:R_P_limit_m2_v} shows very clearly that this failure is not accurately predicted by the static constraints from \Cref{subsec:stat_math} alone. Hence, the dynamic estimation (\Cref{subsec:dyn_math}) is vital. Variation in $\ensuremath{v_{\mathrm{on},P}}$ strengthens this point further, since different methods of estimating the dynamic behavior leads to important changes regarding the agreement of the simulations and the derived analytical constraints. On the upper end of the $\ensuremath{v_{\mathrm{on},P}}$ range, \Cref{eqn:vonp_wmax} ({evaluated using $R_Q=R_{Q,3}$ for $V_{P\mathrm{f}}$, \Cref{eqn:vpf}}) provides good congruence with our simulations, whereas \Cref{eqn:vonp_wmax} (evaluated using $R_Q=R_{Q,2}$ for $V_{P\mathrm{f}}$, \Cref{eqn:vpf}) represents a more conservative estimation. In contrast, evaluating \Cref{eqn:vonp_wmax} using the theoretical minimum $R_Q=R_{Q,1}=R_{\mathrm{min},Q}$ in \Cref{eqn:vpf}, does not yield a good estimation. On the lower end of the $\ensuremath{v_{\mathrm{on},P}}$ range, simulation results indicate some failures for $\ensuremath{v_{\mathrm{on},P}}\leq -0.84\,\mathrm{V}$ ($+20\%$). This behavior cannot be explained by any of the constraints from \Cref{sec:math}.
According to the simulation results (\Cref{subsec:postprocess}, \Cref{fig:full_results_variation_von_voff}), these specific failures only occur when $\ensuremath{v_{\mathrm{on},Q}}\geq -0.7\,\mathrm{V} (\pm 0\%)$, which leads us to believe that the reason for failure is the $20\%$ mismatch between $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$. Regarding both, $\ensuremath{v_{\mathrm{off},P}}$ and $\ensuremath{v_{\mathrm{off},Q}}$, there are almost no failures as expected, except for a (minor) failure during initialization for $\ensuremath{v_{\mathrm{off},Q}}$ at $+50\%$.\par
In terms of $R_P$ variation, the simulation results suggest that $\ensuremath{R_{\mathrm{off},P}}$ can lie within the uncertain range between logic thresholds while the \gls{imply} operation still outputs correct results. This stands to reason since the thresholds are artificial limits not governed by the circuit behavior. Further, $\ensuremath{R_{\mathrm{on},P}}$ is fine up to the lowest simulated value of $\ensuremath{R_{\mathrm{off},P}}$, because at that point $\ensuremath{R_{\mathrm{off},P}}>\ensuremath{R_{\mathrm{on},P}}$ changes to $\ensuremath{R_{\mathrm{off},P}}<\ensuremath{R_{\mathrm{on},P}}$, and hence the operation fails. \par
Combining all the simulation results and their respective analytical constraints, we can identify the areas in which the circuit is most likely to operate correctly. These are the areas highlighted in \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}. \Cref{eqn:vonq_wmin,eqn:vonp_wmax} and their respective dependencies, \Cref{eqn:VQi,eqn:wmin,eqn:vpf} (evaluated using $R_Q=R_{Q,3}$), are recommended for estimating the valid ranges of $\ensuremath{R_{\mathrm{off},P}}$ versus $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{on},Q}}\}$; whereas the static constraints {\Cref{eqn:vonq_case1,eqn:vonq_case3,eqn:Roffp_case1,eqn:Ronp_case3}} are sufficient for $\ensuremath{R_{\mathrm{on},P}}$ versus $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{on},Q}}\}$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m1_v.pdf}
\caption{Analytical constraints and logic thresholds for the 1/3 scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},v_P, v_Q\}$.}
\label{fig:R_P_limit_m1_v}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{R_P_limits_m2_R.pdf}
\caption{Analytical constraints and logic thresholds for the TTL scheme plotted over a range of memristor parameters $\{\ensuremath{R_P}, \ensuremath{R_Q}\}$.}
\label{fig:R_P_limit_m2_R}
\end{figure}
\subsubsection{Variation in resistance limits}\label{par:r_var}
There are no static constraints limiting $\ensuremath{R_{\mathrm{on},Q}}$ or $\ensuremath{R_{\mathrm{off},Q}}$. Therefore, only logic thresholds and the dynamic estimation of \Cref{eqn:vonq_wmin} can be applied. The latter depends on \Cref{eqn:VQi,eqn:wmin} and is evaluated in two ways: First, varying $\ensuremath{R_{\mathrm{on},Q}}$, but not $\ensuremath{R_{\mathrm{off},Q}}$; and second varying $\ensuremath{R_{\mathrm{off},Q}}$, but not $\ensuremath{R_{\mathrm{on},Q}}$. It is interesting to see that -- for any of the three schemes of \Cref{tab:logic_thresholds} -- the logic thresholds limit the operating areas (blue and purple) much more than the actual analytical constraints. Simulation results for $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{off},P}}$ are identical to \Cref{fig:R_P_limit_m2_v}, however, $\ensuremath{R_{\mathrm{off},Q}}$ cannot reach as low as $\ensuremath{R_{\mathrm{off},P}}$ without causing a failure. This is solely due to the chosen logic thresholds, as an \gls{imply} output of $\ensuremath{R_{\mathrm{off},Q}}<R_\mathrm{OL}$ is considered as failure. \par
Overall, resistance variation does not seem to hold as much potential for failures as variation in threshold voltage(s) does. \Cref{eqn:vonq_wmin} and its dependencies, \Cref{eqn:VQi,eqn:wmin}, can be used to identify valid parameter ranges, but -- based on our simulation results -- it is most likely not necessary. This is true for all three logic threshold schemes listed in \Cref{tab:logic_thresholds}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{R_P_limits_m2_k.pdf}
\caption{Analytical constraints and logic thresholds for the TTL scheme plotted over a range of memristor parameters $\{\ensuremath{R_P},k_P,k_Q\}$. }
\label{fig:R_P_limit_m2_k}
\end{figure}
\subsubsection{Variation in switching speed}\label{par:k_var}
The dynamic constraint in \Cref{eqn:vonq_wmin} can be used to extract limits of $\ensuremath{k_{\mathrm{on},Q}}$, while \Cref{eqn:vonp_wmax} provides the basis for the analysis of $\ensuremath{k_{\mathrm{on},P}}$. \Cref{fig:R_P_limit_m2_k} shows the plotted equations and logic thresholds. \Cref{eqn:vonp_wmax} (evaluated using \Cref{eqn:vpf}, where $R_Q=R_{Q,1}$) is omitted, as well as all constraints containing $\ensuremath{k_{\mathrm{off},P}}$ and $\ensuremath{k_{\mathrm{off},Q}}$, since they are far outside of the plotted range.
The graph in \Cref{fig:R_P_limit_m2_k} shows that $\ensuremath{k_{\mathrm{on},P}}$ is hardly restricted by any constraint. Only at relatively high values, greater than $+50\%$ variation, \Cref{eqn:vonp_wmax} (evaluated using \Cref{eqn:vpf} with $R_Q=R_{Q,2}$) comes into effect, but cannot be compared to simulation results, as our simulated range ends at $+50\%$, in compliance with our methodology (\Cref{subsec:method}). In contrast, \Cref{eqn:vonq_wmin} provides a reasonable constraint for $\ensuremath{k_{\mathrm{on},Q}}$. Nonetheless, our simulated range only reaches down to $-50\%$ and thus results cannot be compared to the constraint. The other two logic threshold schemes show similar behavior. As before, the colored areas indicate the merged, predicted functional range of both, $\ensuremath{k_{\mathrm{on},P}}$ and $\ensuremath{k_{\mathrm{on},Q}}$.
\par
In conclusion, switching speed $k$ of both memristors can vary at least by $\pm50\%$ without performance issues, according to our simulation. Analytical constraints suggest that there is a lower boundary for $\ensuremath{k_{\mathrm{on},Q}}$ at approximately $2\,\mathrm{mm/s}$ ($-80\%$).
\section{Simulation -- Crossbar}\label{sec:sim_xbar}
\subsection{Setup}\label{subsec:xbar_circuit}
Analogous to the single \gls{imply} gate simulation setup (\Cref{subsec:sim_circuit}), the circuit in \Cref{fig:imply_gate} is the basis for the crossbar simulation. A complete $128\times 128${} cell 1T1R crossbar circuit was used. The \gls{imply} gate is formed by two memristors arbitrarily located within the crossbar. Each memristor has its own access device, in our case an ideal switch, and is connected to adjacent cells via resistors that model the nanowire resistances. The ideal switch is modeled using an on-resistance of $1\,\mathrm{\mu\Omega}$ and an off-resistance of $100\,\mathrm{M\Omega}$. Line resistances were chosen to be $10\,\mathrm{\Omega}$ each, according to the worst case in~\cite{sneak_paths_closed_form}. \Cref{fig:xbar_cell} shows the structure of a single cell.
\begin{figure}
\centering
\begin{circuitikz}[font=\Large,scale=.6,transform shape]
\node[rotate=0] at(-.7,0) {\dots};
\node at (-1.9,0) {word $y$};
\node[rotate=90] at(2.5,3.3) {\dots};
\node at (2.5,3.8) {bit $x$};
\node[rotate=0] at (3.2,0) {\dots};
\node[rotate=90] at (2.5,-.7) {\dots};
\draw(2.5,2.8) to[short,-*] (2.5,2.5) to [R,l=$R_{xy/x(y+1)}$] (2.5,0) to[short] (2.5,-.3)
(-.3,0) to[short,-*] (0,0) to[R,l_=$R_{xy/(x+1)y}$] (2.5,0) to[short](2.8,0);
\draw (0,0) to[memristor,*-,l=$M_{xy}$] (1.5,1.5) to[short] (1.7,1.7) to[short] (1.9,2.15) (2,2) to [short](2.5,2.5);
\node[rotate=45] at (1.5,2.3) {$T_{xy}$};
\end{circuitikz}
\caption{Structure of a single cell within the 1T1R crossbar, including line resistances.}
\label{fig:xbar_cell}
\end{figure}
Circuit parameters of the \gls{imply} gate are identical to \Cref{subsec:sim_circuit}, \Cref{tab:parameters}. Bit-line drivers are attached at the top and bottom for symmetry. The readout strategy described in \Cref{sec:xbar_fundamental} was implemented. Analog transient simulations were conducted in Cadence Spectre. The method of parameter variation is the same as defined for the single gate in \Cref{subsec:method}, except that only relative parameter variations ($\pm 50\%$) were conducted for the crossbar.
\begin{figure}[b]
\centering
\begin{tikzpicture}
\begin{axis}[
ybar,
ymin=0,
grid=major,
xtick distance=0.25,
width=0.6\linewidth,
height=.35\linewidth,
xlabel=$s$,
ylabel={$n$ per bin},
font=\footnotesize,
]
\addplot +[hist={bins=100}] table [y index=0] {data/init_states.csv};
\end{axis}
\end{tikzpicture}\vspace{-3mm}
\caption{Histogram of initial (normalized) device states, $s$, within the $128\times 128${} crossbar, plotted using 100 bins.}
\label{fig:xbar_initial_histogram}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=.85\textwidth]{img/corner_analysis_multiplot_special_xbar.pdf}
\caption{Crossbar results summary for different degrees of variation in $\ensuremath{v_\mathrm{on}}, \ensuremath{v_\mathrm{off}}$ of $P$ and $Q$. Logic thresholds for `1' and `0' were set according to the TTL threshold scheme (\Cref{tab:logic_thresholds}). Memristor $P$ was at position $\{0,0\}$, while $Q$ was at the center, $\{63,63\}$.}
\label{fig:xbar_full_results_variation_von_voff}
\end{figure*}
\subsection{Methodology}\label{subsec:xbar_method}
\gls{imply} gates can be formed by any two memristors in the crossbar. Both, the worst case scenario in terms of parasitic resistance between the two memristors forming a gate, and the worst case voltage drop, were considered. Hence, four separate simulations were conducted with $P$ and $Q$ at different $\{\text{bit},\text{word}\}$ positions.
\begin{enumerate}
\item Memristor $P$ at position $\{0,0\}$, $Q$ at position $\{127,127\}$
\item Memristor $P$ at position $\{127,127\}$, $Q$ at position $\{0,0\}$
\item Memristor $P$ at position $\{0,0\}$, $Q$ at position $\{63,63\}$
\item Memristor $P$ at position $\{63,63\}$, $Q$ at position $\{0,0\}$
\end{enumerate}
Instead of using idealized ($s=0$ or $s=1$) or manually fixed initial memristor states, each cell is assigned a different initial state during (automated) netlist generation. The states are generated via Octave and follow a Gaussian distribution which has been cut in half as shown in \Cref{fig:xbar_initial_histogram}. Although this approach requires a greater effort, it represents a more realistic scenario than ideal initial states.
\begin{figure*}[]
\centering
\defSingle gate{Single gate}
\defCrossbar{Crossbar}
\def\barsep{-.7}
\def\groupysep{-4}
\def\groupxsep{13}
\begin{tikzpicture}[scale=.6,transform shape,font=\large]
\begin{scope}[shift={(0,0)}]
\node[] at (-8,0) {Single gate};
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{k_{\mathrm{off},P}}/1\,\mathrm{\frac{nm}{s}}$} -- (4,0)node[right]{$\ensuremath{k_{\mathrm{on},P}}/1\,\mathrm{\frac{mm}{s}}$};
\fill[green] (-1.25,-.08) rectangle (1.25,.08);
\fill[green] (-3.25,-.08) rectangle (-2.75,.08);
\draw[] (-3.0,.1) -- (-3.0,.7) node[above]{\ensuremath{k_\mathrm{off}}};
\draw[] (0,.1) -- (0,.7) node[above]{\ensuremath{k_\mathrm{on}}};
\draw[thick] (-2.5,-.15) -- (-2.5,.15);
\foreach \x/\t in {-3.5/-1.0,-3.0/,-2.5/0.0,-2/,-1.5/4.0,-1/,-.5/8.0,0/,.5/12.0,1/,1.5/16.0,2/,2.5/20.0,3/,3.5/24.0}{
\draw[] (\x,-.1) -- (\x,.1) node[above,fill=white,inner sep=1pt] {\t};
}
\end{scope}
\begin{scope}[shift={(0,\barsep)}]
\node[] at (-8,0) {Crossbar};
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{k_{\mathrm{off},P}}/1\,\mathrm{\frac{nm}{s}}$} -- (4,0)node[right]{$\ensuremath{k_{\mathrm{on},P}}/1\,\mathrm{\frac{mm}{s}}$};
\fill[green] (-1.25,-.08) rectangle (1.25,.08);
\fill[green] (-3.25,-.08) rectangle (-2.75,.08);
\fill[red] (.5,-.08) rectangle (1.25,.08);
\fill[orange] (.25,-.08) rectangle (.5,.08);
\fill[red] (-.5,-.08) rectangle (-1.25,.08);
\fill[orange] (-.25,-.08) rectangle (-.5,.08);
\fill[red] (-2.9,-.08) rectangle (-2.75,.08);
\fill[orange] (-2.9,-.08) rectangle (-2.95,.08);
\fill[red] (-3.15,-.08) rectangle(-3.25,.08);
\fill[orange] (-3.15,-.08) rectangle (-3.10,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\draw[thick] (-2.5,-.15) -- (-2.5,.15);
\end{scope}
\begin{scope}[shift={(0,2*\barsep)}]
\node[] at (-8,0) {Single gate};
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{k_{\mathrm{off},Q}}/1\,\mathrm{\frac{nm}{s}}$} -- (4,0)node[right]{$\ensuremath{k_{\mathrm{on},Q}}/1\,\mathrm{\frac{mm}{s}}$};
\fill[green] (-1.25,-.08) rectangle (1.25,.08);
\fill[green] (-3.25,-.08) rectangle (-2.75,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\draw[thick] (-2.5,-.15) -- (-2.5,.15);
\end{scope}
\begin{scope}[shift={(0,3*\barsep)}]
\node[] at (-8,0) {Crossbar};
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{k_{\mathrm{off},Q}}/1\,\mathrm{\frac{nm}{s}}$} -- (4,0)node[right]{$\ensuremath{k_{\mathrm{on},Q}}/1\,\mathrm{\frac{mm}{s}}$};
\fill[green] (-1.25,-.08) rectangle (1.25,.08);
\fill[green] (-3.25,-.08) rectangle (-2.75,.08);
\fill[red] (.5,-.08) rectangle (1.25,.08);
\fill[orange] (.25,-.08) rectangle (.5,.08);
\fill[red] (-.5,-.08) rectangle (-1.25,.08);
\fill[orange] (-.25,-.08) rectangle (-.5,.08);
\fill[red] (-3.225,-.08) rectangle (-3.175,.08);
\fill[orange] (-3.25,-.08) rectangle (-3.225,.08)
(-3.15,-.08) rectangle (-3.175,.08);
\fill[red] (-2.825,-.08) rectangle (-2.775,.08);
\fill[orange] (-2.775,-.08) rectangle (-2.75,.08)
(-2.85,-.08) rectangle (-2.825,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\draw[thick] (-2.5,-.15) -- (-2.5,.15);
\end{scope}
\begin{scope}[shift={(\groupxsep,0)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},P}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},P}}/1\,\mathrm{mV}$};
\fill[red] (-3.675,-.08) rectangle (-.25,.08);
\fill[orange] (-2.7,-.08) rectangle (-1.3,.08);
\fill[green] (-2.35,-.08) rectangle (-1.65,.08);
\fill[green] (2.0,-.08) rectangle (3,.08);
\draw (-2,.1) -- (-2,.7) node[above]{$\ensuremath{v_\mathrm{on}}$};
\draw (2.5,.1) -- (2.5,.7) node[above]{$\ensuremath{v_\mathrm{off}}$};
\draw[thick] (1.5,-.15) -- (1.5,.15);
\foreach \x/\t in {-3.5/-1.0,-3.0/,-2.5/-0.8,-2/,-1.5/-0.6,-1/,-.5/-0.4,0/,.5/-0.2,1/,1.5/0.0,2/,2.5/10,3/,3.5/20.0}{
\draw[] (\x,-.1) -- (\x,.1) node[above,fill=white,inner sep=1pt] {\t};
}
\end{scope}
\begin{scope}[shift={(\groupxsep,\barsep)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},P}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},P}}/1\,\mathrm{mV}$};
\fill[red] (-3.675,-.08) rectangle (-.25,.08);
\fill[orange] (-2.35,-.08) rectangle (-1.65,.08);
\fill[green] (-2-.06,-.08) rectangle (-2+.06,.08);
\fill[green] (2.0,-.08) rectangle (3,.08);
\fill[orange](-2+.06,-.08)rectangle(-1.65,.08);
\fill[red] (2.4,-.08) rectangle(2.45,.08);
\fill[orange] (2.45,-.08) rectangle (2.48,.08)
(2.4,-.08) rectangle(2.37,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\draw[thick] (1.5,-.15) -- (1.5,.15);
\end{scope}
\begin{scope}[shift={(\groupxsep,2*\barsep)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},Q}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},Q}}/1\,\mathrm{mV}$};
\fill[red] (-3.675,-.08) rectangle (-.25,.08);
\fill[orange] (-2.35,-.08) rectangle (-2.0,.08);
\fill[green] (-2.0-.03,-.08) rectangle (-.25,.08);
\fill[green] (2.0,-.08) rectangle (3,.08);
\fill[red] (2.9,-.08) rectangle (3,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\draw[thick] (1.5,-.15) -- (1.5,.15);
\end{scope}
\begin{scope}[shift={(\groupxsep,3*\barsep)}]
\draw[stealth-stealth] (-4,0) node[left]{$\ensuremath{v_{\mathrm{on},Q}}/1\,\mathrm{V}$} -- (4,0)node[right]{$\ensuremath{v_{\mathrm{off},Q}}/1\,\mathrm{mV}$};
\fill[red] (-3.675,-.08) rectangle (-.25,.08);
\fill[orange] (-2.35,-.08) rectangle (-2.0,.08);
\fill[green] (-2.0-.06,-.08) rectangle (-.25,.08);
\fill[green] (2.0,-.08) rectangle (3,.08);
\fill[orange](-1.3-.03,-.08) rectangle(-2+.06,.08);
\fill[red] (-1.65-.03,-.08) rectangle (-.25,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\draw[thick] (1.5,-.15) -- (1.5,.15);
\end{scope}
\begin{scope}[shift={(0,\groupysep)}]
\node[] at (-8,0) {Single gate};
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{off},P}}/1\,\mathrm{k\Omega}$};
\fill[green] (-3,-.08) rectangle (2,.08);
\draw(-.5,.1) --(-.5,.7) node[above]{$\ensuremath{R_\mathrm{off}}$};
\foreach \x/\t in {-3.5/400,-3.0/,-2.5/600,-2/,-1.5/800,-1/,-.5/1000,0/,.5/1200,1/,1.5/1400,2/,2.5/1600,3/,3.5/}{
\draw[] (\x,-.1) -- (\x,.1) node[above,fill=white,inner sep=1pt] {\t};
}
\end{scope}
\begin{scope}[shift={(0,\groupysep+\barsep)}]
\node[] at (-8,0) {Crossbar};
\draw[-stealth] (-4,0)-- (4,0)node[right]{$\ensuremath{R_{\mathrm{off},P}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2.5,-.08) rectangle (2,.08);
\fill[orange] (-2.5,-.08) rectangle (-2.75,.08);
\fill[red] (-2.75,-.08) rectangle (-3,.08);
\fill[red] (1.75,-.08) rectangle (2,.08);
\fill[orange] (1.5,-.08) rectangle (1.75,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\end{scope}
\begin{scope}[shift={(\groupxsep,\groupysep)}]
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{on},P}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2,-.08) rectangle (3,.08);
\draw (.5,.1) -- (.5,.7) node[above]{$\ensuremath{R_\mathrm{on}}$};
\foreach \x/\t in {-3.5/2,-3.0/,-2.5/4,-2/,-1.5/6,-1/,-.5/8,0/,.5/10,1/,1.5/12,2/,2.5/14,3/,3.5/16}{
\draw[] (\x,-.1) -- (\x,.1) node[above,fill=white,inner sep=1pt] {\t};
}
\end{scope}
\begin{scope}[shift={(\groupxsep,\groupysep+\barsep)}]
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{on},P}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2,-.08) rectangle (3,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\end{scope}
\begin{scope}[shift={(0,\groupysep+2*\barsep)}]
\node[] at (-8,0) {Single gate};
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{off},Q}}/1\,\mathrm{k\Omega}$};
\fill[green] (-1,-.08) rectangle (2,.08);
\fill[orange] (-1,-.08) rectangle (-1.5,.08);
\fill[red] (-1.5,-.08) rectangle (-3,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\end{scope}
\begin{scope}[shift={(0,\groupysep+3*\barsep)}]
\node[] at (-8,0) {Crossbar};
\draw[-stealth] (-4,0)-- (4,0)node[right]{$\ensuremath{R_{\mathrm{off},Q}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2,-.08) rectangle (2,.08);
\fill[orange] (-2,-.08) rectangle (-2.5,.08);
\fill[red] (-2.5,-.08) rectangle (-3,.08);
\fill[red] (1.75,-.08) rectangle (2,.08);
\fill[orange] (1.5,-.08) rectangle (1.75,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\end{scope}
\begin{scope}[shift={(\groupxsep,\groupysep+2*\barsep)}]
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{on},Q}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2,-.08) rectangle (3,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\end{scope}
\begin{scope}[shift={(\groupxsep,\groupysep+3*\barsep)}]
\draw[-stealth] (-4,0) -- (4,0)node[right]{$\ensuremath{R_{\mathrm{on},Q}}/1\,\mathrm{k\Omega}$};
\fill[green] (-2,-.08) rectangle (3,.08);
\foreach \x in {-3.5,-3.0,...,3.5}{
\draw[] (\x,-.1) -- (\x,.1);
}
\end{scope}
\end{tikzpicture}
\caption{Comparison of single gate and (combined) crossbar simulation results. A range of $\pm 50\%$ around the nominal value is plotted for each parameter. The results are color-coded: Green for correct \gls{imply} output, red for false output and orange for ranges inbetween, that are not covered by the simulation.}
\label{fig:xbar_vs_single}
\end{figure*}
\subsection{Results analysis}\label{subsec:xbar_analysis}
In this section we compare the crossbar simulation results against the single gate results.
As before, to be efficient, results are represented using our technique introduced in \Cref{subsec:postprocess}. \Cref{fig:xbar_full_results_variation_von_voff} shows a complete set of crossbar simulations for the parameters $\{\ensuremath{v_{\mathrm{on},P}},\ensuremath{v_{\mathrm{off},P}},\ensuremath{v_{\mathrm{on},Q}},\ensuremath{v_{\mathrm{off},Q}}\}$.
\Cref{fig:xbar_vs_single} depicts the combined, i.e. worst case, results of all crossbar simulation setups explained in \Cref{subsec:xbar_method}, and the results of the single gate simulation, where the TTL threshold scheme was applied. The bars and color coding are identical to \Cref{fig:R_P_limit_m2_v,fig:R_P_limit_m2_R,fig:R_P_limit_m2_k,fig:R_P_limit_m0_v,fig:R_P_limit_m1_v}, \Cref{subsec:sim_analysis}. While the conclusions from \Cref{subsec:sim_analysis} remain true, unless noted otherwise, here we highlight the differences.
\subsubsection{Variation in voltage threshold}\label{subsubsec:xbar_v_var}
Given that the circuit is in a crossbar architecture, an increased number of errors due to threshold voltage variation can be expected in the crossbar simulation, when compared to the single gate simulation. Surprisingly, however, it is not significantly worse. \par
There are three main differences: First, the initialization failure of $\ensuremath{v_{\mathrm{off},Q}}$ (initially shown in \Cref{fig:R_P_limit_m2_v}) does not arise in the crossbar simulation. However, there were initialization failures in the crossbar for $8\,\mathrm{mV}\leq\ensuremath{v_{\mathrm{off},P}}\leq 9\,\mathrm{mV}$ ($-20\%$ to $-10\%$). Having said that, as $\ensuremath{v_\mathrm{off}}$ is of minor interest to the \gls{imply} operation, this can neither be considered an improvement, nor a degradation compared to the single gate. Second, results indicate failures if both $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$ are above $-0.63\,\mathrm{V}\ (-10\%)$ at the same time.
Based on the single gate simulation results (\Cref{subsec:sim_analysis}) and our recommendation to use \Cref{eqn:vonp_wmax} -- in combination with $R_Q=R_{Q,3}$ in \Cref{eqn:vpf} -- for device variability evaluation, this failure is predictable. As for the exact reason of this error, we assume that it is due to the increased state drift in $P$, as $|\ensuremath{v_{\mathrm{on},P}}|$ is so low. In terms of operational range, the valid values for $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$ are drastically restricted to the nominal value $\ensuremath{v_\mathrm{on}}$, as shown in \Cref{fig:xbar_vs_single}. It is only then that correct operations can be guaranteed.
However, if $\ensuremath{v_{\mathrm{on},P}} < -0.63\,\mathrm{V}\ (-10\%)$ is ensured, a much greater range for $\ensuremath{v_{\mathrm{on},Q}}$ is admissible, similar to the case of the single gate in \Cref{subsec:sim_analysis}. Finally, the third difference is that the \gls{imply} operation fails for $\ensuremath{v_{\mathrm{on},P}}<-0.77\,\mathrm{V}\ (+10\%)$ while $\ensuremath{v_{\mathrm{on},Q}}=\ensuremath{v_\mathrm{on}}$, as compared to $+20\%$ in the single gate simulation. Thus, the tolerable mismatch between $\ensuremath{v_{\mathrm{on},P}}$ and $\ensuremath{v_{\mathrm{on},Q}}$ shrinks to $10\%$ within the crossbar. \par
Apart from these differences the results of both simulations are identical, although \Cref{fig:xbar_vs_single} might not reveal it at the first look. This means that the proposed constraints for $\ensuremath{v_\mathrm{on}}$ and $\ensuremath{v_\mathrm{off}}$ from \Cref{sec:math} can be applied to get a basic understanding of threshold voltage variability within crossbar architectures. \par
\subsubsection{Variation in resistance limits}\label{subsubsec:xbar_r_var}
Varying the resistance limits of the memristors within the crossbar reveals some interesting results, as we can see in \Cref{fig:xbar_vs_single}. While \gls{imply} operations in the single gate simulation fail for $\ensuremath{R_{\mathrm{off},Q}}\leq 800\,\mathrm{k\Omega}\ (-20\%)$, the crossbar simulation shows correct results down to $\ensuremath{R_{\mathrm{off},Q}}=700\,\mathrm{k\Omega}\ (-30\%)$. We believe that this is due to the readout strategy applied to the crossbar, since the measured $R_Q$ after executing Case~3 (\Cref{tab:truth_table_imply}) is almost $1\,\mathrm{M\Omega}$ in a majority of the $-30\%$ simulation runs. Failures start occuring below $\ensuremath{R_{\mathrm{off},Q}}\leq 600\,\mathrm{k\Omega}\ (-40\%)$. The range between $-30\%$ and $-40\%$ variation is not explicitly covered by our simulation steps. \par
Furthermore, false \gls{imply} results within the crossbar come about at the upper and lower end of our simulated $\ensuremath{R_{\mathrm{off},Q}}$ range, as well as at the upper and lower end of the simulated $\ensuremath{R_{\mathrm{off},P}}$ range. This is a combined effect, since those errors only occur if both, $\ensuremath{R_{\mathrm{off},P}}\leq 500\,\mathrm{k\Omega}\ (-50\%)$ and $\ensuremath{R_{\mathrm{off},Q}}\geq 1.5\,\mathrm{M\Omega}\ (+50\%)$, or vice versa, are present at the same time. Interpreting this scenario based on the 1/2 or TTL logic thresholds from \Cref{tab:logic_thresholds}, one can see that if either $\ensuremath{R_{\mathrm{off},P}}$ or $\ensuremath{R_{\mathrm{off},Q}}$ are below $500\,\mathrm{k\Omega}$, they are not interpreted as logic `0', but logic `1'. Thus, they do not fulfill Case~1 of the truth table, where $p=0$ and $q=0$ must be true. Applying the 1/3 logic threshold scheme, an off-resistance of $500\,\mathrm{k\Omega}$ yields an undefined logic state. Therefore, none of the cases in the truth table is fulfilled. Hence, such errors are predicted via logic thresholds alone and do not require further evaluation using the constraints defined in \Cref{sec:math}. \par
Lastly, we should remark that simulation results for $\ensuremath{R_{\mathrm{on},P}}$ and $\ensuremath{R_{\mathrm{on},Q}}$ in the crossbar are identical to the the single gate simulation results.
\subsubsection{Variation in switching speed}\label{subsubsec:xbar_k_var}
Swichting speed variation does not pose a threat to single \gls{imply} gates, as deduced in \Cref{subsec:sim_analysis}. However, based on our simulation results (\Cref{fig:xbar_vs_single}), behavior within a crossbar is very different. For variations in $\ensuremath{k_{\mathrm{on},P}}$ and $\ensuremath{k_{\mathrm{on},Q}}$ larger than $\pm 20\%$, the \gls{imply} operation fails. Further analysis of those failures reveals that it is the mismatch between $P$ and $Q$ which causes most errors.
If either $\Delta\ensuremath{k_{\mathrm{on},P}}\leq -20\%$ while $\Delta\ensuremath{k_{\mathrm{on},Q}}\geq +20\%$, or $\Delta\ensuremath{k_{\mathrm{on},P}}\geq +20\%$ while $\Delta\ensuremath{k_{\mathrm{on},Q}}\leq -20\%$, the operation result is wrong. This mismatch cannot be predicted by our constraints. Further, the simulation indicates failures for variation in $\ensuremath{k_{\mathrm{off},P}}$ larger than $\pm 20\%$, as well as for $\Delta\ensuremath{k_{\mathrm{off},Q}}=\pm 40\%$. As $\ensuremath{k_{\mathrm{off},P}}$ and $\ensuremath{k_{\mathrm{off},Q}}$ are never relevant during \gls{imply}, we infer that these are initialization errors. They can, however, be resolved by using a different initialization scheme than the one we applied. For example, using an additional readout cycle to confirm written initial states. Such a scheme provides feedback to resolve initialization errors before \gls{imply} is executed.
\section{Conclusion}\label{sec:conclusion}
Device variability is one of the main challenges when implementing memristor-based logic. In this paper, we formulated novel constraints based on static switching conditions and state change dynamics. We note that the underlying causes of variation in device parameters are not differentiated by our methodology. Hence, environmental effects (such as temperature) causing parameter variation are taken into account by our constraints, just as process variations are. \gls{imply} operation results after a fixed timestep of execution were used as the metric to assess gate performance. In addition, different logic threshold schemes were considered. The derived constraints were put to the test in an extensive analysis for single gate and $128\times 128${} 1T1R crossbar and their simulation results were compared. An efficient simulation results presentation method was introduced and applied to find critical parameters. \par
As a result of our analysis, variability in threshold voltages, especially $\ensuremath{v_{\mathrm{on},Q}}$, was identified as a major root of concern regarding correct operations. We conclude that the most dominant reasons for failure are predictable by our theoretical analysis for both the single gate and the crossbar. Therefore, our analysis and recommendations can be used for designing a reliable \gls{imply} gate. More specifically, we suggest to choose design parameters away from the borders of the recommended areas. Ideally, this distance should be chosen such that the typical (or maximum) variations, do not lead to crossing the borders of recommended area. Nonetheless, accompanying studies or simulations should be conducted to understand the non-deterministic errors, especially regarding voltage threshold- and switching speed mismatch within the crossbar, as well as state drift phenomena. \par
Lastly, we note that our analysis can be used to decide whether a specific memristor technology and \gls{imply} logic are compatible. To that end, technology parameters need to be assessed based on the constraints for reliable \gls{imply} operations we extracted in this work. Further, considering technology-dependent parameter variation, an acceptable margin from the borders of the operating area must be ensured. Otherwise, chances for failures in \gls{imply} operations are increased. Hence, it would be better to use other technologies to implement the intended \gls{imply}-based circuits, or use other logics to implement the intended functionalities on the given technology.\par
\bibliographystyle{unsrt2authabbrvpp}
|
1,116,691,500,430 | arxiv | \section{\bf Introduction and main results}
\smallskip
\noindent The long-standing famous Nagata conjecture for characteristic $0$ was proved
by Shestakov and Umirbaev \cite{SU1, SU2}, and a strong version
of the Nagata conjecture was proved by Umirbaev and Yu \cite{UY}.
That is, the Nagata automorphism
$(x-2y(y^2+xz)-(y^2+xz)^2z, y+(y^2+xz)z, z)$ (Nagata coordinates $x-2y(y^2+xz)-(y^2+xz)^2z$ and $y+(y^2+xz)z$ respectively)
is (are) wild. In \cite{MSY, UY}, a stronger question (which implies the Nagata conjecture
and the strong Nagata conjecture) was raised:
whether the Nagata automorphism (coordinates) of the polynomial algebra $F[x,y,z]$ can be lifted to an automorphism
(coordinates) of the free associative $F\langle x,y,z\rangle$ over a field $F$? We can also formulate
\
\noindent {\bf The General Lifting Problem.}
Let $\phi=(f_1,\dots, f_n)$ be an automorphism of the polynomial algebra
$F[x_1,\dots,x_n]$ over a field $F$. Does there exists an $F$-automorphism
$\phi^{\prime}=(f_1^{\prime},\dots, f_n^{\prime})$ of the free associative algebra
$F\langle x_1,\dots, x_n\rangle$ such that each $f_i$ is the abelianization
of $f_i^{\prime}$?
\
\noindent For $n=2$, the answer of the above problem is positive,
as due to Jung \cite{J} and van der Kulk \cite{vdK} every automorphism of $F[x,y]$ is
composation of linear and elementary automorphisms which are liftable
to automorphisms of $F\langle x,y\rangle$. Moreover, Makar-Limanov \cite{ML} and Czerniakiewicz \cite{Cz} proved independently that Aut$(F\langle x,y\rangle)$ is actually isomorphic to Aut$(F[x,y])$, which implies
that the lifting is unique.
\
\noindent In this paper we prove the following new result,
which partially answers the question raised in \cite{UY} negatively. The result can be viewed
as the first step to attack the general lifting problem. In
a forthcoming paper \cite{BY}, we will deal the general lifting problem.
\begin{theorem} \label{lifting}
Let $(f,\ g)$ be an wild $F[z]$-automorphism of $F[x,y,z]=F[z][x,y]$.
Then $(f,g,z)$, as
an $F$-automorphism of $F[x,y,z]$, cannot be lifted to an
automorphism of $F\langle x,y,z\rangle$ fixing $z$.
\end{theorem}
\noindent The crucial step to prove Theorem \ref{lifting} is the following
\begin{theorem} \label{main}
Let $(f,g)$ be a wild
$F[z]$-automorphism of $F[x,y,z]=F[z][x,y]$, which can be effectively obtained
as the product of the canonical sequence of uniquely determined
alternative operations (elementary $F(z)$-automorphisms), and the sequence contains an
elementary $F(z)$-automorphism of
the type $(x, y+z^{-k}x^l+\dots)$ or
$(x+z^{-k}y^l+\dots, y)$ where $l>1$. Then $(f,g,z)$, as
an $F$-automorphism of $F[x,y,z]$, cannot be lifted to an
automorphism of $F\langle x,y,z\rangle$ fixing $z$.
\end{theorem}
\begin{corollary}
The Nagata automorphism cannot be lifted to an
automorphism of $F\langle x,y,z\rangle$ fixing $z$.
\end{corollary}
\begin{corollary}
Let $(f,g)$ be a wild
$F[z]$-automorphism of $F[x,y,z]=F[z][x,y]$.
Then neither $f$ nor $g$ can be lifted
to a $z$-coordinate of $F\langle x,y,z\rangle$. In particular,
the Nagata coordinates $x-2y(y^2+xz)-(y^2+xz)^2z$ and $y+(y^2+xz)z$
cannot be lifted to any $z$-coordinate of $F\langle x,y,z\rangle$.
\end{corollary}
\begin{proof}
Suppose $(f,h)$ is an $F[z]$-automorphism, then obviously
$(f,h)$ is the product of $(f,g)$ and an elementary
$F[z]$-automorphism of the type $(x,h_1)$. Therefore
$(f,h)$ is liftable if and only if $(f,g)$ is liftable.
Hence any $F[z]$-automorphism of the type $(f,h)$
is not liftable. Therefore $f$ cannot be lifted to
a $z$-coordinate of $F\langle x,y,z\rangle$. Same for $g$.
\end{proof}
\noindent Crucial to the proof of Theorem \ref{main} is the following new result,
that implies that the automorphism group
${\text{Aut}_Q(Q*_FF\langle x,y\rangle)}$ is tame, which has
its own interests.
\begin{theorem}[on degree increasing process] \label{degreeincrease}
Let $Q$ be an extension field over a field $F$.
A $Q$-automorphism of $Q*_FF\langle x,y\rangle$ can be effectively obtained as the
product of a sequence of
uniquely determined alternating operations (elementary automorphisms) of the following types:
\begin{itemize}
\item $\quad x\to x,\ y\to ryr'+\sum r_{0}xr_{1}x\cdots
r_kxr_{k+1},$
\item $x\to qxq'+q_{0}\sum yq_{1}y\cdots q_kyq_{k+1},\quad y\to y$
\end{itemize}
where $r,q, r_{j}, q_{j}\in Q$.
\end{theorem}
\noindent The following new result of degree estimate is also essential to the proof
of Theorem \ref{main}.
\begin{theorem}[Degree estimate] \label{degreeestimate}
Let $Q$ be an extension field of a field $F$.
Let $A = Q*_FF\langle x_1,\dots, x_n\rangle$ be a co-product
of $Q$ and the free associative algebra $F\langle x_1,\dots, x_n\rangle$ over $F$.
Suppose $f,g\in A$ are algebraically independent over $Q$,
$f^+$ and $g^+$ are algebraically independent over $Q$; or $f^+$ and $g^+$
are algebraically dependent, and neither $f^+$ is
$Q$-proportional to a power $g^+$, nor $g^+$ is
$Q$-proportional to a power $f^+$. Let $P \in
Q*_FF\langle x,y\rangle\backslash Q$.
Then
$$\deg(P(f,g))
\geq
\frac{\deg([f,\ g])}{\deg(fg)}w_{\deg(f), \deg(g)}(P),$$
where the degree is the usual homogeneous degree
with respect to $x_1,\dots,x_n$ and $w_{r,s}$ is the weight degree
with respect to $r,s.$\end{theorem}
\noindent Note that $u$ is {\bf proportional} to $v$ for $u,\ v\in Q*_FF\langle x_1,\dots, x_n\rangle$
means that there exist $p_1,\dots, p_m; q_1,\dots, q_m\in Q$ such that
$u=\Sigma_{i=1}^m p_ivq_i$ (it is important that `proportional'
is not reflexive, i.e. $u$ is proportional to $v$
does not imply $v$ is proportional to $u$), and that $f^+$ is the highest homogeneous form of $f$.
\
\begin{remark}
Theorem \ref{degreeestimate} is still valid for an arbitrary
division ring $Q$ over a field $F$. The proof is almost the same.
When $Q=F(z)$ then the result can be directly deduced from the
degree estimate in \cite{MLY, YuYungChang} via substitution
$\psi:\ x\to P_1(z)xP_2(z); y\to R_1(z)yR_2(z)$ for appropriate
$P_i, R_i\in F[z]$. For any element $\tau$ in $Q*_FF\langle
x,y\rangle$ there exist $P_i, R_i\in F[z]$ such that
$\psi(\tau)\in F\langle x,y,z\rangle$. In the sequel we only use
this special case regarding the lifting problem.
\end{remark}
\section{Proofs}
\noindent{\bf Proof of Theorem \ref{degreeestimate}}
Similar to the proof of the main result of Li and Yu
\cite{YuYungChang}, where Bergman's Lemma \cite{B1, B2} on
centralizers is used. See also Makar-Limanov and Yu \cite{MLY} for
the special case of characteristic $0$, where Bergman's Lemma on
radical \cite{B1, B2} is used.\qed
\
\noindent {\bf Proof of Theorem \ref{degreeincrease}.} Let $\phi=(f,g)$
be a $Q$-automorphism in $\text{Aut}_Q(Q*_FF\langle x,y\rangle)$ which is not linear,
namely,
$$\deg_{x,y}(f)+\deg_{x,y}(g)\ge 3.$$
By Theorem 1.5, we obtain
that either a power of $f^+$ is proportional to $g^+$,
or a power of $g^+$ is proportional to $f^+$. Now the proof is done by
induction.\qed
\
\noindent To prove the main result, we need a few more
lemmas.
\
\noindent {\bf Definition.} Let $D$ be a domain containing a field $K$,\ $E$
the field of fractions of $D$.
A monomial $\in E*_KK\langle
x, y\rangle$ of the following form,
$$...... {ptq} ......\ \ \ $$
where $t\in E\backslash D$,\ $p, q\in\{x, y\}$,\
is called a {\it sandwich monomial}, or just a {\it sandwich} for short.
\
\begin{lemma}[on sandwich preserving] \label{LeSandwich}
In the constructive decomposation in Theorem \ref{degreeincrease}, suppose
a sandwich $... {ptq} ...$ (where $p,q\in\{x,y\}$,\ $t\in
F(z)\backslash F[z]$, appears on some step during the process
of the effective decomposation, then there will be
some sandwich in any future step.
\end{lemma}
\begin{proof}
Let $f$ be the polynomial obtained in the $(n-1)^{-th}$ step of
the effective operation in Theorem \ref{degreeincrease},
$k=\deg(f)$. Take all sandwiches $s_\alpha$ of the maximum total
degree with respect to $x$ and $y$. Let $S=\sum s_\alpha$ be their
sum. Let $T=\sum t_\beta$ be the sum of components (monomials)
$t_\beta$ of $f$ maximum total degree respect to $x$ and $y$. It
is possible that $s_\alpha=t_\beta$ for some $\alpha,\beta$, then
$\deg(s_\alpha)=\deg(t_\beta)$ for all $\alpha,\beta$. In this
case $T=S+D$.
\
\noindent Suppose the $n^{-th}$ step has the following form $x\to
x, y\to y+G(x)$. Let $\bar{G}$ be the sum of monomials in $G$ with
the maximum degree. Then $\bar{G}(x)=\widetilde{G}(x,\dots,x)$
where
$$\widetilde{G}(x_1,\dots,x_n)=\sum_iq_{i,1}x_1q_{i,2}x_2\cdots
x_mq_{i,m+1},\quad q_{ij}\in F(z),$$
\noindent $m$ be the degree
of the $n^{-th}$ step operation (elementary automorphism).
\
\noindent Let $\deg(S)<\deg(T)$. Consider elements of the form
$\widetilde{G}(S,T,\dots,T)$. It is a linear combination of
sandwiches. All of them have the following form
$$q_0s_iq_1t_{2}\cdots
t_{m-1}q_m,\quad q_i\in Q.$$
Their sum is not zero, because for any polynomial of the form
$H=\sum_iq_{i_1}x_1q_{i_2}x_2\cdots x_mq_{i_{m+1}}$ such that
$H(x,\dots,x)\ne 0$ and for any $S, T\notin Q$,\ \
$H(S,T,\dots,T)\ne 0$.
\
\noindent If $\deg(S)=\deg(T)$, we consider elements of the form
$\widetilde{G}(S,S,\dots,S)$. It is a linear combination of
sandwiches. All of them have the following form
$$q_0s_1q_1\cdots
s_{m}q_m,\quad q_i\in Q.$$ \noindent $s_i$ are monomials from $S$.
Their sum is not zero, because for any polynomial of the form
$H=\sum_iq_{i_1}x_1q_{i_2}x_2\cdots x_mq_{i_{m+1}}$ such that
$H(x,\dots,x)\ne 0$ and for any $S\in Q$,\ \ $H(S,\dots,S)\ne 0$.
\
\noindent Now we are going to prove (via degree estimate) that
they cannot cancel out by other monomials (which must be
sandwiches). That is, there are no other sandwiches which are in
this form. They cannot be produced by $H(R_1,\dots,R_m)$ where
$R_i$ are monomials either from $S$ or $T$. This can be easily
seen from the following argument:
\
\noindent Suppose $\deg(S)=\deg(T)$ and $D\ne 0$. Then
if we substitute monomials forming $S$ and $D$ in different
`words', the outcomes would be different. Similarly suppose $\deg(S)<\deg(T)$ and $D\ne 0$,
then substituting monomials forming $S$ and $T$ in different `words',
the outcomes would be different. Suppose $D=0$, i.e. $S=T$. Then
$H(T,\dots,T)=H(S,\dots,S)$. Obviously in this case we need to do nothing.
\
\noindent Suppose we get such a sandwich because an element from
$F(z)\backslash F[z]$ appears between summands of polynomials,
obtained by the the (previous) $(n-1)^{-th}$ step. It means that
`fractional coefficient' in $F(z)\backslash F[z]$ appears in the
position in some term, between two monomials obtained on the
$(n-1)$-th step. Let us describe this situation in more details.
\
\noindent Let
$$x\to \sum v_i,\quad y\to \sum u_i$$
be an
automorphism, obtained on the $(n-1)$-th step. Consider $n$-th
step:
$$x\to x,\ y\to y+\sum_i q_0^ixq_1^i\cdots xq_{n_i}^i.$$
Let
$v_i=a_i\bar{v}_ib_i$ where $a_i, b_i\in Q, \bar{v}_i$ begins with
either $x$ or $y$ and also ends with either $x$ or $y$.
\
\noindent Suppose the leftmost factor $q\in F(z)\backslash F[z]$ corresponding
to the leftmost factor $q$ in monomial $s_i$ in $s$ appears in the
corresponding sandwich $w$.
Then it has the form
$$w=q_0^iv_{\alpha_1}q_1^i\cdots a_{\alpha_k}\bar{v}_{\alpha_k}b_{\alpha_k}q_k^i
a_{\alpha_{k+1}}\bar{v}_{\alpha_{k+1}}b_{\alpha_{k+1}}\cdots
a_{\alpha_{n_i}}\bar{v}_{\alpha_{n_i}}b_{\alpha_{n_i}}q_{n_i}^i$$
and the position $\bar{v}_{\alpha_k}b_{\alpha_k}q_k^i
a_{\alpha_{k+1}}\bar{v}_{\alpha_{k+1}}$ corresponds to the
position of fractional coefficient in the sandwich $s\cdot
\prod_{i=1}^{n-1}v_i$ living inside $s=s_1qs_2$, $s_1$ ends with
$x$ or $y$, $s_2$ begins with $x$ or $y$. Then
$$q=b_{\alpha_k}q_k^i a_{\alpha_{k+1}},
s_1= q_0^iv_{\alpha_1}q_1^i\cdots a_{\alpha_k}\bar{v}_{\alpha_k},
s_2T_{i_2}\cdots T_{i_n}=$$
$$=\bar{v}_{\alpha_{k+1}}b_{\alpha_{k+1}}\cdots
a_{\alpha_{n_i}}\bar{v}_{\alpha_{n_i}}b_{\alpha_{n_i}}q_{n_i}^i.
$$
Only in that case cancellation is possible. Here $T_i$ are
monomial summands of $T$.
\
\noindent Now let us compare the degrees. $\deg(s_1)<\deg(s)$,
$\sum_{i=k}^n\deg(v_i)\le (n_i-k+1)\deg(T)\le (m-1)\deg(T)$. Hence
$$\deg(W)<\deg(s)+(m-1)\deg(T)=\deg(\widetilde{G}(s,T,\dots,T))$$ so any
cancellation is impossible.
\end{proof}
\begin{lemma}[on coefficient improving] \label{LeCoeffImpr}
\noindent a) Let $x'=pxq;\ p,q\in F(z)$, $M_{\vec{q}}(x) =xq_1xq_2\cdots x$,
$q_i'=q^{-1}q_ip^{-1}$. Then $M_{\vec{q}}(x')= pM_{\vec{q'}}(x)q$.
\noindent b) Take the process in Theorem \ref{degreeincrease} without
sandwiches. Then after each step (except the last step), the outcome $(f,g)$
has the following properties:
\begin{itemize}
\item Both $f$ and $g$ are sandwich-free.
\item The left coefficients of $f$ and $g$ belong to $F[z]$. Moreover,
the two coefficients are relatively prime.
\item The right coefficients of $f$ and $g$ belong to $F[z]$. Moreover,
the two coefficients are relatively prime.
\end{itemize}
\noindent Now we can clearly see that the outcome of the last step also has the
above property.
\end{lemma}
\begin{proof} a) is obvious;\ b) is a consequence of a).\end{proof}
\
\begin{lemma} \label{Lefinish}
Let $f\in F(z)*_FF\langle x,y\rangle$,\ $P(u)\in F(z)*F[u]$ such
that each monomial has degree $\ge 2$ respect to $x$ and $y$.
Suppose that one of the coefficients of $P$ has zero right
$z$-degree and one of the coefficients of $f$ has zero right
$z$-degree, and there is no coefficients of $P$ and $f$ with
negative right $z$-degree. Then $P(f)$ has one of the coefficients
with zero right $z$-degree and the degree (respect to $x$ and $y$)
of corresponding term is strictly more then $\deg(f)$.
\end{lemma}
\begin{proof} Consider the highest degree monomials of $P$ and $f$ with zero
right $z$-degree, let $\widetilde{P},\widetilde{f}$ will be their
sums. Let $g$ be sum of terms of $f$ with zero right $z$-degree,
$h$ be the sum of terms of $f$ of maximal degree.
\
\noindent Now consider again the highest degree monomials in $P(u)$ with zero
right $z$-degree and substitute $\widetilde{f}$ on the rightmost
position instead of $u$ and $h$ on other positions of $u$. We
shall get some terms with non-zero sum $T$ (same argument as in
the proof of sandwich lemma). All such terms have zero right
$z$-degree.
\
\noindent It remains to prove that such terms cannot cancel out from
the other
terms. First of all, we need to consider only terms of $P$ with
zero right $z$-degree, other terms can not make any influence.
Second, we have to consider substitutions only of terms with zero
right $z$-degree on the rightmost positions of $u$. Let $V$ be
their sum.
\
\noindent But the sum of highest terms satisfying this conditions is equal to $T$
and $T$ is the highest homogeneous component of $V$, hence $V\ne
0$.
\end{proof}
\begin{corollary} \label{Cofinish}
Let $f$ be a polynomial, $P\in F(z)*F[x]$ such that each monomial
has degree $\ge 2$. Suppose that one of the coefficients of
$P(f)$ has (zero)negative right $z$-degree. Then $P(f)$ has one of
the coefficients with (zero) negative right $z$-degree and degree
of corresponding term is strictly more then $\deg(f)$.
\end{corollary}
\noindent There is just the `dual' left version of lemma \ref{Lefinish} and corollary
\ref{Cofinish}.
\
\noindent As a consequence of the above corollary, we get
\begin{lemma}\label{Lm2new}
In the step $x\to x$, ($z\to z$
because we are working with $z$-automorphisms) $y\to y+x^kz^{-l}$,\
$k>1$ of the degree-strictly-increasing process, applied to the automorphism $x\to x+\mbox{higest\ terms}, y\to
y+\mbox{higest\ terms}$ causes some negative
power(s).
\end{lemma}
\medskip
\noindent In order to prove Theorem \ref{lifting} we need a similar
statement which is also a consequence of the Corollary
\ref{Cofinish}.
\begin{lemma}\label{Lm2newLifting}
In the step $x\to x$, ($z\to z$ because we are working with
$z$-automorphisms) $y\to y+P(x)$,\ such that $P$ has negative
powers of $z$ as left coefficients of some monomial of degree $\ge
2$ in the degree-strictly-increasing process causes some negative
power(s) on any succeeding step.
\end{lemma}
\noindent Lemma \ref{Lefinish}, Lemma \ref{Lm2newLifting} and Corollary \ref{Cofinish} says
that any further step of non-linear operation either contains terms
of negative power with bigger degree, or does not interfere in the
process. Hence they imply the following
\begin{lemma} \label{LeLinindNonCancell}
\noindent a) Consider stage in strictly increasing process of following
form.
$$
x\to T_1+h_1,\quad y\to T_2+h_2
$$
where $T_i$ are sums of the terms with negative powers of $z$ to
the right, $h_i$ -- are sums of the terms without negative powers
of $z$ to the right.
\noindent If $T_i$ are $F(z)$-linear independent, then the negative powers can
not be cancelled in the strictly increasing process.
\
\noindent b) Suppose $T_1$ is the sum of the terms with negative powers of
$z$ to the right, $h_1$ -- is the sum of the terms without
negative powers of $z$ to the right, $T_2$ is the sum of the terms
with zero powers of $z$ to the right, $h_2$ is the sum of the
terms with positive powers of $z$ to the right.
\noindent If $T_i$ are $F(z)$-linear independent, then the
negative powers can not be cancelled in the strictly increasing
process.
\end{lemma}
\noindent In order to prove Theorem \ref{lifting} we need slight
generalization of the previous lemma, which also follows from the
Lemma \ref{Lefinish} and Corollary \ref{Cofinish}.
\begin{proposition} \label{PrLinindNonCancellGen}
Consider stage in strictly increasing process of following form:
$$
x\to T_1+h_1+g_1,\quad y\to T_2+h_2+g_2
$$
where $T_i$ are sums of the terms with negative powers of $z$ to
the right, $h_i$ -- are sums of the terms with zero powers of $z$
to the right, $g_i$ are sums of the terms with positive powers of
$z$ to the right.
\
\noindent If $T_i$ are $F(z)$-linear independent, or wedge product of
vectors
$$(T_1,T_2)\bigwedge_{F[z_l,z_r]} (h_1,h_2)\ne 0,$$
then the negative powers can not be cancelled in the strictly
increasing process. Wedge product is taken respect to left and
right $F(z)$-actions, i.e. as $F[z_l,z_r]$-modula, monomial
(respect to $x, y$, and inner positions of $z$) are considered as
basis vectors.
\end{proposition}
\begin{proof} Pbviously, any linear operation cannot cancel the negative powers of
$z$, but Lemma \ref{Lefinish} and corollary \ref{Cofinish} allows us
to consider only such operations.
\end{proof}
\begin{remark} \label{RemValuations}
Considering the substitutions $z\to z+c$ one can get similar results
for negative powers of $z+c$ (or via considering other valuations
of $F(z)$).
\end{remark}
\begin{lemma} \label{Leverylast}
Consider the step in the strictly increasing process of following form.
$$
x\to T+h_1',\quad y\to U+h_2'
$$
where $T$ is the sum of the terms with negative powers of $z$ to
the right, $U$ the sum of the terms with negative powers of $z$ to
the right, $h_1$ is the sum of the terms without negative powers
of $z$ to the right, $h_2$ is the sum of the terms with positive
powers of $z$ to the right.
\
\noindent If $T$ and $U$ are $F(z)$-linear independent, then the negative
powers cannot be cancelled in the strictly increasing process.
\end{lemma}
\begin{proof} By induction. Input of composition with polynomials
with $x$-degree $\ge 2$ cannot be cancelled (otherwise some negative power
appears in the highest terms, and the $F(z)$-independence preserves).
But the $x$-linear term action only produces the $F(z)$-linear combinations.
\end{proof}
\noindent Consider, for instance, the elementary automorphism $$x\to x,\quad y\to
y+z^nx^k.$$ It can be lifted to an $Q$-automorphism $$x\to x,\quad
y\to y+z^{n_0}x^{k_1}z^{n_1}\cdots x^{k_s}z^{n_s},\quad \sum
k_i=k,\ \sum n_i=n.$$ Though $n<0$, $n_0$ and $n_s$ can still be
non-negative. It is necessary to deal with that kind of situation by the next lemma.
\begin{lemma}\label{LeFinishSandwich}
Consider a elementary mapping
$$x\to x;\quad y\to y+P(x)$$
such that $P(x)$ has a monomial of the following form:
$$z^{k_1}xz^{k_2}x\cdots xz^{k_s}$$
where one of $k_i<0$ for some $i$ such that $1<i<s$. Then if such an
elementary transformation occurs in the strictly increasing process, it must produce some
sandwich.
\end{lemma}
\begin{proof}
First of all, due to the Lemma \ref{LeSandwich}, we may assume without
loss of generality that there exists no sandwiches before this
step.
\
\noindent Consider $z^{k_i}$, the minimum power of $z$, lying
before the variables for all monomials in $P$. Next, consider the
monomials in $P$ of the minimum degree containing $z^{k_i}$ between
$x$'s and among them, i.e. the monomials such that $z^{k_i}$
positioned on the left-most possible position (but then
$i>1$, it should be a sandwich position). Let us denote such
terms $T_i$.
\
\noindent Let $$\varphi(x)=\sum u_i,\quad \varphi(y)=\sum v_i$$
will be an automorphism, obtained by the previous step. Due to the
Lemma \ref{LeSandwich} we may assume that no terms come from $u_i, v_i$
are sandwiches.
\
\noindent Now we consider $u_i$ with minimal right $z$-degree
$n_r$, and among them -- terms with minimal degree (respect to $x$
and $y$). Let $u_j^r$ will be such terms, $u^r=\sum u_j^r$.
Because $x$ is one of the $u_i$, $n_r\le 0$. Similarly we consider
$u_i$ with minimal left $z$-degree $n_l$, terms $u_j^l$ and their
sum $u^l=\sum u_j^l$. We also get $n_l\le 0$.
\
\noindent Now for any monomial $T_j$, consider the element
$$E_{T_j}=q^{(j)}_0x\cdots u^rz^{n_i}u^l\cdots xq^{(j)}_s$$
obtained by replacement of $u^r$ and $u^l$ into the positions of
$x$ surrounding occurrence of $z^{n_i}$ as discussed previously, the
resulting power of $z$ would be equal to $n_r+n_l+n_i\le n_i<0$.
\
\noindent Now $E_{T_j}$ can be presented as a sum $E_{T_j}=\sum M_{E_{T_j}}$,
where $M_{E_{T_j}}$ are monomials. Monomials
$M_{E_{T_j}}$ are sandwiches, they may appear only that way which
was described previously and hence cannot cancell by other
monomials. Hence we must have a sandwich.
\end{proof}
\medskip
\
\newpage
\section{Proofs of the main theorems}
\noindent {\bf Proof of Theorem \ref{main}.}
\noindent Suppose the
automorphism $(f,g)$ can be lifted to a $z$-automorphism
of $F\langle x,y,z\rangle$. Then it induces an automorphism of
$F(z)*_FF\langle x,y\rangle$ and can be obtained by the process described in
the Lemma on coefficient improving.
\
\noindent Then at some steps some negative powers of $z$ appear either
between variables or on the right or on the left and it will be
preserved to the end, due to Lemma \ref{LeSandwich} and Lemma
\ref{LeFinishSandwich}, or Lemmas \ref{Leverylast},
\ref{LeLinindNonCancell}, \ref{Lm2new}.
\
\noindent Hence in the lifted automorphism, there exists
some negative power of $z$. A contradiction.
\qed
\
\noindent {\bf Proof of Theorem \ref{lifting}.}
\noindent Let $(f, g)$ be a wild $F[z]$-automorphism of $F[x,y,z]$ such that
it is not of the type in Theorem \ref{main}. Consider
corresponding strictly increasing process. We shall need few more
statements.
\noindent The following lemma is a consequence of Proposition \ref{PrLinindNonCancellGen}.
\begin{lemma} \label{LeAdjoint}
In the strictly increasing process. Consider the steps with
negative powers of $z$ appearing to the right.
$$\varphi:\ x\to x+P(y), y\to y$$
\noindent Let
$$\psi:\ x\to x, y\to y+Q_1(x)+Q_2(x)$$
where $\deg(Q_1)=1$, each term of $Q_2$ has degree $\ge 2$ and does not contain negative powers of $z$. Then
$\psi=\psi_1\circ\psi_2$ where $\psi_1:\ x\to x, y\to y+Q_1(x)$,
$\psi_2:\ x\to x, y\to y+Q_2(x)$ and $\varphi\psi_2\varphi^{-1}$
has no negative powers of $z$ to the right.
\end{lemma}
\noindent Lemma \ref{LeAdjoint} together with its left analogue and remark
\ref{RemValuations} imply following statement:
\begin{proposition}\label{PrAdjoint}
In the strictly increasing process, consider the step with
appearing coefficients not in $F[z]$.
$$\varphi:\ x\to\ x+P(y), y\to y$$
Let
$$\psi:\ x\to x, y\to y+Q_1(x)+Q_2(x)$$
where $\deg(Q_1)=1$, each term of $Q_2$ has degree $\ge 2$ and does not contain negative powers of $z$ . Then
$\psi=\psi_1\circ\psi_2$ where $\psi_1:\ x\to x, y\to y+Q_1(x)$,
$\psi_2:\ x\to x, y\to y+Q_2(x)$ and $\varphi\circ\psi_2\circ\varphi^{-1}$
is a $z$-automorphism of $F\langle x,y,z\rangle$.
\end{proposition}
\begin{proof}
Consider set of elements from $F(z)$ which are coefficients of our
monomials. If all valuations of $F(z)$ centered in finite points
are positive, then they belong to $F[z]$ and we are done. Due to
symmetry, it is enough to consider right coefficients and due to
substitution $z\to z+a$ just valuation centered in zero. Then
by Lemma \ref{LeAdjoint}, we are done.
\end{proof}
\begin{proof}
It is easy to see that $\psi\circ\varphi\circ\psi^{-1}$ has following form:
$x\to
x+c_1R(a'_{21}x+a'_{22}y), y\to c_2R(a'_{21}x+a'_{22}y)$,
where $a'_{ij}=\alpha a_{ij}\in F[z]$ are relatively prime, $\alpha\in F[z]$ is the least common multiple of the denominators of $a_{21}, a_{22}\in F[z]$ and $c_1, c_2\in F[z]$ such that
$c_1a_{21}+c_2a_{22}=0$. Choose $r, s\in F[z]$ such that $ra'_{21}+sa'_{22}=1$.
\
\noindent Acting the linear automorphism $x\to rx+sy, y\to
a'_{21}x+a'_{22}y$ over $F[z]$ to $\psi\circ\varphi\circ\psi^{-1}$, we get an automorphism of the following form: $x\to
rx+sy+tR(a'_{21}x+a'_{22}y), y\to a'_{21}x+a'_{22}y$, which is elementarily
equivalent to $x\to rx+sy, y\to a'_{21}x+a'_{22}y$. Hence $\psi\circ\varphi\circ\psi^{-1}$ is tame.
\end{proof}
\noindent The next proposition is well-known from linear algebra.
\begin{proposition} \label{PrLinzTame}
Let $(f,g)$ is a $z$-automorphism
of $F[z][x,y]$ linear in both $x$ and $y$. Then
it is a tame $z$-automorphism.
\end{proposition}
\noindent Now we are ready to complete the proof of Theorem 1.1. Suppose a
$z$-automorphism $\varphi=(f, g)$ of $F[z][x,y]$ can be lifted to an automorphism of $F[z]*_FF\langle x,y\rangle$
(i.e. an automorphism of $F\langle x,y,z\rangle$ fixing $z$),
which is decomposed into
product of elementary one according to strictly increasing
process. The coefficients of elementary operation can be in
$F(z)\backslash F[z]$ only for linear terms (see Lemma
\ref{Lm2newLifting} and Remark \ref{RemValuations}) and
conjugating non-linear elementary step with respect to the automorphisms
corresponding to these terms are $z$-tame. Hence $\varphi$ is a
product of $z$-tame automorphisms and $z$-automorphisms linear in both $x$ and
$y$. Now we are done by Proposition \ref{PrLinzTame}.
\
\noindent By carefully looking through the above proofs, we actually
obtained the following
\begin{theorem}\label{decomposation}
An automorphism $(f,g)$ in $\text{Aut}_{F[z]}F\langle x,y,z\rangle$,
can be canonically decomposed as product of the following type
of automorphisms:
\noindent i) Linear automorphisms in $\text{Aut}_{F[z]}F\langle x,y,z\rangle$;\
\noindent ii) Automorphisms which can be obtained by an
elementaty automorphism in
$\text{Aut}_{F[z]}F\langle x,y,z\rangle$ conjugated by a linear automorphism
in
\noindent $\text{Aut}_{F(z)}F(z)*_FF\langle x,\ y\rangle$.
\end{theorem}
\noindent Theorem \ref{decomposation} opens a way to obtain stably tameness
of $\text{Aut}_{F[z]}F\langle x,y,z\rangle$, which will be done in a separate
paper \cite{BY2}.
\
\section{\bf Acknowledgements}
\noindent Jie-Tai Yu would like to thank Shanghai University and Osaka University
for warm hospitality and stimulating atmosphere during his visit, when part
of the work was done. The authors thank Vesselin Drensky and
Leonid Makar-Limanov for their helpful comments and suggestions.
\smallskip
\noindent
\smallskip
|
1,116,691,500,431 | arxiv | \section{Introduction}
Recommender systems (RS) are a popular strategy to enable personalized users' experience \cite{ricci}.
Historical data (e.g., browsing activity and ratings) and product characteristics (e.g., title and description) are well-recognized data sources to train RSs.
Product information is often augmented with \emph{Knowledge Graphs} (KGs)~\cite{cao-etal-2018-neural,10.1145/2926718}.
These KGs include \emph{entities} (e.g., users, movies, actors) and \emph{relations} between entities (e.g., an actor starred a movie).
Integrating KGs within RSs has led to a gain in recommendation utility \cite{kgat, ripple}, especially under sparse data and cold-start scenarios \cite{9251221}.
Their inclusion is essential to make RS explainable and turn recommendation into a more transparent social process \cite{explainable-recsys-survey,Tintarev2007}.
Notable recommendation methods based on KGs include \emph{path reasoning methods}
~\cite{cfkg,10.1609/aaai.v33i01.33015329,ma2019jointly,kprn,ni-etal-2019-justifying,musto2021generating,pgpr,Song2019EkarAE,leveraging-demostrations}.
To guide RS training, they rely on paths that model high-order relations between users and products in the KG,
and identify those deemed as relevant between already experienced products and products to recommend.
Such paths are also used to create explanations, through explanation templates or text generation.
In the movie domain, the path “user$_1$ watched movie$_1$ directed director$_1$ directed$^{-1}$ movie$_2$” might lead to
the template-based explanation “movie$_2$ is recommended to you because you watched movie$_1$ also directed by director$_1$”.
Path reasoning methods are in contrast to regularization methods, which weight product characteristics based on their importance for a given recommendation
but do not provide any explanation~\cite{cke,deepcon,sequentialmemorykg,he2020mining, ripple, kgat}.
An abundance of KGs were proposed for recommendation, along with path reasoning methods, to produce both recommendations and explanations~\cite{arrieta2019explainable}.
However, evaluation protocols were heterogeneous (e.g., different train-test splits) and limited to a narrowed set of evaluation data sets and metrics.
Prior works often showed that a novel method led to a higher recommendation utility, compared to (non) knowledge-aware baselines.
None of the them deeply analyzed beyond utility goals (e.g., coverage, serendipity) nor monitored consumer (i.e., end users) and provider fairness.
Hence, it remains unclear whether path reasoning methods emphasize any trade-off between goals unexplored so far.
Being the landscape convoluted and polarized to utility, there is a need for a common evaluation ground to understand how and when each method can be adopted.
In this paper, we conduct a replicability study (different team and experimental setup) on unexplored evaluation perspectives relevant to path reasoning methods.
In a first step, we scanned the proceedings of top-tier conferences and journals, identifying seven relevant papers.
We tried to replicate the original methods based on the released source code, but only three of them were replicable.
In a second step, we defined a common evaluation protocol, including two public data sets (movies; music), two sensitive attributes (gender; age),
and sixteen metrics pertaining to four perspectives (recommendation utility; beyond utility goals; explanation quality; fairness).
We evaluated path reasoning methods under this protocol and compared them against other knowledge-aware methods.
Results reveal that, despite of an often similar utility, path reasoning methods differ in the way they meet other recommendation goals.
Our study calls for a broader evaluation of these methods and a more responsible adoption.
\section{Research Methodology}
In this section, we describe the collection process for path reasoning methods, the steps for their replication, and the common evaluation protocol.
\subsection{Papers Collection}
To collect existing path reasoning methods, we systematically scanned the recent proceedings of top-tier information retrieval events
(CIKM, ECIR, ECML-PKDD, FAccT, KDD, RecSys, SIGIR, WSDM, WWW, UMAP)
and journals edited by top-tier publishers (ACM, Elsevier, IEEE, Springer).
The adopted keywords combined a technical term between ``\emph{path reasoning recommender systems}'' and ``\emph{explainable recommender system}''
and a non-technical term between ``\emph{explainable AI}'' and ``\emph{knowledge enabled AI}''.
We marked a paper as relevant if (a) it addressed recommendation, (b) it proposed a KG-based method, and (c) the method could produce reasoning paths.
Papers on other domains or tasks, e.g., non-personalized rankings or mere entity prediction tasks (w/o any recommendation) were excluded.
We also excluded knowledge-aware methods unable to yield reasoning paths, although we will use some representatives of this class for comparison.
Seven relevant papers were selected for our study (Table \ref{tab:papers}).
We attempted to replicate the method of each relevant paper, relying as much as possible on the original source code.
To obtain it, we first tried to search for the source code repository into the original paper and on the Web.
As a last resort, we sent an e-mail to the original authors.
We considered a method to be replicable in case a fully working version of the source code was obtained
and needed minor changes to accept another data set and extract recommendations (and reasoning paths).
Three out of the seven relevant papers were replicable with a reasonable effort.
As per the non-replicable ones, three did not provide any source code\footnote{
Note that the source code of these papers might appear soon online as an effect of our e-mails to the original authors. We leave their replication as a future work.
}. The other one included unavailable external dependencies \cite{10.1162/dint_a_00013}.
\begin{table}[!t]
\caption{Path reasoning methods deemed as relevant in our study.}
\label{tab:papers}
\setlength{\tabcolsep}{6pt}
\resizebox{1\linewidth}{!}{
\begin{tabular}{llc|ccccc}
\hline
\textbf{Method} & \textbf{Year} & \textbf{Status}$^1$ & \multicolumn{5}{c}{\textbf{Experimental Setting}} \\
& & & \textbf{Data Sets}$^2$ & \textbf{Split Size}$^3$ & \textbf{Split Method}$^4$ & \textbf{Recommendation$^5$} & \textbf{Explanation$^5$} \\
\hline
\texttt{PGPR} \cite{pgpr} & 2019 & $RE$ & $AZ$ & $70$-$00$-$30$ & $Rand$ & NDCG, R, HR, P & - \\
\texttt{EKAR} \cite{Song2019EkarAE} & 2019 & $\overline{RE}$ & $ML$, $LFM$, $DB$ & $60$-$20$-$20$ & $Rand$ & NDCG, HR & - \\
\texttt{CAFE} \cite{cafe} & 2020 & $RE$ & $AZ$ & $70$-$00$-$30$ & $Rand$ & NDCG, R, HR, P & - \\
\texttt{UCPR} \cite{usercentric} & 2021 & $RE$ & $ML$, $AZ$ & $60$-$20$-$20$ & $Rand$ & NDCG, R, HR, P & PPC\\
\texttt{MLR} \cite{10.1145/3485447.3512083} & 2022 & $R\overline{E}$ & $AZ$ & $70$-$00$-$30$ & $Rand$ & NDCG, R, HR, P & - \\
\texttt{PLM-Rec} \cite{10.1145/3485447.3511937} & 2022 & $\overline{RE}$ & $AZ$ & $60$-$20$-$20$ & $Time$ & NDCG, R, HR, P & - \\
\texttt{TAPR} \cite{10.1145/3531267} & 2022 & $\overline{RE}$ & $AZ$ & $60$-$10$-$30$ & $Rand$ & NDCG, R, HR, P & - \\
\hline
\multicolumn{8}{l}{$^1$ \textbf{Status} $RE$ : Replicable and Extensible; $R\overline{E}$ : Replicable but not Extensible; $\overline{RE}$ : Not Replicable nor Extensible.}\tabularnewline
\multicolumn{8}{l}{$^2$ \textbf{Data Set} $AZ$ : Amazon \cite{amazon_dataset}; $ML$ : MovieLens 1M \cite{ml1m}; $LFM$ : LastFM \cite{lastfm-dataset}; $DB$ : DBbook2014 \cite{ktup}.}\tabularnewline
\multicolumn{8}{l}{$^3$ \textbf{Split Size} reports the percentage of data for training, validation, and test, respectively.}\tabularnewline
\multicolumn{8}{l}{$^4$ \textbf{Split Method}. $Rand$ : Random based; $Time$ : Time based.}\tabularnewline
\multicolumn{8}{l}{$^5$ \textbf{Metrics} $R$ : Recall; $HR$ : Hit Ratio $P$ : Precision; $PPC$ : Path Pattern Concentration}\tabularnewline
\end{tabular}
}
\vspace{-4mm}
\end{table}
\subsection{Methods Replication}
For each relevant paper, we analyzed the rationale of the proposed method and the characteristics of the experimental setting, as summarized in Table \ref{tab:papers}.
\texttt{PGPR} \cite{pgpr} (original source code: \url{https://github.com/orcax/PGPR})
was based on the idea of training a reinforcement learning (RL) agent for finding paths.
During training, the agent starts from a user and learns to reach the correct products, with high rewards.
During inference, the agent directly walks to correct products for recommendation, without enumerating all the paths between users and products.
The original experiments were done on four AZ data sets \cite{amazon_dataset} and on a KG built from product metadata and reviews.
\texttt{EKAR} \cite{Song2019EkarAE} (original source code not available)
modeled the task as a Markov decision process on the user-item-entity graph and used deep RL to solve it.
The user-item-entity graph is treated as the environment, from which the agent gets a sequence of visited nodes and edges.
Based on the encoded state, a policy network outputs the probability distribution over the action space.
Finally, a positive reward is given if the agent successfully finds those products consumed by the target users in the training set.
The novelty lays in using an LSTM for the policy network and a reward function that makes training stable and encourages agent exploration.
Only this study included data sets from three diverse domains: movies (ML1M), music (LFM), and books (DB).
\texttt{CAFE} \cite{cafe} (original source code: \url{https://github.com/orcax/CAFE})
follows the coarse-to-fine paradigm.
Given the KG, a user profile is created to capture user-centric patterns in the coarse stage.
To conduct multi-hop path reasoning guided by the user profile, the reasoner is decomposed into an inventory of neural reasoning modules.
Then, these modules are combined based on the user profile, to efficiently perform path reasoning.
Original experiments followed the PGPR experimental setting (same data sets, data split, and evaluation metrics).
\texttt{UCPR} \cite{usercentric} (original source code: \url{https://github.com/johnnyjana730/UCPR/})
introduces a multi-view structure leveraging not only local sequence reasoning information, but also a view of the user’s demand portfolio.
The user demand portfolio, built in a pre-processing phase and updated via a multi-step refocusing, makes the path selection process adaptive and effective.
The original experimental setting covered the movie (ML1M) and e-commerce domains (AZ).
This study was the only one assessing an explanation quality property, i.e., to what extent the KG relation type differs among the selected paths.
\texttt{MLR} \cite{10.1145/3485447.3512083} (source code shared by e-mail, but external dependencies missing)
is another RL framework that leverages both ontology-view and instance-view KGs to model multi-level user interests.
Through the Microsoft Concept Graph (MCG) \cite{10.1162/dint_a_00013},
the method creates various conceptual levels (e.g., Prada is an Italian luxury fashion brand).
The reasoning is then performed by navigating through these multiple levels with an RL agent.
The authors provided the source code, but the MCG was no longer online and the provided KG dump referred only to the originally used AZ data sets.
\texttt{PLM-Rec} \cite{10.1145/3485447.3511937} (original source code not available),
given a KG, extracts training path sequences under different hop constraints.
By leveraging augmentations of language features with semantics, the method obtains a series of training data sequences.
A transformer-based decoder is then used to train an auto-regressive path language model.
This method could limit previous methods' recall bias in terms of KG connectivity.
Only AZ data sets were used in the experiments.
\texttt{TAPR} \cite{10.1145/3531267} (original source code not available)
proposed another path reasoning approach based on RL, characterized by the incorporation of a temporal term in the reward function.
This temporal term guides a temporal-informed search for the agent, to capture recent trends of user's interests.
Original results (on AZ data sets) showed a gain in utility compared to PGPR, although the model was evaluated using a random split, which is not ideal for time-aware models.
\begin{table}[!t]
\caption{Interaction and knowledge information for the two considered data sets.}
\label{tab:data-stats}
\vspace{-3mm}
\centering
\setlength{\tabcolsep}{3pt}
\parbox{.4\textwidth}{
\centering
\footnotesize
\resizebox{1\linewidth}{!}{
\begin{tabular}{lrr}
\hline
\textbf{Interaction} & \textbf{ML1M} & \textbf{LFM1M} \\
\hline
Users & 6,040 & 4,817 \\
Products & 2,984 & 12,492 \\
Interactions & 932,295 & 1,091,275 \\
Density & 0.05 & 0.01 \\
Gender (Age) Groups & 2 (7) & 2 (7) \\
\hline
\end{tabular}}
}
\hfill
\parbox{.45\textwidth}{
\centering
\footnotesize
\renewcommand{\thesubfigure}{ii-z}
\resizebox{1\linewidth}{!}{
\begin{tabular}{lrr}
\hline
\textbf{Knowledge} & \textbf{ML1M} & \textbf{LFM1M} \\
\hline
Entities (Types) & 13,804 (12) & 17,492 (5) \\
Relations (Types) & 193,089 (11) & 219,084 (4) \\
Sparsity & 0.0060 & 0.0035 \\
Avg. Degree Overall & 28.07 & 25.05 \\
Avg. Degree Products & 64.86 & 17.53 \\
\hline
\end{tabular}}
}
\vspace{-4mm}
\end{table}
\subsection{Evaluation Protocol}
To ensure evaluation consistency and uniformity across methods, given the heterogeneous original experimental settings, we mixed replication and reproduction
\cite{10.1007/978-3-030-99736-6_37}, but use only the term ``replicability'' for convenience throughout this paper.
Specifically, we relied on the source code provided by the original authors to run their methods, and our own data and source code to (a)
pre-process the input data sets as per their requirements and (b) compute evaluation metrics based on the recommendations and reasoning paths they returned.
\noindent{\bf Data Collection.}
We conducted experiments on two data sets: MovieLens (ML1M) \cite{ml1m} and LastFM (LFM1B) \cite{lastfm-dataset}.
Given our interest in the fairness perspective, we selected data sets that provide (or make it possible to collect) users and providers' demographic attributes.
We therefore discarded other data sets, such as the Amazon ones \cite{amazon_dataset}, where this was not reasonably possible.
The selected data sets are all public and vary in domain, extensiveness, and sparsity, providing novel insights on the generalizability of the replicated path reasoning methods under a common ground, with respect to their original settings (see Table \ref{tab:papers}).
For ML1M, we used the KG generated in \cite{ktup} from DBpedia, while we generated the KG from the Freebase dump extracted by \cite{kb4rec} for LFM1B.
\noindent{\bf Data Preprocessing.}
Concerning the ML1M data set, both gender and age sensitive attributes for the consumers, but not for the providers, were originally provided in \cite{ml1m}.
Being directors considered as movie providers in prior work \cite{DBLP:journals/umuai/BorattoFM21}, we relied on their sensitive attribute labels collected in that study.
In LFM1B, gender and age labels were attached only to a small subset of end users.
We therefore discarded all those users whose sensitive attributes were not available.
Given that the original papers included only data sets far smaller than LFM1B and that our preliminary experiments uncovered a low scalability for those methods\footnote{
Solving substantial scalability issues goes beyond the scope of our replicability study.},
we then uniformly sampled a subset of the filtered LFM1B, ensuring that users (products) had at least 20 (10) interactions. {\color{black}We will refer to this data subset as LFM1M throughout the paper and results.}
This sampling allowed us to obtain a data set size comparable to ML1M and avoid cold-start scenarios, which are not our focus.
Since providers' sensitive attributes were not attached to the original data set (in music RS, artists are commonly considered as providers), we crawled them from Freebase and released them with our study.
Both KGs were pre-processed as performed in \cite{10.1145/3523227.3547374} to make triplets uniformly formatted.
{\color{black}More specifically, we only consider triplets composed of a product as the entity head and an external entity as the entity tail, to obtain a common ground data set for the analysis of both knowledge-aware and path-based methods. Considering triplets having external entities or products as the head and tail entities would have required to craft additional meta-paths (needed for path-based methods) compared to the reproduced studies, going beyond the scope of our work. In addition, to control sparsity, we removed relations having a type represented in less than 3\% of the total number of triplets}.
Concerning the user-product interactions, we discarded products (and their interactions) which are not present in the KG.
Pre-processed data set statistics are collected in Table \ref{tab:data-stats}.
\noindent{\bf Data Preparation and Split.}
For each data set, we first sorted the interactions of each user chronologically.
{\color{black}We then performed a training-validation-test split, following a time-based hold-out strategy, with the 60\% oldest interactions in the training set, the following 20\% for validation, and the 20\% most recent ones as the test set.
The aforementioned pre-processed data sets were used to train, optimize, and test each benchmarked model.
This allowed us to carry out the evaluation procedure in a realistic setting, in which the trends that might determine interaction patterns are non-stationary and evolve over time}.
\noindent{\bf Comparative Knowledge-aware Models.}
Path reasoning methods belong to a subclass of the knowledge-aware recommendation class.
To better contextualize our study, we therefore decided to provide comparisons (when interesting) against two knowledge-aware models based on knowledge embeddings,
namely CKE \cite{cke} and CFKG \cite{cfkg}, and a knowledge-aware model based on propagation, namely KGAT \cite{kgat}.
These three models, unable to provide reasoning paths to users, were replicated and evaluated under the same protocol\footnote{
For conciseness, we did not include non-knowledge-aware methods (e.g., BPR), which were compared against path reasoning methods under some metrics (e.g., NDCG) in studies like \cite{Balloccu2022ReinforcementRR}. Nevertheless, this is an important aspect for future work.}.
For conciseness, we do not explain their replication in detail and refer the reader to our repository.
\noindent{\bf Hyper-parameter Fine-tuning.}
Given a data set and a model, we selected the best hyper-parameters setting via a grid search that involved those hyperparameters (and their values) found to be sensitive in the original papers.
In certain cases, given our findings from preliminary experiments, we extended the grid of values to better adhere to the characteristics of the data set at hand.
Full details on the hyper-parameters and their values in our grid search are reported in our repository.
Models obtained via different hyper-parameter settings were evaluated on the validation set, selecting the one achieving the highest NDCG.
\noindent{\bf Evaluation Metrics Computation.}
Given a model and a data set, we monitored recommendation utility, beyond utility objectives, explanation quality, and both consumer and provider fairness, on recommended lists with the well-known size of $k=10$ (e.g., \cite{10.1007/978-3-030-99736-6_37}), based on the corresponding test set.
We describe each metric in Table~\ref{tab:metrics-summary} and refer to the repository for implementation details.
Concerning recommendation utility for consumers, we monitored the Normalized Discounted Cumulative Gain (NDCG) \cite{WangWLHL13},
using binary relevance scores and a base-2 logarithm decay,
and the Mean Reciprocal Rank (MRR) \cite{Craswell2009}.
Differently from recall and accuracy, NDCG takes into account the position of the relevant products in the recommended list.
MMR instead considers the position of the first relevant product only, giving us a perspective different than NDCG.
In our work, we focused also on four well-known beyond utility goals \cite{10.1145/2926720}.
We monitored the extent to which the generated recommendations cover the catalog of available products (coverage).
High coverage may increase users' satisfaction and the sales.
Another goal, diversity, was found to be relevant for human understanding \cite{GEDIKLI2014367} and content acceptance \cite{10.1145/290941.291025}.
We computed it as the percentage of distinct product categories in the recommended list.
Further, serendipity measures recommendation surprise\cite{10.1145/963770.963772}.
Given our offline setting, we compared the recommendations with those of a baseline model, i.e., a most popular recommender \cite{10.1007/978-3-540-78197-4_5}.
The more the recommendations differ between the benchmarked and the baseline model, the higher the serendipity.
Finally, we estimated novelty as the inverse of product popularity (as per the received ratings), assuming that products with low popularity are more likely to be surprising \cite{Zhou2010SolvingTA}.
\begin{table}[!t]
\caption{Evaluation metrics covered in our replicability study.}
\label{tab:metrics-summary}
\vspace{-3mm}
\resizebox{1\linewidth}{!}{
\begin{tabular}{lllll}
\hline
\textbf{Perspective} & \textbf{Metric} & \textbf{Acronym} & \textbf{Range} & \textbf{Description} \\
\hline
\multirow{2}{*}{\makecell[l]{\textbf{Consumers} \\ \texttt{Utility}}} & \makecell[l]{Normalized Discounted \\ Cumulative Gain} & NDCG & {[}0, 1{]} & The extent to which the recommended products \\
& & & & are useful for the user (1 means more useful). \\
& Mean Reciprocal Rank & MRR & {[}0, 1{]} & The extent to which the first recommended product \\
& & & & is useful for the user (1 means more useful). \\
\hline
\multirow{4}{*}{\makecell[l]{\textbf{Consumers} \\ \texttt{Beyond} \\ \texttt{Utility}}} & Coverage & COV & (0, 1) & The percentage of products overall \\
& & & & recommended at least once (1 means high coverage). \\
& Diversity & DIV & (0, 1{]} & The percentage of product categories covered \\
& & & & in the recommended list (1 means high diversity).\\
& Novelty & NOV & (0, 1) & Inverse of the popularity of products recommended \\
& & & & to a user (1 means low popularity, so high novelty). \\
& Serendipity & SER & {[}0, 1{]} & The percentage of the recommended products not \\
& & & & suggested also by a baseline (1 means more unexpected). \\
\hline
\multirow{8}{*}{\makecell[l]{\textbf{Consumers} \\ \texttt{Explanation} \\ \texttt{Quality}}} & Fidelity & FID & {[}0, 1{]} & The percentage of the recommended products that can \\
& & & & be explained (1 means all products can be explained). \\
& Linking Interaction Recency & LIR & {[}0, 1{]} & The recency of the past interaction in the paths \\
& & & & accompanying recommended products (1 means recent). \\
& Linking Interaction Diversity & LID & (0, 1{]} & The number of distinct past interactions in the paths \\
& & & & accompanying recommended products (1 means different). \\
& Shared Entity Popularity & SEP & {[}0, 1{]} & The popularity of the shared entity in the paths \\
& & & & accompanying recommended products (1 means popular). \\
& Shared Entity Diversity & SED & (0, 1{]} & The number of distinct shared entities in the paths \\
& & & & accompanying recommended products (1 means different). \\
& Path Type Diversity & PTD & (0, 1{]} & The percentage of distinct path types within paths \\ & & & & accompanying recommended products (1 means different). \\
& Path Type Concentration & PTC & {[}0, 1{]} & The extent to which the distinct path types \\
& & & & representation is equally balanced (1 means balanced). \\
\hline
\makecell[l]{\textbf{Providers} \\ \texttt{Utility}} & Exposure & EXP & {[}0, 1{]} & \makecell[l]{Exposure of the items of a given provider in the \\ recommended list (1 means high exposure).}\\
\hline
\end{tabular}}
\vspace{-6mm}
\end{table}
With regard to explanation quality, we considered the proportion of explainable products in a recommended list (fidelity) \cite{10.1145/3219819.3220072}.
In addition, we monitored reasoning paths properties concerning recency, popularity, and diversity \cite{10.1145/3477495.3532041}.
Recent linking interactions can help the user to better catch the explanation.
The linked interaction recency measures the recency of the past interaction presented in the explanation path,
whereas the linked interaction diversity monitors how many distinct past interactions are present.
The second perspective is related to shared entities, assuming that more popular shared entities have a higher chance of being familiar to the user.
The shared entity popularity measures the popularity (node degree in the KG) of the shared entity in an explanation path.
Conversely, the shared entity diversity monitors the distinct shared entities.
Finally, path type diversity focuses on how many distinct path types are included.
Path type concentration monitors whether path types are equally balanced.
With the increasing importance received by fairness, we also assessed fairness with respect to a notion of demographic parity \cite{10.1007/978-3-030-99736-6_37,DBLP:conf/sigir/GomezZBSM21}.
For consumer fairness, given a metric, we computed the average value of that metric for each demographic group and monitored the absolute pairwise difference between groups\footnote{
Our data includes sensitive attributes pertaining to the gender (Male, Female) and age (Under 18, 18-24, 25-34, 35-44, 45-49, 50-55, 56$+$), as per the data set labels.
}.
Concerning provider fairness, we computed the average exposure given to products of providers in a given demographic group \cite{DBLP:journals/umuai/BorattoFM21}.
Again, we finally computed the average absolute pairwise difference between provider groups.
\section{Experimental Results}
Our study aimed to investigate multiple evaluation perspectives of path reasoning methods, by answering to the following research questions:
\vspace{-1.5mm}
\begin{enumerate}[label=\textbf{RQ\arabic*},leftmargin=10mm]
\item Do path reasoning methods trade recommendation utility and/or beyond utility objectives for explanation power?
\item To what extent can path reasoning methods produce explanations for all the recommended products, depending on the recommended list size?
\item How does the quality of the selected paths vary among path reasoning methods, based on the path type and characteristics?
\vspace{-4mm}
\end{enumerate}
\subsection{Trading Recommendation Goals for Explanation Power (RQ1)}
\label{sec:rq1}
In a first analysis, we investigated whether there exists any substantial difference in recommendation utility and beyond utility objectives between the considered path reasoning methods (PGPR, CAFE, UCPR) and relevant knowledge-aware but not explainable methods (KGAT, CKE, CFKG). {\color{black}
To assess statistical significance, t-tests were carried out for each metric, considering the two categories of methods as the two separate groups, under each data set. This allowed us to discern behavior also with respect to the sparsity of the KG.
Although the total sample size (six methods) is rather small, this setup made it possible to notice some preliminary characteristics of the two method classes. Still, further studies on a broader set of methods should be run to assess results generalizability more.
P-values obtained for each test are reported in Table~\ref{tab:rq1-pvalue}.}
Figure~\ref{fig:rq1-radar} depicts our evaluation results in terms of utility (NDCG, MMR) and beyond utility goals (serendipity, diversity, novelty, and coverage). We also report provider fairness estimates, whereas we will discuss consumer fairness later on. Even though Figure~\ref{fig:rq1-radar} makes the comparison easier, we refer to Table \ref{tab:rq1} for specific values.
Concerning recommendation utility, path reasoning methods achieved comparable scores (0.26 to 0.28 NDCG; 0.18 to 0.21 MRR) to knowledge-aware non-explainable methods (0.26 to 0.29 NDCG; 0.21 to 0.23 MRR) in ML1M.
These observations did not hold in LFM1M, where path reasoning methods led to recommendations of lower utility (0.15 to 0.34 NDCG; 0.09 to 0.27 MMR) than knowledge-aware baselines (0.13 to 0.40 NDCG; 0.10 to 0.34 MMR).
{\color{black} However, in both cases and under both data sets, no statistical differences were found in terms of recommendation utility between method classes (all p-values were greater than $0.05$).
}
\begin{figure*}[!t]
\centering
\includegraphics[width=.75\textwidth]{rec-quality-radarplot.pdf}
\vspace{-3mm}
\caption{Comparison on recommendation utility and beyond utility goals [RQ1].}
\label{fig:rq1-radar}
\vspace{2mm}
\end{figure*}
\begin{table}[!t]
\vspace{-2mm}
\caption{Metric scores for recommendation utility and beyond utility goals [RQ1].}
\vspace{-3mm}
\resizebox{1\linewidth}{!}{
\begin{tabular}{l|l|l|l|l|l|l|l||l|l|l|l|l|l|l}
\hline
\multicolumn{1}{c}{\textbf{Method}} & \multicolumn{7}{c}{\textbf{ML1M}} & \multicolumn{7}{c}{\textbf{LFM1M}} \\
& NDCG $\uparrow$ & MMR $\uparrow$ & SER $\uparrow$ & DIV $\uparrow$ & NOV $\uparrow$ & PF$^1$ $\downarrow_0$ & COV $\uparrow$ & NDCG $\uparrow$ & MMR $\uparrow$ & SER $\uparrow$ & DIV $\uparrow$ & NOV $\uparrow$ & PF$^1$ $\downarrow_0$ & COV $\uparrow$ \\
\hline
\texttt{CKE} & \textbf{0.29} & \textbf{0.23} & 0.26 & 0.10 & \textbf{0.93} & 0.19 & \underline{0.70} &
\textbf{0.40} & \textbf{0.34} & \underline{0.82} & 0.18 & 0.88 & 0.18 & \textbf{0.91} \\
\texttt{CFKG} & \underline{0.26} & 0.21 & 0.11 & 0.11 & 0.92 & 0.25 & 0.16 & 0.13 &
0.10 & 0.04 & 0.27 & 0.86 & \underline{0.34} & 0.02 \\
\texttt{KGAT} & \textbf{0.29} & \textbf{0.23} & 0.29 & 0.10 & \textbf{0.93} & 0.19 & \textbf{0.75} & \underline{0.37} & \underline{0.31} & 0.79 & 0.19 & \textbf{0.88} & 0.18 & \underline{0.89} \\
\hline
\texttt{PGPR} & 0.28 & 0.21 & 0.78 & 0.42 & 0.93 & 0.27 & 0.42 & 0.31 & 0.25 & 0.81 & 0.54 & 0.82 & 0.32 & 0.20 \\
\texttt{UCPR} & \underline{0.26} & 0.20 & 0.53 & \underline{0.42} & \textbf{0.93} & 0.22 & 0.25 & 0.34 & 0.27 & \textbf{0.94} & \underline{0.57} & \underline{0.87} & 0.22 & 0.41 \\
\texttt{CAFE} & \underline{0.26} & 0.18 & \underline{0.63} & \textbf{0.44} & \textbf{0.93} & \textbf{0.36} & 0.21 & 0.15 & 0.09 & 0.75 & \textbf{0.58} & 0.84 & \textbf{0.36} & 0.11 \\
\hline
\multicolumn{15}{l}{For each dataset: best result in \textbf{bold}, second-best result \underline{underlined}. $\; \;$ $^1$ \textbf{Metrics} $PF$: Provider Fairness.}\tabularnewline
\end{tabular}
}
\label{tab:rq1}
\end{table}
\begin{table}[!t]
\vspace{-2mm}
\caption{T-test p-values to assess statistically significant differences between path-based and knowledge-aware methods across evaluation metrics [RQ1].}
\vspace{-3mm}
\centering
\resizebox{.6\linewidth}{!}{
\begin{tabular}{l|r|r|r|r|r|r|r}
\hline
\textbf{Data Set$^1$} & NDCG & MMR & SER & DIV & NOV & PF & COV \\
\hline
\textbf{ML1M} & 0.33 & 0.08 & \textbf{0.01} & \textbf{0} & 0.422 & 0.21 & 0.33 \\
\hline
\textbf{LFM1M} & 0.77 & 0.65 & 0.38 & \textbf{0} & 0.16 & 0.38 & 0.34 \\
\hline
\multicolumn{8}{l}{P-values below $0.05$ are reported in \textbf{bold}.} \tabularnewline
\end{tabular}
}
\label{tab:rq1-pvalue}
\vspace{-3mm}
\end{table}
\begin{figure*}[!t]
\centering
\includegraphics[width=.8\textwidth]{rec-quality-cfairness-radarplot.pdf}
\vspace{-3mm}
\caption{Disparate impacts between groups (gender and age) on recommendation utility, beyond utility objectives, and provider fairness. The lower it is, the fairer [RQ1].}
\label{fig:rq1-fairness-radar}
\vspace{-4mm}
\end{figure*}
With regard to beyond utility objectives, path reasoning methods achieved substantially higher serendipity (0.53 to 0.78 ML1M; 0.75 to 0.94 LFM1M) than knowledge-aware non-explainable baselines (0.11 to 0.29 ML1M; 0.04 to 0.82 LFM1M).
Interestingly, it was confirmed that path reasoning methods tend to perform worse in LFM1M than ML1M.
{\color{black} Furthermore, our statistical tests show that, on ML1M, the two method classes performed differently in terms of serendipity (p-value equal to 0.011), with path reasoning methods showing higher serendipity than the knowledge aware methods, on average.
Similar patterns were found for diversity. Path reasoning methods led to a higher diversity on average (respectively
for path-reasoning and knowledge-aware methods, 0.56 and 0.21 on LFM1M; 0.43 and 0.10 on ML1M), on both ML1M and LFM1M (p-value $\approx 0$). }
{\color{black} Conversely, on coverage, the best path reasoning method showed a decrease of 44\% on ML1M and 54.9\% on LFM1M than the best knowledge-aware method. This might be due to the low number of paths in the KG available to the path-reasoning methods. Except for CFKG, there is evidence that knowledge-aware methods should be preferred in case someone aims to optimize for coverage.
}
Finally, novelty scores were similar between the two classes of methods.
From a provider fairness perspective, path reasoning methods led to a fairer exposure of provider groups (0.22 to 0.36 ML1M; 0.22 to 0.36 LFM1M), compared to the other family (0.19 to 0.25 ML1M; 0.18 to 0.34 LFM1M).
Surprisingly, CFKG reported the second best provider fairness score, despite of its low recommendation utility.
On the other hand, consumer fairness estimates according to the considered evaluation metrics\footnote{
Differences between the demographic groups achieving the best and worst score on avg. were all statistically significant under t-test (if applicable) or h-test otherwise.
} are collected in Figure \ref{fig:rq1-fairness-radar}.
Being the patterns comparable across data sets and demographic groups, we describe only the results obtained in ML1M and the gender groups.
For the latter, all the models presented some yet low levels of unfairness in term of utility (both NDCG and MRR).
Path reasoning methods (PGPR, UCPR, CAFE) achieved higher levels of unfairness on coverage, respectively 0.084, 0.045, 0.032, compared to baseline methods (KGAT reached the highest coverage unfairness, with 0.06).
For the other metrics, we uncovered small yet comparable differences.
\vspace{-1mm}
\hlbox{Findings RQ1}{
Path reasoning methods trade recommendation utility and coverage for explanation power, especially in LFM1M.
Conversely, they resulted in higher estimates on other beyond utility objectives and provider fairness than knowledge-aware non-explainable baselines.
}
\vspace{-1mm}
\subsection{Producing Explanations for All Recommended Products (RQ2)}
In a second analysis, we were interested in understanding the extent to which path reasoning methods can produce explanations
for all the recommended products across recommended lists of different sizes.
This property, fidelity, is essential for a method which yields reasoning paths (and produces explanations).
Table \ref{tab:rq3} shows fidelity scores for the two data sets on lists of size 10, 20, 50 and 100.
Concerning PGPR, paths were attached to almost every product of the recommended list, until the size of 50.
Under a size of 100, fidelity remarkably decreased to 78\%. This decay in fidelity was exacerbated in LFM1M.
In the latter data set, already with a size of 20, only 74\% of recommended products were explained.
With regard to UCPR, we observed a similar but reversed pattern, compared to PGPR, on the two considered data sets.
UCPR was challenged to produce explanations even under a size of 10 on ML1M (only 61\% of products were explained).
Conversely, in LASTFM, the same model obtained a higher fidelity than PGPR, with 99\% of explained products under a size of 10.
Surprisingly, CAFE was able to provide a reasoning path for each recommended product until a list of size 100.
It should however be noted that, to make this happen, the list size must be specified in advance during training.
CAFE indeed automatically adapts the size of the neighbourhood to search around, according to the list size.
Hence, this method would be the best choice when the list size is known in advance, constant, or up to a certain limit.
Although fidelity could be controlled, doing this led to a smaller NDCG for CAFE (see Section \ref{sec:rq1}).
\vspace{-2.5mm}
\hlbox{Findings RQ2}{
Path reasoning methods show very different patterns in terms of fidelity.
CAFE's fidelity is high and stable across data sets and recommended list sizes.
On the other hand, PGPR provides higher but rapidly decaying fidelity in ML1M than LFM1M, viceversa for UCPR.
}
\begin{table}[!t]
\centering
\caption{Explanation fidelity analysis across cut-offs $k=\{10,20,50,100\}$ [RQ2].}
\label{tab:rq3}
\vspace{-2mm}
\setlength{\tabcolsep}{6pt}
\resizebox{.6\linewidth}{!}{
\begin{tabular}{l|rrrr|rrrr}
\hline
\textbf{Method} & \multicolumn{4}{c}{\textbf{ML1M}} & \multicolumn{4}{c}{\textbf{LFM1M}} \\
& 10 & 20 & 50 & 100 & 10 & 20 & 50 & 100 \\
\hline
\texttt{PGPR} & 1.00 & 0.99 & 0.99 & 0.78 & 0.98 & 0.74 & 0.31 & 0.15 \\
\texttt{CAFE} & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
\texttt{UCPR} & 0.61 & 0.34 & 0.14 & 0.07 & 0.99 & 0.98 & 0.68 & 0.35\\
\hline
\end{tabular}}\\
\vspace{-3mm}
\end{table}
\subsection{Differences on Explanation Quality (RQ3)}
\label{sec:rq2}
In a final analysis, we investigated how the quality of the selected paths (and so of the resulting explanations) varied based on the path characteristics.
To this end, Table \ref{tab:rq2} collects seven explanation path quality perspectives (LIR, LID, SEP, SED, PTD, PTC, PPC) for each reasoning path method and data set.
Concerning the recency dimension, we did not observe any substantial difference in linked interaction recency among the three methods,
with the maximum (minimum) value 0.44 (0.34) achieved by PGPR (CAFE) on ML1M (similarly on LFM1M).
Whereas, in terms of linked interaction diversity, it can be interestingly noted that PGPR (0.84 ML1M; 0.77 LFM1M) and UCPR (0.82 ML1M; 0.84 LFM1M) led to higher diversity than CAFE.
Moving to the popularity perspective and, the shared entity popularity in particular,
CAFE was able to obtain the highest SEP in both data sets (0.75 ML1M; 0.77 LFM1M),
meaning that it had the tendency to yield paths with more popular shared entities.
Compared to CAFE, PGPR and UCPR showed instead substantially lower values.
Estimates on SED were very high (0.92 to 1 ML1M; 0.78 to 0.98 LFM1M) for all the methods.
These methods had hence the ability to include a good variety of shared entities in their reasoning paths.
Patterns regarding path types were particularly interesting.
Specifically, both path type diversity and path type concentration were higher in CAFE and UCPR than PGPR.
This highlights that the explanations of the former were, on average, richer in terms of path types (e.g., starred by, directed by),
while PGPR's explanations were limited, on average, to a narrow set of different path types.
\begin{table}[!t]
\centering
\vspace{-2mm}
\caption{Explanation quality analysis [RQ3].}
\label{tab:rq2}
\vspace{-2mm}
\resizebox{1\linewidth}{!}{
\begin{tabular}{l|l|l|l|l|l|l|l||l|l|l|l|l|l|l}
\hline
\textbf{Model} & \multicolumn{7}{c}{\textbf{ML1M}} & \multicolumn{7}{c}{\textbf{LFM1M}} \\
& LIR $\uparrow$ & LID $\uparrow$ & SEP $\uparrow$& SED $\uparrow$& PTD $\uparrow$& PTC $\uparrow$& PPC $\uparrow$& LIR $\uparrow$& LID $\uparrow$& SEP $\uparrow$& SED $\uparrow$& PTD $\uparrow$& PTC $\uparrow$& PPC $\uparrow$\\
\hline
\texttt{PGPR} & \textbf{0.44} & \textbf{0.84} & \underline{0.43} & \underline{0.99} & 0.12 & \underline{0.03} & 0.12 & \textbf{0.49} & \underline{0.77} & 0.61 & \underline{0.94} & \underline{0.30} & 0.05 & 0.24 \\
\texttt{CAFE} & 0.34 & 0.16 & \textbf{0.75} & \textbf{1.00} & \textbf{0.33} & \textbf{0.73} & \textbf{0.37} & \textbf{0.49} & 0.25 & \textbf{0.77} & \textbf{0.98} & 0.25 & \textbf{0.62} & \textbf{0.50} \\
\texttt{UCPR} & \underline{0.40} & \underline{0.82} & 0.35 & 0.92 & \underline{0.24} & 0.01 & \underline{0.24} & \textbf{0.49} & \textbf{0.84} & \underline{0.67} & 0.78 & \textbf{0.42} & \underline{0.24} & \underline{0.34}\\
\hline
\multicolumn{15}{l}{For each dataset: best result in \textbf{bold}, second-best result \underline{underlined}.}\tabularnewline
\end{tabular}
}
\end{table}
\begin{figure*}[!t]
\centering
\includegraphics[scale=0.42]{path-quality-cfairness-radarplot.pdf}
\vspace{-2mm}
\caption{Disparate impacts between groups (gender and age) on explanation quality
pertaining to recency, popularity, and diversity. The lower it is, the fairer [RQ3].}
\label{fig:rq2-fairness-radar}
\vspace{-3mm}
\end{figure*}
Figure \ref{fig:rq2-fairness-radar} depicts pairwise differences in the average score of a given metric between demographic groups (consumer fairness).
Again, under the same conditions of RQ1, all the differences were statistically valid.
For conciseness, we discuss only the results for gender groups.
PGPR and CAFE showed very low unfairness estimates across all metrics,
with all scores lower than 0.01 in UCPR and 0.02 in PGPR on both data sets.
Differently, UCPR emphasized unfairness on PTD, SED and PPC.
Remarkably, the strongest disparate impact was reported on FID (0.12), PTD and PPC (0.05) for both data sets, and SED (0.03). {\color{black} We conjecture that these estimates of unfairness could be driven by data imbalance across the different demographic groups.}
\vspace{-1mm}
\hlbox{Findings RQ3}{
Path reasoning methods often yield substantially different paths in terms of recency, popularity, and diversity.
Although they exist, no remarkable disparate impacts on explanation quality were found.
}
\vspace{-3mm}
\section{Discussion and Conclusion}
In this section, we connect the main findings coming from the individual experiments
and present the implications and limitations of our replicability study.
In the first analysis, we analyzed how methods able to produce explanations (path reasoning) compared against knowledge-aware non-explainable baselines in terms of utility, beyond utility objectives, and provider fairness. {\color{black}Results show that these two classes of methods diverge slightly in terms of utility, although explanations may have a persuasive effect which could not be captured offline. Further studies should investigate how explanations impact user decisions and consequently utility. Considering beyond accuracy objectives, we observed that path reasoning methods, due to their internal mechanics, tend to favor serendipity and diversity.} At the same time, the methodological decisions made to produce explanations make the methods more sensible to the KG structure, consequently resulting in low product coverage for the benchmarked methods. Results also show that all methods (including baselines) emphasize some levels of unfairness in almost all perspectives, especially utility. Our study calls for debiasing methods that consider multiple perspectives in the knowledge-aware setup.
In the second analysis, we analyzed whether path reasoning methods can produce explanations across various recommended list sizes. What emerged is that some models (e.g., UCPR) are more sensible to the data and KG composition, which influence their capability of producing reasoning paths even under short recommended lists. {\color{black} This limitation could be avoided by making specific model design choices. For example, CAFE operates with a pre-defined search space for each user to deliver reasoning paths for each recommended product (although this design choice might affect recommendation utility).}
In the last analysis, we went beyond the ability of just producing reasoning paths, focusing on their quality.
Several studies highlighted the benefits of explanations \cite{Tintarev2007}.
Recent studies also showed that path properties (e.g., recency, popularity, and diversity) can influence the user perception of explanations \cite{10.1145/3477495.3532041}.
Results show that not all of these goals can be met at the same time.
For instance, PGPR fails to produce diverse explanations in ML1M, whereas CAFE yields explanations based on a tiny set of past user interactions. Future studies should address this aspect through in- and post-processing methods and look at other explanation perspectives (e.g., persuasiveness, trust, and efficiency).
Overall, our analyses showed that replicating research in this area is still a challenging task.
In future work, we plan to explore in detail the impact of KG characteristics on the considered perspectives, as well as
devise novel path reasoning methods robust to the KG structure and effective on multiple objectives.
\bibliographystyle{splncs04}
|
1,116,691,500,432 | arxiv | \section{Introduction}
\label{Introduction}
Nowadays, the ubiquitous nature of mobile and wearable devices has allowed users to access a multitude of new applications, services and content. More and more personal related information is stored on (or accessed via) personal devices such as smart phones, which enhances users' experience and convenience,
and creates new opportunities for both, consumers and service providers. However, such access of multitude applications via personal devices also brings new
challenges for service providers that must now secure access from a wide variety of devices~\cite{Sagiroglu2013}. Moreover, there is a continuous growth of mobile malware and other mobile security threats. Thus, it is important these mobile devices to be equipped with reliable means of authentication and authorization.
However, usually, these mobile and wearable devices have limited computational and interaction capabilities. Furthermore, because these devices are small, light, and easy to carry, there is also an associated risk in that they are susceptible to loss and theft, and easier to break. The use of context information (such as the user's current location, his typical behavior, etc.) may also trigger privacy concerns. Moreover, due to the increased prevalence of wearable and mobile applications, users nowadays expect a frictionless customer experience, making minimum effort. Taking into account these characteristics, the way users are authenticated and granted access to a wide range of online services and content becomes more challenging.
\begin{figure}[t]
\centering
\includegraphics[width=0.85\textwidth]{trends01}
\caption{Collaborative, frictionless and adaptive mulfi-factor authentication with many mobile devices.}
\label{fig:trends01}
\end{figure}
Ideally, users' devices will jointly and continuously operate in the background to establish the identity of the individual by continuously monitoring the context and detecting unusual deviations, as depicted in Figure~\ref{fig:trends01}. The advantage is that this will move the verification of the additional factors away from the user, making it transparent, and thereby greatly improving the convenience for the user, but posing important privacy challenges when sensitive context information is used, the addressing of which is an important aspect. The objective of pursuing a collaborative multi-device approach is that it can be less vulnerable against malicious users or unauthorized access after theft or loss of a device. Systems that support such user experience are called frictionless authentication systems~\cite{MustafaSECURWARE2017}.
In this paper we provide an overview of the emerging trends, research challenges and opportunities in such frictionless authentication systems that allow users to authenticate themselves using their devices to service providers without intentionally performing any specific authentication-related actions, such as entering a password.
The rest of this paper is structured as follows. In Section~\ref{sec:stateofpractice}, we review the current state of practice in mobile and multi-factor authentication, as well as risk-adaptive solutions. Emerging trends on collaborative and behavioral are highlighted in Section~\ref{sec:emerging}. Section~\ref{sec:challenges} reviews challenges and opportunities for further research. We conclude the paper in Section~\ref{sec:conclusions}.
\section{State-of-Practice in Authentication}
\label{sec:stateofpractice}
Before highlighting emerging trends in frictionless authentication systems, we will briefly review current best practices and the state-of-the-art in multi-factor authentication.
\subsection{Mobile and Multi-Factor Authentication}
Weak passwords are a major cause of data and security breaches~\cite{Jakobsson2013}. With dictionary attacks and optimized password cracking tools, users with simple or short (i.e., less than 8 characters) passwords are easy prey, especially if they use the same password for various services. Additionally, complex passwords are difficult to enter on mobile and wearable devices. This illustrates the generally acknowledged conception that passwords are problematic. Therefore, efforts are ongoing to replace password-based authentication with better alternatives~\cite{Bhargav-Spantzel2006,Bonneau2012,Grosse2013,Guidorizzi2013}. With multi-factor authentication, users authenticate with a combination of authentication factors, i.e., knowledge, intrinsic (biometrics) and possession. Biometric factors like speaker recognition, fingerprints, iris or retina scans cannot be forgotten, but may require expensive equipment to implement. Furthermore, such solutions require storing biometric templates, which can also be compromised and which are often cumbersome to revoke.
An interesting alternative to multi-factor mobile authentication is the Pico, a concept introduced by Stajano~\cite{Stajano2011}. The Pico is a dedicated
hardware token to authenticate the user to a myriad of remote servers; it is designed to be very secure while remaining quasi-effortless for users. The authentication process is based on the use of public-key cryptography and certificates, making common attacks on passwords (such as sniffing, phishing, guessing, and social engineering) impossible. Although being an interesting proposal, an actual implementation is currently lacking.
Leveraging on these recent initiatives, dynamic, multi-factor, collaborative and context-based authentication could further improve the current state-of-the-art on mobile authentication, finding an optimal balance between cost, user-convenience and security and privacy. Early work in this direction was presented in~\cite{Preuveneers2015} in which the authors presented SmartAuth, a scalable context-aware authentication framework built on top of OpenAM, a state-of-practice Identity and Access Management (IAM) suite (see Figure~\ref{fig:riskauthn}). It uses adaptive and dynamic context fingerprinting based on Hoeffding trees~\cite{Domingos:2000:MHD:347090.347107} to continuously ascertain the authenticity of a user's identity.
However, existing solutions that exploit context information often depend on a single device. Especially for mobile devices, a simple device or browser fingerprint is hardly unique and can easily be intercepted and spoofed by an attacker~\cite{Spooren2015}.
\begin{figure*}[t]
\centering
\includegraphics[width=0.85\textwidth]{riskauthn}
\caption{Risk-adaptive step-up authentication leveraging context and behaviometrics adopted within contemporary Identity and Access Management systems.}
\label{fig:riskauthn}
\end{figure*}
\subsection{Risk-based Access Control and Enabling Technologies}
Authentication is a basic building block of practically all business models. As mobile devices and wearables continue to proliferate and become part of the
user's expanded computing environment - fundamentally changing the way people access services and content - there is an associated security risk in that these devices are susceptible to loss and theft because they are small, light, and easy to carry.
The latest trend in access control models is Risk-Adaptive Access Control (RAdAC) where access decisions depend on dynamic risk assessments. There is a
large body of knowledge on this topic in the scientific literature~\cite{Li2013,Molloy:2012,SHAIKH2012447,Ni:2010,Santos2014,
BARACALDO2013237,KHAMBHAMMETTU201386,Kandala2011}, and risk-based authentication and access controls are being adopted in contemporary identity and access
management solutions, such as SecureAuth IdP 8.0, RSA SecurID Risk-Based Authentication, CA Technologies and ForgeRock's OpenAM 14. Contextual
information (device fingerprints, user location, time zone, IP address, time of day and other parameters) is used to evaluate the risk of users attempting to
access a resource, but the approach is often based on weighted score functions or meaningless user-defined risk thresholds.
\section{Emerging Trends}
\label{sec:emerging}
\subsection{Collaborative Authentication}
Authentication means solely based on possession factors bear the risk that the unique possession factor could be lost or stolen, hence compromising the security of the authentication system. Combining these schemes with other authentication factors, such as passwords or PINs, could improve the security, but at the cost of user-friendliness. Furthermore, one still needs to take into account the typical attacks on knowledge-based authentication factors, such as
PIN guessing or phishing attacks. An interesting alternative are collaborative authentication schemes, where multiple devices jointly authenticate to a remote server or within a device-to-device setting. To limit the cost, the combination of wearables and the user's smartphone would be preferred. Such collaborative authentication schemes overcome the security problems of using a single possession factor during the authentication process as an adversary would have to steal multiple wearables to successfully impersonate a user, while still offering user-friendliness. Moreover, by using wearables the user is carrying anyhow, one avoids the need of employing external hardware authentication tokens, which could be quite costly.
The concept of collaborative authentication is to transform a challenge-response protocol with a single prover and verifier, to a challenge-response protocol with multiple collaborating provers and a single verifier. To mitigate the threat of wearables being stolen or lost, and the fact that the set of wearables is dynamic (the user is not always carrying the same set of wearables), threshold-based cryptography is used. The aim of threshold cryptography is to protect a key by sharing it amongst a number of entities in such a way that only a subset of minimal size, namely a threshold $t+1$, can use the key. No information about the key can be learnt from $t$ or less shares. Shamir~\cite{Sha79} was the first to introduce this concept of secret sharing. Feldman~\cite{Feldman1987} extended this concept by introducing verifiable secret sharing. Pedersen~\cite{Pedersen92} then used this idea to construct the first Distributed Key Generation (DKG) protocol. Shoup~\cite{shoup2000} showed how signature schemes such as RSA could be transformed into a threshold-based variant.
To increase the resilience in a threshold-based authentication scheme, the number of devices included in the threshold scheme should be maximized. Therefore, Simoens~et~al.~\cite{Simoens2010} presented a new DKG protocol and demonstrated how this allows wearables not capable of securely storing secret shares to be incorporated. Peeters~et~al.~\cite{Peeters2012} used this idea to propose a threshold-based distance bounding protocol. A gap that remains to be filled is a threshold-based mobile authentication scheme, where the secret keying material is distributed among a set of personal wearables. For recent developments in continuous authentication, we refer the reader to~\cite{patel2016continuous}.
\subsection{Behaviometrics}
A recent trend in the area of continuous authentication is the use of behaviometrics. DARPA hosted the Active Authentication
program~\cite{guidorizzi2013security} in which various kinds of behavioral biometrics, i.e., metrics that measure human behavior to recognize or verify the identity of a person, are investigated. Several studies have investigated the application of using behaviometrics in order to provide an authentication method that is (a) \textit{continuous}, during an entire user session, and (b) \textit{non-intrusive}, since the normal user interaction with the system is
analyzed. It has been demonstrated that a user identity can be recognized and verified by means of several behaviometrics, such as keystroke dynamics, mouse movements (together with display resolution)~\cite{de2013mouse}, gait analysis~\cite{DBLP:conf/dbsec/hammePJ17}, CPU and RAM usage~\cite{deutschmann2013continuous}, accelerometer~\cite{DBLP:conf/essos/GoethemSPJ16} and battery fingerprints of mobile devices~\cite{DBLP:journals/mis/SpoorenPJ17}, stylometry~\cite{calix2008stylometry}, web browsing behavior~\cite{abramson2013user}, etc. An overview of techniques can be found in these works~\cite{KARNAN20111565,Deutschmann2013,Saevanee2012} and survey~\cite{wang2009behavioral}. A key challenge will be to investigate which
combination of behaviometrics will deliver a sufficient low number of false positives (mistakenly granted access = security concern) and false negatives
(mistakenly denied access = user experience concern) such that the risk is acceptable given the circumstances.
\section{Challenges and Opportunities}
\label{sec:challenges}
A frictionless authentication system is a complex system, involving multiple devices and sensors that interact with each other. This complexity makes such systems also a very flexible kind of authentication system. Nonetheless, several challenges and research opportunities remain. Authentication systems are usually characterized by the following interacting dimensions (see Figure~\ref{fig:tradeoff}):
\begin{itemize}
\item[-] \textit{Security}, which refers to how difficult it is for an impostor to be falsely authenticated.
\item[-] \textit{Usability}, which describes how easy and convenient it is for genuine users to be authenticated.
\item[-] \textit{Privacy}, which describes how any private information about the user being used are securely stored and/or processed by the system.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=0.40\columnwidth]{tradeoff}
\caption{Security, privacy and usability trade-offs in frictionless authentication.}
\label{fig:tradeoff}
\end{figure}
Security and usability are usually a trade-off in most authentication systems. For instance, False Acceptance and False Rejection Rates (FAR and FRR, respectively) are usually depicted in a ROC curve in biometric systems, and the lower the FAR is the higher the FRR is, where FAR is related to security, and FRR is related to usability. Hence, authentication systems are characterized by a specific security-usability trade-off. Regarding privacy, it can be also related to the security and usability of an authentication system. For instance, biometric systems based on protected templates, with a superior privacy protection when compared to their unprotected counterparts, usually provide an inferior set of working points regarding usability and security. In addition, the disclosure of a biometric template can lead to a security problem, unless appropriate revocation mechanisms are incorporated.
Active authentication systems involve multiple devices and sensors that interact with each other. This complexity also makes a frictionless authentication system a very flexible and powerful kind of system, which can be dynamically adapted to different usage scenarios, security-usability trade-offs, and overcome situations in which other types of authentication mechanisms would normally fail. In what follows, we expose different challenges and opportunities related to these three dimensions, \textit{security}, \textit{usability} and \textit{privacy}, and specific to frictionless authentication systems
\subsection{Security}
Regarding security, active authentication systems based on multiple behaviometrics and/or biometrics can provide increased security, since they are intrinsically multi-factor, and each employed behavioural modality makes them more difficult to spoof. However, the authentication decision will be based on the outcome of the classification and/or clustering algorithms. Such algorithms are usually not 100\% accurate~\cite{wang2009behavioral}, and in some cases the templates must be retrained by discarding old data to account for changes in the user's behaviour. This creates an opportunity for an attacker to impersonate a legitimate user by manipulating input data to compromise the learning process (i.e., a poisoning attack).
A specific security concern in continuous authentication systems is related to the enrollment. The enrollment phase establishes the identity of the subject within the authentication systems. Typically, this is based on credentials or certificates. However, with behavioral and context-dependent authentication, the enrollment phase becomes far more challenging, especially when using a collaborative authentication relying on multiple mobile and wearable devices. In the case of other biometrics, this can be done by ensuring the identity of the user during the enrollment phase by other means. However, since the enrollment in behaviometrics is done in an uncontrolled environment, the enrollment can also pose a threat to security, since it may be easier to inject artificial data to the system. Furthermore, behavioral authentication systems relying on machine learning methods require a time-consuming training step on an individual basis before they become effective.
\subsection{Usability}
Regarding usability, the frictionless nature of continuous authentication makes these systems one of the most convenient and easy to use modalities, since the user does not even need to learn how to use the authentication system, and the authentication process is transparent, potentially providing a smooth user experience. Furthermore, the availability of different sensors and modalities opens the opportunity to provide a very flexible authentication mechanism, where the system can implement different security/usability trade-offs for controlling the access to different functionalities or services. However, this also poses a challenge regarding the design of template protection techniques, since this flexibility may increase significantly the complexity of the system.
\subsection{Privacy}
Another key challenge with frictionless authentication systems is addressing the privacy concerns which arise when user behaviour analytics on sensitive data is used to continuously authenticate against online services. \textit{Honest but curious} service providers can use the keystrokes $-$ collected for behavioral authentication purposes $-$ to reconstruct the original text typed by the users. In addition, accelerometer data could be used by the same kind of adversary to reconstruct the whole history of a user's location. Furthermore, continuous authentication can also use physiological biometric measurements, whose implications regarding privacy are well known. Hence, employing the adequate biometric template protection mechanisms and appropriately imposing data minimality principles in the system design is even more important in continuous authentication.
\section{Conclusions}
\label{sec:conclusions}
There is a continuous quest for stronger authentication systems that at the same time offer a frictionless experience towards users of mobile and wearable devices. Context and behavioral information are nowadays being adopted in the enterprise marketplace as part of an adaptive authentication strategy that better serves the needs of the mobile consumer in diverse situational circumstances. However, irrespective of the technological advances to have multiple mobile and wearable devices collaborate to authenticate a user, the adoption of frictionless authentication will only be successful when the right balance
between usability, security and privacy can be found that meets the demands of a diverse set of users.
\bibliographystyle{IEEEtran}
|
1,116,691,500,433 | arxiv | \section{\label{sec1} Introduction}
Quantum walk(QW) is an extension of the classical random walk, in which the walker
is controlled by a quantum mechanical variable \cite{a2,PhysRevA.48.1687}.
The quantum interference from different walk paths brings about QW novel and
interesting features. An motivation for work on QW is developing new quantum algorithms \cite{PhysRevA.67.052307,r1,r2},
aiming at an exponential speed-up compared to classical algorithms for a certain classes of problem \cite{a23,k1,am1}.
On the physical side, QW is a valuable model for studying evolution process
from simple quantum protocol: discrete-time QW \cite{a11} and continuous-time QW
in one dimension (1D) \cite{a24}. Deep studies involve multiple coins
\cite{a4}, multiple walkers \cite{a6}, multiple dimensions \cite{a3}, and multiple coins with different
degrees of entanglement \cite{px1,a8,a9}. A recent resurgence of interest in the QW
has come about to a significant extent because experimental capability has
caught up with many of these theoretical proposals, particularly to create
walks with complex coin protocols and in two phase-space dimensions
\cite{Do:05,PhysRevLett.104.050502,e1,PhysRevA.81.052322,a33,PhysRevLett.103.090504,PhysRevLett.104.100503,a32,e2,Schreiber06042012}.
However, all of these QWs implementation are restricted to rather small
numbers of steps, and some have inherent limits to the size of their
phase space.
Despite the large volume of work performed on the QW, we have not been
able to find any that address the properties of the system at long evolution
times, meaning at large values of the step number $N$. As the challenges
in quantum computing move toward the large scale, it would appear that there
is a need to understand the behavior of quantum algorithms in the large-$N$
regime. The aim of this paper is to analyze QWs at large $N$ to establish
their properties, both universal and specific, and thus effectively to gauge
the flow and concentration of information in one particular set of algorithms.
We will consider in detail the 1D QW, where we have performed calculations
up to $N = 1000000$ steps, to establish the scaling characteristics,
spatial information content, universality, and limiting form of the
probability distribution. We will then turn to different types of possible
QW in 2D, where we use our 1D knowledge both directly and to benchmark
the additional forms of behavior that emerge, particularly entanglement,
accelerated diffusion, and semi-classical limiting distributions.
In this paper we perform a detailed analysis of the discrete-time 1D QW and two
fundamental 2D QWs. By calculating probabilities for large numbers $N$ of
steps, we investigate the destructive interference, the scaling properties,
the frequency content, and the combination of QWs. The structure of the
article is as follows. In Sec.~II we review the classical random walk and
a simple, symmetrical model for understanding the discrete-time QW in
1D, including its analytical solution. Section III presents our numerical
results for 1D QWs up to large $N$ and their analysis in both real and
Fourier space. In Sec.~IV we discuss two typical 2D QWs as a theoretical basis of Sec.~V.
Borrowing from the understanding developed in 1D, in Sec.~V we
provide the complete numerical characterization of these two 2D QWs.
Section VI provides a brief summary.
\section{\label{sec2} One-dimensional Random Walks}
\subsection{\label{sec2a} Classical Random Walk}
A walker standing at the origin of a line flips an unbiased coin and steps
to the right if a head comes up or to the left if the result is a tail. After
many flips, and taking a fixed number of steps based the coin state, the
location of the walker is an unknown, random position, but the probability
distribution of this position is a definite statistical quantity. After $N$
steps, the probability of the walker being at position $x$ is
\begin{equation}
P_N(x) = \frac{1}{2^N} C_N^{\frac{N-x}{2}} \;\; = \;\; \left( \frac{1}{2} \right)^N
\frac{N!}{\left( \frac{N+x}{2} \right) \! \left( \frac{N-x}{2} \right)!},
\label{eq01}
\end{equation}
i.e.~this process, the discrete-time 1D classical random walk, follows a
binomial distribution. The walker's position $x$ may only be an even (odd)
integer when $N$ is even (odd). In the continuum limit, which is approached
for sufficiently large $N$, the distribution is Gaussian (Fig.~\ref{fig:1dcrw}).
Its standard deviation, which represents a mean propagation distance in a
sample of many walkers, is $\sigma = \sqrt{N}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{1-1dcrw.eps}
\caption{Probability distribution for a 1D classical random walk with
$N = 100$.}
\label{fig:1dcrw}
\end{figure}
\subsection{\label{sec2b} Quantum Walks}
A QW refers to a random walk effected using a quantum device, and as such
requiring a quantum mechanical description.Following the classical formulation of a coin and a walker, the QW differs fundamentally from its classical counterpart in that coin and walker are ``entangled'' in the same quantum
``particle''. The essential consequence of this entanglement is that the propagation
of the quantum walker depends not on its probability (Sec.~IIA) but on its
amplitude. This amplitude is subject to quantum mechanical interference,
which can be constructive or destructive, leading to completely unconventional
forms of probability distribution (Fig.~\ref{fig:1dqrw}).
The numerical results in Sec.\ref{sec3} based on following calculation.
While there are many ways of describing such a walk (Ref.~\cite{a2} for a review),
here we introduce only the case of a discrete time symmetric QW on an open line.
With a view to later application, we denote the two internal ``coin''
states of the walker, or particle, as $\uparrow$ and $\downarrow$.
In a classical walk the coin states are completely separate
($\uparrow$ or $\downarrow$ with probabilities 0 or 1), whereas a quantum
coin can occupy any superposition state $a |\uparrow \rangle + b |
\downarrow \rangle$. Thus the quantum system is described by the wave
function
\begin{equation}
|\psi_N \rangle = \sum\limits_{i \, = -N}^{N} (a_i |\downarrow \rangle
+ b_i |\uparrow \rangle) |i \rangle,
\label{eqrwp}
\end{equation}
where $N$ is number of steps in the walk, $i$ is the position index, and
$|i \rangle$ the corresponding state.
The QW is described by the
evolution of this wave function under the quantum operation for successive
steps. The probability distribution for finding the walker at position $i$
(state $|i \rangle$) is found from the trace over the coin states to
be $P_i = a_i^2 + b_i^2$, with $\sum_i P_i = 1$.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{2-1dqrw.eps}
\caption{Probability distribution for a 1D quantum walk with
$N = 100$.}
\label{fig:1dqrw}
\end{figure}
The evolution has two substeps: (1) A rotation in the coin space, represented
by a unitary operator $U$, which is often taken as the Hadamard transformation
\begin{equation}
U_H = \frac{1}{\sqrt{2}} \left( \!\! \begin{array}{cc} 1 & 1 \\ 1 & -1
\end{array} \!\! \right) \! .
\label{eho}
\end{equation}
(2) A coin-dependent translation of the walker, described by
the shift operator
\begin{equation}
S = |\uparrow \rangle \langle \uparrow |\otimes \sum\limits_i
|i+1 \rangle \langle i |\, + |\downarrow \rangle \langle \downarrow |
\otimes \sum\limits_i |i-1 \rangle \langle i|,
\label{eso}
\end{equation}
The coefficients $a_0$ and $b_0$ of the initial coin state are arbitrary,
which has a profound influence on the symmetry of the probability distribution
of the QW \cite{a2}. To obtain a distribution symmetric under the application
of the Hadamard operator, the initial coin state is written as
\begin{equation}
|\psi_0 \rangle = {\textstyle \frac{1}{\sqrt{2}}} (|\uparrow \rangle +
i |\downarrow \rangle) \otimes |0 \rangle,
\label{002}
\end{equation}
A full step of the QW is
\begin{equation}\label{esh}
|\psi_1 \rangle = S U_H |\psi_0 \rangle = \frac{1+i}{2} |\uparrow
\rangle |1 \rangle + \frac{1-i}{2} |\downarrow \rangle |-1 \rangle
\end{equation}
The probability distribution for a QW of $N = 100$ steps is shown in
Fig.~\ref{fig:1dqrw}. It is manifestly completely different from the
classical random walk (Fig.~\ref{fig:1dcrw}), with clear maxima at values of
$i$ close to $\pm \, 0.7 N$ (revealed at larger $N$ to be $\pm \, N/\sqrt{2})$.
The probability for $i = 0$ is close to 0, indicating that the origin of this
counter-intuitive behavior is an almost complete destructive interference
among the paths of the quantum walker returning to the origin. The standard
deviation of the probability distribution in this QW is $\sigma \propto N$,
indicating a linear spreading rate, which is one of the most important
attributes of this evolution algorithm [Eq.~(\ref{esh})]. We present a
quantitative analysis of the properties of the QRW in Sec.~III. The QW
evolution process acts to propagate the quantum mechanical amplitudes,
preserving the complete information content of the internal states.
There are another two kinds of analytical descriptions
do benefit to understand the evolution of QW.
The difference between the classical and quantum walks can be understood
from the non-commuting nature of the (matrix) quantum operators and its
consequences for the interference of different walker paths. The Hadamard
operator can be decomposed \cite{a10} as $U_H = P + Q$ with $$P = \frac{1}
{\sqrt{2}} \left( \! \begin{array}{cc} 1 & 1 \\ 0 & 0 \end{array} \! \right)
\; {\rm and} \;\; Q = \frac{1}{\sqrt{2}} \left( \! \begin{array}{cc} 0 & 0
\\ 1 & -1 \end{array} \! \right) \! .$$It is easy to see that $P$ determines
motion of the walker to the right and $Q$ to the left. Evolution under the
QW for $N$ steps is represented as $U_H^N = (P + Q)^N$. In the classical
random walk one has $1^N = (p + q)^N$, where $p = q = \frac{1}{2}$ are real numbers representing probabilities.
One may conclude that the non-commutativity of the quantum operators determining different paths
to the same walker position, encodes the interference of amplitudes leading to the entirely unconventional
quantum phenomena reflected in Figs.~\ref{fig:1dcrw} and \ref{fig:1dqrw}. We expand this method to study a novel 2D-QW behaviour in Sec.\ref{sec5b}.
We pay attention to another analytical solution\cite{a11} in order to introduce the concept of Fourier
analysis. The evolution operator has a more concise form in $k$-space (Fourier
space) than in $x$-space (real space), so the initial wave function may be
transformed to and evolved in $k$-space. By inverse Fourier transformation, the real-space wave functions for the
$|\uparrow \rangle$ and $|\downarrow \rangle$ internal states are
\begin{widetext}
\begin{equation}
\begin{split}
\psi_\uparrow (x,t) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \!\!\! dk e^{-ikx} &
\frac{e^{-i\omega_kt} (\sqrt{1 + \cos^2k} + \cos k + ie^{-ik}) + e^{i(\omega_k - \pi)t}
(\sqrt{1 + \cos^2k} - \cos k - ie^{-ik})}{2\sqrt{2}\sqrt{1 + \cos^2k}}, \\
\psi_\downarrow (x,t) = \frac{1}{2\pi} \int_{-\pi}^{\pi} \!\!\! dk e^{-ikx} &
\frac{ i [e^{-i\omega_kt} (\sqrt{1 + \cos^2k} - \cos k - i e^{ik}) + e^{i(\omega_k
- \pi)t} (\sqrt{1 + \cos^2k} + \cos k + i e^{ik})]}{2\sqrt{2}\sqrt{1 +
\cos^2k}}, \\
\end{split}
\label{f01}
\end{equation}
\end{widetext}
and finally the probability of finding the walker at a given position $x$
after a walk of $t$ steps is given by
\begin{equation}
P(x,t) = |\psi_\uparrow (x,t)|^2 + |\psi_\downarrow (x,t)|^2.
\label{f02}
\end{equation}
Eqs.~(\ref{f01}) and (\ref{f02}) contain no dynamics,
because the term $e^{-i\omega_kt}$ provides no more than a compact notation for
the combination of $N$ with $k$, the spatial Fourier variable conjugate to
the actual walker displacement $x$.
A numerical calculation of the exact analytical solution, contained in
Eqs.~(\ref{f01}) and (\ref{f02}), is shown as the red curves and points in
Fig.~\ref{num-ana}, where we compute the probability distribution in both
real space [Fig.~\ref{num-ana}(a)] and Fourier space [Fig.~\ref{num-ana}(b)]
for $N = 100$ and 1000. The blue curves show our numerical calculations
based on method in Eqs.~(\ref{esh})
with the initial state specified in Eq.~(\ref{002}). The results are
identical up to a relative error of $10^{-5}$ caused by the numerical
integration of Eq.~(\ref{f01}). We comment in detail on the forms of
these distributions in Sec.~III.
The importance of the values $\pm \, N/\sqrt{2}$, noted in Sec.~IIB, is
clearly evident in Fig.~\ref{num-ana}(a), and it was deduced in Ref.~\cite{a25}
that the limiting distribution is concentrated in the interval $\left[
-\frac{N}{\sqrt{2}}, \frac{N}{\sqrt{2}} \right]$ as $N \rightarrow
\infty$. We will qualify this statement in Sec.~IIIA. Although the analytical
solution \cite{a11,a25} gives the exact probabilities for any position and
number of steps, in fact the expressions in Eq.~(\ref{f01}), which require
numerical integration over complex quantities, are not easy to compute when
$N$ becomes large. In this regime, direct numerical calculation of the
probability distribution is more straightforward, and we use this approach
in Sec.~III to reveal the properties and structure of the 1D QW at large $N$.
\begin{figure}[!htpb] \center
\includegraphics[width=0.5\textwidth]{3-pxandfk-1.eps}
\caption{(color online) Probability distribution for the 1D QW, comparing
the exact expression of Eqs.~(\ref{f01}) and (\ref{f02}), shown in red,
with numerical calculations based on Eqs.~(\ref{002}) and (\ref{esh})
shown in blue, for walks of $N = 100$ and 1000 steps. (a)
Probability distribution $P(x)$ in real space. (b) Fourier components $F(k)$
of $P(x)$. Insets for $N = 1000$ show (a) the probability oscillations near
$x = 0$ and (b) beats in the Fourier envelope near $k = \pi$ (see text). }
\label{num-ana}
\end{figure}
\section{\label{sec3} Numerical Studies of the 1D QW}
We now consider in detail the properties of the 1D QW introduced in
Sec.~II. The probability distribution obtained from this quantum
algorithm has a number of very striking properties in both real and
Fourier space (Fig.~\ref{num-ana}). Among them are the clear importance
of $\pm \, N/\sqrt{2}$ noted above, the twin-peaked ``envelope function''
of the $P(x)$ distributions with its remarkable zone of destructive
interference around $x = 0$, the width of these peaks, and the rapid
oscillation of the functions (both $P(x)$ and $F(k)$) at high spatial
frequencies within the envelope. We clarify immediately that this
oscillatory behavior is ``real'' in the sense that Figs.~\ref{fig:1dqrw}
and \ref{num-ana} show only the probabilities at even steps 0, $\pm \, 2$,
$\pm \, 4$, $\dots$, with the zero-probability odd steps not shown; the
QRW contains additional oscillations in space between the ``microscale''
of individual steps and the ``macroscale'' of the walk length.
\begin{figure}[t] \center
\includegraphics[width=0.5\textwidth]{4-qrwforN.eps}
\caption{Numerical results for the 1D QW with different numbers $N$ of
steps, scaled to the same horizontal axis. Probabilities are shown only for
even values of the site position $x$, with zero values on odd sites excluded
from the curves.}
\label{1001e}
\end{figure}
\subsection{\label{sec2d2} $N/\sqrt{2}$ Property}
We begin by analyzing the most obvious feature of the QW, which is the
tendency for the probability distribution to peak around 0.7$N$. From
Sec.~IIC it is clear that the factor $1/\sqrt{2}$ plays an important
role in the analytic solution and in physical terms it would appear to
mark the crossover in behavior from a low probability arising due to
destructive interference to a low probability arising simply from the
extremely low likelihood of having more than 85\% of the steps of an
unbiased random walk be in the same direction. In Fig.~\ref{1001e} we
show numerical results for the probability distributions of 1D QWs
with four different values of $N$, with the position axis normalized
by $N$. As $N$ increases, the distributions exhibit both increasingly
oscillatory behavior, which we analyze in Sec.~IIIB, and a peaking of
the envelope function, which becomes sharper as it converges towards
a maximum probability close to step $N/\sqrt{2}$ (Sec.~IIIC).
\begin{figure}[t] \center
\includegraphics[width=0.45\textwidth]{5-x_max.eps}
\caption{(color online) Scaling of the position $x_{\rm max}$ for
maximum probability with total step number $N$ in the 1D QW.}
\label{1001t}
\end{figure}
This convergence in shown in more detail for values up to $N = 1000000$
in Fig.~\ref{1001t}. The macroscopic feature is indeed a convergence
towards $N/\sqrt{2}$. Further, on a relative scale the distribution
appears to tend towards $\delta$-functions centered at $\pm N/\sqrt{2}$.
However, we caution that this is not the complete story and we consider the
asymmetric envelope shape in Sec.~IIIC. The point of maximum probability is
in fact $x_{\rm max} = 70684$ for $N = 100000$ and $x_{\rm max} = 707050$
for $N = 1000000$, while the exact value of $1/\sqrt{2}$ is 0.707106.
Thus in fact there are still more than 25 even steps of finite probability
separating $x_{\rm max}$ from $N/\sqrt{2}$ for $N = 1000000$ [see
Figs.~\ref{qob}(i) and \ref{qob}(j)]. The probability at $x = N/\sqrt{2}$,
shown in Fig.~\ref{1004t}, is always close to one half of $P(x_{\rm max})$,
and this point marks the approximate crossover where $P(x,N)$ changes from
algebraic to exponential decay with $N$ (below). There are always points
of finite $P(x)$ beyond $x = N/\sqrt{2}$ and the more exact statement of the
result of Ref.~\cite{a25} is that the normalized support converges to the
interval $\left[ -\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right]$ as $N
\rightarrow \infty$. The probability of a walker passing beyond this interval
vanishes more rapidly than interference effects can cause it to grow, and the
net consequence of the destructive interference between paths is to ``pile
up'' the probability close to (but mostly below) the limits of the interval.
Further insight into the nature of the (upper) envelope function may be
obtained by considering the probabilities at different representative
positions on the normalized $x$-axis, as shown in Fig.~\ref{1004t}.
The probabilities $P_N(0)$, $P_N(N/2\sqrt{2})$, and $P_N(N/2)$ all fall
linearly with $1/N$, suggesting a constant weight if binned into intervals
whose width scales with $N$. It is worth noting that the weight at position
zero, which has the maximum number of interfering paths, does not
vanish completely due to destructive interference in any finite-length
QW. By contrast, the probability at positions $x = x_{\rm max}$ and $x =
N/\sqrt{2}$ scales as $P \propto N^{-2/3}$ [specifically, $P(x_{\rm max}) =
1.8 N^{-2/3}$ and $P(N/\sqrt{2}) = 0.44 P(x_{\rm max})$], accounting for the
sharpening of the distribution peaks with increasing $N$. We return to the
question of the peak shape in Sec.~IIIC. As noted in the preceding paragraph,
beyond $x = N/\sqrt{2}$ there is a very abrupt change in the form of $P(N)$
to an exponential decay, as shown in Fig.~\ref{1004t} for the point
$x = 0.7072N$ and in the sudden loss of oscillatory behavior in
Figs.~\ref{qob}(i) and \ref{qob}(j); this we will also analyze in more
detail in Sec.~IIIC.
\begin{figure}[t] \center
\includegraphics[width=0.45\textwidth]{6-N-dependence.eps}
\caption{Representative probability data from different points on the
distribution, $x = 0$, $x = N/2\sqrt{2}$, $x = N/2$, $x = x_{\rm max}$,
$x = N/\sqrt{2}$, and $x = 0.7072N$, for 1D QWs of step lengths from
$N = 1000$ to $N = 1000000$. Where relevant, all data are taken from
the upper envelope of the distribution.}
\label{1004t}
\end{figure}
\subsection{\label{sec2d3} Oscillatory Behavior}
We turn next to the question of the oscillatory behavior of the probability
distribution within its envelope function. We stress again that this has
nothing to do with the period-2 oscillation created by the fact that walkers
alternate between odd and even sites at successive steps of the walk. We
begin by showing in Fig.~\ref{qob} the qualitative nature of the oscillations
in the real-space probability distribution function for selected regions of
the interval, using different values of $N$ to highlight their universal
nature.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{7-oscillations.eps}
\caption{Illustration of probability distributions in different parts of the
interval for 1D QWs of different step numbers $N$. First 15 oscillation
peaks for (a) $N = 1000$ and (b) $N = 10000$. Central region of the
distribution for (c) $N = 4000$ and (d) $N = 10000$, showing apparent
beating. Detail of the envelopes in the region of the beating structure
for (e) $N = 4000$ and (f) $N = 10000$. Detail within the envelope of the
beat structure for (g) $N = 4000$ and (h) $N = 10000$, showing the most
rapid oscillations in the distribution. Probability maxima for (i) $N =
100000$ and (j) $N = 1000000$. }
\label{qob}
\end{figure}
\begin{table*}[bt]
\centering
\caption{Numbers of peaks in the probability distribution $P(x)$ found within
windows of width 100 steps for QWs of $N = 1000$, 10000, and 100000.}
\label{1005t}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline interval ($N$ = 1000) & [0,100] & [100,200] & [200,300] & [300,400] &
[400,500] & [500,600] & [600,700] \\
\hline number of peaks & 2 & 5 & 8 & 12 & 17 & 23 & 17 \\
\hline interval ($N$ = 10000) & [500,600] & [1500,1600] & [2500,2600] &
[3500,3600] & [4500,4600] & [5500,5600] & [6500,6600] \\
\hline number of peaks & 2 & 5 & 8 & 12 & 17 & 23 & 17 \\
\hline interval ($N$ = 100000) & [500,600] & [15500,15600] & [25500,25600] &
[35500,35600] & [45500,45600] & [55500,55600] & [65500,65600] \\
\hline number of peaks & 2 & 5 & 8 & 12 & 17 & 23 & 16 \\
\hline interval ($N$ = 10000) & [5000,5100] & [5500,5600] & [5600,5700] &
[5700,5800] & [5800,5900] & [5900,6000] & [6000,6100] \\
\hline number of peaks & 20 & 23 & 24 & 25 & 25 & 24 & 23 \\
\hline interval ($N$ = 100000) & [50000,50100] & [50000,50200] & [52000,52100]
& [54000,54100] & [56000,56100] & [58000,58100] & [60000,60100] \\
\hline number of peaks & 20 & 20 & 21 & 22 & 24 & 25 & 23 \\
\hline
\end{tabular}
\end{table*}
One of the most important properties of the oscillations is that their
effective wavelength, in terms of the fundamental step size, appears to
decrease towards larger values of $x$. At the center of the distribution,
as shown in Figs.~\ref{qob}(a) and \ref{qob}(b), they have relatively long
wavelengths, but towards the edges these become shorter [Figs.~\ref{qob}(c)
and \ref{qob}(d)] until in the region $x \approx 0.6N$ they have only twice
the fundamental length [Figs.~\ref{qob}(g) and \ref{qob}(h)]. Towards the
center of each half of the distribution, some beat-like structures develop
[Figs.~\ref{qob}(e) and \ref{qob}(f)]. As the peak is approached, the
frequencies drop and the oscillations vanish suddenly at $x = N/\sqrt{2}$
[Sec.~IIIA, Figs.~\ref{qob}(i) and \ref{qob}(j)]. We draw attention to the
fact that the distribution also has an effective lower envelope function,
in that the probability is never zero on any even points and in fact is
very much larger than the size of the oscillations close to $x = 0$
[Figs.~\ref{qob}(a) and \ref{qob}(b)], but we do not analyze this further
here.
To quantify the nature of the oscillations, we count the numbers of peaks
in the probability distribution within different intervals and for QWs
of different $N$. First of all, by counting the total number of peaks it
is clear (Fig.~\ref{10051t}) that this is linearly proportional to
(approximately 8.5\% of) $N$. Subdividing the QW into intervals of fixed
length and counting the peaks in each of these gives the results shown in
Table \ref{1005t}. By comparing these peak counts horizontally, meaning
for different intervals within walks of the same $N$, a steady increase
in frequency becomes apparent out to $x \approx 0.6N$, from very
long-wavelength (periods of 50 or more steps) oscillations near $x = 0$
to extremely rapid (period-4) ones when $x$ is a signifcant fraction of
$1/\sqrt{2}$. We remind the reader here that a period of 4 in a system
where odd sites have probability zero is essentially a max-min-max-min-$\dots$
structure within the envelope [Figs.~\ref{qob}(g) and \ref{qob}(h)]. Thus
spatial information about the QW is truly contained on all length scales.
By comparing the peak counts vertically, meaning for different values of
$N$, it becomes apparent that corresponding regions have precisely the same
frequencies, with the maximum frequency occurring in the region around $x =
0.58N$. Thus the spatial modulation of the QW is a quantity independent of
$N$; although QWs of different $N$ cannot be called self-similar, they do
share similarities in particular aspects.
\begin{figure}[b] \center
\includegraphics[width=0.48\textwidth]{8-numberofpeaks.eps}
\caption{Total number of peaks in the probability distribution for QWs
from $N = 100$ to 100000.}
\label{10051t}
\end{figure}
Other forms of similarity and scaling appear in particular segments of the
probability distribution. Considering first the region close to $x = 0$,
Fig.~\ref{qob}(a) shows 15 peaks in the region $[0,300]$ for $N = 1000$.
The corresponding region for $N = 10000$, which is $[0,3000]$, contains
148 peaks, confirming the conclusion drawn above that the frequencies
are the same in corresponding areas for differing $N$, leading to the
overall linearity in $N$ of the peak number (Fig.~\ref{10051t}). However,
by counting the first 15 peaks in the probability distribution for $N =
10000$, shown in Fig.~\ref{qob}(b), they fill the interval $[0,960]$,
indicating a $\sqrt{N}$ scaling of the maximum wavelengths around the
center of the distribution. We clarify that this is not in contradiction
with Table \ref{1005t}, where the representative low-$x$ intervals are
taken at different finite values of $x$.
This type of scaling may also be observed in other regions of the QW
probability distribution. Focusing next on the special structures noted
above, Figs.~\ref{qob}(c) and Figs.~\ref{qob}(d) show three of these, which
we find quite reproducibly in the region around $x = N/2$. More accurately,
these macroscopic dips of the distribution envelope appear around $x =
0.36N$, $x = 0.45N$, and $x = 0.58N$. The interval around $0.58N$, which is
also the region with maximum oscillation frequency, shows a particularly
remarkable beating structure [Figs.~\ref{qob}(e) and \ref{qob}(f)], with
multiple points where the upper and lower envelopes meet. Figures \ref{qob}(g)
and \ref{qob}(h) show the maximal frequency regime and parts of the beating
envelopes in the fullest detail, and Table \ref{2001t} contains the
corresponding information for the beating interval for QWs of $N = 1000$,
10000, and 100000. Again it is clear that a factor-10 increase in $N$ causes
only a factor $\sqrt{10}$ magnification of the width of the beating structure
and, given the fixed maximal frequency in this interval, of the number of
peaks it contains.
\begin{table}[b]
\centering
\caption{Position, width, and number of peaks in the distinctive beating
structure around $x = 0.58N$ for walks of $N = 1000$, 10000, and 100000 steps.}
\begin{tabular}{|c|c|c|c|}
\hline N & region & width & number of peaks \\
\hline 1000 & [546,604] & 58 & 14 \\
\hline 10000 & [5682,5860] & 176 & 44 \\
\hline 100000 & [57452,58016] & 566 & 141 \\
\hline
\end{tabular}\label{2001t}
\end{table}
We close our discussion of the scaling of peak widths by considering the
probability oscillations around $x = N/\sqrt{2}$. By counting the width of
the region covered by the last 10 peaks up to and including the peak of
maximum probability, we find that this scales according to $N^{1/3}$
(Fig.~\ref{qop}). Similarly, the full width at half maximum height (FWHM)
of the leading peak in $P(x)$ also scales with $N^{1/3}$, and hence this
last peak retains its aspect ratio when $x$ is normalized by $N$. Thus we
demonstrate the presence of algebraic scaling in the probability oscillations
over the full distribution.
\begin{figure}[t] \center
\includegraphics[width=0.45\textwidth]{9-widthofpeaks.eps}
\caption{Width in $x$ of different numbers of peaks in the probability
distribution for QWs over the full range of $N$ values studied. The width
of the first 10 peaks from $x = 0$ (black) scales with $\sqrt{N}$; the width
of the last 10 peaks up to the maximum (red) and also the FWHM of the tallest
peak (green) both scale with $N^{1/3}$.}
\label{qop}
\end{figure}
\subsection{\label{sec2d6} Peak Shape}
Next we consider the shape and scaling of the asymmetric envelope of peaks in
the probability distribution to ascertain its functional form, $P(x,N)$. The
dependence on $N$ is largely contained in Fig.~\ref{1004t} and we make these
results more systematic here. Concerning the dependence on $x$, we take the
results of Sec.~IIIA as a demonstration that the ``crossover'' region just
beyond the maximum peak becomes a set of vanishing measure at large $N$; to
fit the envelope function, meaning the set of points extracted from the full
data set that fall close to the upper edge of $P(x)$, for the region $x \le
x_{\rm max}$, we assume that it diverges at $x = \pm \, N/\sqrt{2}$ for large
values of $N$. Before considering the QW, we recall that the functional form
of the probability distribution for a classical random walk (Sec.~IIA) becomes
a Gaussian at large $N$, with the form
\begin{equation}
P(x) = P_0 + A e^{-\frac{(x - b)^2}{2 \sigma^2}},
\label{egf}
\end{equation}
where $\sigma = \sqrt{N}$, $A = 1/\sqrt{2 \pi N}$, and $P_0 = 0 = b$ for a
normalized and centered distribution. This is an exponential function whose
characteristic width scales with $\sqrt{N}$.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{10-a-fit.eps}
\includegraphics[width=0.45\textwidth]{10-b-fit.eps}
\includegraphics[width=0.45\textwidth]{10-c-fit.eps}
\caption{Fit to the envelope of the probability distribution, $P_e (x)$, for
a QW of $N = 1000000$ steps. (a) Fit for the interval $400000 \le x \le
x_{\rm max}$ using the algebraic form of Eq.~(\ref{01e}) with $B = 1.884$,
$A = 1.487$, $b = 707144$, and $c = 0.4918$ (see text). (b) Fit for the
interval $0 \le x \le 200000$ using the algebraic form of Eq.~(\ref{02e})
with $B' = 0.6152$, $A' = 1.0017$, and $c' = 2.074$ (see text). (c) Fit to
the probability distribution $P(x)$ for the interval $x > N/\sqrt{2}$ using
the exponential form of Eq.~(\ref{03e}) with $P_0 = 0$, $b = N/\sqrt{2}$,
$d = 3.2$, and $A = 0.8$ (see text).}
\label{fit01}
\end{figure}
By contrast, for the QW with large $N$ we find an excellent fit to an
algebraic form,
\begin{equation}
P_e (x) = P_0 + a (b - x)^{-c}
\label{01e}
\end{equation}
for each half of the distribution, with $b = \pm \, N/\sqrt{2}$. To quantify
the extent of the validity of such a fit, we examine the data on logarithmic
axes [inset, Fig.~\ref{fit01}(a)], finding that a single power-law provides
a robust description of the envelope for the entire region $0.4N \le x \le
x_{\rm max}$. For the three free parameters, our results as $N \rightarrow
\infty$ (in practice, up to $N = 1000000$) indicate that the constant $P_0 =
- B/N$ with $B = 1.884$. Fits performed using only two remaining free
parameters, and illustrated in the main panel of Fig.~\ref{fit01}(a) for
the case $N = 1000000$, allow us to deduce for the large-$N$ limit that
the prefactor approaches $a = A/\sqrt{N}$ with $A = 1.5$ and the exponent
approaches $c = 0.5$. We conclude that to an excellent approximation the
envelope function follows a square-root form in $x$ measured away from
$x = \pm \, N/\sqrt{2}$, and this determines the algebraic form of the
peak width. The behavior of the prefactor $A$ ensures that $P_e (x)
\propto 1/N$ across all of this range, consistent with the results of
Fig.~\ref{1004t} but excluding the final peak. We remind the reader that
the envelope function $P_e(x)$ is not a quantity obeying a normalization
law as the true distribution $P(x)$ does.
Below $x = 0.4N$, the shape of the envelope begins to deviate from the
universal square-root form [inset, Fig.~\ref{fit01}(a)]. To ascertain its
shape close to the center of the distribution, we instead apply a fit of the
form
\begin{equation}
P_e (x) = P_0 + a' x^{c'}
\label{02e}
\end{equation}
for each half of the distribution. Again the data on logarithmic axes [inset,
Fig.~\ref{fit01}(b)] show an excellent fit to a single set of parameters over
a broad region, $0 \le x \le 0.2N$, where $P_0 = B'/N$ with $B' = 0.615$,
$a' = A'/N^{c'+1}$ with $A' = 1$, and $c' = 2$. The results are shown in
the main panel of Fig.~\ref{fit01}(b). Thus the $P \propto 1/N$ form is
maintained, the universal behavior of the envelope around its center is a
quadratic dependence on $x$, and there is a relatively significant constant
contribution that reflects directly the incomplete nature of destructive
interference in the central region of the QW. We regard the intermediate
regime $0.2N \le x \le 0.4N$ as a crossover zone between the two limiting
forms [Eqs.~(\ref{01e}) and (\ref{02e})] and do not consider it further.
We close our discussion of the probability distributon $P(x,N)$ by considering
the region $x > N/\sqrt{2}$. Here there are no longer any oscillations
[Figs.~\ref{qob}(i) and \ref{qob}(j)] and $P(x,N)$ is the ``envelope.''
In Sec.~IIIA [Fig.~\ref{1004t}] we showed that the dependence on $N$
crosses very rapidly to an exponential decay around $x = N/\sqrt{2}$. A
complete fit of the data in this regime reveals the form
\begin{equation}
P(x) = P_0 + a e^{- d \frac{(x - b)^{1.5}}{N^{0.5}}},
\label{03e}
\end{equation}
with $P_0 = 0$, $b = N/\sqrt{2}$, $d = 3.2$, and $a = A N^{-2/3}$ with $A =
0.8$, i.e.~an exponential and pseudo-Gaussian behavior but with unconventional
alterations to the exponents in both $x$ and $N$. The effectiveness of this
fit is shown in Fig.~\ref{fit01}(c), which also highlights how rapidly the
probability falls away in a short distance beyond $x = N/\sqrt{2}$.
\subsection{\label{sec2r} Fourier Transformation of the 1D-QW }
In Sec.~IIIB we explored the rich spatial frequency information contained
in the oscillations of the probability distribution $P(x)$. The QW contains
oscillations on all length scales from ultra-short wavelengths around $x =
0.58 N$ to long-wavelength oscillations scaling as $\sqrt{N}$ around $x = 0$,
with similar algebraic scaling around $x = x_{\rm max}$. These differing spatial
frequencies can even combine to create highly reproducible beating structures
in certain regions. All of this information should be reflected in the Fourier
transform of $P(x)$, which we discussed from an analytical point of view in
Sec.~IIC.
Here we perform a discrete Fourier transformation on the data sets for 1D
QWs of all lengths $N$, finding results for the Fourier components, $F(k)$,
of the type shown already in Fig.~\ref{num-ana}(b) for $N = 100$ and $N =
1000$. $F(k)$ possesses a primary peak with amplitude $F(0) = 1$ at $k = 0$,
flanked by two secondary peaks with negative components at $k \simeq \frac{5
\pi}{N}$, and then shows an oscillatory form between positive and negative
values of the Fourier components. The oscillations are again contained within
a decaying envelope function, which we find to be identical at positive and
negative values, and the $k$-space periodicity of the oscillation is remarkably
constant across the range $- \pi < k \le \pi$. This result is a clear
reflection of the fact that spatial information is present in the probability
distribution on all length scales, and the mixture of positive and negative
components across the range of $k$ is manifest in complex mixing phenomena
such as the beating structure. However, beyond the large $k = 0$ component
there are no special spatial frequencies appearing in the distribution. We
comment here that the constant component $F(0) = 1$ is simply the sum of all
data in real space, and therefore is the result expected for a normalized
distribution.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{11-peaksof1dFT.eps}
\caption{(color online) (a) Number of peaks in Fourier space for QWs of
lengths $N = 100$ to 100000. (b) Length of the envelope function of the
final beat below $k = \pi$, in units of $Nk/2\pi$ (black), and the number
of peaks in this region (red), shown as a function of $\sqrt{N}$.}
\label{fourier01c}
\end{figure}
We begin our quantitative analysis of the Fourier transform $F(k)$ by
counting its peaks. Figure \ref{fourier01c}(a) confirms that the total
number of peaks in the Fourier spectrum scales linearly with $N$, as in
real space and again with a constant of proportionality of order 8\%
(more precisely, 1/12) for each half of the transformed distribution.
As in real space, we may also count the peaks in particular parts of
the distribution to investigate their scaling form. The most striking
feature of the Fourier transformed data for large values of $N$ is a
reappearance of beating phenomena between the upper and lower envelope
functions. In complete consistency with the results of Sec.~IIIB, where
the beating structures were observed in the region of the distribution
with the highest spatial frequencies, the Fourier-space beats are
clearest close to $k = \pi$. In Fig.~\ref{fourier01a} we illustrate this
property with the $k$-axis rescaled to $Nk/2\pi$ to better reflect the
number of Fourier components in the data set. As in Sec.~IIIB, we may
characterize the structure of the beat pattern by considering the length
of the final beat and the number of peaks it contains, which are tabulated
in Table \ref{tfour} and illustrated in Fig.~\ref{fourier01c}(b). From the
latter it is clear once again that the beat structures scale according to
$\sqrt{N}$.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{12-fftcompare.eps}
\caption{Fourier components of the probability distribution close the maximum
frequency, shown rescaled to $Nk/2\pi$. In this regime the envelope function
shows clear beating behavior.}
\label{fourier01a}
\end{figure}
\begin{table}[b]
\centering
\caption{Characterization of oscillation frequencies in Fourier space. The
length of the last beat is quoted in units where the interval of the Fourier
components is rescaled to $(- Nk/2\pi,Nk/2\pi]$.}
\begin{tabular}{|c|c|c|c|}
\hline N & total peaks & peaks in last beat & length of last beat \\
\hline 1000 & 167 & 9 & 45 \\
\hline 4000 & 667 & 18 & 87 \\
\hline 10000 & 1667 & 29 & 140 \\
\hline 40000 & 6667 & 58 & 276 \\
\hline
\end{tabular}
\label{tfour}
\end{table}
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{13-a-1dft-N-dependence.eps}
\includegraphics[width=0.45\textwidth]{13-b-1dft-k-dependence.eps}
\includegraphics[width=0.45\textwidth]{13-c-1dft-k-dependence.eps}
\caption{Fit to the envelope of the Fourier transform of the probability
distribution, $F_e (k)$, for QWs of $N = 1000$ to 100000 steps. (a)
$N$-dependence of $F_e (k)$ for four different values of $k$, showing a
clear algebraic decay with exponent 1/2. (b) Fit using the algebraic form
of Eq.~(\ref{04e}), demonstrating its relevance for the interval $0 \le k
\le 0.2 \pi$ and returning the exponent $c = 1/2$. (c) Fit using the
algebraic form of Eq.~(\ref{05e}), demonstrating its relevance for the
interval $0.4 \pi \le k \le \pi$ and returning the exponent $c' = 1$. }
\label{fit02}
\end{figure}
Proceeding as in Sec.~IIIC, we consider the possibility of a universal
fit to the envelope function $F_e(k,N)$, whose $k$-dependence is valid
for all values of $N$. The $N$-dependence of $F_e (k,N)$ is shown in
Fig.~\ref{fit02}(a) for values of $k$ from across the full range, and
the constant slope gives the clear result $F_e (k,N) \propto 1/\sqrt{N}$.
Mindful of the fact that the real-space envelope changes form between the
limits of small and large $x$, we consider the functional forms
\begin{equation}
F(k) = \frac{A}{\sqrt{N}} \left( \frac{k}{\pi} \right)^{-c}
\label{04e}
\end{equation}
and
\begin{equation}
F(k) = \frac{A'}{\sqrt{N}} \left( 1 - \frac{k}{\pi} \right)^{c'}
\label{05e}
\end{equation}
for small and large $k$. As shown in Figs.~\ref{fit02}(b) and \ref{fit02}(c),
Eq.~(\ref{04e}) with exponent $c = 1/2$ provides an excellent fit at small
$k$, out to $k \approx 0.2 \pi$, and Eq.~(\ref{05e}) with $c'= 1$ an excellent
fit for all $k$ values in the upper half of the range. Thus we find the
$k$-dependence of the envelope of Fourier components to be algebraic over
the whole range, with an inverse square-root decay away from $k = 0$ crossing
over to a linear decrease as $k$ approaches $\pi$.
In summary, the Fourier transform of the QW probability distribution contains
all of the same information in a complementary form. It is bounded by upper
and lower envelope functions with the same algebraic decay. It demonstrates
that spatial frequencies are present on all scales from the inverse step
length to the inverse system size, with no special internal period(s) but
with distinctive beating structures on a length scale of $\sqrt{N}$. While
the QW does not satisfy the strict definition of self-similarity (no fractal
structures appear), many of its features are similar and scale-invariant
across the full range of $N$ values. Thus the simple quantum evolution
algorithm of Sec.~IIB contains a very rich variety of spatial information.
\section{\label{sec4} Two-Dimensional Quantum Walks}
\subsection{\label{sec4a} Classical Random Walk in 2D}
As noted in Sec.~I, it is the advent of experiments capable of realizing
a 2D QW \cite{Schreiber06042012} that has caused the resurgence of interest,
both experimental and theoretical, in the field. Before discussing the range of properties exhibited by QWs in 2D,
it is helpful to review the classical case. In principle there are
two ways to generalize the unbiased 1D classical random walk to 2D.
\noindent
1) Adopting a square grid, the walker is equipped only with a
two-face coin and therefore flips it once to step to $(x \pm 1,y)$, then
a second time to arrive at $(x \pm 1,y \pm 1)$. Now the conventional $N$-step 1D binomial
distribution for $x$ or $y$ is recovered by summing over the probabilities
in the orthogonal direction and it is easy to show for any $N$ that
$P(x,y) = P(x)P(y)$. In the large-$N$ limit, the distribution approaches
\begin{equation}
P(x,y) = {\textstyle \frac{1}{2 \pi N}}e^{-\frac{x^2 + y^2}{2N}}
\label{ep2dcrw2}
\end{equation}
and the mean distance of the walker from the origin after N steps is $\langle r \rangle = \sqrt{N}$.
\noindent
2) Remaining on a square grid, the walker at site $(x,y)$ has equal
probabilities of 1/4 (equivalent to a four-face coin) to move to any of
the points $(x \pm 1, y \pm 1)$. It results in an identical distribution to case (1).
A further generalization is to consider a continuous space in which
the walker takes unit-length steps at any random angle $0 \le \phi < 2\pi$.
The walk approaches perfect circular symmetry ($r^2 = x^2 + y^2$) at large values of $N$.
\subsection{\label{sec4b} Two Types of 2D QW}
With a view to experimental realization, 2D QW also has two-face coin scheme
and four-face coin scheme.
A four-face coin $H_4$ can be constructed by
product the Hadamard operators, i.e. $H_4=U_H\otimes U_H$(as well as corresponding initial state).
It evolves to a distribution shown in Fig.~\ref{fig:four} for a walk of 100 steps.
We note immediately that this 2D QW is a directly expand of 1D QW, in which $P(x,y)=P(x)P(y)$.
One will understand this equivalence by invoking the matrix identity
\begin{equation}
(A_1B_1)\otimes(A_2B_2)=(A_1\otimes A_2)(B_1\otimes B_2),
\label{otimes1}
\end{equation}
$A_1, A_2$ represent the Hadamard operator $U_H$ and $B_1, B_2$ for the initial state
$|\uparrow\rangle + i|\downarrow\rangle$.
The two-face-cion scheme lead to a completely different 2D distribution, shown in
Fig.~\ref{fig:2dqrw2cc} for a walk of 100 cycles. Specifically, an evolution
protocol using the same two-face coin twice in each complete cycle to
generate successive steps in the $x$ and $y$ directions. The first flip
of the coin selects the direction of $\pm x$ and the second $\pm y$.
A full cycle can be represented as $\hat U = S_y Y (S_x Y)$ and the quantum
state after $N$ evolution cycles as $|\psi_N \rangle = (\hat U)^N |\psi_0
\rangle$.This probability distribution has several properties in common with the 1D QW,
including strong probability peaks far from the center, strong destructive
interference everywhere near $(x,y) = (0,0)$, and oscillatory behavior
around the locus of maxima. However, the qualitative features of this QW
are strikingly different from those of the four-face-coin walk, most notably
in that the peaks are rotated by 45$^{\rm o}$, they occur right at the edge of
the system, and here the locus of maxima does show some tendencies towards a
circular shape, even if the peaks upon it remain strongly anisotropic and
four-fold symmetric. The location of the probability maxima is really very
anomalous, in that the minimum of the destructive interference is truly
obtained when a walker makes all $N$ of its steps in the same direction
for one lattice orientation, but precisely $N/2$ in each direction for
the other orientation.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{14-2d-unentangle.eps}
\caption{Probability distribution of 2D QW effected using a four-face
coin selecting steps in the $\pm$($x$+$y$) and $\pm$($x$-$y$) directions,
for $N = 100$ steps.}
\label{fig:four}
\end{figure}
\begin{figure}[htpb]
\centering
\includegraphics[width=0.5\textwidth]{15-2d-entangle.eps}
\caption{Probability distribution of 2D QW effected using a two-face coin
twice in each cycle to select steps in the $\pm x$ and $\pm y$ directions,
for $N = 100$ cycles.}
\label{fig:2dqrw2cc}
\end{figure}
Fig.~\ref{fig:2dqrw2cc} can also be realized by a four-face coin(Grove walk\cite{a4}).
In one of the most notable recent studies \cite{a8,a9}, Di Franco and
coauthors proved the equivalence of the spatial probability distributions
between the 2D QW using a single two-face coin, which these authors termed the
``alternate quantum walk'' (AQW) and the Grover walk. The AQW is actually a 2D QW
in which the two direction are maximal entangled.
The Grover coin can be expressed in the form
\begin{equation}
G_4 = \frac{1}{2} \left( \! \begin{array}{cccc} -1 & 1 & 1 & 1 \\
1 & -1 & 1 & 1 \\ 1 & 1 & -1 & 1 \\ 1 & 1 & 1 & -1 \end{array} \! \right) \! ,
\label{004}
\end{equation}
Noted that the distribution and spreading rates are effected by the initial state \cite{a3,a4}.
Eq.\ref{e104} is the one providing Grove walk a maximum spread rate
\begin{equation}
|\psi_0^m \rangle = {\textstyle \frac{1}{2}} (|\uparrow \rangle -
|\rightarrow \rangle - |\leftarrow \rangle + |\downarrow \rangle)
|0_x,0_y \rangle.
\label{e104}
\end{equation}
We conclude here the difference between the two type of 2D distribution,
is not distinguished by two-face coin or four-face coin, but whether the two orthogonal directions are entangled.
Fig.~\ref{fig:four} can also be realized using two uncorrelated two-face coins.
Some authors have studied decoherence \cite{a5} and localization \cite{a7}
in 2D QWs using the concept of two coins, and others have quantified the
effects of entanglement by a partial or complete swapping of the two coins
after each step of the walk \cite{a6}.
The method of generating a QW by using the same two-face coin twice, to
which we refer henceforth as the AQW \cite{a8,a9}, is an important one for
a number of reasons. First and foremost is that two-state systems are much
easier to find, or to produce, in any physical realization of a quantum
walker, and hence are much more relevant for experimental implementation.
Secondly, this maximally entangled protocol contains further unconventional
phenomena, which we discuss in Sec.~V. Further, coin entanglement provides
a valuable and completely general means of constructing and perhaps also of
realizing high-dimensional QWs using only two-face coins.
We conclude this subsection by contrasting the two QWs generated by
unentangled and by maximally entangled coins. The unentangled
system appears to show a perfectly square, factorized probability distribution
(a result we demonstrate in Sec.~V) characteristic of the 1D QW, with
maxima at positions converging to $(\pm \, N/\sqrt{2}, \pm \, N/\sqrt{2})$.
The entangled case shows a circular ``probability front'' and significant
complexity in the interference pattern within it. In the sense that the
unentangled situation can be discussed as accelerated diffusion from the
classical random walk to the quantum walk as a result of destructive interference,
so the maximally entangled walk appears to show a still further accelerated
diffusion. Indeed, it achieves a situation where information propagates to
the very edge of the system with high probability, which is a quite
remarkable consequence of near-perfect destructive interference among
trajectories in the center region. A possible interpretation of this
result may be found in preservation of the information content of the
coin, because the degree of two-coin entanglement is ``complete'' in
the sense that the walk can be generated using only one coin.
\section{\label{sec5} Numerical Studies of 2D QWs}
For an exact characterization of the 2D QWs introduced in Sec.~IVB, we
turn now to a numerical investigation of their probability distributions.
We begin by contrasting in Table \ref{label101} the analytical results
for the probabilities of the two walks for $N = 6$ steps.
\begin{table}[b]
\caption{Probability distributions of the 2D QW for $N = 6$ steps, shown
as $4096 P(x,y)$. On the left is the four-face-coin QW (equivalent to two
unentangled coins) and on the right the AQW (equivalent to a single coin,
or two maximally entangled coins). For the interpretation of these numbers
we note that in the 1D QW with $N = 6$ one obtains $64 P(x) = [1, 18,
9, 8, 9, 18, 1]$.}
\begin{equation*}
\begin{tabular}{|c|ccccccc||ccccccc|}
\hline \backslashbox{y\!\!}{\!\!x} & -6\! & -4\! & -2\! & 0\! & 2\! & 4\!
& 6 & -6\! & -4\! & -2\! & 0\! & 2\! & 4\! & 6 \\
\hline
6 & 1 \!& 18\!& 9 \!& 8 \!& 9 \!& 18\!& 1 & 1 \!& 26\!&125\!&200\!&125\!&
26\!& 1 \\
4 & 18\!&324\!&162\!&144\!&162\!&324\!& 18& 26\!& 68\!& 50\!&208\!& 50\!&
68\!& 26 \\
2 & 9 \!&162\!& 81\!& 72\!& 81\!&162\!& 9 &125\!& 50\!& 89\!& 40\!& 89\!&
50\!&125 \\
0 & 8 \!&144\!& 72\!& 64\!& 72\!&144\!& 8 &200\!&208\!& 40\!& 64\!& 40\!&
208\!&200 \\
-2& 9 \!&162\!& 81\!& 72\!& 81\!&162\!& 9 &125\!& 50\!& 89\!& 40\!& 89\!&
50\!&125 \\
-4& 18\!&324\!&162\!&144\!&162\!&324\!& 18& 26\!& 68\!& 50\!&208\!& 50\!&
68\!& 26 \\
-6& 1 \!& 18\!& 9 \!& 8 \!& 9 \!& 18\!& 1 & 1 \!& 26\!&125\!&200\!&125\!&
26\!& 1 \\
\hline
\end{tabular}
\end{equation*}\label{label101}
\end{table}
This exercise makes clear that the two QWs are radically different from
the outset. For the unentangled case, it is easy to see the result one may
already suspect from Fig.~\ref{fig:four}, that
the probability distribution of the unentangled 2D QW, $P(x,y) = P(x)P(y)$,
is a direct product of two 1D QWs in the $x$ and $y$ directions. We will
shortly demonstrate it numerically for large values of $N$. We remind the
reader that this result is not exactly intuitive, given that the walk steps
are each made in the $x$$\pm$$y$ direction of the lattice and as such would
appear to entangle the two directions completely; however, this result also
emerges from the 2D classical random walk (Sec.~IVA). For the entangled
case, the probability distribution has no direct relation to the 1D QW and
indeed already shows the concentration of probability along the $x$ and $y$
directions rather than along the diagonals.
\subsection{\label{sec5a} Cross-Sections of the Probability Distribution}
\begin{figure}[t]
\includegraphics[width=0.25\textwidth]{17-unentangle.eps}\includegraphics[width=0.25\textwidth]{17-entangle.eps}
\caption{Schematic illustration of 1D slices taken through the 2D
probability distribution for the two 2D QWs considered, whose data are
shown in Fig.~\ref{compare00}. Slice A denotes the $x$-direction, B the
diagonal, and C is the edge of data. In panel (a), where the blue square
denotes the region of maximum probability, slice C is taken at $x = 0.7 N$;
in panel (b), where the maximum probability is found along the blue circle,
slice C is taken at the true edge ($x = N$).} \label{compare0}
\end{figure}
For our numerical calculation of the characteristics of the different 2D
QWs, we have computed the probability distributions for both walks up to
$N = 1000$ steps, whose illustration requires a 1000$\times$1000 grid. To
show the results in a manner more quantitative than is possible in
Figs.~\ref{fig:four} and \ref{fig:2dqrw2cc}, we consider slices through the
2D probability data taken along the $x$ axis, along the diagonal $x$$+$$y$,
and along the ``edge'' of the data set, as shown in Fig.~\ref{compare0}.
The results of this process are illustrated in Fig.~\ref{compare00} for walks
of $N = 100$ steps. For the non-entangled case, panel A1 proves the numerical
identity with the 1D QW, $P(x,0) = P(x)P(0)$, which can be compared with
Fig.~\ref{fig:1dqrw} using the result for $P(0)$ with $N = 100$. This being
the case, panel C1 is very easy to interpret and for any parallel cut would
have the same functional form with any prefactor from $P(x)$. Panel B1 can
be expected to satisfy $P(x,x) = P^2(x)$, and therefore to have an envelope
function of the form $(N/\sqrt{2} - x)^{-1}$ across the outer half of the
distribution, a result we demonstrate below. By contrast, rather little is
known about the distribution $P(x,y)$ for the AQW and its understanding will
require applying the techniques of Sec.~III, for small data sets, to the
panels A2, B2, and C2 of Fig.~\ref{compare0}. Qualitatively, in A2, the
horizontal slice through the probability maxima, we observe an apparent
envelope function with no oscillations and a divergence towards $x = \pm N$
with an unknown power. In B2 we observe a complex oscillatory pattern with
significant probability extending beyond its peak value. In C2 we observe a
remarkably classical-looking probability distribution at the edge of the
system, where we remind the reader that the peak is the absolute maximum
anywhere in the walk.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{18-2dcutting.eps}
\caption{Probability distributions on 1D slices through the 2D QW data
(see Fig.~\ref{compare0}) for walks of $N = 100$. Panels A1, B1, and C1
are for the unentangled QW and A2, B2, and C2 for the AQW.}
\label{compare00}
\end{figure}
We wish to characterize the AQW by the power-law form of its probability
envelope function around the peak values of the distribution. For this we
analyze the 2D probability slices following Eq.~(\ref{01e}), but because
our largest system size in 2D is $N = 1000$, the numerical results do not
give particularly reliable curve fits. Although the divergence of $P(x)$
at $x = N/\sqrt{2}$ (at $x = N$ for slice A2) is not achieved with any
accuracy for these values of $N$, we enforce this value in all cases other
than slice B2 to reduce the arbitrariness in the fitting parameters. We
show in Fig.~\ref{compare001} the probability on logarithmic axes as a
function of $(b-x)/N$ for fixed values $b = 0.707N$ in cases A1 and B1,
$b = N$ in case A2, and the fitted value $b = 0.8N$ in case B2. As expected,
the probabilities for panels A1 and B1 have gradients close to $-1/2$ and $-1$
[$c = 1/2$ and $c = 1$ in Eq.~(\ref{01e})], respectively, over the bulk of the
range, and the accuracy to which the data fall on a straight line benchmarks
the method for $N = 1000$. For panel A2, the data are remarkably clean and
give a clear gradient parameter $c = 1$, indicating that the envelope function
of the peak in the maximally entangled walk satisifes the form $P(x,0) = P_0
+ a/(1 - x/N)$ to high accuracy. For panel B2, a slice that does not include
the main peaks but only the circular edge of local probability maxima, we find
a result close to $c = 1/2$, although here the envelope is poorly defined and
the errors significant.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{19-2d-logs.eps}
\caption{(color online) Scaling properties of the envelope function of the
2D probability distribution for $N = 500$ (black) and $N = 1000$ (red) on
the data slices shown in Fig.~\ref{compare00}, panels A1, B1, A2, and B2. }
\label{compare001}
\end{figure}
While our estimate of the functional form of the $x$-dependence of
the AQW is somewhat approximate, the $N$-dependence of the probability
distribution is beyond doubt. At every corresponding position, the probability
falls by a factor of 4 for every doubling of $N$. This result is illustrated
by the black ($N = 500$) and red ($N = 1000$) lines in Fig.~\ref{compare001}.
Thus we propose that the appropriate fit to the probability data on these
four slices is given by the algebraic form
\begin{equation}
y = \frac{a_1}{N^2} + \frac{a_2}{N^{d}} (b - x)^{-c},
\label{2dfunc}
\end{equation}
with $c + d = 2$ and the values of $c$ as deduced from Fig.~\ref{compare001}.
We may conclude that the entangled 2D QW has algebraic scaling properties
similar to the 1D-case. While its probability distribution shows no
oscillations in the $x$ and $y$ directions, in the $x$$\pm$$y$ directions
it oscillates in a manner not dissimilar to the 1D QW (panel B2 in
Fig.~\ref{compare00}). In Fig.~\ref{cuttingb}, we show that the number of
peaks in this slice is again proportional to the step number $N$, and with
a constant of proportionality again of order 8.5\% (Sec.~IIIB), although we
point out that for $x \sim N/\sqrt{2}$, where the 1D QW shows its strongest
peak, this distribution has its deepest valley.
\subsection{\label{sec5b}Edge of the Probability Distribution}
Next we consider the probability distribution at the edge of the walk for
the AQW, shown in panel C2 of Fig.~\ref{compare00}. As remarked above,
this QW has the highly anomalous feature that its maximum probability
occurs when the walker takes the maximum number of steps in the same
direction along one of the two axes, but returns to the center of the
other axis. Further, a visual inspection of the 1D slice through this
maximum suggests that the distribution on this edge may be a Gaussian,
or a related function similar to the 1D classical random walk.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{20-peaksof2d-diag.eps}
\caption{Number of peaks in the probability distribution of the AQW (maximally
entangled two-coin QW) in the direction $x$$+$$y$, shown in panel B2 of
Fig.~\ref{compare00}, as a function of step number $N$. }
\label{cuttingb}
\end{figure}
To investigate whether a quantum walk can lead to a classical results, we
employ the decomposition of the unitary evolution matrix into the matrices
$P$ and $Q$ introduced in Sec.~\ref{sec2b}. For the 2D QW we require
matrices $P_x$ for steps to the left, $Q_x$ for right, $P_y$ for up, and
$Q_y$ for down. Exploiting the AQW equivalence of the maximally entangled
two-coin QW, we use the same coin alternately for the $x$ and $y$
directions. Because our aim is to understand the probability distribution
on the right-hand edge of the system (equivalent to Fig.~\ref{compare0}(b),
slice C), every $x$-direction operation for the walker must be a $Q_x$ matrix
and not $P_x$. A complete cycle of the walk may then be either $R = P_y Q_x$
or $S = Q_y Q_x$, where
\begin{equation}
\begin{split}
& R = P_yQ_x = \frac{1}{2} \left( \!\! \begin{array}{cc} 1 & 1 \\ 0 & 0
\end{array} \!\! \right) \left( \!\! \begin{array}{cc} 0 & 0 \\ 1 & -1
\end{array} \!\! \right) = \frac{1}{2} \left( \!\! \begin{array}{cc}
1 & -1 \\ 0 & 0 \end{array} \!\! \right) \! , \\
& S = Q_y Q_x = \frac{1}{2} \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1
\end{array} \!\! \right) \left( \!\! \begin{array}{cc} 0 & 0 \\ 1 & -1
\end{array} \!\! \right) = \frac{1}{2} \left( \!\! \begin{array}{cc}
0 & 0 \\ 1 & -1 \end{array} \!\! \right) \! ,
\end{split}
\end{equation}
and thus
\begin{equation}
\begin{split}
& R R = \frac{1}{4} \left( \!\! \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array}
\!\! \right) \left( \!\! \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array} \!\!
\right) = \frac{1}{4} \left( \!\! \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array}
\!\! \right) = \frac{1}{2} R, \\
& R S = \frac{1}{4} \left( \!\! \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array}
\!\! \right) \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array} \!\!
\right) = \frac{1}{4} \left( \!\! \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array}
\!\! \right) = \frac{1}{2} R, \\
& S R = \frac{1}{4} \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array}
\!\! \right) \left( \!\! \begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array} \!\!
\right) = \frac{1}{4} \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array}
\!\! \right) = \frac{1}{2} S, \\
& S S = \frac{1}{4} \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array}
\!\! \right) \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array} \!\!
\right) = \frac{1}{4} \left( \!\! \begin{array}{cc} 0 & 0 \\ -1 & 1 \end{array}
\!\! \right) = \frac{1}{2} S,
\end{split}
\label{p02}
\end{equation}
i.e.~only the left-most operator in the string of steps is important.
With this result it is possible to calculate the entire edge distribution
analytically for any value of $N$. We illustrate the process for a walk
of $N = 4$ steps.
The paths arriving at position $(N,0)$ are $RRSS$,
$RSSR$, $RSRS$, $SSRR$, $SRSR$, and $SRRS$, which are divided between
the coin states $|\uparrow \rangle$ and $|\downarrow \rangle$ according to
\begin{equation}
\begin{split}
& U_{|\uparrow \rangle}^{(N,0)} = RRSS + RSSR + RSRS = \frac{1}{16} \left(
\!\! \begin{array}{cc} 3 & -3 \\ 0 & 0 \end{array} \!\! \right) \! , \\
& U_{|\downarrow \rangle}^{(N,0)} = SSRR + SRSR + SRRS = \frac{1}{16} \left(
\!\! \begin{array}{cc} 3 & -3 \\ 0 & 0 \end{array} \!\! \right) \! ,
\end{split}\nonumber
\end{equation}
whence
\begin{equation}
P_{|\uparrow \rangle}^{(N,0)} = \left| \frac{3-3i}{16\sqrt{2}} \right| =
\frac{9}{256}, \;\; P_{|\downarrow \rangle}^{(N,0)} \; = \; \left| \frac{3-3i}
{16\sqrt{2}} \right| = \frac{9}{256}, \nonumber
\end{equation}
and finally
\begin{equation}
P^{(N,0)} = P_{|\uparrow\rangle}^{(N,0)} + P_{|\downarrow\rangle}^{(N,0)} =
\frac{18}{256}.
\label{epaqwe0}
\end{equation}
Paths arriving at position $(N,2)$ are $RRRS$, $RRSR$, $RSRR$, and $SRRR$,
yielding
\begin{equation}
\begin{split}
& U_{|\uparrow \rangle}^{(N,2)} = RRRS + RRSR + RSRR = \frac{1}{16} \left(
\!\! \begin{array}{cc} 3 & -3 \\ 0 & 0 \end{array} \!\! \right) \! , \\
&U_{|\downarrow \rangle}^{(N,2)}= SRRR =\frac{1}{16} \left( \!\! \begin{array}{cc}
0 & 0 \\ -1 & 1 \end{array} \!\! \right), \\ & P_{|\uparrow \rangle}^{(N,2)} =
\left| \frac{3-3i}{16\sqrt{2}} \right| = \frac{9}{256}, \;\; P_{|\downarrow
\rangle}^{(N,2)} \; = \; \left| \frac{-1+i}{16\sqrt{2}} \right| = \frac{1}{256},
\end{split}\nonumber
\end{equation}
and thus
\begin{equation}
P^{(N,2)} = P_{|\uparrow \rangle}^{(N,2)} + P_{|\downarrow \rangle}^{(N,2)} =
\frac{10}{256}.
\label{epaqwe2}
\end{equation}
The only path arriving at position $(N,4)$ is $RRRR$, leading to
\begin{eqnarray}
U_{|\uparrow \rangle}^{(N,4)} & = & RRRR \; = \; \frac{1}{16} \left( \!\!
\begin{array}{cc} 1 & -1 \\ 0 & 0 \end{array} \!\! \right) \! , \nonumber \\
P^{(N,4)} & = & P_{|\uparrow \rangle}^{(N,4)} = \left| \frac{1-i}{16\sqrt{2}}
\right| = \frac{1}{256}.
\label{epaqwe4}
\end{eqnarray}
This analytical solution demonstrates that the quantum mechanical
interference leading to the probability on the edge of the system is
constructive everywhere, with none of the paths interfering destructively.
Regions at the center of the edge simply have the most paths, and this is
the origin of what we can call the ``semi-classical'' result that the
probability is maximal at the center of the edge, leading to a distribution
similar in appearance to the binomial distribution of the classical random
walk. In fact the degree of destructive interference everywhere else in the
2D AQW is such that these maxima on the edges are the global maxima, meaning
that the walker is effectively pushed to the maximal number of steps in order
that it does not ``destroy itself'' by interference in the maximally entangled
two-coin walk.
\begin{table}[b]
\caption{Probabilility tables for the first 7 steps of the binomial
distribution (top), omitting a prefactor of $1/2^N$ at each level $N$
of the table, and for the pseudobinomial distribution (bottom) achieved
on the edges of the 2D AQW, omitting a prefactor of $1/4^N$.}
\begin{equation*}
\begin{array}{cc}
\begin{tabular}{|ccccccccccccccc|}
\hline & & & & & & & 1 & & & & & & & \\
\hline & & & & & & 1 & & 1 & & & & & &\\
\hline & & & & & 1 & & 2 & & 1 & & & & & \\
\hline & & & & 1 & & 3 & & 3 & & 1 & & & & \\
\hline & & & 1 & & 4 \ & & 6 \ & & 4 \ & & 1 & & & \\
\hline & & 1 & & 5 & & 10 & & 10 & & 5 & & 1 & & \\
\hline & 1 & & 6 & & 15 & & 20 & & 15 & & 6& & 1 &\\
\hline 1 & & 7 & & 21 & & 35 & & 35 & & 21& & 7 & & 1 \\
\hline
\hline & & & & & & & 1 & & & & & & & \\
\hline & & & & & & 1 & & 1 & & & & & & \\
\hline & & & & & 1 & & 2 & & 1 & & & & & \\
\hline & & & & 1 & & 5 & & 5 & & 1 & & & & \\
\hline & & & 1 & & 10 & & 18 & & 10 & & 1 & & & \\
\hline & & 1 & & 17 & & 52 & & 52 & & 17 & & 1 & & \\
\hline & 1 & & 26 & & 125 & & 200 & & 125 & & 26& & 1 & \\
\hline 1 & & 37 & & 261 & & 625 & & 625 & & 261& & 37 & & 1 \\
\hline
\end{tabular}
\end{array}\end{equation*}\label{label203}
\end{table}
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{21-1dcrw-2dedge.eps}
\caption{Comparison between the AQW edge probability distribution and a
binomial (classical random walk) distribution for values up to $N = 1000$
steps, where the binomial is very accurately Gaussian. (a) Standard
deviation $\sigma$. (b) Prefactor $A$.} \label{fitcompare}
\end{figure}
A quantitative examination of the coefficients of the AQW edge reveals
that, despite being peaked at the center, they are not the same as the 1D
classical random walk, as shown in Table \ref{label203}. The probability
distribution at position $x$ on the edge of an $N$-step AQW can in fact be
expressed exactly as
\begin{eqnarray}
P_N (N,x) & = & {\textstyle {\frac{1}{4}}}[P_{N-1}^2 (x-1) + P_{N-1}^2 (x+1)]
\nonumber \\ & = & \frac{1}{4^N} \left( \left[ C_{N-1}^{\frac{N-x-1}{2}} \right]^2
+ \left[ C_{N-1}^{\frac{N-x+1}{2}} \right]^2 \right) \! ,
\label{eq02}
\end{eqnarray}
where $C_n^i$ is a binomial coefficient (Sec.~IIA). This remarkable result,
which we term a ``pseudobinomial'' distribution, encodes the emergence of
the physics of classical probabilities in the highly entangled QW. The
binomial coefficients appearing in the penultimate step before measurement
(Table \ref{label203}) arise as a consequence of the complete constructive
interference of paths, as illustrated in Eqs.~(\ref{epaqwe0}) to
(\ref{epaqwe4}) and discussed in the preceding paragraph.
To demonstrate the form of the AQW edge probability at large $N$, we fit both
binomial [Eq.~(\ref{eq01})] and pseudobinomial distributions to the Gaussian,
as specified in Eq.~(\ref{egf}) and again with $P_0 = 0 = b$, to compare
their forms and to extract their standard deviations $\sigma$. As shown in
Fig.~\ref{fitcompare}(a), while the binomial distribution gives the result
$\sigma = \sqrt{N}$, the standard deviation of the AQW edge probability is
$\sigma = \sqrt{N/2}$, i.e.~the distribution is narrower and the diffusion,
or spreading, rate of the walk slower (more localized) by a factor of
$1/\sqrt{2}$. Concerning the normalization prefactor $A$
[Fig.~\ref{fitcompare}(b)], the binomial approaches the Gaussian result
$A = 1/\sqrt{2 \pi N}$, but the AQW edge distribution does not, following
instead a dependence $A \propto 1/N$. While the probability of the classical
random walk of course sums to unity, determining $A$ for a true Gaussian,
the probability distribution at the edge of the AQW is not normalized due
to the weight in the interior of the walk, and hence the edge probability
is found to decay more rapidly than a true Gaussian. The emergence of such
a pseudobinomial distribution in a QW is yet another example of the rich
physics contained in a deceptively simple quantum evolution algorithm.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{22-2dFT-unentangle-a.eps}
\includegraphics[width=0.5\textwidth]{22-2dFT-entangle-b.eps}
\caption{2D Fourier transform $F(k_x,k_y)$ of the probability distributions
$P(x,y)$ (Fig.~\ref{fig:2dqrwc}) for (a) the four-face-coin or unentangled
2D QW and (b) the 2D AQW or maximally entangled two-coin QW, both for
$N = 100$.}
\label{ft2d}
\end{figure}
\subsection{\label{sec5c} Fourier Transformation of 2D QWs}
For further insight into the structure of the AQW probability distribution,
we compute the Fourier transform $F(k_x,k_y)$ of $P(x,y)$ for both the 2D
QWs we consider. Figure \ref{ft2d}(a) shows the 2D Fourier transform of
the unentangled (four-face-coin) QW, whose product structure is again
clearly visible. It was shown in Sec.~III that the 1D QW has spatial
frequency information at all scales up to the inverse step size ($k = \pi$)
and this is clear in the finite components of $F(k_x,k_y)$ up to the edge
of Fourier space. The AQW, shown in Fig.~\ref{ft2d}(b), is again quite
different, showing both an intrinsically 2D character and an apparent
concentration of weight closer to the center of Fourier space.
To analyze these distributions in a quantitative manner, we present the
data in the form of 1D slices. Figure \ref{FTcutting1} is completely
analogous to Fig.~\ref{compare00}, with the unentangled QW in the left
panels, the entangled one on the right, A denoting a horizontal slice
through the center of the Fourier distribution, B a diagonal slice, and
C the edge. We illustrate the qualitative features of the data using QWs
of $N = 100$ steps, but for our numerical analysis of the properties of
the Fourier transforms we use values of $N$ up to 1000.
\begin{figure}[t]
\includegraphics[width=0.5\textwidth]{23-2dcutting-FT.eps}
\caption{Cross-sections of the 2D Fourier transform $F(k_x,k_y)$ of the
probability distributions [Fig.~\ref{compare00}] for the two 2D QWs of
Sec.~IV, calculated with $N = 100$ and shown for the slices studied in
Sec.~VA (Fig.~\ref{compare0}).}
\label{FTcutting1}
\end{figure}
In panel A1 of Fig.~\ref{FTcutting1}, which shows $F(k_x,0)$ for the
unentangled QW, we observe the factorized form $F(k_x)F(0) = F(k_x)$
expected from Sec.~VA and shown in Fig.~\ref{num-ana}(b). Panel B1 shows
$F(k_x,k_x) = F^2(k_x)$, with only positive components and a correspondingly
steeper decay of the envelope. Panel C1, showing $F(\pi,k_y)$, is identical
to A1 up to a multiplicative prefactor, which is small and happens to be
negative at $k_x = \pi$. Turning to the AQW, where the probability
distribution cannot be factorized, the slice $F(k_x,0)$ in panel A2 is
not dissimilar to panel B1, in that all of the Fourier components are
positive and they decay significantly more rapidly than those of the 1D
QW. However, their oscillatory form shows a very precise odd/even
modulation, which is not evident in the 1D QW. The diagonal slice
$F(k_x,k_x)$ is shown in panel B2 and confirms both the odd/even
modulation and the concentration of Fourier amplitude near the center
of the system. Finally, panel C2 has no readily discernible structure,
reflecting the fact that the AQW edge is non-oscillatory and thus
dramatically different from all the other QW distribution slices (Sec.~VB).
Following Sec.~IIID, we investigate two properties of the Fourier
transform slices, namely their envelope and their oscillations. To
characterize the decay of the envelope function, we follow the procedure
shown in Fig.~\ref{fit02} and use logarithmic axes. As shown in
Fig.~\ref{FTcutting2}(a), and as expected from Sec.~IIID, the envelope
functions for panels A1 and B1 yield algebraic decay exponents close to
$c = 1/2$ and $c = 1$, respectively, for the Fourier components around
$k = 0$; the accuracy with which the data adhere to a straight line again
benchmarks the accuracy one may expect from data up to $N = 1000$. The data
from slice A2 form by far the best-quality set in Fig.~\ref{FTcutting2}(a)
and the algebraic decay exponent is unequivocally $c = 1/2$, meaning that
the spatial periodicity content of the AQW is qualitatively similar to
that of the unentangled QW at low frequencies. We take this opportunity
to remind the reader that there is {\it a priori} no direct connection
between the real- and reciprocal-space exponents of the envelope functions
(respectively around the peak of $P(x)$ and around $k = 0$) and the two
QWs present an example pair with different exponents in real space,
describing the peak shape, but the same exponent in reciprocal space,
describing the frequency content. The data from slice B2 exhibit the
lowest-quality envelope in the figure, but still show a strong qualititive
similarity to slice B1, with $c = 1$. Again this highlights the complexity
of the spatial frequency content of the AQW in its different directions
and indicates that the entanglement inherent in the AQW also entangles the
spatial information of the different lattice directions.
For completeness, we show in Fig.~\ref{FTcutting2}(b) the Fourier components
of the four slices at high $k$, where once again slices A1 and B1, with
respective gradients $c' = 1$ and $c' = 2$ benchmark the accuracy of the
approach. Again we observe that slices A2 and B2 for the AQW are qualitatively
similar, with the caveat that the data for slice A2 show evidence of a
distinctively different intermediate regime. We stress that the difference
in amplitude of the Fourier components between the unentangled QW and the
AQW, which reaches two orders of magnitude over the large-$k$ half of the
range for slices A1 and A2, is the most meaningful measure of strong
quantitative differences between the two QWs.
\begin{figure}[t]
\includegraphics[width=0.48\textwidth]{24-FFT-cut-a.eps}
\includegraphics[width=0.48\textwidth]{24-FFT-cut-b.eps}
\caption{Fourier transform of the 2D probability distributions for slices
A1, B1, A2, and B2 shown in Fig.~\protect{\ref{compare00}}, (a) as a function
of $k$, showing algebraic decay of the small-$k$ components, and (b) as a
function of $1 - k/\pi$, showing different power-law dependences at large $k$.}
\label{FTcutting2}
\end{figure}
Concerning the oscillations visible in the Fourier transforms, in the
unentangled case it is known from Sec.~III that the 2D QW contains
spatial frequency information on all length scales and that beating
phenomena become visible both in real space around the maximum frequency
and as a result in Fourier space near $k = \pm \pi$. For the AQW, our data
(Figs.~\ref{FTcutting1} and \ref{FTcutting2}) show that the high-frequency
spatial components are very significantly weaker, meaning that lattice-scale
oscillations are of little relevance, and this suggests that beating (which
we are unable to find up to $N = 1000$) is unlikely to be present. We believe
that the extra strength of even harmonics in the AQW results from the even
step number.
We close our analysis of the Fourier-space information contained in a QW
by summarizing a complementary perspective, named the ``dispersion relation''
approach in Ref.~\cite{PhysRevA.87.022336}. In Sec.~IIC we presented the
exact solution of the 1D QW by considering its nature in Fourier space,
finding that the Hadamard operator (\ref{eho}) gave rise to a dispersion
relation for a quantity $\omega_k = \sin^{-1} [\sin k/\sqrt{2}]$, specified
in terms of $k$. Although $\omega_k$ is largely a simplifying notation, as
there is no concept of a separate ``time'' (step number) and space in a
QW, it does result in a compact description in higher dimensions and it
has the added advantage of reflecting different degrees of coin entanglement
in a transparent way. When the AQW in 2D is transformed into Fourier space
\cite{PhysRevA.87.022336}, one may express the operator as
\begin{equation}
G_4 = \frac{1}{2} \! \left( \! \begin{array}{cccc}
\! - e^{ik_1} e^{ik_2} \! & \!\! e^{ik_1} e^{ik_2} \! & \!\! e^{ik_1} e^{ik_2} \! &
\!\! e^{ik_1} e^{ik_2} \! \\ \! e^{ik_1} e^{-ik_2} \! & \!\! - e^{ik_1} e^{-ik_2} \! &
\!\! e^{ik_1} e^{-ik_2} \! & \!\! e^{ik_1} e^{-ik_2} \! \\ \! e^{-ik_1} e^{ik_2} \! &
\!\! e^{-ik_1} e^{ik_2} \! & \!\! - e^{-ik_1} e^{ik_2} \! & \!\! e^{-ik_1} e^{ik_2} \!
\\ \! e^{-ik_1} e^{-ik_2} \! & \!\! e^{-ik_1} e^{-ik_2} \! & \!\! e^{-ik_1} e^{-ik_2} \!
& \!\! - e^{-ik_1}e^{-ik_2} \! \end{array} \!\!\! \right) \nonumber
\label{eg4}
\end{equation}
and deduce the eigenvalues
\begin{equation}
\begin{aligned}
\lambda_k^{1\pm} & = \pm 1, \\
\lambda_k^{2\pm} & = - {\textstyle \frac{1}{2}} [\cos(k_1 + k_2) + \cos(k_1 -
k_2) \\ & \;\; \pm \sqrt{(\cos(k_1 + k_2) + \cos(k_1 - k_2))^2 - 4} ].
\end{aligned}
\label{ee01}
\end{equation}
Because $\lambda_k = e^{i\omega_k}$, the dispersion relations for the four
eigenmodes of the AQW take the form
\begin{equation}
\begin{aligned}
\omega_k^{1+} & = 0, \;\;\;\; \omega_k^{1-} = \pi, \\
\omega_k^{2\pm} & = \pi \mp {\textstyle \frac{1}{2}} (\cos k_1 + \cos k_2),
\end{aligned}
\label{ee02}
\end{equation}
allowing a considerable simplification of the process [Eq.~(\ref{f01})] of
generating the probability distribution in this case.
\section{\label{sec7} Summary}
Quantum walks produce probability distributions entirely different
from the well-known classical ``drunkard's walk.'' In most cases, the
distribution is controlled completely by predominantly destructive
interference between the paths returning to the center, with the result
that the positions of maximal probability are pushed out towards the edges
of the walk. This results in the ``linear diffusion'' property of a quantum
walker, which has made the QW evolution algorithm so valuable in studies of
quantum computing.
However, as quantum computing approaches large-scale implementation, it is
necessary to understand the nature of quantum algorithms at large scales.
By this is meant an exact account of their properties, information content,
limiting behavior, and scaling characteristics at long evolution times.
Although analytical studies of different quantum walks in one and two
dimensions have revealed certain characteristics of their evolution and
interference, in particular their dependence on the initial state and the
entanglement of the quantum coins generating them, to date there has been
rather little consideration of the situation at large step number $N$.
By calculating the probability distributions for one- and two-dimensional
quantum walks up to $N = 1000000$ in the former case and $N = 1000$
in the latter, we have revealed a number of properties and scaling
characteristics. In one dimension, we verify that the probability approaches
peaks at $x/N = \pm \, 1/ \! \sqrt{2}$ at large $N$, which is the transition
region between destructive interference and vanishing probability for all
steps to be made in the same direction. The normalized support converges to
the region $[- 1/ \! \sqrt{2},1/ \! \sqrt{2}]$ and the envelope of the
distribution peaks has a square-root decay, i.e.~an algebraic form. Within
this envelope, the probability shows systematic oscillations on all length
scales, with the number of probability peaks always the same fraction of $N$.
The frequency of these oscillations increases from the inverse system size to
the inverse step length as a function of distance from the center of the walk.
The different frequencies show complex beating phenomena in the regime where
the oscillations are most rapid. These properties are revealed in a
complementary fashion by taking the spatial Fourier transform of the
distribution. All of these features are universal for walks of all $N$
values, giving them very strong similarities, but not the property of
self-similarity (there are no fractal structures in the simple walks
studied here).
In two dimensions there is not one quantum walk, or even one
algorithm, but a spectrum of protocols capable of generating quantum
evolution in a plane of phase space. We study two examples
that in fact represent the limiting cases of the entanglement between
two orthogonal directions: unentangled 2D QW and the maximally entangled alternate quantum walk(AQW).
In our numerical studies of these two limits, the understanding developed
in one dimension gives a complete account of the unentangled quantum walk,
whose two-dimensional probability distribution factorizes exactly into two
one-dimensional functions. By contrast, the maximally entangled case exhibits
strong correlations between the two orthogonal directions, damping of the
oscillatory behavior, and the extraordinary feature that the maximum of the
probability distribution is pushed all the way to the system edge by the
dominance of destructive interference. We provide an analytical description
of the edge distribution, showing that all paths arriving at the system
edges interfere constructively and proving that its functional form is a
type of pseudo-binomial, which is semi-classical in the sense of approaching
a Gaussian dependence on the spatial coordinate at large $N$.
As two-dimensional quantum walks become an experimental science,
our analytical and numerical studies demonstrate that even the simplest
algorithms for quantum evolution contain a rich variety of physical
phenomena and potential for technological application.
\acknowledgments
The authors gratefully acknowledge helpful discussions with Professor B.Normand.
Work at Renmin University was supported by the National Natural Science Foundation
of China (NSFC) under Grant No.~11174365 and by the National Basic Research
Program of China (NBRPC) under Grant No.~2012CB921704. PX was supported by
the NSFC under Grant Nos.~11174052 and 11474049, by the NBRPC under Grant
No.~2011CB921203, by the Open Fund from the State Key Laboratory of Precision
Spectroscopy of East China Normal University, and by the CAST Innovation Fund.
\bibliographystyle{apsrev4-1}
|
1,116,691,500,434 | arxiv | \section{Gordon's integral: Introduction}
\noindent Among the important integrals in theoretical and mathematical physics is W. Gordon's integral \cite{gordon}, see also \cite{karule,landau,saad2003,tarasov2003},
\begin{align}\label{integral1}
\operatorname {J}_c^{j(\pm p)}(b,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c\pm p;zx)dx\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq 0,-1,-2,\dots;~p\geq 0;~j=0,\pm 1,\pm 2,\dots),
\end{align}
where ${}_1F_1$ is the confluent hypergeometric function ${}_1F_1(b;c;z)=\sum_{k=0}^\infty{(b)_k\,z^k}/[(c)_k\,{k!}]$ in which $(b)_k=b(b+1)\dots(b+n-1)=\Gamma(b+k)/\Gamma(b)$ is the Pochhammer symbol defined in terms of Gamma function. The massive uses of this integral and the subclasses of it span large volume of research papers and monographs \cite{karule,landau,opps2009,saad2003,tarasov2003}.
It was proven (Lemma 1 in \cite{saad2003}) that, for $c+j>0$ and $|w|+|z|<|\lambda|$,
\begin{align}\label{integral2}
\operatorname {J}_c^{j(\pm p)}(b,b';\lambda,w,z)&
=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\,F_2\left(\begin{matrix}c+j;&b,&b\rq{}\\
~&c,&c\pm p\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right),
\end{align}
where the second Appell function reads (\cite{appell}, equation (2))
\begin{align}\label{def1}
\setlength\arraycolsep{1pt}
F_2\left(\begin{matrix}a;&b,&b'\\
~&c,&c'\end{matrix};w,z\right)&\equiv F_{2}\left( a; b,b^{\prime};c,c^{\prime};w,z\right) =\sum_{m=0}^{\infty}\sum
_{p=0}^{\infty}\frac{\left( a\right)_{m+p}\left(b\right)_{m}\left(
b^{\prime}\right)_{p}}{\left(c\right)_{m}\left(c^\prime\right)_{p}}\frac{w^{m}\,z^{p}}{m!\,p!},\quad (c,c^\prime \neq 0,-1,\dots; |w|+ |z| <1).
\end{align}
Exact analytical expressions of this integral by means of more elementary functions are given in the present work, where many subclasses are analysed and evaluated in simplified expressions allow for faster computations.
\section{Gordon's integral: Closed form expressions}
\noindent By means of the double integral representation of the second Appell function (\cite{appell}, equation 7; see also \cite{saad2003}), for $c,c-p\neq 0,-1,-2,\dots$, $j,p=0,1,2,\dots,~|w|+|z|<1$, it follows, for $j\geq p$, that
\begin{align}\label{integral4}
\operatorname {J}_c^{j(\pm p)}(b,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c\pm p;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j-b'}(\lambda-z)^{b'}}\sum_{k=0}^{j\mp p} \dfrac{(-j\pm p)_k(b')_k}{(c\pm p)_k~k!}\left(1-\dfrac{\lambda}{z}\right)^{-k}F_1\left(b,c+j-b',b'+k;c;\dfrac{w}{\lambda},\dfrac{w}{\lambda-z}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq 0,-1,\dots;~p\geq 0;~j=0,\pm 1,\dots;~|w|+|z|<\lambda),
\end{align}
particularly, for $p=j= 0,1,2,\dots$,
\begin{align}\label{integral5}
\operatorname {J}_c^{jj}(b,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c+ j;zx)dx=\dfrac{\Gamma(c+j)\,F_1\left(b,c+ j-b',b';c;\dfrac{w}{\lambda},\dfrac{w}{\lambda-z}\right)}{\lambda^{c+j-b'}(\lambda-z)^{b'}},\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,\dots;~j=0,\pm 1,\dots;~|w|+|z|<\lambda),\notag\\
\operatorname {J}_c^{jj}(c+j,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(c+j;c;wx){}_1F_1(b';c+ j;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)(\lambda-w)^{b'-c-j}}{(\lambda-z-w)^{b'}}F_1\left(-j,c+ j-b',b';c;\dfrac{w}{w-\lambda},\dfrac{w}{w+z-\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,\dots;~j=0,\pm 1,\dots;~|w|+|z|<\lambda),\notag\\
\operatorname {J}_c^{jj}(c+j,c;\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(c+j;c;wx){}_1F_1(c;c+ j;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{(\lambda-w)^{j}(\lambda-z-w)^{c}}F_1\left(-j,j,c;c;\dfrac{w}{w-\lambda},\dfrac{w}{w+z-\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,\dots;~j=0,\pm 1,\dots;~|w|+|z|<\lambda),
\end{align}
where $F_1$ is the first Appell function (\cite{appell}, equation (1)). By mean of (\cite{slater}, formula (8.3.5))
\begin{align}\label{def6}
F_1(a;b,b';c;w,z)&=(1-w)^{-a}F_1\left(a,c-b-b',b';c;\dfrac{w}{w-1},\dfrac{z-w}{1-w}\right),
\end{align}
it follows, for $j\geq p$,
\begin{align}\label{integral7}
\operatorname {J}_c^{j(\pm p)}(b,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c\pm p;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j-b-b'}(\lambda-w)^b(\lambda-z)^{b'}}\sum_{k=0}^{j\mp p}\dfrac{(-j\pm p)_k(b')_k}{(c\pm p)_k\,k!}\left(1-\dfrac{\lambda}{z}\right)^{-k}\notag\\
&\times \sum_{r=0}^{j+k}\dfrac{(b)_r\,(-j-k)_r}{(c)_r\,r!}\left(1-\dfrac{\lambda}{w}\right)^{-r}{}_2F_1\left(b+r,b'+k;c+r;\dfrac{w\,z}{(\lambda-z)(\lambda-w)}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq 0,-1,-2,\dots;~ |w|+|z|<\lambda),
\end{align}
where for $p=j= 0,1,2,\dots$
\begin{align}\label{integral8}
\operatorname {J}_c^{jj}(b,b';&\lambda,w,z)=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c+ j;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j-b-b'}(\lambda-w)^b(\lambda-z)^{b'}}
\sum_{r=0}^{j}\dfrac{(b)_r\,(-j)_r}{(c)_r\,r!}\left(1-\dfrac{\lambda}{w}\right)^{-r}{}_2F_1\left(b+r,b';c+r;\dfrac{w\,z}{(\lambda-z)(\lambda-w)}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c+j\neq 0,-1,-2,\dots;~ |w|+|z|<\lambda).
\end{align}
Setting $b'=c+j$, equation \eqref{integral4} yield
\begin{align}\label{integral9}
\operatorname {J}_c^{j(\pm p)}(b,c+j;\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(c+j;c\pm p;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{(\lambda-z)^{c+j-b}(\lambda-z-w)^b}\sum_{k=0}^{j\mp p} \dfrac{(-j\pm p)_k(c+j)_k}{(c\pm p)_k~k!}\left(1-\dfrac{\lambda}{z}\right)^{-k}{}_2F_1\left(b,-j-k;c;\dfrac{w}{w+z-\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq 0,-1,-2,\dots;~p\geq 0;~j=0,\pm 1,\pm 2,\dots;~|w|+|z|<\lambda),
\end{align}
By means of the Kummer's first transformation
$
{}_1F_1(b;c;z)=e^z{}_1F_1(c-b;c;-z),
$
and the series representation
\begin{equation}\label{def10}
F_2(a; b,b';c,c'; x,y)= \sum_{m=0}^{\infty} {(a)_{m}\,(b)_m\over (c)_m\,m!}\, x^m\, {}_2F_1(a+m,b';c',y),
\end{equation}
it easily follows
\begin{align}\label{integral11}
\operatorname {J}_c^{j(\pm p)}(c+j,b;\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(c+j;c;wx){}_1F_1(b;c\pm p;zx)\,dx\notag\\
&=\dfrac{\Gamma(c+j)}{(\lambda-w)^{c+j}}\sum_{k=0}^j\dfrac{(-j)_k(c+j)_k}{(c)_k\,k!}\left(1-\dfrac{\lambda}{w}\right)^{-k}{}_2F_1\left(\begin{matrix}b,&c+j+k\\
~&c\pm p\end{matrix};\frac{z}{\lambda-w}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq0,-1,\dots;~|w|+|z|<\lambda).
\end{align}
By means of the identity (\cite{brychkov2008}, formula 5.14.3)
\begin{equation}\label{def12}
\sum_{k=0}^n{n\choose k} \dfrac{(-z)^k}{(b)_k}{}_1F_1(a;b+k;z)={}_1F_1(a-n;b;z),
\end{equation}
it follows that
\begin{align}\label{integral13}
\operatorname {J}_c^{j(\pm p)}(c-j,b;\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(c-j;c;wx){}_1F_1(b;c\pm p;zx)\,dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\sum_{k=0}^j\dfrac{(-j)_k(c+j)_k}{(c)_k\,k!}\left(\dfrac{w}{\lambda}\right)^kF_2\left(\begin{matrix}c+j+k,&c,&b\\
~&c+k&c\pm p\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq0,-1,\dots;~|w|+|z|<\lambda).
\end{align}
By means of the identity (\cite{brychkov2008}, formula 5.14.1)
\begin{equation}\label{def14}
\sum_{k=0}^n(-1)^k{n\choose k} \dfrac{(b-a)_k}{(b)_k}{}_1F_1(a;b+k;z)=\dfrac{(a)_n}{(b)_n}{}_1F_1(a+n;b+n;z),
\end{equation}
it follows that
\begin{align}\label{intgral15}
\operatorname {J}_{c+n}^{(j-n)(\pm p-n)}(b+n,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b+n;c+n;wx){}_1F_1(b';c\pm p;z\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\dfrac{(c)_n}{(b)_n}\sum_{k=0}^n\dfrac{(-n)_k(c-b)_k}{(c)_k\,k!}
F_2\left(\begin{matrix}c+j;&b,&b\rq{}\\
~&c+k,&c\pm p\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;n=0,1,\dots;~c+n,c\pm p\neq0,-1,\dots;~|w|+|z|<\lambda).
\end{align}
By means of the identity (\cite{brychkov2008}, formula 5.14.5)
\begin{equation}\label{def16}
{}_1F_1(b+n;c+n;w)=\dfrac{(c-1)_n(c)_n}{(b)_n(-w)^n}\sum_{k=0}^n\dfrac{(-n)_k(1-c)_k}{(2-c-n)_k\, k!}{}_1F_1(b,c-k,w)
\end{equation}
it follows that
\begin{align}\label{integral17}
\operatorname {J}_{c+n}^{(j-n)(\pm p-n)}&(b+n,b';\lambda,w,z)=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b+n;c+n;wx){}_1F_1(b';c\pm p;z\,x)\,dx\notag\\
&=\dfrac{(c-1)_n(c)_n}{(-w)^n\,(b)_n}\dfrac{\Gamma(c+j-n)}{\lambda^{c+j-n}}\sum_{k=0}^n\dfrac{(-n)_k(1-c)_k}{(2-c-n)_k\,k!}
F_2\left(\begin{matrix}c+j-n;&b,&b\rq{}\\
~&c-k,&c\pm p\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right),\notag\\
&(c\neq0,\pm 1;\dots,c+j>0;~\lambda>0;n=0,1,\dots;~c+n,c\pm p\neq0,-1,\dots;~|w|+|z|<\lambda).
\end{align}
By means of the identity (\cite{brychkov2008}, formula 5.14.6)
\begin{equation}\label{def18}
{}_1F_1(b+n;c;w)=\dfrac{(b-c+1)_n}{(b)_n}\sum_{k=0}^n(-1)^k{n\choose k}\dfrac{(1-c)_k}{(b-c+1)_k}{}_1F_1(b,c-k,w)
\end{equation}
it follows
\begin{align}\label{integral19}
\operatorname {J}_{c}^{j(\pm p)}(b+n,b';\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b+n;c;wx){}_1F_1(b';c\pm p;z\,x)\,dx\notag\\
&=\dfrac{(b-c+1)_n}{(b)_n}\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\sum_{k=0}^n\dfrac{(-n)_k(1-c)_k}{(b-c+1)_k\,k!}
F_2\left(\begin{matrix}c+j;&b,&b\rq{}\\
~&c-k,&c\pm p\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right)\notag\\
&(c\neq 0,\pm 1,\dots,c+j>0;~\lambda>0;n=0,1,\dots;~c,c\pm p\neq0,-1,\dots;~|w|+|z|<\lambda).
\end{align}
By means of the identity (\cite{brychkov2008}, formula 5.14.7)
\begin{equation}\label{def20}
{}_1F_1(b-n;c-n;w)=\dfrac{(w)^n}{(1-c)_n}\sum_{k=0}^n{n\choose k}\dfrac{(1-c)_k}{w^k}{}_1F_1(b,c-k,w)
\end{equation}
it follows
\begin{align}\label{integral21}
\operatorname {J}_{c-n}^{(j+n)(\pm p+n)}&(b-n,b';\lambda,w,z)=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b-n;c-n;wx){}_1F_1(b';c\pm p;z\,x)\,dx\notag\\
&=\dfrac{w^n\Gamma(c+j+n)}{\lambda^{c+j+n}(1-c)_n}
\sum_{k=0}^n\dfrac{(-n)_k(1-c)_k}{k!\,(1-c-j-n)_k}
\left(\dfrac{\lambda}{w}\right)^k
F_2\left(\begin{matrix}c+j-k+n;&b,&b\rq{}\\
~&c-k,&c\pm p\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right)\notag\\
&(c\neq 0,\pm 1,\dots,c+j>0;~\lambda>0;n=0,1,\dots;~c-n,c\pm p\neq0,-1,\dots;~|w|+|z|<\lambda).
\end{align}
Since (\cite{prudnikov}, formula 7.2.4.68)
\begin{equation}\label{def22}
F_2(a;b,b;c,c;z,-z)={}_4F_3\left(\dfrac{a}{2},\dfrac{a+1}{2},b,c-b;\dfrac{c}{2},\dfrac{c+2}{2},c;z^2\right).
\end{equation}
it follows that
\begin{align}
\operatorname {J}_c^{j0}(b,b;\lambda,w,-w)&=\int\limits_0^\infty x^{c+j-1}\,e^{-\lambda x}\,{}_1F_1(b;c;wx){}_1F_1(b;c;-wx)\,dx=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}{}_4F_3\left(\begin{matrix}b,&c-b,&\dfrac{c+j}{2},&\dfrac{c+j+1}{2}\\
c,&\dfrac{c}{2},&\dfrac{c+1}{2}\end{matrix};\frac{w^2}{\lambda^2}\right),\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,-2,\dots,~|w|<\lambda),\label{integral23}\\
\operatorname {J}_c^{10}(b,b;\lambda;w,-w)&=\int\limits_0^\infty x^{c}\,e^{-\lambda x}\,{}_1F_1(b;c;w\,x)\,{}_1F_1(b;c;-w\,x)\,dx=\dfrac{\Gamma(c+1)}{\lambda^{c+1}}{}_3F_2\left(\begin{matrix}b,&c-b,&\dfrac{c}{2}+1\\
c,&\dfrac{c}{2},\end{matrix};\frac{w^2}{\lambda^2}\right),\notag\\
&(c>-1;~\lambda>0;~|w|<\lambda),\label{integral24}\\
\operatorname {J}_c^{10}\left(\dfrac{c}{2},\dfrac{c}{2};\lambda;w,-w\right)&=\int\limits_0^\infty x^{c}\,e^{-\lambda x}\,{}_1F_1\left(\dfrac{c}{2};c;w\,x\right)\,{}_1F_1\left(\dfrac{c}{2};c;-w\,x\right)\,dx=\dfrac{\Gamma(c+1)}{\lambda^{c+1}}{}_2F_1\left(\begin{matrix}\dfrac{c}{2},&\dfrac{c}{2}+1\\ \\
c,\end{matrix};\frac{w^2}{\lambda^2}\right),\notag\\
&(c>-1;~\lambda>0;~|w|<\lambda).\label{integral25}
\end{align}
On other hand, by means of (\cite{prudnikov}, formula 7.2.4.68)
\begin{equation}\label{def26}
F_2(a;b,c-b;c,c;z,z)=(1-z)^{-a}{}_4F_3\left(\dfrac{a}{2},~\dfrac{a+1}{2},~b,~c-b;~\dfrac{c}{2},~\dfrac{c+2}{2},~c;\dfrac{z^2}{(1-z)^2}\right),
\end{equation}
it follows that
\begin{align}
\operatorname {J}_c^{j0}(b,c-b;\lambda,z,z)&=\int\limits_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;z\,x){}_1F_1(c-b;c;z\,x)\,dx=\dfrac{\Gamma(c+j)}{(\lambda-z)^{c+j}}{}_4F_3\left(\begin{matrix}b,&c-b,&\dfrac{c+j}{2},&\dfrac{c+j+1}{2}\\
c,&\dfrac{c}{2},&\dfrac{c+1}{2}\end{matrix};\frac{z^2}{(\lambda-z)^2}\right),\notag\\
&(c+j>0;\lambda>0;~c\neq 0,-1,-2,\dots,~|w|<\lambda),\label{integral27}\\
\operatorname {J}_c^{10}(b,c-b;\lambda,z,z)&=\int\limits_0^\infty x^{c}\,e^{-\lambda x}\,{}_1F_1(b;c;z\,x)\,{}_1F_1(c-b;c;z\,x)\,dx=\dfrac{\Gamma(c+1)}{(\lambda-z)^{c+1}}{}_3F_2\left(\begin{matrix}b,&c-b,&\dfrac{c}{2}+1\\
c,&\dfrac{c}{2},\end{matrix};\frac{z^2}{(\lambda-z)^2}\right),\label{integral28}\\
&(c>-1;~\lambda>0;~|z|<|\lambda|),\notag\\
\operatorname {J}_c^{10}\left(\dfrac{c}{2},\dfrac{c}{2};\lambda,z,z\right)&=\int\limits_0^\infty x^{c}\,e^{-\lambda x}\left[{}_1F_1\left(\dfrac{c}{2};c;z\,x\right)\right]^2dx=\dfrac{\Gamma(c+1)}{(\lambda-z)^{c+1}}{}_2F_1\left(\begin{matrix}\dfrac{c}{2},&\dfrac{c}{2}+1\\
c,\end{matrix};\frac{z^2}{(\lambda-z)^2}\right),\notag\\
&(c>-1;~\lambda>0;~|z|<\lambda).\label{integral29}
\end{align}
By means of the identity (\cite{opps2009}, Theorem 3, formula 29)
\begin{align}\label{def30}
F_2(\sigma;\alpha_1,\alpha_2;\beta_1,\beta_2+n;w,z)&= \frac{(\beta_2)_n}{(\beta_2-\alpha_2)_n}\sum\limits_{k=0}^n(-1)^k{n\choose k}{(\alpha_2)_k\over (\beta_2)_k} F_2(\sigma;\alpha_1,\alpha_2+k;\beta_1,\beta_2+k;w,z)\notag\\
&\left(|w|+|y|<1; n=0,1,2,\dots;\beta_1,\beta_2\neq
0,-1,-2,\dots;\beta_2>\alpha_2\right),
\end{align}
it easily follow
\begin{align}\label{integral31}
\operatorname {J}_c^{jp}(b,b';\lambda;w,z)
&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c+ p;zx)dx\notag\\
&=\frac{\Gamma(c+j)\,(c)_p}{\lambda^{c+j}(c-b')_p}\sum\limits_{k=0}^p\dfrac{(-p)_k\,(b')_k}{k!\,(c)_k} F_2\left(c+j;b,b'+k;c,c+k;\frac{w}{\lambda},\frac{z}{\lambda}\right),\notag\\
(c+j>0;p\geq 0;&\lambda>0;;~c \neq 0,-1,-2,\dots;~if~~c-b' (negative~integer), b'-c\geq p~|w|+|z|<\lambda),
\end{align}
Further by means of (\cite{opps2009}, Theorem 3, formula 29)
\begin{align}\label{def32}
F_2\left(\begin{matrix}\sigma;&\alpha_1,&\alpha_2\\
~&\beta_1,&\beta_2-n\end{matrix};w,z\right)
&=\frac{1}{\left[\prod\limits_{i=0}^n(\beta_2-i)\right]}\sum\limits_{k=0}^n{n\choose k}\left[\prod\limits_{j=0}^{n-k}(\beta_2-j)\right]{(\sigma)_k(\alpha_2)_k\over (\beta_2)_k}z^k
F_2\left(\begin{matrix}\sigma+k;&\alpha_1,&\alpha_2+k\\
~&\beta_1,&\beta_2+k\end{matrix};w,z\right),\notag\\
&\left(\beta_2\neq i, i=0,1,\dots,n;\beta_1,\beta_2-n\neq 0,-1,-2,\dots;n=0,1,2,\dots;|w|+|z|<1
\right),
\end{align}
it follows
\begin{align}\label{integral33}
\operatorname {J}_c^{j(-p)}(b,b';\lambda;w,z)&
=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(b';c-p;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\sum\limits_{k=0}^p
\dfrac{(-p)_k\,(b')_k\,(c+j)_k}{(c)_k\,(c-p)_k\,k!}\left(-\dfrac{z}{\lambda}\right)^k
F_2\left(\begin{matrix}c+k+j;&b,&b'+k\\
~&c,&c+k\end{matrix};\frac{w}{\lambda},\frac{z}{\lambda}\right),\notag\\
&(c+j>0;~p\geq 0;\lambda>0;~c\neq 0,-1,-2,\dots;~c-p\neq 0,-1,-2,\dots;~|w|+|z|<\lambda)
\end{align}
whence
\begin{align}\label{integral34}
\operatorname {J}_c^{j(-p)}(b,c;\lambda;w,z)&
=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(c;c-p;zx)dx\notag\\
&=\dfrac{\Gamma(c+j){(\lambda-z)^{b-c-j}}}{(\lambda-w-z)^b}
\sum\limits_{k=0}^p
\dfrac{(-p)_k(c+j)_k}{(c-p)_k\,k!}\left(\dfrac{z}{z-\lambda}\right)^k
{}_2F_1\left(\begin{matrix}-k-j,&b\\
~&c\end{matrix};\dfrac{w}{w+z-\lambda}\right)\notag\\
&(c+j>0;~p\geq 0;\lambda>0;~|w|+|z|<\lambda;~c\neq 0,-1,-2,\dots;~c-p\neq 0,-1,-2,\dots).
\end{align}
The following identities are straightforward consequences of the pervious integrals:
\begin{align}\label{integral35}
\operatorname {J}_c^{j0}(c+j,0;\lambda,w,0)&=\int_0^\infty x^{c+j-1}\,e^{-\lambda\, x}{}_1F_1(c+j;~c;~w\,x)\,dx=\dfrac{\Gamma(c+j)}{(\lambda-w)^{c+j}}{}_2F_1\left(\begin{matrix}-j,&c+j\\
~&c\end{matrix};\frac{w}{w-\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,-2,\dots,~|w|<\lambda),
\end{align}
\begin{align}\label{integral36}
\operatorname {J}_c^{j(\pm p)}(0,b;\lambda,0,z)&=\int_0^\infty x^{c+j-1}\,e^{-\lambda x}\,{}_1F_1(b;c\pm p;z\,x)\, dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}{}_2F_1\left(\begin{matrix}c+j,&b\\
~&c\pm p\end{matrix};\frac{z}{\lambda}\right)=\dfrac{\Gamma(c+j)}{\lambda^{\pm p-b+ c}(\lambda-z)^{b\mp p+j}}{}_2F_1\left(\begin{matrix}\pm p-j,&c\pm p-b\\
~&c\pm p\end{matrix};\frac{z}{\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c,c\pm p\neq 0,-1,-2,\dots;j=0,\pm 1,\dots;~p=0,1,2\dots;~|z|<\lambda),
\end{align}
\begin{align}\label{integral37}
\operatorname {J}_c^{jj}(0,b;\lambda,0,z)&=\int_0^\infty x^{c+j-1}\,e^{-\lambda x}\,{}_1F_1(b;c+j;z\,x)\, dx=\dfrac{\Gamma(c+j)}{\lambda^{c-b+j}(\lambda-z)^b},\notag\\
&(c+j>0;\lambda>0;j=0,\pm 1,\dots;~|z|<\lambda),
\end{align}
\begin{align}\label{integral38}
\operatorname {J}_c^{j0}(0,b;\lambda,0,z)&=\int_0^\infty x^{c+j-1}\,e^{-\lambda x}\,{}_1F_1(b;c;zx)\, dx=\dfrac{\Gamma(c+j)}{\lambda^{c+j-b}(\lambda-x)^b}
\left[1
+\dfrac{b\,z}{c\left(\lambda-z\right)}\sum_{k=1}^j{}_2F_1\left(\begin{matrix}-j+k,&b+1\\ \\
~&c+1\end{matrix};\dfrac{z}{z-\lambda}\right)
\right],\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,-2,\dots;~|z|<\lambda).
\end{align}
\begin{align}\label{integral39}
\operatorname {J}_c^{00}(0,b;\lambda,0,z)&=\int_0^\infty x^{c-1}\,e^{-\lambda x}\,{}_1F_1(b;c;zx)\, dx=\dfrac{
\lambda^{b-c}\, \Gamma(c)}{(\lambda-z)^{b}},\qquad(c>0;~\lambda>0;~|z|<\lambda).
\end{align}
\section{Gordon's integral and confluent hypergeometric polynomials}
\noindent In the case of $\alpha=-n$, the confluent hypergeometric function ${}_1F_1(\alpha;\beta;z)$ reduces to $n$-degree polynomial in $z$, namely ${}_1F_1(-n;\beta;z)=\sum_{k=0}^n{(-n)_k\,z^k}/((\beta)_k\,k!),~n=0,1,\dots.
$.
Thus,
\begin{align}\label{integral40}
\operatorname {J}_c^{j(\pm p)}(b,-n;\lambda,w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(-n;c\pm p;zx)dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j-b}(\lambda-w)^b}
\sum_{k=0}^n\dfrac{(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1(-j-k,b;c;\dfrac{w}{w-\lambda}),\notag\\
&(c+j>0;~\lambda>0;~c,~c\pm p\neq 0,-1,\dots;p=0,1,\dots;~|w|<|\lambda|),
\end{align}
where a direct differentiation of both sides with respect to $z$ yields
\begin{align}\label{integral41}
\operatorname {J}_c^{(j+m)(\pm p+m)}(b,m-n;\lambda,w,z)&=\int_0^\infty x^{c+j+m-1}e^{-\lambda x}{}_1F_1(b;c;wx){}_1F_1(m-n;c\pm p+m;zx)\,dx\notag\\
&=\dfrac{(-1)^m\Gamma(c+j)(c\pm p)_m}{(-n)_mz^m\lambda^{c+j-b}(\lambda-w)^b}
\sum_{k=m}^n\dfrac{(-k)_m(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1(-j-k,b;c;\dfrac{w}{w-\lambda}),\notag\\
&(m\leq n; c+j+m>0;~\lambda>0;~c,c,~c\pm p+m\neq 0,-1,\dots;~|w|<|\lambda|).
\end{align}
Further, setting $b=-m$ in equation \eqref{integral40} implies
\begin{align}\label{integral42}
\operatorname {J}_c^{jp}(-m,-n;\lambda;w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-m;c;w\,x)\,{}_1F_1(-n;c\pm p;z\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\sum_{k=0}^m\dfrac{(c+j)_k(-m)_k}{(c)_k\,k!}\left(\dfrac{w}{\lambda}\right)^k{}_2F_1(-n,c+j+k;c\pm p;\dfrac{z}{\lambda})\notag\\
&\equiv\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\sum_{k=0}^n\dfrac{(c+j)_k(-n)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1(-m,c+j+k;c;\dfrac{w}{\lambda}),\notag\\
&(c+j>0,~\lambda>0;~c,c\pm p\neq 0,-1,\dots;j=0,\pm 1,\dots; n,m=0,1,\dots),
\end{align}
where a direct differentiation of both sides with respect to $w$ yields
\begin{align}\label{integral43}
\int_0^\infty x^{c+j+l-1}&e^{-\lambda\, x}{}_1F_1(l-m;c+l;w\,x)\,{}_1F_1(-n;c\pm p;z\,x)\,dx\notag\\
&=\dfrac{(-1)^l\Gamma(c+j)(c)_l}{\lambda^{c+j}w^l(-m)_l}\sum_{k=l}^m\dfrac{(c+j)_k(-m)_k(-k)_l}{(c)_k\,k!}\left(\dfrac{w}{\lambda}\right)^k{}_2F_1(-n,c+j+k;c\pm p;\dfrac{z}{\lambda}),\notag\\
&(l\leq m;c+j+l>0,~\lambda>0;~c+l,c\pm p\neq 0,-1,\dots;j=0,\pm 1,\dots; n,m=0,1,\dots),
\end{align}
with a further differentiation of both sides with respect to $z$ yields
\begin{align}\label{integral44}
\int_0^\infty& x^{c+j+k+s-1}e^{-\lambda\, x}{}_1F_1(l-m;c+l;w\,x)\,{}_1F_1(s-n;c\pm p+s;z\,x)\,dx\notag\\
&=\dfrac{(-1)^l\Gamma(c+j)(c)_l}{\lambda^{c+j+s}w^l(-m)_l}{}\sum_{k=l}^m\dfrac{(c+j+k)_s(c+j)_k(-m)_k(-k)_l}{(c)_k\,k!}\left(\dfrac{w}{\lambda}\right)^k{}_2F_1(s-n,c+j+k+s;c\pm p+s;\dfrac{z}{\lambda})\notag\\
&(s\leq n;l\leq m;c+j+l+s>0,~\lambda>0;~c+l,c\pm p+s\neq 0,-1,\dots;s,l,j=0,\pm 1,\dots; n,m=0,1,\dots).
\end{align}
If $z=\lambda$, equation \eqref{integral40} reads
\begin{align}\label{integral45}
\operatorname {J}_c^{j(\pm p)}(-m,-n;\lambda;w,\lambda)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-m;c;w\,x)\,{}_1F_1(-n;c\pm p;\lambda\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)(\pm p-j)_n}{\lambda^{c+j}(\pm p+c)_n}{}_3F_2(-m,c+j,1+j\mp p;c,1+j-n\mp p;\dfrac{w}{\lambda}),\notag\\
(c+j>0,~\lambda>0;~c,c\pm p&\neq 0,-1,\dots;j=0,\pm 1,\dots; n,m=0,1,\dots;1+j-n\pm p\neq 0,-1,\dots),
\end{align}
and if $w=\lambda$
equation \eqref{integral40} reads
\begin{align}\label{integral46}
\operatorname {J}_c^{j(\pm p)}(-m,-n;\lambda;\lambda,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-m;c;\lambda\,x)\,{}_1F_1(-n;c\pm p;z\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)(-j)_m}{\lambda^{c+j}(c)_m}{}_3F_2(-n,c+j,1+j;c\pm p,1+j-m;\dfrac{z}{\lambda}),\notag\\
(c+j>0,~\lambda>0;~c,c\pm p&\neq 0,-1,\dots;j=0,\pm 1,\dots; n,m=0,1,\dots;1+j-m\neq 0,-1,\dots).
\end{align}
Further if $w=\lambda$, equation \eqref{integral45} reads
\begin{align}\label{integral47}
\operatorname {J}_c^{jp}(-m,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-m;c;\lambda\,x)\,{}_1F_1(-n;c\pm p;\lambda\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)\,(\pm p-j)_n}{\lambda^{c+j}\,(\pm p+c)_n}\,{}_3F_2(-m,c+j,1+j\mp p;c,1+j-n\mp p;1)\notag\\
&=\dfrac{\Gamma(c+j)(-j)_m}{\lambda^{c+j}(c)_m}{}_3F_2(-n,c+j,1+j;c\pm p,1+j-m;1),\notag\\
(c+j>0,~\lambda>0;~c,c\pm p&\neq 0,-1,\dots;j=0,\pm 1,\dots; n,m=0,1,\dots;1+j-n\pm p,1+j-m\neq 0,-1,\dots),
\end{align}
whenece
\begin{align}\label{integral48}
\operatorname {J}_c^{j0}(-m,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-m;c;\lambda\,x)\,{}_1F_1(-n;c;\lambda\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)\,(-j)_n}{\lambda^{c+j}\,(c)_n}\,{}_3F_2(-m,c+j,1+j;c,1+j-n;1),\notag\\
&\equiv\dfrac{\Gamma(c+j)\,(-j)_m}{\lambda^{c+j}\,(c)_m}\,{}_3F_2(-n,c+j,1+j;c,1+j-m;1),\notag\\
(c+j>0,~\lambda>0;~c\neq 0,-1,&\dots;j=0,\pm 1,\dots; n,m=0,1,\dots;1+j-n,1+j-m\neq 0,-1,\dots).
\end{align}
If $j=0$, equation \eqref{integral47}
\begin{align}\label{integral49}
\operatorname {J}_c^{0p_\pm}(-n,-m;\lambda,\lambda,\lambda)&=\int_0^\infty x^{c-1}e^{-\lambda x}{}_1F_1(-n;c;\lambda\,x){}_1F_1(-m;c\pm p;\lambda x)\,dx=\dfrac{\Gamma(c)}{\lambda^{c}}\dfrac{m!}{(m-n)!} \dfrac{ (\pm p)_{m-n}}{(c\pm p)_m},\notag\\
(m\geq n;~c>0,~\lambda>0;&~c\pm p\neq 0,-1,\dots;j=0,\pm 1,\dots; n,m=0,1,\dots).
\end{align}
If $m=n$, equation \eqref{integral47}
\begin{align}\label{integral50}
\operatorname {J}_c^{j0}(-n,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}\left[{}_1F_1(-n;c;\lambda\,x)\,\right]^2\,dx=\dfrac{\Gamma(c+j)\,(-j)_n}{\lambda^{c+j}\,(c)_n}\,{}_3F_2(-n,c+j,1+j;c,1+j-n;1),\notag\\
(c+j>0,~\lambda>0;~c&\neq 0,-1,\dots;j=0,\pm 1,\dots; n=0,1,\dots;1+j-n\neq 0,-1,\dots).
\end{align}
The condition $1+j-n\neq 0,-1,-2,\dots$ in \eqref{integral50} can be softened using the identity
\begin{align}\label{eq51}
(-j)_n\,{}_3F_2(-n,c+j,1+j;c,1+j-n;1)=n!\,{}_3F_2(-n,-j,j+1;c,1;1),
\end{align}
to yield
\begin{align}\label{integral52}
\operatorname {J}_c^{j0}(-n,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}\left[{}_1F_1(-n;c;\lambda\,x)\right]^2\,dx=\dfrac{\Gamma(c+j)\,n!}{\lambda^{c+j}\,(c)_n}\,{}_3F_2(-n,-j,1+j;c,1;1),\notag\\
&(c+j>0;~\lambda>0;~c\neq 0,-1,-2,\dots;j=0,\pm 1,\pm 2,\dots; n=0,1,2,\dots).
\end{align}
and thus
\begin{align}\label{integral53}
\operatorname {J}_c^{10}(-n,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c}\,e^{-\lambda\, x}\left[{}_1F_1(-n;c;\lambda\,x)\right]^2\,dx=\dfrac{\Gamma(c)\,n!}{\lambda^{c+1}\,(c)_n}\left(c+2n\right),~~(c>0;~\lambda>0).
\end{align}
From the symmetric property $j\longleftrightarrow -j-1$ of ${}_3F_2(-n,-j,1+j;c,1;1)$, it also follow
\begin{align}\label{integral54}
\operatorname {J}_c^{(-j-1)0}(-n,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c-j-2}e^{-\lambda\, x}\left[{}_1F_1(-n;c;\lambda\,x)\right]^2\,dx=\dfrac{\Gamma(c-j-1)\,n!}{\lambda^{c-j-1}\,(c)_n}\,{}_3F_2(-n,-j,1+j;c,1;1),\notag\\
&(c-j-2>0;~\lambda>0;~c\neq 0,-1,-2,\dots;j=0,\pm 1,\pm 2,\dots; n=0,1,2,\dots)
\end{align}
whence
\begin{align}\label{integral55}
\operatorname {J}_c^{(-j-1)0}(-n,-n;\lambda;\lambda,\lambda)
&=\dfrac{\Gamma(c-j-1)\lambda^{2j+1}}{\Gamma(c+j)}\operatorname {J}_c^{j0}(-n,-n;\lambda;\lambda,\lambda),
\end{align}
for example
\begin{align}\label{integral56}
\operatorname {J}_c^{(-2)0}(-n,-n;\lambda;\lambda,\lambda)&=\dfrac{\Gamma(c-2)\lambda^{3}}{\Gamma(c+1)}\operatorname {J}_c^{10}(-n,-n;\lambda;\lambda,\lambda)=\dfrac{\Gamma(c-2)n!}{c\lambda^{c-2}\,(c)_n}\left(c+2n\right).
\end{align}
Further recurrence relations of this type are developed in the appendix. Note, from equations \eqref{integral42} and \eqref{integral45}, it follows
\begin{align}\label{eq57}
\sum_{k=0}^n\dfrac{(-n)_k\,(c+j)_k}{(c\pm p)_k\,k!}{}_2F_1\left(-m,c+j+k;c;\dfrac{w}{\lambda}\right)=\dfrac{(\pm p-j)_n}{(\pm p+c)_n}\,{}_3F_2\left(-m,c+j,1+j\mp p;c,1+j-n\mp p;\dfrac{w}{\lambda}\right).
\end{align}
An important class of W. Gordon's integral occur in the case of $w=k_1,z=k_2$, and $\lambda=(k_1+k_2)/2$, namely,
\begin{align}\label{integral58}
\operatorname {J}_c^{j(\pm p)}\left(-n,-m;\dfrac{k_1+k_2}{2},k_1,k_2\right)&=\int_0^\infty x^{c+j-1}e^{-(k_1+k_2) x/2}{}_1F_1(-n;c;k_1\, x){}_1F_1(-m;c\pm p;k_2\, x)dx\notag\\
&=\dfrac{2^{c+j}\Gamma(c+j)}{(k_1+k_2)^{c+j}}\,F_2\left(\begin{matrix}c+j;&-n,&-m\\
~&c,&c\pm p\end{matrix};\frac{2k_1}{k_1+k_2},\frac{2k_2}{k_1+k_2}\right),\notag\\
&(k_1+k_2>0; c+j>0;c,c\pm p\neq 0,\pm 1,\dots)
\end{align}
equivalently,
\begin{align}\label{integral59}
\operatorname {J}_c^{j(\pm p)}&\left(-n,-m;\dfrac{k_1+k_2}{2},k_1,k_2\right)=\int_0^\infty x^{c+j-1}e^{-(k_1+k_2) x/2}{}_1F_1(-n;c;k_1\, x){}_1F_1(-m;c\pm p;k_2\, x)dx\notag\\
&=\left\{ \begin{array}{ll}
\dfrac{\Gamma(c+j)(\pm p-j)_m}{k_1^{c+j}(\pm p+c)_m}{}_3F_2(-n,c+j,1+j\mp p;c,1+j-m\mp p;1),&\mbox{if $k_1= k_2,$} \\ \\
\dfrac{2^{c+j}\Gamma(c+j)}{(k_1+k_2)^{c+j}}\left(\dfrac{k_1-k_2}{k_1+k_2}\right)^m
\sum_{i=0}^{\min\{j\mp p,m\}} \dfrac{(-j\pm p)_i(-m)_i}{(c\pm p)_i~i!}\left(\dfrac{2k_2}{k_2-k_1}\right)^{i}\\
\times F_1\left(-n,c+j+m,i-m;c;\dfrac{2k_1}{k_1+k_2},\dfrac{2k_1}{k_1-k_2}\right), &\mbox{if $k_1\neq k_2$.}
\end{array} \right.
\end{align}
and further equivalent to
\begin{align}\label{integral60}
\operatorname {J}_c^{j(\pm p)}&\left(-n,-m;\dfrac{k_1+k_2}{2},k_1,k_2\right)=\int_0^\infty x^{c+j-1}e^{-(k_1+k_2) x/2}{}_1F_1(-n;c;k_1\, x){}_1F_1(-m;c\pm p;k_2\, x)dx\notag\\
&=\left\{ \begin{array}{l}
\dfrac{\Gamma(c+j)(\pm p-j)_m}{k_1^{c+j}(\pm p+c)_m}{}_3F_2(-n,c+j,1+j\mp p;c,1+j-m\mp p;1),\mbox{if $k_1= k_2,$} \\ \\
\dfrac{(-1)^n2^{c+j}\Gamma(c+j)}{(k_1+k_2)^{c+j}}\left(\dfrac{k_1-k_2}{k_1+k_2}\right)^{m+n}
\sum_{i=0}^{\min\{j\mp p,m\}} \dfrac{(-j\pm p)_i(-m)_i}{(c\pm p)_i~i!}\left(\dfrac{2k_2}{k_2-k_1}\right)^{i}\\
\times\sum\limits_{r=0}^{j+i}\dfrac{(-n)_r(-j-i)_r}{(c)_r\,r!}\left(\dfrac{2k_2}{k_2-k_1}\right)^r {}_2F_1\left(r-n,i-m;c+r;\dfrac{-4k_1k_2}{(k_1-k_2)^2}\right), \mbox{\qquad\qquad if $k_1\neq k_2$.}
\end{array} \right.
\end{align}
In particular
\begin{align}\label{integral61}
\operatorname {J}_c^{j0}&\left(-n,-m;\dfrac{k_1+k_2}{2},k_1,k_2\right)=\int_0^\infty x^{c+j-1}e^{-(k_1+k_2) x/2}{}_1F_1(-n;c;k_1\, x){}_1F_1(-m;c;k_2\, x)dx\notag\\
&=\left\{ \begin{array}{l}
\dfrac{\Gamma(c+j)(-j)_m}{k_1^{c+j}(\pm p+c)_m}{}_3F_2(-n,c+j,1+j;c,1+j-m;1),, \qquad\qquad (k_1= k_2, m\leq j),\\ \\
\dfrac{(-1)^n2^{c+j}\Gamma(c+j)}{(k_1+k_2)^{c+j}}\left(\dfrac{k_1-k_2}{k_1+k_2}\right)^{m+n}
\sum_{i=0}^m \dfrac{(-j)_i(-m)_i}{(c)_i~i!}\left(\dfrac{2k_2}{k_2-k_1}\right)^{i}\\
\times\sum\limits_{r=0}^{j+i}\dfrac{(-n)_r(-j-i)_r}{(c)_r\,r!}\left(\dfrac{2k_2}{k_2-k_1}\right)^r {}_2F_1\left(r-n,i-m;c+r;\dfrac{-4k_1k_2}{(k_1-k_2)^2}\right), \quad\quad (k_1\neq k_2, m\leq j),
\end{array} \right.
\end{align}
and
\begin{align}\label{integral62}
\operatorname {J}_c^{00}\left(-m,-n;\dfrac{k_1+k_2}{2};k_1,k_2\right)&=\int_0^\infty x^{c-1}e^{-(k_1+k_2)x/2}{}_1F_1(-m;c;k_1\,x)\,{}_1F_1(-n;c;k_2\,x)\,dx\notag\\
&=\dfrac{2^c\Gamma(c)}{(k_1+k_2)^{c}}\sum_{k=0}^m\dfrac{(-m)_k}{k!}\left(\dfrac{2k_1}{k_1+k_2}\right)^k{}_2F_1(-n,c+k;c;\dfrac{2k_2}{k_1+k_2}),\notag\\
&(c>0,~\lambda>0;~c\neq 0,-1,-2,\dots; n,m=0,1,2,\dots),
\end{align}
where, generally,
\begin{align}\label{integral63}
\operatorname {J}_c^{00}(-m,-n;\lambda;w,z)&=\int_0^\infty x^{c-1}e^{-\lambda\, x}{}_1F_1(-m;c;w\,x)\,{}_1F_1(-n;c;z\,x)\,dx\notag\\
&=\dfrac{\Gamma(c)}{\lambda^{c}}\sum_{k=0}^m\dfrac{(-m)_k}{k!}\left(\dfrac{w}{\lambda}\right)^k{}_2F_1(-n,c+k;c;\dfrac{z}{\lambda}),\notag\\
&(c>0,~\lambda>0;~c\neq 0,-1,-2,\dots; n,m=0,1,2,\dots),
\end{align}
from which the classical orthogonality property of the confluent hypergeometric functions follows, namely,
\begin{align}\label{integral64}
\operatorname {J}_c^{00}(-m,-n;\lambda;\lambda,\lambda)&=\int_0^\infty x^{c-1}e^{-\lambda\, x}{}_1F_1(-m;c;\lambda\,x)\,{}_1F_1(-n;c;\lambda\,x)\,dx=\dfrac{\Gamma(c)\,n!}{\lambda^c\,(c)_n}\delta_{nm},\notag\\
&(c>0,~\lambda>0;~c\neq 0,-1,-2,\dots; \delta_{nm}=0~if~ n\neq m,\delta_{nm}=1~if~ n= m),
\end{align}
using
$\sum_{k=0}^m{(-m)_k(-k)_n}/{k!}=n!\delta_{nm}$.
The same conclusion also follows from equation \eqref{integral48} using the fact that
$$\lim_{j\rightarrow 0}\, (-j)_m\,{}_3F_2(-n,c+j,1+j;c,1+j-m;1)=n!\,\delta_{nm}.
$$
If in equation \eqref{integral1}, $b=b'=-n$ and $p=0$, it follows
\begin{align}\label{integral65}
\operatorname {J}_c^{j0}(-n,-n;\lambda;w,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-n;c;w\,x)\,{}_1F_1(-n;c;z\,x)\,dx\notag\\
&=\dfrac{n!\,\Gamma(c+j)}{\lambda^{c+j}(c)_n}\sum_{k=0}^n\dfrac{(c+j)_k(-n)_k}{(c)_k\,k!}\left(\dfrac{z}{\lambda}\right)^kP_n^{(c-1,j+k-n)}\left(1-\dfrac{2w}{\lambda}\right),
\end{align}
where $P_n^{(\alpha,\beta)}(z)$ is the Jacobi polynomial of order $\alpha$, $\beta$ and degree $n$ in $z$.
The relation
$ P_n^{(a,b)}(-1)={(-1)^n}(b+1)_n/{n!}$ reduce the equation \eqref{integral65} to
\begin{align}\label{integral66}
\operatorname {J}_c^{j0}(-n,-n;\lambda;\lambda,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda\, x}{}_1F_1(-n;c;\lambda\,x)\,{}_1F_1(-n;c;z\,x)\,dx\notag\\
&=\dfrac{\Gamma(c+j)(-j)_n}{\lambda^{c+j}\,(c)_n}\,{}_3F_2\left(-n,c+j,1+j;c,1+j-n;\dfrac{z}{\lambda}\right),\notag\\
&(c+j>0,~\lambda>0;~c\neq 0,-1,\dots; n=0,1,\dots;1+j-n\neq 0,-1,\dots),
\end{align}
as expected.
From equation \eqref{integral66}, it follows
\begin{align}\label{integral67}
\operatorname {J}_c^{n0}(-n,-n;\lambda;\lambda,z)=&\int_0^\infty x^{c+n-1}e^{-\lambda\, x}{}_1F_1(-n;c;\lambda\,x)\,{}_1F_1(-n;c;z\,x)\,dx=\dfrac{(-1)^n\,\Gamma(c)\,n!}{\lambda^{c+n}}\,{}_3F_2\left(-n,c+n,1+n;c,1;\dfrac{z}{\lambda}\right),\notag\\
&(c+n>0,~\lambda>0;~c\neq 0,-1,\dots).
\end{align}
For $n\geq m$
\begin{align}\label{integral68}
\operatorname {J}_c^{(n-m)(\pm p)}(-n,-m;\lambda,\lambda,z)&=\int_0^\infty x^{c+n-m-1}e^{-\lambda x}{}_1F_1(-n;c;\lambda\,x){}_1F_1(-m;c\pm p;z\,x)\,dx\notag\\
&=\dfrac{(-1)^{m+n}\,\Gamma(c)\,n!}{\lambda^{c+n-m}(c\pm p)_m}\left(\dfrac{z}{\lambda}\right)^m,\quad (c+n-m>0;\lambda>0;c,c\pm p\neq 0,-1,\dots;p\geq 0).
\end{align}
The following integral follows immediately
\begin{align}\label{integral69}
\operatorname {J}_c^{j(\pm p)}(0,-n;\lambda,0,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(-n;c\pm p;z\,x)dx=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}
{}_2F_1\left(-n,c+j;c\pm p;\dfrac{z}{\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c\pm p\neq 0,-1,\dots;p=0,1,\dots)
\end{align}
whence
\begin{align}\label{integral70}
\operatorname {J}_c^{j(\pm p)}(0,-n;\lambda,0,\lambda)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(-n;c\pm p;\lambda x)dx=\left\{ \begin{array}{ll}
\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\dfrac{(\pm p-j)_n}{(c\pm p)_n}, &\mbox{ if $j\mp p\geq n,$} \\
0 ,&\mbox{ if $j\mp p< n,$}
\end{array} \right.\notag\\
&(c+j>0;~\lambda>0;~c\pm p\neq 0,-1,\dots;p=0,1,\dots),
\end{align}
and if $p=j$,
\begin{align}\label{integral71}
\operatorname {J}_c^{jj}(0,-n;\lambda,0,z)&=\int_0^\infty x^{c+j-1}e^{-\lambda x}{}_1F_1(-n;c+j;z\, x)dx
=\dfrac{\Gamma(c+j)}{\lambda^{c+j}}\left(1-\dfrac{z}{\lambda}\right)^n,\quad( c+j>0;\lambda>0).
\end{align}
and
\begin{align}\label{integral72}
\operatorname {J}_c^{j0}(0,-n;\lambda,0,\lambda)&=\int_0^\infty x^{c+n-1}e^{-\lambda x}{}_1F_1(-n;c;\lambda\, x)dx=\dfrac{(-1)^n\,n!\,\Gamma(c)}{\lambda^{c+n}},\quad (c+n>0;~\lambda>0; ~c\neq 0,-1,\dots).
\end{align}
\section{Goron\rq{}s Integral and special functions}
\noindent The generalized Laguerre polynomials are defined, for integer $n$, in terms of confluent hypergeometric functions by
\begin{align}\label{def73}
L_n^\lambda(z)=\dfrac{(\lambda+1)_n}{n!}{}_1F_1(-n;\lambda+1;z),
\end{align}
thus,
\begin{align}\label{integral74}
\int_0^\infty x^{c+j-1}&e^{-\lambda x}
L_n^{c\pm p -1}(z\,x) {}_1F_1(b;c;w\,x)dx=\dfrac{\Gamma(c+j)(c\pm p)_n}{n!\lambda^{c+j}}
\sum_{k=0}^n\dfrac{(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1\left(c+j+k,b;c;\dfrac{w}{\lambda}\right)\notag\\
&(c+j>0;~\lambda>0;~c,~c\pm p\neq 0,-1,\dots;p=0,1,\dots;~|w|<|\lambda|),
\end{align}
whence, if $b=0$,
\begin{align}\label{integral75}
\int_0^\infty x^{c+j-1}&e^{-\lambda x}
L_n^{c\pm p -1}(z\,x)dx=\left\{ \begin{array}{ll}
\dfrac{\Gamma(c+j)(c\pm p)_n}{n!\,\lambda^{c+j}}
{}_2F_1\left(-n,c+j;c\pm p;\dfrac{z}{\lambda}\right), &\mbox{ if $z\neq \lambda,~c\pm p\neq 0,-1,\dots,$} \\
\dfrac{\Gamma(c+j)(\pm p-j)_n}{\lambda^{c+j}\, n!} ,&\mbox{ if $z=\lambda ,~j\mp p\geq n,$}
\end{array} \right.\notag\\
&(c+j>0;~\lambda>0;~p=0,1,\dots).
\end{align}
\noindent From equation \eqref{integral75}, it follows
\begin{align}\label{integral76}
\int_0^\infty x^{c}\,e^{-\lambda x}\,
L_n^{c }(\lambda x)\,dx&= \left\{ \begin{array}{ll}
\dfrac{\Gamma(c+1)}{\lambda^{1+c}}, &\mbox{ if $n=0$}, \\ \\
0, &\mbox{ if $n\geq 1;c>-1,\lambda>0,~ n=0,1,\dots$},
\end{array} \right.\notag\\ \notag\\
\int_0^\infty x^{c+j}\,e^{-\lambda x}\,
L_n^{c }(\lambda x)\,dx&= \left\{ \begin{array}{ll}
\dfrac{\Gamma(c+j+1)\, (-j)_n}{\lambda^{j+c+1}\, n!}, &\mbox{ if $n<j$}, \\ \\
(-1)^n\dfrac{\Gamma(c+n+1)}{\lambda^{n+c+1}} ,&\mbox{ if $n=j$}, \\ \\
0, &\mbox{ if $n> j; c+j>-1, ~\lambda>0,~n=0,1,\dots,$}
\end{array} \right.\notag\\ \notag\\
\int_0^\infty x^{c}\,e^{-\lambda x}\,
L_n^{c- p }(\lambda x)\,dx&= \left\{ \begin{array}{ll}
\dfrac{\Gamma(c+1)(-p)_n}{\lambda^{c+1}\, n!},&\mbox{ if $n<p$},\\ \\
(-1)^n\dfrac{\Gamma(c+1)}{\lambda^{c+1}}, &\mbox{ if $n=p$}, \\ \\
0, &\mbox{ if $n> p;c>-1,\lambda>0,c-p\geq 0$.}
\end{array} \right.
\end{align}
Since
$$\dfrac{d^m}{dz^m}L_n^\lambda(az)=(-a)^mL_{n-m}^{\lambda+m}(az)$$
it easily follows, for $n\geq m$ and $m=0,1,2,\dots$, that
\begin{align}\label{integral77}
\int_0^\infty x^{c+j+m-1}&e^{-\lambda x}
L_{n-m}^{c\pm p+m-1}(zx){}_1F_1(b;c;wx)dx\notag\\
&=\dfrac{(c\pm p)_n\Gamma(c+j)}{n!\,z^m\,\lambda^{c+j}}
\sum_{k=m}^n\dfrac{(-k)_m(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1(c+j+k,b;c;\dfrac{w}{\lambda}).
\end{align}
and, for $\mu=0,1,2,\dots, m\leq n$,
\begin{align}\label{integral78}
\int_0^\infty &x^{c+j+m+\mu-1}e^{-\lambda x}
L_{n-m}^{c\pm p+m-1}(zx){}_1F_1(b+\mu;c+\mu;w\,x)dx\notag\\
&=\dfrac{\Gamma(c+j)(c+p)_n}{n!\,z^m\,\lambda^{c+j+\mu}}
\sum_{k=0}^n\dfrac{(-k)_m(-n)_k(c+j)_k(c+j+k)_\mu}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1(c+j+k+\mu,b+\mu;c+\mu;\dfrac{w}{\lambda}).
\end{align}
On other hand,
\begin{align}\label{integral79}
\int_0^\infty x^{c+j-1}e^{-\lambda x}
L_n^{c\pm p -1}(zx)L_m^{c-1}(w\, x)dx&=\dfrac{(c)_m(c\pm p)_n\Gamma(c+j)}{m!\, n!\,\lambda^{c+j}}
\sum_{k=0}^n\dfrac{(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1\left(c+j+k,-m;c;\dfrac{w}{\lambda}\right),\notag\\
&(c+j>0;~\lambda>0;~c,~c\pm p\neq 0,-1,\dots;p=0,1,\dots).
\end{align}
and by direct differentiation $s$-times, with respect to $w$, of both sides
\begin{align}\label{integral80}
\int_0^\infty& x^{c+j+s-1}e^{-\lambda x}
L_n^{c\pm p -1}(zx)L_{m-s}^{c+s-1}(w\, x)dx\notag\\
&=\dfrac{(-1)^s(c)_m(c\pm p)_n\Gamma(c+j)}{m!\, n!\,\lambda^{c+j+s}}
\sum_{k=0}^n\dfrac{(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\dfrac{(c+j+k)_s(-m)_s}{(c)_s}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1\left(c+j+k+s,s-m;c+s;\dfrac{w}{\lambda}\right),\notag\\
&(m\geq s; c+j+s>0;~\lambda>0;~c+s,~c\pm p\neq 0,-1,\dots;p=0,1,\dots).
\end{align}
and further differentiation of both sides $\mu$-times, with respect to $z$,
\begin{align}\label{integral81}
\int_0^\infty& x^{c+j+s+\mu-1}\,e^{-\lambda x}\,
L_{n-\mu}^{c\pm p+\mu -1}(zx)\,L_{m-s}^{c+s-1}(w\, x)\,dx\notag\\
&=\dfrac{(-m)_s (c)_m(c\pm p)_n\Gamma(c+j)}{(-1)^s\,m!\, n!\,z^\mu\,\lambda^{c+j+s}(c)_s}
\sum_{k=\mu}^n\dfrac{(-k)_\mu(c+j+k)_s(-n)_k(c+j)_k}{(c\pm p)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1(c+j+k+s,s-m;c+s;\dfrac{w}{\lambda}),\notag\\
&(s\leq m; ~\mu\leq n; ~c+j+s+\mu>0;~\lambda>0;~c,~c\pm p\neq 0,-1,\dots;p=0,1,\dots).
\end{align}
If $w=\lambda$, equation \eqref{integral79} reads
\begin{align}\label{integral82}
\int_0^\infty x^{c+j-1}e^{-\lambda x}
L_n^{c\pm p -1}(zx)L_m^{c-1}(\lambda\, x)dx
&=\dfrac{(-j)_m(c\pm p)_n\Gamma(c+j)}{m!\, n!\,\lambda^{c+j}}
{}_3F_2\left(-n,c+j,1+j;c\pm p,1+j-m;\dfrac{z}{\lambda}\right),\notag\\
&(j\geq m;~c+j>0;~\lambda>0;~c,~c\pm p\neq 0,-1,\dots;p=0,1,\dots).
\end{align}
and if $p=0$ it yields
\begin{align}\label{integral83}
\int_0^\infty x^{c+j-1}e^{-\lambda x}
L_n^{c+j -1}(zx)L_m^{c-1}(\lambda\, x)dx
&=\dfrac{(-j)_m\Gamma(c+j+n)}{m!\, n!\,\lambda^{c+j}}
{}_2F_1\left(-n,1+j;1+j-m;\dfrac{z}{\lambda}\right),\notag\\
&(j\geq m;~c+j>0;~\lambda>0).
\end{align}
and by taken limit of both sides as $j\rightarrow 0$
\begin{align}\label{integral84}
\int_0^\infty x^{c-1}e^{-\lambda x}
L_n^{c -1}(zx)L_m^{c-1}(\lambda\, x)dx
&=\dfrac{(-1)^{m}\,z^m\Gamma(c+n)(-n)_{m}}{m!\,n!\,\lambda^{c+n}\,(\lambda-z)^{m-n}}, \quad (n\geq m;~c>0;\lambda>0;|z|<\lambda),
\end{align}
thus
\begin{align}\label{integral85}
\int_0^\infty x^{c-1}e^{-\lambda x}
L_n^{c-1}(\lambda x)L_m^{c-1}(\lambda\, x)dx
&=\dfrac{(c)_n\Gamma(c)}{m!\, \lambda^{c}}
\delta_{m,n},\qquad
(c>0;~\lambda>0;n,m=0,1,\dots).
\end{align}
By means of
$H_{2n}(\sqrt{z})=(-1)^n\,(2n)!{}_1F_1(-n;0.5;z)/n!$, it follows using \eqref{integral74} that
\begin{align}\label{integral86}
\int_0^\infty x^{j-\frac12}e^{-\lambda x}&\, L_{n}^{\pm p-\frac12}(zx)\, H_{2n}(\sqrt{wx})\, dx
=\dfrac{(-1)^n(2n)!(\pm p+\frac12)_n\Gamma(j+\frac12)}{(n!)^2\lambda^{j+\frac12}}\notag\\
&\times \sum_{k=0}^{n}\dfrac{(-n)_k(j+\frac12)_k}{(\pm p+\frac12)_k\,k!}\left(\dfrac{z}{\lambda}\right)^k{}_2F_1\left(j+k+\dfrac12,-n;\dfrac12;\dfrac{w}{\lambda}\right),\quad (j> 1/2;p,n=0,1,\dots),
\end{align}
from which it follows
\begin{align}\label{integral87}
\int_0^\infty x^{j-\frac12}e^{-\lambda x}&\, L_{n}^{\pm p-\frac12}(zx)\, H_{2n}(\sqrt{\lambda x})\, dx
=\dfrac{(-1)^n(2n)!(-j)_n(\pm p+\frac12)_n\Gamma(j+\frac12)}{(n!)^2\,(\frac12)_n\,\lambda^{j+\frac12}}\notag\\
&\times {}_3F_2\left(j+\frac12,1+j,-n;1+j-n,\pm p+\frac12;\dfrac{z}{\lambda}\right),\quad (j>1/2;\lambda>0;~~p,n=0,1,\dots)
\end{align}
However, by means of
\begin{align}\label{def88}
\lim_{j\rightarrow 0}\,(-j)_n\,\,{}_3F_2\left(-n,\frac12+j,j+1;\pm p+\frac12,j+1-n;1\right)=\dfrac{(2n)!}{4^n(\pm p+\frac12)_n}
\end{align}
it easily follows that
\begin{align}\label{integral89}
\int_0^\infty x^{-\frac12}e^{-\lambda x}\, L_{n}^{\pm p-\frac12}(\lambda x)\, H_{2n}(\sqrt{\lambda x})\, dx
&=\dfrac{(-1)^n((2n)!)^2\sqrt{\pi}}{4^n(n!)^2\,(\frac12)_n\,\sqrt{\lambda}},\quad (\lambda>0, p: ~arbitrary).
\end{align}
Note also, if $c=1/2$ and $p=1$, it easily follows
\begin{align}\label{integral90}
\int_0^\infty x^{j-1}&e^{-\lambda x} H_{2m}(\sqrt{wx}) H_{2n+1}(\sqrt{z\,x})
\,dx\notag\\
&=(-1)^{m+n}\dfrac{(2m)!}{m!}\dfrac{2\sqrt{z}(2n+1)!}{n!}\dfrac{\Gamma(\frac12+j)}{\lambda^{\frac12+j}}\sum_{k=0}^m\dfrac{(\frac12+j)_k(-m)_k}{(\frac12)_k\,k!}\left(\dfrac{w}{\lambda}\right)^k{}_2F_1(-n,\frac12+j+k;\frac32;\dfrac{z}{\lambda}),\notag\\
&(j>0;~\lambda>0;~m,n=0,1,\dots),
\end{align}
For $j>n$ and $z=\lambda$
\begin{align}\label{integral91}
\int_0^\infty x^{j-1}&e^{-\lambda x} H_{2m}(\sqrt{wx}) H_{2n+1}(\sqrt{\lambda\,x})
\,dx=(-1)^{m+n}\dfrac{(2m)!}{m!}\dfrac{2(2n+1)!}{n!}\dfrac{\Gamma(\frac12+j)}{\lambda^{j}}\dfrac{(1-j)_n}{(\frac32)_n}
{}_3F_2(j+\dfrac12,j,-m;\dfrac12,j-n;\dfrac{w}{\lambda})
\end{align}
and for $c=1/2$ and $p=0$
\begin{align}\label{integral92}
\int_0^\infty x^{j-\frac12}&e^{-\lambda x}
H_{2m}(\sqrt{wx}) H_{2n}(\sqrt{z\,x})
\,dx=(-1)^{m+n}\dfrac{(2m)!}{m!}\dfrac{(2n)!}{n!}\dfrac{\Gamma(\frac12+j)}{\lambda^{\frac12+j}}\sum_{k=0}^m\dfrac{(\frac12+j)_k(-m)_k}{(\frac12)_k\,k!}\left(\dfrac{w}{\lambda}\right)^k{}_2F_1(-n,\frac12+j+k;\frac12;\dfrac{z}{\lambda})\end{align}
We may remark that all the above results involving ${}_1F_1$ can be rewritten in the representation using the
Whittaker function because of the following relationship:
$$
{}_1F_1(a,b,z)=e^{z/2}z^{-b/2}M_{(b-2a)/2,(b-1)/2}(z)
$$
\section*{Acknowledgements}
\noindent This work was
supported by the grant No. GP249577 from the Natural
Sciences and Engineering Research Council of Canada.
|
1,116,691,500,435 | arxiv | \section{INTRODUCTION}
\label{sec:intro}
Because Lyman $\alpha$ emission is easily quenched by dust,
the Lyman Alpha Emitting galaxies (LAEs)
are often characterized as protogalaxies experiencing their
first burst of star formation \citep{hum96}.
However, the differing behavior of Lyman $\alpha$ and
continuum photons encountering dust and neutral gas
makes it possible for older galaxies to exhibit
Lyman $\alpha$ emission when morphology and kinematics favor
the escape of these photons
\citep[e.g.][]{haimans98}.
Hence the LAEs could instead represent an older population with actively
star-forming regions.
LAEs offer the chance
to probe the bulk of the high-redshift galaxy luminosity function
as the strong emission line allows spectroscopic confirmation of
objects dimmer than the continuum limit $R\leq 25.5$.
Previous studies of LAEs
at $z\sim3$ have concentrated on
known
overdensities
\citep{steideletal00,hayashinoetal04,venemansetal05} or
searches for Lyman Alpha emission near known Damped Lyman Alpha
absorption systems
(\citealp{fynboetal03}, see \citealp{wolfegp05} for a review).
Blank fields, i.e. those not previously known to contain
unusual objects or overdensities, have been studied at
$z=3.1$ and
$z=3.4$, covering
468 arcmin$^2$ \citep{ciardulloetal02} and
70 arcmin$^2$ \citep{cowieh98,huetal98}, respectively.
Significant work has been done in recent years on large blank fields
at higher redshifts to
study the LAE luminosity function at
$z=3.7$ \citep{fujitaetal03},
$z=4.5$ \citep{huetal98},
$z=4.9$ \citep{ouchietal03,shimasakuetal03},
$z=5.7$ (\citealp{martins04};\citealp{malhotrar04} and references therein)
and $z=6.5$ \citep{malhotrar04}.
Spectroscopically confirmed samples are small, including
31 LAEs at $z=3.1$ \citep{venemansetal05},
18 at $z=4.5$ \citep[LALA,][]{dawsonetal04}
27 at $z=5.7$ \citep{huetal04,ouchietal05},
and 9 at $z=6.6$ \citep{taniguchietal05}.
The current investigation expands upon the
blank-field survey of \citet{ciardulloetal02} by covering twice the
area to a narrow-band detection limit one magnitude deeper.
The study of Lyman Alpha Emitting galaxies at $z\simeq 3.1$
is a major goal
of the Multiwavelength Survey
by Yale-Chile (MUSYC, \citealp{gawiseretal06a},
\url{http://www.astro.yale.edu/MUSYC}). The Extended Chandra Deep
Field South (ECDF-S) has been targeted with deep narrow-band imaging
and multi-object spectroscopy,
complemented by deep broad-band $UBVRIzJK$ and
public Chandra+ACIS-I imaging. These multiwavelength data make it
possible to study the physical nature of
LAEs and to distinguish star formation from AGN as the source of
their emission.
We assume a $\Lambda$CDM cosmology
consistent with WMAP results \citep{bennettetal03} with
$\Omega_m=0.3, \Omega_\Lambda=0.7$ and
$H_0 = 70 h_{70}$ km s$^{-1}$ Mpc$^{-1}$.
All magnitudes
are given in the AB95 system \citep{fukugitaetal96}.
\section{OBSERVATIONS}
\label{sec:obs}
Our narrow-band imaging
of the ECDF-S
was obtained using the
NB5000\AA\ filter (50\AA\ FWHM)
with CTIO4m+MOSAIC-II on several nights from 2002 to 2004 for
a total of 29 hours of exposure time.
Our
$UBVRI$ imaging results from combining public images taken with
ESO2.2m+WFI by the ESO Deep Public Survey
and COMBO-17 teams
\citep{erbenetal05,hildebrandtetal05,arnoutsetal01,wolfetal04}. Our
$z'$ imaging was taken with CTIO4m+MOSAIC-II on January
15, 2005.
Details of our optical images will be presented in E. Gawiser et al.
(in prep.). Our $JK$ images of ECDF-S were obtained with
CTIO4m+ISPI on several nights during 2003-2004 and will be described
in E.N. Taylor et al. (in prep.).
The final images
cover $31.5'\times31.5'=992$ arcmin$^2$
centered on the Chandra Deep Field South
and were processed through
the MUSYC photometric pipeline to generate
APCORR (corrected aperture) fluxes and uncertainties as described in
\citet{gawiseretal06a}. Table \ref{tab:depths} gives our source
detection depths.
\begin{table}[h!]
\begin{center}
\caption{5$\sigma$ Point Source Detection Limits for MUSYC ECDF-S Images in
AB magnitudes.\label{tab:depths}}
\begin{tabular}{ccccccccc}
\tableline\tableline
NB5000 & U & B & V & R & I & z$'$ & J & K\\
\tableline
25.5 & 26.0 & 26.9 & 26.4 & 26.4 & 24.6 & 23.6 & 22.7 & 22.0\\
\tableline
\end{tabular}
\end{center}
\end{table}
Multi-object spectroscopy of 23 LAE candidates
was performed with the IMACS instrument on the
Magellan-Baade telescope on Oct. 26-27, 2003, Oct. 7-8, 2004, and
Feb. 4-7, 2005. The 300 line/mm grism was used with 1.2$''$ slitlets
to cover $4000-9000$\AA\ at a resolution of 7.8\AA\ .
Details of our spectroscopy will be given in P.~Lira et al. (in prep).
\section{CANDIDATE SELECTION}
\label{sec:colors}
The greatest challenge in selecting Lyman Alpha Emitting galaxies
at $z\simeq 3.1$
is to minimize contamination from $z\simeq 0.34$
galaxies exhibiting emission lines in [O II]3727\AA\ .
These interlopers can be avoided by requiring a
high equivalent width ($>$150\AA\ in the observed frame)
which eliminates all but the
rarest [O II] emitters \citep{terlevichetal91,sternetal00}.
Contamination from
[O III]5007\AA\ is minimal at these wavelengths, as the volume for
extragalactic emitters is small, and Galactic planetary nebulae
are very rare at such high Galactic latitude ($b=-54$).
Selecting LAEs requires an estimate of the continuum
at the wavelength of the narrow-band filter, so we tested
weighted sums of the $B$ and $V$ flux densities and found that
$f_\nu^{BV}=(f_\nu^B+f_\nu^V)/2$ minimizes the scatter in
predicting the NB5000 flux density of typical objects.
The ``narrow-band excess'' in magnitudes, $BV - NB5000$, was
then used to select the candidate LAEs
with $(BV - NB5000)>1.5$,
corresponding to EW$_{obs}^{Ly\alpha}>150$\AA\ .
When the
broad-band fluxes are small,
significant errors in the
equivalent width estimate may result, and a small fraction
of the numerous objects without emission lines,
i.e. with $(BV-NB5000) \simeq 0$,
could enter the
``narrow-band excess'' sample. To avoid both types of interlopers,
we calculated a formal uncertainty in the $BV-NB5000$ color in
magnitudes, $\sigma_{BV-NB}$,
and required candidate LAEs to have ($BV-NB5000$)$ - \sigma_{BV-NB} >1.5$
and ($BV-NB5000$)$ - 3 \sigma_{BV-NB} > 0$.
The latter criterion is similar to the color excess requirement
of \citet{bunkeretal95}, but our color uncertainties are object-specific
and account for variation in image depth across the field
\citep[see][]{gawiseretal06a}.
To make spectroscopic confirmation feasible, we also required
NB5000$<$25.0, implying an
emission line flux $\geq 2.5\times10^{-17}$ ergs cm$^{-2}$ s$^{-1}$.
Visual inspection to eliminate false narrow-band detections caused by
CCD defects or cosmic ray residuals resulted in 40 candidate
LAEs. 23 of these candidates
have been observed spectroscopically, yielding
18 confirmations where
the Lyman $\alpha$ emission line was clearly detected
in both the two-dimensional
and extracted spectra and no other emission lines were visible across
the full optical spectrum.
We tested the procedure by observing lower-equivalent width objects
and found several [O II]3727 emitters; all of these interlopers exhibit
clear emission lines in H$\beta$, [O III]4959,5007
and H$\alpha$. Five of the LAE candidate spectra
fail to show
emission lines.
\begin{figure}[h!]
\vspace{-0.2in}
\includegraphics[angle=0,scale=0.35]{f1.eps}
\caption{
Histogram of $R$-band magnitudes for candidate LAEs (thin
histogram) and the subset of
confirmed LAEs (thick histogram), with
typical
spectroscopic limit of $R=25.5$ marked with
dashed vertical line. Objects with negative $R$ fluxes were
assigned $R=30$.
\label{fig:maghist}
}
\end{figure}
\section{RESULTS}
\label{sec:results}
Figure \ref{fig:maghist} shows the distribution of candidate
and confirmed LAE $R$-band continuum magnitudes versus
the ``spectroscopic'' Lyman break galaxy (LBG)
limit of $R\leq 25.5$ \citep{steideletal03}. Our
study of LAEs is able to observe objects much dimmer than this,
with a median
magnitude $R\sim27$.
36/40 candidates and 15/18 confirmed LAEs have $R>25.5$,
showing the efficacy of LAE selection in identifying
objects from the bulk of the high-redshift galaxy luminosity function.
Figure \ref{fig:uvr} shows
$UV_{corr}R$ colors of our LAE candidates versus the LBG
selection region determined by \citet{gawiseretal06a},
where $V_{corr}$ refers
to the $V$-band magnitude after subtracting the flux contributed by the
Lyman $\alpha$ emission lines.
Only 2 out of 18 confirmed
LAEs fall within the $R\leq25.5$ ``spectroscopic''
LBG sample,
but 16 out of 18
fall within
the color selection region. About half of our candidate LAEs would meet
the $R<27$ magnitude limit of the ``photometric'' LBG sample explored
by \citet{sawickit05}, and these objects should comprise 5\%
of their sample.
\vspace{-0.1 in}
\begin{figure}[h!]
\includegraphics[angle=0,scale=0.38]{f2.eps}
\caption{
$UVR$ color-color plot of confirmed LAEs (solid circles),
candidate LAEs with spectroscopy but no confirmed redshift
(open circles) and candidate LAEs without spectroscopy (plusses)
versus distribution
of the entire 84,410 object optical catalog (dots).
The polygonal region in
the upper left is the Lyman break galaxy
selection region.
\label{fig:uvr}
}
\end{figure}
In order to investigate the full SED of the LAEs, which are too dim
to obtain individual detections in our NIR photometry, we
measured stacked fluxes for the confirmed sample
and show the results of SED modelling in Fig. \ref{fig:sed}.
\citet{bruzualc03}
population synthesis models were used, with
constant star formation rate,
a \citet{salpeter55}
initial mass function from
0.1M$_\odot$ to 100M$_\odot$,
solar metallicity
and \citet{calzettietal97} dust reddening
\citep[e.g.][]{forsterschreiberetal04,vandokkumetal04}.
Uncertainties in the stacked photometry were determined using bootstrap
resampling and are close to the formal errors calculated
from the reported APCORR flux uncertainties. Parameter uncertainties
were computed via a Monte Carlo analysis where the stacked fluxes were
varied within their uncertainties to yield a probability
distribution of best-fit
parameters.
The age of the stellar population is weakly constrained and has been
restricted to the physically reasonable range 10 Myr $\leq t_* \leq$ 2 Gyr.
The best-fit parameters shown in Fig. \ref{fig:sed}
correspond to minimal dust extinction,
significant
star formation rates
(5$\leq$SFR$\leq$23 $h_{70}^{-2}$ M$_\odot$yr$^{-1}$
at 95\% confidence)
and low stellar mass
(the 95\% confidence upper limit is $M_*=8.5\times10^9 h_{70}^{-2}$M$_\odot$).
The LAEs
appear to have much less dust and stellar mass than
the $\sim500$ Myr old, $A_V\simeq1$, $\sim2\times10^{10}$M$_\odot$ Lyman break galaxy
population \citep{shapleyetal01} or the
$\sim2$ Gyr old, $A_V\simeq2.5$, $\sim10^{11}$M$_\odot$ Distant Red Galaxy
population \citep{forsterschreiberetal04}.
The star formation rates of the confirmed LAEs
inferred
from their Lyman $\alpha$ luminosities
average 5$h_{70}^{-2}$ M$_\odot$yr$^{-1}$
and from their rest-frame UV continuum luminosity densities
average 9$h_{70}^{-2}$ M$_\odot$yr$^{-1}$.
The consistency of these values with the best-fit SFR from SED modelling
implies
minimal dust extinction.
\vspace{-0.1 in}
\begin{figure}[h!]
\includegraphics[angle=0,scale=0.5]{f3.eps}
\caption{
$UBVRIzJK$ broad-band photometry (average flux density of
stacked sample)
of confirmed LAEs
along with best-fit model from SED fitting (solid) with
model parameters listed. The dotted curve shows a maximally
old model
with stellar population age fixed to
2 Gyr (the age of the universe at $z=3.1$),
$A_V=0.1$, SFR=7$h_{70}^{-2}$ M$_\odot$ yr$^{-1}$
and $M_* = 1.1\times 10^{10} h_{70}^{-2}$ M$_\odot$.
\label{fig:sed}
}
\end{figure}
To check for AGN contamination of our LAE candidate sample, we
have looked for Chandra detections of these objects. One LAE candidate
has an X-ray detection in the catalogs of \citet{viranietal05}
and \citet{lehmeretal05b}, with a 0.5-8keV luminosity of
10$^{44}$ erg s$^{-1}$.
No other candidates showed individual detections, so we removed
this object and performed a stacking
analysis \citep[e.g.][]{rubinetal04,lehmeretal05a}
which resulted in a non-detection of the entire population.
Using the conversion between SFR and X-ray flux given by \citet{ranallietal03},
the upper limit
on the average star formation rate per object is
200$h_{70}^{-2}$ M$_\odot$ yr$^{-1}$, which
is clearly consistent with the observed SFR.
None of our LAE spectra show broad emission line widths
($> 1000$ km s$^{-1}$)
that would be
inconsistent with the energetics of star formation.
We therefore expect that very few LAE candidates contain
luminous AGN which dominate their Lyman $\alpha$ or continuum emission.
\section{DISCUSSION}
\label{sec:discussion}
Our survey covers $31.5'\times31.5'\times(\Delta z=0.04)$ or
$59\times59\times38h_{70}^{-3}$Mpc$^3$, yielding an LAE
number density of
$3\pm1\times10^{-4}$$h_{70}^3$Mpc$^{-3}$,
equivalent to $4000\pm 1600$ deg$^{-2}$ per unit
redshift.
The survey volume was computed using the filter bandpass
FWHM=50{\AA}, and the five candidates without confirmed redshifts
were assumed to be LAEs.
The error bars account for variations in the LAE abundance
within our survey volume
caused by large-scale structure assuming a bias of 2.
The true uncertainties could
be bigger given the large fluctuations in density observed
for LAEs at $z=4.9$ by \citet{shimasakuetal04}.
Combining the measured number density and using the best-fit
star formation rate per object of 6$h_{70}^{-2}$ M$_\odot$ yr$^{-1}$,
we find a cosmic star formation
rate density of $2\times10^{-3}h_{70}$ M$_\odot$ yr$^{-1}$Mpc$^{-3}$.
This is significantly less than the LBG SFR density \citep{steideletal99},
but it underestimates the total LAE contribution due to our requirements
of high
equivalent width and relatively bright
$NB5000$ flux designed to select
a pure sample amenable to spectroscopic confirmation.
A detailed calculation of the LAE luminosity function at $z\simeq 3.1$,
which can be integrated to give a fuller estimate of the SFR density,
will be given in C.~Gronwall et al. (in prep).
The number density, stellar masses, star formation rates, and median
UV continuum fluxes found for LAEs are within a factor of three of those
predicted by \citet{ledelliouetal05,ledelliouetal06}; the agreement is
even better when our equivalent width threshold is accounted for.
The only strong disagreement seen versus these models is their claimed escape
fraction of 0.02 for Lyman $\alpha$ photons versus our lower limit of
0.2 (and best-fit of 0.8)
implied by the comparison of star formation rates determined
from the observed Lyman $\alpha$ luminosities and SED modelling. This
discrepancy could be resolved by using a larger escape fraction and
a standard IMF instead of the top-heavy IMF assumed in the models.
Our determination that $z=3.1$ LAEs are
predominantly blue contrasts with the results of
\citet{stiavellietal01} and \citet{pascarelleetal98}
that LAEs in blank fields at $z\simeq 2.4$
are typically red,
($B-I$)$\simeq1.8$.
This differs from the median value of ($V_{corr}-z$) $\simeq 0.1$ for
our spectroscopically confirmed LAEs and the
median color ($V-I$) $\simeq 0.1$
measured by \citet{venemansetal05}.
The difference seems unlikely to be caused by
evolution in the
LAE population from $z=3.1$ to $z=2.4$ given the small increase
in the age of the universe.
At $z=4.5$, LALA \citep{malhotrar02} reported that a majority of LAE
candidates had EW$_{rest} > 240$\AA\ , providing
evidence
of a top-heavy IMF possibly caused by Population III stars,
although equivalent widths this high could also result from
highly anisotropic radiative transfer due to the differing effects
of dust and gas on Lyman $\alpha$ and UV-continuum photons.
This photometric measurement
is sensitive to considerable scatter when the sample is
selected in the narrow-band and the broad-band imaging
is shallow, as broad-band non-detections can receive extremely
large implied equivalent
widths, and this is guaranteed to occur for any spurious narrow-band
detections. Indeed, when
$2\sigma$ upper limits on the continuum flux were used,
only 10\% of their $z=4.5$ sample had such high EWs, and
$\sim 20$\% of the confirmed objects have EW$_{rest} > 240$\AA\
\citep{dawsonetal04}. We do not find equivalent widths
this high for any of our LAE candidates at $z=3.1$.
The difference might reveal evolution in the
LAE population or could be the result of small number statistics.
\vspace{-0.2 in}
\acknowledgments
We acknowledge the referee, Andrew Bunker,
for helpful comments that improved this Letter.
We thank James Rhoads and Masami Ouchi for valuable conversations.
We are grateful for
support from Fundaci\'{o}n Andes, the FONDAP Centro de Astrof\'{\i}sica,
and the Yale Astronomy Department.
This material is based upon work supported by the National Science
Foundation under Grant. Nos. AST-0201667,0137927,0071238, and 0302030
awarded to E.G., C.G., R.C. and J.F. respectively.
This work was supported in part by NASA grant HST-GO-09525.13-A.
Facilities:CTIO(MOSAIC II),LCO(IMACS)
|
1,116,691,500,436 | arxiv | \section{Introduction}\label{sec:intro}
In this paper we propose a (1) general factorization framework (GFF) that (2) works also on implicit feedback data; (3) integrates recommendation context into the model; (4) and is flexible enough to employ the various underlying relationships of dimensions allowing us to adequately model preferences. We first argue why we consider these design points important.
Recommender systems \citep{RicciRSH} are information filtering tools that help users in information overload to find relevant content. Here we focus on the class of latent factor based collaborative filtering (CF) methods that gained popularity due to their good \emph{accuracy} and \emph{scalability} \citep{KorenRSH11}. They capture the users' preferences by uncovering latent features that explain the observed user--item ratings using factor models.
In most practical scenarios, however, users do not rate content/items explicitly: one can only observe the users' interactions\footnote{User purchased an item or viewed an product page, etc. Interactions also called events or transactions.} with items---retrieved from web logs, for instance---as they use the system. This type of feedback is termed implicit feedback, also called one-class CF in the literature, and contains unary data, i.e. recorder user--item interactions.
Implicit feedback data contains less information on user preferences than explicit feedback.
Explicit feedback requires the active contribution of the users to state their preferences on items they consumed or familiar with; thus it can directly encode both the positive and negative opinions. Implicit feedback is less accurate and negative feedback is missing. Firstly, user interactions can only be \emph{interpreted} as positive feedback; this can be inaccurate when, e.g. user is disappointed with a purchased item, or clicks on an article due to clickbaiting. Secondly, direct negative preferences are completely missing: the lack of an interaction typically means that the user was not aware of the item's existence.\footnote{We use the classic notion of implicit and explicit feedback here. In some cases explicit feedback can also be positive only, called also unary rating, typical is the voting scenario, such as Facebook likes, or Google's $+1$, \cite{RicciRSH11}. However our focus is the easily collectable implicit feedback, that is unary data. While one can (and must) infer negative signs of preference from such data, e.g., by considering missing feedback or using additional information such as time spent on page, negative and positive preferences are not explicitly distinguished.} The fact, however, that implicit feedback is always available, highlights the importance of methods working on such data for real-world recommendation applications.
Context-aware recommendation systems (CARS) refine recommendations by considering additional information, available to the system. They extend the dualistic user--item modeling concept and consider additional information that may influence the user preferences at recommendation. Such data are together termed \emph{contextual information}, or briefly \emph{context} \citep{AdomaviciusRecsys08}. One class of CARS uses latent factor methods (see e.g. \citet{KaratzogluRecsys10,RendleWSDM2010,itals_ecml,Shi_tfmap,rendle2012factorization}). Most of the latent factor based CARS however work only on explicit feedback problems that strongly limits their real-world applicability.
Factorization methods are characterized by three components, see e.g. \citep{BellkorICDM07,brismf,Salak08}. (1) A loss function that is to be minimized by the algorithm. The loss is a function of predicted preferences (or ratings) and usually (but not necessarily) contains the difference between actual and predicted preferences (or ratings). (2) An optimization method that iteratively optimizes the value of the latent factors in order to minimize the loss function. (3) A preference model that describes how preferences are estimated. For example: BRISMF \citep{brismf} optimizes for minimal root mean squared error (RMSE) loss, with a stochastic gradient descent (SGD) optimizer under a preference model where the preference of a user on an item is predicted by the dot product of the user's and the item's feature vector; BPR \citep{bpr} optimizes for maximal BPR criteria using SGD optimization strategy, with the same preference model as BRISMF; iTALS \citep{itals_ecml} optimizes for a weighted RMSE loss by using ALS optimization strategy with a preference model where the preference of a user on an item under context is the N-way dot product of the user's the item's and the context-states' feature vectors\footnote{More precisely: the sum of elements in the elementwise product of corresponding vectors.}; Factorization Machines \citep{rendle2012factorization} optimizes for minimal RMSE loss, with one of three optimization strategies (SGD, coordinate descent or Bayesian inference using Markov Chain Monte Carlo (MCMC)) with a pairwise interaction preference model, i.e. the sum of dot products between the feature vectors of every pairs of entities (e.g. user with item, user with context-state and item with context-state).
Different CARS apply various loss functions and optimization strategies. However preference modeling under context is less explored. Most methods use either the N-way or the pairwise interaction model. In the former, the preference is predicted by an N-way dot product between interacting entities; the latter calculates the model as the sum of pairwise dot products between every pair of interacting entities. As additional (context) dimensions are introduced beyond users and items, the space of possible preference models and the importance of proper modeling largely increase. We argue that the lack of proper exploration of this area is due to the lack of flexible tools in which one can experiment with various models without being required to implement a specific algorithm for each model. We therefore created the General Factorization Framework (GFF), a single, flexible algorithm that takes the preference model as an input and computes latent feature matrices for the input dimensions. GFF allows us to easily experiment with various linear models on any context-aware recommendation task, be it explicit or implicit feedback based. We believe that GFF opens up a new research path in preference modeling under context.
The following properties were important at the design of GFF.
\begin{enumerate}
\item No restriction on context: GFF works on any context-aware recommendation problem independently of the number and the meaning of context dimensions.
\item Large preference model class: the only restriction on the preference model is that it must be linear in the dimensions of the problem\footnote{Meaning that a dimension can not directly interact with itself in the model}
. This intuitive restriction does not restrict the applicability to real-world problems.
\item Data type independence: besides the practically more useful implicit case, explicit problems can be also addressed by simply changing the weighting scheme in the loss function.
\item Flexibility: the weighting scheme of GFF is very flexible, enabling to incorporate extra knowledge through the weights such time decay, dwell time dependent weighting, missing not at random hypotheses and more.
\item Scalability: GFF scales well both in terms of the number of interactions in the training set and in the number of features. This makes it applicable in real life recommender systems.
\end{enumerate}
The rest of the paper is organized as follows. Section~\ref{sec:datamodel} introduces dataspace models for context-enriched data. Building on this, the basic version of GFF is introduced in Section~\ref{sec:gff-basic}. In Section~\ref{sec:models} we demonstrate the usefulness of GFF by experimenting with different models on a 4 dimensional context-aware problem.
The results clearly imply that proper preference modeling is important is this field.
We compare the results of some models in GFF to state-of-the-art methods in Section~\ref{sec:comp-other}. GFF is further extended in Section~\ref{sec:gff-extended} to be fully compliant to the Multidimensional Dataspace Model. Extended GFF allows the seamless incorporation of information into the factorization framework beyond context, like item metadata, social networks, session information, etc. Preliminary experiments show great potential of this capability. Finally, Section~\ref{sec:conclusion} summarizes this work and hints on future research.
\section{Data model}\label{sec:datamodel}
In this section we briefly review data models for the representation of context-aware data. The focus is on the representation of the input, that is users, items, context; the target attribute (e.g. rating, preference) can be added in a straightforward way. One of the most extensive data models for this task is the Multidimensional Dataspace Model (MDM, \citet{AdomaviciusACMTIS05}). In MDM the dataspace is the Cartesian product of $N_D$ dimensions: $DS=D_1\times D_2\times\cdots\times D_{N_D}$. Each dimension contains one or more attributes: $D_i=A_{i,1}\times A_{i,2}\times\cdots\times A_{i,N_i}$. The data model is very similar to that of relational databases. It is usually also required that the values of an attribute come from a set of atomic and nominal attributes. Therefore continuous variables should be discretized and the order between attribute values is disregarded. The data -- usually in the form of transactions -- is the subset of every possible combination of the attribute values of all attributes of all dimensions.
We give an example for representing data in MDM. Let $D_1=U$ be the dimension for users, $D_2=I$ the dimension for items, and $D_3=L$ the dimension for locations, thus the dataspace is every possible combination of users, items and locations, i.e. $DS=U\times I\times L$. Let us describe the users by their ID, gender and age; the items by their ID and genres; and the location by the city. Note the following: (1) The data model does not require using the IDs for users/items. However in the classical recommendation scenario the system recommends individual items to individual users. Therefore IDs should be present to distinguish them. If the subject of the recommendation is not an item but one item property, the ID can be omitted. (2) If an item can belong to only one genre, then the item dimension has one attribute that contains this information. If an item has multiple genres then either the combination of genres are the attribute values for a single genre attribute or a binary attributes are required for each genre (e.g. IsAction, IsComedy, etc.) that contain one if the item belongs to that genre.
Factorization methods usually use a simplified version of MDM. There are several ways to simplify MDM, here we review the major ones.
Generic factorization methods -- such as Factorization Machines (FM) \citep{rendle2012factorization} or iTALS(x) (\citet{itals_ecml,italsx_infocomm}) -- restrict the number of attributes to one per dimension, however, do not limit the number of dimensions. We refer to this data model as Single Attribute MDM (SA-MDM). Most information can be equally represented in SA-MDM as MDM by just ignoring the grouping of attributes by the dimensions. The main conceptual difference is that interactions between attributes of the same dimension (e.g. item IDs and item genres) cannot be captured. By ``converting'' all attributes to dimensions we lose the information of this grouping and thus assume extra interactions. This may result in much more interactions (and therefore complexity), especially if multi-valued attributes, like genre or category, is decomposed to many binary attributes.
The data model can be further simplified by setting a limit on the number of dimensions as well. Matrix factorization (e.g. \citet{HuICDM08,bpr,brismf}) limits the number of dimensions to two (one for users, one for items) and several tensor factorization methods work on only three dimensional data (e.g. \citet{RendleWSDM2010,Shi_tfmap}).
An other interesting variant of MDM is when the number of dimensions is fixed, but the number of attributes in a dimension is not. Prominent examples using such data model are SVDFeature \citep{Chen12JMLR}, SVD++ \citep{Koren2008KDD} and NSVD1 \citep{paterek2007improving}. They use two fixed dimensions: users and items. SVDFeature sets no restrictions on the number and meaning of attributes for neither the users nor the items. SVD++ requires one of the dimensions to contain a single ID attribute only while the other dimension consists of an ID and several other attributes. Usually the user dimension is restricted to the ID and the additional attributes in this case are binary attributes for all item IDs that are set to one if the user intereacted with the given item. NSVD1 also restricts one of the dimensions to an ID attribute, while the other consists of binary entities of descriptor entities. The descriptor entities are either metadata tokens or users that rated the given item.
GFF is designed to be fully compliant with MDM. The framework has two levels: basic GFF builds on SA-MDM (see Section~\ref{sec:gff-basic}), while the extended version also incorporates multiple attributes per dimensions (see Section~\ref{sec:gff-extended}.
\section{Basic GFF}\label{sec:gff-basic}
GFF is a general modeling framework --- inspired by the latent factor CF approach --- which (1) efficiently integrates context data into the preference model; (2) allows experimentation with non-traditional models for more accurate preference estimation.
The basic framework relies on SA-MDM (see Section~\ref{sec:datamodel}). In recommendation problems, the main goal is the modeling of user preferences on items, therefore one dimension is dedicated for the \emph{users} and one dedicated for the \emph{items}. We use one ID attribute in these dimensions. Other dimensions contain context data that helps modeling user preferences. Context can be the location or time of the interaction, the device on which the interaction was performed, or any other parameters that may influence the user preference, including weather, referral's link, search keyword, etc. Since SA-MDM is used, each context dimension contains exactly one attribute. The preference model is solely learnt from sample \emph{events} (also called transactions).
Inspired by factorization methods, we assign a feature vector of length $K$ to each possible value of each attribute. We refer to these values as entities. For instance, the possible user IDs are entities. Therefore each attribute is represented as a feature matrix ($M^{(i)}\in \mathbb{R}^{K\times S_i}$, where $S_i$ is the number of entities in the $i^{\rm{th}}$ dimension), assembled from the feature vectors of entities of the attribute. Since each dimension consists of exactly one attribute, dimensions are also represented by this feature matrix.
SA-MDM compliant data can be arranged into an $N_D$ dimensional tensor $R$. The values in the tensor are the preferences for the given combination of entities (i.e. a user-item-context combination). In case of explicit feedback data, the preferences are ratings. Typically the data space is very sparse, few ratings are observed, others are missing. Our focus is the implicit case and therefore $R$ is filled with binary preference information: if a combination of entities occurred in the training data then the corresponding cell is set to 1, otherwise to 0.
\begin{equation}
r_{i_1,\ldots,i_{N_D}}=
\begin{cases}
1,& \text{if } t_{i_1,\ldots,i_{N_D}}\in T\\
0, & \text{otherwise}
\end{cases}
\end{equation}
Since the missing feedback is clearly a weaker signal of negative preference than the presence of positive feedback we construct a $\mathcal{W}(i_1,\ldots,i_{N_D})$ weight function that assigns a real value to every possible entity combination. In practice, the construction of $\mathcal{W}(\cdot)$ depends on the problem, and can also affect the complexity of the training. In order to be able to train the model efficiently we restrict $\mathcal{W}(\cdot)$ as follows:
\begin{equation}\label{eq:weight}
\begin{gathered}
\mathcal{W}: ({i_1,\ldots,i_{N_D}}) \rightarrow \mathbb{R}\\
\mathcal{W}({i_1,\ldots,i_{N_D}})=\begin{cases}
w^1(i_1,\ldots,i_{N_D}) \gg w^0({i_1,\ldots,i_{N_D}}), &\text{if } t_{i_1,\ldots,i_{N_D}}\in R\\
w^0(i_1,\ldots,i_{N_D})=\prod_{j=1}^{N_D}{\left(\mu^{(j)}v^{(j)}_{i_j}+\gamma^{(j)}\right)}, &\text{otherwise}
\end{cases}
\end{gathered}
\end{equation}
Where $w^1(i_1,\ldots,i_{N_D})$ is the weight of entity combinations of the training set and $w^0(i_1,\ldots,i_{N_D})$ is the weight of missing entity combinations. Both weight functions depend on the actual entities. Note that we require $w^0(\cdot)$ to be factorized by the dimensions. $v^{(j)}_{i_j}$ is a weight for the $(i_j)^{\rm{th}}$ entity in the $j^{\rm{th}}$ dimension. This weight can depend on any property of the entity. $\mu^{(j)}$ and $\gamma^{(j)}$ are constants for the $j^{\rm{th}}$ dimension. Therefore the weight by a given dimension can be either a constant or depend on a property of the actual entity. Although this sufficiently generic weight function class enables using different weighting schemes, we leave the exploration of its effect to future research.
For the sake of simplicity, in this paper we use a simple weight function by setting $\mu^{(j)}=0$ and $\gamma^{(j)}=1$ for all $j$, and setting $w^1(i_1,\ldots,i_{N_D})=\alpha\cdot \#(i_1,\ldots,i_{N_D})$. That is $w^0(\cdot)=w_0=1$ for every entity combination and $w^1(\cdot)$ is proportional with the number of occurrences of said combination in the training set. This basic weighting assumes that entity combinations are missing at completely random \citep{MissingData} and that it is more important to accurately predict for entity combinations with actual feedback than for ones with no feedback. This weighting scheme is the generalization of the concept introduced in \citep{HuICDM08}.\footnote{Note that by setting $w^0=0$ and $w^1=1$ and using ratings in $R$ we get the standard explicit setting in $N_D$ dimensions.}
\begin{equation}\label{eq:simpleweight}
\begin{gathered}
\mathcal{W}(i_1,\ldots,i_{N_D})=\begin{cases}
w^1(i_1,\ldots,i_{N_D})=\alpha\cdot\#(i_1,\ldots,i_{N_D}) \gg w^0_{i_1,\ldots,i_{N_D}}, &\text{if } t_{i_1,\ldots,i_{N_D}}\in R\\
w^0(i_1,\ldots,i_{N_D})=w_0=1, &\text{otherwise}
\end{cases}
\end{gathered}
\end{equation}
We define the loss as the weighted sum of squared loss:\footnote{We omit regularization for clearer presentation, but $\ell_2$ regularization is used in the actual algorithm.}
\begin{equation}\label{eq:loss}
L=\sum_{i_1=1,\ldots,i_{N_D}=1}^{S_1,\ldots,S_{N_D}}{\mathcal{W}(i_1,\ldots,i_{N_D})(\hat{r}_{i_1,\ldots,i_{N_D}}-r_{i_1,\ldots,i_{N_D}})^2}
\end{equation}
The main novelty in GFF is that the preference model, i.e. the computation of $\hat{r}_{i_1,\ldots,i_{N_D}}$ is an input of the algorithm. This allows us to experiment with \emph{any linear models} beyond the usual ones. In the general framework, a preference model is a linear model of the feature vectors such that: (1) a model consists of sums of Hadamard (or elementwise) products; (2) each product contains at least two feature vectors; (3) in a product each feature vector belongs to a different attribute (linearity); (4) constant importance weights can be applied to each product.\footnote{Omitted from the deduction for clearer presentation.}
\begin{equation}\label{eq:model}
\begin{aligned}
\hat{r}_{i_1,\ldots,i_{N_D}}=1^T\big(&M^{(\sigma_1)}_{\pi_1}\circ\ldots\circ M^{(\sigma_{p_1})}_{\pi_{p_1}}+\ldots+M^{(\sigma_{p_{q-1}+1})}_{\pi_{p_{q-1}+1}}\circ\ldots\circ M^{(\sigma_{p_q})}_{\pi_{p_q}}\big)
\end{aligned}
\end{equation}
where $\sigma_k\in[1\ldots {N_D}]$ and $\pi_k=i_j$ if $\sigma_k=j$. Biases can be included in the feature vectors and are not presented here separately due to clearer presentation. The model basically consists of selected interactions between members of a subset of dimensions.
\subsection{Training with ALS(-CG)}
Recall that the framework is designed to work also for implicit feedback, thus we need an optimization method that can efficiently handle the implicit setting. Methods that work for the explicit case can not be applied directly for the implicit case due to scalability issues that arise with the handling of missing feedback. One way to deal with this is by sampling the missing feedback thus easily averting scalability issues. The other possibility is to smartly decompose computations into independently computable parts that can be shared through computations. We follow the latter route.
We use an Alternating Least Squares (ALS) method. In ALS only one matrix is updated at a time and all the other matrices are fixed. The optimization of the loss function is done through finding the optimal values in one feature matrix, given the others.
The two main advantages of ALS are (1) that it does not use sampling, therefore it is usually more accurate and converges faster; (2) the computations of the feature vectors -- with linear models -- are independent from each other and thus can be easily parellelized on multi-core or multiprocessor systems. The main problem with ALS is that it requires a least squares step for each feature vector computation and thus scales cubically in $K$ that makes it hard to train high factor models. Therefore we approximate the solution of the least squares problem through conjugate gradient (CG) optimization. We derive the algorithm up to efficiently computing the least squares problem where we apply CG to solve it. See earlier work \citep{cgcd_arxiv} on how to apply this learning strategy effectively.
We use the loss function from equation~\ref{eq:loss} and insert the general linear factorization model of equation~\ref{eq:model} into it with the weighting scheme described in equation~\ref{eq:simpleweight}.
Without the loss of generality, we demonstrate the calculation of $M^{(i)}$ on the $M^{(1)}$ matrix. For clearer presentation, the members of the model (equation~(\ref{eq:model})) are grouped into two based on whether a column of $M^{(1)}$ is part of them:\footnote{To avoid more complex notation, we assume that the columns of $M^{(1)}$ are the first members in the products where they are present.}
\begin{equation}\label{eq:ordered}
\begin{gathered}
\hat{r}_{i_1,\ldots,i_{N_D}}=\\
=\underbrace{\big(M^{(\sigma_2)}_{\pi_2}\circ\ldots\circ M^{(\sigma_{p_1})}_{\pi_{p_1}}+\ldots+M^{(\sigma_{p_{k-1}+2})}_{\pi_{p_{k-1}+2}}\circ\ldots\circ M^{(\sigma_{p_k})}_{\pi_{p_k}}\big)^T}_{\left(\mathcal{Q}_1\right)^T}M^{(1)}_{i_1}+\\[-1mm]
+\underbrace{\big(M^{(\sigma_{p_k+1})}_{\pi_{p_k+1}}\circ\ldots\circ M^{(\sigma_{p_{k+1}})}_{\pi_{p_{k+1}}}+\ldots+M^{(\sigma_{p_{q-1}+1})}_{\pi_{p_{q-1}+1}}\circ\ldots\circ M^{(\sigma_{p_{q}})}_{\pi_{p_q}}\big)^T}_{\left(\mathcal{Q}_2\right)^T}1
\end{gathered}
\end{equation}
When recomputing $M^{(1)}$, every other matrix is fixed, thus $L$ is convex in the elements of $M^{(1)}$. The minimum is reached when $\partial L/\partial M^{(1)}$ is zero. The columns of $M^{(1)}$ can be computed separately, because the derivative is linear in them. Each column is computed similarly, therefore only the steps for $M^{(1)}_1$ (the first column of $M^{(1)}$) are shown:
\begin{equation}\label{eq:der}
\begin{gathered}
\frac{\partial L}{\partial M^{(1)}_1}=\underbrace{-2\sum_{i_2=1,\ldots,i_{N_D}=1}^{S_2,\ldots,S_{N_D}}{r_{1,i_2,\ldots,i_{N_D}}\mathcal{W}(1,i_2,\ldots,i_{N_D})\mathcal{Q}_1}}_{\mathcal{O}}+\\[-1mm] +\underbrace{2\sum_{i_2=1,\ldots,i_{N_D}=1}^{S_2,\ldots,S_{N_D}}{w_0\hat{r}_{1,i_2,\ldots,i_{N_D}}\mathcal{Q}_1}}_{\mathcal{I}_2=\mathcal{I}+\mathcal{J}M^{(1)}_1}+ \\[-1mm] +\underbrace{2\sum_{i_2=1,\ldots,i_{N_D}=1}^{S_2,\ldots,S_{N_D}}{(\mathcal{W}(1,i_2,\ldots,i_{N_D})-w_0)\hat{r}_{1,i_2,\ldots,i_{N_D}}\mathcal{Q}_1}}_{\mathcal{I}_1=\mathcal{I}'+\mathcal{J}'M^{(1)}_1}
\end{gathered}
\end{equation}
We introduce $\mathcal{O}$, $\mathcal{I}_1=\mathcal{I}'+\mathcal{J}'M^{(1)}_1$ and $\mathcal{I}_2=\mathcal{I}+\mathcal{J}M^{(1)}_1$ to simplify further equations. $\mathcal{O}$ is the weighted sum of $\mathcal{Q}_1$ type vectors from equation~(\ref{eq:ordered}) over all possible configurations involving the first entity of the first dimension. The weights are the products of corresponding elements of the preference tensor $R$ and the value of the weighting function $\mathcal{W}$ for that setting. Due to the values of the preferences, most of the members of this sum are zero. Both $\mathcal{I}_1$ and $\mathcal{I}_2$ are the sum of a coefficient matrix multiplied by the vector we seek (i.e. $M^{(1)}_1$ in this case) and a vector. The difference is that these parts of $\mathcal{I}_2$ (i.e. $\mathcal{I}$ and $\mathcal{J}$) are the same for every column of $M^{(1)}$ (and therefore can be precomputed); while those of $\mathcal{I}_1$ (i.e. $\mathcal{I}'$ and $\mathcal{J}'$) are not.
$\mathcal{O}$, $\mathcal{I}'$ and $\mathcal{J}'$ can be computed efficiently (see section~\ref{sec:complex}), however the naive computation of $\mathcal{I}$ and $\mathcal{J}$ is expensive. Therefore we further transform $\mathcal{I}_2$. With the expansion of $\hat{r}_{1,\ldots,i_{N_D}}$ (substituting (\ref{eq:ordered}) with $i_1=1$):
\begin{equation}\label{eq:brtrf}
\mathcal{I}_2=2w_0\sum_{i_2=1,\ldots,i_{N_D}=1}^{S_2,\ldots,S_{N_D}}{\mathcal{Q}_1(\mathcal{Q}_1)^TM^{(1)}_1+\mathcal{Q}_1(\mathcal{Q}_2)^T1}
\end{equation}
Expanding either $\mathcal{Q}_1(\mathcal{Q}_1)^T$ or $\mathcal{Q}_1(\mathcal{Q}_2)^T$ results in sums of matrix products, where the arguments are the elementwise products of multiple feature vectors:
\begin{equation}\label{eq:expr}
\sum_{i_2=1,\ldots,i_{N_D}=1}^{S_2,\ldots,S_{N_D}}{\left(M^{(j_1)}_{i_{j_1}}\circ\ldots\circ M^{(j_m)}_{i_{j_m}}\right)\left(M^{(l_1)}_{i_{l_1}}\circ\ldots\circ M^{(l_t)}_{i_{l_t}}\right)^T}
\end{equation}
where $j_i\neq j_k$ if $i\neq k$, $l_i\neq l_k$ if $i\neq k$, $j_i\in[2\ldots n]$ and $l_k\in[2\ldots n]$. With rearranging this expression, only the following types of quantities are needed to be computed
\begin{equation}
\begin{aligned}
\label{eq:trf}
&\text{(a)}\enskip C^{(j)}=\sum_{i=1}^{S_{j}}{M^{(j)}_{i}\left(M^{(j)}_{i}\right)^T},\\
&\text{(b)}\enskip O^{(l)}=\sum_{i=1}^{S_{l}}{M^{(l)}_{i}},\\
&\text{(c)}\enskip S_k,
\end{aligned}
\end{equation}
where (a) $C^{(j)}\in\mathbb{R}^{K\times K}$ is the covariance matrix of the feature vectors of the $j^{\rm{th}}$ feature matrix; (b) $O^{(l)}\in\mathbb{R}^{K}$ is the sum of the feature vectors of the $l^{\rm{th}}$ feature matrix; (c) $S_k\in\mathbb{R}$ is the domain size. (\ref{eq:expr}) can be computed from (a), (b) and (c) using (1) elementwise product of $\mathbb{R}^{K\times K}$ matrices; (2) elementwise product of $\mathbb{R}^{K}$ vectors; (3) matrix product of $\mathbb{R}^{K}$ vectors; (4) matrix--scalar multiplication. Note that $S_k$ is a fix value during the training process, and $C^{(j)}$ and $O^{(j)}$ only changes after the $j^{\rm{th}}$ feature matrix is recomputed. Therefore these quantities can be precomputed and should be updated only once per epoch.
After $\mathcal{O}$, $\mathcal{I}'$, $\mathcal{J}'$, $\mathcal{I}$ and $\mathcal{J}$ from equation~(\ref{eq:der}) are computed, $\frac{\partial L}{\partial M^{(1)}_1}=0$ can be solved for $M^{(1)}_1$. Instead the least squares solver (LS), we use an approximate conjugate gradient solver to get the new value of the feature vector. Algorithm~\ref{alg:hicode} shows the high level pseudocode of the training.
\begin{algorithm}[!h]
\caption{ALS-based learning of the general framework on implicit data}\label{alg:hicode}
\textbf{Input:} {$T$: training data; MODEL: the description of the desired model $K$: number of features; $E$: number of epochs; $\lambda$: regularization coefficient} \newline
\textbf{Output:} {$\{M^{(i)}\}_{i=1,\ldots, N_D}$} $K\times S_i$ sized low rank matrices \newline
\textbf{procedure} {\hbox{\ }Train{($T$, MODEL, $K$, $E$, $\lambda$)}}
\begin{algorithmic}[1]
\For{$i=1,\ldots,N_D$}
\State $M^{(i)} \leftarrow $ Random $K\times S_i$ sized matrix
\State $C^{(i)} \leftarrow \sum_{k=1}^{S_{i}}{M^{(i)}_{k}\left(M^{(i)}_{k}\right)^T}$ and $O^{(i)} \leftarrow \sum_{k=1}^{S_{i}}{M^{(i)}_{k}}$
\EndFor
\For{$e=1,\ldots,E$}
\For{$i=1,\ldots,N_D$}
\State Compute the shared parts $\mathcal{I}$ and $\mathcal{J}$
\For{$j=1,\ldots,S_i$}
\State Compute $\mathcal{O}$, $\mathcal{I}'$ and $\mathcal{J}'$
\State Add regularization
\State Solve $\frac{\partial L}{\partial M^{(i)}_j}=0$ for $M^{(i)}_j$
\EndFor
\State $C^{(i)} \leftarrow \sum_{k=1}^{S_{i}}{M^{(i)}_{k}\left(M^{(i)}_{k}\right)^T}$ and $O^{(i)} \leftarrow \sum_{k=1}^{S_{i}}{M^{(i)}_{k}}$
\EndFor
\EndFor
\State \textbf{return} $\{M^{(i)}\}_{i=1,\ldots,N_D}$
\end{algorithmic}
\textbf{end procedure}
\end{algorithm}
Yet we neglected regularization and biases. Regularization can be done by adding a $K\times K$ sized diagonal matrix to $\mathcal{J}+\mathcal{J}'$ (i.e. to the coefficient matrix of $M^{(i)}_j$) just before computing the feature vector. The model (\ref{eq:model}) can be extended with biases by adding $\sum_{i=1}^{N_D}\sum_{j=1}^{S_i}{v_{i,j}b_{i,j}}$ to it, where $b_{i,j}$ is the bias value for the $j^{\rm{th}}$ entity of the $i^{\rm{th}}$ attribute and $v_{i,j}$ is the weight of the bias. The training of this biased model can also be done efficiently (with complexity of the non-biased $K+1$-feature model's).
\subsection{Complexity of training}\label{sec:complex}
The complexity of one epoch (i.e. computing each matrix once) is $O(N_DN^+|O|K^2+\sum_{i=1}^{N_D}{S_i}K^3)$ with a naive LS solver (see Table~\ref{tab:complex} for breakdown). This is reduced to $O(N_DN^+|O|K+\sum_{i=1}^{N_D}{S_i}K^2)$ with a carefully implemented CG solver. Since $|O|N_DN^+ \gg \sum_{i=1}^{N_D}{S_i}$ and $K$ is small ($K\in[20\ldots300]$), the first term dominates. Therefore the algorithm scales \emph{linearly} with both the number of transactions and $K$ in practice (see Section~\ref{sec:runtimes} for empirical results on running times).
\begin{table*}[!ht]
\centering
{
\caption{Complexity of computations}\label{tab:complex}
\begin{tabular}{llp{0.5\hsize}}
\toprule
\multicolumn{1}{c}{\textbf{Task}} & \multicolumn{1}{c}{\textbf{Complexity}} & \multicolumn{1}{c}{\textbf{Comments}} \\
\midrule
\multicolumn{3}{c}{\textbf{Computations required per columns of $M^{(1)}$}} \\
\midrule
$\mathcal{O}$, $\mathcal{I}'$ and $\mathcal{J}'$ & $O(N_1^+K^2|O|)$ & $N_1^+$ is the number of training events, where the value of the $A^{(1)}$ attribute is $a^{(1)}_1$, and $|O|$ is the complexity of the model (i.e. the number of vector operations to compute $\hat{r}$). This is possible due to the definition of $c$ weights and $r$ preferences, as most of the members in the sums of $\mathcal{O}$, $\mathcal{I}'$ and $\mathcal{J}'$ are in fact zeroes. \\
Solving for $M^{(1)}$ & $O(K^3)$ & Using the naive LS solver. \\
\midrule
\multicolumn{3}{c}{\pbox{10cm}{\textit{Total complexity of the above for all columns of $M^{(1)}$: $O(N^+K^2|O|+S_1K^3)$, ($N^+$ is the number of transactions)}}} \\
\midrule
\multicolumn{3}{c}{\textbf{Computations once per computing $M^{(1)}$}} \\
\midrule
Computing $\mathcal{I}$ and $\mathcal{J}$ & $O(|O|K^2)$ & Assembled from members described in equation~(\ref{eq:trf}): $C^{(j)}$ and $O^{(j)}$. These need to be recomputed when $M^{(j)}$ changes. \\
Recomputing $C^{(1)}$ and $O^{(1)}$ & $O(S_1K^2)$ & Computed after finishing the recomputation of $M^{(1)}$. \\
\midrule
\multicolumn{3}{c}{\textit{Total complexity of an epoch: $O(N_DN^+|O|K^2+\sum_{i=1}^{N_D}{S_i}K^3)$}} \\
\bottomrule
\end{tabular}
}
\end{table*}
\subsection{Special cases}
We now show that standard factorization algorithms are special cases of GFF. In standard 2D MF for implicit feedback \citep{HuICDM08}, the preference of user $u$ on item $i$ is predicted as product of user and an item features: $\hat{r}_{u,i}=1^T\left(M^{(1)}_u\circ M^{(2)}_i\right)$. iTALS -- the context-aware tensor factorization model \citep{itals_ecml} -- with 3 dimensions predicts the preference of user $u$ on item $i$ under context-state $c$ as $\hat{r}_{u,i,c}=1^T\left(M^{(1)}_u\circ M^{(2)}_i\circ M^{(3)}_c\right)$, the product of each features. Its modification -- iTALSx -- does the same by using $\hat{r}_{u,i,c}=1^T\left(M^{(1)}_u\circ M^{(2)}_i\right)+1^T\left(M^{(2)}_i\circ M^{(3)}_c\right)+1^T\left(M^{(2)}_i\circ M^{(3)}_c\right)$. If ratings are used in $R$ with $w^0(\cdot)=0$, $w^1(\cdot)=1$ and $\hat{r}_{u,i}=1^T\left(M^{(1)}_u\circ M^{(2)}_i\right)$, we get the classic ALS MF algorithm \citep{BellkorICDM07}. SVD++ \citep{Koren2008KDD} can be also derived from GFF if explicit compliant weighting and preferences are used and the items rated by the users are included through binary attributes. The model should be set accordingly to the SVD++ model. However we recommend using the extended framework (see Section~\ref{sec:gff-extended}) instead of using many binary attributes, because of increased training times.
\section{Modeling preferences under context}\label{sec:models}
In this section we demonstrate the usefulness of the GFF by examining several models therein. Recall that the preference model is an input of the framework that allows experimentation with novel models without implementing a specific algorithm. Therefore we can examine novel models and compare them and also to the traditional N-way and pairwise interaction models.
\subsection{Experimental setup}\label{sec:expsetup}
We used five genuine implicit feedback data sets to evaluate our algorithm. Three of them are public (LastFM 1K, \citep{lastfm1k}; TV1, TV2, \citep{tv1_tv2}), the other two are proprietary (Grocery, VoD). The properties of the data sets are summarized in Table~\ref{tab:data}. The column ``Multi'' shows the average multiplicity of user--item pairs in the training events.\footnote{This value is 1.0 at TV1 and TV2. This is possibly due to preprocessing by the original authors that removed duplicate events.} The train--test splits are time-based: the first event in the test set is after the last event of the training set. The length of the test period was selected to be at least one day, and depends on the domain and the frequency of events. We used the artists as items in LastFM.
\begin{table}
\centering
\caption{Main properties of the data sets}\label{tab:data}
\medskip
{\small
\begin{tabular}{@{}l@{\hskip2mm}l@{\hskip4mm}r@{\hskip2mm}r@{\hskip2mm}r@{\hskip2mm}r@{\hskip4mm}r@{\hskip2mm}l@{}}
\toprule
\multirow{2}{*}{\textbf{Dataset}} & \multirow{2}{*}{\textbf{Domain}}& \multicolumn{4}{c}{\textbf{Training set}}& \multicolumn{2}{c}{\textbf{Test set}} \\
&& \textbf{\#Users}& \textbf{\#Items}& \textbf{\#Events}& \textbf{Multi}& \textbf{\#Events} & \textbf{Length}\\
\midrule
Grocery & E-grocery & 24947 & 16883 & 6238269 & 3.0279 & 56449 & 1 month \\
TV1 & IPTV & 70771 & 773 & 544947 & 1.0000 & 12296 & 1 week \\
TV2 & IPTV & 449684 & 3398 & 2528215 & 1.0000 & 21866 & 1 day \\
LastFM & Music & 992 & 174091 & 18908597 & 21.2715 & 17941 & 1 day \\
VoD & IPTV/VoD & 480016 & 46745 & 22515406 & 1.2135 & 1084297 & 1 day \\
\bottomrule
\end{tabular}}
\end{table}
We focus on topN recommendations. For a given user--context configuration setting all items are ranked by their predicted preference ($\hat{r}$). The first $N=20$ items of the list are used for recommendations.
Our primary evaluation metric is recall@20,\footnote{We also measured recall@10 and recall@5 (not shown); the relation between different models are the same.} defined as the ratio of relevant recommended items and relevant items. The reason for using recall@N is twofold: (1) we found that in live recommender systems recall usually correlates well with click-through rate (CTR), that is, an important online metric for recommendation success. (2) As described in \citep{itals_ecml}, recall@20 is a good proxy of estimating recommendation accuracy offline for real-world applications; similar finding is available in \citep{Liu:2012EBR}.\footnote{If we have no highlighted items in the recommendations (i.e. all recommended items are equal), then it makes sense to disregard the order of the recommended items. Whether this is true is determined by both the interface and the recommendation logic. For example, if we want to show more items or more diverse itemset to a user during a session while still giving relevant recommendations, we can randomize the top $N$ recommendation and recommend the first $K$ of this random order. This way we can overcome showing users the same $K$ items multiple times and have a higher chance for clicking. The goal of the system is to recommend items that the user likes. The @20 comes from a very average setting of recommending 5 items (from a randomized pool of top 20 items) per page and the user having 4--6 page views in a session. Of course these numbers are highly varied in different applications, but we still think that this is a realistic proxy for a real recommender as it can get.} Note also that recall is event based, while ranking based metrics like MAP and NDCG are query based. The inclusion of context changes the query set of the test data, therefore the comparison by query based metrics is unfair.
The hyperparameters of the algorithm, such as regularization coefficients, were optimized on a part of the training data (validation set). Then the algorithm was trained on the whole training data (including the validation set) and recall was measured on the test set. The number of epochs was set to 10 in all cases, because (1) we found that methods converge fairly well in at most 10 epochs; (2) the time of the training should be also considered and 10 epochs is usually a good trade-off between time and accuracy in practical settings. The number of features was generally set to $K=80$ that is considered to be between low and high factor models and is usually a good setting in practice. However we conduct an experiment regarding the effect of $K$ on the performance of selected models.
\subsubsection{A context-aware problem}
Our aim is to apply GFF to the implicit feedback based context-aware recommendation problem and find models that generally perform well. The area of context-aware problems is wide, as any additional information to the user--item interaction can be considered as context. In compliance with SA-MDM, we assume that the context dimensions are event contexts, meaning that their value is not determined solely by the user or the item; rather it is bound to the transaction. E.g. the time of the transaction is an event context, while the genres of the item is not.
Here we choose a general CA setup and use the \emph{time} and \emph{the order of the transactions} to derive context variables that are relevant and thus help improving recommendation accuracy. Implicit feedback data does not typically contain many other event context variables: some contexts, like \emph{mood}, require to be explicitly stated, while others, like \emph{location, device}, are specific to domains. Thus, \emph{seasonality} and \emph{sequentiality} are applied as contexts of the transaction. Therefore the problem we use as an example consists of four dimensions, a transaction is a 4-tuple that contains (1) the user, (2) the item, (3) the time band (based on the timestamp), (4) and the previously consumed item by the same user.
\textbf{Seasonality:} Many application areas of recommender systems exhibit the seasonality effect, because periodicity can be observed in many human activities. Therefore seasonal data is an obvious choice for context \citep{LiuCAMRA10}. First we have to define the length of the season. Within a season we do not expect repetitions in the aggregated behavior of users, but we expect that at the same time offset in different seasons, the aggregated behavior of the users will be similar. The length of the season depends on the data. Once we have this, within seasons we need to create \emph{time bands} (bins) that are the possible context-states. Time bands specify the time resolution of a season, which is also data dependent. We can create time bands with equal or different length. In the final step, events are assigned to time bands according to their time stamp.
For \emph{Grocery} we defined a week as the season and the days of the week as the time bands. The argument here is that people usually do shopping on weekly or biweekly basis and that shopping habits differ on weekends and weekdays. One day was used as season for the other four data sets with 4 hour intervals. We note that one can optimize the lengths and distribution of time bands but this is beyond the scope of the current paper.
\textbf{Sequentiality:} In some domains, like movies or music, users consume similar items. In other domains, like electronic gadgets or e-commerce in general, they avoid items similar to what they already consumed and look for complementary products. Sequential patterns can be observed on both domain types. Sequentiality was introduced in \citep{itals_ecml} and uses the previously consumed item by the user as a context for the actual item. This information helps in the characterizations of repetitiveness related usage patterns and sequential consumption behavior.
During evaluation we fix the sequential context to the item that was targeted by the last transaction of the user in the training set. Thus we do not use information from the test data during the evaluation. The other way (i.e. constantly update the context value based on test events) would be valid as well and would result in better results. Because the test data spans over a short period of time that generally contains a few purchasing sessions for the users, preferences thus can be accurately predicted also from this information.
\subsection{Preference models}
First, we introduce a highly simplified notation for preference models. We denote the four dimensions by $U$, $I$, $S$ and $Q$ for users, items, seasonality and sequentiality respectively. The models consists of selected interactions between selected dimensions. An interaction is denoted by putting the dimensions after one another. E.g. $UI$ is the user--item interaction, $USI$ is the user--item--seasonality interaction and so on. A model usually contains more than one interaction. Table~\ref{tab:expl} shows examples of this notation.
\begin{table}
\centering
\caption{Examples of the simplified notation system}\label{tab:expl}
\medskip
{\small
\begin{tabular}{ll}
\toprule
\multirow{2}{*}{$UI$} & Vanilla MF model (user--item interactions): \\
& $\hat{r}_{u,i}=1^T\left(M^{(U)}_u\circ M^{(I)}_i\right)$ \\
\midrule
\multirow{2}{*}{$USQI$} & N-way model with all 4 dimensions (tensor factorization): \\
& $\hat{r}_{u,i,s,q}=1^T\left(M^{(U)}_u\circ M^{(I)}_i\circ M^{(S)}_s\circ M^{(Q)}_q\right)$ \\
\midrule
\multirow{2}{*}{$UI+US+IS$} & Pairwise interaction model with 3 dimensions (U, I, S): \\
& $\hat{r}_{u,i,s}=1^T\left(M^{(U)}_u\circ M^{(I)}_i+M^{(U)}_u\circ M^{(S)}_s+M^{(I)}_i\circ M^{(S)}_s\right)$ \\
\bottomrule
\end{tabular}}
\end{table}
There are 11 different possible interactions with 4 dimensions therefore the number of possible preference models is $2^{11}-1=2047$. Removing the ones that do not contain $U$ or $I$, we still get 2018 potential models. In the field of context-aware recommendations state-of-the-art factorization methods use two models. The pairwise interaction model ($UI+US+IS+UQ+IQ+SQ$ with all 4 dimensions) (\citet{RendleWSDM2010,rendle2012factorization,italsx_infocomm}) assumes pairwise interaction between each pair of dimensions. On the other hand the N-way model ($UISQ$ with all 4 dimensions) (\citet{itals_ecml,Shi_tfmap}) assumes that the preferences can be best described by the joint interaction of all dimensions. \citep{rendle2012factorization} also mentions the generalization of the pairwise interaction model, coined d-way interaction model (e.g. the 3-way interaction model: $UI+US+IS+UQ+IQ+SQ+USI+UQI+USQ+ISQ$ with all 4 dimensions). This model includes all interactions between subsets of dimensions up to $d$ size. The authors argue that such a model is slow to train and usually does not result in more accurate recommendations.
We approach the preference modeling from the perspective of the context-aware recommendation task. In this setting the users initiate transactions with the items. Additional variables (context) may or may not influence user behavior, therefore not all possible interactions should be considered for preference modeling. We focus on the ones where either the user, the item or both interact with a context. Interactions where contexts interact with each other are disregarded (see Section~\ref{sec:independence} for additional justification), except for $SQ$ that we only keep for compatibility's sake with the pairwise model. Therefore we get to the followings:
\begin{itemize}[noitemsep]
\item \textbf{$\boldsymbol{UI}$:} Interaction between users and items, the classic CF model.
\item \textbf{$\boldsymbol{USI}$, $\boldsymbol{UQI}$, $\boldsymbol{USQI}$:} The context value dependent reweighting of the user--item relation, i.e. the context influences how the users interact with items. More context dimensions can be used for reweighting. But the more we use, the more sensitive it becomes to noise and more latent features are required for filtering this out \citep{italsx_infocomm}.
\item \textbf{$\boldsymbol{US}$, $\boldsymbol{UQ}$:} The user--context interaction produces a context dependent user bias that does not play role during the ranking but has noise filtering properties during training. We allow only one context in these interactions, because additional contexts would assume that different context dimensions interact somehow.
\item \textbf{$\boldsymbol{IS}$, $\boldsymbol{IQ}$:} The item--context interaction results in a context dependent item bias that helps in ranking as well as in learning. Only one context is allowed in these interactions.
\item \textbf{$\boldsymbol{SQ}$:} Interactions between the two context dimensions. Required for the traditional pairwise model.
\end{itemize}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{fig/model_hierarchy}
\caption{Hierarchy of the models}
\label{fig:models}
\end{figure*}
The models we used are depicted on Figure~\ref{fig:models}. Models on the right side follow the pairwise interaction scheme, while models on the left are of the N-way flavor. \emph{Traditional models} -- that were used also earlier -- are indicated with orange background and black text and \emph{novel models} are with green background and white text. The models are sorted to layers based on the dimensions used. In 2D there is only the classical $UI$ model of CF. With the inclusion of one context dimension (either $S$ or $Q$) the N-way and the pairwise philosophy of preference modeling diverges. There are only a few novel models with three dimensions and we only selected those that we coined \emph{interaction model}. Things get interesting with all four dimensions where one can create many novel models. We selected the following novel models for experimentation:
\begin{itemize}[noitemsep]
\item \textbf{Interaction model ($\boldsymbol{UI+USI+UQI}$):} This model is the composite of the base behavior of the users ($UI$) and their context-influenced modification of this behavior ($USI$ and $UQI$). This model assumes that the preferences of the users can be divided into context independent and dependent parts. In the latter the user--item relation is reweighted by a context dependent weight vector. $USQI$ is not included due to the noisiness of reweighting by more than one weight vector simultaneously.
\item \textbf{Context interaction model ($\boldsymbol{USI+UQI}$):} Preferences in this model are modeled by solely context dependent parts, i.e. it assumes that user--item interactions strongly depend on the context and this dependency affects the whole interaction rather than solely the items or users.
\item \textbf{Reduced pairwise model ($\boldsymbol{UI+US+IS+UQ+IQ}$):} This model is a minor variation of the traditional pairwise model with the exclusion of the interaction between context dimensions ($SQ$). The interaction with context is done separately by users and items, i.e. it does not affect the whole user--item relation.
\item \textbf{User bias model ($\boldsymbol{UI+US+UQ}$):} Here it is assumed that only the user interacts with the other dimensions. This results in a model where the user--item relation is supported by context dependent user biases. Note that during recommendation the user biases are constant, thus do not affect the ranking. However they might filter out some context related noise during training.
\item \textbf{Item bias model ($\boldsymbol{UI+IS+IQ}$):} This model assumes that the effect of context can described by context dependent item biases (e.g. items are popular under certain conditions). The item biases affect the ranking as well as filter context related noise during training.
\item \textbf{A complex model ($\boldsymbol{UI+US+IS+UQ+IQ+USI+UQI}$):} This model is the composite of the reduced pairwise and the interaction model. It can be also treated as a reduced 3-way interaction model from which the context-context interactions are omitted.
\end{itemize}
Note, that we restricted our model space to those where exactly one feature matrix belongs to each dimension. In GFF it is possible to use several set of features for selected dimensions. By doing so it is possible to decouple the modeling of different effects from each other. For example user and item interaction with a certain context dimension can be modeled separately by using two sets of feature for the context dimension. This is a far reaching research direction that is out of the scope of this paper, but nonetheless made available by GFF.
\subsection{Results}
\begin{table}[!h]
\centering
{
\caption{Recall@20 values for different models within the framework. Differences are statistically significant at $p=0.05$. Traditional models are with gray background. Best results are typeset bold.}\label{tab:results}
\begin{tabular}{lrrrrr}
\toprule
\textbf{Model} & \textbf{Grocery} & \textbf{TV1} & \textbf{TV2} & \textbf{LastFM} & \textbf{VoD} \\
\midrule
\pbox{4.7cm}{$USI+UQI$ \\ (context interaction model)} & 0.1504 & \textbf{0.1551} & 0.2916 & 0.1984 & 0.1493 \\[0.25cm]
\pbox{4.7cm}{$UI+USI+UQI$ \\ (interaction model)} & \textbf{0.1669} & 0.1482 & \textbf{0.3027} & \textbf{0.2142} & \textbf{0.1509} \\[0.25cm]
\rowcolor{LGray} \pbox{4.7cm}{$USQI$ \\ (N-way model)} & 0.1390 & 0.1315 & 0.2009 & 0.1906 & 0.1268 \\[0.25cm]
\pbox{4.7cm}{$UI+US+IS+UQ+IQ$ \\ (reduced pairwise model)} & 0.1390 & 0.1352 & 0.2388 & 0.1884 & 0.0569 \\[0.25cm]
\pbox{4.7cm}{$UI+US+UQ$ \\ (user bias model)} & 0.1619 & 0.0903 & 0.1399 & 0.1993 & 0.0335 \\[0.25cm]
\pbox{4.7cm}{$UI+IS+IQ$ \\ (item bias model)} & 0.1364 & 0.1266 & 0.2819 & 0.1871 & 0.1084 \\[0.25cm]
\rowcolor{LGray} \pbox{4.7cm}{$UI+US+IS+UQ+IQ+SQ$ \\ (pairwise interaction model)} & 0.1388 & 0.1344 & 0.2323 & 0.1873 & 0.0497 \\[0.25cm]
\pbox{4.7cm}{$UI+US+IS+UQ+IQ+USI+UQI$ \\ (complex model example)} & 0.1389 & 0.1352 & 0.2427 & 0.1866 & 0.0558 \\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:results} shows the accuracy in terms of recall@20 of two traditional models and the six novel models we just introduced.
There exists a novel model with all five datasets that performs better than the both traditional models. 4 out of 5 cases the interaction model ($UI+USI+UQI$) is the best and it is the second best in the remaining one case. Thus this model is not only intuitively sound but also performs well that underpins its assumptions on preference modeling. The context interaction model ($USI+UQI$) come second in 3, and third in 2 cases. Interestingly the user bias model ($UI+US+UQ$) is the second best in 2 out 5 cases while worst one in the other 3 cases. This can be explained by the differences between the repetitiveness of the datasets. Highly repetitive datasets affected more heavily by sequentiality and benefit from the noise filtering property of the $UQ$ member.As sequentiality is more closely related to user behavior than to the items, $UQ$ is much more effective than $IQ$.
The reduced pairwise model is better than the full pairwise interaction model in all cases, however the difference is negligible in 3 out of 5 cases. But the difference is $\sim14.44\%$ by VoD and $\sim2.8\%$ by TV2 dataset. Finally, note that the complex model generally does not improve over the reduced pairwise model considerably and is always worse than the interaction and the context interaction models. Three way interactions contribute to the score in a lesser way, because features being generally less than 1 and thus three way products give smaller values. This causes the context dependent biases to be more prominent initially, thus the features are set accordingly to optimize the bias values. This confirms the observations by \citep{rendle2012factorization} finding the d-way interaction model no more useful than the pairwise interaction model. However this problem might be tackled by using two sets of features for $S$ and $Q$, separately for the three way interactions and context dependent biases.
\begin{table}[!h]
\centering
{
\caption{Improvements over traditional models}\label{tab:summary}
\begin{tabular}{llcp{4cm}}
\toprule
\textbf{Dataset} & \textbf{Best model} & \textbf{Improvement} &\hfill \textbf{Models better than traditional ones (out of 6)}\\
\midrule
Grocery & $UI+USI+UQI$ & +20.14\% & \multicolumn{1}{c}{3} \\
TV1 & $USI+UQI$ & +15.37\% & \multicolumn{1}{c}{2} \\
TV2 & $UI+USI+UQI$ & +30.30\% & \multicolumn{1}{c}{5} \\
LastFM & $UI+USI+UQI$ & +12.40\% & \multicolumn{1}{c}{3} \\
VoD & $UI+USI+UQI$ & +19.02\% & \multicolumn{1}{c}{2} \\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:summary} summarizes the improvements by novel models over the traditional ones. The best novel model (interaction model in 4/5 and context interaction model 1/5) outperforms the best traditional model by 12--30\% in terms of recall@20. This difference is significant. Besides, there are several novel models for each dataset that outperform the traditional models by more than $2\%$. These include the context interaction model and models specifically good for the data (e.g. the user bias model for Grocery and LastFM).
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{fig/numfactors}
\caption{Model accuracy versus number of factors.}
\label{fig:numfactors}
\end{figure*}
So far the number of features was fixed at $K=80$. However this parameter can significantly affect the relation of models. In earlier work \citep{italsx_infocomm} we compared the pairwise and the N-way model on two 3 dimensional context-aware problems. We found that pairwise models perform better with lower number of factors, but N-way models improve more rapidly as $K$ increases. This is due to low factor models blurring different aspects of the entities together thus making the reweighting of the N-way model more difficult if not impossible.
Figure~\ref{fig:numfactors} depicts recall@20 for different values of $K$ ranging from 40 to 400. Five models were selected for this experiment: the well performing interaction ($UI+USI+UQI$) and context interaction ($USI+UQI$) model; the traditional N-way ($USQI$) and pairwise ($UI+US+IS+UQ+IQ+SQ$) model; and the reduced pairwise model ($UI+US+IS+UQ+IQ$). The results are presented on the LastFM dataset. At $K=40$ $USI+UQI$ and $USQI$ are clearly worse than the other models and the reduced pairwise model is even slightly better than $UI+USI+UQI$ and the pairwise model. By $K=400$ the context interaction model is leading slightly (within $2\%$) compared to the interaction and the traditional N-way models. The pairwise and reduced pairwise models on the other hand lag behind by more than $15\%$. We can observe that as $K$ increases, the accuracy of models with members of higher order of interactions increase more rapidly. The N-way model improves the fastest and would probably outperform other models if the $K$ is sufficiently high. On the other hand, larger $K$ values (at or beyond 400) require longer training time and even more importantly, comes with longer recommendation times, therefore their practical use is limited.
It is also worth noting that $UI+USI+UQI$ performs more stable than $USI+UQI$ or the N-way model. This is due to the $UI$ part (i.e. the context independent user--item relation) stabilizing the prediction. We conclude that the interaction model ($UI+USI+UQI$) performs well not just for $K=80$ but for practically important $K$ values in general.
\subsection{Context-context interactions}\label{sec:independence}
We discussed that interactions between context dimensions should be excluded from the model. Comparing the pairwise and reduced pairwise model showed that modeling such interactions does not increase accuracy and sometimes even degrades it. From a recommendation task perspective context dimensions never interact with each other. They can influence the users' behavior (also via their active/inactive status), and through the users they affect the consumption pattern of items as well. One could argue that other context dimensions are also affected in a similar way. However recall that not all dimensions are equal and the main focus in recommendation is to recommend items to the users (under different contexts). Even if certain context dimensions correlate there is no direct interaction between them.
We also argue that context dimensions should be independent from each other. The context-aware recommendation task becomes harder \citep{nguyen2014gaussian} and slower \citep{Rendle2013VLDB} as the number of dimensions increase. Therefore context dimensions should ideally capture different aspects of the data rather than describing the same or highly correlated characteristics in different ways.
The context dimensions of the example setting ($S$ and $Q$) are fairly independent from each other. To quantify the independence of two context dimensions $C^{(1)}$ and $C^{(2)}$, the following probability distributions can be approximated from the training data: $P(C^{(1)})=\left\{P(C^{(1)}=c^{(1)}_i)\right\}$ and $P_j(C^{(1)})=\left\{P(C^{(1)}=c^{(1)}_i|C^{(2)}=c^{(2)}_j)\right\}$. The average Kullback--Leibler divergence between $P(C^{(1)})$ and $P_j(C^{(1)}|C^{(2)})$ for all $j$ can be then computed. Small average KL divergence means that $P(C^{(1)})$ can be used in the place of $P_j(C^{(1)}|C^{(2)})$ distributions. In other words knowing the state in $C^{(2)}$ gives us low information on the state in $C^{(1)}$.
This experiment was executed with $C^{(1)}=C^{(2)}=S$ (totally dependent context dimensions); $C^{(1)}=S$, $C^{(2)}=Q$ (sequentiality's information on seasonality); $C^{(1)}=Q$, $C^{(2)}=S$ (seasonality's information on sequentiality); $C^{(1)}=S$, $C^{(2)}=S'$; $C^{(1)}=S'$, $C^{(2)}=S$, where $S'$ is seasonality with the same season as $S$, but uses different time bands. The results are shown in table~\ref{tab:kl}. It is obvious that seasonality has little information on sequentiality and vice versa, therefore these context dimensions hardly correlate. This explains why the full pairwise model performs worse than the reduced pairwise model.
\begin{table*}[!ht]
\centering
{
\caption{Average KL divergences from $P(C^{(1)})$ to $P_j(C^{(1)}|C^{(2)})$}\label{tab:kl}
\begin{tabular}{llllll}
\toprule
\multirow{3}{*}{\textbf{Data set}} & \multicolumn{5}{c}{\textbf{Average $D_{KL}\left(P_j(C^{(1)}|C^{(2)})||P(C^{(1)})\right)$}} \\
& $C^{(1)}=S$ & $C^{(1)}=S$ & $C^{(1)}=S$ & $C^{(1)}=Q$ & $C^{(1)}=S'$ \\
& $C^{(2)}=S$ & $C^{(2)}=Q$ & $C^{(2)}=S'$ & $C^{(2)}=S$ & $C^{(2)}=S$ \\
\midrule
Grocery & 3.2574 & 0.0696 & 2.2997 & 0.0695 & 2.5238 \\
TV1 & 3.1032 & 0.0189 & 1.7235 & 0.0171 & 1.5203 \\
TV2 & 2.8132 & 0.0811 & 2.7979 & 0.0947 & 2.7707 \\
LastFM & 2.6376 & 0.0030 & 2.6162 & 0.0976 & 2.5618 \\
VoD & 2.6300 & 0.0262 & 1.8024 & 0.0547 & 2.1650 \\
\bottomrule
\end{tabular}
}
\end{table*}
\subsection{Training time}\label{sec:runtimes}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{fig/runtimes}
\caption{Training times of models}
\label{fig:runtimes}
\end{figure*}
Figure~\ref{fig:runtimes} shows the time of one epoch (i.e. computing each feature matrix once) for selected models for different values of $K$ on the VoD dataset. The experiments were carried out using a single core of a multi-core CPU machine. Note that the computation can be easily parallelized therefore these training times can be greatly reduced in practice. As stated in Section~\ref{sec:complex}, the running time scales linearly with the number of features for $K$ in the practically useful range. There is a difference between the actual time of training for different models as it also depends on the complexity of the model. The complexity of the model is the number of operations required to compute the preference model. If the set of dimensions is fixed, the scaling in the model complexity is linear. In accordance with this the N-way model is the fastest and the pairwise model is the slowest from the selected ones. Also note that modeling the useless $SQ$ interaction also slows down the training.
\section{Comparison with state-of-the-art algorithms}\label{sec:comp-other}
In this section we compare GFF with other methods. The qualitative comparison focuses on pointing out key differences between GFF and other factorization algorithms. Although the main advantage of GFF is not necessarily that it can outperform other methods, but rather its flexibility (regarding the model and weighting); we also include a quantitative comparison with widely accepted algorithms such as Factorization Machines (FM) \citep{rendle2012factorization} and Bayesian Personalized Ranking (BPR) \citep{bpr}.
\subsection{Qualitative comparison with factorization methods}
\subsubsection{Factorization Machines}
Rendle et.\ al proposed factorization machines (FM; \citep{rendle2012factorization}) as a general factorization method. It is for rating prediction (explicit flavor). Implicit feedback problem can be tackled through subsampling negative feedback. Each rating is associated with different attributes, for example the user who rated, the item that was rated, the context of the rating, metadata of the item, etc. The preference model is a full (weighted) pairwise model: the prediction score is given by the sum of pairwise interaction scores between every pair of dimensions.\footnote{We rephrased here the feature matrix based introduction of the original paper.} The weight of a certain interaction is determined by the weight of the two corresponding attributes; this is an input of the algorithm. It builds on the SA-MDM datamodel just like basic GFF, therefore it handles composite dimensions through binary variables as dimensions. This solution has two drawbacks: (1) it significantly increases the training time; (2) and a lot of unnecessary interactions are modeled between these binary attributes. The authors proposed a partitioning method to overcome this problem in \citep{Rendle2013VLDB}, which basically results in excluding certain interactions from the pairwise model. The latent feature vectors can be learnt by several learning methods: stochastic gradient descent (SGD), coordinate descent\footnote{A certain version of ALS, which optimizes for one parameter at a time.}, adaptive SGD and a Bayesian inference using Markov Chain Monte Carlo (MCMC). The latter is advised as the best one of the four. The implementation of FM is available in libFM.\footnote{\url{http://libfm.org}}
The key differences between GFF and FM are as follows: (1) FM uses a subset of all possible pairwise interactions between dimensions, while GFF can use arbitrary linear preference model. (2) FM handles implicit feedback through subsampling the missing (negative) feedback and is mainly an explicit method. GFF smartly decomposes computations therefore does not need to sample implicit feedback and with the proper weighting it can either be an implicit or an explicit method. (3) Both basic GFF and FM builds on SA-MDM, however the extended GFF (see Section~\ref{sec:gff-extended}) is fully compliant with the more extensive MDM. (4) The optimization strategy of the two methods differ.
\subsubsection{SVDFeature}
Chen et.\ al proposed another framework, coined SVDFeature, that uses a subset of the FM model \citep{chen2012feature,Chen12JMLR}. Basically it assigns each attribute either to the user or to the item as a property. A feature vector is defined for each property (including the item and the user itself), and the feature vector of the item (or user) is the weighted sum of the feature vectors of its properties. The rating is predicted by the scalar product of these aggregated feature vectors. In other words, it uses a partial pairwise model that only keeps the interactions between item and user attributes. The authors claim that doing so the training time decreases drastically compared to that of FM, and the interactions dropped are mostly useless (such as interactions between metadata terms of the items). Our experiments also show that leaving out useless interactions results in more accurate models. SVDFeature can incorporate either explicit or implicit feedback as it uses a ranking loss function. The model is learned using SGD. The datamodel is a dimension restricted MDM with only 2 dimensions, one for users and one for items.
The key differences between GFF and SVDFeature are as follows: (1) SVDFeature uses a fixed model, GFF takes the preference model as an input. (2) The methods use different data models, although the data model of SVDFeature is a special case of the data model of the extended GFF. (3) SVDFeature uses (pairwise) ranking loss, GFF uses pointwise ranking loss. (4) The optimization strategies differ.
Due to the incompatibility between the data model of the basic GFF and SVDFeature, and the perception of context -- i.e. a context should be assigned to either the items or the users -- no direct quantitative comparison is possible.
\subsubsection{Other implicit context-aware factorization algorithms}
iTALS \citep{itals_ecml} and iTALSx \citep{italsx_infocomm} are general factorization algorithms that use the N-way and pairwise models respectively. The key difference to GFF is that GFF does not use a fixed model. By setting the appropriate preference model, iTALS and iTALSx are special cases of GFF.
TFMAP \citep{Shi_tfmap} is a tensor factorization algorithm for three dimensional context-aware problems that minimizes a listwise ranking loss function with SGD on a fixed 3-way model. GFF is much more flexible as TFMAP restricts not just the model class, but also the number of dimensions. The loss function and the optimization strategy of the two methods also differ.
\subsection{Quantitative comparison}
Although we argue that the main novelty and the importance of GFF is allowing experimentation with novel models without requiring specific implementations, a quantitative comparison to Factorization Machines (state-of-the-art in context-aware factorization) and to Bayesian Personalized Ranking (state-of-the-art in handling implicit feedback) is included in this section. Both FM and BPR require the missing (negative) feedback to be sampled. We followed the steps of \citet{nguyen2014gaussian} and sampled a negative example to each positive example by replacing the item of the positive example with an item that has never occurred in the training set with the same user and context values. For FM we assigned ratings 1 and 0 to positive and negative feedbacks, respectively.
FM was trained using MCMC that is encouraged by the authors of the method. The number of factors was set to $K=80$ and the number of iterations was set to $10$, because of practical requirements in the training time. Also, the method converged fairly well in $10$ epochs. There were no additional hyperparameters to be optimized by FM.
The number of features and iterations was set to $K=80$ and $10$ respectively for BPR as well. The regularization coefficients and learning rate were optimized in the same way we optimized hyperparameters for GFF.
\begin{table}[!h]
\centering
{
\caption{Comparison of GFF models to LibFM and BPR}\label{tab:comp}
\begin{tabular}{lccccc}
\toprule
\multirow{2}{*}{Dataset} & \multicolumn{3}{c}{GFF} & \multirow{2}{*}{LibFM} & \multirow{2}{*}{BPR} \\
& N-way & Pairwise & Best non-traditional && \\
\midrule
Grocery & 0.1390 & 0.1388 & \textbf{0.1669} & 0.0912 & 0.1412 \\
TV1 & 0.1315 & 0.1344 & 0.1551 & \textbf{0.1683} & 0.1365 \\
TV2 & 0.2009 & 0.2323 & 0.3027 & \textbf{0.3081} & 0.1957 \\
LastFM & 0.1906 & 0.1873 & \textbf{0.2142} & 0.0652 & 0.2002 \\
VoD & 0.1268 & 0.0497 & \textbf{0.1509} & 0.1151 & 0.0539 \\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:comp} shows the results (recall@20). For GFF, the pairwise, the N-way and the best non-traditional model (either interaction or context interaction model) was included. GFF outperforms FM in 3 out of 5 cases, performs very similarly in 1 case and underperforms in 1 case. GFF outperforms BPR in all cases.
\begin{figure*}[!h]
\centering
\includegraphics[width=\textwidth]{fig/comptimes}
\caption{Training times of FM and GFF models on the LastFM dataset.}
\label{fig:comptimes}
\end{figure*}
W.r.t. running times we compared FM and GFF. BPR was not included because it does not deal with context and therefore has an unfair advantage. The training time of FM was measured by both libFM's inner logging as well as from external code and the two values were very similar. For this measurement we did not provide a test set for libFM in order to exclude the computation of the test error. Figure~\ref{fig:comptimes} depicts the results on the LastFM dataset. GFF was twice as fast with the pairwise model and even faster with the interaction model. Due to the need of subsampling the negative feedback, FM trains on twice as much examples for the same problem. This increases the time required for training significantly. Note that the results were achieved on a single core of a multi-core CPU, and the training times of GFF can be greatly reduced if multiple cores are used in parallel.
\section{Extension -- MDM compliant GFF}\label{sec:gff-extended}
In this section we lift the restrictions imposed by SA-MDM and extend GFF to allow more attributes per dimensions and thus make it fully compliant with the Multidimensional Dataspace Model. More attributes per dimensions are useful for including multi-value properties of the interacting entities, e.g. tags associated with the items; session behavior; the social network of the users; etc. Such information could be also included in SA-MDM through several dimensions with a single binary attribute. Each attribute describes if a property (e.g.\ a tag) applies to the entity (e.g.\ item) that participates in the transaction. The main drawback of this method that it results in many dimensions and therefore significantly increases training times.
In our solution we intend to handle properties of entities together. This is achieved by bundling their binary attributes into one dimension in accordance with MDM. This admittedly restricts the space of possible preference models by excluding interactions between these attributes. Analogously as for context interactions, we can also argue to exclude property interactions -- between properties of the same kind, since they are irrelevant from the recommendation point of view.
Our solution is inspired by NSVD1 \citep{paterek2007improving} and is as follows.
\begin{enumerate}
\item A dimension should be defined with entities (i.e. different values of the context variable) that are associated with the properties;
\item Each property is represented by an attribute whose value denotes the strength of the attribute for a given entity. A (sparse) mixing matrix ($W\in\mathbb{R}^{S^{(P)}\times S^{(E)}}$, where $S^{(P)}$ and $S^{(E)}$ is the number of properties and entities, respectively) is formed from the values of the attributes.
\item A feature vector is assigned to each property.
\item Since each entity is the weighted sum of its properties, the feature vector of an entity is a weighted sum of the feature vectors of its properties' feature vectors. This allows the learning algorithm to be unchanged for dimensions with single attributes, because the feature vectors can be computed for the entities that directly participate in the transaction. The feature vectors of the entities can be computed using matrix multiplication: $M^{(E)}=M^{(P)}W$.
\end{enumerate}
Since the derivative of the loss function w.r.t.\ the properties' features is not linear in the columns of $M^{(P)}$, an approximative solution is required. We chose to update the properties' feature vectors as if they were independent. To ensure convergence, after training some of the properties' feature vectors, the model should be updated before continuing. Since the update is fast, it can be done after the computation of each vector. Moreover, the update of feature vectors can be parallelized. This method can be still slow if the average number of properties assigned to entities is high.
An other way is to apply two-phase learning similarly to \citep{PilaRecsys09}. The first phase computes $M^{(E)}$ using a normal ALS step. In the second phase $M^{(P)}$ is computed from $M^{(E)}$ and $W$. The finishing step is to compute $M^{(E)}=M^{(P)}W$ from the new $M^{(P)}$, thus the following ALS steps remain consistent. Naturally, the two-phase learning is less accurate, therefore we stick to the direct optimization when possible.
Two examples are shown below on how this extension can be used.
\subsection{Item metadata as attributes}
CBF is often combined with CF to create hybrid algorithms that outperform both of them. E.g.\ item metadata helps overcoming the item cold-start problem in CF \citep{burke2007hybrid}. Here we show how to include item metadata into a model using the extended GFF.
Let us assume that the relevant item metadata is tokenized, preprocessed. From there the outline of the solution is followed.
\begin{enumerate}
\item We create an \emph{item dimension}, its entities are the items, to which we will assign the metadata attributes.
\item Each metadata token is represented by an attribute that indicates the strength of a token for the items. If the item is not associated with the token, the value of the attribute is set to 0 for that item. $W$ is created from these values.
\item A feature vector is assigned to each token.
\item The feature vectors of the items now can be computed as $M^{(I)}=M^{(M)}W$, where $M^{(M)}$ is the feature matrix of the metadata attributes.
\end{enumerate}
\subsection{Session information}
Different sessions of the same user are usually treated uniformly by recommender systems, assuming that user preference does not change across sessions. Session information, however, can be of great help in identifying what the user is currently interested in. This information can further refine recommendations and is exceptionally useful in domains where users likely have broader interests (e.g.\ e-commerce, news sites).
As the context of the transaction, let us assign all items visited during the session but the actual one. Thus the whole session is assigned to each transaction. We exclude the actual item from the session context, since this is the prediction target. Following the outline:
\begin{enumerate}
\item Each transaction will be a separate entity, thus the dimension will consists of all of the transactions. The sessions can not be used as entities, because the associated attributes are different by each transaction of the session since the actual item is omitted.
\item Each item in assigned with an attribute. The attribute is either binary (i.e. the item belongs to the session or not) or weighted by the occurrences of the item in the session. $W$ is created assigning items to each event.
\item Each item is assigned with a feature vector.
\item The feature vectors of the events now can be computed as $M^{(E)}=M^{(X)}W$, where $M^{(X)}$ is the feature matrix of the items. Note that $M^{(X)}$ is a different matrix than the feature matrix of the item dimension $M^{(I)}$.
\end{enumerate}
\subsection{Experimental evaluation}
Initial experiments were done with the extended GFF to incorporate item metadata and session information into the factorization model. The general settings are the same as in Section~\ref{sec:expsetup}. A user session is defined as a sequence of events of a user where the largest gap between two consecutive timestamp is less than 20 minutes. Item metadata consists of the tokenized title, description and category string of the items. The data was filtered for too common and rare tokens. For both context, the weights were $\ell_2$ normalized on an entity by entity basis. The experiments were run on the Grocery dataset, because here the usage of sessions is justified and we have the necessary metadata available.
Session context is denoted by $X$, metadata is by $M$ in our simplified notation. The following models were compared to the classic CF model ($UI$):
\begin{itemize}[noitemsep]
\item \textbf{$\boldsymbol{XI}$:} Interactions between items and the session. Basically this model guesses the actual item based on the other items in the session.
\item \textbf{$\boldsymbol{UI+XI}$:} The classic user--item interaction refined by the actual session.
\item \textbf{$\boldsymbol{UM}$:} The items are replaced by the sum of their metadata in the classic CF model.
\item \textbf{$\boldsymbol{UI+UM}$:} Two aspects of the items are used to model interaction with users, their entity and the sum of their metadata.
\item \textbf{$\boldsymbol{XM}$:} Interaction between other items on the session and the metadata of the actual item.
\end{itemize}
\begin{table}[!h]
\centering
{
\caption{Results for the extended framework on Grocery}\label{tab:extended}
\begin{tabular}{lcc}
\toprule
Model & Recall@20 & Improvement \\
\midrule
$UI$ & 0.1013 & N/A \\
$XI$ & 0.2248 & $+$121.97\% \\
$UI+XI$ & 0.2322 & $+$129.36\% \\
$UM$ & 0.0614 & \hphantom{1}$-$39.34\% \\
$UI+UM$ & 0.2166 & $+$113.87\% \\
$XM$ & 0.2154 & $+$112.77\% \\
\bottomrule
\end{tabular}
}
\end{table}
Table~\ref{tab:extended} summarizes the results. Note that data from the test set is needed for predicting with session in the form of other items of the test session. The results suggest that session information is very important for recommending with the Grocery dataset. $XI$ gives strong result in and of itself and is further improved by mixing in the $UI$ interaction as well. While metadata does not perform well in the place of items, they complements the basic $UI$ model well and is also useful for averting the item cold-start problem.
\section{Conclusion \& future work}\label{sec:conclusion}
In this paper we introduced a general factorization framework, GFF. The novelty of this framework over existing algorithms is its flexibility. It works both on explicit and implicit feedback data, and can incorporate any recommendation context, but even more importantly, the preference model is an input of the algorithm. This allows experimentation with novel preference models without implementing algorithms for every new model separately. The framework optimizes for a weighted square loss function and is very flexible in terms of weighting schemes. This allows us to use the framework for either explicit or implicit feedback based problems and even assumptions on the nature of the missing feedback can be included. The learning is done by a well scaling ALS-CG learner. The computations are smartly decomposed, therefore no sampling of the missing feedback data is required.
In Section~\ref{sec:models} we demonstrated the usefulness of GFF on a four dimensional context-aware recommendation problem. The experimentation showed that certain models capture the preferences of users on items under context much better than the traditional N-way or pairwise models. From the investigated models, we identified the so-called \emph{context interaction model} to be generally useful. This model is the composite of the user--item interaction, refined by context specific user--item interactions ($UI+USI+UQI$ in our simplified notation); to our best knowledge despite its intuitiveness this model was never used before for recommendation. We also found that modeling useless interactions -- such as those between context dimensions -- in fact worsens recommendation accuracy as well as increases the time required for training. The novel models in this framework are generally also more accurate than state-of-the-art algorithms.
GFF was further extended in Section~\ref{sec:gff-extended} to be fully compliant with the Multidimensional Dataspace Model and be able to handle multiple attributes in a dimension. This extension allows for the incorporation of additional data, such as metadata or session information into GFF models. Initial experiments showed that these information can significantly improve the accuracy of the recommendations.
GFF opens up several research paths. While we found a model that works generally well in a common example setting and we had success with the same model is some similar scenarios, the optimal preference modeling for novel tasks is still an open question. Also, we completely ignored models where certain dimensions have multiple sets of features. We think that there is great potential in such models if used properly. A loosely connecting but nonetheless important path is the characterization of context dimensions, i.e.\ determining their quality and their usefulness in $UCI$, $UC$ and $IC$ like interactions prior to training. GFF can help this research by allowing easy evaluation of different context dimensions and models.
GFF can be also improved. We would like to generalize a pairwise ranking loss function and allow for its optimization as an alternative of the current pointwise ranking loss while maintaining the efficiency and scaling of the training. Another potential improvement could be a meta-learner over GFF suggesting the best model for a context-aware recommendation problem and refining it during the training.
\section*{Acknowledgements}
The work leading to these results has received funding from the European Union's Seventh Framework Programme (FP7/2007-2013) under CrowdRec Grant Agreement n$^\circ$ 610594.
\bibliographystyle{spbasic}
|
1,116,691,500,437 | arxiv | \section{Introduction}
With the end of the {\em Spitzer} cold phase and the widespread availability of 8~$\mu$m and 24~$\mu$m bands observations in the archives, the availability of relations to determine the total infrared (TIR) emission from these wave bands by themselves is crucial to efficiently exploit the archives. In addition, the {\em Herschel} Space Observatory observes dust emission at rest-frame 24~$\mu$m and longer wavelengths for galaxies redshifted to $z\simeq1.5$ in the PACS 60~$\mu$m and longward bands with a spatial resolution as good as the {\em Spitzer} 24~$\mu$m band, for instance. Thus, estimating the total infrared flux from these bands becomes crucial for measuring total infrared fluxes using {\em Herschel} data. Finally, the advent of new instrumentation in the coming years, such as the James Webb Space Telescope (JWST) or the Atacama Large Millimeter Array (ALMA) will also open a new window on the infrared emission of nearby and distant galaxies. For instance, the 18~$\mu$m JWST/MIRI band and the 350~$\mu$m ALMA band will probe the rest-frame 8~$\mu$m and the 160~$\mu$m emission of $z\simeq1.2$ galaxies.
Determining the total infrared emission using the 8-160~$\mu$m {\em Spitzer} bands with equation 4 from \cite{dale2002a} or equation 22 from \cite{draine2007a} yields a better estimate of the total infrared flux than using a single wave band as a proxy for the total infrared flux, but they necessitate 3 and 4 infrared bands respectively. Indeed, these relations necessitate using the much lower resolution 70~$\mu$m and 160~$\mu$m data, which have resolutions of 18\arcsec\ and 40\arcsec, respectively. The poorer resolution of the 70~$\mu$m and 160~$\mu$m bands strongly constrains the scales on which total infrared fluxes can be measured, even for local galaxies. Attempts to derive a relation to estimate the total infrared emission from the 8~$\mu$m and 24~$\mu$m bands have been made by \cite{calzetti2005a} using NGC~5194 (M51) data, \cite{perez2006a} using NGC~3031 (M81) data, and \cite{thilker2007a} using M33 data. For each galaxy, the scatter around the relation is about 40\%. \cite{rieke2009a} recently showed that using the 24~$\mu$m band only provided good results. Using the 24~$\mu$m and the 70~$\mu$m bands \cite{papovich2002a} have also provided an estimate of the TIR emission with a similar uncertainty. However, these relations may be applicable only to galaxies with similar metallicities and star formation rate intensities (Calzetti et al. 2010, submitted; Li et al. 2010, in preparation). Metallicity variations have been associated with variations in mid- and far-infrared colors, as has been observed in metal-poor galaxies \citep[e.g.][]{engelbracht2005a, engelbracht2008a}. Indeed, the \cite{calzetti2005a} relation, which was derived using data from a very metal-rich galaxy, underestimates the total infrared emission by a factor of a few in metal-poor dwarf galaxies \citep{cannon2005a,cannon2006a,cannon2006b}.
In this article we derive relations to estimate the total infrared emission using just one or two of the {\em Spitzer} bands, with a strong focus on deriving the total infrared flux from just the higher resolution, shorter wavelength 8~$\mu$m and 24~$\mu$m bands. To do so, we use regions within a subset of nearby face-on spiral galaxies from the Spitzer Nearby Galaxies Survey \citep[SINGS,][]{kennicutt2003a} that are resolved in the {\em Spitzer} 160~$\mu$m band as well as integrated galaxy luminosities for galaxies in the SINGS, the Local Volume Legacy survey (LVL) and the \cite{engelbracht2008a}, hereafter E08, samples. In section \ref{sec:sample}, we describe the samples of galaxies and the data processing. In section \ref{sec:results}, we present the results, and we discuss them in section \ref{sec:discussion}. Finally we summarize our results and conclude in section \ref{sec:conclusion}.
\section{Sample and data}
\label{sec:sample}
We use several samples to derive the TIR emission both from galaxy subregions and galaxies. We use the subset of SINGS galaxies selected by \cite{bendo2008a} to study galaxy subregions. The requirements to perform this study are similar to theirs: the galaxies must have major axes of 5\arcmin\ or greater so that galaxy substructure is resolved; the inclination must be no more than 60$^o$ from face-on or less so that spatial variations can be seen with little overlap due to projection effects; and spatially resolved oxygen abundance data must be available. Galaxies which have compact infrared emission (NGC~1512, NGC~4826 [M64]), where muxbleed effects at 8.0~$\mu$m caused problems with interpreting the data (NGC~1097, NGC~1566 and NGC~4736 [M94]), or where very bright foreground stars caused problems with interpreting the IRAC data (NGC~3561) were excluded. NGC~3938 and NGC~4579 (M58), which were in the \cite{bendo2008a} sample, were not used here. The optical spectrum of NGC~3938 is very noisy, which makes the oxygen abundance measurements unreliable, while the optical spectrum of NGC~4579 is strongly affected by an AGN that causes problems when determining the oxygen abundances (Moustakas et al. 2010, in preparation). The final sample consists of 13 galaxies: NGC~0628 (M74), NGC~0925, NGC~2403, NGC~3031 (M81), NGC~3184, NGC~3351 (M95), NGC~4254 (M99), NGC~4321 (M100), NGC~4725, NGC~5055 (M63), NGC~5194 (M51), NGC~6946 and NGC~7793.
Observations and data processing information are provided by \citet{bendo2008a}. The extraction of flux densities for subregions in these galaxies was performed using the method described by \cite{bendo2008a}. First, the data were convolved with kernels developed by \cite{gordon2008a} to match their PSF to that of the 160~$\mu$m data. The entire analysis is performed on the data smoothed to the 160~$\mu$m image resolution. Next the stellar continuum emission was subtracted from the 8.0~$\mu$m and 24~$\mu$m bands using the relations
\begin{equation}
L_\nu\left(PAH~8~\mu m\right)=L_\nu\left(8~\mu m\right)
-0.232\times L_\nu\left(3.6~\mu m\right)
\end{equation}
\begin{equation}
L_\nu\left(24~\mu m\right)=L_\nu\left(24~\mu
m~obs\right)-0.032\times L_\nu\left(3.6~\mu m\right)
\end{equation}
given by \cite{helou2004a}. The images were then rebinned into 45\arcsec\ pixels, and pixels with low S/N and pixels strongly affected by artifacts or foreground stars were masked out. The processed images of NGC~3031 are shown in Figure~\ref{fig:M81} as an example. The flux densities in these 45\arcsec\ pixels are then used in the analysis. See \cite{bendo2008a} for additional details. The physical resolution ranges from 0.7~kpc to 3.6~kpc with a mean of $2.0\pm1.0$~kpc. The oxygen abundances have been calculated for each 45\arcsec\ region using the abundance gradients from Moustakas et al. (2010, in preparation), assuming that the gradient in each galaxy is azimuthally symmetric. They range from 8.29 to 8.93 with $\left<12+\log O/H\right> = 8.60\pm0.13$ averaging the estimates from the \cite{kobulnicky2004a} (hereafter KK04) and the \cite{pilyugin2005a} (hereafter PT05) estimators. Indeed these estimators are representative of the maximum and minimum oxygen abundance and as such bracket the actual abundance. A similar method was used for the same reasons by \cite{calzetti2007a}. Unless specified otherwise we use this average throughout the paper to estimate the metallicity of galaxy subregions.
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{figures/M81.ps}
\caption{Processed PAH 8, 24, 70, and 160~$\mu$m images of NGC~3031. The intensity of the color is proportional to the flux. Each pixel has an angular resolution of 45$\arcsec$. North is up and east is left.\label{fig:M81}}
\end{figure}
We use integrated galaxy data from the SINGS \citep{dale2007a} and LVL \citep{lee2008a,dale2009a} samples as well as the E08 sample as three additional data sets in this analysis. Although only half of the LVL galaxies currently have oxygen abundances available (Marble et al. 2010, submitted), these galaxies along with the SINGS and E08 galaxies allow us to probe a larger range of metallicities than the SINGS galaxies subregions. The SINGS sample contains galaxies with a metallicity from 8.02 to 8.99 with $\left<12+\log O/H\right> = 8.64\pm0.21$ averaging the KK04 and PT05 estimators \citep{calzetti2007a}. Finally, the E08 sample ``includes well-known starbursting or star-forming galaxies from the literature''. It probes a large range of oxygen abundances, from 7.31 to 8.85, using electron temperature measurements and O3N2 \citep{pettini2004a} for the most metal-rich galaxies. The sampling is not even and is mainly concentrated on lower-mid abundances: $\left<12+\log O/H\right> = 8.28\pm0.33$ \citep{engelbracht2008a}. Galaxies which were not detected in one or more bands are excluded from this study. In addition, all galaxies are selected with S/N$>$5. The final sample contains 57 of the 75 SINGS galaxies, 179 of the 258 LVL galaxies and 48 of the 66 E08 galaxies.
\section{Methods to derive the relations}
\label{sec:methods}
As a first step, we estimate the total infrared luminosity from the PAH 8~$\mu$m to 160~$\mu$m Spitzer bands. To do so we use equation 22 from \cite{draine2007a} -- a newer derivation than \cite{dale2002a} -- based on models that have been calibrated using results from {\em Spitzer}:
\begin{equation}
\label{eqn:draine}
\begin{array}{rcl}
L\left(TIR\right)&=&0.95 L\left(PAH~8\mu m\right)+1.15 L\left(24~\mu m\right)\\&&+L\left(70~\mu m\right)+L\left(160~\mu m\right),
\end{array}
\end{equation}
where $L=\nu L_\nu$. The difference with equation 4 of \cite{dale2002a} is minimal, $0.03\pm0.03$ dex (Figure~\ref{fig:metal-draine-dale} and section \ref{sec:discussion}).
To derive the relations to determine the total infrared luminosity from the combination of {\em Spitzer} bands we proceed in two ways. The first way is to perform a linear fit on the colors. For instance we calculate the coefficients $a$ and $b$ of a relation of the form: $\log\left(L(TIR)/L(24~\mu m)\right)=a+b\times\log\left(L(PAH~8~\mu m)/L(24~\mu m)\right)$. For easier use, we provide the relations under the form: $\log L(TIR)=\log L(24~\mu m)+a+b\times\log\left(L(PAH~8~\mu m)/L(24~\mu m)\right)$. We use distance independent quantities -- the ratio of two luminosities -- in order to avoid the well known correlation bias induced by luminosity versus luminosity relations. Indeed luminosity versus luminosity correlations are observed whether or not the data are actually correlated due to the multiplication of the flux density by the square of the distance. The drawback of such a relation is that the luminosities in the two {\em Spitzer} bands are not taken into account independently but are tied through the parameter $b$. To alleviate this limitation, we have performed fits in another distance independent way using the luminosity per unit area $\Sigma$. The relations obtained are of the form: $\log \Sigma(TIR)=a+b\times\log \Sigma(PAH~8~\mu m)+c\times\log \Sigma(24~\mu m)$ for instance. One drawback of this formulation is that it requires the target galaxy to be resolved -- which is often not the case for distant galaxies, especially in mid- and far- infrared bands -- and to know its distance.
In addition to deriving the relations, we also study the influence of the metallicity and the star formation intensity (by way of the total infrared luminosity per unit area). To do so we perform the aforementioned fit dividing the sample into 5 bins each containing the same number of data points in $12+\log O/H$ or $\Sigma(TIR)$. Evaluating the slope change as a function of the abundance allows us to study in detail its influence on the determination of the TIR luminosity. Finally, we also provide relations of the form $\log \Sigma(TIR)=a+b\times\log \Sigma(PAH~8~\mu m)+c\times\log \Sigma(24~\mu m)+d\times\left(12+\log O/H\right)$ to correct for the effect of the metallicity on the estimation of the total infrared luminosity per unit area.
\section{Results}
\label{sec:results}
The relations obtained through the methods described in the previous section are listed in Tables~\ref{tab:relations}, \ref{tab:relations-sigma} and \ref{tab:relations-sigma-oxy}. We present the results hereafter.
\begin{deluxetable*}{cccccc}
\tablecolumns{6} \tablewidth{0pc} \tablecaption{TIR estimations from luminosities\label{tab:relations}}
\tablehead{\colhead{y}&\colhead{$x_1$}&\colhead{$x_2$}&\colhead{a}&\colhead{b}&\colhead{$\sigma$}}
\startdata
$L\left(TIR_{SUB}\right)$&$L\left(24~\mu m\right)$&$L\left(PAH~8\mu m\right)/L\left(24~\mu m\right)$&$1.012\pm0.009$&$0.510\pm0.021$&$0.096$\\
$L\left(TIR_{SINGS}\right)$&$L\left(24~\mu m\right)$&$L\left(PAH~8\mu m\right)/L\left(24~\mu m\right)$&$0.997\pm0.023$&$0.494\pm0.071$&$0.142$\\
$L\left(TIR_{LVL}\right)$&$L\left(24~\mu m\right)$&$L\left(PAH~8\mu m\right)/L\left(24~\mu m\right)$&$1.141\pm0.010$&$0.120\pm0.024$&$0.132$\\
$L\left(TIR_{E08}\right)$&$L\left(24~\mu m\right)$&$L\left(PAH~8\mu m\right)/L\left(24~\mu m\right)$&$0.887\pm0.034$&$0.400\pm0.074$&$0.185$\\
$L\left(TIR_{SUB}\right)$&\nodata&$L\left(PAH~8\mu m\right)$&$4.580\pm0.108$&$0.888\pm0.003$&$0.080$\\
$L\left(TIR_{SINGS}\right)$&\nodata&$L\left(PAH~8\mu m\right)$&$6.696\pm0.532$&$0.836\pm0.015$&$0.113$\\
$L\left(TIR_{LVL}\right)$&\nodata&$L\left(PAH~8\mu m\right)$&$8.519\pm0.504$&$0.782\pm0.015$&$0.218$\\
$L\left(TIR_{E08}\right)$&\nodata&$L\left(PAH~8\mu m\right)$&$7.075\pm0.790$&$0.829\pm0.022$&$0.182$\\
$L\left(TIR_{SUB}\right)$&\nodata&$L\left(24~\mu m\right)$&$4.961\pm0.109$&$0.887\pm0.003$&$0.082$\\
$L\left(TIR_{SINGS}\right)$&\nodata&$L\left(24~\mu m\right)$&$3.810\pm0.975$&$0.923\pm0.028$&$0.194$\\
$L\left(TIR_{LVL}\right)$&\nodata&$L\left(24~\mu m\right)$&$1.838\pm0.326$&$0.979\pm0.010$&$0.157$\\
$L\left(TIR_{E08}\right)$&\nodata&$L\left(24~\mu m\right)$&$3.487\pm1.022$&$0.924\pm0.029$&$0.212$\\\hline
$L\left(TIR_{SUB}\right)$&$L\left(70~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(70~\mu m\right)$&$0.662\pm0.003$&$0.423\pm0.010$&$0.059$\\
$L\left(TIR_{SINGS}\right)$&$L\left(70~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(70~\mu m\right)$&$0.583\pm0.017$&$0.300\pm0.032$&$0.063$\\
$L\left(TIR_{LVL}\right)$&$L\left(70~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(70~\mu m\right)$&$0.486\pm0.010$&$0.158\pm0.011$&$0.067$\\
$L\left(TIR_{E08}\right)$&$L\left(70~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(70~\mu m\right)$&$0.478\pm0.018$&$0.198\pm0.023$&$0.055$\\
$L\left(TIR_{SUB}\right)$&$L\left(70~\mu m\right)$&$L\left(24~\mu m\right)/L\left(70~\mu m\right)$&$0.789\pm0.013$&$0.351\pm0.020$&$0.083$\\
$L\left(TIR_{SINGS}\right)$&$L\left(70~\mu m\right)$&$L\left(24~\mu m\right)/L\left(70~\mu m\right)$&$0.497\pm0.050$&$0.078\pm0.075$&$0.101$\\
$L\left(TIR_{LVL}\right)$&$L\left(70~\mu m\right)$&$L\left(24~\mu m\right)/L\left(70~\mu m\right)$&$0.583\pm0.034$&$0.287\pm0.043$&$0.089$\\
$L\left(TIR_{E08}\right)$&$L\left(70~\mu m\right)$&$L\left(24~\mu m\right)/L\left(70~\mu m\right)$&$0.415\pm0.020$&$0.183\pm0.039$&$0.073$\\
$L\left(TIR_{SUB}\right)$&\nodata&$L\left(70~\mu m\right)$&$1.210\pm0.142$&$0.981\pm0.004$&$0.095$\\
$L\left(TIR_{SINGS}\right)$&\nodata&$L\left(70~\mu m\right)$&$-1.057\pm0.580$&$1.042\pm0.016$&$0.101$\\
$L\left(TIR_{LVL}\right)$&\nodata&$L\left(70~\mu m\right)$&$-1.210\pm0.222$&$1.045\pm0.006$&$0.076$\\
$L\left(TIR_{E08}\right)$&\nodata&$L\left(70~\mu m\right)$&$-1.098\pm0.409$&$1.040\pm0.011$&$0.078$\\\hline
$L\left(TIR_{SUB}\right)$&$L\left(160~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(160~\mu m\right)$&$0.454\pm0.005$&$0.283\pm0.010$&$0.050$\\
$L\left(TIR_{SINGS}\right)$&$L\left(160~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(160~\mu m\right)$&$0.599\pm0.037$&$0.370\pm0.069$&$0.125$\\
$L\left(TIR_{LVL}\right)$&$L\left(160~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(160~\mu m\right)$&$0.458\pm0.019$&$0.027\pm0.023$&$0.118$\\
$L\left(TIR_{E08}\right)$&$L\left(160~\mu m\right)$&$L\left(PAH~8~\mu m\right)/L\left(160~\mu m\right)$&$0.831\pm0.038$&$0.310\pm0.083$&$0.182$\\
$L\left(TIR_{SUB}\right)$&$L\left(160~\mu m\right)$&$L\left(24~\mu m\right)/L\left(160~\mu m\right)$&$0.616\pm0.005$&$0.332\pm0.006$&$0.031$\\
$L\left(TIR_{SINGS}\right)$&$L\left(160~\mu m\right)$&$L\left(24~\mu m\right)/L\left(160~\mu m\right)$&$0.714\pm0.014$&$0.436\pm0.019$&$0.047$\\
$L\left(TIR_{LVL}\right)$&$L\left(160~\mu m\right)$&$L\left(24~\mu m\right)/L\left(160~\mu m\right)$&$0.751\pm0.013$&$0.447\pm0.017$&$0.055$\\
$L\left(TIR_{E08}\right)$&$L\left(160~\mu m\right)$&$L\left(24~\mu m\right)/L\left(160~\mu m\right)$&$0.748\pm0.010$&$0.465\pm0.024$&$0.070$\\
$L\left(TIR_{SUB}\right)$&$L\left(160~\mu m\right)$&$L\left(70~\mu m\right)/L\left(160~\mu m\right)$&$0.408\pm0.002$&$0.390\pm0.006$&$0.030$\\
$L\left(TIR_{SINGS}\right)$&$L\left(160~\mu m\right)$&$L\left(70~\mu m\right)/L\left(160~\mu m\right)$&$0.437\pm0.007$&$0.623\pm0.030$&$0.052$\\
$L\left(TIR_{LVL}\right)$&$L\left(160~\mu m\right)$&$L\left(70~\mu m\right)/L\left(160~\mu m\right)$&$0.395\pm0.003$&$0.552\pm0.015$&$0.040$\\
$L\left(TIR_{E08}\right)$&$L\left(160~\mu m\right)$&$L\left(70~\mu m\right)/L\left(160~\mu m\right)$&$0.393\pm0.025$&$0.854\pm0.054$&$0.054$\\
$L\left(TIR_{SUB}\right)$&\nodata&$L\left(160~\mu m\right)$&$-1.278\pm0.093$&$1.047\pm0.003$&$0.057$\\
$L\left(TIR_{SINGS}\right)$&\nodata&$L\left(160~\mu m\right)$&$1.683\pm0.851$&$0.965\pm0.024$&$0.159$\\
$L\left(TIR_{LVL}\right)$&\nodata&$L\left(160~\mu m\right)$&$1.642\pm0.263$&$0.965\pm0.008$&$0.114$\\
$L\left(TIR_{E08}\right)$&\nodata&$L\left(160~\mu m\right)$&$1.747\pm1.004$&$0.971\pm0.028$&$0.215$
\enddata \tablecomments{Coefficient for the fit $\log y=\log x_1+a+b\times \log x_2$. $\sigma$ is the standard deviation of the data points around the best fit line. The subscripts refer to the sample.}
\end{deluxetable*}
\begin{deluxetable*}{ccccccc}
\tablecolumns{7} \tablewidth{0pc} \tablecaption{TIR estimations from luminosities per unit area\label{tab:relations-sigma}}
\tablehead{\colhead{y}&\colhead{$x_1$}&\colhead{$x_2$}&\colhead{a}&\colhead{b}&\colhead{c}&\colhead{$\sigma$}}
\startdata
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 5.692\pm 0.124$&$ 0.433\pm 0.014$&$ 0.425\pm 0.014$&$ 0.062$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 2.085\pm 0.380$&$ 0.525\pm 0.067$&$ 0.442\pm 0.070$&$ 0.133$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 5.408\pm 0.416$&$ 0.174\pm 0.020$&$ 0.693\pm 0.026$&$ 0.105$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 6.795\pm 0.780$&$ 0.257\pm 0.053$&$ 0.566\pm 0.050$&$ 0.122$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $ 5.982\pm 0.170$&$ 0.845\pm 0.005$&\nodata&$ 0.086$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $ 2.766\pm 0.476$&$ 0.944\pm 0.014$&\nodata&$ 0.175$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $13.085\pm 0.663$&$ 0.629\pm 0.021$&\nodata&$ 0.233$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $ 7.771\pm 1.515$&$ 0.799\pm 0.045$&\nodata&$ 0.242$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $ 6.418\pm 0.169$&$ 0.842\pm 0.005$&\nodata&$ 0.087$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $ 1.714\pm 0.540$&$ 0.981\pm 0.016$&\nodata&$ 0.192$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $ 4.220\pm 0.466$&$ 0.904\pm 0.014$&\nodata&$ 0.127$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $ 8.083\pm 0.898$&$ 0.783\pm 0.027$&\nodata&$ 0.151$ \\\hline
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 3.994\pm 0.070$&$ 0.425\pm 0.006$&$ 0.476\pm 0.006$&$ 0.033$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 0.627\pm 0.207$&$ 0.302\pm 0.034$&$ 0.697\pm 0.037$&$ 0.063$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 1.978\pm 0.295$&$ 0.178\pm 0.011$&$ 0.777\pm 0.016$&$ 0.063$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 0.085\pm 0.433$&$ 0.194\pm 0.023$&$ 0.817\pm 0.028$&$ 0.054$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 4.593\pm 0.137$&$ 0.405\pm 0.015$&$ 0.483\pm 0.016$&$ 0.063$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $-0.018\pm 0.319$&$ 0.069\pm 0.076$&$ 0.946\pm 0.078$&$ 0.100$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 1.078\pm 0.400$&$ 0.310\pm 0.045$&$ 0.675\pm 0.049$&$ 0.089$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 1.260\pm 0.698$&$ 0.216\pm 0.047$&$ 0.760\pm 0.061$&$ 0.072$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $ 3.847\pm 0.176$&$ 0.902\pm 0.005$&\nodata&$ 0.082$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $-0.118\pm 0.299$&$ 1.017\pm 0.009$&\nodata&$ 0.101$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $ 0.042\pm 0.417$&$ 1.010\pm 0.013$&\nodata&$ 0.100$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $-0.617\pm 0.673$&$ 1.028\pm 0.020$&\nodata&$ 0.087$ \\\hline
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-1.794\pm 0.180$&$ 0.156\pm 0.014$&$ 0.908\pm 0.018$&$ 0.046$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 1.474\pm 0.365$&$ 0.408\pm 0.068$&$ 0.567\pm 0.071$&$ 0.119$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 0.957\pm 0.632$&$ 0.040\pm 0.028$&$ 0.946\pm 0.042$&$ 0.118$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 0.396\pm 1.709$&$ 0.300\pm 0.092$&$ 0.713\pm 0.122$&$ 0.182$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-0.641\pm 0.094$&$ 0.283\pm 0.006$&$ 0.753\pm 0.008$&$ 0.028$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 0.823\pm 0.134$&$ 0.435\pm 0.019$&$ 0.562\pm 0.019$&$ 0.046$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 1.177\pm 0.228$&$ 0.452\pm 0.017$&$ 0.535\pm 0.019$&$ 0.055$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 1.955\pm 0.602$&$ 0.482\pm 0.025$&$ 0.482\pm 0.035$&$ 0.067$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-1.357\pm 0.070$&$ 0.325\pm 0.006$&$ 0.727\pm 0.007$&$ 0.024$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 0.051\pm 0.153$&$ 0.634\pm 0.030$&$ 0.377\pm 0.030$&$ 0.051$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-0.798\pm 0.145$&$ 0.571\pm 0.013$&$ 0.465\pm 0.013$&$ 0.034$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-1.570\pm 0.635$&$ 0.843\pm 0.050$&$ 0.215\pm 0.055$&$ 0.075$ \\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $-3.288\pm 0.127$&$ 1.107\pm 0.004$&\nodata&$ 0.049$ \\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $ 0.791\pm 0.438$&$ 0.989\pm 0.013$&\nodata&$ 0.152$ \\
$\Sigma\left(TIR_{LVL}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $ 0.457\pm 0.496$&$ 0.999\pm 0.015$&\nodata&$ 0.120$ \\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $-2.001\pm 1.697$&$ 1.081\pm 0.050$&\nodata&$ 0.203$
\enddata \tablecomments{Coefficient for the fit $\log y=a+b\times \log x_1+c\times \log x_2$. $\sigma$ is the standard deviation of the data points around the best fit line. The subscripts refer to the sample.}
\end{deluxetable*}
\begin{deluxetable*}{cccccccc}
\tablecolumns{8} \tablewidth{0pc} \tablecaption{TIR estimations from luminosities per unit area and the oxygen abundance\label{tab:relations-sigma-oxy}}
\tablehead{\colhead{y}&\colhead{$x_1$}&\colhead{$x_2$}&\colhead{a}&\colhead{b}&\colhead{c}&\colhead{d}&\colhead{$\sigma$}}
\startdata
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 5.667\pm 0.138$&$ 0.431\pm 0.015$&$ 0.426\pm 0.014$&$ 0.009\pm 0.022$&$ 0.062$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 3.996\pm 0.909$&$ 0.637\pm 0.081$&$ 0.339\pm 0.081$&$-0.256\pm 0.112$&$ 0.127$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8\mu m\right)$&$\Sigma\left(24~\mu m\right)$& $ 7.214\pm 0.790$&$ 0.336\pm 0.066$&$ 0.508\pm 0.057$&$-0.134\pm 0.071$&$ 0.118$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $ 6.122\pm 0.189$&$ 0.854\pm 0.007$&\nodata&$-0.051\pm 0.030$&$ 0.086$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $ 6.283\pm 0.828$&$ 0.972\pm 0.013$&\nodata&$-0.514\pm 0.106$&$ 0.146$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8\mu m\right)$&\nodata& $ 8.893\pm 1.271$&$ 0.882\pm 0.041$&\nodata&$-0.471\pm 0.099$&$ 0.197$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $ 5.711\pm 0.186$&$ 0.805\pm 0.007$&\nodata&$ 0.222\pm 0.028$&$ 0.084$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $-0.147\pm 1.050$&$ 0.970\pm 0.017$&\nodata&$ 0.260\pm 0.127$&$ 0.185$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(24~\mu m\right)$&\nodata& $ 7.522\pm 0.982$&$ 0.777\pm 0.027$&\nodata&$ 0.092\pm 0.068$&$ 0.148$\\\hline
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 3.606\pm 0.075$&$ 0.393\pm 0.006$&$ 0.488\pm 0.006$&$ 0.122\pm 0.011$&$ 0.031$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 0.045\pm 0.541$&$ 0.266\pm 0.046$&$ 0.732\pm 0.048$&$ 0.069\pm 0.060$&$ 0.063$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 0.690\pm 0.461$&$ 0.241\pm 0.028$&$ 0.773\pm 0.031$&$-0.082\pm 0.030$&$ 0.050$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 3.497\pm 0.138$&$ 0.326\pm 0.014$&$ 0.513\pm 0.014$&$ 0.308\pm 0.018$&$ 0.056$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $-1.941\pm 0.505$&$ 0.054\pm 0.065$&$ 0.949\pm 0.067$&$ 0.268\pm 0.059$&$ 0.086$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(70~\mu m\right)$& $ 0.834\pm 0.684$&$ 0.215\pm 0.045$&$ 0.755\pm 0.058$&$ 0.078\pm 0.031$&$ 0.067$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $ 2.450\pm 0.161$&$ 0.827\pm 0.006$&\nodata&$ 0.453\pm 0.022$&$ 0.069$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $-2.037\pm 0.491$&$ 1.004\pm 0.008$&\nodata&$ 0.271\pm 0.059$&$ 0.086$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(70~\mu m\right)$&\nodata& $-1.042\pm 0.682$&$ 1.021\pm 0.019$&\nodata&$ 0.079\pm 0.038$&$ 0.083$\\\hline
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-1.566\pm 0.170$&$ 0.164\pm 0.013$&$ 0.939\pm 0.017$&$-0.178\pm 0.015$&$ 0.043$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$&$ 4.522\pm 0.537$&$ 0.476\pm 0.052$&$ 0.519\pm 0.054$&$-0.430\pm 0.065$&$ 0.088$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(PAH~8~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 2.514\pm 1.504$&$ 0.449\pm 0.084$&$ 0.590\pm 0.105$&$-0.357\pm 0.079$&$ 0.151$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-0.626\pm 0.088$&$ 0.274\pm 0.006$&$ 0.790\pm 0.008$&$-0.115\pm 0.009$&$ 0.026$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 1.463\pm 0.250$&$ 0.419\pm 0.018$&$ 0.582\pm 0.019$&$-0.094\pm 0.032$&$ 0.043$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(24~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $ 2.020\pm 0.590$&$ 0.472\pm 0.025$&$ 0.504\pm 0.037$&$-0.055\pm 0.032$&$ 0.065$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-1.350\pm 0.070$&$ 0.328\pm 0.006$&$ 0.721\pm 0.008$&$ 0.011\pm 0.009$&$ 0.024$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-0.385\pm 0.329$&$ 0.658\pm 0.034$&$ 0.351\pm 0.034$&$ 0.060\pm 0.040$&$ 0.050$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(70~\mu m\right)$&$\Sigma\left(160~\mu m\right)$& $-1.632\pm 0.645$&$ 0.856\pm 0.054$&$ 0.198\pm 0.061$&$ 0.026\pm 0.038$&$ 0.075$\\
$\Sigma\left(TIR_{SUB}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $-3.199\pm 0.117$&$ 1.151\pm 0.005$&\nodata&$-0.180\pm 0.013$&$ 0.045$\\
$\Sigma\left(TIR_{SINGS}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $ 2.848\pm 0.792$&$ 1.005\pm 0.013$&\nodata&$-0.301\pm 0.099$&$ 0.141$\\
$\Sigma\left(TIR_{E08}\right)$&$\Sigma\left(160~\mu m\right)$&\nodata& $-1.507\pm 1.659$&$ 1.113\pm 0.051$&\nodata&$-0.189\pm 0.092$&$ 0.194$
\enddata \tablecomments{Coefficient for the fit $\log y=a+b\times \log x_1+c\times \log x_2+d\times\left(12+\log O/H\right)$. $\sigma$ is the standard deviation of the data points around the best fit line. The subscripts refer to the sample.}
\end{deluxetable*}
\subsection{TIR estimates from the combination of the 8 and 24~$\mu$m bands}
\label{ssec:8-24}
As PAH and hot dust are significant contributors to the total infrared luminosity in metal-rich galaxies \citep{draine2007a}, the 8~$\mu$m and 24~$\mu$m observations permit an accurate determination of the TIR luminosities. Furthermore, as their spatial resolution is significantly better than the one of the 70~$\mu$m and the 160~$\mu$m bands, resolving local galaxies is possible.
\subsubsection{Metallicity effects\label{sssec:metallicity}}
It is well known that the presence of dust carriers influencing the 8~$\mu$m band emission is directly affected by the metallicity \citep{engelbracht2005a,rosenberg2006a,wu2006a,madden2006a,jackson2006a,draine2007a}. The large number of galaxy subregions in the sample studied here allows us to take into account accurately this parameter in the determination of the total infrared emission from the PAH 8~$\mu$m and 24~$\mu$m bands. In addition, the integrated SINGS and the E08 galaxies have oxygen abundances that are averaged over all the galaxy.
The relations between the $L(24~\mu m)/L(TIR)$ and $L(PAH~8\mu m)/L(24~\mu m)$ colors for all the samples are presented in Figure~\ref{fig:PAH-24-TIR}. In order to clearly show the difference in the behavior of integrated galaxies and galaxy subregions, we plot them together. For comparison we also show the relations for NGC~5194, NGC~3031, and M33 determined by \cite{calzetti2005a}, \cite{perez2006a}, and \cite{thilker2007a} respectively. The NGC~5194 relation by \cite{calzetti2005a} is offset from the other relations and also from the observations as it has a much higher $L(24~\mu m)/L(TIR)$ ratio. The reason is that the NGC~5194 flux densities were measured after removing a local background, whereas for NGC~3031 and other data, flux densities were measured after removing only a global background. The difference therefore could be an indication of the relative contribution of the diffuse large scale infrared emission compared to point-like sources. The parameters of the best fitting relations are given in Table~\ref{tab:relations}, separately for each sample and the galaxy subregions. In addition, we provide relations between the total infrared luminosity per unit area as a combination of the luminosity per unit area in the PAH 8~$\mu$m and 24~$\mu$m bands in Table~\ref{tab:relations-sigma} and we also take into account the oxygen abundance in Table~\ref{tab:relations-sigma-oxy}.
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{figures/pah_ov_24_vs_24_ov_TIR.ps}
\caption{Ratio of the 24~$\mu$m to the total infrared emission -- as calculated using equation \ref{eqn:draine} -- versus the ratio of the PAH 8~$\mu$m to 24~$\mu$m emission. The ``x'' symbol represents the data for the subregions in the sample of face-on galaxies. The color of the ``x'' symbols represent the oxygen abundances as given by Moustakas et al. (2010, in preparation). The filled circles represent integrated galaxy data for the SINGS sample, the empty circles represent integrated galaxy data for the LVL sample, and the filled triangles represent integrated galaxy data for the E08 sample. The solid lines represent various relations: the yellow one for the relation found in NGC~5194 by \cite{calzetti2005a}, the magenta one for the relation found in NGC~3031 by \cite{perez2006a}, the purple one for the relation found in M~33 by \cite{thilker2007a}, the blue one for LVL galaxies, the red one for SINGS galaxies, the green one for E08 galaxies, and the black one for galaxy subregions. For the relations published in the literature, the $L(PAH~8~\mu m)/L(24~\mu m)$ range over which they were derived are displayed with a solid line, and a dashed line out of these bounds (P. P\'erez-Gonz\'alez and D. Thilker, private communications).\label{fig:PAH-24-TIR}}
\end{figure}
First of all in Figure~\ref{fig:PAH-24-TIR} we notice that despite the scatter, different relations can be seen for subregions with different abundances. Indeed, we see that the slope is clearly shallower for the lower oxygen abundance subregions which describe a lower branch in the diagram. Interestingly, the low metallicity trend is followed by a part of the SINGS and most of the LVL samples. Even though the LVL sample spans a significant range of metallicities, it is chiefly constituted of dwarf galaxies. As a consequence, LVL galaxies are statistically more metal-poor than SINGS ones and therefore exhibit for a significant number of them little or no PAH emission. So in this particular case the data follow a trend similar to the one for the lowest oxygen abundance galaxy subregions (note that $12+\log O/H>8.29$ for the subregions). For most of the LVL galaxies, the $L(PAH~8\mu m)/L(24~\mu m)$ ratio is primarily dependent on the PAH 8~$\mu$m luminosity which itself is dependent on the metallicity. That is, the $L(24~\mu m)/L(TIR)$ ratio is hardly affected by a change of the $L(PAH~8\mu m)/L(24~\mu m)$ color. Quantitatively, we see in Table~\ref{tab:relations} that the slope is indeed much shallower for SINGS and especially LVL galaxies, which show little dependence on $L(PAH~8\mu m)/L(24~\mu m)$ as could be expected. The subregions sample has the highest mean slope due to being constituted of spiral galaxies only. The behavior of the E08 sample shows that the metallicity is not the only parameter driving the correlation between $L(24~\mu m)/L(TIR)$ and $L(PAH~8\mu m)/L(24~\mu m)$. Most of the sample seems to follow the trend set by higher metallicity subregions describing the upper branch in the diagram. As this sample has been specifically constituted with star-forming and starburst galaxies, the chief parameter that drives the variation could be the intensity of star formation. We will probe its effect in section \ref{sssec:SFR-intensity}.
In Table~\ref{tab:relations-sigma-oxy} we estimate the total infrared luminosity per unit area as a function of the luminosity per unit area in the PAH 8~$\mu$m and 24~$\mu$m bands as well as the oxygen abundance. We see that combining the PAH 8~$\mu$m and 24~$\mu$m bands, the contribution of the metallicity is smaller than when estimating from the PAH 8~$\mu$m band only. The reason is that the combination of these two bands is an indirect measure of the metallicity which permits to obtain a more accurate measurement than when no information on the metallicity is available. Indeed, taking into account the oxygen abundance, averaging over the subregions, SINGS and E08 samples, the scatter is reduced by 0.003 dex when combining the PAH 8~$\mu$m and 24~$\mu$m bands but 0.025 dex when estimating the total infrared emission from the one in the PAH 8~$\mu$m band only.
In order to study how the fit parameters evolve as a function of the oxygen abundance, in Figure~\ref{fig:fit-subregions}, we plot the best fit parameters for different bins of oxygen abundances following the method described earlier in section \ref{sec:methods}.
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{figures/origin.ps}
\includegraphics[width=\columnwidth]{figures/slope.ps}
\caption{Top: y-intercept $a$ (left) and slope $b$ (right) of the linear best fit: $\log L(TIR) = \log L(24~\mu m)+a+b\times\log \left(L(8~\mu m)/L(24~\mu m)\right)$ versus the oxygen abundance. The data points are calculated dividing the sample in 5 bins each containing the same number of data points. Here only subregions are taken into account.\label{fig:fit-subregions}}
\end{figure*}
Three different trends can be seen with increasing oxygen abundance: 1. at low oxygen abundance, the slope is shallow, 2. at a higher oxygen abundance, the slope is steeper, this shows the greater dependence on the PAH emission, 3. finally at the highest oxygen abundance, the slope decreases slightly. Considering that $L\left(24~\mu m\right)/L\left(TIR\right)$ is a function of the intensity of the illuminating radiation field \citep{dale2001a,draine2007a}, the observed trend reflects the results obtained by other authors \citep[e. g.][]{calzetti2005a,bendo2006a,bendo2008a} that the relative strength of PAH emission decreases as the strength of the illuminating radiation field increases. This could be because of PAH destruction or the inhibition of PAH emission at 8~$\mu$m in regions with intense or hard radiation fields. The ratio $L(PAH~8\mu m)/L(24~\mu m)$ may drop off more gradually in high metallicity galaxies because of increased dust extinction. The photons that most strongly affect PAHs in those systems can only travel a short distance in the ISM, so the drop off in $L(PAH~8\mu m)/L(24~\mu m)$ with increasing radiation field intensity is more gradual. In low metallicity systems, however, the photons that affect PAH emission can propagate much further through the ISM, so $L(PAH~8\mu m)/L(24~\mu m)$ decreases much more quickly as the intensity of the radiation field (and $L(24~\mu m)/L(TIR)$) increases.
\subsubsection{Star formation intensity effect\label{sssec:SFR-intensity}}
As mentioned in section \ref{sssec:metallicity}, the metallicity cannot explain that some low-metallicity galaxies follow the trend set by rather high-metallicity subregions. This hints that another parameter is playing a role in driving the correlation.
In Figure~\ref{fig:PAH-24-TIR-SFR} we plot the $\log \left(L(PAH~8\mu m)/L(24~\mu m)\right)$ ratio color-coded by $\Sigma\left(TIR\right)$, the total infrared luminosity per unit area, which is a proxy to the star formation intensity at high metallicities. In Figure~\ref{fig:fit-subregions-SFR} the fit parameters for the subregions are plotted versus $\Sigma\left(TIR\right)$.
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{figures/pah_ov_24_vs_24_ov_TIR_SFR_intensity.ps}
\caption{The axis and the shapes of the symbols are the same as in Figure~\ref{fig:PAH-24-TIR}. The symbols are color-coded as a function of $\log \Sigma\left(TIR\right)$ (W\ kpc$^{-2}$).\label{fig:PAH-24-TIR-SFR}}
\end{figure}
We see that the two branches in Figure~\ref{fig:PAH-24-TIR-SFR} are separated by $\Sigma\left(TIR\right)$. Indeed, most galaxies and galaxy subregions that have a low $\Sigma\left(TIR\right)$ tend to be on the lower branch, the $L(24~\mu m)/L(TIR)$ ratio is mostly independent of the $L(PAH~8\mu m)/L(24~\mu m)$ ratio. Conversely, whole galaxies and galaxy subregions that have a higher $\Sigma\left(TIR\right)$ are on the upper branch. The $L(24~\mu m)/L(TIR)$ ratio becomes dependent on the $L(PAH~8\mu m)/L(24~\mu m)$ ratio on the upper branch compared to the lower one. That is, for higher $\Sigma\left(TIR\right)$, $L(24)$ increases as the infrared spectral energy distribution is hotter. This dichotomy could be due to the ``loss'' of infrared in low metallicity objects. However, most importantly we note that for galaxies that have a $L(PAH~8\mu m)/L(24~\mu m)$ ratio typically lower than $-0.5$~dex, the $L(24~\mu m)/L(TIR)$ ratio depends almost exclusively on $\Sigma\left(TIR\right)$. Galaxies that have a $L(PAH~8\mu m)/L(24~\mu m)$ ratio typically lower than $-0.5$~dex happen to have a low metallicity and as such are expected to be predominantly on the lower branch. However, the weakness or even the lack of PAH emission in these galaxies implies that the $L(24~\mu m)/L(TIR)$ ratio is very sensitive to $\Sigma\left(TIR\right)$ at low metallicities.
Figure~\ref{fig:fit-subregions-SFR}, made using the same method as for Figure~\ref{fig:fit-subregions}, confirms quantitatively the results described above. At low $\Sigma\left(TIR\right)$ the slope is shallow but increases to reach a local maximum around $\log \Sigma\left(TIR\right)\simeq34.3$ before decreasing slightly. We note that unsurprisingly the behavior is globally similar to the one determined for the oxygen abundance. Indeed, lower metallicity galaxies are more transparent as they contain less dust and therefore the oxygen abundance and $\Sigma\left(TIR\right)$ are partly correlated.
\begin{figure*}[!ht]
\includegraphics[width=\columnwidth]{figures/origin_intensity.ps}
\includegraphics[width=\columnwidth]{figures/slope_intensity.ps}
\caption{y-intercept $a$ (left) and slope $b$ (right) of the linear best fit: $\log L(TIR) = \log L(24~\mu m)+a+b\log\left( L(8~\mu m)/L(24~\mu m)\right)$ versus $\Sigma\left(TIR\right)$.\label{fig:fit-subregions-SFR}}
\end{figure*}
The observed trend also accounts for the relation determined by \cite{calzetti2005a} (yellow line in Figure~\ref{fig:PAH-24-TIR-SFR}) since those authors have fitted star-forming regions in M51 removing the contribution to TIR from the dust heating evolved populations, maximizing the effect of the star formation intensity.
\subsubsection{Quality of the correlations}
An important point to note is that the scatter around the best fit tends to be larger for whole galaxies compared to subregions. This is likely due to the diversity in the samples. It clearly shows that those samples present an important variety of infrared properties \citep[e.g.][]{dale2009a}.
We also estimate the TIR luminosity per unit area from the 24~$\mu$m band, that has been shown to be an accurate SFR estimator by \cite{rieke2009a}. This is particularly useful in case no other mid- or far-infrared wavelength is available. The scatter around the best fit is 0.087~dex, yielding an estimate of the TIR luminosity with an accuracy slightly over 20\%. Unsurprisingly, both luminosities per unit area are an increasing function of the oxygen abundance (Pearson correlation coefficient $r=0.45$ for the 24~$\mu$m emission and $r=0.53$ for the total infrared emission). The slope of the correlation -- 0.842, shallower than what was found by \cite{rieke2009a} -- shows that the 24~$\mu$m by itself is not a linear tracer of the TIR luminosity. Indeed, at higher luminosities, an increasing fraction of the infrared emission can be accounted for by the hot dust traced by the 24~$\mu$m band.
\subsubsection{Summary}
It appears that both the metallicity and $\Sigma\left(TIR\right)$ play a role in the determination of the TIR emission from the 8~$\mu$m and 24~$\mu$m bands. While those two parameters are intertwined (a low metallicity galaxy tends to be more transparent and hence have a lower TIR luminosity), they are the dominant parameters in different regimes.
For $L(PAH~8\mu m)/L(24~\mu m)<-0.5$~dex, $L(PAH~8\mu m)/L(24~\mu m)$ depends only on $\Sigma\left(TIR\right)$.
For $L(PAH~8\mu m)/L(24~\mu m)>-0.5$~dex two branches can be seen. For $\Sigma\left(TIR\right)\lesssim34.3$~dex, $L(24~\mu m)/L(TIR)$ is very weakly dependent on $L(PAH~8\mu m)/L(24~\mu m)$, higher metallicity galaxies and subregions tend to have a higher $L(PAH~8\mu m)/L(24~\mu m)$ ratio this being due to the increasingly stronger PAH emission with the metallicity up to a ratio of $\sim0.7$~dex. For higher metallicity galaxies and subregions, an upper branch is described, the $L(24~\mu m)/L(TIR)$ ratio increasing with $\Sigma\left(TIR\right)$ while the $L(PAH~8\mu m)/L(24~\mu m)$ ratio decreases, probably due to a combination of both the heating of the very small grains which strongly increases the 24~$\mu$m luminosity and possible a destruction of PAH in intense radiation fields. In other words, the main parameters are the metallicity which drives the strength of the PAH depending on the presence of dust carriers, and the star formation intensity which drives the temperature of the dust controlling the luminosity at 24~$\mu$m.
\subsection{TIR estimates that include the 70~$\mu$m band}
Here we demonstrate how to estimate the TIR luminosity combining the 70~$\mu$m and 24~$\mu$m bands. The fits are presented in Figure~\ref{fig:70} and the numerical relations in Tables~\ref{tab:relations}, \ref{tab:relations-sigma} and \ref{tab:relations-sigma-oxy}.
\begin{figure*}[!htbp]
\includegraphics[width=\columnwidth]{figures/24_ov_70_vs_70_ov_TIR.ps}
\includegraphics[width=\columnwidth]{figures/24_ov_70_vs_70_ov_TIR_SFR_intensity.ps}
\caption{$\log \left(L(70~\mu m)/L(TIR)\right)$ versus $\log \left(L(24~\mu m)/L(70~\mu m)\right)$. The symbols and the color-coding of the lines are the same as in Figure~\ref{fig:PAH-24-TIR}. The color codes the oxygen abundance (left) and $\Sigma\left(TIR\right)$ (right). \label{fig:70}}
\end{figure*}
We see that $L(70~\mu m)/L(TIR)$ depends significantly on the metallicity: low metallicity regions tend to have a high $L(70~\mu m)/L(TIR)$ ratio while higher metallicity regions tend to have a lower $L(70~\mu m)/L(TIR)$ ratio. Indeed, low metallicity galaxies tend to have a bluer $L(70~\mu m)/L(160~\mu m)$ color. This may demonstrate that the color temperature of the ``cool'' ($<$30~K) dust increases as metallicity decreases. This effect has by shown by \cite{helou1986a,engelbracht2008a} for instance. This could occur because the lower metallicity results in lower dust extinction, so a greater fraction of the dust that is present in low-metallicity systems is heated by strong radiation fields. In contrast, dust extinction in high metallicity systems would tend to make the radiation field appear less intense through most of the ISM. Consequently, most of the dust in high metallicity systems would be heated by a weaker radiation field and would appear cooler than the dust in low metallicity systems. The dependence of the fit parameters on $\Sigma\left(TIR\right)$ is weak. A similar result has been found by Calzetti et al. (2010, submitted).
In the relation between $L(24~\mu m)/L(70~\mu m)$ and $L(70~\mu m)/L(TIR)$, the integrated galaxy colors exhibit a shallower slope. Interestingly, about 31\% of LVL and E08 galaxies have $L(70~\mu m)/L(TIR)$ values greater than $-0.3$~dex but only 12\% of the SINGS galaxies and a mere 0.1\% of the galaxy subregions have $L(70~\mu m)/L(TIR)$ values that are this high.
The TIR luminosity per unit area is well correlated with the 70~$\mu$m one, and the scatter around the best fit is slightly smaller than when using the 24~$\mu$m band only (0.082 vs 0.087). This is due to the fact that large dust grains appear warmer in low metallicity galaxies. The slope -- 0.902 -- slightly under 1 makes the 70~$\mu$m is sub-linear tracer of the TIR luminosity which traces the star formation rate for galaxy subregions but nearly linear for galaxy samples. See Calzetti et al. (2010, submitted) and Li et al. (2010, in preparation) for the use of the 70~$\mu$m band as a star formation tracer. Taking into account the oxygen abundance in Table~\ref{tab:relations-sigma-oxy} improves the precision of the relation. Indeed, as stated earlier, the dust temperature is a function of the metallicity. Providing the oxygen abundance permits to indirectly take into account the dust temperature and therefore correct the estimation of the total infrared emission from the 70~$\mu$m band only.
\subsection{TIR estimates that include the 160~$\mu$m band}
The 160~$\mu$m band probes the cold dust, which makes it a good proxy to the TIR luminosity in galaxies where it is the dominant source of infrared emission \citep[see for example][]{bendo2008a}. In Figure~\ref{fig:160-24}, we show the correlation between $\log \left(L(24~\mu m)/L(160\mu m)\right)$ and $\log \left(L(160\mu m)/L(TIR)\right)$. In Table \ref{tab:relations} we see that the scatter around the best fit is smaller than for the 24~$\mu$m and the 70~$\mu$m.
\begin{figure*}[!htbp]
\includegraphics[width=\columnwidth]{figures/24_ov_160_vs_160_ov_TIR.ps}
\includegraphics[width=\columnwidth]{figures/24_ov_160_vs_160_ov_TIR_SFR_intensity.ps}
\caption{$\log \left(L(160~\mu m)/L(TIR)\right)$ versus $\log \left(L(24~\mu m)/L(160~\mu m)\right)$. The symbols and the color-coding of the lines are the same as in Figure~\ref{fig:PAH-24-TIR}. The color codes the oxygen abundance (left) and $\Sigma\left(TIR\right)$ (right).\label{fig:160-24}}
\end{figure*}
Interestingly, neither the metallicity nor $\Sigma\left(TIR\right)$ have a significant effect on the parameters of the fit between $L(24~\mu m)/L(160\mu m)$ and $L(160\mu m)/L(TIR)$ as we can see in Figure~\ref{fig:160-24}. This proves to be an advantage when the metallicity of the galaxy is not known or very uncertain. The 160~$\mu$m luminosity per unit area is tightly correlated with the TIR luminosity per unit area. Unsurprisingly the slope is slightly superlinear (1.107) and the scatter around the best fit is small (0.049 dex) making the 160~$\mu$m the best MIPS {\em Spitzer} band to trace the TIR emission along with the 70~$\mu$m band. The scatter around the best fit is significantly larger for the SINGS, LVL and E08 samples however the slope is almost completely linear, especially in the case of the SINGS and LVL samples. Once again, this is likely due to the diversity of the populations constituting the samples. Taking into account the oxygen abundance slightly reduces the scatter when estimating the TIR emission from the 160~$\mu$m one. As for the 70~$\mu$m band the reason is that the metallicity gives an indication of the possible dust temperature.
As the 70~$\mu$m and 160~$\mu$m are major contributors to the TIR \citep[][Fig.~11c, f]{dale2009a}, their temperature should be a good indicator of the TIR emission. In Figure~\ref{fig:160-70}, we show the correlation between $\log \left(L(70~\mu m)/L(160\mu m)\right)$ and $\log \left(L(160\mu m)/L(TIR)\right)$.
\begin{figure*}[!htbp]
\includegraphics[width=\columnwidth]{figures/70_ov_160_vs_160_ov_TIR.ps}
\includegraphics[width=\columnwidth]{figures/70_ov_160_vs_160_ov_TIR_SFR_intensity.ps}
\caption{$\log \left(L(70~\mu m)/L(160~\mu m)\right)$ versus $\log \left(L(160~\mu m)/L(TIR)\right)$. The symbols and the color-coding of the lines are the same as in Figure~\ref{fig:PAH-24-TIR}. The color codes the oxygen abundance (left) and $\Sigma\left(TIR\right)$ (right).\label{fig:160-70}}
\end{figure*}
We observe that there is a tight non-linear relation between the temperature of the dust as traced by the $L(70~\mu m)/L(160\mu m)$ ratio. At low $L(70~\mu m)/L(160\mu m)$, which indicates that the dust is very cold, the bulk of the TIR is accounted for by the 160~$\mu$m. A larger fraction of the total infrared emission is accounted for by the 70~$\mu$m band as the dust temperature increases. It appears that unlike the estimation from the 8~$\mu$m and 24~$\mu$m only for instance, the parameters of the correlation do not seem to depend on the the metallicity or $\Sigma\left(TIR\right)$ in any significant way.
\section{Discussion}
\label{sec:discussion}
The relations derived in this paper are dependent on the measurement of the total infrared luminosities and oxygen abundances.
We have estimated the TIR luminosity using equation \ref{eqn:draine} from \cite{draine2007a}. In the context of this study, equation 4 from \cite{dale2002a} yields very similar results. For all galaxy subregions, $\left<\log(TIR_{Draine}/TIR_{Dale})\right>=0.01\pm0.02$ as can be seen in the left panel of Figure~\ref{fig:metal-draine-dale}. The mean offset is similar for integrated SINGS and LVL galaxies data but slightly higher for the E08 sample: $\left<\log(TIR_{Draine}/TIR_{Dale})\right>=0.03\pm0.03$. The difference in the estimate of the TIR luminosity is small compared to the internal scatter within the subregions and the integrated galaxies which spans about 1 dex in $L(24~\mu m)/L(TIR)$. Therefore it should not affect the results significantly.
\begin{figure*}[!htbp]
\includegraphics[width=\columnwidth]{figures/draine_dale.ps}
\includegraphics[width=\columnwidth]{figures/comp_metal.ps}
\caption{Left: difference of the TIR luminosity estimated by the \cite{dale2001a} and \cite{draine2007a} relations versus the luminosity estimated from the \cite{draine2007a} relation. Right: difference between the oxygen abundances derived from the PT05 and the KK04 methods versus the latter one. The symbols and the colors are the same as in Figure~\ref{fig:PAH-24-TIR}.\label{fig:metal-draine-dale}}
\end{figure*}
The method used to calculate the oxygen abundance is another source of uncertainty. The oxygen abundance estimates between the PT05 and KK04 are offset by $0.60\pm0.10$ dex. The shapes of the slope and y-intercept curves versus the oxygen abundance are not identical, similarly to what was done in Figure~\ref{fig:fit-subregions}. There are two reasons to this. First of all, the two estimators are not in linear relation. Then, the abundance gradients derived by Moustakas et al. (2010, in preparation) are different for the two estimators as can be seen in the right panel of Figure~\ref{fig:metal-draine-dale}. The consequence is that each subregion has a different oxygen abundance offset between the two estimators. The difference in the fit parameters between the two oxygen abundance estimators for a given observation provides the typical uncertainties due to the abundance estimators. However, the precise evaluation of the uncertainties would require resolved spectroscopic observations at a $\sim6$\arcsec\ resolution. If we take the difference of the extrema of the fit parameters using the KK04 and PT05 estimators, the typical uncertainty on the slope can be evaluated to about $\pm0.2$ and is the main uncertainty in this study. Even though the uncertainty is important, it is significantly smaller than the range spanned by the slope, $\sim1.2$ for the PT05 method and $~0.8$ for the KK04 one.
Even if the relations derived here are distance independent, the physical scale encompassed by a pixel ranges from 0.7~kpc to 3.6~kpc in galaxy subregions. Even for the closest galaxy, each pixel encompasses several star forming regions and the ISM of the galaxy. However, for the more distant galaxies the mixing of the two components increases with the physical scale encompassed. To check whether this effect could induce a distance dependent bias in our analysis, we plot in Figure~\ref{fig:test-res-effect} the ratio of the luminosity in each infrared band to the total infrared luminosity for each pixel, as a function of the distance of the galaxy.
\begin{figure}[!ht]
\includegraphics[width=\columnwidth]{figures/test_resolution_effect.ps}
\caption{Ratio of the luminosity in the PAH 8~$\mu$m (red), 24~$\mu$m (green), 70~$\mu$m (blue), 160~$\mu$m (cyan) bands for each pixel as a function of the distance of the galaxy.\label{fig:test-res-effect}}
\end{figure}
We see that there is no obvious trend with the distance.
\section{Conclusions}
\label{sec:conclusion}
Using data for spatially resolved subregions within 13 face-on spiral galaxies as well as integrated luminosities for the SINGS, LVL, and E08 galaxies, we have derived new relations to estimate the total infrared luminosity based on using only one or two {\em Spitzer} bands, particularly the 8.0~$\mu$m and 24~$\mu$m bands. Relations incorporating 8.0~$\mu$m data vary significantly with oxygen abundance and especially with $\Sigma\left(TIR\right)$. However, TIR estimates that do not include 8~$\mu$m data are less dependent on oxygen abundances. In particular, the relations between the TIR emission and the 70~$\mu$m or 160~$\mu$m bands are relatively independent of oxygen abundances compared to the 8~$\mu$m and 24~$\mu$m ones.
|
1,116,691,500,438 | arxiv | \section{Introduction}
We consider a unique continuation (or data assimilation) problem for the Helmholtz equation
\begin{align}\label{helholtz_intro}
\Delta u + k^2 u = -f,
\end{align}
and introduce a stabilized finite element method (FEM) to solve the problem computationally.
Such methods have been previously studied for Poisson's equation in \cite{BurmanSIAM}, \cite{BurmanComptes} and \cite{BHL2018}, and for the heat equation in \cite{BurmanOksanen}.
The main novelty of the present paper is that our method is robust with respect to the wave number $k$, and we prove convergence estimates with explicit dependence on $k$, see \cref{L2error} and \cref{H1error} below.
An abstract form of a unique continuation problem is as follows.
Let $\omega \subset B \subset \Omega$ be open, connected and non-empty sets in $\R^{1+n}$ and suppose that $u \in H^2(\Omega)$ satisfies (\ref{helholtz_intro}) in $\Omega$.
Given $u$ in $\omega$ and $f$ in $\Omega$, find $u$ in $B$.
This problem is non-trivial since no information on the boundary $\p \Omega$ is given.
It is well known, see e.g. \cite{IsakovBook},
that if $\overline{B \setminus \omega} \subset
\Omega$ then the problem is conditionally H\"older stable: for all $k \ge 0$ there are $C > 0$ and $\alpha \in (0,1)$
such that for all $u \in H^2(\Omega)$
\begin{align}\label{stability_intro}
\norm{u}_{H^1(B)}
\le
C (\norm{u}_{H^1(\omega)} +
\norm{\Delta u + k^2 u}_{L^2(\Omega)})^\alpha \norm{u}_{H^1(\Omega)}^{1-\alpha}.
\end{align}
If $B \setminus \omega$ touches the boundary of $\Omega$, then one can only expect logarithmic stability, since it was shown in the classical paper \cite{John} that the optimal stability estimate for analytic continuation from a disk of radius strictly less than 1 to the concentric unit disk is of logarithmic type, and analytic functions are harmonic.
In general, the constants $C$ and $\alpha$ in \eqref{stability_intro} depend on $k$, as can be seen in \cref{wkb} given in \ref{appendix}. However, under suitable convexity assumptions on the geometry and direction of continuation it is possible to prove that in (\ref{stability_intro}) both the constants $C$ and $\alpha$ are independent of $k$, see the uniform estimate in \cref{cor_Holder} below, which is closely related to the so-called increased stability for unique continuation
\cite{Isakov}. Obtaining optimal error bounds in the finite element approximation crucially depends on deriving estimates similar to (\ref{stability_intro}), with weaker norms in the right-hand side, as in \cref{cor_Holder_impr} below, or in both sides, by shifting the Sobolev indices one degree down, as in \cref{shifted3b} below.
In addition to robustness with respect to $k$,
an advantage of using stabilized FEM for this unique continuation problem is that--when designed carefully--its implementation does not require information on the constants $C$ and $\alpha$ in (\ref{stability_intro}), or any other quantity from the continuous stability theory, such as a specific choice of a Carleman weight function.
Moreover, unlike other techniques such as Tikhonov regularization or quasi-reversibility, no auxiliary regularization parameters need to be introduced.
The only asymptotic parameter in our method is the size of the finite element mesh, and in particular, we do not need to saturate the finite element method with respect to an auxiliary parameter as, for example, in the estimate (34) in \cite{Bourgeois}.
Throughout the paper, $C$ will denote a positive constant independent of the wave number $k$ and the mesh size $h$, and which depends only on the geometry of the problem. By $A \lesssim B$ we denote the inequality $A \le C B$, where $C$ is as above.
For the well-posed problem of the Helmholtz equation with the Robin boundary condition
\begin{equation}\label{helmholtzbvp}
\Delta u + k^2 u = -f \quad \text{in } \Omega \quad \text{and} \quad \p_n u + \mathrm{i} ku = 0 \quad \text{on } \p\Omega,
\end{equation}
the following sharp bounds
\begin{equation}\label{well-posed1}
\norm{\nabla u}_{L^2(\Omega)} + k\norm{u}_{L^2(\Omega)} \le C \norm{f}_{L^2(\Omega)}
\end{equation}
and
\begin{equation}\label{well-posed2}
\norm{u}_{H^2(\Omega)} \le C k \norm{f}_{L^2(\Omega)}
\end{equation}
hold for a star-shaped Lipschitz domain $\Omega$ and any wave number $k$ bounded away from zero \cite{Spence2016}. The error estimates that we derive in \cref{stabilizedfem}, e.g. $\norm{u-u_h}_{H^1(B)} \le C (h k)^\alpha \norm{u}_*$ in \cref{H1error}, contain the term
\begin{equation}\label{starnorm}
\norm{u}_* = \norm{u}_{H^2(\Omega)} + k^2 \norm{u}_{L^2(\Omega)},
\end{equation}
which corresponds to the well-posed case term $k \norm{f}_{L^2(\Omega)}$.
It is well known from the seminal works \cite{BabuskaRev, IhlenburgBabuska1, IhlenburgBabuska2} that the finite element approximation of the Helmholtz problem is challenging also in the well-posed case due to the so-called pollution error. Indeed, to observe optimal convergence orders of $H^1$- and $L^2$-errors the mesh size $h$ must satisfy a smallness condition related to the wave number $k$, typically for piecewise affine elements, the condition $k^2 h \lesssim 1$.
This is due to the dispersion error that is most important for low order approximation spaces. The situation improves if higher order polynomial approximation is used. Recently, the precise conditions for optimal convergence when using $hp$-refinement ($p$ denotes the polynomial order of the approximation space) were shown in \cite{Melenk}.
Under the assumption that the solution operator for Helmholtz problems is polynomially bounded in $k$, it is shown that quasi-optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least $O(\log k)$.
Another way to obtain absolute stability (i.e. stability without, or under mild, conditions on the mesh size) of the approximate scheme is to use stabilization. The continuous interior penalty stabilization (CIP) was introduced for the Helmholtz problem in \cite{Wu}, where stability was shown in the $k h \lesssim 1$ regime, and was subsequently used to obtain error bounds for standard piecewise affine elements when $k^3 h^2 \lesssim 1$. It was then shown in \cite{BurmanHelmholtz} that, in the one dimensional case, the CIP stabilization can also be used to eliminate the pollution error, provided the penalty parameter is appropriately chosen. When deriving error estimates for the stabilized FEM that we herein introduce, we shall make use of the mild condition $k h \lesssim 1$. To keep down the technical detail we restrict the analysis to the case of piecewise affine finite element spaces, but the extension of the proposed method to the high order case follows using the stabilization operators suggested in \cite{BurmanSIAM} (see also \cite{BurmanChapter} for a discussion of the analysis in the ill-posed case).
From the point of view of applications, unique continuation problems often arise in control theory and inverse scattering problems. For instance, the above problem could arise when the acoustic wave field $u$ is measured on $\omega$
and there are unknown scatterers present outside $\Omega$.
\section{Continuum stability estimates}\label{continuum_estimates}
Our stabilized FEM will build on certain variations of the basic estimate (\ref{stability_intro}),
with the constants independent of the wave number,
and we derive these estimates in the present section.
The proofs are based on a Carleman estimate that is a variation of \cite[Lemma 2.2]{Isakov}
but we give a self-contained proof for the convenience of the reader.
In \cite{Isakov} the Carleman estimate was used to derive a so-called increased stability estimate under suitable convexity assumptions on the geometry. To be more precise, let $\Gamma \subset \partial \Omega$ be such that $\Gamma \subset \partial \omega$ and $\Gamma$ is at some positive distance away from $\partial \omega \cap \Omega$. For a compact subset $S$ of the open set $\Omega$, let $P(\nu; d)$ denote the half space which has distance $d$ from $S$ and $\nu$ as the exterior normal vector. Let $\Omega(\nu;d) = P(\nu; d) \cap \Omega$ and denote by $B$ the union of the sets $\Omega(\nu;d)$ over all $\nu$ for which $P(\nu; d) \cap \partial \Omega \subset \Gamma$. This geometric setting is exemplified by \cref{domain_conv} and it is illustrated in a general way in Figures 1 and 2 of \cite{Isakov} where $B$ is denoted by $\Omega(\Gamma;d)$. Under these assumptions it was proven that
\begin{align}
\label{increased_stability}
\norm{u}_{L^2(B)}
\le
C F + C k^{-1} F^\alpha \norm{u}_{H^1(\Omega)}^{1-\alpha},
\end{align}
where $F = \norm{u}_{H^1(\omega)} +
\norm{\Delta u + k^2 u}_{L^2(\Omega)}$
and the constants $C$ and $\alpha$ are independent of $k$.
Here $F$ can be interpreted as the size of the data in the unique continuation problem and the $H^1$-norm of $u$
as an a priori bound.
As $k$ grows, the first term on the right-hand side of (\ref{increased_stability}) dominates the second one,
and the stability is increasing in this sense.
As our focus is on designing a finite element method,
we prefer to measure the size of the data in the weaker norm
$$
E = \norm{u}_{L^2(\omega)} +
\norm{\Delta u + k^2 u}_{H^{-1}(\Omega)}.
$$
Taking $u$ to be a plane wave solution to (\ref{helholtz_intro}) suggests that
\begin{align*}
\norm{u}_{L^2(B)}
\le
C k E + C E^\alpha \norm{u}_{L^2(\Omega)}^{1-\alpha},
\end{align*}
could be the right analogue of (\ref{increased_stability})
when both the data and the a priori bound are in weaker norms.
We show below, see \cref{shifted3b},
a stronger estimate with only the second term on the right-hand side.
\cref{lem_carleman_eq} below captures the main step of the proof of our Carleman estimate. This is an elementary, but somewhat tedious, computation that establishes an identity similar to that in \cite{Liu} where the constant in a Carleman estimate for the wave equation was studied.
For an overview of Carleman estimates see \cite{LeRousseau,TataruRev}, the classical references are \cite[Chapter 17]{HormanderVol3} for second order elliptic equations, and \cite[Chapter 28]{HormanderVol4} for hyperbolic and more general equations.
In the proofs, the idea is to use an exponential weight function $e^{\ell(x)}$ and study the expression
$$
\Delta (e^\ell w) = e^\ell \Delta w + \text{lower order terms},
$$
or the conjugated operator $e^{-\ell} \Delta e^\ell$.
A typical approach is to study commutator estimates for the real and imaginary part of the principal symbol of the conjugated operator,
see e.g. \cite{LeRousseau}. This can be seen as an alternative way to estimate the cross terms appearing in the proof of \cref{lem_carleman_eq}.
Sometimes semiclassical analysis is used to derive the estimates, see e.g. \cite{LeRousseau}. This is very convenient when the estimates are shifted in the Sobolev scale, and we will use these techniques in \cref{sec_shifting} below.
\subsection{A Carleman estimate and conditional H\"older stability}
Denote by $(\cdot, \cdot)$, $|\cdot|$, $\div$, $\nabla$ and $D^2$ the inner product, norm, divergence, gradient and Hessian with respect to the Euclidean structure in $\Omega \subset \R^{1+n}$.
(Below, \cref{lem_carleman_eq} and \cref{cor_ptwise_Carleman} are written so that
they hold also when $\Omega$ is a Riemannian manifold
and the above concepts are replaced with their Riemannian analogues.)
\begin{lemma}
\label{lem_carleman_eq}
Let $k \ge 0$.
Let $\ell, w \in C^2(\Omega)$ and $\sigma \in C^1(\Omega)$.
We define $v = e^\ell w$, and
\begin{align*}
a = \sigma - \Delta \ell, \quad
q = k^2 + a + |\nabla \ell|^2, \quad
b = -\sigma v - 2(\grad v, \grad \ell), \quad
c = (|\grad v|^2 - q v^2) \grad \ell.
\end{align*}
Then
\begin{align*}
&e^{2 \ell} (\Delta w + k^2 w)^2/2
=
(\Delta v + q v)^2/2 + b^2/2
\\&\quad
+ a |\nabla v|^2 + 2 D^2 \ell(\grad v, \grad v)
+ \left(-a |\grad \ell|^2 + 2 D^2 \ell (\grad \ell, \grad \ell)\right)v^2
- k^2 a v^2
\\&\quad
+ \div(b \nabla v + c) + R,
\end{align*}
where
$
R = (\grad \sigma , \grad v)v + \left(\div (a \grad \ell) - a\sigma \right) v^2.
$
\end{lemma}
A proof of this result is given in \ref{appendix}. In the present paper we use \cref{lem_carleman_eq} only with the choice $\sigma =
\Delta \ell$, or equivalently $a=0$, but the more general version of the lemma is useful when non-convex geometries are considered. In fact, instead of using a strictly convex function $\phi$ as in \cref{cor_ptwise_Carleman} below, it is possible to use a function $\phi$ without critical points, and convexify by taking $\ell = \tau e^{\alpha \phi}$ and $\sigma =
\Delta \ell + \alpha \lambda \ell$ for suitable constants $\alpha$ and $\lambda$. In the present context this will lead to an estimate that is not robust with respect to $k$, but we will use such a technique in the forthcoming paper \cite{BNO2018}.
\begin{corollary}[Pointwise Carleman estimate]
\label{cor_ptwise_Carleman}
Let $\phi \in C^3(\Omega)$ be a strictly convex function without critical points, and choose $\rho > 0$ such that
\begin{align*}
D^2 \phi (X, X) \ge \rho |X|^2, \quad X \in T_x \Omega,\ x \in \Omega.
\end{align*}
Let $\tau > 0$ and $w \in C^2(\Omega)$.
We define $\ell = \tau \phi$, $v = e^\ell w$, and
\begin{align*}
b = -(\Delta \ell) v - 2(\grad v, \grad \ell), \quad
c = (|\grad v|^2 - (k^2 + |\nabla \ell|^2) v^2) \grad \ell.
\end{align*}
Then
\begin{align*}
e^{2 \tau\phi}
\left( (a_0 \tau - b_0) \tau^2 w^2 +
(a_1 \tau - b_1)|\nabla w|^2 \right)
+ \div(b \nabla v + c)
\le e^{2 \tau\phi} (\Delta w + k^2 w)^2/2,
\end{align*}
where the constants $a_j, b_j > 0$, $j=0,1$,
depend only on $\rho$, $\inf\limits_{x \in \Omega} |\nabla \phi(x)|^2$ and $\sup\limits_{x \in \Omega} |\grad (\Delta \phi(x))|^2$.
\end{corollary}
\begin{proof}
We employ the equality in \cref{lem_carleman_eq}
with $\ell = \tau \phi$ and $\sigma = \Delta \ell$.
With this choice of $\sigma$, it holds that $a = 0$.
As the two first terms on the right-hand side of the equality are positive, it is enough to consider
\begin{align*}
&2 D^2 \ell(\grad v, \grad v) + 2 D^2 \ell (\grad \ell, \grad \ell) v^2 + R
\\&\quad\ge
2 \rho \tau |\grad v|^2 + 2 \rho \tau^3 |\grad \phi|^2 v^2
- \tau |\grad (\Delta \phi)| |\grad v| |v|.
\end{align*}
The claim follows by combining this with
\begin{align*}
|\grad v|^2 = e^{2\tau \phi} |\tau w \grad \phi + \grad w|^2
\ge e^{2\tau \phi} \frac13 |\grad w|^2 - e^{2\tau \phi} \frac12 |\grad \phi|^2 \tau^2 w^2,
\end{align*}
and
\begin{align*}
\tau |\grad (\Delta \phi)| |\grad v| |v|
\le C (|\grad v|^2 + \tau^2 |v|^2).
\end{align*}
\end{proof}
The above Carleman estimate implies an inequality that is similar to the three-ball inequality, see e.g. \cite{Alessandrini}. The main difference is that here the foliation along spheres is followed in the opposite direction, i.e. the convex direction.
When continuing the solution inside the convex hull of $\omega$ as in \cite{Isakov}, we consider for simplicity a specific geometric setting defined in \cref{cor_Holder} below and illustrated in \cref{geometry}. The stability estimates we prove below in \cref{cor_Holder} and \cref{cor_Holder_impr}, and \cref{shifted3b} also hold in other geometric settings in which $B$ is included in the convex hull of $\omega$ and $B \setminus \omega$ does not touch the boundary of $\Omega$, such as the one in \cref{domain_conv}. We prove this in \cref{ex_convex_square}.
\begin{figure}[h]
\centering
\resizebox{0.5\textwidth}{!}
\begin{tikzpicture}
\filldraw[color=gray] (-2,0) arc [radius=2, start angle=180, end angle=360]
-- (3,0) arc [radius=3, start angle=0, end angle=-180]
-- cycle;
\draw[dashed] (-5,0) -- (5,0);
\filldraw (0,0) circle (2pt) node[align=center, above] {$0$};
\draw (0,0) circle [radius=2];
\draw (0,0) circle [radius=3];
\draw (2.1,0) node[align=right, above] {$r$};
\draw (3.1,0) node[align=right, above] {$R$};
\draw (-4.5,0) node[align=right, below] {$H$};
\filldraw [gray] (0,2.5) circle (2pt) node[align=center, above] {$y$};
\draw (0,2.5) circle [radius=3.4];
\draw (0,-2.75) node[align=right, above] {$\omega$};
\draw (0,-1.75) node[align=right, above] {$B$};
\end{tikzpicture}
\caption{The geometric setting in \cref{cor_Holder}.}
\label{geometry}
\end{figure}
We use the following notation for a half space
\begin{align*}
H &= \{(x^0, \dots, x^n) \in \R^{1+n};\ x^0 < 0\}.
\end{align*}
\begin{corollary}
\label{cor_Holder}
Let $r > 0$, $\beta > 0$, $R > r$ and $\sqrt{r^2 + \beta^2} < \rho < \sqrt{R^2 + \beta^2}$. Define
$y = (\beta, 0, \dots,0)$ and
$$
\Omega = H \cap B(0, R),
\quad
\omega = \Omega \setminus \overline{B(0,r)},
\quad
B = \Omega \setminus \overline{B(y, \rho)}.
$$
Then there are $C > 0$ and $\alpha \in (0,1)$
such that for all $u \in C^2(\Omega)$ and $k \ge 0$
$$
\norm{u}_{H^1(B)}
\le
C (\norm{u}_{H^1(\omega)} +
\norm{\Delta u + k^2 u}_{L^2(\Omega)})^\alpha \norm{u}_{H^1(\Omega)}^{1-\alpha}.
$$
\end{corollary}
\begin{proof}
Choose $\sqrt{r^2 + \beta^2} < s < \rho$
and observe that $\p \Omega \setminus B(y,s) \subset \overline \omega$.
Define $\phi(x) = |x-y|^2$.
Then $\phi$ is smooth and strictly convex in $\overline{\Omega}$,
and it does not have critical points there.
Choose $\chi \in C_0^\infty(\Omega)$ such that
$\chi = 1$ in $\Omega \setminus (B(y,s) \cup \omega)$ and set $w = \chi u$.
\cref{cor_ptwise_Carleman} implies that for large $\tau > 0$
\begin{align}
\label{Holder_step1}
\int_{\Omega} (\tau^3 w^2 + \tau |\nabla w|^2) e^{2 \tau \phi} dx
\le C \int_{\Omega} (\Delta w + k^2 w )^2 e^{2 \tau \phi} dx,
\end{align}
a result also stated, without a detailed proof, in \cite[Exercise 3.4.6]{IsakovBook}. The commutator $[\Delta, \chi]$ vanishes outside $B(y,s) \cup \omega$
and $\phi < s^2$ in $B(y,s)$.
Hence the right-hand side of (\ref{Holder_step1})
is bounded by a constant times
\begin{align}
\label{cylinders_step2}
&\int_{\Omega} |\Delta u + k^2 u|^2 e^{2 \tau \phi} dx
+ \int_{B(y,s) \cup \omega} |[\Delta, \chi]u|^2 e^{2 \tau \phi} dx
\\\notag&\quad\le
Ce^{2 \tau (\beta+R)^2} (\norm{\Delta u + k^2 u}_{L^2(\Omega)}^2
+ \norm{u}_{H^1(\omega)}^2)
+ C e^{2\tau s^2}
\norm{u}_{H^1(B(y,s))}^2.
\end{align}
The left-hand side of (\ref{Holder_step1})
is bounded from below by
\begin{align}
\label{cylinders_step3}
&\int_{B}
\left( \tau |\nabla u|^2
+ \tau^3 |u|^2 \right) e^{2 \tau \phi}\, dx
\ge
e^{2 \tau \rho^2} \norm{u}_{H^1(B)}^2.
\end{align}
The inequalities (\ref{Holder_step1})-(\ref{cylinders_step3}) imply
$$
\norm{u}_{H^1(B)}
\le
e^{q\tau}
\left(\norm{\Delta u + k^2 u}_{L^2(\Omega)} +
\norm{u}_{H^1(\omega)} \right)
+ e^{-p \tau} \norm{u}_{H^1(\Omega)},
$$
where $q=(\beta + R)^2 - \rho^2$ and $p = \rho^2 - s^2 > 0$.
The claim follows from \cite[Lemma 5.2]{LeRousseau}.
\end{proof}
\begin{corollary}\label{cor_Holder_impr}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Then there are $C>0$ and $\alpha\in(0,1)$ such that
$$
\norm{u}_{H^1(B)}
\le
C k (\norm{u}_{L^2(\omega)} +
\norm{\Delta u + k^2 u}_{H^{-1}(\Omega)})^\alpha (\norm{u}_{L^2(\Omega)} + \norm{\Delta u + k^2 u}_{H^{-1}(\Omega)})^{1-\alpha}.
$$
\end{corollary}
\begin{proof}
Let $\omega_1 \subset \omega \subset B \subset \Omega_1 \subset \Omega$, denote for brevity by $\mathcal{L}$ the operator $\Delta + k^2$, and consider the following auxiliary problem
\begin{align*}
\mathcal{L}w &= \mathcal{L}u \quad \text{in } \Omega_1 \\
\p_n w + \mathrm{i} kw &= 0 \quad \text{on } \p\Omega_1,
\end{align*}
whose solution satisfies the estimate \cite[Corollary 1.10]{Spence2016}
$$
\norm{\nabla w}_{L^2(\Omega_1)}+k\norm{w}_{L^2(\Omega_1)} \leq C k\norm{\mathcal{L}u}_{H^{-1}(\Omega_1)},
$$
which gives $$
\norm{w}_{H^1(\Omega_1)} \leq C k \norm{\mathcal L u}_{H^{-1}(\Omega)}.
$$
For $v = u - w$ we have $\mathcal{L}v = 0$ in $\Omega_1$. The stability estimate in \cref{cor_Holder} used for $\omega_1,B,\Omega_1$ reads as
$$
\norm{v}_{H^1(B)}
\le
C \norm{v}^\alpha_{H^1(\omega_1)} \norm{v}_{H^1(\Omega_1)}^{1-\alpha},
$$
and the following estimates hold
\begin{align*}
\norm{u}_{H^1(B)}
&\le
\norm{v}_{H^1(B)} + \norm{w}_{H^1(B)}\\
&\le
C( \norm{u}_{H^1(\omega_1)} + \norm{w}_{H^1(\omega_1)} )^\alpha ( \norm{u}_{H^1(\Omega_1)} + \norm{w}_{H^1(\Omega_1)} )^{1-\alpha} + C k \norm{\mathcal{L}u}_{H^{-1}(\Omega)}
\\
&\le
C( \norm{u}_{H^1(\omega_1)} + k \norm{\mathcal{L}u}_{H^{-1}(\Omega)} )^\alpha ( \norm{u}_{H^1(\Omega_1)} + k \norm{\mathcal{L}u}_{H^{-1}(\Omega)} )^{1-\alpha}.
\end{align*}
Now we choose a cutoff function $\chi \in C^{\infty}_0(\omega)$ such that $\chi = 1$ in $\omega_1$ and $\chi u$ satisfies
$$
\mathcal{L}(\chi u)=\chi \mathcal{L}u +[\mathcal{L},\chi]u,\quad \p_n (\chi u) + ik(\chi u) = 0 \text{ on } \p\omega.
$$
Since the commutator $[\mathcal{L},\chi]$ is of first order, using again \cite[Corollary 1.10]{Spence2016} we obtain
\begin{align*}
\norm{u}_{H^1(\omega_1)} &\le \norm{\chi u}_{H^1(\omega)}
\le Ck \left( \norm{ [\mathcal{L},\chi]u }_{H^{-1}(\omega)} + \norm{ \chi \mathcal{L}u }_{H^{-1}(\omega)} \right) \\
&\le Ck \left( \norm{u}_{L^2(\omega)} + \norm{\mathcal{L}u}_{H^{-1}(\omega)} \right)
\end{align*}
The same argument for $\Omega_1 \subset \Omega$ gives
$$
\norm{u}_{H^1(\Omega_1)} \le Ck( \norm{u}_{L^2(\Omega)} + \norm{\mathcal{L}u}_{H^{-1}(\Omega)}),
$$
thus leading to the conclusion.
\end{proof}
\subsection{Shifted three-ball inequality}
\label{sec_shifting}
\def\h{\hbar}
\def\scl{\text{scl}}
In this section we prove
an estimate as in \cref{cor_Holder},
but with the Sobolev indices shifted down one degree,
and our starting point is again the Carleman estimate
in \cref{cor_ptwise_Carleman}.
When shifting Carleman estimates, as we want to keep track of the large parameter $\tau$, it is convenient to use the semiclassical version of pseudodifferential calculus.
We write $\h > 0$ for the semiclassical parameter
that satisfies $\h = 1/\tau$.
The semiclassical (pseudo)differential operators are (pseudo)differential operators where, roughly speaking, each derivative is multiplied
by $\h$, for the precise definition see Section 4.1 of \cite{Zworski}.
The scale of semiclassical Bessel potentials is defined by
$$
J^s = (1-\h^2 \Delta)^{s/2}, \quad s \in \R,
$$
and the semiclassical Sobolev spaces by
$$
\norm{u}_{H_\scl^s(\R^n)} = \norm{J^s u}_{L^2(\R^n)}.
$$
Then a semiclassical differential operator of order $m$
is continuous from $H_\scl^{m+s}(\R^n)$ to $H_\scl^s(\R^n)$,
see e.g. Section 8.3 of \cite{Zworski}.
We will give a shifting argument that is similar to that in Section 4 of \cite{DKSU}.
To this end, we need the following pseudolocal and commutator estimates for semiclassical pseudodifferential operators,
see e.g. (4.8) and (4.9) of \cite{DKSU}.
Suppose that $\psi,\chi \in C_0^\infty(\R^n)$
and that $\chi = 1$ near $\supp(\psi)$,
and let $A, B$ be two semiclassical pseudodifferential
operators of orders $s, m$, respectively.
Then for all $p,q,N \in \R$, there is $C>0$
\begin{align}
\norm{(1-\chi) A (\psi u)}_{H_\scl^{p}(\R^n)}
&\le C \h^N \norm{u}_{H_\scl^{q}(\R^n)}, \label{pseudolocality}
\\
\norm{[A,B] u}_{H_\scl^{p}(\R^n)}
&\le C \h \norm{u}_{H_\scl^{p+s+m-1}(\R^n)}. \label{commutator}
\end{align}
Both these estimates follow from the composition calculus, see e.g. \cite[Theorem 4.12]{Zworski}.
Let $\phi$ be as in \cref{cor_ptwise_Carleman}
and set $\ell = \phi / \h$ and $\sigma = \Delta \ell$ in \cref{lem_carleman_eq}. Then
\begin{align*
(e^{\phi / \h} \Delta e^{-\phi / \h} v + k^2 v)^2 / 2
&\ge
2 \h^{-1} D^2 \phi(\nabla v, \nabla v)
+ 2 \h^{-3} D^2 \phi(\nabla \phi, \nabla \phi) v^2
\\&\quad
+ \div(b \nabla v + B) + \h^{-1} (\nabla \Delta \phi, \nabla v) v
\end{align*}
Write $P = e^{\phi / \h} \h^2 \Delta e^{-\phi / \h}$
and let $v \in C_0^\infty(\Omega')$ where $\Omega' \subset \R^n$ is open and bounded, and $\overline \Omega \subset \Omega'$. Then, rescaling by $\h^4$,
\begin{align*}
C \norm{P v + \h^2 k^2 v}_{L^2(\R^n)}^2
\ge \h \norm{\h \nabla v}_{L^2(\R^n)}^2 + \h \norm{v}_{L^2(\R^n)}^2 - C \h^2 \norm{v}_{H_\scl^1(\R^n)}^2,
\end{align*}
and for small enough $\h > 0$ we obtain
\begin{align*}
\sqrt \h \norm{v}_{H_\scl^1(\R^n)} \le C \norm{P v + \h^2 k^2 v}_{L^2(\R^n)}.
\end{align*}
Now the conjugated operator $P$ is a semiclassical differential operator,
\begin{align*}
Pu = e^{\phi / \h} \h^2 \div \grad (e^{-\phi / \h} u)
= \h^2 \Delta u - 2 (\nabla \phi, \h \nabla u)
- \h (\Delta \phi) u + |\nabla \phi|^2 u.
\end{align*}
Let $\chi, \psi \in C_0^\infty(\Omega')$ and suppose that $\psi=1$ near $\Omega$ and $\chi = 1$ near $\supp(\psi)$. Then for $v \in C_0^\infty(\Omega)$,
$$
\norm{v}_{H_\scl^{1+s}(\R^n)}
\le \norm{\chi J^s v}_{H_\scl^{1}(\R^n)}
+ \norm{(1-\chi) J^s \psi v}_{H_\scl^{1}(\R^n)}
\le C \norm{\chi J^s v}_{H_\scl^{1}(\R^n)}
$$
where we used the pseudolocality (\ref{pseudolocality}) to absorb the second term
on the right-hand side by the left-hand side.
We have
\begin{align}\label{shift_step1}
\sqrt \h \norm{v}_{H_\scl^{1+s}(\R^n)}\le C\sqrt \h \norm{\chi J^s v}_{H_\scl^1(\R^n)}
\le C \norm{(P + \h^2 k^2) \chi J^s v}_{L^2(\R^n)},
\end{align}
and using the commutator estimate (\ref{commutator}), we have
\begin{align*
\norm{[P,\chi J^s] v}_{L^2(\R^n)}
\le C \h \norm{v}_{H^{1+s}_\scl(\R^n)}.
\end{align*}
This can be absorbed by the left-hand side of (\ref{shift_step1}). Thus
\begin{align*
\sqrt \h \norm{v}_{H_\scl^{1+s}(\R^n)}\le C
\norm{\chi J^s (P + \h^2 k^2) v}_{L^2(\R^n)}
\le C \norm{(P + \h^2 k^2) v}_{H_\scl^{s}(\R^n)}.
\end{align*}
Take now $s=-1$ and let the cutoff $\chi$ and the weight $\phi$ be as in the proof of \cref{cor_Holder}, with the additional condition on $\chi$ such that there is
$\psi \in C_0^\infty(B(y,s) \cup \omega)$
satisfying $\psi = 1$ in $\supp([P,\chi])$.
Let $u \in C^\infty(\R^n)$ and set $w = e^{\phi/\h} u$. Then the previous estimate becomes
\begin{align*}
\sqrt \h \norm{\chi w}_{L^2(\R^n)}
\le C \norm{(P + \h^2 k^2) \chi w}_{H_\scl^{-1}(\R^n)}.
\end{align*}
We have
$$
\norm{[P,\chi] w}_{H_\scl^{-1}(\R^n)}
=
\norm{[P,\chi] \psi w}_{H_\scl^{-1}(\R^n)}
\le C \h \norm{\psi w}_{L^2 (\R^n)}.
$$
Using the norm inequality $\norm{\cdot}_{H_\scl^{-1}(\R^n)} \le C \h^{-2} \norm{\cdot}_{H^{-1}(\R^n)}$, we thus obtain
\begin{align*}
\sqrt \h \norm{\chi e^{\phi/\h} u}_{L^2(\R^n)}
&\le
C \norm{\chi(e^{\phi / \h} \Delta e^{-\phi / \h} + k^2) w}_{H_\scl^{-1}(\R^n)} + C \h \norm{\psi w}_{L^2 (\R^n)}
\\&\le
C \h^{-2} \norm{\chi e^{\phi/\h} (\Delta u + k^2 u)}_{H^{-1}(\R^n)} + C \h \norm{\psi e^{\phi/\h} u}_{L^2 (\R^n)}
\end{align*}
Using the same notation as in the proof of \cref{cor_Holder}, due to the choice of $\psi$ we get
$$
e^{\rho^2/\h} \norm{u}_{L^2(B)}
\le
C e^{(\beta+R)^2/\h}
\left( \h^{-\frac72} \norm{\Delta u + k^2 u}_{H^{-1}(\Omega)} + \h^{\frac12} \norm{u}_{L^2(\omega)} \right)
+ C e^{s^2/\h} \h^{\frac12} \norm{u}_{L^2(\Omega)},
$$
for small enough $\h > 0$. Absorbing the negative power of $\h$ in the exponential, and using \cite[Lemma 5.2]{LeRousseau}, we conclude the proof of the following result.
\begin{lemma}\label{shifted3b}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Then there are $C>0$ and $\alpha\in(0,1)$ such that
\begin{equation*}
\norm{u}_{L^2(B)}
\le
C (\norm{u}_{L^2(\omega)} +
\norm{\Delta u + k^2 u}_{H^{-1}(\Omega)})^\alpha \norm{u}_{L^2(\Omega)}^{1-\alpha}.
\end{equation*}
\end{lemma}
\section{Stabilized finite element method}\label{stabilizedfem}
We aim to solve the unique continuation problem for the Helmholtz equation
\begin{equation}\label{helmholtz}
\Delta u + k^2u = -f \text{ in } \Omega, \quad u=q|_{\omega},
\end{equation}
where $\omega \subset \Omega \subset \R^{1+n}$ are open, $f\in H^{-1}(\Omega)$ and $q\in L^2(\omega)$ are given.
Following the optimization based approach in \cite{BurmanSIAM, BHL2018} we will make use of the continuum stability estimates in \cref{continuum_estimates} when deriving error estimates for the finite element approximation.
\subsection{Discretization}
Consider a family $\mathcal{T}=\{\mathcal{T}_h\}_{h>0}$ of triangulations of $\Omega$ consisting of simplices such that the intersection of any two distinct ones is either a common vertex, a common edge or a common face. Also, assume that the family $\mathcal{T}$ is quasi-uniform.
Let
$$V_h = \{u\in C(\bar{\Omega}): u|_K \in \P_1(K), K\in \mathcal{T}_h\}$$
be the $H^1$-conformal approximation space based on the $\P_1$ finite element and let
$$
W_h = V_h \cap H^1_0(\Omega).
$$
Consider the orthogonal $L^2$-projection $\Pi_h:L^2(\Omega) \to V_h$, which satisfies
\begin{align*}
(u-\Pi_h u,v)_{L^2(\Omega)} &= 0,\quad u\in L^2(\Omega),\, v\in V_h, \\
\norm{\Pi_h u}_{L^2(\Omega)} &\le \norm{u}_{L^2(\Omega)},\quad u\in L^2(\Omega),
\end{align*}
and the Scott-Zhang interpolator $\pi_h:H^1(\Omega) \to V_h$, that preserves vanishing Dirichlet boundary conditions. Both operators have the following stability and approximation properties, see e.g. \cite[Chapter 1]{ErnBook},
\begin{align}
\norm{i_h u}_{H^1(\Omega)} &\le C \norm{u}_{H^1(\Omega)}, &u\in H^1(\Omega), \label{H1stab} \\
\norm{u-i_h u}_{H^m(\Omega)} &\le C h^{k-m} \norm{u}_{H^k(\Omega)}, &u\in H^k(\Omega) \label{interp},
\end{align}
where $i= \pi,\Pi,\, k=1,2$ and $m=0,k-1$.
The regularization on the discrete level will be based on the $L^2$-control of the gradient jumps over elements edges using the jump stabilizer
\begin{equation*}
\J (u,u) = \sum_{F\in\mathcal{F}_h} \int_F h \llbracket n \cdot \nabla u \rrbracket ^2 ds,\quad u\in V_h,
\end{equation*}
where $\mathcal{F}_h$ is the set of all internal faces, and the jump over $F\in\mathcal{F}_h$ is given by
$$
\llbracket n \cdot \nabla{u} \rrbracket_F = n_1 \cdot \nabla{u}|_{K_1} + n_2 \cdot \nabla{u}|_{K_2},
$$
with $K_1,K_2\in \mathcal{T}_h$ being two simplices such that $K_1 \cap K_2 = F$, and $n_j$ the outward normal of $K_j,\, j=1,2$. The face subscript is omitted when there is no ambiguity.
\begin{lemma}
There is $C>0$ such that all $u\in V_h,\, v\in H^1_0(\Omega),\, w\in H^2(\Omega)$ and $h>0$ satisfy
\begin{align}
(\nabla u,\nabla v)_{L^2(\Omega)} &\le C \J(u,u)^{1/2} (h^{-1}\norm{v}_{L^2(\Omega)} + \norm{v}_{H^1(\Omega)}),\label{jumpineq1} \\
\J(i_h w,i_h w) &\le C h^2 \norm{w}^2_{H^2(\Omega)},\quad i\in \{\pi,\Pi\}.\label{jumpineq2}
\end{align}
\begin{proof}
See \cite[Lemma 2]{BurmanOksanen} when the interpolator is $\pi_h$. Since this proof uses just the approximation properties of $\pi_h$, it holds verbatim for $\Pi_h$.
\end{proof}
\end{lemma}
Adopting the notation
$$
a(u,z)=(\nabla u,\nabla z)_{L^2(\Omega)},\quad G_f(u,z)=a(u,z)-k^2(u,z)_{L^2(\Omega)} - \pair{f,z},\quad G=G_0,
$$
we write for $u\in H^1(\Omega)$ the weak formulation of $\Delta u + k^2u = -f$ as
$$
G_f(u,z)=0, \quad z\in H^1_0(\Omega).
$$
Our approach is to find the saddle points of the Lagrangian functional
\begin{equation*}
L_{q,f}(u,z) = \frac{1}{2} \norm{u-q}^2_{\omega} + \frac{1}{2}s(u,u)-\frac{1}{2}s^*(z,z)+G_f(u,z),
\end{equation*}
where $\norm{\cdot}_\omega$ denotes $\norm{\cdot}_{L^2(\omega)}$, and $s$ and $s^*$ are stabilizing (regularizing) terms for the primal and dual variables that should be consistent and vanish at optimal rates. The stabilization must control certain residual quantities representing the data of the error equation. The primal stabilizer will be based on the continuous interior penalty given by $\J$. It must take into account the zeroth order term of the Helmholtz operator. The dual variable can be stabilized in the $H^1$-seminorm. Notice that when the PDE-constraint is satisfied, $z=0$ is the solution for the dual variable of the saddle point, thus the stabilizer $s^*$ is consistent. Hence we make the following choice
$$
s(u,u) = \J(u,u) + \norm{hk^2 u}^2_{L^2(\Omega)},\quad s^* = a.
$$ For a detailed presentation of such discrete stabilizing operators we refer the reader to \cite{BurmanSIAM} or \cite{BurmanChapter}.
We define on $V_h$ and $W_h$, respectively, the norms
$$
\norm{u}_V = s(u,u)^{1/2}, \quad u\in V_h, \quad \norm{z}_W = s^*(z,z)^{1/2},\quad z\in W_h,
$$
together with the norm on $V_h \times W_h$ defined by
$$
\tnorm{(u,z)}^2 = \norm{u}^2_V + \norm{u}^2_{\omega} + \norm{z}^2_W.
$$
The saddle points $(u,z)\in V_h \times W_h$ of the Lagrangian $L_{q,f}$ satisfy
\begin{equation}\label{weakform}
A[(u,z),(v,w)] = (q,v)_{\omega} + \pair{f,w},\quad (v,w)\in V_h \times W_h,
\end{equation}
where $A$ is the symmetric bilinear form
\begin{equation*}
A[(u,z),(v,w)] = (u,v)_{\omega} + s(u,v) + G(v,z) - s^*(z,w) + G(u,w).
\end{equation*}
Since $A[(u,z),(u,-z)] = \norm{u}^2_{\omega} + \norm{u}^2_V + \norm{z}^2_W$ we have the following inf-sup condition
\begin{equation}\label{inf_sup}
\sup_{(v,w)\in V_h \times W_h} \frac{A[(u,z),(v,w)]}{\tnorm{(v,w)}} \ge \tnorm{(u,z)}
\end{equation}
that guarantees a unique solution in $V_h \times W_h$ for (\ref{weakform}).
\subsection{Error estimates}
We start by deriving some lower and upper bounds for the norm $\norm{\cdot}_V$. For $u_h \in V_h,\, z\in H^1_0(\Omega)$, we use (\ref{jumpineq1}) to bound
\begin{align*}
G(u_h,z) &= (\nabla u_h,\nabla z)_{L^2(\Omega)} - k^2(u_h,z)_{L^2(\Omega)} \\
&\le C \J(u_h,u_h)^{1/2} (h^{-1}\norm{z}_{L^2(\Omega)} + \norm{z}_{H^1(\Omega)}) + k^2 \norm{u_h}_{L^2(\Omega)} \norm{z}_{L^2(\Omega)},
\end{align*}
and hence
\begin{equation}\label{lowerv}
G(u_h,z) \le C \norm{u_h}_V (h^{-1}\norm{z}_{L^2(\Omega)} + \norm{z}_{H^1(\Omega)}).
\end{equation}
For $u \in H^2(\Omega)$, from (\ref{jumpineq2}) and the stability of the $L^2$-projection
\begin{equation*}
\norm{\Pi_h u}^2_V = \J(\Pi_h u,\Pi_h u) + \norm{hk^2 \Pi_h u}^2_{L^2(\Omega)} \le C(h^2 \norm{u}^2_{H^2(\Omega)} + \norm{hk^2 u}^2_{L^2(\Omega)})
\end{equation*}
implies
\begin{equation}\label{upperv}
\norm{\Pi_h u}_V \le C h (\norm{u}_{H^2(\Omega)} + k^2 \norm{u}_{L^2(\Omega)}) = C h \norm{u}_*,
\end{equation}
where $\norm{u}_*$ is defined as in (\ref{starnorm}).
\begin{lemma}\label{tnormerror}
Let $u\in H^2(\Omega)$ be the solution to (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to (\ref{weakform}). Then there exists $C>0$ such that for all $h\in(0,1)$
$$
\tnorm{(u_h-\Pi_h u,z_h)} \le C h \norm{u}_*.
$$
\begin{proof}
Due to the inf-sup condition (\ref{inf_sup}) it is enough to prove that for $(v,w)\in V_h \times W_h$,
$$
A[(u_h-\Pi_h u,z_h),(v,w)] \le C h \norm{u}_* \tnorm{(v,w)}.
$$
The weak form of (\ref{helmholtz}) implies that
\begin{equation*}
A[(u_h-\Pi_h u,z_h),(v,w)] = (u-\Pi_h u,v)_\omega + G(u-\Pi_h u,w) - s(\Pi_h u,v).
\end{equation*}
Using (\ref{interp}) we bound the first term to get
$$
(u-\Pi_h u,v)_\omega \le C h^2 \norm{u}_{H^2(\Omega)} \norm{v}_\omega.
$$
For the second term we use the $L^2$-orthogonality property of $\Pi_h$, and (\ref{interp}) to obtain
\begin{equation*}
G(u-\Pi_h u,w) = (\nabla (u-\Pi_h u), \nabla w)_{L^2(\Omega)}
\le C h \norm{w}_W \norm{u}_{H^2(\Omega)},
\end{equation*}
while for the last term we employ (\ref{upperv}) to estimate
$$
s(\Pi_h u,v) \le \norm{\Pi_h u}_V \norm{v}_V \leq C h \norm{u}_* \norm{v}_V.
$$
\end{proof}
\end{lemma}
\begin{theorem}\label{L2error}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Let $u\in H^2(\Omega)$ be the solution to (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to (\ref{weakform}). Then there are $C>0$ and $\alpha \in (0,1)$ such that for all $k,\, h>0$ with $k h \lesssim 1$
$$
\norm{u-u_h}_{L^2(B)} \le C (h k)^\alpha k^{\alpha-2} \norm{u}_*.
$$
\begin{proof}
Consider the residual $\pair{r,w}=G(u_h-u,w)=G(u_h,w)-\pair{f,w},\, w\in H^1_0(\Omega)$. Taking $v=0$ in (\ref{weakform}) we get $G(u_h,w) = \pair{f,w} + s^*(z_h,w),\, w\in W_h$ which implies that
\begin{align*}
\pair{r,w} &= G(u_h,w) - \pair{f,w} - G(u_h,\pi_h w) + G(u_h,\pi_h w) \\
&= G(u_h,w-\pi_h w) - \pair{f,w-\pi_h w} + s^*(z_h,\pi_h w),\quad w\in H^1_0(\Omega).
\end{align*}
Using (\ref{lowerv}) and (\ref{interp}) we estimate the first term
\begin{align*}
G(u_h,w-\pi_h w) &\le C \norm{u_h}_V (h^{-1} \norm{w-\pi_h w}_{L^2(\Omega)} + \norm{w-\pi_h w}_{H^1(\Omega)}) \\
&\le C \norm{u_h}_V \norm{w}_{H^1(\Omega)} \le C h \norm{u}_* \norm{w}_{H^1(\Omega)},
\end{align*}
since, due to \cref{tnormerror} and (\ref{upperv})
\begin{equation*}
\norm{u_h}_V \le \norm{u_h-\Pi_h u}_V + \norm{\Pi_h u}_V
\le C h \norm{u}_*.
\end{equation*}
The second term is bounded by using (\ref{interp})
$$
\pair{f,w-\pi_h w} \le \norm{f}_{L^2(\Omega)} \norm{w-\pi_h w}_{L^2(\Omega)} \le C h \norm{f}_{L^2(\Omega)} \norm{w}_{H^1(\Omega)}
$$
and the last term by using \cref{tnormerror} and the $H^1$-stability (\ref{H1stab})
$$
s^*(z_h,\pi_h w) \le \norm{z_h}_W \norm{\pi_h w}_W \le C h \norm{u}_* \norm{w}_{H^1(\Omega)}.
$$
Hence the following residual norm estimate holds
$$
\norm{r}_{H^{-1}(\Omega)} \le C h (\norm{u}_* + \norm{f}_{L^2(\Omega)}) \le C h \norm{u}_*.
$$
Using the continuum estimate in \cref{shifted3b} for $u-u_h$ we obtain the following error estimate
$$
\norm{u-u_h}_{L^2(B)} \le C (\norm{u-u_h}_{L^2(\omega)} + \norm{r}_{H^{-1}(\Omega)})^\alpha \norm{u-u_h}_{L^2(\Omega)}^{1-\alpha}.
$$
By (\ref{interp}) and \cref{tnormerror} we have the bounds
\begin{align*}
\norm{u-u_h}_{L^2(\omega)} &\le \norm{u-\Pi_h u}_{L^2(\omega)} + \norm{u_h-\Pi_h u}_{L^2(\omega)} \\
&\le C h \norm{u}_{H^1(\Omega)} + C h \norm{u}_* \\
&\le C h \norm{u}_*
\end{align*}
and
\begin{align*}
\norm{u-u_h}_{L^2(\Omega)} &\le \norm{u-\Pi_h u}_{L^2(\Omega)} + \norm{u_h-\Pi_h u}_{L^2(\Omega)} \\
&\le C h^2 \norm{u}_{H^2(\Omega)} + C h^{-1} k^{-2} \norm{u_h-\Pi_h u}_V \\
&\le C ((h^2+k^{-2})\norm{u}_{H^2(\Omega)} + \norm{u}_{L^2(\Omega)}) \\
&\le C k^{-2} \norm{u}_*
\end{align*}
thus leading to the conclusion.
\end{proof}
\end{theorem}
\begin{theorem}\label{H1error}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Let $u\in H^2(\Omega)$ be the solution to (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to (\ref{weakform}). Then there are $C>0$ and $\alpha \in (0,1)$ such that for all $k,\, h>0$ with $k h \lesssim 1$
$$
\norm{u-u_h}_{H^1(B)} \le C (h k)^\alpha \norm{u}_*.
$$
\end{theorem}
\begin{proof}
We employ a similar argument as in the proof of \cref{L2error} with the same estimates for the residual norm and the $L^2$-errors in $\omega$ and $\Omega$, only now using the continuum estimate in \cref{cor_Holder_impr} to obtain
\begin{align*}
\norm{u-u_h}_{H^1(B)} &\le C k ( \norm{u-u_h}_{L^2(\omega)} + \norm{r}_{H^{-1}(\Omega)})^\alpha ( \norm{u-u_h}_{L^2(\Omega)} + \norm{r}_{H^{-1}(\Omega)})^{1-\alpha}
\\
&\le C k h^\alpha (k^{-2}+h)^{1-\alpha} \norm{u}_*,
\end{align*}
which ends the proof.
\end{proof}
Let us remark that if we make the assumption $k^2 h \lesssim 1$ then the estimate in \cref{H1error} becomes
$$
\norm{u-u_h}_{H^1(B)} \le C (h k^2)^\alpha k^{-1} \norm{u}_*,
$$
and combining \cref{L2error} and \cref{H1error} we obtain the following result.
\begin{corollary}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Let $u\in H^2(\Omega)$ be the solution to (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to (\ref{weakform}). Then there are $C>0$ and $\alpha \in (0,1)$ such that for all $k,\, h>0$ with $k^2 h \lesssim 1$
$$
k\norm{u-u_h}_{L^2(B)} + \norm{u-u_h}_{H^1(B)} \le C (h k^2)^\alpha k^{-1} \norm{u}_*.
$$
\end{corollary}
Comparing with the well-posed boundary value problem (\ref{helmholtzbvp}) and the sharp bounds (\ref{well-posed1}) and (\ref{well-posed2}), we note that the $k^{-1} \norm{u}_*$ term in the above estimate is analogous to the well-posed case term $\norm{f}_{L^2(\Omega)}$.
\subsection{Data perturbations} The analysis above can also handle the perturbed data
$$
\tilde{q} = q + \delta q,\quad \tilde{f} = f + \delta f,
$$
with the unperturbed data $q,\, f$ in (\ref{helmholtz}), and perturbations $\delta q \in L^2(\omega),\, \delta f \in H^{-1}(\Omega)$ measured by
$$
\delta (\tilde{q},\tilde{f}) = \norm{\delta q}_\omega + \norm{\delta f}_{H^{-1}(\Omega)}.
$$
The saddle points $(u,z)\in V_h \times W_h$ of the perturbed Lagrangian $L_{\tilde{q},\tilde{f}}$ satisfy
\begin{equation}\label{pweakform}
A[(u,z),(v,w)] = (\tilde{q},v)_{\omega} + \pair{\tilde{f},w},\quad (v,w)\in V_h \times W_h.
\end{equation}
\begin{lemma}\label{ptnormerror}
Let $u\in H^2(\Omega)$ be the solution to the unperturbed problem (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to the perturbed problem (\ref{pweakform}). Then there exists $C>0$ such that for all $h\in(0,1)$
$$
\tnorm{(u_h-\Pi_h u,z_h)} \le C (h \norm{u}_* +\delta(\tilde{q},\tilde{f})).
$$
\begin{proof}
Proceeding as in the proof of \cref{tnormerror}, the weak form gives
\begin{align*}
A[(u_h-\Pi_h u,z_h),(v,w)] &= (u-\Pi_h u,v)_\omega + G(u-\Pi_h u,w) - s(\Pi_h u,v)
\\
&+ (\delta q,v)_\omega + \pair{\delta f,w}.
\end{align*}
We bound the perturbation terms by
\begin{align*}
(\delta q,v)_\omega + \pair{\delta f,w} &\le \norm{\delta q}_\omega \norm{v}_\omega + C\norm{\delta f}_{H^{-1}(\Omega)} \norm{w}_W \\
&\le C \delta(\tilde{q},\tilde{f}) \tnorm{(v,w)}
\end{align*}
and we conclude by using the previously derived bounds for the other terms.
\end{proof}
\end{lemma}
\begin{theorem}\label{L2error_perturbed}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Let $u\in H^2(\Omega)$ be the solution to the unperturbed problem (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to the perturbed problem (\ref{pweakform}). Then there are $C>0$ and $\alpha \in (0,1)$ such that for all $k,\, h>0$ with $k h \lesssim 1$
$$
\norm{u-u_h}_{L^2(B)} \le C (h k)^\alpha k^{\alpha-2} ( \norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f}) ).
$$
\begin{proof}
Following the proof of \cref{L2error}, the residual satisfies
$$
\pair{r,w} = G(u_h,w-\pi_h w) - \pair{f,w-\pi_h w} + s^*(z_h,\pi_h w) + \pair{\delta f, \pi_h w},\quad w\in H^1_0(\Omega)
$$
and
$$
\norm{r}_{H^{-1}(\Omega)} \le C (\norm{u_h}_V + h\norm{f}_{L^2(\Omega)} + \norm{z_h}_W + \norm{\delta f}_{H^{-1}(\Omega)}).
$$
Bounding the first term in the right-hand side by \cref{ptnormerror} and (\ref{upperv})
\begin{equation*}
\norm{u_h}_V \le \norm{u_h-\Pi_h u}_V + \norm{\Pi_h u}_V
\le C (h \norm{u}_* +\delta(\tilde{q},\tilde{f}))
\end{equation*}
and the third one by \cref{ptnormerror} again, we obtain
$$
\norm{r}_{H^{-1}(\Omega)} \le C h(\norm{u}_* + \norm{f}_{L^2(\Omega)}) + C \delta(\tilde{q},\tilde{f}) \le C (h\norm{u}_* + \delta(\tilde{q},\tilde{f})).
$$
The continuum estimate in \cref{shifted3b} applied to $u-u_h$ gives
$$
\norm{u-u_h}_{L^2(B)} \le C \left( h \norm{u}_* + \delta(\tilde{q},\tilde{f}) \right)^\alpha \norm{u-u_h}_{L^2(\Omega)}^{1-\alpha},
$$
where $\norm{u-u_h}_{L^2(\omega)}$ was bounded by using \cref{ptnormerror} and (\ref{interp}). Then the bound
\begin{align*}
\norm{u-u_h}_{L^2(\Omega)} &\le \norm{u-\Pi_h u}_{L^2(\Omega)} + \norm{u_h-\Pi_h u}_{L^2(\Omega)} \\
&\le C ( h^2 \norm{u}_{H^2(\Omega)} + h^{-1} k^{-2} \norm{u_h-\Pi_h u}_V )\\
&\le C ( h^2 \norm{u}_{H^2(\Omega)} + k^{-2} \norm{u}_* + h^{-1} k^{-2} \delta(\tilde{q},\tilde{f}) ) \\
&\le C k^{-2} (\norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f}))
\end{align*}
concludes the proof.
\end{proof}
\end{theorem}
\begin{theorem}\label{H1error_perturbed}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Let $u\in H^2(\Omega)$ be the solution to the unperturbed problem (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to the perturbed problem (\ref{pweakform}). Then there are $C>0$ and $\alpha \in (0,1)$ such that for all $k,\, h>0$ with $k h \lesssim 1$
$$
\norm{u-u_h}_{H^1(B)} \le C (h k)^\alpha ( \norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f}) ).
$$
\begin{proof}
Following the proof of \cref{L2error_perturbed}, we now use \cref{cor_Holder_impr} to derive
\begin{align*}
\norm{u-u_h}_{H^1(B)} &\le C k ( \norm{u-u_h}_{L^2(\omega)} + \norm{r}_{H^{-1}(\Omega)})^\alpha ( \norm{u-u_h}_{L^2(\Omega)} + \norm{r}_{H^{-1}(\Omega)})^{1-\alpha}
\\
&\le C k \left(h\norm{u}_* + \delta(\tilde{q},\tilde{f})\right)^{\alpha} \left((k^{-2}+h)(\norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f}))\right)^{1-\alpha}
\\
&\le C k h^\alpha (k^{-2}+h)^{1-\alpha} (\norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f})),
\end{align*}
which ends the proof.
\end{proof}
\end{theorem}
Analogous to the unpolluted case, if $k^2 h \lesssim 1$ the above result becomes
$$
\norm{u-u_h}_{H^1(B)} \le C (h k^2)^\alpha k^{-1} (\norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f})),
$$
and combining \cref{L2error_perturbed} and \cref{H1error_perturbed} gives the following.
\begin{corollary}
Let $\omega \subset B \subset \Omega$ be defined as in \cref{cor_Holder}. Let $u\in H^2(\Omega)$ be the solution to the unperturbed problem (\ref{helmholtz}) and $(u_h,z_h)\in V_h \times W_h$ be the solution to the perturbed problem (\ref{pweakform}). Then there are $C>0$ and $\alpha \in (0,1)$ such that for all $k,\, h>0$ with $k^2 h \lesssim 1$
$$
k\norm{u-u_h}_{L^2(B)} + \norm{u-u_h}_{H^1(B)} \le C (h k^2)^\alpha k^{-1} (\norm{u}_* + h^{-1} \delta(\tilde{q},\tilde{f})).
$$
\end{corollary}
\section{Numerical examples}
We illustrate the above theoretical results for the unique continuation problem (\ref{helmholtz}) with some numerical examples. Drawing on previous results in \cite{BurmanSIAM}, we adjust the stabilizer in (\ref{weakform}) with a fixed stabilization parameter $\gamma>0$ such that $s(u,v) = \gamma \J(u,v) + \gamma h^2k^4 (u,v)_{L^2(\Omega)}$. The error analysis stays unchanged under this rescaling. Various numerical experiments indicate that $\gamma = 10^{-5}$ is a near-optimal value for different kinds of geometries and solutions. The implementation of our method and all the computations have been carried out in FreeFem++ \cite{FreeFem}. The domain $\Omega$ is the unit square, and the triangulation is uniform with alternating left and right diagonals, as shown in \cref{mesh}. The mesh size is taken as the inverse square root of the number of nodes.
In the light of the convexity assumptions in \cref{continuum_estimates}, we shall consider two different geometric settings: one in which the data is continued in the convex direction, inside the convex hull of $\omega$, and one in which the solution is continued in the non-convex direction, outside the convex hull of $\omega$.
In the convex setting, given in \cref{domain_conv}, we take
\begin{equation}\label{convex}
\omega = \Omega \setminus [0.1,0.9] \times [0.25,1],\quad
B = \Omega \setminus [0.1,0.9] \times [0.95,1]
\end{equation}
for continuing the solution inside the convex hull of $\omega$. This example does not correspond exactly to the specific geometric setting in \cref{cor_Holder}, but all the theoretical results are valid in this case as proven in \cref{ex_convex_square} below.
\begin{example}\label{ex_convex_square}
Let $\omega\subset B \subset \Omega$ be defined by \eqref{convex} (\cref{domain_conv}). Then the stability estimates in \cref{cor_Holder}, \cref{cor_Holder_impr} and \cref{shifted3b} hold true.
\begin{proof}
Consider an extended rectangle $\tilde{\Omega} \supset \Omega$ such that the unit square $\Omega$ is centred horizontally and touches the upper side of $\tilde{\Omega}$, and $\tilde{\omega} \supset \omega$ and $\tilde{B} \supset B$ are defined as in \cref{cor_Holder}. Choose a smooth cutoff function $\chi$ such that $\chi=1$ in $\Omega \setminus \omega$ and $\chi=0$ in $\tilde{\Omega} \setminus \Omega$.
Applying now \cref{cor_Holder} for $\tilde{\omega},\, \tilde{B},\, \tilde{\Omega}$ and $\chi u$ we get
\begin{align*}
\norm{u}_{H^1(B\setminus \omega)} &\le C \norm{\chi u}_{H^1(\tilde{B}\setminus \tilde{\omega})}
\le C (\norm{\chi u}_{H^1(\tilde{\omega})} +
\norm{\Delta (\chi u) + k^2 \chi u}_{L^2(\tilde{\Omega})})^\alpha \norm{\chi u}_{H^1(\tilde{\Omega})}^{1-\alpha}
\\& \le C (\norm{u}_{H^1(\omega)} +
\norm{\Delta u + k^2 u}_{L^2(\Omega)})^\alpha \norm{u}_{H^1(\Omega)}^{1-\alpha},
\end{align*}
where we have used that the commutator $[\Delta,\chi]u$ is supported in $\omega$.
A similar proof is valid for the estimates in \cref{cor_Holder_impr} and \cref{shifted3b}.
\end{proof}
\end{example}
We will give results for two kinds of solutions: a Gaussian bump centred on the top side of the unit square $\Omega$, given in \cref{ex_gauss}, and a variation of the well-known Hamadard solution given in \cref{ex_hadamard}.
\begin{example}\label{ex_gauss} Let the Gaussian bump
\begin{equation*}
u = \exp \left( -\frac{(x-0.5)^2}{2\sigma_x} - \frac{(y-1)^2}{2\sigma_y} \right),\quad \sigma_x=0.01,\sigma_y=0.1,
\end{equation*}
be a non-homogeneous solution of (\ref{helmholtz}), i.e. $f = -\Delta u - k^2 u$ and $q=u|_{\omega}$.
\end{example}
\begin{figure}
\centering
\includegraphics[draft=false, width=0.4\textwidth]{mesh.eps}
\caption{Mesh example.}
\label{mesh}
\end{figure}
\cref{gauss_conv_10} shows that for \cref{ex_gauss}, when $k=10$, the numerical results strongly agree with the convergence rates expected from \cref{L2error} and \cref{H1error}, and \cref{tnormerror}, i.e. sub-linear convergence for the relative error in the $L^2$- and $H^1$-norms, and quadratic convergence for $\J(u_h,u_h)$. Although in \cref{gauss_conv_50} we do obtain smaller errors and better than expected convergence rates when $k=50$, various numerical experiments indicate that this example's behaviour when increasing the wave number $k$ is rather a particular one. For oscillatory solutions, such as those in \cref{ex_hadamard}, with fixed $n$, or the homogeneous $u = \sin(k x /\sqrt{2}) \cos(k y /\sqrt{2})$, we have noticed that the stability deteriorates when increasing $k$.
In the non-convex setting we let
\begin{equation}\label{non-convex}
\omega = (0.25,0.75) \times (0,0.5),\quad
B = (0.125,0.875) \times (0,0.95),
\end{equation}
and the concentric disks
\begin{equation}\label{disk}
\omega = D((0.5,0.5),0.25),\quad
B = D((0.5,0.5),0.45),
\end{equation}
respectively shown in \cref{domain_nonconv} and \cref{domain_disk}, and we notice from \cref{gauss_nonconv} that the stability strongly deteriorates when one continues the solution outside the convex hull of $\omega$, as the error sizes and rates worsen.
We test the data perturbations by polluting $f$ and $q$ in (\ref{helmholtz}) with uniformly distributed values in $[-h,h]$, respectively $[-h^2,h^2]$, on every node of the mesh. It can be seen in \cref{gauss_perturbation} that the perturbations are visible for an $O(h)$ amplitude, but not for an $O(h^2)$ one.
\begin{figure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[draft=false, width=\textwidth]{gauss_convex.eps}
\caption{Convex direction (\ref{convex}).}
\label{domain_conv}
\end{subfigure}
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[draft=false, width=\textwidth]{gauss_nonconvex.eps}
\caption{Non-convex direction (\ref{non-convex}).}
\label{domain_nonconv}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.32\textwidth}
\includegraphics[draft=false, width=\textwidth]{gauss_disk.eps}
\caption{Non-convex direction (\ref{disk}).}
\label{domain_disk}
\end{subfigure}
\caption{Computational domains for \cref{ex_gauss}.}
\end{figure}
\begin{figure}
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{rates_gauss_c_10.eps}
\caption{$k=10$. Circles: $H^1$-error, rate $\approx 0.64$; squares: $L^2$-error, rate $\approx 0.66$; down triangles: $h^{-1} \J(u_h,u_h)$, rate $\approx 1$; up triangles: $\norm{z}_W$, rate $\approx 1.3$.}
\label{gauss_conv_10}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{rates_gauss_c_50.eps}
\caption{$k=50$. Circles: $H^1$-error, rate $\approx 1.02$; squares: $L^2$-error, rate $\approx 2$; down triangles: $h^{-1} \J(u_h,u_h)$, rate $\approx 1$; up triangles: $\norm{z}_W$, rate $\approx 2$.}
\label{gauss_conv_50}
\end{subfigure}
\caption{Convergence in $B$ for \cref{ex_gauss} in the convex direction (\ref{convex}).}
\end{figure}
\begin{figure}
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{rates_gauss_nc_10.eps}
\caption{Non-convex direction (\ref{non-convex}).}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{rates_gauss_disk_10.eps}
\caption{Non-convex direction (\ref{disk}).}
\end{subfigure}
\caption{Convergence in $B$ for \cref{ex_gauss}, $k=10$.}
\label{gauss_nonconv}
\end{figure}
\begin{figure}
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{gauss_u_pert_fqh_10.eps}
\caption{Perturbation $O(h)$.}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{gauss_u_pert_fqh2_10.eps}
\caption{Perturbation $O(h^2)$.}
\end{subfigure}
\caption{Convergence in $B$ when perturbing $f$ and $q$ in \cref{ex_gauss} for (\ref{convex}), $k=10$.}
\label{gauss_perturbation}
\end{figure}
Let us recall that the stability estimates for the unique continuation problem are closely related to those for the notoriously ill-posed Cauchy problem, see e.g. \cite{Alessandrini} or \cite{Isakov}. It is of interest to consider the following variation of a well-known example due to Hadamard, since this example can be used to show that conditional H\"older stability is optimal for the unique continuation problem.
\begin{example}\label{ex_hadamard} Let $n \in \N$ and consider the Cauchy problem
\begin{align*}
\begin{cases}
\Delta u + k^2 u = 0 \quad &\text{in } \Omega=(0,\pi)\times (0,1), \\
u(x,0) = 0 \quad &\text{for } x\in [0,\pi], \\
u_y (x,0) = \sin(nx) \quad &\text{for } x\in [0,\pi],
\end{cases}
\end{align*}
whose solution for $n>k$ is given by $u = \frac{1}{\sqrt{n^2-k^2}} \sin(nx) \sinh(\sqrt{n^2-k^2} y)$, for $n=k$ by $u = \sin(kx) y$, and for $n<k$ by $u = \frac{1}{\sqrt{k^2-n^2}} \sin(nx) \sin(\sqrt{k^2-n^2}y)$.
\end{example}
It can be seen in \cref{hadamard_conv_fig} that the convergence rates agree with the ones predicted for the convex setting
\begin{equation}\label{hadamard_conv}
\omega = \Omega \setminus [\pi/4,3\pi/4] \times [0,0.25],\quad
B = \Omega \setminus [\pi/4,3\pi/4] \times [0,0.95],
\end{equation}
i.e. sub-linear convergence for the relative error in the $L^2$- and $H^1$-norms, and quadratic convergence for $\J(u_h,u_h)$, although one can notice that the values of the jump stabilizer $\J(u_h,u_h)$ visibly increase compared to \cref{ex_gauss}.
When continuing the solution in the non-convex direction, the stability strongly deteriorates and for coarse meshes the numerical approximation doesn't reach the convergence regime, as it can be seen in \cref{hadamard_nonconv_fig} for the non-convex setting
\begin{equation}\label{hadamard_nonconv}
\omega = (\pi/4,3\pi/4) \times (0,0.5),\quad
B = (\pi/8,7\pi/8) \times (0,0.95).
\end{equation}
\begin{figure}[t]
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{rates_had_c_m12_k10.eps}
\caption{Convex direction (\ref{hadamard_conv}). Circles: $H^1$-error, rate $\approx 0.94$; squares: $L^2$-error, rate $\approx 0.83$; down triangles: $h^{-1} \J(u_h,u_h)$, rate $\approx 1$; up triangles: $\norm{z}_W$, rate $\approx 1.6$.}
\label{hadamard_conv_fig}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[draft=false, width=\textwidth]{rates_had_nc_m12_k10.eps}
\caption{Non-convex direction (\ref{hadamard_nonconv}). Circles: $H^1$-error; squares: $L^2$-error; down triangles: $h^{-1} \J(u_h,u_h)$; up triangles: $\norm{z}_W$.}
\label{hadamard_nonconv_fig}
\end{subfigure}
\caption{Convergence in $B$ for \cref{ex_hadamard}, $k=10$, $n=12$.}
\end{figure}
|
1,116,691,500,439 | arxiv | \section{Summary:}
While achieving a compression ratio of 2.0 bits$\slash$base ,the new
algorithm codes non-N bases\footnote{``non-N bases" refers to bases excpet N.Thus A,T,G or C. \ N stands for unknown base.} in fixed length.It dramatically reduces the time
of coding and decoding than previous DNA compression algorithms and some
universal compression programs.
\section{Availability:}http://grandlab.cer.net/topic.php?TopicID=50
\section{Contact:} \href {[email protected]}{[email protected]} {[email protected]} {[email protected]}
\end{abstract}
\section{Introduction}
File compression reduces file redundancy in order to represent more information in less signs in accordance with information theory \citep{Shannon1948}.As specified algorithm for image,audio and video are devised,it is necessary to devise the algorithm specified for DNA compression since huge amounts of DNA sequences needs to be stored and communicated to a large number of people.\citep{ACM2005} \citep{CS_BS}Although some universal compressors\citep{LZ77} are used in bioinformatics field,new DNA sequence compressors are being devised,such as Biocompress\citep{Biocompress} ,Biocompress-2 \citep{Biocompress2}, GenCompress \citep{GenCompress},Cfact \citep{Cfact}, DNACompress\citep{Repeat},CTW-LZ \citep{CTWLZ} and GeNML \citep{ACM2005}.
But they have a big problem,too slow execution.We improve our
LUT\citep{LUT} and use new file structure to identify different types of segment.The most advantage of this algorithm is fast
execution and easy implementation.The compression and decompression
speed is much faster than many newly-devised DNA-specified and well-known universal compression
algorithms.Since the
compression ratio is not much higher than existing ones and the compression
speed is impressively fast,our algorithm is an applicable algorithm
for fast DNA sequence compression,especially for database records compression.
\begin{methods}
\section{methods}
\subsection{Coding non-N bases}
non-N bases have four prossibilities:A,T,G or C.Each of them corresponds to a unique combination of two binary numbers.
We code them as A to 00,T to 01,G to 10,C to 11.Thus,we take 1 Byte(8 bits)to store 4 bases.
\subsection{File format of compressed file}
We will begin discussing file structure with the definition of ``section",a DNA segment.``section" contains a serie of successive Ns and ends at the last non-N base ahead the next serie of successive Ns.
``section" is the basic element to which we consider in compression and decompression.
Each DNA section corresponds to a ``file section" which contains the information of both N and non-N bases in this section.
Each file section starts with an 8 Bytes head.The first 4 Bytes records the amount of N bases whereas the following 4 Bytes records the number of non-N bases in this section.This means that each section corresponds to a real DNA segment which has at most $2^{32}$ N bases and $2^{32}$ non-N bases respectively.
The coded values of non-N bases locate after the head.The coded information is written into destination file Byte by Byte. Considering the number of non-N bases in a section may not be a multiple of 4,the second 4 Bytes in head provides accordance for decompression program about how many bit values are effective and where the next section begins.
\subsection{Compression algorithm}
The compression program reads characters from source file and writes coded binary values into destination file,restricted by the file format defined above.
Steps of compression algorithm is as below.
\begin{enumerate}
\item Preserve 8 Bytes at the beginning of file section.
\item Count the number of Ns in a successive N bases segment(To a sequence starts from non-N bases,this value is 0.) until the first non-N base is encountered.Write the number of Ns into the first 4 Bytes of the section head.
\item Code all following successive non-N bases into destination file while count their number until the next N is encountered.Write the number of non-N bases into the second 4 Bytes of the section head.
\item Move the file writing pointer to the beginning of next Byte in destination file.
\item Repeat all the above until the end-of-file is encounted.
\end{enumerate}
\subsection{Decompression algorithm}
Steps of decompression algorithm is as below.
\begin{enumerate}
\item Read the head(the first 8 Bytes) to obtain information about how many Ns are in this section and how many non-N bases are effective.
\item Write Ns into destination file,the decompressed file,according to the number written in the first 4 Bytes of the head.
\item Read the following 4 Bytes to determind how many bits should be decoded then and where the next section begins.The next section begins from the most nearby next Byte of compressed file.
\item Decode effective bits whose amount is recorded in last 4 Bytes of this section's head.Move reading pointer to the next section.
\item Repeat all the above from the beginning of the next section until the end-of-file is encounted.
\end{enumerate}
\subsection{Algorithm implementation}
The C++ and C source codes of algorithm implementation are avaliable at the website provided in Abstract of this paper.
\end{methods}
\section{experiments}
\label{experiment}
Experiments are operated to test our algorithm. Codes for testing the algorithm are
continually revising.\citep{OurCode}These tests are performed on a computer whose CPU is AMD Duron 750MHz and operating system is
MagicLinux 1.2 (Linux Kernel 2.6.9) without swap partition. Testing programs are
executed at multiuser text mode and compiled by gcc 3.3.2 with optimization level O3.
The file system is ext3.Files are stored on a 4.3 GB Quantum Fireball hard disk with 5400 RPM.
Table \ref{size} compares compression ratio while table \ref{time} compares running time.
\begin{table}[!ht]
\processtable{Comparison on compression ratio \label{size}}
{\begin{tabular}{lccccc}\toprule
sequence & size & ours & DNA & Gzip &bzip2\\\midrule
atatsgs & 9647 & 2.0068 & -- & 2.1702 & 2.15\\
atef1a23 & 6022 & 2.0113 & -- & 2.0379 & 2.15\\
atrdnaf & 10014 & 2.0068 & -- & 2.2784 & 2.15\\
atrdnai & 5287 & 2.0125 & -- & 1.8846 & 1.96\\
chmpxx & 121024 & 2.0005 & 1.6716 & 2.2821 & 2.12\\
chntxx & 155939 & 2.0004 & 1.6127 & 2.3349 & 2.18\\
hehcmvcg & 229354 & 2.0003 & 1.8492 & 2.3278 & 2.17\\
hsg6pdgen & 52173 & 2.0013 & -- & 2.2444 & 2.07\\
humdystrop & 38770 & 2.0018 & 1.9116 & 2.3633 & 2.18\\
humghcsa & 66495 & 2.0010 & 1.0272 & 2.0655 & 1.31\\
humhdabcd & 58864 & 2.0011 & 1.7951 & 2.2399 & 2.07\\
humhprtb & 56737 & 2.0012 & 1.8165 & 2.2670 & 2.09\\
mmzp3g & 10833 & 2.0065 & -- & 2.3225 & 2.13\\
mpomtcg & 186609 & 2.0004 & 1.8920 & 2.3291 & 2.17\\
mtpacg & 100314 & 2.0007 & -- & 2.2922 & 2.12\\
vaccg & 191737 & 2.0004 & 1.7580 & 2.2520 & 2.09\\
xlxfg512 & 19338 & 2.0035 & -- & 1.8310 & 1.80\\
chr10(rice) & 22432531 & 2.0000 & -- & 2.4498 & 2.3033\\
Average & -- & 2.0031 & 1.7037 & 2.3224 & 2.0674\\ \botrule
\end{tabular}}{Compress ratio of other algorithms are cited from their original papers.As the compression ratio of newly-devised algorithms are similiar,we take DNACompress as an example."ours" refers to our algorithm.DNA stands for DNACompress.The unit of file size is bit rather than Byte.}
\end{table}
\begin{table}[!ht]
\processtable{Comparison on running time \label{time}}
{\begin{tabular}{lccccc}\toprule
sequence & Gzip(s)&encode(CLK) & decode(CLK) & encode(s) & decode(s)\\ \midrule
atatsgs & 0.013 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
atef1a23 & 0.011 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
atrdnaf & 0.014 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
atrdnai & 0.010 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
chmpxx & 0.105 & 10000 & 10000 & 0.01 & 0.01 \\
chntxx & 0.135 & 20000 & 20000 & 0.02 & 0.02\\
hehcmvcg & 0.198 & 30000 & 30000 & 0.03 & 0.03\\
hsg6pdgen & 0.044 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
humdystrop & 0.037 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
humhdabcd & 0.050 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
humghcsa & 0.055 & 10000 & 10000 & 0.01 & 0.01\\
humhprtb & 0.049 & $<$10000 & $<$10000 & 0.01 & 0.01\\
mmzp3g & 0.014 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
mpomtcg & 0.100 & 20000 & 30000 & 0.02 & 0.03\\
mtpacg & 0.088 & 10000 & 10000 & 0.01 & 0.01\\
vaccg & 0.164 & 30000 & 20000 & 0.03 & 0.02\\
xlxfg512 & 0.018 & $<$10000 & $<$10000 & $<$0.01 & $<$0.01\\
chr10(rice) & 9.5 & 3460000 & 3510000 & 3.46 & 3.51\\ \botrule
\end{tabular}}{``Gzip" includes the total of time elapsed in both
compression and decompression by Gzip.More experiments indicate that bzip2 takes more time to perform same operation.Following four fields list the time elapsed in compression and decompression respectively.``encode" means compression while ``decode" means decompression.Each operation is evaluated in two units,CPU clock and second.}
\end{table}
\section{Discussion}
The performance of a compression algorithm has two sides,the compression ratio and the running time.Many newly-devised DNA compression algorithms focus on compression ratio while ignore the running time.But the time occupation of obtaining a little lower compression ratio is very high.
Many of them run 100 times slower than universal compression algorithm,according to Table 2 of Chen's paper \citep{Repeat}.Our algorithm runs many times faster than Gzip which is 100 times faster than newly-devised algorithms.Considering the compression ratio and running time both advance traditional compressors(Gzip and bzip2) considerablly,our algorithm is a wise choice of replacing them.It is more useful in those fields which need fast running,such as database.
|
1,116,691,500,440 | arxiv | \section{Introduction}
The Sloan Digital Sky Survey (SDSS)~\citep{Yo00} is an imaging and
spectroscopic survey that has now mapped over $ 1/4$ of the sky. It
has already proven to be a powerful tool for the identification of
Galactic substructure. For example, \citet{Ne02} and \citet{Ya03}
identified a ring at low Galactic latitude, often called ``the
Monoceros Ring''. It spans over 100$^\circ$ in the sky~\citep[see
also][]{Ib03,Ro03}, but its progenitor remains unclear~\citep{Pe05}.
\citet{Od01} used SDSS data to find the spectacular 10$^\circ$ tidal
tails around the sparse and disrupting Galactic globular cluster Pal
5. More recently still, \citet{Be06a} found $4.5^\circ$ tails around
the high-latitude globular cluster NGC 5466, while \cite{Gr06a}
discovered a $63^\circ$ tail that presumably arises from a so far
unidentified globular cluster.
The best known example of a stream is the tidally stripped stars and
globular clusters associated with the Sagittarius dwarf spheroidal. A
panorama of the the Sagittarius stream in the Northern hemisphere was
recently obtained by \cite{Be06b}, who mapped out the stars satisfying
$g-r < 0.4$ in almost all of SDSS Data Release 5 (DR5). This color
plot of the high-latitude Galactic northern hemisphere has been dubbed
the ``Field of Streams''. In addition to the features associated with
the Sagittarius dSph, the plot exhibits extensive substructure. The
purpose of this paper is to analyze the ``Orphan Stream'' -- so named
for its lack of obvious progenitor. This is a striking feature
discovered by~\citet{Be06b} in the Field of Streams, and independently
detected in public SDSS data by~\citet{Gr06b}.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=8cm]{f1.eps}
\caption{\label{fig:vasa} Left: A false color RGB composite of density
of stars with $20.0 < r < 22.0$. Blue corresponds to $0.0<g-r \le0.2$, green
$0.2 < g-r \le 0.4$ and red $0.4 < g-r \le 0.6$. Right: The
Sagittarius and Monoceros structures are marked, together with the
on-stream and off-stream fields along the Orphan Stream. Also shown is
the location of the recently-discovered probable globular cluster
Segue 1~\citep{Be06c}. The black curves mark the limits of the DR5
spectroscopic footprint.}
\end{center}
\end{figure*}
\section{Morphology of the Orphan Stream}
SDSS imaging data are produced in five photometric bands, namely $u$,
$g$, $r$, $i$, and $z$~\citep[see e.g.,][]{Fu96,Ho01,Sm02,Gu06}. The
data are automatically processed through pipelines to measure
photometric and astrometric properties and to select targets for
spectroscopic follow-up~\citep{Lu99,St02,Pi03,Iv04,AM06}. To correct
for Galactic reddenning, we use the maps of~\citet*{Sc98}. All the
magnitudes in the paper are reddenning corrected. Data Release 5
covers $\sim 8000$ square degrees around the Galactic North Pole, and
3 strips in the Galactic southern hemisphere.
The left panel of Figure~\ref{fig:vasa} shows the Orphan Stream in an
RGB composite image. It has been constructed using all SDSS DR5
stars~\footnote{A small number of Data Release 6 stars are used to
ensure continuous photometric coverage of the Orphan Stream
(c.f. Grillmair 2006b, who used Data Release 4 only).} with $20.0 < r
< 22.0$, with blue for stars with $0.0< g-r \le 0.2$, green for stars
with $0.2 < g-r \le 0.4$ and red for stars with $0.4 < g-r \le 0.6$.
The Orphan Stream is clearly visible running roughly from top to
bottom at right ascensions $\alpha \approx 150^\circ-165^\circ$. It
may be traced over nearly $50^\circ$ of arc in DR5. Some familiar
objects are marked in the right panel of Figure~\ref{fig:vasa},
including the bifurcated Sagittarius stream, the Monoceros Ring, the
newly discovered globular cluster Segue 1~\citep{Be06c} and the two
distant dwarf spheroidal galaxies Leo I and Leo II. On moving from
lower to higher declinations, the stream becomes less detectable.
\begin{figure}
\begin{center}
\includegraphics[width=15cm]{f2.eps}
\caption{\label{fig:hess} Left: Hess diagram for all stars in DR5. The
typical errors in color are shown as a column of error bars on the
left of the plot. The vertical line shows the color cut to constrain
the population to blue stars. Right: Isochrones from \citet{Gi04},
with different colors corresponding to different metallicities. Four
representative ages are shown for each metallicity (1, 5, 10 and 14
Gyrs, left to right). Masks based on the ridgelines of M92 (solid),
M13 (dotted) and M71 (dashed) from \cite{Cl05} are also shown.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{f3a.eps}
\includegraphics[width=7cm]{f3b.eps}
\caption{\label{fig:stellarpops} Left: For each field, the excess of
stars in the Orphan Stream is shown as a function of age for different
metallicities. Each point on the curve gives the maximum number of
excess stars over all distance moduli. The peak of each curve gives
the maximal signal for a given metallicity and is marked with its
corresponding distance. It is apparent that there is a trade-off
between age, metallicity and distance. Although there is a
degeneracy, old metal-poor populations perform slightly better than
young metal-rich ones. The filled circles show the signal picked up by
the ridgeline masks for the three clusters M92, M13 and M71. Right:
For three metal-poor populations ([Fe/H] $= -2.3, -1.7, -1.3$), we
show the excess stars as a function of distance modulus for the age at
the peak of the curve in the left panel. Also shown as solid and
dotted lines are the curves using the ridgelines for M92 and M13. The
inset shows the average of the five curves, together with a Gaussian
fit in red. The number gives the central value of the Gaussian, which
is our best estimate for the distance modulus, and hence the distance
in kpc.}
\end{center}
\end{figure}
\section{Distances and Stellar Populations of the Orphan Stream}
Our aim is to constrain the distance and the stellar content of the
Orphan Stream by comparison with CMDs of populations of known age and
metallicity. For this purpose, we create masks (or color-magnitude
boxes) based on the $r$ versus $g\!-\!r$ ridgeline of globular
clusters and theoretical isochrones by shifting them both horizontally
and vertically. The mask is applied to select stars from the fields
on and off the stream shown in the right panel of
Figure~\ref{fig:vasa}. The signal is assessed by measuring the
difference in the number of stars between the on and off streams as a
function of distance modulus.
The left panel of Figure~\ref{fig:hess} shows a Hess diagram using all
the stars in DR5. From blue to red, the main components of this CMD
are the halo, thick disk and thin disk of the Milky Way. The vertical
line shows a color cut at $g-r = 0.75$ used to minimise contamination
from the disc stars. The right panel shows a sequence of theoretical
isochrones computed by ~\citet{Gi04} for the SDSS photometric system.
Different metallicities are distinguished by different colors as shown
in the key to the panel. Although we investigate a range of ages, only
the isochrones corresponding to 1, 5, 10 and 14 Gyrs are displayed.
To compare the model predictions with observations, we also use the
illustrated masks based on the ridgelines of the old (14 Gyrs)
clusters M92 ([Fe/H] = $-2.2$), M13 ([Fe/H] = $-1.6$) and M71 ([Fe/H]
= $-0.7$). These are produced from the data of \citet{Cl05} and chosen
to span a representative range of metallicities.
The Orphan Stream (and parts of the Sagittarius stream) shown in
Figure~\ref{fig:vasa} are blue-green in our RGB scheme. This
corresponds to $g-r \sim 0.3$, which is typical for the old, metal-poor
halo. Most of the stars are redwards of $g-r \sim 0.2$. The stars that
are bluewards could be either blue stragglers/blue horizontal branch
stars, or main sequence turn-off stars scattered by large photometric
errors.
It is well-known that, given a color-magnitude diagram, simultaneously
fitting for age, metallicity and distance modulus is
degenerate. However, we have some clues as to the likely nature of the
solution. The length and the width of the Orphan Stream suggests that
it is dynamically old. If star formation ceased after the interaction
that produced the tails, then the stars in the Stream are expected to
be old. Also, when viewed from the Sun, tidal streams deviate from a
great circle. The deviation is controlled by $\langle D \rangle
/R_{\rm GC}$, where $\langle D \rangle$ is the stream's average
heliocentric distance and $R_{\rm GC}$ is the offset of the Sun from
the Galactic Center (here taken as 8 kpc). This parallactic effect
gives an average heliocentric distance to the Stream of $\sim 15 \pm
5$ kpc.
The results of applying the masks to the photometric data are shown in
Figure~\ref{fig:stellarpops}. In the left column, each panel
corresponds to a different field. For a given metallicity and age, the
masks are applied at different distance modulus and the number of
excess stars in the on-stream field is calculated. The maximum value
fixes a distance modulus. The number of excess stars at this distance
modulus is used to build up the curve. The different colored lines
correspond to different metallicities. Even though there is a
degeneracy, the distances to which models of different metallicities
converge are broadly consistent. For example, in Field 1, most of the
distances are between 20 and 30 kpc, whereas in Field 5, they are
mostly above 30 kpc. These values are somewhat higher than the $15
\pm5$ kpc obtained from the parallactic effect, but consistent given
the uncertainties and given the strong assumptions in the latter
calculation. The M92 and M13 masks perform similarly to the
corresponding isochrones, although there are some small discrepancies.
It is reassuring that the results based on theoretical models are
supported by those based on data.
Even though young stellar population seemingly perform quite well in
Figure~\ref{fig:stellarpops}, this is not really the case. From the
turn-off color of the Orphan Stream (see e.g., Figure~\ref{fig:vasa}),
we can exclude young populations with a turn-off bluer than $g-r \sim
0.2$. Young isochrones are so blue that even the metal-rich ones
extend blue enough to overlap with the Orphan Stream CMD and hence
give a signal in Figure~\ref{fig:stellarpops}.
The general conclusion is that the old, metal-poor masks perform
better, and in some instances, significantly better than the
metal-rich masks. Given this, we take the three metal-poor models,
namely [Fe/H] $= -2.2, -1.7, -1.3$, and identify the age at which the
signal is maximised. These masks are now used to investigate the
behavior of the signal as a function of distance modulus, shown in the
panels in the right column. Also shown are the curves based on the
ridgelines of M92 (solid) and M13 (dotted). They all show a similar
performance and so we average them as displayed in the inset (solid
black curve). A Gaussian model is then fit to this curve to give a
central value, its uncertainty and a dispersion. These numbers are
recorded in Table~\ref{tab:orphan}. The uncertainty in the mean gives
a lower bound to the true distance error, whilst the dispersion
overestimates the error, as the curves in the insets are a convolution
of the true error distribution with the mask.
\citet{Gr06b} has also recently analyzed the stars of the Orphan
Stream. Grillmair's detection method uses M13 as a template for the
stellar population of the stream. His Figure 3 shows a clear upper
main sequence and sub-giant branch -- which appear to be located
somewhat blueward of the M13 ridgeline. This is consistent with a
stream composed of a somewhat younger and/or metal-poor stellar
population than M13. In turn, this is in agreement with the blue color
of the Orphan Stream in our Figure~\ref{fig:vasa}, suggesting that our
detection method is picking up primarily turn-off and upper main
sequence stars.
\begin{table}
\centering
\caption{Positions, distance moduli and distances of the Orphan Stream.}
\label{tab:orphan}
\begin{tabular}{@{}ccccc}
Field & $\alpha$ & $\delta$ & $m-M$ & $D$ \\ \hline
1 & $162.1^\circ$ & $-0.5^\circ$ & $(16.5 \pm 0.1) \pm 0.7$ & $(20
\pm 1)^{+7}_{-5}$ kpc \\
2 & $158.9^\circ$ & $8.5^\circ$ & $(16.5 \pm 0.1) \pm 0.9$ &
$(20\pm 1)^{+10}_{-7}$ kpc \\
3 & $155.4^\circ$ & $17.0^\circ$ & $(17.1 \pm 0.1) \pm 0.7$ &
$(26\pm 1)^{+10}_{-7}$ kpc \\
4 & $152.3^\circ$ & $25.0^\circ$ & $(17.5 \pm 0.1) \pm 0.8$ &
$(32\pm 1)^{+13}_{-10}$ kpc \\
5 & $149.4^\circ$ & $32.0^\circ$ & $(17.5 \pm 0.1) \pm 0.9$ &
$(32\pm 1)^{+15}_{-12}$ kpc \\
\hline
\end{tabular}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=10cm]{f4.eps}
\caption{\label{fig:bhbs} Differential histograms of number of stars
with $g-r < 0.1$ as a function of apparent magnitude in different
fields.}
\end{center}
\end{figure}
Figure~\ref{fig:bhbs} shows the results of a search for blue
horizontal branch (BHB) stars in the Orphan Stream. These are
identified with the cut $g-r < 0.1$, and as usual the difference
between on-stream and off-stream fields is computed. BHB stars have an
absolute magnitude of $M_r = 0.75$ (see the right panel of
Figure~\ref{fig:hess}). Such a BHB population would be detected by
peaks in the differential number distribution in the apparent
magnitude range $16< r <18$ for the five fields inspected. There is no
evidence for such a signal. However, blue stragglers have an absolute
magnitude $M_r$ of between 2 and 4 and there are hints of peaks in
fields 1 and 5 for $r \approx 20$. This seems consistent, given the
distance moduli in Table~\ref{tab:orphan}.
\section{Absolute Magnitude and Width of the Orphan Stream}
Figure~\ref{fig:dist} shows cross-sections across the stream in a
coordinate system which has been rotated so the stream lies along the
$y$-axis. The cross-section contains only those stars with $g\!-\!r<
0.4$ and $21.0 < r <22.0$. We estimate that the thickness (FWHM) of
the Orphan Stream is $\sim 2^\circ$ in projection (or $\mathrel{\spose{\lower 3pt\hbox{$\sim$} 650$ pc
assuming a conservative distance of $\sim 20$ kpc), which would make
it broader than all known globular cluster streams, but smaller than
the branches of the Sagittarius stream and the Monoceros Ring. This
suggests that the progenitor was a low luminosity dwarf satellite
galaxy, rather than a globular cluster.
Taking the total profile (upper curve in Figure~\ref{fig:dist}), we
fit a polynomial to estimate the background and then compute the
luminosity of the stream. The number of excess stars in the $45^\circ$
arc of the stream is $\sim 4110$ within $160^\circ < x < 164^\circ$.
Hence, the total apparent magnitude of the stream in these stars is
around $r \sim 12$ (assuming an average $r$ magnitude of 21.0). Given
the stream FWHM of $\sim 2^\circ$, the average density of these stars
is $\sim 45.6$ per square degree, which translates to an average
surface brightness of the stream of $\sim 34\fm6$ per square
arcsec. To correct for other stars, we use the data on the luminosity
function of M92 derived from CFHT observations (Clem 2006, private
communication). This gives the average surface brightness of the
stream as $\sim 32\fm4$ per square arcsec and the total $r$ magnitude
as $\sim 9.8$. Assuming the distance really is $\sim 20$ kpc gives an
absolute magnitude of the $45^\circ$ arc as $M_r \sim -6.7$ Assuming
plausibly that there ought to be at least the same again on the other
side of the progenitor, then we can boost the total to at least $M_r
\sim -7.5$ for the stream stars alone. This number must be augmented
by the (unknown) contribution from the progenitor nucleus, if it is
still intact.
\begin{figure}
\begin{center}
\includegraphics[height=8cm]{f5.eps}
\caption{\label{fig:dist} Profiles in stars with $g\!-\!r< 0.4$ and
$21.0 < r <22.0$ across the stream in a rotated coordinate system in which
the stream lies along the $y$-axis. Notice that the peak of the
profile barely shifts as we move along the stream. }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[height=8cm]{f6a.eps}
\includegraphics[height=8cm]{f6b.eps}
\caption{\label{fig:velocities} Upper: Excess number of stars in the
on- and off-stream fields in the DR5 spectroscopic catalogue using the
M92 (solid) and M13 (dotted) ridgeline masks. Even though the
catalogue is much smaller, there is a reasonably clear detection of
the Orphan Stream at a distance modulus of $\sim 17$. Lower: Velocity
histograms of stars selected with the M92 and M13 ridgeline masks at
the distance moduli satisfying $16.5 \le m-M \le 17.5$ in Field 1 and
$17.5 \le m-M \le 18.5$ in Field 5. Black represents on-stream and red
off-stream field stars.}
\end{center}
\end{figure}
\newpage
\section{Velocities of the Orphan Stream}
The spectroscopic portion of SDSS DR5 covers $5713$ deg$^{2}$
~\citep{Ad06}. The outline of the spectroscopic footprint is shown as a
black line in Figure~\ref{fig:vasa}. It does not cover Fields 3 and 4
of the Orphan Stream. The number of stars in the spectroscopic database
is $\sim 215\,000$. Only stars brighter than $g \sim 20$ were
targeted, and the selection algorithm is strongly non-uniform. A
large number of spectra of standard stars were taken for calibration
purposes. Other targets include K giants and dwarfs, F turn-off and
subdwarf stars, as well as BHBs.
The accuracy of the radial velocities measured from the $R = 2000$
spectra obtained by the SDSS spectrographs vary from 7 km/s for the
brighter stars ($r \sim 14$) to on the order of 20 km/s for the
fainter stars ($r \sim 20$)~\citep{Al06}, based on empirical
tests. Information on the stellar atmospheric parameters (T$_{\rm
eff}$, $\log$ g , [Fe/H]) for the DR5 stars with adequate spectra are
presently being obtained by application of an automated spectroscopy
analysis pipeline~\citep{Bee06}. This information will be utilized in
future discussions of the Orphan Stream, once a comparison of the
pipeline-derived atmospheric parameters with independently obtained
high-resolution spectroscopy has been completed~\citep{Si05}.
To test whether there is a signal of the Orphan Stream in the
spectroscopic database, we apply the masks of M92 and M13 discussed
earlier to the on- and off-stream fields. Given the faint magnitude
cut-off, there are no turn-off stars belonging to the Stream in the
database. So, our earlier algorithm is slightly changed to include a
color cut designed to pick up giant branch stars. It is reassuring to
see the peaks in the numbers of excess stars in the upper panel of
Figure~\ref{fig:velocities} at similar distance moduli to those
reported in Table~\ref{tab:orphan} for Fields 1 and 5 (no signal was
detected in Field 2).
We now select stars using the masks of M92 and M13 placed at the
distance moduli indicated by the peaks in the upper panels. Histograms
of the velocities of these stars for the on and off stream fields are
shown in black and red in the lower panels. A possible signature of
the Orphan Stream is a peak in black without a corresponding peak in
red. The largest peak in Field 1 is at $\sim -40$ kms$^{-1}$ and there
is no signal in red. The largest peak in Field 5 is at $\sim 100$
kms$^{-1}$ and similarly there is a low signal in red. Bearing in mind
the sparsity of the data, these detections are suggestive rather than
conclusive.
\begin{figure*}[t]
\begin{center}
\includegraphics[height=5cm]{f7.eps}
\caption{\label{fig:complexa} The Orphan Stream in Galactic
coordinates ($\ell,b$). Superposed on the map are the HI column
densities of the Complex A association of High Velocity Clouds from
Wakker (2001). Shown in red is a Galactocentric great circle fit
assuming a orbit radius of $\sim 25$ kpc. }
\end{center}
\end{figure*}
\begin{figure}[t]
\begin{center}
\includegraphics[height=8cm]{f8.eps}
\caption{\label{fig:wyn} Polar paths of objects passing close to
the pole of the Orphan Stream (marked by a filled circle).}
\end{center}
\end{figure}
\section{The Great Circle of the Orphan Stream}
As pointed out by~\citet{Ly95}, tidal streams lie on great circles
when viewed from the Galactic Center. The pole of the best fitting
great circle is at $\ellg \approx 42^\circ, \bg \approx 55^\circ$,
where ($\ellg, \bg$) are Galactic coordinates centered on the Galactic
Center. Figure~\ref{fig:complexa} shows the stream in Galactic
coordinates ($\ell,b$), together with the best fitting great
circle. Superposed on the figures are the contours of HI column
density of an association of High Velocity Clouds (HVCs) known as
Complex A, taken from \cite{Wa01}. This is a stream of HI enshrouding
seven clouds (A0 to AVI) and stretching $\sim 30^\circ$ on the
sky. The arc of neutral gas in Complex A runs along the same great
circle as the optical stream. Complex A has a distance bracket 4.0 to
10.1 kpc, based on the presence or absence of absorption lines in the
spectra of the stars AD UMa and PG 0859+593~\citep{Wa96,Wo99}.
Although Complex A is closer than the optical stream, it may still be
associated and simply lie on a different wrap of the same orbit. The
velocities of the gas clouds in Complex A range from $-140$ to $-190
\;\kms$.
Very close to the great circle of the Orphan Stream and behind Complex
A is the recently discovered, disrupting, dwarf spheroidal galaxy,
Ursa Major II, or UMa II~\citep[][ see also Grillmair (2006) who noted
it as a stellar overdensity]{Zu06}. This object lies at Galactic
coordinates $\ell = 152.5^\circ$, $b= 37.4^\circ$. Its color-magnitude
diagram exhibits a well-defined main sequence turn-off, from which its
distance is estimated to be $\sim 30$ kpc, comparable to the distance
of the Orphan Stream as it fades from view. Its radial velocity has
not yet been measured.
\citet{Ly95} developed a method to identify possible globular clusters
associated with a stream. Every possible pole of an object lies at
right angles to its position vector (reckoned from the Galactic
Center). The possible poles sweep out a great circle. Objects that can
lie on the same orbit are identified as intersections in the paths of
the poles of great circles. In practice, it is useful to plot the
polar paths in the coordinates $X = \sqrt{ 1 -\sin \bg}\cos \ellg$ and
$Y = \sqrt{ 1 -\sin \bg}\sin \ellg$ (Lambert's zenithal equal area
projection). Using the online table of globular cluster data provided
by Harris (1996), we show the polar paths of some objects possibly
associated with the Orphan Stream in Figure~\ref{fig:wyn}. The pole
of the Orphan Stream lies at $(X \approx 0.31, Y \approx 0.28)$ and is
marked as the filled circle. It is close to the intersections of
Ruprecht 106, Palomar 1, Arp 2, Terzan 7 and the recently discovered
Segue 1~\citep{Be06c}. Pal 1 and Rup 106 have often been noted as
peculiar. Both are young halo globular clusters. From isochrone
fitting, \citet{Ro98} estimated that Pal 1 has an age of between 6.3
and 8 Gyr and a metallicity [Fe/H] $\approx -0.6 \pm 0.2$. Rup 106 is
also younger than typical halo globular clusters, by about 3 to 5 Gyr,
and is very metal poor~\citep{Fr97}. \citet{Pr05} noted that its
$\alpha$ element ratios are significantly lower than Galactic field
stars of similar metallicity. The anomalous properties of Rup 106 had
earlier led \citet{Li92} to propose that it had been accreted from the
Large Magellanic Cloud. Although this is probably not the case, the
idea that the young halo globular clusters may have been accreted from
elsewhere -- possibly from now defunct dwarf galaxies -- has occurred
to a number of investigators~\citep[e.g.,][]{Ly95,Be00,Pr05}. Terzan
7 and Arp 2 are also young halo globular clusters~\citep{Bu94}, but
their association with the Orphan Stream seems more speculative. Both
have already been claimed as part of the Sagittarius stream on the
basis of distance, kinematics and chemical composition~\citep[see
e.g.,][]{Sb05}.
\section{Conclusions and Summary}
The Orphan Stream is a $\sim 50^\circ$ arc of stars that was detected
in the Field of Streams by \citet{Be06b}. It was also discovered in
public SDSS data by \citet{Gr06b}, who applied a matched filter
technique to build a color-magnitude diagram and used this to estimate
the average heliocentric distance. His analysis is based on a good
match between the stellar population of the Orphan Stream and the
globular cluster M13. In this paper, we have presented and analyzed
new observational data on the Orphan Stream, providing continuous
coverage of the Stream through the area where it crosses the
Sagittarius stream.
We carried out a detailed analysis of the available photometric data
and used it to study the stellar populations in the Stream. Both
theoretical isochrones and observational data on three globular
clusters -- M92, M13 and M71 -- were used to build CMD masks. There is
a degeneracy between age, metallicity and distance which cannot be
broken with the existing photometric data. However, there is a strong
indication that the stellar content of the Stream is old and
metal-poor -- similar to, but not identical to, the globular clusters
M92 and M13. A search for blue horizontal branch population was
carried out. Although this did not yield a positive detection,
nonetheless there appears to be a possible blue straggler population
that is associated with the Stream.
We presented evidence for the detection of a distance gradient along
the Stream. The low declination fields are at a heliocentric distance
of $20^{+7}_{-5}$ kpc. At higher declinations, the Stream moves
farther away from us. The last photometric detection of the Stream is
at $32_{-12}^{+15}$ kpc. This is close to the estimated distance of
the newly-discovered dwarf spheroidal galaxy UMa~II, suggesting that
the Orphan Stream may be physically associated with it. \citet{Fe06}
have recently carried out N body simulations of the disruption of
UMa~II and show that its tidal tails match the observational data
available on the Orphan Stream.
Kinematic data can play a crucial role in understanding the nature
of the Orphan Stream. Accordingly, we searched the spectroscopic
database associated with SDSS DR5. This provides radial velocities for
only about $\sim 2 \%$ of all the stars in DR5. Even though there are
very few candidate Orphan Stream stars with spectroscopic data, we
have detected a tentative velocity signal in two fields. At the
celestial equator, the stream is moving towards us at $\sim 40$
kms$^{-1}$. At high declinations, it is moving away from us at $\sim
100$ kms$^{-1}$.
The Orphan Stream lies on the same great circle as Complex A, a linear
association of High Velocity Clouds, as well as a number of globular
clusters, including Ruprecht 106 and Palomar 1. The recently
discovered extended globular cluster Segue 1 is also very close in
position and distance. All this is consistent with a picture in which
a satellite galaxy merged with the Milky Way long ago. In this
scenario, the Orphan Stream, UMa II, and the young halo globular
clusters were torn off as tidal debris during the merging. Complex A
could be neutral gas that was stripped from a gas-rich dwarf irregular
progenitor, perhaps at a disk-crossing. Alternatively, a large galaxy
can shock and compress ionised gas in the halo, which can then
cool~\citep{Ka66}, leaving a trail of neutral gas in its wake. If so,
then the progenitor must have been massive, with a mass well in excess
of the total mass of Complex A, which is $\sim 10^5 \msun$.
\acknowledgments We particularly wish to thank Bart Wakker and James
Clem for sending us data on Complex A and M92 respectively. Funding
for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science
Foundation, the U.S. Department of Energy, the National Aeronautics
and Space Administration, the Japanese Monbukagakusho, the Max Planck
Society, and the Higher Education Funding Council for England. The
SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the
American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, Cambridge University, Case Western Reserve
University, University of Chicago, Drexel University, Fermilab, the
Institute for Advanced Study, the Japan Participation Group, Johns
Hopkins University, the Joint Institute for Nuclear Astrophysics, the
Kavli Institute for Particle Astrophysics and Cosmology, the Korean
Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos
National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the
Max-Planck-Institute for Astrophysics (MPA), New Mexico State
University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States
Naval Observatory, and the University of Washington.
|
1,116,691,500,441 | arxiv | \section{Introduction}
Let $V$ be a $2n$-dimensional vector space over the finite field $\F_q$. A {\it spread} $\cS$ of $V$ is a collection of $n$-dimensional subspaces that partitions the nonzero vectors in $V$. The members of $\cS$ are the {\it components}, and $V$ is the {\it ambient space}. The {\it kernel} is the subring of $\Gamma L(V)$ that fixes each component, and it is a finite field containing $\F_q$. The {\it dimension} of $\cS$ is the common value of the dimensions of its components over the kernel. The {\it automorphism group} $\Aut(\cS)$ is the subgroup of $\Gamma L(V)$ that maps components to components. The incidence structure $\Pi(\cS)$ with point set $V$ and line set $\{W+v:\,W\in\cS,\,v\in V\}$ and incidence being inclusion is a translation plane. The kernel or dimension of $\Pi(\cS)$ is that of $\cS$ respectively. Andre \cite{andre_tr} has shown that $\Aut(\cS)$ is the translation complement of the plane $\Pi(\cS)$ and each finite translation plane can be obtained from a spread in this way. Two spreads $\cS$ and $\cS'$ of $V$ are {\it isomorphic} if $\cS'=\{g(W):\,W\in\cS\}$ for some $g\in\Gamma L(V)$, and isomorphic spreads correspond to isomorphic planes.
An affine plane is called {\it flag-transitive} if it admits a collineation group which acts transitively on the flags, namely, the incident point-line pairs. Throughout this paper, we will only consider finite planes. Wagner \cite{wagner} has shown that finite flag-transitive planes are necessarily translation planes, so the plane must have prime power order and can be constructed from a spread $\cS$ with ambient space $V$ of dimension $2n$ over $\F_q$ for some $n$ and $q$. The affine plane $\Pi(\cS)$ constructed from a spread $\cS$ is flag-transitive if and only if $\Aut(\cS)$ is transitive on the components. Foulser has determined all flag-transitive groups of finite affine planes in \cite{foulser,foulser0}. The only non-Desarguesian flag-transitive affine planes with nonsolvable collineation groups are the nearfield planes of order $9$, the Hering plane of order $27$ \cite{hering}, and the L$\ddot{u}$neburg planes of even order \cite{luneburg}, cf. \cite{ls,kantor_h}. In the solvable case, Foulser has shown that with a finite number of exceptions, which are explicitly described, a solvable flag transitive group of a finite affine plane is a subgroup of a one-dimensional Desarguesian affine plane.
Kantor and Suetake have constructed non-Desuarguesian flag-transitive affine planes of odd order in \cite{kantor_odd,ks,suetake, suetake2}, and we will refer to these planes as the Kantor-Suetake family. The dimension two case is also due to Baker and Ebert \cite{be_const}. Kantor and Williams have constructed large numbers of flag-transitive affine planes of even order arising from symplectic spreads in \cite{kantor_even,kwnew}. The dimensions of these planes over their kernels are odd. It remains open whether there is a non-Desarguesian flag-transitive affine plane of even order whose dimension over its kernel is even and greater than $2$. Prince has completed the determination of all the flag-transitive affine planes of order at most $125$ in \cite{prince}, and there are only the known ones.
Except for the L$\ddot{u}$neburg planes and the Hering plane of order $27$, all the known finite non-Desarguesian flag-transitive affine planes have a translation complement which contains a linear cyclic subgroup that either is transitive or has two equal-sized orbits on the line at infinity. Under a mild number-theoretic condition involving the order and dimension of the plane (see Lemma \ref{lemma_gcd} below), it can be shown that one of these actions must occur. We call flag-transitive planes of the first kind {\it $\mathcal{C}$-planes} and those of the second kind {\it $\mathcal{H}$-planes}, and call the corresponding spreads of {\it type $\mathcal{C}$} and {\it type $\mathcal{H}$} respectively. There has been extensive study on these two types of planes in the literature. In the case the plane has odd order and dimension two or three over its kernel, it has been shown that the known examples are the only possibilities for either of these two types, see \cite{be_last,be_baer2,be_baer,be_2dim,be_x,ebert_sur}. The classification takes a geometric approach by making use of the intersection of the corresponding spread with the orbits of a certain Singer subgroup and considering relevant Baer subgeometry partitions.
In the study of finite flag-transitive projective planes, deep results from finite group theory are invoked and considerable progress has been made towards a complete classification, cf. \cite{kantor_proj,Kthas_proj, tz_proj}. In contrast, the affine case is more of a combinatorial flavor, and a complete classification seems far out of reach. In this paper, we show that there is a broader connection between flag-transitive affine planes and other combinatorial objects than that is previously known. This will lead us to new characterization results on such planes by making use of the deep results already obtained in other circumstances. To be specific, in Section 3 we develop a new approach to the study of such planes by associating them with planar functions and permutation polynomials in the odd order and even order case respectively. In the odd order case, this new approach allows us to characterize the Kantor-Suetake family by making use of Menichetti's classification of generalized twisted fields in \cite{meni1,meni2}. In particular, the cases of dimension two and three over their kernels follow as a consequence. In Section 4 we will consider the nuclei of the associated commutative semifields and study planar functions that correspond to rank two commutative semifields by the classification results of such semifields by Blokhuis, Lavrauw and Ball in \cite{rtcs_char1,rtcs_char2}. In the even order case, we will develop a technique to study permutation polynomials of DO type by quadratic forms and characterize such planes that have dimensions up to four over their kernels in Section 5. This is the first characterization result in the even order case to our knowledge.
\section{Preliminaries}
A finite {\it presemifield} $S$ is a finite ring with no zero-divisors such that both the left and right distributive laws hold. If further it contains a multiplicative identity, then we call $S$ a {\it semifield}. The additive group of a finite presemifield is necessarily an elementary abelian $p$-group for some prime $p$, so it is conventional to identify $(S,+)$ as the additive group of the finite field $\F_q$ with $q=|S|$ elements. Two presemifields $S_1=(\F_q,+,\star)$ and $S_2=(\F_q,+,\ast)$ are {\it isotopic} if there exist three linear bijections $L,\,M,\,N$ from $(\F_q,+)$ to itself such that $M(x)\ast N(y)=L(x\star y)$ for all $x,y\in \F_q$. In this case, we call $S_1$ an {\it isotope} of $S_2$ and vice versa. For a presemifield $S=(\F_q,+,*)$, fix any $e\ne 0$ and define a new multiplication $\circ$ by $(x*e)\circ(e*y)=x*y$. Then $S'=(\F_q,+,\circ)$ is a semifield with multiplicative identity $e*e$ that is isotopic to $S$.
Let $S=(\F_q,+,*)$ be a commutative semifield. Its {\it middle nucleus} $\cN_m(S)$ and {\it nucleus} $\cN(S)$ are defined respectively as follows:
\begin{align*}
\cN_m(S)=\{\alpha\in \F_q:\,(x*\alpha)*y=x*(\alpha*y) \textup{ for all } x,y\},\\
\cN(S)=\{\alpha\in \F_q:\,(\alpha*x)*y=\alpha*(x*y) \textup{ for all } x,y\}.
\end{align*}
Both $\cN_m(S)$ and $\cN(S)$ are finite fields, and we can regard $S$ as a vector space over $\cN_m(S)$. For more details, please refer to \cite{handbook} or the surveys in \cite[Chapter 9]{handbookff} and \cite{fs_cur}.
A {\it rank two commutative semifield} (RTCS for short) is a commutative semifield that is of rank at most two over its middle nucleus. A finite field is a RTCS by definition, and other known examples include: Dickson semifields \cite{dickson_r2}, Cohen-Ganley semifields \cite{CG} and the Penttila-Williams semifield \cite{rtcs_pw}. They have close connections with many central objects in finite geometry, cf. \cite{rtcs_sur,fs_cur}. The following approach to RTCS is due to Cohen and Ganley \cite{CG}.
\begin{thm}\label{thm_rtcs_def}
Let $q$ be odd, and fix $t\in \F_{q^2}\setminus\F_q$. Let $f,\,g:\,\F_q\mapsto\F_q$ be two functions. The algebraic system $S(g,f):=(\F_{q^2},+,\circ)$ with multiplication
\begin{equation}\label{eqn_rtcs_def}
(xt+y)\circ (ut+v)=(xv+yu+g(xu))t+yv+f(xu)
\end{equation}
is a RTCS if and only if $f,\,g$ are linear and $g(x)^2+4xf(x)$ is a nonsquare for $x\ne 0$.
\end{thm}
In this manner, the Dickson semifield can be described as $S(g,f)$ with $g(z)=0$ and $f(z)=mz^\sigma$, where $q$ is odd, $\sigma$ is an automorphism of $\F_q$, and $m$ is a nonsquare. It is clear that a different choice of $t$ in the theorem lead to an isotopic semifield. In the following theorem we collect some characterization results of Dickson semifields among all RTCSs.
\begin{thm}[\cite{rtcs_char1,rtcs_char2}]\label{thm_rtcs}
Let $S$ be a RTCS of order $p^{2n}$, $p$ an odd prime. If the nucleus $\cN(S)=\F_q$, and $p^n=q^s$, then $S$ is isotopic to either a finite field or a Dickson semifield if $p>2n^2-(4-2\sqrt{3})n+(3-2\sqrt{3})$, $q\ge 4s^2-8s+2$ or $s=3$.
\end{thm}
A polynomial $f(X)\in\F_{q}[X]$ is {\it reduced} if $\deg(f)\le q-1$. It is well known that any function $f:\,\F_q\mapsto\F_q$ is uniquely representable by a reduced polynomial in $\F_q[X]$, i.e., there exists a unique reduced polynomial $g(X)\in\F_q[X]$ such that $f(x)=g(x)$ for all $x\in\F_q$. We will write $f(X)$ and $f(x)$ when we regard $f$ as a polynomial and a function respectively. A polynomial of the form $L(X)=\sum_{i=0}^{m}a_iX^{q^i}$ with coefficients $a_i$'s in $\F_{q^n}$ is called a {\it $q$-polynomial} over $\F_{q^n}$. If $q$ is not specified in the context, then it is also called a {\it linearized polynomial}. If $L(X)$ is a reduced linearized polynomial over $\F_{q^n}$, then the map $x\mapsto L(x)$ is $\F_q$-linear if and only $L(X)$ is a $q$-polynomial. If a subspace $U$ is in the kernel of $L$, then $L_U(X)=\prod_{a\in U}(X-a)$ is a linearized polynomial, and there is a linearized polynomial $R$ such that $L(X)=R(L_U(X))$, cf. \cite[Theorem 3.52, Theorem 3.62]{ff}. For a $q$-polynomial $L(X)=\sum_{i=0}^{n-1}d_iX^{q^i}$, define its {\it adjoint} polynomial as $\tilde{L}(X)=\sum_{i=0}^{n-1}d_i^{q^{n-i}}X^{q^{n-i}}$, and its associated matrix as
\begin{equation}\label{eqn_mat}
M:=\begin{pmatrix}
d_0& d_1 & \cdots &d_{n-1} \\
d_{n-1}^q&d_0^q &\cdots &d_{n-2}^q \\
\vdots&\vdots&\ddots& \vdots\\
d_{1}^{q^{n-1}} &d_2^{q^{n-1}} &\cdots&d_0^{q^{n-1}}
\end{pmatrix}.
\end{equation}
The following lemma is well-known, and we sketch a proof.
\begin{lemma}\label{lemma_M_rank} If $L(X)=\sum_{i=0}^{n-1}d_iX^{q^i}\in\F_{q^n}[X]$, then the kernel of the $\F_q$-linear map $x\mapsto L(x)$ from $\F_{q^n}$ to itself has dimension $n-\rank(M)$, where $M$ is as defined in \eqref{eqn_mat}.
\end{lemma}
\begin{proof}Take $\{t_0,t_1,\cdots,t_{n-1}\}$ to be a basis of $\F_{q^n}$ over $\F_q$, and let $T$ be the $n\times n$ matrix whose $(i,j)$-th entry is $t_j^{q^i}$ for $0\le i,j\le n-1$. By \cite[Lemma 3.51]{ff}, $T$ is invertible. It is routine to check that
$(T^tMT)_{i,j}=\tr_{\F_{q^n}/\F_q}(t_iL(t_j))$ and $(MT{\bf x}^t)_i=L(x)^{q^i}$,
where ${\bf x}=(x_0,x_1,\cdots,x_{n-1})\in\F_q^n$ and $x=x_0t_0+x_1t_1+\cdots +x_{n-1}t_{n-1}\in \F_{q^n}$. It follows that $L(x)=0$ if and only if ${\bf x}^t$ is in the null space of $T^tMT$. The matrix $T^tMT$ has all its entries in $\F_q$, so its rank over $\F_q$ and $\F_{q^n}$ are the same. The conclusion now follows.
\end{proof}
A function $f:\,\F_{q}\mapsto\F_q$ is a {\it planar function} if $x\mapsto f(x+a)-f(x)-f(a)$ is a permutation of $\F_q$ for any $a\ne 0$. It is known that there are no planar functions in even characteristic. A Dembowski-Ostrom (or DO) polynomial over a field of characteristic $p$ is a polynomial of the shape $\sum_{i,j}a_{ij}X^{p^i+p^j}$. If a function $f:\,\F_q\mapsto\F_q$ is representable by a DO polynomial, we call $f$ a function of DO type. Two planar functions $f,g$ of DO type are {\it equivalent} if there exist linearized polynomials $L_1,\,L_2$ such that $f(x)=L_1(g(L_2(x)))$ for all $x\in\F_q$. It is shown in \cite{coulter} that there is a close connection between commutative presemifield and planar functions of DO type: if $(\F_q,+,*)$ is a commutative presemifield, then $x\mapsto x*x$ is a planar function of DO type over $\F_q$; conversely, if $f$ is a planar function of DO type, then $S_f=(\F_q,+,*)$ is a commutative presemifield, where the multiplication is $x*y=f(x+y)-f(x)-f(y)$. Weng and Hu have given a characterization of planar functions in terms of the images of $f$ in \cite[Theorem 2.3]{weng}.
\begin{lemma}\label{lemma_2to1}
Let $f:\,\F_q\mapsto\F_q$ be a DO polynomial. Then $f$ is a planar function if and only if $f$ is $2$-to-$1$, namely, every nonzero element has $0$ or $2$ preimages.
\end{lemma}
We shall need the following simple lemma.
\begin{lemma}\label{lemma_L_perm}
Let $q$ be an odd prime power and $w$ be a nonsquare of $\F_q$. If $L_1(X)$ and $L_2(X)$ are linearized polynomials such that $Q(x)=L_1(x)^2-wL_2(x)^2$ is a planar function over $\F_q$, then at least one of $L_1$ and $L_2$ is a permutation polynomial.
\end{lemma}
\begin{proof}
If neither $L_1$ nor $L_2$ is a permutation, then there exists nonzero elements $u_1,u_2$ such that $L_1(u_1)=L_2(u_2)=0$. However, this leads to that $Q(u_1+u_2)-Q(u_1)-Q(u_2)=0$, which contradicts the planarity property.
\end{proof}
A {\it quadratic space} is a pair $(Q,\,V)$ where $V$ is a finite dimensional vector space over $\F_q$ and $Q:\,V\mapsto\F_q$ satisfies that: (1) $Q(\lambda v)=\lambda^2Q(v)$ for all $\lambda\in\F_q$ and $v\in V$; (2) $B_Q(x,y):=Q(x+y)-Q(x)-Q(y)$ is a bilinear form. If we fix a basis $\{e_1,\cdots,e_n\}$ of $V$ over $\F_q$, then $f(x_1,\cdots, x_n)=Q(\sum_{i=1}^nx_ie_i)$ is a {\it quadratic form} in $n$ indeterminants over $\F_q$. A different choice of basis yields an equivalent form, and the {\it rank} of $Q$ is the minimum number of variables in a quadratic form induced from $Q$ in this way. The {\it radical} of a quadratic space $(Q,\,V)$ is
\[
\rad(Q):=\{v\in V:\,B_Q(v,x)=0 \textup{ for all } x\in V\}.
\]
In the following theorem we collect the facts about quadratic forms with $q$ even that we shall need, cf. \cite[Theorem 7.2.9]{handbookff} and Section 6.2 of \cite{ff}.
\begin{thm}\label{thm_N0}
Let $q$ be even and $(Q,V)$ be a quadratic space over $\F_q$. Fix an element $d\in\F_q$ such that $X^2+X+d$ is irreducible over $\F_q$. Write $n:=\dim(V)$, $r=\dim(\rad(Q))$, and denote by $N_0$ the number of $v\in V$ such that $Q(v)=0$. Then $n-r=2s$ is even and there is a basis of $V$ such that the resulting quadratic form is one of the following:
\begin{enumerate}
\item[(i)] $x_1x_2+x_3x_4+\cdots+x_{2s-1}x_{2s}$ (hyperbolic),
\item[(ii)] $x_1^2+x_1x_2+dx_2^2+x_3x_4+\cdots+x_{2s-1}x_{2s}$ (elliptic),
\item[(iii)] $x_0^2+x_1x_2+\cdots+x_{2s-1}x_{2s}$ (parabolic),
\end{enumerate}
and $N_0=q^{n-1}+(q-1)q^{r+s-1}\epsilon$, where $\epsilon=1,\,-1,\,0$ in (i), (ii) and (iii) respectively. Moreover, (iii) occurs if and only if $Q(\rad(Q))\ne \{0\}$.
\end{thm}
\begin{corollary}\label{cor_rank}
Let $q$ be even and $(Q,V)$ be a quadratic space over $\F_q$ with $\dim_{\F_q}V=n$. Then the number of $v\in V$ such that $Q(v)=0$ is $q^{n-1}$ if and only if $Q$ has odd rank.
\end{corollary}
\begin{proof}
In Theorem \ref{thm_N0}, $Q$ has even rank in case (i) and (ii), and has odd rank in case (iii).
\end{proof}
For $n\ge 1$, we define the trace function $\tr_{\F_{q^n}/\F_q}:\,\F_{q^n}\mapsto\F_q$ by $\tr_{\F_{q^n}/\F_q}(x)=x+x^q+\cdots+x^{q^{n-1}}$. If a function $f:\,\F_{q^n}\mapsto\F_{q^n}$ satisfies that $f(\lambda x)=\lambda^df(x)$ for some positive integer $d$ and all $\lambda\in\F_q$ and $x\in\F_{q^n}$, then we call $f(X)$ a {\it homogeneous} polynomial of {\it degree} $d$ over the subfield $\F_q$. Such a polynomial $f$ naturally induces a map $\bar{f}:\,\F_{q^n}^*/\F_q^*\mapsto \F_{q^n}^*/\F_q^*$ such that $\bar{f}(\bar{x}):=\overline{f(x)}$ for each $\bar{x}=x\F_q^*$. In the case $\bar{f}$ is a bijection, we say that $f$ induces a permutation of $\F_{q^n}^*/\F_q^*$ or simply say that $f$ permutes $\F_{q^n}^*/\F_q^*$.
\begin{lemma}\label{lemma_perm_rank} Let $Q(X)=\sum_{i,j}a_{ij}X^{q^i+q^j}\in\F_{q^n}[X]$ with $q$ even. Then $Q(X)$ is a permutation polynomial of $\F_{q^n}$ if and only if $Q_y(x)=\tr_{\F_{q^n}/\F_q}(yQ(x))$ has odd rank for $y\ne 0$.
\end{lemma}
\begin{proof}
If $Q(X)$ is a permutation polynomial of $\F_{q^n}$, then the size of $\{x:\,Q_y(x)=0\}$ is equal to that of $\{z:\,\tr_{\F_{q^n}/\F_q}(yz)=0\}$, which is $q^{n-1}$ for $y\ne 0$. By Corollary \ref{cor_rank}, the quadratic form $Q_y$ has odd rank for each $y\ne 0$. This proves the necessary part.
Now assume that $Q_y$ has odd rank for each $y\ne 0$. Since $q$ is even and $Q(X)$ is homogeneous of degree $2$ over $\F_q$, it is easy to show that $Q(X)$ is a permutation polynomial if and only if it induces a permutation of the quotient group $\F_{q^n}^*/\F_q^*$. We regard $\F_{q^n}$ as a vector space over $\F_q$, and identify the elements of $\F_{q^n}^*/\F_q^*$ with the projective points of $\PG(n-1,q)$ in the natural way. Each hyperplane of $\PG(n-1,q)$ is of the form $\{x\F_{q}^*:\,\tr_{\F_{q^n}/\F_{q}}(yx )=0\}$ for some $y\ne 0$, and $Q(x)\F_q^*$ lies on the hyperplane if and only if $Q_y(x)=0$. For each element $g\in \F_{q^n}^*/\F_{q}^*$, we define $n(g):=\{x\in \F_{q^n}^*/\F_{q}^*:\, Q(x)\F_q^*=g\}$.
For a set $U$ of projective points, write $n(U):=\sum_{g\in U}n(g)$. If $H$ is a hyperplane, then $n(H)=\frac{q^{n-1}-1}{q-1}$ by Corollary \ref{cor_rank}.
For a fixed point $g\in\PG(n-1,q)$, let $H_1,\cdots, H_{\frac{q^{n-1}-1}{q-1}}$ be the set of hyperplanes containing it. As a multiset, $\bigcup_iH_i$ covers the point $g$ exactly $\frac{q^{n-1}-1}{q-1}$ times and each other point $\frac{q^{n-2}-1}{q-1}$ times. We thus have
\begin{align*}
q^{n-2}\cdot n(g)+\frac{q^{n-2}-1}{q-1}\cdot\sum_{h\in \PG(n-1,q)}n(h)=\sum_in(H_i)=\left(\frac{q^{n-1}-1}{q-1}\right)^2.
\end{align*}
On the other hand, $\sum_{h\in \PG(n-1,q)}n(h)=\frac{q^n-1}{q-1}$. It follows that $n(g)=1$ for each $g$. Hence $Q(X)$ induces a permutation of $\F_{q^n}^*/\F_q^*$, and so it permutes $\F_{q^n}$. This proves the sufficiency part.
\end{proof}
The above lemma describes how to study the permutation behavior of a polynomial of DO type via quadratic forms. This is the technique that we will apply in Section 5.
\section{A function approach to $\cC$-planes and $\cH$-planes}\label{sect_function}
Let $q$ be a prime power, $n\ge 2$ be an integer, and regard the finite field $\F_{q^{2n}}$ as a $2n$-dimensional vector space over $\F_q$. Let $\gamma$ be a primitive element of $\F_{q^{2n}}$ and let $\sigma:\,x\mapsto x^p$ be the Frobenius automorphism. For each $a\in\F_{q^{2n}}^*$, we use $\Theta(a)$ for the linear map from $\F_{q^{2n}}$ to itself that maps $x$ to $ax$. The group $\la\Theta(\gamma)\ra$ is a Singer group, and $\Gamma L(1,q^{2n})=\la\Theta(\gamma)\ra\rtimes \Aut(\F_{q^{2n}})$. Let $\cS$ be a spread with ambient space $(\F_{q^{2n}},+)$ and kernel $\F_q$. The associated affine plane $\Pi(\cS)$ is flag-transitive if and only if $\Aut(\cS)$ is transitive on the components. If further $\Aut(\cS)$ is solvable, then it is isomorphic to a subgroup of $\Gamma L(1,q^{2n})$. After taking an isomorphic spread if necessary, we assume that $\Aut(\cS)$ is a subgroup of $\la\Theta(\gamma)\ra\rtimes \Aut(\F_{q^{2n}})$.
If the Singer subgroup $\Aut(\cS)\cap \la\Theta(\gamma)\ra$ has order $\frac{q^n+1}{2}$ or $q^n+1$ respectively, then we call $\cS$ a spread of {\it type $\cH$} or {\it type $\cC$} and the corresponding plane $\Pi(\cS)$ a {\it $\cH$-plane} or a $\cC$-plane respectively. All the known flag-transitive affine planes with a solvable full collineation group are either $\cC$-planes or $\cH$-planes, and there are no other such planes of order at most $125$. The following result summarizes Lemma 1 and the subsequent comments in \cite{ebert_sur}, restricting to the solvable case.
\begin{lemma}\label{lemma_gcd}
Let $q=p^e$ with $p$ prime and $n\ge2$ be an integer. If $\cS$ is a spread of $(\F_{q^{2n}},+)$ with kernel $\F_q$ such that $\Aut(\cS)$ is solvable and transitive on the components, then $\cS$ is either of type $\cH$ or of type $\cC$ provided that $\gcd(\frac{q^n+1}{2},ne)=1$ in the case $p$ is odd and $\gcd(q^n+1,ne)=1$ in the case $p=2$.
\end{lemma}
The above lemma indicates that $\cC$-planes and $\cH$-planes are the only possibilities under a mild number theoretical condition in the solvable case, and it is an open problem whether the $\gcd$ condition can be dropped. In this paper, we will focus on $\cC$-planes and $\cH$-planes. {\it Throughout the paper, we fix the following notation}. Take $\beta$ to be an element of order $(q^n+1)(q-1)$. Let $\cS$ be a spread of type $\cH$ or type $\cC$ such that $\Aut(\cS)\cap \la\Theta(\gamma)\ra=\la\Theta(\beta^2)\ra$ or $\la\Theta(\beta)\ra$ respectively. Let $W$ be a component of $\cS$, so that $\cS=\{g(W):
\,g\in \Aut(\cS)\}$. Since the number of $\F_{q^n}$-subspaces of $\F_{q^{2n}}$ is $q^n+1$, there exists $\delta\in\F_{q^{2n}}\setminus\F_{q^n}$ such that $W\cap\F_{q^n}\cdot\delta=\{0\}$. From the decomposition $\F_{q^{2n}}=\F_{q^n}\oplus\F_{q^n}\cdot\delta$, we can write the $\F_q$-subspace $W$ as follows:
\begin{equation}\label{eqn_W}
W=\{x+\delta\cdot L(x):\,x\in\F_{q^n}\},
\end{equation}
where $L(X)\in\F_{q^n}[X]$ is a reduced $q$-polynomial. We also define
\begin{equation}\label{eqn_Q}Q(X):=(X+\delta L(X))\cdot(X+\delta^{q^n} L(X)),
\end{equation}
which is a DO polynomial over $\F_{q^n}$. The following is our key lemma.
\begin{lemma}\label{lemma_key} Take notation as above, and let $W$ be the $\F_q$-subspace in \eqref{eqn_W}.
\begin{enumerate}
\item If $q$ is odd, then the orbit of $W$ under the group $\la\Theta(\beta^2)\ra$ forms a partial spread if and only if $Q(x)$ is a planar function over $\F_{q^n}$.
\item If $q$ is odd, then the orbit of $W$ under the group $\la\Theta(\beta)\ra$ forms a spread if and only if $x\mapsto Q(x)$ induces a permutation of $\F_{q^n}^*/\F_q^*$.
\item If $q$ is even, then the orbit of $W$ under the group $\la\Theta(\beta)\ra$ forms a spread if and only if $x\mapsto Q(x)$ is a permutation of $\F_{q^n}$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove (1). The orbit of $W$ under $\la\Theta(\beta^2)\ra$ forms a partial spread if and only if the following holds: $y+\delta L(y)=\beta^{2i}(x+\delta L(x))\ne 0$ occurs only in the case $\beta^{2i}\in\F_q^*$ and $y=\beta^{2i}x$.
First assume that we get a partial spread from $W$ as described. If $Q(x)=Q(y)$ for $xy\ne 0$, then $s_1^{q^n+1}=1$, where $s_1=\frac{y+\delta L(y)}{x+\delta L(x)}$. Since $s_1\in\la \beta^2\ra$, it follows from our assumption that $s_1\in\F_q^*$ and $y=s_1x$. Since $\gcd(q^n+1,q-1)=2$, we get $s_1^2=1$, i.e., $s_1=\pm 1$. We thus have shown that $x\mapsto Q(x)$ is $2$-to-$1$, and it follows that $Q(x)$ is planar by Lemma \ref{lemma_2to1}.
Conversely, assume that $Q(x)$ is a planar function. If $y+\delta L(y)=\beta^{2i}(x+\delta L(x))\ne 0$, then taking norm we get $Q(y)=\beta^{2(q^n+1)i}Q(x)=Q(\beta^{(q^n+1)i}x)$. It follows from Lemma \ref{lemma_2to1} that $y=\pm \beta^{(q^n+1)i}x$. Plugging this into $y+\delta L(y)$, we get $\pm \beta^{(q^n+1)i}=\beta^{2i}$. This gives that $\beta^{2i(q^n-1)}=1$, i.e., $\beta^{2i}\in\F_{q^n}^*$. It is easy to show that $\la \beta^2\ra\cap \F_{q^n}^*=\F_q^*$, so $\beta^{2i}\in\F_q^*$. The conclusion now follows.
We next prove (2). Since $Q(\lambda x)=\lambda^2Q(x)$ for $\lambda\in\F_q$, the map $x\mapsto Q(x)$ induces a function from $\F_{q^n}^*/\F_q^*$ to itself. Observe that $\la\beta\ra$ is the set of elements in $\F_{q^{2n}}^*$ whose relative norm to $\F_{q^n}$ is in $\F_q^*$. The orbit of $W$ under $\la\Theta(\beta)\ra$ forms a spread if and only if the following holds: $y+\delta L(y)=\beta^{i}(x+\delta L(x))\ne 0$ occurs only in the case $\beta^{i}\in\F_q^*$ and $y=\beta^{i}x$.
First assume that we get a spread from $W$ as described. If $Q(x)/Q(y)\in\F_{q}^*$ for $xy\ne 0$, then $y+\delta L(y)=s_1(x+\delta L(x))$ for some $s_1\in\la\beta\ra$. It follows that $s_1\in\F_q^*$ and $y=s_1x$. Hence $Q(x)$ permutes $\F_{q^n}^*/\F_q^*$.
Conversely, assume that $Q(x)$ permutes $\F_{q^n}^*/\F_q^*$. If $y+\delta L(y)=\beta^{i}(x+\delta L(x))\ne 0$, then taking norm we get $Q(y)/Q(x)=\beta^{i(1+q^n)}\in\F_{q}^*$. It follows that $y/x\in\F_q^*$, and the conclusion follows.
The claim (3) can be proved similarly, and we omit the details.
\end{proof}
\begin{corollary}
Let $W$ be the $\F_q$-subspace in \eqref{eqn_W}, and assume that $q$ and $n$ are odd. If the orbit of $W$ under $\la\Theta(\beta^2)\ra$ forms a partial spread, then its orbit under $\la\Theta(\beta)\ra$ forms a spread of type $\cC$.
\end{corollary}
\begin{proof}
By the previous lemma, we see that $Q(x)$ is a planar function, and so $x\mapsto Q(x)$ is $2$-to-$1$ by Lemma \ref{lemma_2to1}. Denote by $D$ the image set $\{Q(x):\,x\ne 0\}$, and write $E$ for its complement in $\F_{q^n}^*$. For a nonsquare $\lambda\in\F_q^*$, we have $E=\lambda \cdot D$ by \cite[Proposition 3.6]{weng}. In particular, $Q(y)/Q(x)\in\F_q^*$ implies that $Q(y)/Q(x)=u^2$ for some $u\in\F_q^*$. The $2$-to-$1$ property of $Q$ then gives that $y/x=\pm u\in\F_q^*$. This shows that $Q(x)$ permutes $\F_{q^n}^*/\F_q^*$. The claim then follows from claim (2) in Lemma \ref{lemma_key}.
\end{proof}
\begin{thm}
There is no type $\mathcal{C}$ spread with ambient space $(\F_{q^{2n}},+)$ and kernel $\F_q$ when $n$ is even and $q$ is odd.
\end{thm}
\begin{proof}
We take notation introduced preceding Lemma \ref{lemma_key} and prove by contradiction. By claim (2) in Lemma \ref{lemma_key}, $x\mapsto Q(x)$ induces a permutation of $\F_{q^n}^*/\F_q^*$.
By \cite[Lemma 3.3]{weng}, $\tr_{\F_{q^n}/\F_q}(Q(x))$ is a nondegenerate quadratic form over $\F_q$. By \cite[Theorem 6.26]{ff},
the number $N_0=\#\{x\in\F_{q^n}:\,\tr_{\F_{q^n}/\F_q}(Q(x))=0\}$ is equal to $q^{n-1}\pm(q-1)q^{n/2-1}$. On the other hand, $N_0=\#\{y\in\F_{q^n}:\,\tr_{\F_{q^n}/\F_q}(y)=0\}=q^{n-1}$ by the fact that $Q(x)$ permutes $\F_{q^n}^*/\F_q^*$. This contradiction completes the proof.
\end{proof}
In the case $q$ is odd, we may further restrict the form of $W$.
\begin{lemma}\label{lemma_delta}
Let $\cS$ be a spread such that $\Aut(\cS)\cap \la\Theta(\gamma)\ra$ contains $\la\Theta(\beta^2)\ra$, and let $W$ be a component.
If $q$ is odd and $\delta^{q^n-1}=-1$, then $W$ intersects $\F_{q^n}$ or $\F_{q^n}\cdot\delta$ trivially.
\end{lemma}
\begin{proof}
From the decomposition $\F_{q^{2n}}=\F_{q^n}\oplus\F_{q^n}\cdot\delta$, we can write $W=\{L_1(x)+\delta L_2(x):\,x\in\F_{q^n}\}$ for some $q^n$-polynomials $L_1(X),\,L_2(X)$. The argument in Lemma \ref{lemma_key} shows that $Q(x)=L_1(x)^2-\delta^2L_2(x)^2$ is planar. By Lemma \ref{lemma_L_perm}, at least one of $L_1,\,L_2$ is a permutation, and correspondingly $W$ intersects one of $\F_{q^n}$ and $\F_{q^n}\cdot\delta$ trivially.
\end{proof}
\begin{remark}In Lemma \ref{lemma_delta}, if $\Aut(\cS)$ is transitive, then $\cS'=\{g(\delta\cdot W):\,g\in \Theta(\delta)\cdot\Aut(\cS)\cdot\Theta(\delta)^{-1}\}$ is a spread isomorphic to $\cS$ and $\Aut(\cS')$ contains $\la\Theta(\beta^2)\ra$. So after replacing $W$ by $\delta\cdot W$ if necessary, we always assume that $\delta^{q^n-1}=-1$ in \eqref{eqn_W} in the case $q$ is odd.
\end{remark}
The Kantor-Suetake family constitutes a major part of the known non-Desarguesian flag-transitive affine planes of odd order. As a first application of our new approach, we give a characterization of this important family based on the following result of Menichetti. Please refer to \cite{albert_tw,handbook} for details on generalized twisted fields.
\begin{thm}\label{thm_men}\cite{meni1,meni2}
Let $S$ be a finite semifield of prime dimension $n$ over the nucleus $\F_q$. Then there is an integer $\nu(n)$ such that if $q\ge \nu(n)$ then $S$ is isotopic to a finite field or a generalized twisted field. Moreover, we have $\nu(3)=0$.
\end{thm}
By \cite[Proposition 11.31]{handbook}, which is essentially due to Albert \cite{albert_iso1,albert_iso2}, a generalized twisted field that has a commutative isotope must be isotopic to the commutative presemifield defined by a planar function $x^{1+p^\alpha}$ over $\F_{p^{e}}$, where $1\le \alpha \le n-1$ and $e/\gcd(e,\alpha)$ is odd. The following result characterizes planar functions whose associated presemifield is isotopic to a commutative twisted field or a finite field, see \cite[Corollaries 3.9, 3.10]{coulter}.
\begin{lemma}\label{lemma_iso} Let $p$ be an odd prime and $q=p^e$. Let $f$ be a planar function of DO type over $\F_q$ and $S_f=(\F_q,+,*)$ be the associated presemifield with $x*y=f(x+y)-f(x)-f(y)$. There exist linearized permutation polynomials $M_1$ and $M_2$ such that
\begin{enumerate}
\item if $S_f$ is isotopic to a finite field, then $f(M_2(x))\equiv M_1(x^2)$ for $x\in\F_{q}$;
\item if $S_f$ is isotopic to a commutative twisted field, then $f(M_2(x))=M_1(x^{p^\alpha+1})$ for $x\in\F_{q}$, where $\alpha$ is an integer such that $1\le\alpha\le e-1$ and $e/\gcd(e,\alpha)$ is odd.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{lemma_ps} Let $n$ be an odd prime, $\nu(n)$ be as in Theorem \ref{thm_men}, and let $q$ be an odd prime power such that $q\ge \nu(n)$. Suppose that the orbit of $W=\{x+\delta\cdot L(x):\,x\in\F_{q^n}\}$ under $\la\Theta(\beta^2)\ra$ forms a partial spread, where $\delta^{q^n-1}=-1$ and $L(X)$ is a $q$-polynomial over $\F_{q^n}$. Then $W=\{\alpha\cdot(x+u\delta x^{q^i}):\, x\in\F_{q^n}\}$ for some $\alpha\in\F_{q^{2n}}^*$, $u\in\F_{q^n}$ and $0\le i\le n-1$.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma_key}, $Q(x)=x^2-\delta^2L(x)^2$ is a planar function. By Theorem \ref{thm_men}, the associated presemifield $S_Q$ is isotopic to a finite field or a commutative twisted field under the conditions in the lemma. By Lemma \ref{lemma_iso}, there are reduced linearized permutation polynomials $M_1,M_2$ such that
\begin{equation}\label{eqn_M12ff}
M_1(X^2)\equiv M_2(X)^2-\delta^2 L(M_2(X))^2 \pmod{X^{q^n}-X},
\end{equation}
or
\begin{equation}\label{eqn_M12tw}
M_1(X^{1+p^\alpha})\equiv M_2(X)^2-\delta^2 L(M_2(X))^2 \pmod{X^{q^n}-X},
\end{equation}
where $q=p^e$ with $p$ prime, $1\le\alpha\le ne-1$ and $\frac{ne}{\gcd(ne,\alpha)}$ is odd. Write $M_2(X)=\sum_{i=0}^{ne-1}a_iX^{p^i}$, $L(M_2(X))\equiv\sum_{i=0}^{ne-1}b_iX^{p^i}\pmod{X^{q^n}-X}$, and set $I=\{i:\,a_i\ne 0\}$, $J=\{i:\, b_i\ne 0\}$.
Comparing coefficients of $x^{2p^i}$ in \eqref{eqn_M12tw}, we see that $0=a_i^2-\delta^2 b_i^2$ for $0\le i\le ne-1$. Since $\delta^2$ is a nonsquare in $\F_{q^n}^*$, all the $a_i$'s and $b_j$'s are zero. This contradicts the assumption that $M_2$ is a permutation polynomial, so \eqref{eqn_M12tw} can not occur. It remains to check \eqref{eqn_M12ff}. We look at the coefficients of $x^{p^i+p^j}$, $0\le i<j\le ne-1$, on both sides and get $a_ia_j=\delta^2 b_ib_j$. If $|I|\ge 2$ or $|J|\ge 2$, then it is easy to deduce that $I=J$. If both $I$ and $J$ have size at most $1$, then $L$ is a monomial and $W$ takes the desired form with $\alpha=1$. We assume that $I=J$ and they have size at least two below. In this case, for any distinct $i,j\in I$, exactly one of $\{a_ib_i^{-1},\,a_jb_j^{-1}\}$ is a square and the other is a nonsquare, since $\delta^2$ is a nonsquare in $\F_{q^n}^*$. This is only possible when $|I|=2$. Therefore,
$M_2(X)=aX^{p^k}+bX^{p^\ell}$, $L(M_2(X))=cX^{p^k}+dX^{p^\ell}$ for some $0\le k<\ell\le ne-1$ and $a,b,c,d\in\F_{q^n}^*$ such that $ab=\delta^2 cd$. Since $x\mapsto M_2(x)$ is a permutation, the elements of $W$ can be written as
\[
M_2(x)+\delta\cdot L(M_2(x))=(a+\delta c)\cdot(y+\delta da^{-1}y^{p^{\ell-k}}),\;y=x^{p^k}.
\]
Since $W$ is a $\F_q$-linear subspace, $p^{\ell-k}$ is a power of $q$.
This completes the proof.
\end{proof}
\begin{remark}\label{remark_size} Let $q$ be an odd prime power, $w$ be a nonsquare of $\F_q$, and $N>1$ be an integer. Let ${\bf a}=(a_0,a_1\cdots,a_{N-1})$ and ${\bf b}=(b_0,b_1,\cdots,b_{N-1})$ be two sequences consisting of elements in $\F_q$, and define their supports as $I_1=\{i:\,a_i\ne 0\}$, $I_2=\{i:\,b_i\ne 0\}$ respectively. As in the proof of Lemma \ref{lemma_ps}, we can show that: if $a_ia_j=wb_ib_j$ for any distinct $i,\,j$, then either both $|I_1|$ and $|I_2|$ have size at most one or $I_1=I_2$ and both have size $2$.
\end{remark}
\begin{thm}\label{thm_typeC}
Let $n$ be an odd prime, $\nu(n)$ be as in Theorem \ref{thm_men}, and let $q$ be an odd prime power such that $q\ge \nu(n)$. A type $\cC$ spread $\cS$ of $ (\F_{q^{2n}},+)$ with kernel $\F_q$ is isomorphic to the orbit of
$W=\{x+\delta\cdot x^{q^i}:\,x\in\F_{q^n}\}$ under $\la\Theta(\beta)\ra$ for some $\delta$ and $i$ such that $\delta^{q^n-1}=-1$, $1\le i\le n-1$ and $\gcd(i,n)=1$.
\end{thm}
\begin{proof}
Let $W$ be a component of $\cS$ such that $\cS$ is the orbit of $W$ under the Singer subgroup $\la\Theta(\beta)\ra$. By the remark following Lemma \ref{lemma_delta}, up to isomorphism $W=\{x+\delta' L(x):\,x\in\F_{q^n}\}$ for some $q$-polynomial $L(X)$ and $\delta'$ such that $\delta'^{q^n-1}=-1$. By Lemma \ref{lemma_ps}, $W=\{\alpha\cdot(x+u\delta' x^{q^i}):\, x\in\F_{q^n}\}$ for some $\alpha\in\F_{q^{2n}}^*$ and $u\in\F_{q^n}^*$. Its orbit under $\la\Theta(\beta)\ra$ is a spread isomorphic to the one described in the theorem with $\delta=u\delta'$. The kernel contains the fixed subfield of $x\mapsto x^{q^i}$, so we have $\gcd(i,n)=1$.
\end{proof}
\begin{thm}\label{thm_typeH}
Let $n$ be an odd prime, $\nu(n)$ be as in Theorem \ref{thm_men}, and let $q$ be an odd prime power such that $q\ge \nu(n)$. A type $\cH$ spread $\cS$ of $(\F_{q^{2n}},+)$ with kernel $\F_q$ is isomorphic to the orbit of $\{x+\delta x^{q^k}:\,x\in\F_{q^n}\}$ under the group $A$ generated by $\la\Theta(\beta^2)\ra$ and $\psi:\,z\mapsto \eta z^{q^n}$, where $1\le k\le n-1$, $\gcd(k,n)=1$, $\delta^{q^n-1}=-1$, $\eta^{(1+q^n)(q^k-1)}=1$ and $\eta$ is a nonsquare.
\end{thm}
\begin{proof} Write $q=p^e$ with $p$ prime. By Lemma \ref{lemma_ps}, under the conditions in the theorem a spread $\cS$ of type $\cH$ is isomorphic to the orbit of $W=\{x+\delta\cdot x^{\tau}:\,x\in\F_{q^n}\}$ under $\Aut(\cS)$, where $\delta^{q^n-1}=-1$, $\tau=q^k$ with $1\le k\le n-1$ and $\gcd(k,n)=1$, and $\Aut(\cS)\cap\la\Theta(\gamma)\ra=\la\Theta(\beta^2)\ra$.
There exists $\psi\in\Aut(\cS)$ that permutes the two $\la\Theta(\beta^2)\ra$-orbits by the transitivity of $\Aut(\cS)$ on the components, so the spread $\cS$ consists of the orbit of $W$ under the subgroup $A:=\la\Theta(\beta^2),\,\psi\ra$. By elementary group theory,
\[
\Aut(\cS)/\la\Theta(\beta^2)\ra\cong \Aut(\cS)\cdot\la\Theta(\gamma)\ra/\la\Theta(\gamma)\ra\le
\la\Theta(\gamma)\ra\rtimes \Aut(\F_{q^{2n}})/\la\Theta(\gamma)\ra
\cong \Aut(\F_{q^{2n}}).
\]
Since any odd power of $\psi$ also permutes the two $\la\Theta(\beta^2)\ra$-orbits, we can assume that the order of $\overline{\psi}\in\Aut(\cS)/\la\Theta(\beta^2)$ is a power of $2$. Write $\psi(z)=\eta z^\sigma$, where $\sigma=p^\ell$ ($1\le \ell\le 2ne-1$) and $\eta\in\F_{q^{2n}}^*$. We now derive conditions on $\psi$ to guarantee that the orbit of $W$ under the subgroup $A$ forms a spread. To be specific, we need to make sure the following hold:
\begin{enumerate}
\item $\psi^2(W)=\beta^{2i}\cdot W$ for some $i$, since $\psi^2$ stabilizes the $\la\Theta(\beta^2)\ra$-orbits;
\item $\psi(W)$ intersects each of $\beta^{2i}\cdot W$, $0\le i\le \frac{q^n-1}{2}$, trivially.
\end{enumerate}
We fist consider the condition (1). Assume that $\psi^2(W)=\beta^{2i}\cdot W$ for some $i$. This means that for each $x\in\F_{q^n}$, there exists $y\in\F_{q^n}$ such that
$\eta^{1+\sigma}\left(x+\delta x^\tau\right)^{\sigma^2}=\beta^{2i}(y+\delta y^\tau).$
Write $\eta^{1+\sigma}\beta^{-2i}=u+v\delta$ for some $u,v\in\F_{q^n}$. Expanding and comparing the coefficients of the basis $\{1,\delta\}$, we get
$y=ux^{ \sigma^2}+v\delta^{\sigma^2+1}x^{\tau\sigma^2}$, $ y^\tau=u\delta^{\sigma^2-1}x^{\tau\sigma^2}+vx^{ \sigma^2}$.
Canceling out $y$, we see that $(u^\tau-u\delta^{\sigma^2-1})x^{\tau\sigma^2}+v^\tau\delta^{\tau(\sigma^2+1)}x^{\tau^2\sigma^2}
-vx^{\sigma^2}=0$ for all $x$. Hence, the reduced polynomial
$(u^\tau-u\delta^{\sigma^2-1})X^{\tau\sigma^2}+v^\tau\delta^{\tau(\sigma^2+1)}X^{\tau^2\sigma^2}
-vX^{ \sigma^2}\pmod{X^{q^n}-X}$ is the zero polynomial.
Since $\tau\ne 1$ and $n$ is odd, the reduced monomial $X^{\sigma^2}\pmod{X^{q^n}-X}$ occurs only once. This gives $v=0$, and the remaining coefficient gives $u^{\tau-1}=\delta^{\sigma^2-1}$. Raising both sides of $\eta^{1+\sigma}\beta^{-2i}=u$ to the $\frac{1}{2}(q^n+1)(\tau-1)$-st power, we get $\eta^{(q^n+1)(\tau-1)(1+\sigma)/2}=u^{\tau-1}$, so it must hold that
\begin{equation}\label{eqn_eta_del}
\eta^{(1+q^n)(\tau-1)(1+\sigma)/2}=\delta^{\sigma^2-1}.
\end{equation}
Conversely, if \eqref{eqn_eta_del} is true, then (1) holds with $\beta^{2i}=\eta^{ (1-q^n) (1+\sigma)/2}$ by direct check. To summarize, we have shown that (1) holds if and only if \eqref{eqn_eta_del} holds.
We claim that $\gcd(e,\ell)=r$, $\delta_0^{(p^r-1)/2}=-1$ and $\eta^{(1+q^n)(\tau-1)}=\delta^{2(\sigma-1)}$, where $r=\gcd(e,2\ell)$, and $\delta_0=\delta^{2(q^n-1)/(q-1)}$. By raising both sides of \eqref{eqn_eta_del} to the $\left(\frac{q^n-1}{q-1}\right)$-th power we get $\delta_0^{(p^{2\ell}-1)/2}=1$. It follows from $\gcd(p^{2\ell}-1,p^e-1)=p^r-1$ that $\delta_0^{p^r-1}=1$. On the other hand, $\delta^2$ is a nonsquare in $\F_{q^n}^*$, so $\delta_0^{(p^e-1)/2}=-1$.
Now, we see that $\delta_0$ has order dividing $\gcd(\frac{p^{2\ell}-1}{2},p^r-1)$ which is equal to either $p^r-1$ or $(p^r-1)/2$. The latter case will not occur, since it would lead to the contradiction $\delta_0^{(p^e-1)/2}=1$. Hence $p^r-1$ divides $\frac{p^{2\ell}-1}{2}$, which is the case only if $2\ell/r$ is even, i.e., $r|\ell$. This shows that $r=\gcd(e,\ell)$ and $\delta_0^{(p^r-1)/2}=-1$. It is clear that $g:=\gcd\left(\frac{q^n-1}{q-1},\frac{\sigma+1}{2}\right)$ divides $\gcd(p^{ne}-1,p^{2\ell n}-1)=p^{\gcd(ne,2\ell n)}-1=p^{nr}-1$, so it also divides $\gcd(p^{n\ell}-1,p^\ell+1)=2$. On the other hand, $\frac{q^n-1}{q-1}$ is odd, so $g=1$. It follows from \eqref{eqn_eta_del} that $\eta^{(1+q^n)(\tau-1)}=\delta^{2(\sigma-1)}$.
We now consider the condition (2). Recall that $\la\beta\ra$ is the set of elements of $\F_{q^{2n}}^*$ whose relative norm to $\F_{q^n}$ is in $\F_q^*$. The condition amounts to that $\eta^{1+q^n}Q(x)^\sigma=\lambda^2Q(y)$ does not hold for any $x,y\in\F_{q^n}^*$ and $\lambda\in\F_q^*$, where $Q$ is as defined in \eqref{eqn_Q}. By expanding $\eta^{1+q^n}Q(x)^\sigma=\lambda^2Q(y)$ and rearranging terms, we get $Y=\delta^2Y^\tau$ with $Y=\eta^{1+q^n}x^{2\sigma}-\lambda^2y^2\in\F_{q^n}$. Here we have made use of the fact that $\eta^{(1+q^n)(\tau-1)}=\delta^{2(\sigma-1)}$. If $Y\ne 0$, then $Y^{1-\tau}$ is a square while $\delta^2$ is not, which is impossible. We thus must have $Y=0$. It is clear that $Y=0$ has a solution $(x,y,\lambda)\in\F_{q^n}^*\times\F_{q^n}^*\times\F_q^*$ if and only if $\eta^{1+q^n}$ is a square in $\F_{q^n}^*$. To summarize, (2) holds only if $\eta^{1+q^n}$ is a nonsquare in $\F_{q^n}^*$.
Let $e'$ be the highest power of $2$ that divides $e$. It follows from $r=\gcd(e,\ell)=\gcd(e,2\ell)$ that $e/r$ is odd, i.e., $e'|r$. Hence $\ell$ is a multiple of $e'$. Recall that the order of $\overline{\psi}$ is a power of $2$, so $\ell$ is a multiple of $ne$. On the other hand, $0<\ell<2ne$ implies that $\ell=ne$. In this case, $r=e$, and the conditions reduce to the same as stated in the theorem.
The sufficiency part of the theorem is shown in \cite{kantor_odd,ks}.
\end{proof}
The spreads described in Theorem \ref{thm_typeC} and Theorem \ref{thm_typeH} are due to Kantor and Suetake \cite{kantor_odd,ks}. The case $n=3$ has been characterized by Baker and Ebert et al in a series of papers \cite{be_last,be_baer2,be_baer,be_x} by a geometric approach. It is well-known that a two-dimensional finite semifield is a finite field \cite{dickson_2dim}, so the same arguments in this section can be applied to characterize the case $n=2$ which has been dealt with in \cite{be_2dim}.
\section{Planar functions of the form $L(x)^2-wx^2$}
In Section \ref{sect_function}, we have shown that $\cC$-planes and $\cH$-planes of odd order have close connections with planar functions of the form $X^2-\delta^2 L(X)^2$ over $\F_{q^n}$, where $\delta^{q^n-1}=-1$. In this section, we study the properties of the commutative semifields associated with such planar functions and determine the planarity of functions of particular forms. Throughout this section, we shall fix the following notation. For convenience, we write alternatively $Q(X)=L(X)^2-wX^2$, where $w(=\delta^{-2})$ is a nonsquare in $\F_{q^n}$ and $L(X)$ is a reduced $q$-polynomial such that
\begin{equation}\label{eqn_L_ker}
\{\lambda\in\F_{q^n}:\,L(\lambda x)=\lambda L(x) \textup{ for all } x \in\F_{q^n}\}=\F_q,
\end{equation}
Assume that $Q(X)$ is a planar function over $\F_{q^n}$. Then
\[
x\mapsto\,\frac{1}{2}\left(Q(x+1)-Q(x)-Q(1)\right)=L(1)L(x)-wx
\]
is a $\F_q$-linear bijection of $\F_{q^n}$ by definition. Let $L_1$ be its inverse under composition, which is also $\F_q$-linear. Denote by $S_Q=(\F_{q^n},+,\circ)$ the associated semifield with multiplication
\begin{equation}\label{eqn_circ}
x\circ y=L(L_1(x))L(L_1(y))-wL_1(x)L_1(y),
\end{equation}
The multiplicative identity is $e_Q=L(1)^2-w$, and $\cN(S_Q)$ contains $\F_q\cdot e_Q$. We have $(ae_Q)\circ z=az$ for $a\in \F_q$ and $z\in\F_{q^n}$. Let $q^m$ and $q^r$ be the sizes of $\cN_m(S_Q)$ and $\cN(S_Q)$ respectively.
We now explicitly construct a commutative isotope of $S_Q$ that has $\F_{q^m}$ as the middle nucleus. For each $x\in S_Q$, let $R_x$ be the $\F_q$-linear map over $(\F_{q^n},+)$ such that $R_x(z)=z\circ x$. For $u\in \cN_m(S_Q)$ and $i\ge 1$, we use $u^{\circ i}$ for the product of $i$ copies of $u$ under the multiplication $\circ$, and set $u^{\circ 0}:=e_Q$. Let $z'$ be a fixed primitive element of $\F_{q^m}$ with minimal polynomial $X^m+\sum_{i=0}^{m-1}c_iX^i$ over $\F_q$. The map $a\mapsto ae_Q$ is a field isomorphism between $(\F_q,+,\cdot)$ and $(\F_q\cdot e_Q,+,\circ)$, and it naturally extends to a ring isomorphism between their polynomial rings. Therefore, there exists a primitive element $z\in\cN_m(S_Q)$ such that $z^{\circ m}+\sum_{i=0}^{m-1}(c_ie_Q)\circ z^{\circ i}=0$. Let $\{f_1=e_Q,f_2\cdots ,f_{n/m}\}$ and $\{f_1'=1,f_2'\cdots ,f_{n/m}'\}$ be a basis of $(\F_{q^n},+)$ over $\cN_m(S_Q)$ and $\F_{q^m}$ respectively. Now define a $\F_q$-linear map $\psi:\,(\F_{q^n},+)\mapsto (\F_{q^n},+)$ such that
\[
\psi(z'^if_j')=z^{\circ i}\circ f_j,\quad 0\le i\le m-1 , \;1\le j\le n/m.
\]
Here, $\F_q$-linearity means that $\psi(\lambda x)=\lambda\psi(x)$ for $\lambda\in\F_q$ and $x\in\F_{q^n}$. The map $\psi$ is a bijection and has the properties: (1) The restriction $\psi|_{\F_{q^m}}$ is a field isomorphism between $\F_{q^m}$ and $\cN_m(S_Q)$; (2) $\psi(ax)=\psi(a)\circ \psi(x)$, i.e., $\psi^{-1}R_{\psi(a)}\psi(x)=ax$, for $a\in\F_{q^m}$ and $x\in\F_{q^n}$. By the definition of the nucleus, $R_{\psi(y)}R_{\psi(a)}=R_{\psi(a)}R_{\psi(y)}$ for $a\in \F_{q^r}$ and $y\in \F_{q^n}$, so $\psi^{-1}R_{\psi(y)}\psi$ is $\F_{q^r}$-linear for each $y$ by (2). On the other hand, $\psi^{-1}(\psi(y)\circ\psi(x))=\psi^{-1}R_{\psi(y)}\psi(x)$ is symmetric in $x,y$, so it is $\F_{q^r}$-linear in both $x$ and $y$. Therefore, $\psi^{-1}(\psi(y)\circ\psi(x))=x\ast_K y$, where $x\ast_K y=\sum_{i,j}c_{ij}x^{q^{ri}}y^{q^{rj}}$ for some constants $c_{ij}$'s such that $c_{ij}=c_{ji}$. It follows that $\psi(x)\circ \psi(y)=\psi(x\ast_Ky)$ and also $a*_Kx=ax$ for $a\in\F_{q^m}$ by (2). The algebraic system $S_K:=(\F_{q^n},+,\ast_K)$ is a semifield isotopic to $S_Q$, and its middle nucleus is $\F_{q^m}$.
We write $K(X)=\sum_{i,j=0}^{n/r-1}c_{ij}X^{q^{ri}+q^{rj}}$, so that $K(x)=x\ast_K x$ for $x\in\F_{q^n}$. Let $M_1(X),M_2(X)$ be the reduced $q$-polynomials s.t. $M_1(x)=L(L_1(\psi(x)))$ and $M_2(x)=L_1(\psi(x))$ for $x\in\F_{q^n}$ respectively. Applying the map $x\mapsto L(1)L(x)-wx$ to both sides of $M_2(x)=L_1(\psi(x))$, we get $\psi(x)=L(1)M_1(x)-wM_2(x)$. Now $\psi(x)\circ \psi(x)=\psi(x\ast_Kx)$ takes the form
\begin{equation}\label{eqn_M}
M_1(X)^2-wM_2(X)^2\equiv L(1)M_1(K(X))-wM_2(K(X))\pmod{X^{q^n}-X}.
\end{equation}
\begin{lemma}\label{lemma_N}
Let $q$ be an odd prime power, $n\ge 2$ be an integer, and $w$ be a nonsquare in $\F_{q^n}$. Suppose that $L(X)$ is a reduced $q$-polynomial over $\F_{q^n}$ such that \eqref{eqn_L_ker} holds. If $Q(x)=L(x)^2-wx^2$ is planar over $\F_{q^n}$, then the semifield $S_Q=(\F_{q^n},+,\circ)$ with multiplication as defined in \eqref{eqn_circ} is either isotopic to a finite field or has nucleus equal to $\F_q$.
\end{lemma}
\begin{proof}
We use the notation introduced preceding the lemma, and write $s=n/r$. The semifield $S_Q$ is isotopic to a finite field if and only $r=n$, so we assume that $1<r<n$ and try to derive a contradiction. First we introduce some notation for the proof. For a reduced $q$-polynomial $f(X)\in\F_{q^n}[X]$, we have a unique decomposition
\[
f(X)=f_0(X)+f_1(X^{q})+\cdots+f_{r-1}(X^{q^{r-1}}),
\]
where the $f_i$'s are reduced $q^r$-polynomials. We call it the {\it $q^r$-decomposition} of $f$, and call $f_i$ the {\it $i$-th component}. For $t\in\{1,2\}$, write $M_t(X)=\sum_{i=0}^{n-1}a_{i,t}X^{q^i}$, and define
\[
I_t:=\{0\le i\le r-1:\,\textup{one of }a_{i,t},\,a_{i+r,t},\cdots,a_{i+(s-1)r,t}\textup{ is nonzero} \}.
\]
We comment that $i\in I_t$ if and only if $M_t$ has a nonzero $i$-th component in its $q^r$-decomposition. Since $M_2$ is a permutation and $L$ is not the zero polynomial, both $I_1$ and $I_2$ are nonempty. Observe that for $i\not\equiv j\pmod{r}$, the coefficient of $X^{q^i+q^j}$ is zero on the right hand side of \eqref{eqn_M}, so we have $a_{i1}a_{j1}=wa_{i2}a_{j2}$ from the left hand side. We can pick two subsequences of length $r$ whose supports are $I_1$ and $I_2$ from the coefficients of $M_1(X)$ and $M_2(X)$ respectively, satisfying the conditions in Remark \ref{remark_size}. It follows that each of $I_1$ and $I_2$ has size at most $2$ and $I_1=I_2$ when one of them has size $2$.
We first consider the case $|I_1|=|I_2|=1$. In this case, we have $M_t(X)=U_t(X^{q^{r_t}})$ for $t\in\{1,2\}$, where $U_1$ and $U_2$ are $q^r$-polynomials, and $0\le r_1,\,r_2\le r-1$. Set $r_3:=r_1-r_2\pmod{r}$. Since $M_1(X)\equiv L(M_2(X))\pmod{X^{q^n}-X}$, the $q^r$-decomposition of $L(X)$ has exactly one nonzero component, namely the $r_3$-rd.
We have $r_3\ne 0$: otherwise $L(X)$ is $q^r$-polynomial, contradicting \eqref{eqn_L_ker}. By comparing exponents of the monomials in \eqref{eqn_M}, we get
$M_1(x)^2=L(1)M_1(K(x))$ and $M_2(x)^2=M_2(K(x))$
for $x\in\F_{q^n}$. Applying $L$ to both sides of the second equation, we get $L(M_2(x)^2)=M_1(K(x))$. Recall that $x\mapsto M_2(x)=L_1(\psi(x))$ is a permutation. By setting $z:=M_2(x)$ and combining $L(M_2(x)^2)=M_1(K(x))$ with the first equation, we get $L(z)^2=L(1)L(z^2)$. It follows that $L(X)$ is a monomial and $S_Q$ is isotopic to a finite field, contradicting our assumption.
We next consider the case $I_1=I_2=\{r_1,r_2\}$, where $0\le r_1<r_2\le r-1$. In this case, we have the $q^r$-decomposition $M_1(X)=A(X^{q^{r_1}})+B(X^{q^{r_2}})$, $M_2(X)=C(X^{q^{r_1}})+D(X^{q^{r_2}})$, where $A,B,C,D$ are reduced $q^r$-polynomials neither of which is the zero polynomial. Plugging them into \eqref{eqn_M} and again by comparing exponents of monomials, we get
\begin{align}
A(X^{q^{r_1}})B(X^{q^{r_2}})&=wC(X^{q^{r_1}})D(X^{q^{r_2}}),\label{eqn_extra}\\
A(X^{q^{r_1}})^2-wC(X^{q^{r_1}})^2&\equiv L(1)A(K(X)^{q^{r_1}})-wC(K(X)^{q^{r_1}})\pmod{X^{q^n}-X},\label{eqn_extra2}\\
B(X^{q^{r_2}})^2-wD(X^{q^{r_2}})^2&\equiv L(1)B(K(X))^{q^{r_2}}-wD(K(X)^{q^{r_2}})\pmod{X^{q^n}-X}.\label{eqn_extra3}
\end{align}
The equation \eqref{eqn_extra} holds without modulo $X^{q^n}-X$ since both sides have degree at most $2q^{n-1}\le q^n-1$. If the coefficient of $X^{q^{r\ell}}$ in $B(X)$ is nonzero, then by considering the coefficients of the monomials $\{X^{q^{r_2+r\ell}+q^{r_1+ri}}: \,0\le i\le s-1\}$ on both sides of \eqref{eqn_extra}, we see that $A$ and $C$ differs by a constant. That is, $A(X)=\lambda C(X)$, $B(X)=\lambda^{-1}wD(X)$ for some $\lambda\ne 0$. Canceling $A$ and $B$ from \eqref{eqn_extra2} and \eqref{eqn_extra3} by substitution, we get
\begin{align}
(\lambda L(1)-w)\cdot C(K(x)^{q^{r_1}})= (\lambda^2-w)\cdot C(x^{q^{r_1}})^2, \label{eqn_CD}\\
(\lambda^2-\lambda L(1))\cdot D(K(x)^{q^{r_2}})=(\lambda^2-w)\cdot D(x^{q^{r_2}})^2.\label{eqn_CD2}
\end{align}
Since $w$ is a nonsquare, $\lambda^2-w\ne 0$. It follows that $\lambda L(1)-w$ and $\lambda^2-\lambda L(1)$ are both nonzero. By expansion using the $q^m$-decomposition of $M_1,\,M_2$, we have
\[
M_1(x)^2-wM_2(x)^2=(\lambda^2-w)\cdot(C(x^{q^{r_1}})^2-\lambda^{-2}wD(x^{q^{r_2}})^2).
\]
The map $x\mapsto Q(M_2(x))=M_1(x)^2-wM_2(x)^2$ is a planar function of DO type equivalent to $Q$, so at least one of $C,\,D$ is a permutation polynomial by Lemma \ref{lemma_L_perm}. If $C$ is a permutation polynomial, then we substitute $x+y,\,x,\,y$ into \eqref{eqn_CD} and take their linear combination to get $c_1 C((x\ast_Ky)^{q^{r_1}})=\left(c_1 C(x^{q^{r_1}})\right)\cdot \left(c_1 C(y^{q^{r_1}})\right)$, where $c_1=\frac{\lambda^2-w}{\lambda L(1)-w}$. Here, we have used the fact that $K(x+y)-K(x)-K(y)=2(x*_Ky)$. This shows that $S_K$ is isotopic to a finite field, contradicting our assumption. The case $D$ is a permutation polynomial leads to the same contradiction. This completes the proof.
\end{proof}
The size of the nucleus is an invariant under isotopism and provides a measure for the non-associativity of a commutative semifield. Lemma \ref{lemma_N} suggests that the associated semifield $S_Q$ of a planar function $Q(x)$ of the form described in the lemma behaves in two extremes: the size of its nucleus is either the maximum possible or the minimum possible. The planar functions that have associated semifields isotopic to a finite field have been characterized in \cite{coulter}, and those of the prescribed form is implicitly described in Lemma \ref{lemma_ps}. This provides some evidence to the conjecture that the planar functions of this form are all known. There is not much that we can say about the middle nucleus. In \cite{chk}, the authors have studied the equivalent forms of planar functions whose corresponding commutative semifields have specified nuclei. We deal with the planar functions of our special form in the following lemma.
\begin{lemma}\label{lemma_equiv}
Let $q$ be odd, $n\ge 2$ be an integer, and $w$ be a nonsquare in $\F_{q^n}$. Let $L(X)$ be a reduced $q$-polynomial over $\F_{q^n}$ such that \eqref{eqn_L_ker} holds. Assume that $Q(X)=L(X)^2-wX^2$ is planar over $\F_{q^n}$ and the semifield $S_Q=(\F_{q^n},+,\circ)$ with multiplication as in \eqref{eqn_circ} has a middle nucleus of size $q^m$ with $1\le m<n$. Then $Q(X)$ is equivalent to either
\begin{enumerate}
\item[(i)] $A(X)^2-w'X^{2q^k}$, with $\gcd(k,m)=1$, $w'$ a nonsquare and $A(x)$ a $q^m$-polynomial, or
\item[(ii)] $\left(L(1)X+T_1(\Delta)+R(\Delta)\right)^2-w\left(X+T_0(\Delta)+L(1)w^{-1}R(\Delta)\right)^2$, where $\Delta=X^{q^m}-X$, $T_0,\,T_1$ are $q^m$-polynomial and $R$ is a nonzero $q$-polynomial.
\end{enumerate}
\end{lemma}
\begin{proof}
Take the same notation as in the proof of Lemma \ref{lemma_N}, and write $s=n/m$. Let $M_1(X)=\sum_{i=0}^{s-1}f_i(X^{q^i})$ and $M_2(X)=\sum_{i=0}^{s-1}g_i(X^{q^i})$ be their $q^m$-decompositions respectively, where the $f_i$'s and $g_i$'s are reduced $q^m$-polynomials. Set $I_1=\{i:\,f_i\ne 0\}$ and $I_2=\{i:\,g_i\ne 0\}$. Both $I_1$ and $I_2$ are non-empty subsets of $\{0,1,\cdots, s-1\}$.
For $a\in\F_{q^m}$, we have $K(a)=a*_Ka=a^2$, so \eqref{eqn_M} gives that $M_1(a)^2-wM_2(a)^2=L(1)M_1(a^2)-wM_2(a^2)$. In other words,
\begin{equation}\label{eqn_M_new}
M_1(X)^2-wM_2(X)^2\equiv L(1)M_1(X^2)-wM_2(X^2)\pmod{X^{q^m}-X}.
\end{equation}
It is similar to \eqref{eqn_M12ff} and the same argument using Remark \ref{remark_size} show that either
\begin{enumerate}
\item $M_1(X)\equiv c_1X^{q^k}+c_2X^{q^l}$ and $M_2(X)\equiv c_3X^{q^k}+c_4X^{q^l}\pmod{X^{q^m}-X}$, or
\item $M_1(X)\equiv c_1'X^{q^u}$ and $M_2(X)\equiv c_2'X^{q^v}\pmod{X^{q^m}-X}$,
\end{enumerate}
where $0\le k, l,u,v\le m-1$, $k<l$, the $c_i$'s and $c_i'$'s are constants with $c_1c_2=wc_3c_4\ne 0$. Notice that $c_2'\ne 0$, since $M_2(X)$ is a permutation polynomial over $\F_{q^n}$. If $c_1'=0$, then we set $u=v$ for uniform treatment. The monomials in $M_t(X)\pmod{X^{q^m}-X}$ correspond to the components of $M_t(X)$ in its $q^m$-decomposition for $t\in\{1,2\}$, so $k,l\in I_1\cap I_2$ in the case (1), $v\in I_2$ in the case (2), and $u\in I_1$ in the case (2) if $c_1'\ne 0$.
In the case (2), after plugging them back into \eqref{eqn_M_new} and comparing coefficients we get $c_1'=L(1)$ and $c_2'=1$ if $u\ne v$ and $c_1'^2-wc_2'^2=L(1)c_1'-wc_2'$ if $u= v$. Now consider the special case $c_1'=0$. We deduce that $c_2'=1$, $M_2(a)=a^{q^u}$ for $a\in\F_{q^m}$, so $M_2$ maps $\F_{q^m}$ to $\F_{q^m}$ bijectively. On the other hand, $L(M_2(a))=M_1(a)=c_1'a^{q^u}=0$ for all $a\in\F_{q^m}$. This shows that $\F_{q^m}$ lies in the kernel of $L$. Therefore, there exists a $q$-polynomial $R$ such that $L(X)=R(X^{q^m}-X)$, so $Q(X)$ is of the second form in the lemma. We thus assume that $c_1'\ne 0$ in the case (2).
For $a\in\F_{q^m}$ and $x\in\F_{q^n}$, we plug $x+a$, $x$ and $a$ into \eqref{eqn_M} and take their linear combination to get $M_1(a)M_1(x)-wM_2(a)M_2(x)=L(1)M_1(ax)-wM_2(ax)$. Here, we have used the fact $K(x)=x*_Kx$ and $a*_Kx=ax$ for $a\in\F_{q^m}$. Since both $M_1(X)$ and $M_2(X)$ are reduced, we have $M_1(a)M_1(X)-wM_2(a)M_2(X)=L(1)M_1(aX)-wM_2(aX)$. By expanding it using the $q^m$-decompositions and comparing exponents of monomials, we get
\begin{equation}\label{eqn_figi}
(M_1(a)-L(1)a^{q^i})f_i(X^{q^i})=w(M_2(a)-a^{q^i})g_i(X^{q^i}),\quad 0\le i\le s-1.
\end{equation}
If $i$ is an integer such that $0\le i\le s-1$ and neither $M_1(X)-L(1)X^{q^i}$ nor $M_2(X)-X^{q^i}$ is zero modulo $X^{q^m}-X$, then we claim that $i$ is either in both of $I_1$ and $I_2$ or in neither of them. There exists $a\in\F_{q^m}$ such that $M_1(a)-L(1)a^{q^i}\ne 0$ by the assumption, so it follows from \eqref{eqn_figi} that $f_i$ is equal to $g_i$ multiplied by a constant. Similarly, $g_i$ is equal to $f_i$ multiplied by a constant, so the claim follows.
After these preparations, we are now ready to handle each case separately.
In the case (1), we claim that $I_1=I_2=\{k,l\}$, $M_2(X)=g_k(X^{q^k})+g_l(X^{q^l})$ and $M_1(X)=c_1c_3^{-1}g_k(X^{q^k})+c_2c_4^{-1}g_l(X^{q^l})$. For each $0\le i\le s-1$, neither $M_1(X)-L(1)X^{q^i}$ nor $M_2(X)-X^{q^i}$ is zero modulo $X^{q^m}-X$, so $I_1=I_2$ by the preceding claim. For $i\in I_1=I_2$, there is a nonzero constant $d_i$ such that $g_i=d_if_i$ and plugging it in \eqref{eqn_figi} we get
\begin{equation}\label{eqn_a_pf}
(L(1)-wd_i)a^{q^i}=M_1(a)-wd_iM_2(a)=(c_1-wd_ic_3)a^{q^k}+(c_2-wd_ic_4)a^{q^l}.
\end{equation}
It is straightforward to check that the right hand side has at least one nonzero coefficient, so $L(1)-wd_i\ne 0$. It follows that $i=k$ or $i=l$. Comparing the coefficients of $a^{q^l}$ in \eqref{eqn_a_pf} in the case $i=k$ gives $d_k=w^{-1}c_4^{-1}c_2=c_3c_1^{-1}$. Similarly, we get $d_l=c_4c_2^{-1}$ in the case $i=l$. This proves the claim. We then compute that
\[
Q(M_2(x))=M_1(x)^2-wM_2(x)^2=(1-wd_k^2)g_k(x^{q^k})^2+(1-wd_l^2)g_l(x^{q^l})^2.
\]
Since $Q(M_2(x))$ is a planar function equivalent to $Q(x)$, one of $g_k$ and $g_l$ must be a permutation polynomial by Lemma \ref{lemma_L_perm}. Consider the case $g_k$ is a permutation polynomial. With $y=g_k(x^{q^k})^{q^{l-k}}$ we have $x^{q^k}=h_k(y^{q^{k-l}})$ for some $q^m$-polynomial $h_k$, and so $g_l(x^{q^l})=g_l(h_k(y^{q^{k-l}})^{q^{l-k}})=A(y)$ for some $q^m$-polynomial $A$. We now have $Q(M_2(x))=(1-wd_k^2)y^{2q^{k-l}}+(1-wd_l^2)A(y)^2$ with $A$ a $q^m$-polynomial, and so $Q(x)$ is equivalent to one of the first form in the lemma. Since the nucleus of $S_Q$ is $\F_q$ by Lemma \ref{lemma_N}, we have $\gcd(k-l,m)=1$. The case $g_l$ is a permutation polynomial is dealt with similarly.
In the case (2) with $u\ne v$, we claim that $I_1=\{u\}$ and $I_2=\{v\}$. Recall that $c_1'=L(1)\ne 0$, $c_2'=1$, $u\in I_1$ and $v\in I_2$ in this case. For each $i\in\{0,1,\cdots,s-1\}\setminus\{u,v\}$, neither $M_1(X)-L(1)X^{q^i}$ nor $M_2(X)-X^{q^i}$ is zero modulo $X^{q^m}-X$ in this case, so $I_1\setminus\{u,v\}=I_2\setminus\{u,v\}$. If $i\in I_1\setminus\{u,v\}=I_2\setminus\{u,v\}$, then there exists a nonzero constant $d_i$ such that $g_i=d_if_i$, and \eqref{eqn_figi} reduces to $(L(1)-wd_i)a^{q^i}=c_1'a^{q^u}-wd_ic_2'a^{q^v}$. This is impossible since $c_1'\ne 0$ and $wd_ic_2'\ne 0$. Hence $I_1,I_2$ are both subsets of $\{u,v\}$. By setting $i=v$ in \eqref{eqn_figi}, we get $f_v=0$. Similarly, $g_u=0$. This proves the claim. By the same argument in the previous case, $Q(x)$ is equivalent to one of the first form in the lemma.
Finally, consider the case (2) with $u=v$ and $c_1'\ne 0$. In this case, $u$ is in both $I_1$ and $I_2$. As before we have $I_1\setminus\{u\}=I_2\setminus\{u\}$. If $I_1=I_2=\{u\}$, then from the fact $M_2(X)$ is a permutation polynomial and $M_1(X)=L(M_2(X))\pmod{X^{q^n}-X}$ we deduce that $L$ is a $q^m$-polynomial, contradicting \eqref{eqn_L_ker}.
Hence $I_1\setminus\{u\}=I_2\setminus\{u\}\ne\emptyset$. For $t\in I_1\setminus\{u\}$, there exists a constant $d_t\ne 0$ such that $g_t=d_tf_t$ and \eqref{eqn_figi} reduces to $c_1'a^{q^u}-L(1)a^{q^t}=wd_t(c_2'a^{q^u}-a^{q^t})$ for $a\in\F_{q^m}$. By comparing coefficients we get $L(1)=wd_t$ and $c_1'=wd_tc_2'$. Together with $c_1'^2-wc_2'^2=L(1)c_1'-wc_2'$, we deduce that $c_2'=1$, $c_1'=L(1)$ and $d_t=L(1)w^{-1}$. To sum up, we have $M_2(X)-g_u(X^{q^u})=L(1)w^{-1}(M_1(X)-f_u(X^{q^u}))$, $M_1(X)\equiv L(1)X^{q^u}$ and $M_2(X)\equiv X^{q^u}\pmod{X^{q^m}-X}$. The $\F_{q^m}$-linear maps $x\mapsto f_u(x)-L(1)x$ and $x\mapsto g_u(x)-x$ both have $\F_{q^m}$ in their kernels, so there exist $q^m$-polynomials $T_1,T_0$ such that $T_1(X^{q^m}-X)=f_u(X)-L(1)X$ and $T_0(X^{q^m}-X)=g_u(X)-X$. Meanwhile, $M_1(X)-f_u(X^{q^u})$ has $\F_{q^m}$ in the kernel, so there exists a $q$-polynomial $R$ such that it is equal to $R(X^{q^{m+u}}-X^{q^u})$. We thus have $M_1(X)=L(1)X^{q^u}+T_1(X^{q^m}-X)+R(X^{q^{m+u}}-X^{q^u})$, $M_2(X)=X^{q^u}+T_0(X^{q^m}-X)+L(1)w^{-1}R(X^{q^{m+u}}-X^{q^u})$, and $Q(X)$ is equivalent to the second form in this case.
\end{proof}
For the rest of this section, we will only consider the simplest cases as a demonstration of techniques, where the associated semifields are rank two commutative semifields. They correspond to Case (i) in Lemma \ref{lemma_equiv} with $n=2m$. We start with a technical lemma.
\begin{lemma}\label{lemma_Psi} Let $q$ be an odd prime power, $m$ be a positive integer, and take $\zeta\in\F_{q^{2m}}$ such that $\zeta^{q^m-1}=-1$. Let $\Psi:\,\F_{q^{2m}}^2\mapsto\F_{q^m}^3$ be the map defined by
\begin{equation}\label{eqn_Psi}
\Psi(x_0\zeta+x_1,y_0\zeta+y_1):=(x_1y_1,\,x_0y_0,\,x_0y_1+x_1y_0),\quad\forall\, x_0,x_1,y_0,y_1\in\F_{q^m}.
\end{equation}
Then its image set is equal to $\{(A,B,C)\in\F_{q^m}^3:\,C^2-4AB \text{ is a square in $\F_{q^m}$}\}$, and $\Psi(x,y)=(0,0,0)$ if and only if $x=0$ or $y=0$.
\end{lemma}
\begin{proof}
If $(A,B,C)=\Psi(x,y)$ with $x=x_0\zeta+x_1$, $y=y_0\zeta+y_1$ ($x_i,y_i\in\F_{q^m}$), then $A=x_1y_1$, $B=x_0y_0$, $C=x_0y_1+x_1y_0$, and $C^2-4AB=(x_0y_1-x_1y_0)^2$ is a square in $\F_{q^m}$. If further $A=B=C=0$, then it is straightforward to show that at least one of $x$ and $y$ is $0$.
Conversely, suppose that $(A,B,C)\in\F_{q^m}^3$ satisfies that $C^2-4AB$ is a square. We can directly check that $\Psi(\zeta,B\zeta+C)=(A,B,C)$ if $A=0$ and $\Psi(1,C\zeta+A)=(A,B,C)$ if $B=0$. Now assume that $AB\ne 0$, and let $t$ be a solution to $BX^2-CX+A=0$. It is now routine to check that $t\in\F_q^*$ and $\Psi(\zeta+t,B\zeta+At^{-1})=(A,B,C)$.
\end{proof}
\begin{lemma}\label{lemma_specialQ}
Suppose that $m\ge 3$, $1\le k\le m-1$ and $\gcd(k,m)=1$. Let $q$ be odd and $w$ be a nonsquare in $\F_{q^{2m}}$. Then $Q(X)=(X^{q^m}-X)^2-wX^{2q^k}$ is not planar over $\F_{q^{2m}}$.
\end{lemma}
\begin{proof}
Take $\zeta\in\F_{q^{2m}}$ such that $\zeta^{q^m-1}=-1$, and let $\Psi$ be as defined in \eqref{eqn_Psi}. Write $x=x_0\zeta+x_1$, $y=y_0\zeta+y_1$ with $x_i,\,y_i\in\F_{q^m}$, and set $(A,B,C)=\Psi(x,y)$. Then $Q(x+y)=Q(x)+Q(y)$ if and only if
\begin{equation}\label{eqn_ABC0}
-wA^{q^k}+(4\zeta^2 B-w\zeta^{2q^k}B^{q^k})-w\zeta^{q^k} C^{q^k}=0.
\end{equation}
The left hand side is equal to $\frac{1}{2}(Q(x+y)-Q(x)-Q(y))$. By Lemma \ref{lemma_Psi}, $Q(x)$ is planar if and only if there is no triple $(A,B,C)\in\F_{q^m}^3\setminus\{(0,0,0)\}$ such that $C^2-4AB$ is a square and \eqref{eqn_ABC0} holds. To show that $Q(x)$ is not planar, we need to establish the existence of such a triple.
By raising both sides of \eqref{eqn_ABC0} to the $q^m$-th power we get another equation, and together with \eqref{eqn_ABC0} we deduce that
\begin{equation}\label{eqn_AC}
A^{q^k}=2( w^{-q^m}+w^{-1})\zeta^2B-\zeta^{2q^k}B^{q^k},\;
C^{q^k}=2\zeta^{2-q^k}(w^{-1}- w^{-q^m})B,
\end{equation}
We then compute that
\begin{align*}
(C^2&-4AB)^{q^k}=4\zeta^{4-2q^k}(w^{-1}- w^{-q^m})^2B^2-4B^{q^k}\left(2( w^{-q^m}+w^{-1})\zeta^2B-\zeta^{2q^k}B^{q^k}\right)\\
&=4\zeta^{4-2q^k}B^2\left((w^{-1}- w^{-q^m})^2-2( w^{-q^m}+w^{-1})\zeta^{2(q^k-1)}B^{q^k-1}+\zeta^{4(q^k-1)}B^{2(q^k-1)}\right)\\
&=4\zeta^{4-2q^k}B^2\left((w^{-1}+w^{-q^m}-\zeta^{2(q^k-1)}
B^{q^k-1})^2-4w^{-1-q^m}\right).
\end{align*}
Since $\zeta^2$ is a nonsquare in $\F_{q^m}^*$ and $\gcd(q^k-1,q^m-1)=q-1$, we need to find $z\in\F_{q^m}^*$ such that $H(z):=(w^{-1}+w^{-q^m}-\zeta^{2(q^k-1)}z^{q-1})^2-4w^{-1-q^m}$ is a nonsquare. Then by taking $B\in\F_{q^m}$ such that $B^{q^k-1}=z^{q-1}$ and setting $A,\,C$ as in \eqref{eqn_AC}, we get the desired triple. It remains to establish the existence of such an element.
Let $\rho$ be the multiplicative character of order two of $\F_{q^m}^*$, i.e., $\rho(x)=1$ if $x$ is a nonzero square and $\rho(x)=-1$ otherwise. We extend it to $\F_{q^m}$ by setting $\rho(0)=0$. Let $\overline{\F}_{q^m}$ be the algebraic closure of $\F_{q^m}$. If $H(X)=h(X)^2$ for some polynomial $h(X)\in\overline{\F}_{q^m}[X]$, then
\[
(w^{-1}+w^{-q^m}-\zeta^{2(q^k-1)}X^{q-1}+h(X))(w^{-1}+w^{-q^m}-\zeta^{2(q^k-1)}X^{q-1}-h(X))=4w^{-1-q^m}.
\]
It implies that the two factors on the left hand side both have degree $0$, which is impossible. We can now apply \cite[Theorem 6.2.2]{handbookff} to see that
$|\sum_{z\in\F_{q^m}}\rho(H(z))|\le (2(q-1)-1)q^{m/2}$.
It is straightforward to check that $(2(q-1)-1)q^{m/2}<q^m-1$ holds for all $q$ odd and $m\ge 3$, so there exists $z\in\F_{q^m}^*$ such that $H(z)$ is a nonsquare. This completes the proof.
\end{proof}
\begin{thm}\label{thm_Q}
Let $k,m$ be positive integers with $m\ge 3$, $\gcd(k,m)=1$, and let $q=p^e$ with $p$ an odd prime. Suppose that either of the following holds: (1) $q\ge 4n^2-8n+2$, (2) $p>2(em)^2-(4-2\sqrt{3})em+(3-2\sqrt{3})$, (3) $m=3$. Then $Q(X)=(aX+bX^{q^m})^2-wX^{2q^k}$ is planar over $\F_{q^{2m}}$ if and only if $ab=0$, where $a,b\in\F_{q^{2m}}$ and $w$ is a nonsquare in $\F_{q^{2m}}^*$.
\end{thm}
\begin{proof}
Take $\zeta\in\F_{q^{2m}}$ such that $\zeta^{q^m-1}=-1$, and let $\Psi$ be as defined in \eqref{eqn_Psi}. Set $s:=a+b$, $\beta:=(a-b)\zeta$. The case $s=0$ has been handled in Lemma \ref{lemma_specialQ}, so assume that $s\ne 0$. By multiplying $Q(x)$ by a nonzero constant if necessary, we set $s=1$ without loss of generality. If $\beta$ is in $\F_{q^m}$, then $a=\frac{1+\beta\zeta^{-1}}{2}$, $b=\frac{1-\beta\zeta^{-1}}{2}$, $-ab^{-1}=\frac{\beta+\zeta}{\beta-\zeta}=(\beta-\zeta)^{q^m-1}$, and $a^{-2}(\beta-\zeta)^{-2}Q\left((\beta-\zeta) X\right)$ is of the form in Lemma \ref{lemma_specialQ}. Therefore, we only need to consider the case $s=1$ and $\beta\not\in\F_{q^m}$. As in the case of Lemma \ref{lemma_specialQ}, $Q(x)$ is planar if and only if there is no triple $(A,B,C)\in\F_{q^m}^3\setminus\{(0,0,0)\}$ such that $C^2-4AB$ is a square and
\begin{equation}\label{eqn_ABC1}
A-wA^{q^k}+\beta^2 B-w\zeta^{2q^k}B^{q^k}+\beta C-w\zeta^{q^k} C^{q^k}=0.
\end{equation}
The left hand side equals $\frac{1}{2}(Q(x+y)-Q(x)-Q(y))$ in the case $(A,B,C)=\Psi(x,y)$.
Assume that $Q(x)$ is planar. We claim that the $\F_q$-linear map
\[
\Upsilon:\,\F_{q^m}^2\mapsto\F_{q^{2m}},\,(A,C)\rightarrow A-wA^{q^k}+ \beta C-w\zeta^{q^k}C^{q^k}
\]
is a bijection. Otherwise, there exists a pair $(A,C)\ne (0,0)$ such that \eqref{eqn_ABC1} holds with $B=0$. Since $C^2-4\cdot A\cdot 0=C^2$ is trivially a square, this is impossible by our assumption. Therefore, for each $B=u\in\F_{q^m}$, there is a unique pair $(A,C)=(f(u),g(u))$ such that $\Upsilon(f(u),g(u))=-\beta^2 u+w\zeta^{2q^k}u^{q^k}$, where $f(X),g(X)\in\F_{q^m}[X]$. It is clear that both $f$ and $g$ are $\F_q$-linear. By the definition of $f$ and $g$ we have
\begin{align}\label{eqn_ABC}
f(u)-wf(u)^{q^k}+\beta^2 u-w\zeta^{2q^k}u^{q^k}+\beta g(u)-w\zeta^{q^k} g(u)^{q^k}=0,\quad\forall\, u\in\F_{q^m}.
\end{align}
If $f(u)=0$ for some $u\ne 0$, then $(A,B,C)=(0,u,g(u))$ satisfies \eqref{eqn_ABC1} and $C^2-4AB=g(u)^2$ is a square. This contradicts our assumption, so the map $u\mapsto f(u)$ is a bijection.
By the planarity of $Q(x)$, $g(u)^2-4uf(u)$ is a nonsquare for all $u\in\F_{q^m}^*$, so $S(-g,-f)=(\F_{q^{2m}},+,\circ)$ as defined in Theorem \ref{thm_rtcs_def} with $t=\zeta$ is a RTCS. Recall that for $x,y\in\F_{q^{2m}}$ and $(A,B,C)=\Psi(x,y)$, we have
\[
x\circ y=(-g(B)+C)\zeta-f(B)+A.
\]
Let $M$ be the the $\F_q$-linear map such that $M(z_0\zeta+z_1):=\Upsilon(z_1,z_0)$ for $z_0,z_1\in\F_{q^m}$. It is nondegenerate, and with $(A,B,C)=\Psi(x,y)$ we have
\begin{align*}
M(x\circ y)&=\Upsilon(-f(B)+A,-g(B)+C)=-\Upsilon(f(B),g(B))+\Upsilon(A,C)\\
&=\beta^2 B-w\zeta^{2q^k}B^{q^k}+\Upsilon(A,C)=\frac{1}{2}(Q(x+y)-Q(x)-Q(y)).
\end{align*}
Therefore, the semifield $S_Q$ defined by $Q$ is isotopic to $S(-g,-f)$.
Under the conditions in the theorem, $S(-g,-f)$ is isotopic to either a finite field or a Dickson semifield by Theorem \ref{thm_rtcs}. By \cite[Example 2]{CG}, $S(-g,-f)$ is isotopic to a finite field if and only if $g(u)=cu$ and $f(u)=du$, where $c^2-4d$ is a nonsquare. In this case, \eqref{eqn_ABC} becomes
$(d+\beta c+\beta^2)u=w(d +\zeta c+\zeta^{2 })^{q^k}u^{q^k}.$ It holds for all $u$, so $d +\zeta c+\zeta^{2 }=0$ and $d+\beta c+\beta^2=0$. Since $\zeta^2\in\F_{q^m}$, we deduce that $c=0$, $d=-\zeta^2$ and $\beta=\pm\zeta$. It follows from $a+b=1$ and $a-b=\beta\zeta^{-1}$ that $(a,b)=(1,0)$ or $(0,1)$.
From now on, assume that $S(-g,-f)$ is not isotopic to a finite field but is isotopic to a Dickson semifield $S(0,\lambda x^{\theta})=(\F_{q^{2m}},+,\circ')$ as defined in Theorem \ref{thm_rtcs_def} with $t=\zeta$, where $\lambda$ is a nonsquare in $\F_{q^m}$ and $\theta\in\Aut(\F_{q^{2m}})$. By Lemma \ref{lemma_N}, the semifield $S_Q$ and so $S(-g,-f)$ has nucleus $\F_{q}$. It follows that $x^\theta=x^{q^\ell}$ for some $1\le\ell\le m-1$ such that $\gcd(\ell,m)=1$, cf. \cite[Theorem 10.16]{handbook}. By \cite[Theorem 2.6]{coulter}, there exist nondegenerate linear maps $L,N:\,\F_{q^{2m}}\mapsto\F_{q^{2m}}$ and $\alpha\in\F_{q^m}^*$ such that
\begin{equation}\label{eqn_iso}
L(x\circ y)=N(x)\circ' (\alpha N(y)).
\end{equation}
There exist linear maps $L_i:\,\F_{q^m}\mapsto\F_{q^m}$, $1\le i\le 4$, such that $N(x)=(L_1(x_0)+L_2(x_1))\zeta+L_3(x_0)+L_4(x_1)$ for $x=x_0\zeta+x_1$. Write $L_1(X)=\sum_{i=0}^{em-1}a_iX^{p^i}$ and $L_3(X)=\sum_{i=0}^{em-1}b_iX^{p^i}$, with $a_i,b_i\in\F_{q^m}$. We consider \eqref{eqn_iso} in three cases.
\begin{enumerate}
\item In the case $x_1=y_1=0$, it shows that $L(-g(x_0y_0)\zeta-f(x_0y_0))$ is equal to
\begin{equation*}
\lambda\alpha^{q^\ell}L_1(x_0)^{q^\ell}L_1(y_0)^{q^\ell}+\alpha L_3(x_0)L_3(y_0)+\alpha(L_1(x_0)L_3(y_0)
+L_3(x_0)L_1(y_0))\zeta.
\end{equation*}
The coordinates with respect to the basis $\{1,\zeta\}$ in the above expression are polynomials in $x_0y_0$. By comparing coefficients we get $\lambda\alpha^{q^\ell}a_i^{q^\ell}a_j^{q^\ell}+\alpha b_{i+el}b_{j+el}=0$, and $a_{i+el}b_{j+el}+a_{j+el}b_{i+el}=0$ for $i\ne j$. The subscripts are read modulo $em$ here. We observe that $-a_ia_jb_{i+el}b_{j+el}$ is a nonsquare and $-a_{i+el}a_{j+el}b_{i+el}b_{j+el}$ is a square, yielding that $a_ia_{i+el}a_ja_{j+el}$ is a nonsquare in the case $i\ne j$ and the terms involved are nonzero. Assume that $a_ua_v\ne 0$ for $0\le u<v\le em-1$. We claim that none of the elements in $\{a_{u+rel},\,b_{v+sel}:\,0\le r,s\le m-1\}$ is zero. It follows from the first equation with $(i,j)=(u,v)$ that $b_{u+el}b_{v+el}\ne 0$, and follows from the second equation with $(i,j)=(u-e\ell,u)$ and $(i,j)=(v-e\ell,v)$ that none of $a_{u+el},a_{v+el},b_u,b_v$ is zero. The claim then follows by induction. For $0\le i<j\le m-1$, exactly one of $a_{u+iel}a_{u+(i+1)el}$ and $a_{u+jel}a_{u+(j+1)el}$ is a square and the other is a nonsquare by our previous observation. This is impossible when $m\ge 3$. We conclude that $L_1$ is a monomial, and similarly $L_3$ is a monomial. Write $L_1(x_0)=c_1x_0^{p^{i}}$ and $L_3(x_0)=c_3x_0^{p^{i}}$ for some $i$ and constants $c_1,c_3$. Since $N$ is nondegenerate, $(c_1,c_3)\ne(0,0)$. In this case, \eqref{eqn_iso} reduces to
\begin{equation}\label{eqn_x0y0}
-L(g(u)\zeta+f(u))=2\alpha c_1c_3 u^{p^i}\zeta+\alpha c_3^2u^{p^i}+\lambda\alpha^{q^\ell}c_1^{2q^\ell}u^{q^\ell p^i},\;\forall\, u\in\F_{q^m}.
\end{equation}
\item In the case $x_1=y_0=0$, \eqref{eqn_iso} gives that $L(x_0y_1\zeta)$ is equal to
\begin{align*}
\alpha x_0^{p^i}(c_1L_4(y_1) +c_3L_2(y_1))\zeta+\alpha c_3x_0^{p^i}L_4(y_1)+\lambda\alpha^{q^\ell}c_1^{q^\ell}x_0^{p^iq^\ell}L_2(y_1)^{q^\ell}.
\end{align*}
As in the previous case, we know that $\alpha c_3x_0^{p^i}L_4(y_1)+\lambda\alpha^{q^\ell}c_1^{q^\ell}x_0^{p^iq^\ell}L_2(y_1)^{q^\ell}$ is a polynomial in $x_0y_1$. This is possible if and only if there exists constants $c_2,\,c_4$ such that $L_2(x)=c_2x^{p^i}$ and $L_4(x)=c_4x^{p^i}$ for the same $i$. The nondegeneracy of $N$ requires that $c_1c_4-c_2c_3\ne 0$. In this case, \eqref{eqn_iso} takes the form
\begin{equation}\label{eqn_x1y0}
L(u\zeta)=\alpha(c_1c_4+c_2c_3)u^{p^i}\zeta+\alpha c_3c_4u^{p^i}+\lambda \alpha^{q^\ell} c_1^{q^\ell}c_2^{q^\ell}u^{q^\ell p^i},\;\forall\,u\in\F_{q^m}.
\end{equation}
\item In the case $x_0=y_0=0$, \eqref{eqn_iso} gives that
\begin{equation}\label{eqn_x1y1}
L(u)=2\alpha c_2c_4u^{p^i}\zeta+\alpha c_4^2u^{p^i}
+\lambda\alpha^{q^\ell}c_2^{2q^\ell}u^{p^iq^\ell},\;\forall\,u\in\F_{q^m}.
\end{equation}
\end{enumerate}
We compute $L(g(u)\zeta)+L(f(u))$ using \eqref{eqn_x1y0} and \eqref{eqn_x1y1} and add it to
\eqref{eqn_x0y0} to cancel out the left hand side. The coordinate of $\zeta$ on the right hand side gives that
\begin{equation}\label{eqn_c}
2c_2c_4f(u)^{p^i}+(c_1c_4+c_2c_3)g(u)^{p^i}+2c_1c_3u^{p^i}=0.
\end{equation}
We claim that none of $c_1$, $c_2$, $c_3$, $c_4$ is zero. If $c_2c_4=0$, then $c_1c_4+c_2c_3\ne 0$ and \eqref{eqn_c} gives that $g(u)=c_5u$ for some constant $c_5$. The equation \eqref{eqn_ABC} now takes the form
\[
f(u)-wf(u)^{q^k}=-(\beta^2+\beta c_5) u+(w\zeta^{2q^k}+w\zeta^{q^k} c_5^{q^k})u^{q^k}.
\]
We obtain another equation by raising both sides to the $q^m$-th power, and then deduce that both $f(u)$ and $f(u)^{q^k}$ are linear combinations of $u$ and $u^{q^k}$. Since $\gcd(m,k)=1$, this is possible only if $f$ has degree $1$. However, $S(-g,-f)$ is then isotopic to a finite field: a contradiction. Hence $c_2c_4\ne 0$. If $c_1c_3=0$, then with the role of $z=f(u)$ and $u$ interchanged and $g$ considered as a function of $z$, we derive the same contradiction. This proves the claim.
From \eqref{eqn_c}, we see that $f(u)=d_1g(u)+d_2u$ for some constants $d_1,\,d_2\in\F_{q^m}$ with $d_2\ne 0$. Canceling $f$ from \eqref{eqn_ABC} by substitution, we get
\begin{equation}\label{eqn_gonly}
(\beta+d_1)g(u)-w(d_1+\zeta)^{q^k}g(u)^{q^k}=-(d_2+\beta^2)u+w(d_2+\zeta^{2})^{q^k}u^{q^k}.
\end{equation}
We claim that $\frac{w(d_1+\zeta)^{q^k}}{\beta+d_1}\in \F_{q^m}^*$. Otherwise, raising both sides of \eqref{eqn_gonly} to the $q^m$-th power, we get another equation that is linear independent with \eqref{eqn_gonly}. We then deduce that both $g(u)$ and $g(u)^{q^k}$ are linear combinations of $u,u^{q^k}$. This is possible only if $g(u)=c_6u$ for a constant $c_6$, but then $f(u)$ has degree $1$: a contradiction. This proves the claim.
After dividing both sides of \eqref{eqn_gonly} by $\beta+d_1$, the left hand side is in $\F_{q^m}[u]$, so should be the right hand side. This gives that both $\frac{d_2+\beta^2}{d_1+\beta}$ and $ \frac{d_2+\zeta^{2}}{d_1+\zeta} $ are in $\F_{q^m}$. Since $d_2+\zeta^{2}$ is in $\F_{q^m}$ but $d_1+\zeta$ is not, we must have $d_2=-\zeta^2$. We have $d_2+\beta^2\ne 0$, since otherwise $\beta^2=\zeta^2$ and $S_Q$ is isotopic to a finite field as we have shown. Now set $z:=g(u)$. The equation \eqref{eqn_gonly} gives that $u=h_1z+h_2z^{q^k}$ for some constants $h_1,h_2\in\F_{q^m}^*$. In particular, this shows that $u\mapsto z=g(u)$ is a bijection. It follows that $f(u)=h_3z-\zeta^2h_2z^{q^k}$ with $h_3=d_1-\zeta^2 h_1$, and we have
\begin{align*}
g(u)^2-4uf(u)&=z^2-(h_1z+h_2z^{q^k})(h_3z-\zeta^2h_2z^{q^k})\\
&=z^2(h_5+h_6z^{q^k-1}+\zeta^2 h_2^2z^{2(q^k-1)}),
\end{align*}
where $h_5=1-h_1h_3$, $h_6=\zeta^2h_1h_2-h_2h_3$.
Set $H(X):=h_5+h_6X^{q-1}+\zeta^2 h_2^2X^{2(q-1)}$. We have $H(X)=\left(\zeta h_2X^{q-1}+\frac{h_6}{2\zeta h_2}\right)^2+\frac{4\zeta^2h_2^2h_5-h_6^2}{4\zeta^2h_2^2}$.
We directly compute that $4\zeta^2 h_2^2h_5-h_6^2=(4\zeta^2-d_1^2)h_2^2\ne 0$, so $H(X)$ is not a square in $\overline{\F}_{q^m}[X]$ by the same argument in the proof of Lemma \ref{lemma_specialQ}, where $\overline{\F}_{q^m}$ is the algebraic closure of $\F_{q^m}$. The same exponential sum bound there establishes the existence of $z$ such that $h_5+h_6z^{q^k-1}+\zeta^2 h_2^2z^{2(q^k-1)}$ is a square. If we take $u$ such that $z=g(u)$, then $g(u)^2-4uf(u)$ is a nonsquare. This contradiction completes the proof.
\end{proof}
In the proof of Theorem \ref{thm_Q}, we do not directly consider the planarity of $Q$. Instead, we show that the semifield $S_Q$ is a RTCS and make use of the classification results of such semifields obtained in \cite{rtcs_char1,rtcs_char2}. This is in the same spirit as in Section \ref{sect_function}, where we make use of Menichetti's classification of generalized twisted fields. Coulter and Henderson have used this approach to characterize planar functions of certain form over $\F_{q^3}$, cf. \cite{ch_planar}. The list of known commutative semifields is short \cite{helle_sur,fs_cur,cs_zp}, and Minami and Nakagawa \cite{naka} have determined the polynomial forms of certain commutative semifields. It may be of some interest to examine the known commutative semifields one by one to check whether their isotopes will yield planar functions of the form $L(X)^2-wX^2$, but the answer is most probably negative. We need new techniques to prove or disprove the planarity of the functions of interest.
\section{Low-dimensional $\cC$-planes of even order}
In this section, we consider type $\cC$ spreads of even order. Let $q$ be even. Recall that $\beta$ is an element of order $(q^n+1)(q-1)$ in $\F_{q^{2n}}$, and $\Theta(\beta)\in\Gamma L(1,q^{2n})$ is defined by $\Theta(\beta)(x)=\beta x$. A type $\cC$ spread $\cS$ of order $q^n$ with kernel $\F_q$ is isomorphic to the orbit of $W=\{L(x)+\delta x:\,x\in\F_{q^n}\}$ under the group $\la\Theta(\beta)\ra$, where $L(X)$ is a monic reduced $q$-polynomial and $\delta\in\F_{q^{2n}}\setminus\F_{q^n}$.
By Lemma \ref{lemma_key}, $Q(X)=(L(X)+\delta X)(L(X)+\delta^{q^n} X)\in\F_{q^n}[X]$ is a permutation polynomial of $\F_{q^n}$. For each $y\in \F_{q^{n}}$, we define the quadratic form $Q_y(x):=\tr_{\F_{q^n}/\F_q}\left((\delta+\delta^{q^n})^{-1}yQ(x)\right)$, i.e.,
\begin{equation}\label{eqn_Qy}
Q_y(x):=\tr_{\F_{q^n}/\F_q}\left((\delta+\delta^{q^n})^{-1}yL(x)^2+ (\delta^{-1}+\delta^{-q^n})^{-1}y
x^2+yxL(x)\right).
\end{equation}
By Lemma \ref{lemma_perm_rank}, $Q_y$ has odd rank for any $y\ne 0$. Its associated bilinear form is
\[
B_y(u,v)=\tr_{\F_{q^n}/\F_q}(yuL(v)+yvL(u))=\tr_{\F_{q^n}/\F_q}\left((\tilde{L}(yu)+yL(u))v\right),
\]
where $\tilde{L}$ is the adjoint polynomial of $L$. The radical
$\rad(Q_y)=\{u\in \F_{q^n}:\,\tilde{L}(yu)+yL(u)=0\}$. By Theorem \ref{thm_N0}, $Q_y(\rad(Q_y))\ne \{0\}$ for each $y\ne 0$. We will fix these notation throughout this section. We first examine two special cases in the following examples.
\begin{example}\label{ex1}
Assume that $n$ is even, and $L(x)=x^{q^k}$ with $\gcd(n,k)=1$. In this case, the adjoint polynomial $\tilde{L}(X)=X^{q^{n-k}}$. Take $y$ to be a primitive element. Since $\gcd(q^{n-k}-q^k,q^n-1)=q^2-1$ and $\gcd(1-q^{n-k},q^n-1)=q-1$, we have
$\rad(Q_y)=\{u\in \F_{q^n}:\,u^{q^{n-k}-q^k}=y^{1-q^{n-k}}\}=\{0\}$.
Hence, $Q_y$ has rank $n$ and $Q(X)$ is not a permutation polynomial of $\F_{q^n}$.
\end{example}
\begin{example}\label{ex2}
Assume that $n$ is even, and $L(x)=\tr_{\F_{q^n}/\F_q}(x)$. In this case, $\tilde{L}(X)=L(X)$. Set $\Delta:=(\delta^{-1}+\delta^{-q^n})^{-1}$. Take $y\in\F_{q^2}$ such that $y+y^q=\Delta$ if $\Delta\in\F_q^*$, and take $y=\Delta^{-1}$ otherwise. Then $y\not\in\F_q$ and $\rad(Q_y)=\{u:\,\tr_{\F_{q^n}/\F_q}(yu)=\tr_{\F_{q^n}/\F_q}(u)=0\}$.
For $u\in\rad(Q_y)$, we have
$Q_y(u)=\tr_{\F_{q^n}/\F_q}\left(\Delta y\cdot u^2\right)$. It is clear that $ \Delta y \in \F_q+\F_q\cdot y^2$ for the chosen $y$, so $Q_y$ is constantly zero on $\rad(Q_y)$. It follows that $Q(X)$ is not a permutation polynomial of $\F_{q^n}$.
\end{example}
The main objective of this section is to characterize the case $n=3$ and $n=4$ completely. The case $n=2$ can be reduced to either of the above examples with a proper choice of $\delta$. In the case $n=3$, there is the construction by Kantor in \cite{kantor_even}. We start with the case $n=4$. The strategy is to show that there is a trivial $\rad(Q_y)$ except the cases where it can be reduced to the second example above.
\begin{thm}\label{thm_even_4dim}
If $q$ is even, then there is no type $\cC$ spread with ambient space $(\F_{q^8},+)$ and kernel $\F_q$.
\end{thm}
\begin{proof} We continue with the arguments in the beginning of this section.
We only deal with the case $\deg(L)=q^3$ and the other cases can be handled similarly. If $L(X)=X^{q^3}+ax^{q^2}+bx^q+cx$, then by replacing $\delta$ with $\delta+c$ we assume that $c=0$. In this case, $\tilde{L}(X)=b^{q^3}X^{q^3}+a^{q^2}X^{q^2}+X^q$ and $
\tilde{L}(yu)+L(u)y=(b^{q^3}y^{q^3}+y)u^{q^3}+(a^{q^2}y^{q^2}+ay)u^{q^2}+(y^q+by)u^q$.
The associated matrix of this $q$-polynomial as in \eqref{eqn_mat} has determinant $f(y)^2$, where
\[
f(y)=y^{1+q}c_1+y^{1+q^2}c_2+y^{q+q^2}c_1^q+y^{q^3+1}c_1^{q^3}+y^{q+q^3}c_2^q+y^{q^2+q^3}c_1^{q^2},
\]
with $c_1=b^q+a^{1+q}$, $c_2=1+b^{1+q^2}$.
\begin{enumerate}
\item If at least one of $c_1$, $c_2$ is not zero, then $f(Y)$ is a nonzero polynomial with degree less than $q^4$. For $y\in \F_{q^4}$ such that $f(y)\ne 0$, $\rad(Q_y)=\{0\}$ and $Q_y$ has rank $4$.
\item If $c_1=c_2=0$, then $b^{1+q^2}=1$ and $b^q=a^{1+q}$. It follows that $a^{(1+q)(1+q^2)}=1$, i.e., $a$ is a $(q-1)$-st power in $\F_{q^4}^*$. Since $\gcd(q^3-1,q^4-1)=q-1$, there exists $u\in \F_{q^4}^*$ such that $a=u^{q^3-1}$. Then $b=a^{(1+q)q^2}=u^{q^2-1}$, and so $L(x)=u^{-1}\tr_{\F_{q^4}/\F_q}(u^qx)+u^{q-1}x$. With $x'=u^qx$ and $\delta'=1+\delta u^{1-q}$, we have
\[
L(x)+\delta x=u^{-1}(\tr_{\F_{q^4}/\F_q}(x')+\delta' x')
\]
This reduces to the case in Example \ref{ex2}.
\end{enumerate}
In either case, $Q(X)$ is not a permutation polynomial of $\F_{q^4}$. This completes the proof.
\end{proof}
The rest of this section is devoted to the classification of the case $n=3$.
\begin{lemma}\label{lemma_even_3dim}
For $\delta\in\F_{q^6}\setminus \F_{q^3}$,
the map $x\mapsto Q(x)=(\tr_{\F_{q^3}/\F_q}(x)+\delta x)^{1+q^3}$ is a permutation of $\F_{q^3}$ if and only if $\delta^{-1}+\delta^{-q^3}\in\F_q^*$.
\end{lemma}
\begin{proof}
Set $r:=1+\delta^{-1}$, $h:=(\delta^{-1}+\delta^{-q^3})^{-1}$, and set $\tr:= \tr_{\F_{q^3}/\F_q}$ throughout this proof.
By Lemma \ref{lemma_perm_rank} and Theorem \ref{thm_N0}, $Q(X)$ is a permutation polynomial of $\F_{q^3}$ if and only if $Q_y$ is not constantly zero on $\rad(Q_y)$ for each $y\ne 0$, where $Q_y$ is as defined in \eqref{eqn_Qy}. In this case,
\[
Q_y(x)=\tr((\delta+\delta^{q^3})^{-1}y)\cdot\tr(x^2)
+\tr(hyx^2)+\tr(yx)\cdot\tr(x).
\]
If $y\in \F_q^*$, then $\rad(Q_y)=\F_{q^3}$ and $Q_y(x)=y\tr(cx^2)$, with $c=h+1+\tr((\delta+\delta^{q^3})^{-1})$. If $y\not\in \F_q$, then $\rad(Q_y)=\{u:\,\tr(u)=\tr(yu)=0\}=\F_q\cdot (y^q+y^{q^2})$, and
\[
Q_y(y^q+y^{q^2})=y^{1+q+q^2}\cdot\tr\left(h(y^{q-q^2}+y^{q^2-q})\right).
\]
If $h\not\in\F_q^*$, then $ Q_h(h^q+h^{q^2})=0$ and $Q_h(\rad(Q_h))=\{0\}$. Hence we must have $h\in \F_q^*$ in order for $Q$ to be a permutation polynomial, and this proves the necessity part.
Now assume that $h\in \F_q^*$. In this case, it is straightforward to show that $c=\tr(r^{1+q^3})h$ and the minimal polynomial of $rh$ over $\F_{q^3}$ is $X^2+X+r^{1+q^3}h^2=0$. By \cite[Theorem 2.25]{ff}, $\tr_{\F_{q^3}/\F_2}(h^2r^{1+q^3})=1$, which in particular implies that $\tr(h^2r^{1+q^3})=hc\ne 0$. Hence $Q_y(x)=y\tr(cx^2)$ is not constantly zero on $\rad(Q_y)=\F_{q^3}$ if $y\in \F_q^*$.
For each $y\not\in \F_q$, the $\F_q$-linear subspace $\{x\in \F_{q^3}: \,\tr((y^{q-q^2}+y^{q^2-q})x)=0\}$ is spanned by $y$ and $y^{-1}$. It can not contain $\F_q$, since otherwise $y$ would lie in a degree two extension of $\F_q$. It follows that $\tr(y^{q-q^2}+y^{q^2-q})\ne 0$, so $Q_y(\rad(Q_y))\ne\{0\}$ for all $y\not\in \F_q$. This proves the sufficiency part.
\end{proof}
\begin{remark}\label{remark_3dim} Take $\delta\in\F_{q^6}\setminus\F_{q^3}$ such that $\delta^{-1}+\delta^{-q^3}\in\F_q^*$, and
take the decomposition $\F_{q^3}=T_0\oplus \F_q$ as in \cite{kwnew}, where $T_0=\{x\in\F_{q^3}:\tr_{\F_{q^3}/\F_q}(x)=0\}$. Let $W$ be the image of $\F_{q^3}$ under the map $x\mapsto \tr_{\F_{q^3}/\F_q}(x)+\delta x$. Then $W=T_0\cdot\delta\oplus\F_q\cdot(1+\delta)$, and $\{\beta^i\cdot W:\,0\le i\le q^3\}$ forms a spread $\cS$ of type $\cC$, where $\beta$ is an element of order $(q^3+1)(q-1)$. Moreover, the spread $\cS$ is symplectic with respect to the nondegenerate alternating form
$A(x,y)=\tr_{q^6/q}\big((\delta+\delta^{q^3})^{-1}xy^{q^3}\big)$.
\end{remark}
\begin{lemma}\label{thm_even_3dim}
Let $L(X)\in\F_{q^3}[X]$ be a monic reduced $q$-polynomial and $\delta\in\F_{q^6}\setminus\F_{q^3}$. The map $x\mapsto Q(x)=(L(x)+\delta x)^{1+q^3}$ is a permutation of $\F_{q^3}$ if and only if $L(x)+\delta x=u^{-1}\tr_{q^3/\F_q}(u^qx)+\delta' x$ for some $\delta'\not\in\F_{q^3}$ and $u\in\F_{q^3}^*$ such that $\delta'^{-1}+\delta'^{-q^3}\in u^{1-q}\cdot\F_q^*$.
\end{lemma}
\begin{proof} If $L(x)+\delta x=u^{-1}\tr_{q^3/\F_q}(u^qx)+\delta' x$, then with $y=u^qx$ we have $L(x)+\delta x=u^{-1}\left(\tr_{q^3/\F_q}(y)+\delta'u^{1-q} y\right)$, and the sufficient part of the theorem follows from Lemma \ref{lemma_even_3dim}. Therefore, we only need to prove the necessary part.
Assume that $x\mapsto Q(x)$ is a a permutation, and we need to prove that $L(x)+\delta x$ is of the desired form. If $L(x)=x^{q^2}+ax^q+bx$, then by replacing $\delta$ with $\delta+b$ we assume that $b=0$. If $a$ is a nonzero $(q-1)$-st power, then $a=u^{q^2-1}$ for some $u\in\F_{q^3}^*$ and $L(x)+\delta x=u^{-1}\tr_{\F_{q^3}/\F_q}(u^qx)+(u^{q-1}+\delta)x$. Similar to the first paragraph of this proof, we can deduce that $L(x)+\delta x$ is of the desired form by Lemma \ref{lemma_even_3dim}. There are two remaining cases.
\begin{enumerate}
\item In the case $a$ is not a nonzero $(q-1)$-st power, $y\mapsto u=y^{q^2}+a^qy^q$ is a permutation of $\F_{q^3}$ and $L(u)+u\delta=(1+a^q\delta)^{1+q^3}(y^{q^2}+\delta'y^q)^{1+q^3}$, where $\delta'=\frac{\delta+a^{1+q^2}}{\delta a^q+1}$. Therefore, $x\mapsto Q(x)$ is a permutation if and only if the map $x\mapsto (x^{q}+\delta'x)^{1+q^3}$ is.
\item In the case $a=0$, $x^{q^2}+\delta x=\delta\cdot(y^q+\delta^{-1}y)$ for $y=x^{q^2}$. Therefore, $x\mapsto Q(x)$ is a permutation if and only if the map $x\mapsto (x^{q}+\delta^{-1}x)^{1+q^3}$ is.
\end{enumerate}
We now show that $x\mapsto Q(x)=(x^q+\delta x)^{1+q^3}$ is not a permutation of $\F_{q^6}$ for any $\delta\not\in\F_{q^3}$, which will exclude these two cases and conclude the proof. In this case, we have $Q(X)=(X^q+\delta X)(X^q+\delta^{q^3} X)$. By Hermite's criterion for permutation polynomials (cf. \cite[Theorem 7.4]{ff}), $Q(X)^{q^2-1}\pmod{X^{q^3}-X}$ has degree at most $q^3-2$. The polynomial $Q(X)^{q^2-1}$ has degree at most $2q(q^2-1)<2(q^3-1)$, so its coefficient of $X^{q^3-1}$ should be zero. Since $q^2-1=(q-1)q+(q-1)$, we can rewrite $Q(X)^{q^2-1}$ as
\[
X^{2q^2-2}\cdot
\left( X^{2(q^2-q)}+s^qX^{q^2-q}+t^q\right)^{q-1}\cdot
\left( X^{2(q-1)}+sX^{q-1}+t\right)^{q-1},
\]
where $s=\delta+\delta^{q^3}$ and $t=\delta^{1+q^3}$.
The second term in the product contributes monomials that are powers of $X^{q(q-1)}$ and the third term contributes monomials that are powers of $X^{q-1}$. If $q^3-1=2(q^2-1)+q(q-1)i+(q-1)j$ with $0\le i,j\le 2(q-1)$, then we necessarily have $j=q-1$ and $i=q-2$. Thus the coefficient of $X^{q^3-1}$ in $Q(X)^{q^2-1}$ is the product of the coefficient of $X^{q-1}$ in $(X^{2 }+sX +t)^{q-1}$ and that of $X^{ q-2 }$ in $(X^{2 }+s^qX +t^q)^{q-1}$, i.e.,
\begin{equation}\label{eqn_coef}
\sum_{i=0}^{q/2}\binom{q-1}{i,q-1-2i,i}s^{q-1-2i}t^i\cdot \sum_{j=0}^{q/2-1}\binom{q-1}{j,q-2-2j,j+1}s^{q(q-2-2j)}t^{q(j+1)}.
\end{equation}
Here, the numbers $\binom{q-1}{i,j,k}$'s are trinomial coefficients. A straightforward analysis using Lucas' theorem shows that: $\binom{q-1}{i}$ is odd for all $0\le i\le q-2$; $\binom{2i}{i}$ is odd if and only if $i=0$; $\binom{2j+1}{j}$ is odd if and only if $j=2^\ell-1$ for some nonnegative integer $\ell$. Since $\binom{q-1}{i,q-1-2i,i}=\binom{2i}{i}\binom{q-1}{2i}$ and $\binom{q-1}{j,q-2-2j,j+1}=\binom{2j+1}{j}\binom{q-1}{2j+1}$, the quantity in \eqref{eqn_coef} is equal to
\[
s^{q-1}\cdot\sum_{\ell=0}^{e-1}s^{q(q-2^{\ell+1})}t^{2^\ell q}
=s^{q^2+q-1}\cdot\sum_{\ell=0}^{e-1} (ts^{-2})^{2^\ell q},
\]
where $e$ is such that $q=2^e$. Recall that this quantity is zero. Since $s\ne 0$, we have $\sum_{\ell=0}^{e-1}(ts^{-2})^{2^\ell }=0$ and thus $\tr_{\F_{q^3}/\F_2} (ts^{-2})=0$. By \cite[Theorem 2.25]{ff}, there exists $u\in\F_{q^3}$ such that $ts^{-2}=u^2+u$. Set $z=\delta^{q^3-1}$. We have $z\not\in\F_{q^3}$ and $ts^{-2}=(z+z^{-1})^{-1}$ by direct check. The minimal polynomial of $z$ over $\F_{q^3}$ is $(X-z)(X-z^{-1})=X^2+(u^2+u)^{-1}X+1$, but the latter has $\frac{u}{u+1}\in\F_{q^3}$ as a root: a contradiction. This completes the proof.
\end{proof}
As an immediate corollary, we get the following characterization result.
\begin{thm}
Let $q$ be even. The type $\cC$ spreads with ambient space $(\F_{q^6},+)$ and kernel $\F_q$ are isomorphic to those described in Remark \ref{remark_3dim}.
\end{thm}
We end this section with some remarks on the higher dimensional case. Theorem \ref{thm_even_4dim} supports the conjecture that there is no $\cC$-plane of even order and even dimension. The simple nature of the proof in the case $n=4$ suggests that the method may be applicable to larger values of even $n$, and of course new ingredients are needed to prove the conjecture. In the case where $n$ is odd, there are the constructions in \cite{kwnew}, making a characterization in this case a challenging problem. It may be more practical to classify the $\cC$-planes of order $2^p$ with $p$ an odd prime, where no such non-Desargeusian planes are known. The approach in this section provides the first step towards a complete characterization of $\cC$-planes of even order.\\
\noindent{\bf Acknowledgement.} This research was supported by the National Natural Science Foundation of China under Grant 11422112 and Fundamental Research Fund for the Central Universities of China. The author thanks the referees for detailed comments and suggestions that helped to improve the presentation of the paper. The author is indebted to Professor William M. Kantor for numerous helpful comments and suggestions during this project, and he also thanks Professor Gary L. Ebert who brought the problem to his attention in 2011.
|
1,116,691,500,442 | arxiv | \section{Introduction}
In this paper we study existence and uniqueness of rational normal
curves in $\mathbb{P}^n$ passing through a given set of points and
intersecting some codimension two linear spaces in a very natural
way. More precisely, we require the curve to intersect each linear
space in $n-1$ distinct points. In this case, we say that the curve
and the linear space are mutually $(n-1)$-secant. We work over the
field of complex number $\mathbb{C}$ and we consider a {\it rational
normal curve} (briefly a {\it rnc}) as a linearly normal embedding of
$\mathbb{P}^1$.
Our interest in the subject arises from the deeply
intertwined problems of the postulation of schemes and of the dimension
of higher secant varieties, e.g. see the original work of
Terracini \cite{Terracini} and also \cite {Ge}. We give a first
application of our results in this direction in Section
\ref{postulationAPP} where we easily obtain a well known result
by {\AA}dlandsvik \cite{AAdlandsvik} about Segre-Veronese varieties,
recently re-proposed by
Abrescia in \cite{Abrescia}.
The idea of using rational curves in the study of linear systems
and higher secant varieties is classical. Its importance has been
stressed again in the case of double points schemes and higher
secant varieties to Veronese varieties. To explain this, let
$X\subset\mathbb{P}^n$ be a double point scheme supported on $p$ generic
points $P_i$'s, i.e. a scheme with defining ideal
\[
I_X=\left(I_{P_1}\right)^2\cap\ldots\cap \left(I_{P_p}\right)^2
\]
where the $I_{P_i}$'s are the ideals of the $P_i$'s. Then, one
wants to determine the Hilbert function of $X$ in some degree $d$,
say $H(X,d)$. There is an expected value for the Hilbert function
determined by a naive count of conditions, which we will call $h(n,p,d)$. This value is such
that $H(X,d)\leq h(n,p,d)$ and if the points are generic one {\it
expects} equality to hold. In a series of papers, Alexander and
Hirschowitz determined exactly when equality holds, see
\cite{AH95} and \cite{Chandler}. More precisely, $H(X,d)= h(n,p,d)$ in
all but the following cases:
\begin{itemize}
\item $d=2$;
\item $d=4, (n,p)=(2,5),(3,9),(4,14)$;
\item $d=3, (n,p)=(4,7)$.
\end{itemize}
The $d=4$ cases are easily explained by the existence of quadric
hypersurfaces passing through $5,9$ and $14$ points in
$\mathbb{P}^2,\mathbb{P}^3$ and $\mathbb{P}^4$,
respectively.
The $d=3$ case requires a subtler explanation involving rational
normal curves. Given a scheme $X\subset\mathbb{P}^4$ consisting of seven
double points, we do not expect a cubic threefold singular at all
the points to exist, i.e. $h(4,3,7)={4+3\choose 3}=35$. But there
is a rnc $\mathcal{C}$ passing through the points (see Theorem
\ref{castelnuovo}) and the variety of secant lines to
$\mathcal{C}$ is a cubic threefold singular along the curve. Thus,
$H(X,3)<35$ and the variety of secant $\mathbb{P}^6$'s to the 3-uple
embedding of $\mathbb{P}^3$ does not have the expected dimension. For a
more detailed account see \cite{Ge}, the introduction of
\cite{RS00} and \cite{Ci01}.
Our research interest was inspired by the following classical
result
\vskip .25cm
\noindent{\bf Theorem \ref{castelnuovo}.}{\it Given
$n+3$ points in $\mathbb{P}^n$ in generic position, there exists a unique rational
normal curve passing through them.}
\vskip .25cm
This theorem was well known in the late $19^{th}$ century, e.g. it
can be found in works by Bordiga \cite{Bordiga} and Castelnuovo
\cite{Castelnuovo} where it is attributed to Veronese
\cite{Veronese}. We want to mention that even then there were
attempts to generalize this result, but always in a constructive
way. In the sense that the final goal of those attempts was the
synthetic construction of a curve satisfying certain properties.
Recently Graber and Ranestad in \cite{GraberRanestad}, following
Kapranov \cite{Kapranov}, generalized Theorem \ref{castelnuovo}
to d-uple Veronese surfaces and applied their results to the study of line arrangements.
In Section \ref{main} we provide a classical proof of Theorem
\ref{castelnuovo}. We note that it is natural to expect that $n+3$
points determine a finite number of rnc's. In fact, the parameter
space of rnc's in $\mathbb{P}^n$ has dimension $(n+3)(n-1)$. Moreover, a
simple argument on the defining matrix of a rnc shows that the
family of rnc's passing through a given point has dimension
$(n+2)(n-1)$. Thus a point imposes $n-1$ conditions to rnc's.
Theorem \ref{castelnuovo} is nothing more than the proof that
$n+3$ points impose independent conditions.
We can push this kind of argument even further. Take a codimension
two linear space $\Lambda$ in $\mathbb{P}^n$ and consider all rnc's
intersecting it: this is one condition for rnc's. Thus, if we
consider rnc's intersecting $\Lambda$ in $n-1$ points this
incidence condition imposes again $n-1$ conditions. In conclusion,
passing through a fixed point or intersecting a given codimension
two linear space in $n-1$ not fixed points imposes the same number
of conditions to rnc's. Hence, it is natural to look for
generalizations of Theorem \ref{castelnuovo} involving points and
codimension two linear spaces.
As a first step in this direction, one tries to generalize Theorem
\ref{castelnuovo} in $\mathbb{P}^3$. The case of twisted cubic curves was
studied in details in the early $20^{th}$ century and even in this simple case it
is clear that the theorem does not generalize in a straightforward way.
In fact, for a generic choice of four points and two lines,
via the count of conditions above, we expect to find at least
one rnc passing through the points and having the lines as chords. But
this is not the case and such a curve does not exist (see Section
\ref{twisted} for more details).
The classic approaches to the $\mathbb{P}^3$ case use extremely {ad hoc}
arguments which do not generalize easily to higher dimension. We
develop a more general framework where the twisted cubic case and
the general situation can both be studied. As a result of our
analysis we obtain the following
\medskip
\noindent {\bf Theorem \ref{final}.} {\it Let $n,p$ and $l$ be positive
integers such that
$$n\geq
3, \ \ p\geq 1 \mbox{ and} \ \ p+l=n+3 .$$
Choose $p$ points in $\mathbb{P}^n$ and $l$
codimension two linear spaces in generic position. Then, only for
the values
\[
(p,l)=(n+3,0),(n+2,1),(3,n),(2,n+1),(1,n+2)
\]
does there
exist a unique rational normal curve passing through the points and
$(n-1)$-secant to the linear spaces. In the other cases, that is for
$p\geq 4$ and $l\geq 2$, no such curve exists.
}
\medskip
The $(p,l)=(n+3,0)$ case is just
Theorem \ref{castelnuovo}, while the $(p,l)=(n+2,1)$ case
and the non-existence results
are, as far as we know, original (see Propositions
\ref{onePOINTlessPROP} and \ref{donotexist}). The result for
$(p,l)=(3,n)$ is just Steiner's construction, see e.g. \cite[pg.
528]{GH}, but we provide a different proof not using the classical
construction. The cases $(p,l)=(2,n+1),(1,n+2)$ were studied by
Todd in \cite{Todd} and by Veneroni in \cite{Veneroni}. Todd
provides a proof of both the results for $n=4$ which we briefly
sketch in the proofs of Propositions \ref{3points} and
\ref{2points}. In \cite{Todd}, the author also claims that the
results extend for any $n$, but no proof of these facts is given.
We give a complete and independent proof of both these claims in
Propositions \ref{3points} and \ref{2points}. Notice that Theorem
\ref{final} deliberately omits the case $(p,l)=(0,n+3)$. As far as
we know, the only known answer is for $n=3$ and it is given by
Wakeford in \cite{Wakeford} (see Section \ref{twisted}).
The non-existence result deserves a special comment. For $p\geq 4$ and
$l\geq 2$ Proposition \ref{donotexist} states that in $\mathbb{P}^n$ no rnc
exists passing through $p$ generic points and $(n-1)$-secant to
$l$ generic codimension two linear spaces. Thus, for $p+l=n+3, p\geq 4$
and $l \geq 2$ the count of conditions always fails. The proof of this
fact is very simple: one sees that a curve with the required
properties {\it must} be reducible and hence it can not be a rnc.
At this point, one can think to allow degenerations of rnc's and
gain existence also in these cases. But this does not happen. Even
in $\mathbb{P}^3$, allowing degenerations is not enough. In fact, for
$(p,l)=(4,2)$ the degree 3 curves passing through the
points and having the lines as chords split as the union of a
{\it not} intersecting conic and a line. Thus, a degree $3$ curve
of arithmetic genus $-1$. In conclusion, the non-existence is a
fact deeply related to the nature of the problem, not only coming
from a restrictive choice of curves.
We already mentioned an application of our results to the study of
higher secant varieties of some Segre-Veronese varieties. More
precisely, we consider a scheme $Z\subset\mathbb{P}^n$ of $n+2$ double
points union a codimension two linear space. Using rnc's we can
easily show that $H(Z,4)$ is at least one less than expected (see
Lemma \ref{applemma}). We think that this method can be
successfully applied for studying other families of schemes
supported on points and linear spaces and we plan to investigate
this in the future. Another application involves the study of
projective equivalence of some special family of subsets and it is
described in Section \ref{projeq}. In particular, we give a
criterium to establish whether two ordered subset of $\mathbb{P}^n$ each
consisting of $p$ points and $n+3-p$ codimension two linear spaces
are projectively equivalent.
For the convenience of the reader, we give an outline of the
paper. In Section \ref{definition} we give the necessary
definition and we recall some basic facts about rnc's that we will
extensively use. Also, we prove Lemma \ref{generalLEMMA} which is
the technical core of the paper. In Section \ref{twisted} we give
a historical account of the classic results for twisted cubic
curves. Section \ref{main} contains the main results of the paper.
Finally, in Sections \ref{appsection} and \ref{remsection} we
give some applications of our results and we make some final
remarks on the problem.
The authors wish to thank C. Ciliberto, M. Mella and F. Russo for
the interesting discussions on the topic. In particular, F.
Russo's support was crucial in starting and developing this work.
The first author had the occasion of meeting all this people at
the Workshop on Cremona Transformation held at the Politecnico di
Torino in September 2005, organized by G. Casnati, R.
Notari, and M.L. Spreafico. A special thank to the organizers for providing such a
productive scientific occasion.
\section{Notation and basic facts}\label{definition}
A {\it rational normal curve} (a {\it rnc} for short) in $\mathbb{P}^n$ is an
irreducible, reduced, smooth, rational, linearly normal curve.
A linear space $\Lambda\subset\mathbb{P}^n$ is said to be {\it $(n-1)$-secant}
to a rnc $\mathcal{C}$ if $\Lambda\cap \mathcal{C}$ is a set of $n-1$
distinct points, and we will also say that the
curve $\mathcal{C}$ is {\it $(n-1)$-secant} to $\Lambda$.
Let $\mbox{Hilb}^{nt+1}(\mathbb{P}^n)$ be the Hilbert scheme
parameterizing subschemes of $\mathbb{P}^n$ having Hilbert polynomial
$nt+1$. Rational normal curves correspond to the points of a
smooth, irreducible, open subscheme of $\mbox{Hilb}^{nt+1}(\mathbb{P}^n)$
which we denote by $\mathcal{H}$. For more on this see
\cite{MinniRagni} and the references there. We recall that
$\dim\mathcal{H}=(n-1)(n+3)$.
In this paper, we often invoke Bezout type arguments. In
particular, we are interested in showing that a given hypersurface
$X$ contains a rnc $\mathcal{C}$. This is in turn equivalent to
show that a two variable polynomial $F$ of degree $(\deg
X)\cdot(\deg\mathcal{C})$ has too many roots.
We recall that if
$P\in \mathcal{C}$ is a multiple point for $X$, then
$F$ has a multiple root of at least the same multiplicity.
Moreover, if $X$
contains a hyperosculating space (e.g. the tangent space, the
ordinary osculating space, etc. ) to $\mathcal{C}$ in $P$, then
$F$ has again a multiple root of the proper multiplicity.
Given a variety $X$ and a natural number $n$, $(X)^n$ will denote
the product $\underbrace{X\times\ldots\times
X}_{n-\mathrm{times}}$. If $X$ is embedded in $\mathbb{P}^n$ we follow
Harris in \cite[pg. 90]{Harris} and we consider the {\it
$k$-secant map}
\[
(X)^{k+1}\dashrightarrow G(k,\mathbb{P}^n)
\]
mapping $k+1$ generic points of $X$ to the point of the Grassmannian $G(k,\mathbb{P}^n)$
corresponding to the linear space that they
span. In particular, we denote by $\mathbb{S}^{k}X$ the closure of
the image of this map and we call it the {\it abstract variety of
secant $k$-spaces to $X$}.
We recall that a generic $2\times n$ matrix $\mathsf{M}$ of linear forms on
$\mathbb{P}^n$ defines a rnc via its maximal minors.
A generalized row of $\mathsf{M}$ is any row of a matrix conjugate
to $\mathsf{M}$; similarly for a generalized column. Notice that
the zero locus of a generalized row of $\mathsf{M}$ is a point of
the related rnc $\mathcal{C}$ and viceversa. Also, the zero locus
of a generalized column intersect $\mathcal{C}$ in a zero
dimensional scheme of degree $n-1$ and also the converse holds
as shown in the following lemma (for an alternative proof see \cite[pg.
102]{Harris}).
\begin{lem}\label{codim2secant}
Let $\mathsf{M}$ be a $2\times n$ generic matrix of linear forms
and let $\mathcal{C}\subset\mathbb{P}^n$ be the rnc defined as the rank
one locus of $\mathsf{M}$. Then, $\Lambda$ is a codimension
two linear space intersecting $\mathcal{C}$ in a degree $n-1$
scheme if and only if $\Lambda= \{F=G=0\}$ and ${F\choose G}$ is a generalized column of
$\mathsf{M}$.
\end{lem}
\begin{proof}
Let
\[
\mathsf{M}=
\left(
\begin{array}{cccc}
F_1 & F_2 & \ldots & F_n\\
G_1 & G_2 & \ldots & G_n
\end{array}
\right).
\]
If ${F\choose G}$ is a generalized column of $\mathsf{M}$, we can
substitute $\mathsf{M}$ with a conjugate matrix of the form
\[
\left(
\begin{array}{cccc}
F & F_2 & \ldots & F_n\\
G & G_2 & \ldots & G_n
\end{array}
\right).
\]
Delete the first column and consider the rank one locus of the
resulting matrix, call it $X$. Then, $X$ is a degree $n-1$ surface
and $X\cap \Lambda$=$\mathcal{C}\cap \Lambda$. Hence $\{F=G=0\}\cap
\mathcal{C}$ is a zero dimensional scheme of degree $n-1$.
Conversely, assume that $\Lambda$ is a codimension two linear
space intersecting $\mathcal{C}$ in a degree $n-1$ scheme. For
simplicity assume that $\Lambda\cap \mathcal{C}$ is a smooth set of
points, say $\{P_1,\ldots,P_{n-1}\}$. Evaluate $\mathsf{M}$ in
$P_i$ and let $V_i\subset\mathbb{C}^n$ be the space of solution of
the linear system $\mathsf{M}_{|P_i}\underline{\lambda}=0$.
Clearly, $\bigcap_{i=1}^{n-1} V_i\neq 0$ and let
$(\lambda_1,\ldots,\lambda_n)$ be a common solution. Then the generalized column
\[
{F\choose G}=\sum_1^n \lambda_i{F_i\choose G_i}
\]
is such that $\Lambda=\{F=G=0\}$.
\end{proof}
In order to prove our results, we fix some notation and we derive
a crucial technical fact playing a key role in this paper. Given
natural numbers $l$ and $p$, we consider the {\it data} space
\[
\mathcal{D}=G(n-2,\mathbb{P}^n)^l\times(\mathbb{P}^n)^p
\]
parameterizing sets consisting of $l$ codimension 2 linear spaces
and $p$ points in $\mathbb{P}^n$. Notice that
$\dim\mathcal{D}=np+2l(n-1)$. We call an element of $\mathcal{D}$
a {\it datum}. Given a rnc $\mathcal{C}$ and a datum
$\delta=(\Lambda_1,\ldots,\Lambda_l,P_1,\ldots,P_p) \in \mathcal{D}$, we say that
$\mathcal{C}$ {\it satisfies} $\delta$ if
\[P_i\in\mathcal{C} \mbox{ for } 1\leq i\leq p\]
and
\[\Lambda_i\cap \mathcal{C} \mbox{ has degree } n-1 \mbox{ for }
1\leq i\leq l.\]
Then we consider the incidence correspondence
$\Sigma\subset\mathcal{H}\times\mathcal{D}$ defined as
\[
\Sigma=\lbrace (\mathcal{C},\delta): \mathcal{C}\mbox{ satisfies
}\delta\rbrace.
\]
With this notation, we can rephrase our problem about the
existence of rnc's: given a generic datum are there rnc's
satisfying it? If we let $\phi:\Sigma\rightarrow\mathcal{D}$ be
the natural projection, this question reduces to the following: is
$\phi$ dominant?
Finally we can introduce the main technical tool of the paper.
\begin{lem}\label{generalLEMMA}
If $p+l=n+3$ and there exists a datum $\delta\in\mathcal{D}$ such
that $\phi^{-1}(\delta)$ is a finite number of points, then $\phi$ is
dominant. Moreover, if the datum
$\delta=(\Lambda_1,\ldots,\Lambda_l,P_1,\ldots,P_p)$ is such that $\Lambda_i$ is
$(n-1)$-secant to the curve $\phi^{-1}(\delta)|_{\mathcal{H}}$ for
$1\leq i\leq l$, then the same holds for the generic element of
$\mathcal{D}$.
\end{lem}
\begin{proof}
Notation as above. First we will show that $\Sigma$ is
irreducible. Let $\psi:\Sigma\rightarrow\mathcal{H}$ be the
projection map and consider $\mathcal{C}\in\mathcal{H}$. Notice
that
\[
\psi^{-1}(\mathcal{C})_{|\mathbb{P} ^n}\simeq \mathcal{C},
\]
\[
\psi^{-1}(\mathcal{C})_{|G(n-2,\mathbb{P}^n)}\simeq
\mathbb{S}^{n-2}\mathcal{C},
\]
where $\mathbb{S}^{n-2}\mathcal{C}\simeq
\mathcal{C}\times\ldots\times \mathcal{C}=(\mathcal{C})^{n-1}$.
Hence $\psi$ has irreducible fibers all having dimension
$p+(n-1)l$. Thus $\Sigma$ is irreducible and
$\dim\Sigma=p+(n-1)(l+n+3)$. Notice that, as $p+l=n+3$,
$\dim\mathcal{D}=np+2l(n-1)=\dim\Sigma$. Then $\phi$ is readily
seen to be dominant as $\mbox{Im}\phi$ is irreducible and such
that $\dim\Sigma-\dim\mbox{Im}\phi\leq\dim\phi^{-1}(\delta)=0$.
Then, let
\[
\Sigma^\circ=\lbrace(\mathcal{C},(\Lambda_1,\ldots,\Lambda_l,P_1,\ldots,P_p))\in\Sigma
: \mathcal{C}\cap \Lambda_i\mbox{ is not smooth for some }i\rbrace
\]
and notice that $\Sigma^\circ$ is closed and proper in $\Sigma$.
Hence $\dim\Sigma^\circ<\dim\Sigma$ and the second assertion
follows as the fiber of $\phi$ over the generic datum can not be
contained in $\Sigma^\circ$.
\end{proof}
\section{The $\mathbb{P}^3$ case}\label{twisted}
In this section, we briefly illustrate our problem in the well
known case of rnc's in $\mathbb{P}^3$. Twisted cubics have been
thoroughly investigated classically and we recall some of the many
interesting results, which are usually obtained via ad hoc
techniques.
The parameter space of twisted cubic curves has dimension $12$ and
fixing one point in $\mathbb{P}^3$ imposes two conditions. In particular,
from a numerical point of view, the condition of passing through a
fixed point is equivalent to the one of touching a fixed line in
two (not fixed) points. In conclusion, given $p$ points $P_1, \ldots, P_p$ and $l$
lines $\Lambda_1, \ldots, \Lambda_l$ in generic position in $\mathbb{P}^3$, such that $p+l=6$, we expect
to find a finite
number of twisted cubic curves passing through the points and
2-secant to the lines.
For $p=6,l=0$, an answer can be obtained by considering quadric cones.
Namely, let $\mathcal{Q}$ and $\mathcal{Q}'$ be two
quadrics containing the points $P_1,\ldots ,P_6$ and with a double
point in $P_1$ and $P_2$, respectively.
The complete
intersection $\mathcal{Q} \cap \mathcal{Q}'$ is the union of the
line $P_1P_2$ and of a twisted cubic through $P_1,\ldots ,P_6$.
Hence the rnc exists and, by Bezout, it is easy to show that it is unique.
For $p=5,l=1$, we consider again quadrics.
Let $\mathcal{Q}$ and $\mathcal{Q}'$ be two smooth
quadrics containing the points and the line. Then the complete
intersection $\mathcal{Q} \cap \mathcal{Q}'$ is the union of the
given line and of the unique twisted cubic with the required properties.
For $p=4,l=2$, we
expect to find a twisted cubic passing through four given points and
2-secant to two given lines. But such a curve does not exist. To see
this, simply consider the unique quadric $\mathcal{Q}$ containing
the lines and the points $P_1,P_2,P_3$; notice that, by genericity, the
fourth point $P_4$ is not on $\mathcal{Q}$. Clearly, any curve
with the required properties would be contained in $\mathcal{Q}$
by Bezout, and $P_4$ can not be a point of the curve, hence a
contradiction. Thus the naive numeric count can not be blindly
trusted any more.
Now consider the case $p=3,l=3$. Given three points and three
lines, we numerically expect to find a curve passing through the
points and having the lines as chords. But the previous situation
suggests that this might not be the case. Strangely enough, the
numerics works again and the curve exists. To see this, let
$\mathcal{Q}$ be the unique (smooth) quadric containing the two lines
$\Lambda_1,\Lambda_2$,
and the points $P_1,P_2,P_3$. We can assume the lines to be of type $(1,0)$ on
$\mathcal{Q}$. Let $R_1,R_2$ be the points $\mathcal{Q}\cap \Lambda_3$. The vector space of
curves of type $(1,2)$ on $\mathcal{Q}$ has dimension $6$ and
hence there exists a (unique) rnc $\mathcal{C}$ containing
$R_1, R_2, P_1,P_2, P_3$. Thus $\mathcal{C}$ is the
required twisted cubic and the numeric count works again. It is worth of noting that this can
also be seen using the classical projective generation of the rnc
also known as Steiner's construction (see, e.g., \cite{Todd} and
\cite{Harris}).
Next, the case $p=2,l=4$. In \cite{Wakeford}, Wakeford treated
this case using a Cremona transformation. Namely, consider the
linear system of cubics containing the lines. The corresponding
map is a Cremona transformation of type $(3,3)$ mapping the
required twisted cubic curves in lines and viceversa. Existence
and uniqueness follow simply by taking the preimage of a line.
Notice that this is again a constructive method: let $\mathcal{S}$
and $\mathcal{S}'$ be the cubics containing the lines and one of
the points. Then $\mathcal{S}\cap \mathcal{S}'$ splits as the
union of the four lines, the two four secant to them and a
residual twisted cubics with the required properties.
Finally the $p=0,l=6$ case. This has been studied again by
Wakeford in \cite{Wakeford} via a Cremona transformation and a
chords argument. First observe that two twisted cubic can have at most ten common
chords and this is the case if they are generic. To see this use
the linear system of quadrics through the first curve to map
rationally $\mathbb{P}^3$ onto $\mathbb{P}^2$. This map contracts all the chords
of the first curve to points. The second curve maps to a degree
six rational curve whose double points corresponds to common
chords. Hence, ten common chords exist. Now apply the Cremona
given by the linear system of cubic surfaces containing the four generic
lines $\Lambda_1, \ldots, \Lambda_4$: the two extra lines $\Lambda_5, \Lambda_6$ are mapped to twisted cubic
curves having four common chords. The preimages of the remaining six chords give
six twisted cubic curves 2-secant to the six lines $\Lambda_1, \ldots, \Lambda_6$. Notice
that this is the only case in which we have existence but {\it
not} uniqueness. The existence of more than one curve makes the
problem considerably harder in higher dimension and in fact it
still remains unsolved.
Using quite ad hoc and special
arguments, the considerations above give a complete description of the
situation in $\mathbb{P}^3$, which we summarize in the following
\begin{prop}\label{P3}
In $\mathbb{P}^3$ consider $p$ points and $l$ lines in generic position
such that $p+l=6$. Then there exists a rational normal curve
passing through the points and 2-secant to the lines for
\[
(p,l)=(6,0),(5,1),(3,3),(2,4),(1,5),(0,6).
\]
In the case
$(p,l)=(4,2)$ the curve does not exist. Moreover, the curve is
unique in all cases but the $(p,l)=(0,6)$ case, where six such
curves exist.
\end{prop}
\section{General results}\label{main}
In this section we extend to $\mathbb{P}^n$ the results of Proposition \ref{P3}
for every $(p,l)$ such that $p+l=n+3$, $n >3$, and $p \geq 1$. The case
$(p,l) = (0,n+3)$ is still open.
The results of this section follow the
paradigm ``given $p$ points and $l$ lines in {\it generic}
position" then ``some conclusions follows". We mainly use
Lemma \ref{generalLEMMA} and we show that a Zariski non-empty open subset of
the appropriate data space $\mathcal{D}$ exists, and the proper
conclusion holds for all data in that subset.
\begin{thm}\label{castelnuovo} Given $n+3$ points in $\mathbb{P}^n$ in generic position,
there exists a unique rational normal curve passing through them.
\end{thm}
\begin{proof} A constructive proof can be found in \cite[pg. 10]{Harris} and
\cite{Bordiga}. Here we give a classic proof via Cremona
transformations.
Let $P_1, \ldots,P_{n+3}$ be the given points.
We may assume that $P_1, \ldots,P_{n+1}$ are the
coordinate points. Consider the linear system of degree $n$ hypersurfaces having the
coordinate points as singular points of multiplicity $n-1$. If we
denote the map associated to the linear system as $\varphi$, it is
well known that it is a Cremona transformation of type $(n,n)$. In
particular, $\varphi$ maps rnc's through the coordinate points in
lines and viceversa. Hence, the preimage of the unique line
joining $\varphi(P_{n+2})$ and $\varphi(P_{n+3})$ is the required rnc.
\end{proof}
\begin{prop}\label{onePOINTlessPROP}
Consider $n+2$ points in $\mathbb{P}^n$, and a codimension two linear
space in generic position. Then, there exists a unique rational
normal curve passing through the points and $(n-1)$-secant to the
linear space.
\end{prop}
\begin{proof}
We use Lemma \ref{generalLEMMA} and its notation. Thus, to show
existence we have to produce a datum $\delta$ such that
$\phi^{-1}(\delta)$ is a single point.
Let
\[
\mathsf{M}= \left(
\begin{array}{ccc}
F_1 & \ldots & F_n \\
G_1 & \ldots & G_n
\end{array}
\right)
\]
be a generic $2\times n$ matrix of linear forms
and denote by $\mathcal{C}$ the rnc defined by its $2\times 2$
minors.
By the genericity of $\mathsf{M}$, we have that
\begin{itemize}
\item for a generic choice of pairs
$(a_j,b_j)\in\mathbb{C}^2,j=1,\ldots,n+2$, the points $P_j=\{a_j F_1+b_j
G_1=\ldots =a_j F_n+b_j G_n=0\}$ are distinct;
\item $\Lambda=\{F_1=G_1=0\}$ is a codimension two linear space.
\end{itemize}
Now consider the datum
$\delta=(\Lambda,P_1,\ldots,P_{n+2})\in\mathcal{D}$. Clearly, $\delta$
is in the image of $\phi$. We will now show that
$\phi^{-1}(\delta)$ consist of a single
point. Let $\mathcal{Q}_{j}$ be the quadric defined by $\left|\begin{array}{cc}F_1 & F_j \\
G_1 & G_j\end{array}\right|=0,j=2,\ldots,n$. Observe
that $\Lambda \subset \mathcal{Q}_{j}$ and $P_1,\ldots,P_{n+2}\in
\mathcal{Q}_{j}$. Moreover, a simple rank argument yields
\[\bigcap_j \mathcal{Q}_{j}=\Lambda\cup \mathcal{C}.\]
In fact, if a point $P$ lies in this intersection but not in $\Lambda$,
then all the columns of $\mathsf{M}$ evaluated in $P$ are
proportional to the first column and hence $P\in \mathcal{C}$.
Now it is easy to check that any rnc $\mathcal{C}'$ satisfying the datum $\delta$ is
contained in $\mathcal{Q}_{j},j=2,\ldots,n$ by Bezout, and hence
$\mathcal{C}'$ and $\mathcal{C}$ coincide. In conclusion,
$\phi^{-1}(\delta)_{|\mathcal{H}}=\mathcal{C}$ and the map $\phi$
is dominant.
To prove uniqueness, it suffices to repeat the Bezout argument
above.
\end{proof}
\begin{prop}\label{donotexist}
Let $p$ and $l$ be integers such that $p\geq 4$ and $l\geq 2$.
Consider $p$ points and $l$ codimension two linear
spaces in generic position in $\mathbb{P}^n$. Then
no rational normal curve passing through the points and
$(n-1)$-secant to the linear spaces exists.
\end{prop}
\begin{proof}
It is enough to prove the statement for $p=4$ and $l=2$. Let $\Lambda_1$
and $\Lambda_2$ be the linear spaces and let $P_1,\ldots,P_4$ be the
points. We want to show that there exists a quadric containing the
scheme $X=\Lambda_1\cup \Lambda_2\cup P_1\cup P_2\cup P_3$, i.e. we want to
show that $h^0 \mathcal{I}_X(2)>0$. The linear space $\Lambda_1$ imposes
${n\choose 2}$ independent conditions on quadrics and notice that
$\Lambda_1\cap \Lambda_2\simeq\mathbb{P}^{n-4}$. Thus $\Lambda_1\cup \Lambda_2$ imposes
$2{n\choose 2}-{n-2\choose 2}$ conditions. In conclusion
\[
h^0\mathcal{I}_X(2)\geq h^0\mathcal{O}_{\mathbb{P}^n}(2)-\left[2{n\choose
2}-{n-2\choose 2}\right]-3=1.
\]
Let $\mathcal{Q}$ be a quadric containing $X$ and notice that
$P_4\not\in \mathcal{Q}$ by genericity. Suppose that a rational
normal curve with the requires properties exists, say
$\mathcal{C}$. We will show that $\mathcal{Q}\supset \mathcal{C}$,
hence a contradiction. Let $t$ be the degree of the scheme
$\mathcal{C}\cap \Lambda_1\cap \Lambda_2$ and notice that $\mathcal{Q}$ is
singular along the intersection $\Lambda_1\cap \Lambda_2$. Hence the
degree of $\mathcal{Q}\cap \mathcal{C}$ is at least
\[
3+(n-1-t)+(n-1-t)+2t=2n+1
\]
so by Bezout we get $\mathcal{Q}\supset
\mathcal{C}$, a contradiction.
\end{proof}
\begin{prop}\label{3points}
Consider in $\mathbb{P}^n$ three points and $n$ codimension two linear
spaces in generic position. Then, there exists a unique rational
normal curve passing through the points and $(n-1)$-secant to the
linear spaces.
\end{prop}
\begin{proof}
Notation as in Lemma \ref{generalLEMMA}.
Existence is proved if we produce a datum $\delta$ such that
$\phi^{-1}(\delta)$ is a single point.
Let $\mathsf{M}$ and $\mathcal{C}$ be
as in the proof of Proposition \ref{onePOINTlessPROP}.
By the genericity of $\mathsf{M}$, we have that
\begin{itemize}
\item $\{F_1=\ldots=F_n=0\}$ is a point, say $P_1$;
\item $\{G_1=\ldots=G_n=0\}$ is a point, say $P_2$;
\item $\{F_1+G_1=\ldots=F_n+G_n=0\}$ is a point, say $P_3$;
\item $\{F_i=G_i=0\}$ is a codimension two linear space, say $\Lambda_i$,
$i=1,\ldots,n$.
\end{itemize}
Now consider the datum
$\delta=(\Lambda_1,\ldots,\Lambda_n,P_1,P_2,P_3)\in\mathcal{D}$. Clearly,
$\delta$ is in the image of $\phi$. We will now show that
$\phi^{-1}(\delta)$ consist of a single point. For $1\leq i<j\leq
n$, let
$\mathcal{Q}_{ij}$ be the quadric defined by $\left|\begin{array}{cc}F_i & F_j \\
G_i & G_j\end{array}\right|=0$ and notice that
$ \Lambda_i,\Lambda_j \subset \mathcal{Q}_{i,j}$ and $P_1,\ldots,P_3\in
\mathcal{Q}_{i,j}$. It easily follows that any rnc $\mathcal{C}'$ satisfying the
datum $\delta$ is contained in all the $\mathcal{Q}_{ij}$'s by
Bezout. Hence the curves $\mathcal{C}$ and $\mathcal{C}'$ coincide
as they have the same defining ideal. Hence
$\phi^{-1}(\delta)_{|\mathcal{H}}=\mathcal{C}$, thus the map $\phi$
is dominant. This proves the existence of the rnc with the
desired properties. Uniqueness follows by the Bezout
argument above.
\end{proof}
\begin{prop}\label{2points}
Consider in $\mathbb{P}^n$ two points and $n+1$ codimension two linear
spaces in generic position. Then, there exists a unique rational
normal curve passing through the points and $(n-1)$-secant to the
linear spaces.
\end{prop}
\begin{proof} First we will recall the classical proof, only given in $\mathbb{P}^4$, and then we
produce ours.
\vskip .5cm \noindent({\it Veneroni-Todd})
The idea is to use a Cremona transformation, also known as
Veneroni's transformation. We refer to the classical work \cite{Todd} for more
details. Consider the linear system of quartics containing five
given planes in generic position and let
$\varphi:\mathbb{P}^4\dashrightarrow\mathbb{P}^4$ be the corresponding map. This
map can be shown to be a Cremona of type $(4,4)$ and it maps each
rnc 3-secant to each of the planes in a line and vice versa. Given
the points $P_1$ and $P_2$, existence and uniqueness follow by
considering the preimage of the unique line joining $\varphi(P_1)$
and $\varphi(P_2)$.
\vskip .5cm \noindent({\it Complete proof})
We use Lemma \ref{generalLEMMA} and its notation. Thus, the
existence part of the proof is completed if we exhibit a datum
$\delta$ such that $\phi^{-1}(\delta)$ is non-empty and finite.
Let $\mathsf{M}$ and $\mathcal{C}$ be
as in the proof of Proposition \ref{onePOINTlessPROP}.
We have that
\begin{itemize}
\item $\{F_1=\ldots=F_n=0\}$ is a point, say $P_1$;
\item $\{G_1=\ldots=G_n=0\}$ is a point, say $P_2$;
\item $\{ F_i=G_i=0\}$ and $\{\sum F_i=\sum
G_i=0\}$ are codimension two linear spaces, say $\Lambda_i$
(for $i=1,\ldots,n$)
and $\Lambda$, respectively.
\end{itemize}
Now consider the datum
$\delta=(\Lambda_1,\ldots,\Lambda_n,\Lambda,P_1,P_2)\in\mathcal{D}$. Clearly,
$\delta$ is in the image of $\phi$. We will now show that
$\phi^{-1}(\delta)$ consist of a single point. Let $\mathcal{C}'$
be another rnc satisfying the datum $\delta$. By Lemma
\ref{codim2secant} we know that a defining matrix of
$\mathcal{C}'$ can be chosen of the form
\[
\mathsf{M}'= \left(
\begin{array}{ccc}
a _1 F_1+b_1 G_1 & \ldots & a _n F_n+b_n G_n \\
a_1' F_1+b_1' G_1 & \ldots & a _n' F_n+b_n' G_n
\end{array}
\right).
\]
By the genericity of $\mathsf{M}$, $P_1$ annihilates all the $F_i$'s
but none of the $G_i$'s. Hence the vectors $(b_1, \ldots, b_n)$ and $(b'_1, \ldots, b'_n)$
are proportional. Thus a linear combination of the rows of
$\mathsf{M'}$ eliminates all the $G_i$'s in the first row;
similarly for the $F_i$'s in the second row using $P_2$. So that
$\mathsf{M}'$ is conjugate to a matrix of form
\[
\left(
\begin{array}{ccc}
F_1 & \ldots & F_n \\
c_1 G_1 & \ldots & c_n G_n
\end{array}
\right).
\]
Now recall that $\Lambda=\{\sum F_i=\sum G_i=0\}$ intersects both
$\mathcal{C}$ and $\mathcal{C}'$ in a degree $n-1$ scheme. Then,
by Lemma \ref{codim2secant}, the vector space $\langle\sum
F_i,\sum G_i\rangle$ contains both $\sum e_i F_i$ and $\sum e_ic_i
G_i$ for some choice of the $e_i$'s in $\mathbb{C}$. Since
$F_1,\ldots,F_n$ and $\sum G_i$ are linearly independent by the
genericity of $\mathsf{M}$, we have $e_1=\ldots =e_n$.
Similarly by the independence of $G_1,\ldots,G_n$ and
$\sum F_i$, we get $e_1c_1=\ldots =e_nc_n$. These conditions force
$\mathsf{M}$ and $\mathsf{M}'$
to be conjugate, hence $\mathcal{C}$ and $\mathcal{C}'$ coincide,
$\phi^{-1}(\delta)_{|\mathcal{H}}=\mathcal{C}$, and
the map $\phi$ is dominant.
To prove uniqueness, it is enough to repeat the argument
above about the defining matrices.
\end{proof}
\begin{prop} \label{onepoint}
Consider in $\mathbb{P}^n$ a point and $n+2$ codimension two linear
spaces in generic position. Then, there exists a
unique rational normal curve passing through the points and
$(n-1)$-secant to the linear spaces.
\end{prop}
\begin{proof}
\vskip .5cm \noindent({\it Todd})
We only sketch the original proof given in $\mathbb{P}^4$ and we refer to
\cite{Todd} for more details. Let $\Lambda_1,\ldots,\Lambda_6$ be the planes
and $P$ be the point. Consider the linear system of quartics
through $\Lambda_1,\ldots,\Lambda_5$ and let $\varphi$ be the corresponding
rational map. Notice that $\varphi$ is a Cremona of type $(4,4)$.
The rnc's satisfying the data are among the preimages of the lines
through $\varphi(P)$. Since $\varphi$ maps $\Lambda_6$ in a Bordiga
surface, the unique trisecant line through $\varphi(P)$ gives the
desired curve.
\vskip .5cm \noindent({\it Complete proof})
Notation as in Lemma \ref{generalLEMMA}. As usual,
let $\mathsf{M}$ and $\mathcal{C}$ be
as in the proof of Proposition \ref{onePOINTlessPROP}.
We have the following:
\begin{itemize}
\item $\{F_1=\ldots=F_n=0\}$ is a point, say $P$;
\item $\{ F_i=G_i=0\}$ are codimension
two linear spaces, say $\Lambda_i$,
$i=1,\ldots,n$.
\end{itemize}
Then consider a rnc $\mathcal{C}'$, defined by a matrix
$\mathsf{M}'$, and impose that $P\in \mathcal{C}'$ and that
$\Lambda_1,\ldots,\Lambda_n$ intersect also $\mathcal{C}'$ in a degree $n-1$
scheme.
Since $P$ annihilates all the $F_i$'s, but none of the $G_i$'s,
by arguments similar to the ones used in the proof of the previous proposition,
we get
\[
\mathsf{M}'= \left(
\begin{array}{cccc}
F_1 & F_2 & \ldots & F_n \\
G_1 & a _2 F_2+b_2 G_2 &\ldots & a _n F_n+b_n G_n
\end{array}
\right).
\]
Now we choose two extra common secant spaces to
$\mathcal{C}$ and $\mathcal{C}'$ in such a way that the curves are
forced to coincide. The proof is different depending on the parity of $n$.
If $n$ is odd, let $n=2m-1$. Consider the linear spaces $
\Lambda_{n+1}=\{\sum_{i=1}^m F_i=\sum_{i=1}^m G_i=0\}$ and $\Lambda_{n+2}
=\{\sum_{i=m}^{2m-1} F_i=\sum_{i=m}^{2m-1} G_i=0\}$, and require that $\mathcal{C}'$
is $(n-1)$-secant to them. Since $\Lambda_{n+1}$ intersect
$\mathcal{C}'$ in a degree $n-1$ scheme, by Lemma
\ref{codim2secant} there exist constants
$e_1, \ldots, e_n \in \mathbb{C} $ such that
\[
\Lambda_{n+1}=\left \{\sum_{i=1}^n e_i F_i=e_1G_1+\sum_{i=2}^n e_i(a_i
F_i+b_i G_i)=0\right \}
\]
Since the vector space $\langle\sum _{i=1}^m
F_i,\sum _{i=1}^mG_i\rangle$ contains $\sum_{i=1}^n e_i F_i$
and the $n+1$ linear forms $F_1, \ldots, F_n, \sum _{i=1}^mG_i$
are linearly independent, we get $e_1= \ldots= e_m $ and
$e_{m+1}= \ldots= e_n=0 $.
Moreover also $e_1G_1+\sum_{i=2}^n e_i(a_i F_i+b_i G_i)$ is an element of the vector space
above, then
there exist $a$, $b \in \mathbb{C}$ such that
\[
a\sum_{i=1}^m F_i + b\sum_{i=1}^m G_i=G_1+\sum_{i=2}^m (a_i
F_i+b_i G_i).
\]
This equality involves the $2m$ linear forms
$F_1, \ldots, F_m,G_1, \ldots, G_m$, which are
linearly independent. By comparing their
coefficients we get
$a=a_2=\ldots =a_m =0$ , and $b=b_2=\ldots =b_m =1$.
Analogously, imposing that $ \Lambda_{n+2}$ is a
$(n-1)$-secant space to $\mathcal{C}'$, we get $a_i=0$ and $b_i=1$
for all $i$. Thus $\mathsf{M}=\mathsf{M}'$, the curves
coincide, and if we let $\delta=(\Lambda_1,\ldots,\Lambda_{n+2},P)$, then $\phi^{-1}(\delta)$ consists of a
single point.
In the case $n$ even, let $n=2m-2$.Now consider the linear spaces
$\Lambda_{n+1}=\{\sum_{i=1}^m F_i=\sum_{i=1}^m G_i=0\}$ and $
\Lambda_{n+2}=\{F_1+\sum_{i=m}^{2m-2} F_i=G_1+\sum_{i=m}^{2m-2} G_i=0\}$,
and analogously to the $n$ odd case, require that $\mathcal{C}'$
is $(n-1)$-secant to them. Arguing as above,
we get that there exist $a$, $b \in \mathbb{C}$ such that
\[
a\sum_{i=1}^m F_i + b\sum_{i=1}^m G_i=G_1+\sum_{i=2}^m (a_i
F_i+b_i G_i).
\]
Now we may assume that
$G_1=F_1+\ldots+F_m+G_2+\ldots+G_m$.
By comparing the coefficients of the independent linear
forms $F_1, \ldots, F_m, G_2, \ldots, G_m$ we easily get
\[ a_2=\ldots =a_m=0 \mbox{ and } b_2=\ldots =b_m.\]
Since $\Lambda_{n+2}$ intersect the
curve $\mathcal{C}'$ in a degree $n-1$ scheme,
the vector space $\langle
F_1+\sum_{i=m}^{2m-2} F_i,G_1+\sum_{i=m}^{2m-2} G_i
\rangle$ contains $\sum_{i=1}^n e'_i F_i$ and
$e'_1G_1+\sum_{i=2}^n e'_i(a_i F_i+b_i G_i)$
for some $e'_i$'s $\in \mathbb{C}$. Repeating the usual arguments
we get that there exist
$a'$, $b' \in \mathbb{C}$ such that
\begin{eqnarray}
& \nonumber
a'(F_1+F_m+\ldots+F_{2m-2})+b'(G_1+G_m+\ldots+G_{2m-2}) &
\\
&\nonumber = G_1+b_m
G_m+\sum_{i=m+1}^{2m-2} (a_i F_i+b_i G_i) \hfill &
\end{eqnarray}
Since
\[
\{G_1,G_m,\ldots,G_{2m-2},F_{m+1},\ldots,F_{2m-2},F_1+F_m\}
\]
is a set of $n+1$ linearly independent forms, again by comparing their
coefficients we get
$b_m= \ldots =b_{2m-2}=1$ and $a_{m+1}= \ldots =a_{2m-2}=0$.
Summing up these relations with the
previous ones, we obtain $\mathsf{M}=\mathsf{M}'$
and if we let $\delta=(\Lambda_1,\ldots,\Lambda_{n+2},P)$, then $\phi^{-1}(\delta)$
consists of a single
point.
The existence part of the proof is now
completed. To show { uniqueness} it is enough to repeat argue
as above on the defining matrices.
\end{proof}
We summarize the results of Theorem \ref{castelnuovo},
Proposition \ref{onePOINTlessPROP} to \ref{onepoint} in
the following
\begin{thm}\label{final}
Let $n,p$ and $l$ be positive
integers such that
$$n\geq
3, \ \ p\geq 1\mbox{ and} \ \ p+l=n+3 .$$
Choose $p$ points in $\mathbb{P}^n$ and $l$
codimension two linear spaces in generic position. Then, only for
the values
\[
(p,l)=(n+3,0),(n+2,1),(3,n),(2,n+1),(1,n+2)
\]
does there
exist a unique rational normal curve passing through the points and
$(n-1)$-secant to the linear spaces. In the other cases, that is for
$p\geq 4$ and $l\geq 2$, no such curve exists.
\end{thm}
\section{Applications}\label{appsection}
\subsection{Postulation of schemes and defectivity}\label{postulationAPP}
Theorem \ref{final} can be used to produce
schemes that impose less conditions than expected to forms of some degree.
Here we only give an example to show the main ideas. A thorough study will be the subject of a
forthcoming paper.
Let $P_1,\ldots,P_{n+2}$ be $n+2$ generic points in $\mathbb{P}^n$ $(n>2),$
and let $\Lambda$ be a generic codimension two linear space, with defining ideals
$I_{P_1}, \ldots, I_{P_{n+2}}, I_\Lambda$, respectively.
Consider the scheme $X$ having ideal
\[I_X=(I_{P_1})^2\cap \ldots\cap (I_{P_{n+2}})^2\cap (I_\Lambda)^2.\]
It is easy to compute the expected Hilbert function of $X$ in degree $4$, namely
$$h=(n+2)(n+1)+{n+2\choose 4}+2{n+1\choose 3}.$$
The following lemma shows that the scheme $X$ have not the expected postulation:
\begin{lem}\label{applemma}
Notation as above, the scheme $X$ does not impose the
expected number of conditions on degree 4 hypersurfaces, i.e.
$H(X,4)\leq h-1$.
\end{lem}
\begin{proof}
Consider the scheme $X'$ with defining ideal
\[I_{X'}=I_{P_1}\cap (I_{P_2})^2\ldots\cap (I_{P_{n+2}})^2\cap (I_\Lambda)^2.\]
By Theorem \ref{final} we know that there exists a rnc $\mathcal{C}$
through the $P_i$'s and having $\Lambda$ as a $(n-1)$-secant space.
Moreover, any
element $F\in (I_{X'})_4$ vanishes on $\mathcal{C}$ by a standard
Bezout argument as the degree of $\{F=0\}\cap \mathcal{C}$ is
\[ 1+2(n+1)+2(n-1)>4n.\]
Hence, all quartic hypersurfaces through $X'$ have a fixed tangent direction in
$P_1$. This is enough to conclude that $H(X,4)$ is at least one
less than expected.
\end{proof}
As a straightforward application of Lemma \ref{applemma} the next corollary
gives a not trivial
defectiveness result for Segre-Veronese varieties. The same statement
can be deduced by the classification given in \cite{AAdlandsvik},
where {\AA}dlandsvik heavily uses his theory of the
joint of varieties.
Abrescia in \cite{Abrescia} proposes a
simplified proof of this result.
\begin{cor}\label{applcor} Let V be the Segre-Veronese variety $\mathbb{P}^1\times\mathbb{P}^n$ embedded
with bidegree $(2,2)$. Then $V$ is $(n+1)$-defective, i.e. the
secant variety $S^{n+1}(V)$ has not the expected dimension.
\end{cor}
\begin{proof}
The conclusion follows from the previous lemma and the results of
Section 1 in \cite{CGG3}, where the authors relates the
$(n+1)$-defectiveness of $V$ with the Hilbert function in degree
$4$ of schemes consisting of $n+2$ double points and a double
codimension two linear space.
\end{proof}
\subsection{Projectively equivalent subsets}\label{projeq}
It is well known that any two ordered sets of $n+2$ generic points in
$\mathbb{P}^n$ are projectively equivalent, i.e. there is an automorphism
of $\mathbb{P}^n$ mapping one set in the other preserving the order
of the points.
If $\mathbb{X},\mathbb{Y}$ are two ordered sets of $n+3$ points in $\mathbb{P}^n$,
it is
interesting to look for conditions assuring the projective
equivalence. Theorem \ref{castelnuovo} gives the answer:
via the unique rational normal curves passing through the
points of each set, we map $\mathbb{X}$ and $\mathbb{Y}$ into
$\mathbb{P}^1$. Then the question is answered via cross ratios.
For more than $n+3$ points, the problem stays open (see \cite[pg.
8]{Harris}).
If linear spaces and points are taken into account,
as far as we know, there are no similar results. Our Theorem
\ref{final} allows to give an answer for a special family of
subsets of $\mathbb{P}^n$. Namely, the sets consisting of $p$ points and
$l$ codimension two linear spaces in generic
position
with $p+l=n+3, n \geq 3, p\geq 1$.
For example, consider in $\mathbb{P}^3$ a generic set
$A$ consisting of three lines and
three points.
Via the unique rnc through the three points and 2-secant to the three lines,
we obtain a subset
$A'$ of nine points in $\mathbb{P}^1$. Then, another set
$B$ of three lines and
three points is projectively equivalent to $A$ if and only if $A'$ and
$B'$ are so, where $B'$ is similarly constructed. Thus, an answer can be obtained again via cross
ratios.
\section{Final remarks}\label{remsection}
\subsection{The case of $n+3$ codimension two linear spaces}
In this situation we do not have a complete answer to the basic
question: given in $\mathbb{P}^n$ $n+3$ linear spaces of codimension two
in generic position, are there rational normal curves
$(n-1)$-secant to these
spaces? For $n=3$, the answer is
positive (see Proposition \ref {P3}), and we also have a proof
using Lemma \ref{generalLEMMA} and a cubic surface having five
of the six lines as exceptional divisors.
Unfortunately this proof does not extend to
$n>3$.
\subsection{Mixed conditions}
In this paper we generalized the classical Theorem
\ref{castelnuovo} by substituting points with codimension two
linear spaces, motivated by a count of conditions. If we look for
further generalizations, we can again rely on a count of
conditions for inspiration. For example, consider in $\mathbb{P}^{2m+3}$
$m+4$ points, a linear space of dimension $m$, and a linear space
of dimension $m+1$. Then, by counting conditions, we expect to
find a rnc passing through the points, $(m+1)$-secant to the $m$
dimensional space and $(m+2)$-secant to the $m+1$ dimensional
space. Actually, we can prove that such a curve exists and it is
unique. Note that for $m=1$ this result yields a statement similar
to the one of Lemma \ref{applemma}. By this result, analogously to
Corollary \ref{applcor}, we deduce another proof for the
classically known 5-defect of the Segre-Veronese
$\mathbb{P}^2\times\mathbb{P}^3$ embedded with bidegree $(1,2)$ (for a modern
proof see for instance \cite{CaCh}, Theorem 4.3). We are presently
studying in which direction we have to move in order to get a
result as comprehensive as Theorem \ref{final}.
\subsection{The higher dimensional case}
The title of the present paper can be rephrased as: existence
results for 1-dimensional Veronese varieties. Thus, it is
extremely natural to pose questions similar to the ones we
addressed here in the higher dimensional case. In dimension 2,
there is a well known result by Kapranov \cite{Kapranov} about the
existence of Veronese surfaces containing special sets of points.
Recently, Graber and Ranestad re-proposed Kapranov's result and
improved it by considering the existence of Veronese surfaces
``well intersecting" a special configuration of linear spaces.
These are only partial results and the problem remains open even
in the two dimensional case. As far as we know, no relevant
results exist in higher dimension.
\bibliographystyle{alpha}
|
1,116,691,500,443 | arxiv | \section{Introduction}
Texts in scenes play an important role in communication between a machine and its surrounding environments. Automated machine understanding of texts in scenes has a vast range of applications such as navigation assistance for the visually impaired \cite{OrCam}, scene text translation for tourists \cite{GoogleTrans}, etc. It has been a grand challenge for years in the computer vision research community, and it received increasing interests in recent years as observed by a number of scene text reading competitions \cite{ICDAR2013,IncidentialText,SVT,ChineseICDAR}.
\par
Automated reading texts in scenes is a very challenging task due to the large intra-class variations and the small inter-class variations with respect to many non-text objects in scenes. In particular, scene texts often have very different appearance because they may be printed in very different fonts and styles, captured from different distances and perspectives, and have very different cluttered background and lighting conditions. At the same time, texts often have high resemblance to many non-text objects in scenes, e.g. letter 'o'/'O' has similar appearance as many circular objects such as vehicle wheels, letter 'l' has similar appearance as many linear structures such as poles, etc.
\par
In this research, we design a novel scene text proposal technique and integrate it into an end-to-end scene text reading system. Inspired by the pooling layer in deep neural network, a pooling based scene text proposal technique is designed which is capable of grouping image edges into word and text line proposals efficiently. One unique feature of the pooling based proposal technique is that it does not involve heuristic thresholds/parameters such as text sizes, inter-character distances, etc. that are often used in many existing scene text detection techniques \cite{TextFlow,Shij,GVF,TDCNN7}. A novel score function is designed which exploits the histogram of oriented gradients and is capable of ranking proposals according to their probabilities of being text. Preliminary study on max-pooling based proposals has been presented in our prior work \cite{MPT_ICDAR2017}. This paper presents a more comprehensive study by investigating several key proposal parameters such as proposal quality, optimal proposal set-ups, etc. In addition, several new studies are performed from different aspects of edge label assignments, pooling methods and detection of arbitrarily oriented scene texts in different languages. Furthermore, a new score function is designed which is more efficient and robust in proposal ranking. We also integrate the proposed pooling based proposal technique into an end-to-end scene text reading system and study its effects on the scene text reading performance.
\par
The rest of this paper is organized as follows. Section 2 reviews recent works in scene text proposals, scene text detections, and scene text reading. Sections 3 and 4 describe the proposed scene text proposal technique and its application in scene text reading. Section 5 presents experimental results and several concluding remarks are drawn in Section 6.
\section{Related works}
Traditional scene text reading systems consist of a scene text detection step and a scene text recognition step. In recent years, scene text proposal has been investigated as an alternative to the scene text detection, largely due to its higher recall rate which is capable of locating more text regions as compared with the traditional text detection step.
\subsection{Scene text proposal}
The scene text proposal idea is mainly inspired by the success of object proposal in many object detection systems. It has advantage in locating more possible text regions to offer higher detection recall. It's often evaluated according to the recall rate as well as the number of needed proposals - typically the smaller the better at a similar recall level \cite{HowGoodPP}. False-positive scene text proposals are usually eliminated by either a text/nontext classifier \cite{STL,TDCNN3DT} or a scene text recognition model \cite{GomezE2E,TEB} in end-to-end scene text reading systems.
\par
Different scene text proposal approaches have been explored. One widely adopted approach combines generic object proposal techniques with text-specific features for scene text proposal generation. For example, EdgeBoxes \cite{EB} is combined with two text-specific features for scene text proposal generation \cite{TEB}. In another work \cite{JarE2E}, EdgeBoxes is combined with the Aggregate Channel Feature (ACF) and AdaBoost classifiers to search for text regions. In \cite{GomezE2E}, Selective Search \cite{SS} is combined with Maximally Stable Extremal Regions (MSER) to extract texture features for dendrogram grouping. A text-specific symmetry feature is explored in \cite{STL} to search for text line proposals directly, where false text line proposals are removed by training a CNN classifier. Deep features have also been used for scene text proposal due to its superior performance in recent years. For example, inception layers are built on top of the last convolution layer of the VGG16 for generating text proposal candidates in \cite{TDCNN3DT}. The Region Proposal Network (RPN) and Faster R-CNN structure are adopted for scene text proposal generation in \cite{TDCNN7,TextRCNN}.
\par
Most existing scene text proposal techniques have various limitations. For example, the EdgeBoxes based technique \cite{JarE2E} is efficient but often generate a large number of false-positive proposals. The hand-crafted text-specific features rely heavily on object boundaries which are sensitive to image noise and degradation \cite{Shij}. Techniques using heuristic rules and parameters \cite{TEB} do not adapt well across datasets. The deep learning based technique \cite{TDCNN3DT} produces a small number of proposals but the recall rate becomes unstable when the Intersection over Union (IoU) threshold increases. As a comparison, our proposed proposal technique does not leverage heuristic parameters and obtains a high recall rate with a small number of false-positive proposals.
\subsection{Scene text detections}
A large number of scene text detection techniques have been reported in the literature. Sliding window has been widely used to search for texts in scene images \cite{TextFlow,Coarse2FineConv,TaoWang}. However, it usually has a low efficiency because it adopts an exhaustive search process by using multiple windows of different sizes and aspect ratios. Region based techniques have been proposed to overcome the low efficiency constraint. For example, the Maximal Stable External Regions (MSRE) has been widely used \cite{Re1,Re21,Re5,Re13} for scene text detection. In addition, various hand-craft text-specific features have also been extensively investigated such as Stroke Width Transform (SWT) \cite{Re2}, Stroke Feature Transform (SFT) \cite{Re8}, text edge specific features \cite{Shij}, Stroke End Keypoints (SEK), Stroke Bend Keypoints (SBK) \cite{Re20}, and deep features based regions \cite{TDCNN1,TDCNN6,TDCNN8}. Different post-processing schemes have also been designed to remove false positives, e.g heuristic rules based classifier \cite{GVF,Re13,Re15,Re18}, graph processing \cite{TextFlow,Coarse2FineConv}, support vector regression \cite{Shij}, convolutional K-mean\cite{Coarse2FineConv}, distance metric learning \cite{Re1}, AdaBoost \cite{Re5,Re14}, random forest \cite{Re2,Re8}, convolution neural network \cite{TaoWang,Re21}, etc.
\par
With the advance of convolutional neural network (CNN), different CNN models have been exploited for the scene text detection tasks. For example, the DeepText makes use of convolutional layers for deep features extraction and inception layers for bounding boxes predictions \cite{TDCNN3DT} . The TextBoxes \cite{TextBoxes} adopts the Single Shot Multiboxex Detector (SSD) \cite{SSD} to deal with multi-scale texts in scenes. Quadrilateral anchor boxes have also been proposed for detecting tighter scene text boxes \cite{DeepMatchPriorNet}. In addition, direct regression solution has also been proposed \cite{DeepDirectRegress} to remove the hand-crafted anchor boxes. Different CNN based detection and learning schemes have also been explored. For example, some work adopts a bottom-up approach that first detection characters and then group them to words or text lines \cite{TDCNN7,TDCNN4,WordSup}. Some system instead defines a text boundary class for pixel-level scene text detection \cite{WordFence, SelfOrg}. In addition, weakly supervised and semi-supervised learning approach \cite{WeakNet} has also been studied to address the image annotation constraint \cite{WeText}.
\subsection{End-to-end scene text reading}
End-to-end scene text reading integrates detection and recognition into the same system to read texts in scenes. One popular system is a Google-Translation \cite{E2E5} which performs end-to-end scene text reading by integrating a list of techniques including three scene text detection methods, three scene text segmentation and grouping methods, two scene text recognition models, and language models for post-processing. In \cite{E2E4}, sliding window is combined with Histogram of Oriented Gradient feature extraction and Random Ferns Classifier to compute text saliency maps where words are extracted using External Regions (ER) and further re-scored using Support Vector Machine (SVM). In \cite{E2E3}, Adaboost and SVM text classifiers are applied on the extracted text regions using ER to localize scene texts which are further recognized under an Optical Character Recognition (OCR) framework. Similar approach was also adopted in \cite{E2E7}, where Maximal Stable External Regions (MSER) instead of ER is implemented for scene text region localization. In \cite{E2E9}, Stroke Width Transform \cite{Re2} is adopted for scene text region detection and Random Forest is used for character recognition and words are further recognized by component linking, word partition, and dictionary based correction. In \cite{GomezE2E,JarE2E}, potential text regions are first localized using EdgeBox (EB) \cite{EB} or adapted simple selective search for scene text \cite{GomezE2E} and scene texts are further recognized using Jarderberg's scene text recognition model \cite{JarData}.
\par
Quite a number of CNN based end-to-end scene text reading systems have been reported in recent years. In \cite{TaoWang,E2E11}, a CNN based character recognition model is developed where word information is extracted from text saliency map using sliding windows. The same framework has been implemented in \cite{JarTReg}, where a more robust end-to-end scene text reading system is developed by training
a model handling three functions including text and non-text classification, case-insensitive characters recognition, and case-sensitive characters recognition. In \cite{TextBoxes}, an advanced end-to-end scene text reading system is designed where the Single Shot Multiboxes Detector (SSD) is employed for scene text detection and a transcription model proposed in \cite{E2ETrainable} is adopted for recognition. End-to-end trainable scene text reading system has also been proposed which can concurrently produce texts location and text transcription \cite{DTSpotter}
\par
Our developed end-to-end scene text reading system adopts a similar framework as presented in \cite{GomezE2E,JarE2E} that exploits proposals and existing scene text recognition models. One unique feature is that it uses only around one-fifth of the number of proposals that prior proposal based end-to-end systems use thanks to our proposed pooling based proposal technique and gradient histogram based proposal ranking.
\section{Pooling based scene text proposal}
The proposed scene text proposal technique follows the general object proposal framework which consists of two major steps including proposal generation and proposal ranking. For the proposal generation, we design a pooling based technique that iteratively groups image edges into possible words or text lines. Here each edge component could be a part of a single character, several neighbouring characters touching each other, or other non-text objects. Each set of grouped image edges thus forms a proposal which can be represented by a bounding box that covers all grouped edges. For proposal ranking, a scoring function is designed which is capable of ranking the determined proposals according to their probability of being text. The ranking strategy employs the histogram of oriented gradient which first learns a number of text and non-text templates and then ranks proposals according to their distances to the learned templates. Fig. \ref{fig_001} illustrates the framework of our proposed scene text proposal technique.
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{Fig1.pdf}
\end{center}
\caption{The framework of the proposed scene text proposal technique including pooling based scene text proposal generation and proposal ranking based on low-level gradient features. In the output image, proposals are presented for the illustration purpose, where the proposal box in red colour detects the scene text correctly.}
\centering
\label{fig_001}
\end{figure}
\subsection{Proposal generation}
\label{ppgens}
A novel pooling based technique is designed for the scene text proposal generation. The idea is inspired by the pooling layer in the convolution neural network (CNN) which is employed to eliminate insignificant features while shrinking a feature map. Given an image, an edge map is first determined by using the Canny edge detector \cite{Canny}, where each binary edge can be labelled through connected components (CC) analysis. Each binary edge can then be labelled by an unique number indicating the order when it is searched. For example, the first searched binary edge is assigned an unique label number 1, the second with an unique number 2, etc. An initial edge feature map can thus be determined by assigning all pixels of a binary edge with the same number as the component label and all non-edge pixels with a number of zero.
\par
The image edge feature map is then processed iteratively through pooling using a pooling window. Take max-pooling as an example. During each pooling iteration, only the pixel with the largest label number within the pooling window is kept for generating the new edge feature map for the next iteration, and all other pixels with a smaller label number are discarded. The binary edges are therefore shifting to each others iteratively where those closer to each other are grouped first and those farther away are merged later. The iterative edge merging process terminates when there is no zero pixels existing in the edge feature map, meaning that there are no more gaps between the labelled binary edges as illustrated in the `Pooling process' in Fig. \ref{fig_001}. Multiple proposals are accordingly generated with different groups of edges throughout the pooling process.
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{Fig2.pdf}
\end{center}
\caption{Illustration of the max-pooling based scene text proposal generation with a 1-by-3 max pooling window and a horizontal stride of 2 on the synthetic labelled map - more detail on the second row are provided. Zero-padding is performed when the number of column is even. Duplicate proposals are removed, and six proposals are generated including three connected binary edges themselves and three found edge groups.
}
\centering
\label{fig_002_1}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{Fig3.pdf}
\end{center}
\caption{
Two synthetic graphs that explain how horizontal pooling windows and horizontal stride can group non-horizontal text lines: The red-color two-directional arrows illustrate the link-up as determined during the pooling based grouping process. It shows that the proposed technique is able to handle non-horizontal text lines as far as the constituting letters/digits having overlapping in the vertical direction. The graph on the right illustrates when two neighbouring will not be grouped.
}
\centering
\label{fig_002}
\end{figure}
Fig. \ref{fig_002_1} illustrates the max-pooling based proposal generation process by using a synthetic edge map that contains 3 binary edges as labelled by 1, 2, and 3. Taking the second row as an example, the first two edges are grouped to form a proposal after the second pooling iteration and the second and third edges are grouped to form another proposal after the third pooling iteration. Since the first and the third edges are both grouped with the second edge, all three edges are also grouped to form a new proposal. For the first and the third row, a single group of the first and second edges and a single group of three single edges can be derived under the similar idea, respectively. By removing duplicated proposals, six proposals are finally determined including the three single edges, the grouped first and second edges, the grouped second and third edges, and the grouped three edges. It should be noted that zero-padding is implemented at the right side when the studied row has an even number of pixels left.
\par
Though a horizontal pooling window of size 1-by-3 and a horizontal stride of 2 are used, the proposed pooling based proposal technique is able to handle non-horizontal words or text lines as far as the constituting letters/digits having certain overlap in the vertical direction. This is illustrated in the two synthetic graphs in Fig. \ref{fig_002}. As the first graph in Fig. \ref{fig_002} shows, the curved chain of digits 1-6 will be grouped together due to their overlap in the vertical direction. Note digits/letters could be grouped via other neighbouring digits/letters when they have no overlap in the vertical direction. For example, the digits 4 and 5 can be grouped via the digits 2 and 3 even though they have no vertical overlap. Digits/letters will not be grouped when they have no overlap in the vertical direction and also have no neighbouring digits/letters to leverage as illustrated in the second graph in Fig. \ref{fig_002}.
\subsection{Proposal ranking}
Histogram of oriented gradient has been used successfully for the scene text detection and recognition tasks \cite{Wang,Recog4}. The success shows that scene texts actually have certain unique HoG features that can differentiate them from other non-text objects. We therefore adapt HoG for proposal ranking, aiming to exploit the unique text-specific HoG features to rank text proposals to the front of the whole proposal list. Different from the traditional HoG, we extract HoG features from the Canny edge pixels only which we will refer it by Histogram of Oriented Gradient on edges (HoGe) in the ensuing discussion.
\par
In our proposal ranking strategy, a number of text and non-text HoGe templates are first learned from a set of training images to be discussed in \ref{ranking optimization}. Scene text proposals are then scored and ranked according to the distances between their HoGe and the learned text and non-text HoGe templates. The scoring function is defined as follows:
\begin{equation}
s=\frac{d_{nt}}{d_{nt}+d_t}
\end{equation}
where $d_t$ and $d_{nt}$ refer to the distances between the feature vector ($F$) of a detected proposal and the pre-determined text and non-text templates as follows.
\begin{equation}
\begin{split}
d_t = \sum_{i=1}^{n}\lVert (F,T_i) \rVert \\
d_{nt} = \sum_{i=1}^{n}\lVert (F,NT_i) \rVert
\end{split}
\end{equation}
where $n$ denotes the number of text ($T$) and non-text ($NT$) templates, and $\lVert \cdot \rVert$ gives the Euclidean distance between $F$ and a text/non-text feature template. The score function in Eq. 1 is designed based on the observation that the feature vector of a text proposal is usually closer to text templates as compared with non-text templates. The feature vector of a text proposal will thus produce a small $d_t$ and a large $d_{nt}$ which further lead to a high text probability score.
\subsection{Discussion}
\label{optimization}
\subsubsection{Pooling and edge labelling}
\label{PoolingLabeling}
As described in Section \ref{ppgens}, we assign edge labels according to the searching order (from left to right column by column and from top to bottom in each column) and adopt the max-pooling to group text edges within the same line. On the other hand, the proposed technique can work with different edge label assignment and pooling methods. Two new tests are performed for verification. The first test studies two more edge labelling methods that assign edge labels by using the maximum and mean gradient of pixels within an CC, respectively (named by \textit{maxE} and \textit{meanE} in Table \ref{tablePoolingLabeling}). Take the use of \textit{meanE} as an example. It first calculates the mean gradient of each CC and then labels all edge pixels by using the calculated mean gradient directly. The second test studies the min-pooling method that keeps the smallest instead of the largest edge labels (as in max-pooling) falling within the same pooling window.
\par
\begin{table}[!t]
\small
\caption{Recall rates of four variants of the proposed technique on test images of the ICDAR2003 and ICDAR2013 dataset under IoU threshold of 0.8 and four sets of pooling window sizes and stride values in the format of \textit{window height - window width - vertical stride - horizontal stride}.}
\begin{center}
\begin{tabular}{c c c c c}
\hline
\hline
& 1-3-1-2 & 2-3-2-2 & 3-3-2-2 & 3-3-3-3 \\
\hline
\hline
\textit{maxE$\_$maxP} & 78.84 & 77.07 & 71.16 & 56.63 \\
\textit{meanE$\_$maxP}& 79.02 & 76.79 & 71.93 & 56.58 \\
\textit{searL$\_$maxP}& 79.52 & 76.11 & 71.12 & 55.5 \\
\textit{searL$\_$minP}& 78.93 & 76.52 & 71.39 & 56.18 \\
\hline
\hline
\end{tabular}
\end{center}
\label{tablePoolingLabeling}
\end{table}
Table \ref{tablePoolingLabeling} shows the test results on the test images of the ICDAR2003 and ICDAR2013 datasets, where \textit{searL} denotes a labelling method that assigns edge labels according to the edge searching order, \textit{maxP} and \textit{minP} denote max-pooling and min-pooling, respectively. So \textit{maxE$\_$maxP} means that image edges are labelled by using the maximum gradient and pooling is performed by max-pooling. The very close proposal recalls under different window sizes and strides in Table \ref{tablePoolingLabeling} verify that our proposed technique is tolerant to both edge labelling methods and edge label pooling methods.
\subsubsection{Proposal generation}
\label{proposal generation optimization}
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.5in]{Fig6_New.pdf}
\end{center}
\caption{The heat-map presents performances of the proposed proposal technique under different combinations of the pooling window size and the stride values, as evaluated on the ICDAR2003 and ICDAR2013 training datasets. This study shows that the optimal combination is a horizontal pooling window of 1-by-3 and a horizontal stride of 2 for a trade-off of accuracy and processing time}
\centering
\label{fig_008}
\end{figure}
As the optimization of proposal generation targets the best proposal recall, we relax the number of proposals and include all generated proposals while studying the size of the pooling window and strides. We adopt the grid search to study the two key sets of parameters, including a pooling window size (width and height) and stride values (a horizontal stride and a vertical stride). In particular, we vary the size of the pooling window and strides from 1 to 5 which produces 600 (24*25) parameter settings. Note the pooling window size 1-by-1 is not included as it does not perform any grouping operations.
\par
Fig. \ref{fig_008} shows proposal recalls under the 600 parameters setting which are presented by using a heat-map, where each recall is an average of three recalls when three IoU thresholds 0.5, 0.7, and 0.8 are applied. As shown in Fig. \ref{fig_008}, the two numbers at the bottom of each column refer to the row number (the number at the top) and the column number (the number at the bottom) of the pooling window, respectively. The two numbers at the left of each row refer to strides in the vertical direction (the number on the left) and horizontal direction (the number on the right), respectively. We further sort the recalls under the 600 settings and the table on the right shows several best-performing settings. In our implemented system, we take a compromise between recall rate and processing time and select the combination of a 1-by-3 pooling window and strides 1-by-2 in vertical and horizontal directions.
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=3.5in]{Fig_N.pdf}
\end{center}
\caption{The effect of pooling window size on the quality of proposals: Generally, a smaller pooling window will generate the larger number of proposals and have a better chance of capturing the right objects. The number of proposals decreases when pooling window size increases from graph (a) to graph (d) where \textbf{IT} denotes iteration.}
\centering
\label{fig_009}
\end{figure}
Several factors need to be taken into consideration while setting the pooling and strides. The first is the absolute size of the pooling window which defines the minimum distance between neighbouring edges that the pooling based proposal technique could capture. For example, a large pooling window of size 2x4 will not be able to captures distances of 1, 2 and 3 pixels between neighbouring edges whereas a pooling window of size 2x2 is capable of capturing distance as small as 1 pixel only as illustrated in Fig. \ref{fig_009}. The second is specific setting of rows and columns of the pooling window and strides in horizontal and vertical directions. In particular, the increase of coverage/jump in the vertical direction often deteriorate the proposal performance as illustrated in the heat-map in Fig. \ref{fig_008}. One reason could be due to the fact that most text in scenes are positioned in a horizontal direction. In addition, a pooling window with a big span in vertical direction often groups texts with neighbouring non-text objects lying above or below texts. The third is overlap between two consecutive pooling windows which happens when the stride in the horizontal direction is smaller than the width of the pooling window. A smaller stride often produces better recall rate, meaning that overlaps between two consecutive pooling windows helps to produce better proposals. In fact, the proposal performance drops a lot when there are absolutely no overlaps between two consecutive pooling windows as illustrated in the heat-map in Fig. \ref{fig_008}.
\subsubsection{Proposal ranking}
\label{ranking optimization}
\begin{table}[!t]
\small
\caption{Recall rates of the proposed scene text proposal technique (on the training images of the ICDAR2013 and the ICDAR2003 datasets) under different combinations of the number of templates per class (nC) and the template dimension (Dims).}
\begin{center}
\begin{tabular}{c c c c c c c}
\hline
\hline
No. & nC & Dims & Recall\_Aver & Recall\_IoU0.5 & Recall\_IoU0.7 & Recall\_IoU0.8 \\
\hline
\hline
1 & 25 & 120 & 88.75 & 94.96 & 89.67 & 81.61 \\[-0.05cm]
2 & 35 & 120 & 88.75 & 94.71 & 89.67 & 81.86 \\[-0.05cm]
3 & 65 & 140 & 88.75 & 94.71 & 89.67 & 81.86 \\[-0.05cm]
4 & 65 & 170 & 88.75 & 95.21 & 89.42 & 81.61 \\[-0.05cm]
5 & 25 & 180 & 88.75 & 95.21 & 89.42 & 81.61 \\[-0.05cm]
6 & 50 & 130 & 88.66 & 94.71 & 89.67 & 81.61 \\[-0.05cm]
7 & 50 & 140 & 88.66 & 94.71 & 89.67 & 81.61 \\[-0.05cm]
8 & 35 & 180 & 88.66 & 95.21 & 89.42 & 81.36 \\[-0.05cm]
9 & 45 & 120 & 88.58 & 94.96 & 89.42 & 81.36 \\[-0.05cm]
10 & 20 & 130 & 88.58 & 94.96 & 89.42 & 81.36 \\[-0.05cm]
\hline
\hline
\end{tabular}
\end{center}
\label{table5}
\end{table}
We adopt a grid search strategy to investigate the optimal HoGe dimension and the number of text and non-text templates. The dimension of the HoGe feature vector refers to the number of histogram bins within the HoGe which we change from 10 to 180 with a step of 10. The number of text and no-text templates is varied from 5 to 100 with a step of 5. Hence, the full combination of the two sets of parameters thus gives 360 (18x20) settings. Different from the proposal generation optimization, we limit the maximum proposal number at 2000 (a reasonable number by compromising recall and the ensuing computational cost \cite{ObjDT}) for the evaluation of proposal recalls. Under each parameter setting, an average recall is computed for all images within the validation set when three IoU thresholds of 0.5, 0.7 and 0.8 are used. Additionally, 80\% training images of the ICDAR2003 and the ICDAR2013 datasets are used for training and the rest 20\% are used for validation in our study.
\par
Table \ref{table5} shows the first ten best-performing settings of the two parameters which are sorted according to the average recall under the three IoU thresholds. As Table \ref{table5} shows, the recalls are quite close to each other around the best parameter settings. In our implemented system, we select the 25 text/non-text templates and template dimension of 120, i.e., the setting (25, 120), as a compromise of detection recall and detection efficiency.
\section{Automatic scene text reading}
We also develop an end-to-end scene text reading system by integrating the proposed pooling based proposal technique and a state-of-the-art scene text recognition model \cite{JarData} which is trained on generic 90k words list and recognizes words directly. Given an image, a number of scene text proposals are first determined by using the proposed pooling based technique. Each detected proposal is then fed to the word recognition model \cite{JarData} to derive a word recognition score, and it will be discarded if the recognition score is too low or the recognized word is not in the lexicon list. After that, non-maximum-suppression (nms) is applied to keep the proposal with the maximum score and remove those with lower scores. Additionally, a word based nms is also implemented to remove duplicate proposals of the same word. In particular, only a proposal that has the maximum recognition score is kept as the reading output when more than one proposals overlap with each other and produce the same recognized word. More details will be discussed in Section \ref{E2EEval}.
\section{Experiments and results}
\subsection{Experiment setup and evaluation metrics}
Given the very similar performance under different label assignment and pooling methods as described in Section \ref{PoolingLabeling}, we label image edges by their searching order and use the max-pooling in the ensuing evaluations and benchmarking with the state-of-the-arts. In addition, the size of the pooling window is fixed at 1-by-3 and the strides are set at 2 and 1 pixels in the horizontal and vertical directions as described in Section \ref{proposal generation optimization}. Further, 25 feature templates are used for both text and non-text classes and the dimension of the HoGe is fixed at 120 bins as discussed in Section \ref{ranking optimization}.
\par
Three datasets are used in evaluations and comparisons including the focused scene text dataset used in the Robust Reading Competition 2015 (ICDAR2015) \cite{ICDAR2013}, the Street View Text (SVT) \cite{SVT} and the MSRA-TD500 \cite{Re3}. The ICDAR2015 contains 229 training images and 233 testing images, and the SVT contains 101 training images and 249 testing images. The scene text images in both datasets suffer from a wide range of image degradation but most texts are horizontal and printed in English. The MSRA-TD500 contains 500 images including 300 training images and 200 test images, where scene texts are in arbitrary orientations and a mixture of English and Chinese. It is used to show that the proposed technique can work with scene texts in different orientations and languages.
\par
The proposal quality is evaluated by the recall rate, the number of proposal selected and the computation time. The criterion is that a better proposal technique is capable of achieving a higher recall rate with a smaller number of proposals and a lower computation cost. While benchmarking different proposal techniques, the recall rate can be compared by fixing the number of proposals, says 2000 as a widely adopted number \cite{ObjDT}. In addition, the recall rate is also affected by the IoU threshold where a larger IoU usually leads to a lower recall rate. For the scene text reading system, two evaluation criteria are adopted as used in the robust reading competitions \cite{Wang}, namely, the
end-to-end based and the spotting based. The end-to-end based evaluation focuses on alphanumeric words, while the spotting based evaluation targets words consisting of letters only. In particular, a correct word should have at least three characters (otherwise ignored), and only proposals that have over 50\% overlap with corresponding ground truth boxes and contain correctly recognized words are counted as true positives.
\subsection{Comparisons with state-of-the-arts}
The proposed technique (MPT) is compared to several state-of-the-art scene text proposal techniques including Simple Selective Search for Text Proposal (TP) \cite{GomezE2E}, Symmetry Text Line (STL) \cite{STL}, and DeepText (DT) \cite{TDCNN3DT}. In addition, we also compare the MPT with several state-of-the-art generic object proposal methods including EdgeBox (EB) \cite{EB}, Geodesic (GOP) \cite{GOP}, Randomized Prime (RP) \cite{RP}, and Multiscale Combination Grouping (MCG) \cite{MCG}. All these techniques are implemented in Matlab except TP and STL which are implemented in C++. All evaluations are performed on a HP workstation with a Intel Xeon 3.5GHz x 12 CPU and 32GB Ram memory.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=4.7in]{FigNew1.pdf}
\end{center}
\caption{Comparison of the proposed Max-pooling based scene text proposal technique (MPT) with state-of-the-art technique including text-specific proposal techniques: Simple Selective Search for Scene Text Proposal (TP) \cite{GomezE2E}, Symmetry Text Line (STL) \cite{STL}, and Deep Text (DT) \cite{TDCNN3DT} and generic object proposal techniques: EdgeBoxes (EB) \cite{EB}, Geodesic (GOP) \cite{GOP}, Randomized Prime (RP) \cite{RP}, and Multiscale Combination Grouping (MCG) \cite{MCG}. The evaluation is performed on the ICDAR2015 dataset by varying the IoU and the number of selected proposals.}
\centering
\label{fig_004}
\end{figure}
\par
Fig. \ref{fig_004} shows experimental results on the ICDAR2015 dataset. The graph on the left shows proposal recalls when the IoU thresholds changes from 0.5 to 1 with a step of 0.05 and 2000 proposals selected from each image. The graph on the right shows recalls while the number of proposals varies from 1 to 2000 when the IoU threshold is fixed at 0.8. As the graph on the left shows, DT demonstrates competitive recalls when the IoU threshold lies between 0.5 and 0.6, but its recall drops dramatically when the IoU threshold increases. TP and STL are stabler than DT as they both use hand-craft text specific features, but their recalls are lower than the proposed MPT except when the IoU threshold is large than 0.9, which is seldom adopted in real systems. In the right graph, the proposed MPT outperforms most compared techniques when the number of proposals changes. In fact, it even outperforms DT which adopts a deep learning approach. Note that the recalls of DT are only evaluated in the range of 100-500 proposals because it set the maximum proposal number at 500.
\par
\begin{table}[!t]
\small
\caption{Recall (\%) and processing time (in seconds) of the proposed technique (MPT) and state-of-the-art techniques under different IoU on the ICDAR2015 dataset. The Nppb denotes an average number of proposals each technique needs to achieve its presented recalls.}
\begin{center}
\begin{tabular}{c c c c c c}
\hline
\hline
Method & IoU: 0.5 & IoU: 0.7 & IoU: 0.8 & Nppb & times (s) \\[-0.05cm]
\hline
\hline
MPT & \textbf{96.16} & \textbf{89.59} & \textbf{81.83} & 1465 & 3.55\\[-0.05cm]
TP \cite{GomezE2E} & 84.47 & 71.32 & 65.11 & 1907 & 5.17\\[-0.05cm]
STL \cite{STL} & 79.78 & 62.04 & 49.91 & 1034& 361.3\\[-0.05cm]
DT \cite{TDCNN3DT} & 88.5 & 67 & 4 & \textbf{500} & $-$\\[-0.05cm]
EB \cite{EB} & 76.93 & 48.81 & 27.67 & 1968& \textbf{1.02}\\[-0.05cm]
GOP \cite{GOP} & 45.68 & 19.39 & 11.76 &1040 & 4.3\\[-0.05cm]
RP \cite{RP} & 66.91 & 36.95 & 19.3 & 1917 & 10.07\\[-0.05cm]
MCG \cite{MCG} & 56.16 & 36.21 & 27.02 & 550 & 28.92\\[-0.05cm]
\hline
\hline
\end{tabular}
\end{center}
\label{table1}
\end{table}
\begin{table}[!t]
\small
\caption{Recall (\%) and processing time (in seconds) of the proposed technique (MPT) and state-of-the-art techniques under different IoU on the SVT dataset. The Nppb denotes an average number of proposals each technique needs to achieve its presented recalls.}
\begin{center}
\begin{tabular}{c c c c c c}
\hline
\hline
Method & IoU: 0.5 & IoU: 0.7 & IoU: 0.8 & Nppb & times (s) \\[-0.05cm]
\hline
\hline
MPT & \textbf{87.64} & 46.48 & 20.87 & 1780 & 3.19\\[-0.05cm]
TP \cite{GomezE2E} & 74.65 & 42.5 & 21.33 & 1972 & 5.94\\[-0.05cm]
STL \cite{STL} & 77.13 & 31.07 & 10.36 & 1358 & 433.82\\[-0.05cm]
EB \cite{EB} & 76.35 & \textbf{47.45} & \textbf{23.96} & 2000 & \textbf{1.28}\\[-0.05cm]
GOP \cite{GOP} & 52.09 & 18.24 & 6.8 & 1117 & 3.78\\[-0.05cm]
RP \cite{RP} & 62.6 & 27.05 & 12.21 & 2000 & 8.02\\[-0.05cm]
MCG \cite{MCG} & 54.71 & 24.27 & 8.66 & \textbf{557} & 14.97\\[-0.05cm]
\hline
\hline
\end{tabular}
\end{center}
\label{table2}
\end{table}
We also studied the number of needed proposals for good recalls and computational cost. Tables \ref{table1} and \ref{table2} show the experimental results on the test images of the dataset ICDAR2015 and SVT. It can be seen that the proposed MPT outperforms other proposal techniques in most cases for both datasets. TP is also competitive but it requires a larger number of proposals and also higher computational cost. EB is the most efficient and MCG requires a smaller number of proposals but both methods have low recalls under different IoU thresholds.
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.8in]{Fig9.pdf}
\end{center}
\caption{
Performance of the proposed MPT and two state-of-the-art techniques STL and TP on several images in the ICDAR2015 and SVT datasets. The numbers under each image correspond to the position of proposals within the ranked proposal list (the smaller, the better). Note there are more than one good proposal in the proposal list for each ground truth box, and we only show the best proposal for each ground truth box for the illustration purpose.}
\centering
\label{fig_005}
\end{figure}
Fig. \ref{fig_005} illustrates the performance of the proposed MPT and compares it with two state-of-the-art techniques TP and STL (green boxes indicate proposals and red boxes indicate ground-truth). Several sample images are selected from the ICDAR2015 and SVT datasets which suffer from different types of degradations including text size variation (the first images from left), uneven illumination (the second image), ultra-low contrast (the third image), and perspective distortion (the fourth and fifth images). A series of numbers are shown under each image which correspond to the position of each proposal within the ranked proposal list (the smaller, the better). As Fig. \ref{fig_005} shows, the proposed MPT can deal with different types of image degradation and demonstrates superior proposal performance as compared with TP and STL. It should be noted that Fig. \ref{fig_005} only shows good proposals that have over 80\% overlap with ground-truth boxes. In addition, each text ground truth has more than one good proposal and Fig. \ref{fig_005} only shows the proposal which is ranked at the front-most with the smallest index number within the ranked proposal list.
\par
\begin{figure}[t]
\begin{center}
\includegraphics[width=4.8in]{OrientFig1.pdf}
\end{center}
\caption{For the sample images from the MSRA-TD500 dataset \cite{Re3}, the figure shows that the proposed technique can work for scene texts in arbitrary orientations and different languages in most cases (The red oriented rectangles are ground truth boxes and the green oriented rectangles are proposals.). It fails in rare cases when scene texts are vertically oriented as shown in the two images in the last column.}
\centering
\label{Orientfig_001}
\end{figure}
The proposed technique also can detect scene texts in different orientations and languages. We demonstrate this capability using the MSRA-TD500 dataset \cite{Re3} that contains scene texts in arbitrary orientations and a mixture of English and Chinese. Experiments show that recalls of 88.14\%, 83.33\% and 75.77\% are obtained under the IoU thresholds of 0.5, 0.7 and 0.8. These recalls are comparable to those achieved over the ICDAR2015 and SVT datasets (as shown in Tables \ref{table1} and \ref{table2}), where most texts are almost horizontal and printed in English. Fig. \ref{Orientfig_001} shows several sample images from the MSRA-TD500 that capture English and Chinese texts in arbitrary orientations, as well as text proposals by our proposed technique. As Fig. \ref{Orientfig_001} shows, the proposed technique is capable of detecting English and Chinese texts when there exist certain overlaps between neighbouring characters in the vertical direction. It fails when scene texts are vertically oriented as shown in the two images in the last column. Note that a maximum of 2000 proposals are generated in each image and the proposals shown in Fig. \ref{Orientfig_001} are those having the best overlapping with the ground truth.
\par
The superior performance of the MPT is largely attributed to the proposed pooling based grouping strategy that captures the exact text layout and appearance in scenes, i.e. characters are usually closer to each other (as compared with neighbouring non-text objects) forming words and text lines. In fact, the proposed grouping strategy can also handle texts with broken edges as far as they have certain overlap in the vertical direction. As a comparison, the EdgeBox (EB) \cite{EB} makes use of image edges similarly with a much lower recall rate, largely due to different grouping strategies. Besides the proposed grouping strategy, the HoGe based proposal ranking helps to shift scene text proposals to the front of the sorted list which also contributes to the superior performance of the proposed MPT technique when a limited number of proposals are selected.
\subsection{End-to-end and word spotting}
\label{E2EEval}
\begin{table}[!t]
\small
\caption{Word spotting performance of the developed end-to-end scene text reading system (MPT) and other state-of-the-art systems, including proposal based systems (*) and CNN-based systems for the ICDAR2015 dataset and the SVT dataset.}
\begin{center}
\begin{tabular}{c c cccccc}
\hline
\hline
\multirow{3}{*}{Method} & \multirow{3}{*}{SVT-50} & \multicolumn{6}{c}{Words Spotting}\\[-0.05cm]
\cline{3-8}
& & \multicolumn{3}{c}{strong} & \multicolumn{3}{c}{weak}\\[-0.05cm]
\cline{3-8}
& & R & P & F & R & P & F \\[-0.05cm]
\hline
\hline
EB\_Sys$^*$ & 72.84 & 55.26 & 66.81 & 60.49 & 54.79 & 57.48 & 56.10 \\[-0.05cm]
STL\_Sys$^*$ & 75.02 & 61.80 & 85.32 & 71.68 & 61.45 & 81.30 & 69.99 \\[-0.05cm]
TP\_Sys$^*$ & 76.41 & 66.47 & 89.47 & 79.27 & 65.07 & 82.40 & 72.72 \\[-0.05cm]
Jar-E2E\cite{JarE2E} & 68 & 86.68 & 94.64 & 90.49 & - & - & - \\[-0.05cm]
ConvLSTM \cite{E2E11} & $-$ & 84.93 & \textbf{98.91} & 91.39 & 84 & \textbf{97.29} & 90.16\\[-0.05cm]
DeepTextSpotter\cite{DTSpotter} &$-$ & $-$ & $-$ & 92 & $-$ & $-$ & 89 \\[-0.05cm]
TextBoxes \cite{TextBoxes} & 84 & \textbf{90.77} & 97.25 & \textbf{93.90} & 87.38 & 97.02 & \textbf{91.95} \\[-0.05cm]
\hline
\textbf{Proposed MPT} & 84 & 88.20 & 97.55 & 92.64 & \textbf{87.85} & 95.31 & 91.43 \\[-0.05cm]
\hline
\hline
\end{tabular}
\end{center}
\label{T3}
\end{table}
\begin{table}[!t]
\small
\caption{End-to-End performance of the developed scene text reading system (MPT) and other state-of-the-art systems, including proposal based systems (*) and CNN-based systems for the ICDAR2015 dataset.}
\begin{center}
\begin{tabular}{c cccccc}
\hline
\hline
\multirow{3}{*}{Method} & \multicolumn{6}{c}{End-to-end Scene Text Reading}\\[-0.05cm]
\cline{2-7}
& \multicolumn{3}{c}{strong} & \multicolumn{3}{c}{weak} \\[-0.05cm]
\cline{2-7}
& R & P & F & R & P & F \\[-0.05cm]
\hline
\hline
EB\_Sys$^*$ & 52.67 & 65.80 & 58.51 & 52.24 & 56.69 & 54.37 \\[-0.05cm]
STL\_Sys$^*$ & 59.65 & 85.07 & 70.13 & 59.43 & 81.10 & 68.60 \\[-0.05cm]
TP\_Sys$^*$ & 63.90 & 88.12 & 74.08 & 62.70 & 81.21 & 70.77 \\[-0.05cm]
Jar-E2E \cite{JarE2E} & 82.12 & 91.05 & 86.35 & - & - & - \\[-0.05cm]
ConvLSTM \cite{E2E11} & 79.39 & \textbf{96.68} & 87.19 & 79.28 & 94.91 &86.39\\[-0.05cm]
DeepTextSpotter\cite{DTSpotter}& $-$ & $-$ & 89 & $-$ & $-$ & 86 \\[-0.05cm]
TextBoxes\cite{TextBoxes} & \textbf{87.68} & 95.83 & \textbf{91.57} & \textbf{84.51} & \textbf{95.44} & \textbf{89.65} \\[-0.05cm]
\hline
\textbf{Proposed MPT} & 84.08 & 96.25 & 89.76 & 83.86 & 93.89 & 88.59 \\[-0.05cm]
\hline
\hline
\end{tabular}
\end{center}
\label{T4}
\end{table}
Tables \ref{T3} and \ref{T4} compare our developed end-to-end system with several state-of-the-art end-to-end scene text reading systems including several CNN-based: Jar-E2E model \cite{JarE2E}, ConvLSTM \cite{E2E11}, DeepTextSpotter\cite{DTSpotter}, and TextBoxes \cite{TextBoxes} as well as several proposal based: EB\_Sys, TP\_Sys, STL\_Sys which are constructed by combining EB, TP, and STL based scene text proposal techniques with Jarderberg's scene text recognition model. The comparisons are based on precision, recall and f-measure on the ICDAR2015 dataset and the SVT dataset. As the two tables show, the performance of the proposed system is clearly better than other proposal based systems and also comparable to the CNN-based systems. Note that the TextBoxes \cite{TextBoxes} trains two dedicated networks for detection and recognition, and it was trained using a huge amount images including images in the SynthText \cite{SynText} (containing 800,000 images) as well as training images in the ICDAR2011 dataset and the ICDAR2013 dataset. As a comparison, our proposed system was trained using 479 training images in the ICDAR2003 dataset and the ICDAR2013 dataset only.
\par
Fig. \ref{fig_006} shows a number of sample images that illustrate the performance of our developed end-to-end scene text reading system. As Fig. \ref{fig_006} shows, the proposed technique is capable of detecting and recognizing challenging texts with small text size (the first image in the first row), poor illumination and motion blur (the second images in the first and second rows), perspective distortion (the second image in the third row and the third image in second row). The superior scene text reading performance is largely due to the robustness of the proposed scene text proposal technique and the integrated scene text recognition model. Note that the proposed technique may fail when scene texts have ultra-low contrast or are printed in certain odd styles as illustrated in the sample images in the last row.
\begin{figure}
\begin{center}
\includegraphics[width=4.8in]{Fig10.pdf}
\end{center}
\caption{
Several scene text detection and recognition examples where the proposed scene text reading system succeeds (the first three rows) and fails (the last row): The red boxes are ground truth and the green boxes are detection boxes by our proposed technique. The boxes with green-color background give the recognition results (words containing less than three characters are ignored).}
\centering
\label{fig_006}
\end{figure}
\section{Conclusion}
This paper presents a pooling based scene text proposal technique as well as its application to end-to-end scene text reading. The scene text proposal technique is inspired by the CNN pooling layer which is capable of grouping image edges into words and text lines accurately and efficiently. A novel score function is also designed which is capable of ranking generated proposals according to their probabilities of being text and accordingly helps to reduce the number of false-alarm proposals greatly. Further, the proposed proposal technique does not rely on those heuristic thresholds/parameters such as text sizes, inter-character distances, etc. that are widely used in many existing techniques. Extensive experiments show that the pooling based proposal technique achieves superior performance as compared with state-of-the-arts. In addition, the integration of the pooling based proposal technique into an end-to-end scene text reading system also demonstrates state-of-the-art scene text reading performance.
\section*{Authors' Biographies}
\textbf{Dinh} is a PhD student at Sorbonne University – University Pierre and Marie CURIE, France. His current research works are in Image \& Pervasive Access Lab (IPAL, UMI2955, CNRS) and have collaboration with Institute for Infocomm Research (I2R, A-STAR), Singapore. His major research interests are visual understanding and machine learning.
\par
\textbf{Shijian} is an Assistant Professor with School of Computer Science \& Engineering, the Nanyang Technological University, Singapore. His major research interests include image and video analytic, visual intelligence, and machine learning. He published more than 80 international journals and conference papers and co-authored over 10 patents in these research areas.
\par
\textbf{Shangxuan-Tian} is a Senior Researcher at Tencent, China. Previously, he worked as a Research Scientist in the Institute for Infocomm Research, Singapore. He received his Ph.D. degree in School of Computing, National University of Singapore. His research interests include object detection and recognition, text understanding in scene images.
\par
\textbf{Nizar-Ouarti} is Associate Professor at Sorbonne UPMC in France. He is recently in CNRS delegation in the IPAL laboratory in Singapore. He received a PhD of College de France in 2007. He was postdoctoral researcher at INRIA. His topics of interest are ego-motion, computer vision and robotics.
\par
\textbf{Mounir-Mokhtari} is a Professor at Institut MINES TELECOM , France, Director of IPAL-CNRS French-Singaporean joint lab, Singapore, and Research Associate at CNRS-LIRMM Montpellier, France. His background is in human-machine interaction in the domain of Ambient Assistive Living. He has over 100 publications in journals, books and international conferences.
\section*{References}
|
1,116,691,500,444 | arxiv | \section{Introduction}
In the last five decades
pseudo-differential operators
have become a standard tool in the study of partial differential equations.
It is natural to try
to define analogues of the Euclidean pseudo-differential calculus in other settings.
On one hand, while it is always possible to obtain a local calculus on any (connected) manifold,
the question becomes much harder for global calculi.
If, in addition, one requires a notion of symbol,
the quasi inherent context is the one of Lie groups of type~I
where a Plancherel-Fourier analysis is available.
On the other hand,
from the viewpoint of
what can actually be done at the level of operators,
the investigation should start
in the context of Lie groups with polynomial volume-growth
where analysis of integral operators is quite well understood.
Therefore the natural setting to start developing global pseudo-differential calculi
is nilpotent or compact Lie groups, together with their semi-direct products.
The genesis of this idea began quite some time ago;
if a starting line had to be drawn,
it would be in the seventies
with the work of Elias Stein
and his collaborators
Folland, Rotschild, etc.
(see e.g. \cite{folland+stein-1974,rothschild+stein}),
and
continued, in the next decade,
with the work of Beals and Greiner amongst many others.
Their motivation came from the study
of differential operators on CR or contact manifolds,
modelling locally the operators
on homogeneous left-invariant convolution operators on nilpotent groups
(cf \cite{ponge}).
In `practice' and from this motivation,
only nilpotent Lie groups
endowed with some compatible structure of dilations,
i.e. homogeneous groups and, more particularly, graded Lie groups,
are considered.
The latter is also the setting of our present investigation.
Since the seventies,
several global calculi of operators on homogeneous Lie groups
have appeared.
However
they were often calculi of left-invariant operators
with the following notable exceptions
to the authors' knowledge.
Beside Dynin's construction of certain operators
on the Heisenberg group in \cite{dynin,folland-1994},
a non-invariant pseudo-differential calculus on any homogeneous group
was developed in \cite{cggp}
but this is not symbolic since the operator classes are defined via properties of the kernel.
In the revised version of \cite{Tnma},
Taylor describes several (non-invariant) operator calculi
and, in a different direction, he also explains a way to develop symbolic calculi:
using the representations of the group, he defines
a general \emph{quantization}
and the natural \emph{symbols} on any unimodular type I group
(by quantization, we mean a procedure which associates an operator to a symbol).
He illustrates this on the Heisenberg group
and obtains there several important applications for, e.g., the study of hypoellipticity.
He uses the fact that,
because of the properties of the Schr\"odinger representations of the Heisenberg group, a symbol is a family of operators in the Euclidean space,
themselves given by symbols via the Weyl quantization.
Recently,
the attempt at defining suitable classes of Shubin type for these Weyl-symbols led to another
version of the calculus on the Heisenberg group in \cite{bahouri+FK+gallagher_bk2012}\footnote{and in its revised version}.
Recently as well,
using the global quantization procedure noted in \cite{Tnma},
the second author and Turunen
developed a global symbolic calculus on any compact Lie group
in \cite{ruzh+turunen_bk2010}.
They successfully defined symbol classes
so that the quantization procedure makes sense
and the resulting operators form an algebra of operators
with properties `close enough' to the one enjoyed by the Euclidean H\"ormander calculus (in fact, in a later work with Wirth \cite{RTW}, they showed that
the calculus in \cite{ruzh+turunen_bk2010} leads to the usual H\"ormander operator classes on $\mathbb R^{n}$ extended to compact connected manifolds).
Their approach is valid for any compact Lie group
whereas
the calculus of \cite{bahouri+FK+gallagher_bk2012}
is very specific to the Heisenberg group.
The crucial and new ingredient in the definition of symbol classes in \cite{ruzh+turunen_bk2010} was defining \emph{difference operators}
in order to replace the Euclidean derivatives in the Fourier variables.
These difference operators allow expressing the
pseudo-differential behaviour directly on the group.
In our present investigations, we build upon this notion
to study operators in the nilpotent setting.
However it is not possible to extend readily the results of the compact case developed in
\cite{ruzh+turunen_bk2010}
to the nilpotent context.
Some technical difficulties appear because,
for example, the dual of $G$ is no longer discrete
and
the unitary irreducible representations are almost all infinite dimensional.
More problematically there is no Laplace-Beltrami operator
and
one expects to replace it
by a sub-Laplacian on stratified Lie groups
or, more generally, by a positive Rockland operator $\mathcal R$ on graded Lie groups;
such operators are not central.
Hence new technical ideas are needed to develop
a pseudo-differential calculi on graded Lie groups
using the natural quantization
together with the notion of difference operators from \cite{ruzh+turunen_bk2010}.
The results that we have obtained in our investigation of this question so far
were presented in the talk given by the first author at the conference \emph{Fourier analysis and pseudo-differential operators}, Aalto University, 25-30 June, 2012.
They are the following
(here $1\geq \rho\geq \delta \geq 0$):
\begin{description}
\item[(R1)]
The symbol classes form an algebra of operators $\cup_{m\in \mathbb R} S^m_{\rho,\delta}$ stable by taking the adjoint.
\item[(R2)]
Let $\rho\not=0$.
The operators obtained by quantization
from $\cup_{m\in \mathbb R} S^m_{\rho,\delta}$
form an algebra of operators $\cup_{m\in \mathbb R} \Psi^m_{\rho,\delta}$
stable by taking the adjoint.
\item[(R3)]
The set of operators $\cup_{m\in \mathbb R} \Psi^m_{\rho,\delta}$
contains the left-invariant calculus.
\item[(R4)]
The kernels are of Calderon-Zygmund type on homogeneous Lie groups;
in particular our operators of order 0 are more singular than their Euclidean counterparts.
\item[(R5)]
If $\rho\in [0,1)$,
then the operators in $\Psi^0_{\rho,\rho}$ are continuous on $L^2(G)$.
\item[(R6)]
$(\text{\rm Id} +\mathcal R)^{\frac m \nu}\in \Psi^m_{1,0}$, where $\mathcal R$ is a positive Rockland operator
of degree $\nu$, see Section \ref{SEC:RO}.
\item[(R7)]
Positive operators of the calculus satisfy sharp G{\aa}rding inequalities.
\end{description}
As a consequence from Results (R2), (R5) and (R6), if $\rho\not=0$,
any pseudo-differential operator is continuous on the Sobolev spaces
with the loss of derivatives being controlled by the order.
All those properties justify, from our viewpoint,
the choice of vocabulary of pseudo-differential calculi.
Since the conference in Aalto University of June 2012,
these results, together with their complete proof
and other progress made by the authors on the subject have started to be collected in the
monograph \cite{FR}, see also \cite{RF-cras1}.
In this paper, due to the lack of space,
we will state and prove the following parts of Results (1-7).
Result (R1) is proved in Subsection \ref{subsec_1prop_symbols}.
The proof of Result (R2) is given in Subsection \ref{subsec_composition}
but relies on Result (R4) which is stated
in Subsection \ref{subsec_1prop_kernels}
and only partially proved.
In Subsection \ref{subsec_1ex},
Result (R3) is stated and proved
while (R6) is stated in greater generality but not proved in this paper.
The precise statements and proofs of Results (R5) and (R7)
can be found in \cite{FR}.
\medskip
The paper is organised as follows.
In Section \ref{sec_preliminaries},
we explain the precise setting of our investigation for the group,
the Sobolev spaces involved here and the group Fourier transform.
In Section \ref{sec_quantization_S},
we precise our definition of quantization and symbol classes.
In Section \ref{sec_prop_symbol_kernel_op},
we give the properties of the symbols
and of the corresponding kernels and operators stated above.
\medskip
{\it Convention:} All along the paper,
$C$ denotes a constant which may vary from line to line.
We denote by $\lceil r\rceil$ the smallest integer which is strictly greater than the real $r$.
\section{Preliminaries}
\label{sec_preliminaries}
In this section,
we set some notation and recall some known properties
regarding the groups under investigation, the Taylor expansion in this context
and representation theory.
\subsection{The group $G$}
Here we recall briefly the definition of graded nilpotent Lie groups
and their natural homogeneous structure.
A complete description of the notions of graded and homogeneous nilpotent Lie group may be found in \cite[Chapter 1]{folland+stein_bk82}.
We will be concerned with graded Lie groups $G$
which means that $G$ is a connected and simply connected
Lie group
whose Lie Lie algebra $\mathfrak g$
admits an $\mathbb N$-gradation
$\mathfrak g= \oplus_{\ell=1}^\infty \mathfrak g_{\ell}$
where the $\mathfrak g_{\ell}$, $\ell=1,2,\ldots$
are vector subspaces of $\mathfrak g$,
almost all equal to $\{0\}$
and satisfying
$[\mathfrak g_{\ell},\mathfrak g_{\ell'}]\subset\mathfrak g_{\ell+\ell'}$
for any $\ell,\ell'\in \mathbb N$.
This implies that the group $G$ is nilpotent.
Examples of such groups are the Heisenberg group and more generally any stratified groups (which by definition correspond to the case $\mathfrak g_1$ generating the full Lie algebra $\mathfrak g$).
Let $\{X_1,\ldots X_{n_1}\}$ be a basis of $\mathfrak g_1$ (this basis is possibly reduced to $\{0\}$), let
$\{X_{n_1+1},\ldots, X_{n_1+n_2}\}$ a basis of $\mathfrak g_2$
and so on, so that we obtain a basis
$X_1,\ldots, X_n$ of $\mathfrak g$ adapted to the gradation.
Via the exponential mapping $\exp_G : \mathfrak g \to G$, we identify
the points $(x_{1},\ldots,x_n)\in \mathbb R^n$
with the points $x=\exp_G(x_{1}X_1+\cdots+x_n X_n)$ in $G$.
Consequently we allow ourselves to denote by $C(G)$, $\mathcal D(G)$ and $\mathcal S(G)$ etc...
the spaces of continuous functions, of smooth and compactly supported functions or
of Schwartz functions on $G$ identified with $\mathbb R^n$.
This basis also leads to a corresponding Lebesgue measure on $\mathfrak g$ and the Haar measure $dx$ on the group $G$.
The coordinate function $x=(x_1,\ldots,x_n)\in G\mapsto x_j \in \mathbb R$
is denoted by $x_j$.
More generally we define for every multi-index $\alpha\in \mathbb N_0^n$,
$x^\alpha:=x_1^{\alpha_1} x_2 ^{\alpha_2}\ldots x_{n}^{\alpha_n}$,
as a function on $G$.
Similarly we set
$X^{\alpha}=X_1^{\alpha_1}X_2^{\alpha_2}\cdots
X_{n}^{\alpha_n}$ in the universal enveloping Lie algebra $\mathfrak U(\mathfrak g)$ of $\mathfrak g$.
For any $r>0$,
we define the linear mapping $D_r:\mathfrak g\to \mathfrak g$ by
$D_r X=r^\ell X$ for every $X\in \mathfrak g_\ell$, $\ell\in \mathbb N$.
Then the Lie algebra $\mathfrak g$ is endowed
with the family of dilations $\{D_r, r>0\}$
and becomes a homogeneous Lie algebra in the sense of
\cite{folland+stein_bk82}.
The weights of the dilations are the integers $\upsilon_1,\ldots, \upsilon_n$ given by $D_r X_j =r^{\upsilon_j} X_j$, $j=1,\ldots, n$.
The associated group dilations are defined by
$$
r\cdot x
:=(r^{\upsilon_1} x_{1},r^{\upsilon_2}x_{2},\ldots,r^{\upsilon_n}x_{n}),
\quad x=(x_{1},\ldots,x_n)\in G, \ r>0.
$$
In a canonical way this leads to the notions of homogeneity for functions and operators.
For instance
the degree of homogeneity of $x^\alpha$ and $X^\alpha$,
viewed respectively as a function and a differential operator on $G$, is
$[\alpha]=\sum_j \upsilon_j\alpha_{j}$.
Indeed, let us recall
that a vector of $\mathfrak g$ defines a left-invariant vector field on $G$
and more generally
that the universal enveloping Lie algebra of $\mathfrak g$
is isomorphic with the left-invariant differential operators;
we keep the same notation for the vectors and the corresponding operators.
Recall that a \emph{homogeneous norm} on $G$ is a continuous function $|\cdot| : G\rightarrow [0,+\infty)$ homogeneous of degree 1
on $G$ which vanishes only at 0.
Any homogeneous norm satisfies a triangular inequality up to a constant.
Any two homogeneous norms are equivalent.
For example
\begin{equation}
\label{eq_norm_||nuo}
|x|_{\nu_o}:=\left(\sum_{j=1}^n x_j^{2\frac{\nu_o}{\upsilon_j}}\right)^{\frac1{2\nu_o}}
\end{equation}
with $\nu_o$ a common multiple to the weights $\upsilon_1,\ldots,\upsilon_n$.
Various aspects of analysis on $G$ can be developed in a comparable way with the Euclidean setting
\cite{coifman+weiss-LNM71}, sometimes replacing the topological dimension
$$
n:=\dim G =\sum_{\ell=1}^\infty\dim \mathfrak g_\ell
,
$$
of the group $G$ by its homogeneous dimension
$$
Q:=\sum_{\ell=1}^\infty \ell \dim \mathfrak g_\ell
= \upsilon_1 +\upsilon_2 +\ldots +\upsilon_n
.
$$
\subsubsection{Taylor expansions on $G$}
\label{subsubsec_Taylor}
In the setting of graded Lie groups one can obtain the left or right mean value theorem and left or right Taylor expansions adapted to the homogeneous structure \cite[Theorem 1.42]{folland+stein_bk82}. Let us give the statement for left invariance. We will need the following definition:
the (left) Taylor polynomial of homogeneous degree $M$ of a function $f\in C^{M+1}(G)$ at a point $x\in G$ is by definition the polynomial $P_{x,M}^{(f)}$ satisfying
$$
X^\alpha P_{x,M}^{(f)} (0)=
\left\{\begin{array}{ll}
X^\alpha f(x)
& \mbox{whenever}\quad \alpha\in \mathbb N^n \ \mbox{with} \ [\alpha]\leq M ,
\\
0 & \mbox{if} \ [\alpha]>M
.
\end{array}\right.
$$
We also define the remainder to be
$$
R_{x,M}^{(f)}(z):=f(xz) -P_{x,M}^{(f)}(z).
$$
\begin{proposition}[Mean value and Taylor expansion \cite{folland+stein_bk82}]
\label{prop_Taylor}
Let us fix a homogeneous norm $|\cdot |$ on $G$.
\begin{enumerate}
\item (Mean value property) There exist positive group constants $C_0$ and $b$ such that for any function $f\in C^1(G)$, we have
$$
|f(xy) -f(x)| \leq C_0 \sum_{j=1}^n |y|^j \sup_{|z|\leq b |y|} |X_j f(xz)|.
$$
\item (Taylor expansion) For each $M\in \mathbb N_0$ there exist positive group constants $C_M$ such that for any function $f\in C^{M+1}(G)$, we have
$$
\forall y\in G\qquad
|R_{x,M}^{(f)}(y)| \leq C_M
\sum_{\substack{[\alpha] > M\\
|\alpha| \leq \lceil M\rceil}}
|y|^{[\alpha]} \sup_{|z|\leq b^{M+1} |y|} |X^\alpha f(xz)|
,
$$
where $\lceil M\rfloor :=\max\{ |\beta|: \beta\in \mathbb N_0^n\
\mbox{with}\ [\beta]\leq M\}$.
\end{enumerate}
\end{proposition}
The control can be improved in the stratified case
(again see \cite{folland+stein_bk82})
but we present here the more general case of the graded groups.
\begin{remark}\label{rem_vector_valued_taylor}
Proposition \ref{prop_Taylor}
extends easily to functions which are vector valued in a Banach space, replacing the modulus by operator norms.
\end{remark}
The Taylor polynomials can be described in the following way.
Let $(q_\alpha)_{\alpha\in \mathbb N^n}$ be the basis of polynomials
obtained from by the duality
$\langle X,p\rangle:=Xp(0)$ where $X\in \mathfrak U(\mathfrak g)$ and $p$ is a polynomial.
This means that the $q_\alpha$'s are the polynomials satisfying
\begin{equation}
\label{def_qalpha}
\forall \alpha,\beta\in \mathbb N^n_0\qquad
X^{\beta} q_{\alpha}(0)
=
\left\{\begin{array}{ll}
0&\mbox{if}\ \alpha \not = \beta
\\
1&\mbox{if}\ \alpha = \beta
\end{array}
\right. .
\end{equation}
We can then write the Taylor polynomial as
$$
P_{x,M}^{(f)} =\sum_{[\alpha]\leq M} X^\alpha f(x) q_\alpha
.
$$
We will need the following properties of the polynomials $(q_\alpha)$
defined via \eqref{def_qalpha}:
\begin{lemma}
\label{lem_property_qalpha}
\begin{itemize}
\item
Each polynomial $q_\alpha$ is homogeneous of degree
$[\alpha]$.
Moreover, $(q_\alpha)_{[\alpha]=d}$ is a basis of the space of homogeneous polynomials of degree $d$.
\item
For any $\alpha_1,\alpha_2\in \mathbb N_0^n$,
the polynomial
$q_{\alpha_1}q_{\alpha_2}$ can be written as a linear combination
of $q_{\alpha}$ with $[\alpha]=[\alpha_1]+[\alpha_2]$.
\item For any $\alpha\in \mathbb N_0^n$ and $x,y\in G$,
$$
q_\alpha(xy) = \sum_{[\alpha_1] + [\alpha_2] =[\alpha] } c_{\alpha_1,\alpha_2} q_{\alpha_1}(x) q_{\alpha_2}(y) ,
$$
where the coefficients $c_{\alpha_1,\alpha_2}$ are real and, moreover,
$$
c_{\alpha_1,0}=\left\{\begin{array}{ll}
1&\mbox{if}\ \alpha_1 =\alpha\\
0&\mbox{otherwise}
\end{array}\right.
\quad\mbox{and}\quad
c_{0,\alpha_2}=\left\{\begin{array}{ll}
1&\mbox{if}\ \alpha_2 =\alpha\\
0&\mbox{otherwise}
\end{array}\right.
.
$$
\end{itemize}
\end{lemma}
\begin{proof}
Clearly $(q_\alpha)_{[\alpha]=d}$ is the dual basis of $(X^\beta)_{[\beta]=d}$.
The first point follows.
The second point is a direct consequence of the first together with the homogeneity of $q_{\alpha_1}q_{\alpha_2}$.
The third point is a consequence of the homogeneity in $x$ and in $y$ and of
the Baker-Campbell-Hausdorff formula.
\end{proof}
\subsection{The unitary dual and the group Fourier transform}
\label{SEC:udft}
We denote by $\widehat{G}$ the unitary dual of the group $G$, that is,
the set of (strongly continuous) unitary irreducible representations modulo unitary equivalence.
We will often identify a unitary irreducible representation $\pi$ of $G$ and its equivalence class;
we denote the representation Hilbert space by $\mathcal H_\pi$
and the subspace of smooth vectors by $\mathcal H_\pi^\infty$.
The group Fourier transform of a function $f\in L^1(G)$
at $\pi\in \widehat{G}$ is the bounded operator $\widehat f(\pi)$
(sometimes this will be also denoted by $\pi(f)$ for longer expressions) on $\mathcal H_\pi$ given by
$$
(\widehat f(\pi) v_1, v_2)_{\mathcal H_\pi} := \int_G f(x) (\pi(x)^* v_1, v_2)_{\mathcal H_\pi} dx
,\quad v_1,v_2\in \mathcal H_\pi .
$$
One can readily see the equality $\widehat{f_1*f_2}(\pi)=\widehat f_2(\pi) \widehat f_1(\pi)$.
The group Fourier transform of a vector $X\in \mathfrak g$
at $\pi\in \widehat{G}$ is the operator $\pi(X)$ on $\mathcal H_\pi^\infty$ given by
$$
(\pi(X) v_1,v_2)_{\mathcal H_{\pi}} := \partial_{s=0} (\pi(e^{sX} )v_1,v_2)_{\mathcal H_\pi}
,\quad v_1,v_2\in \mathcal H_\pi^\infty .
$$
Setting $\pi(X^\alpha)=\pi(X)^\alpha$,
this yields the definition of the group Fourier transform of any element of $\mathfrak U(\mathfrak g)$.
With this notation we have for any $\alpha\in \mathbb N_0^n$,
$$
\widehat{X^\alpha f}(\pi)=\pi(X)^\alpha \widehat f(\pi)=\pi(X^\alpha)\widehat f(\pi)
\ \mbox{and}\
\widehat{\tilde X^\alpha f}(\pi)=\widehat f(\pi)\pi(X)^\alpha =\widehat f(\pi)\pi(X^\alpha )
\, ,
$$
with the convention for $\alpha=0$ that $\pi(X^0)=\pi(I)=I =\pi(X)^0$.
\medskip
The properties above help in the systematic computations of certain expressions; for example
we see $\pi( \{Xf_1\} *f_2)=\pi(f_2)\pi(X)\pi(f_1)=\pi(\tilde X f_2)\pi(f_1)$
and this is coherent with the direct and more tedious computation $\{Xf_1\} *f_2=f_1 * \{\tilde X f_2\}$.
\subsection{A positive Rockland operator $\mathcal R$}
\label{SEC:RO}
We choose $\mathcal R$ a positive (left) Rockland operator of homogeneous degree $\nu$.
Let us recall that being a Rockland operator means that
$\mathcal R$ is a differential operator on $G$
which is left-invariant and homogeneous of degree $\nu$
and such that for every non-trivial irreducible representation $\pi$ of $G$,
the operator
$\pi(\mathcal R)$ is injective on smooth vectors
(see Section
\ref{SEC:udft} for the definition of $\pi(\mathcal R)$);
being positive means
$$
\forall f\in \mathcal S(G) \qquad
(\mathcal R f,f)_{L^2(G)} \geq 0
.
$$
Here as usual
$$
(f_1,f_2)_{L^2(G)} =\int_G f_1(x) \overline{f_2(x)} dx .
$$
In the stratified case, we choose $\mathcal R= - \mathcal L$
where $\mathcal L$ is the sub-Laplacian
$\sum_{i=1}^{n_1} X_i^2$ (and so $\nu=2$).
In the graded case,
if $\nu_o$ is any common multiple of the weights
$\upsilon_1,\ldots, \upsilon_n$,
then
\begin{equation}
\label{ex_RO}
\sum_{j=1}^n
(-1)^{\frac{\nu_o}{\upsilon_j}}
c_j X_j^{2\frac{\nu_o}{\upsilon_j}}
\quad\mbox{and}\quad
\sum_{j=1}^n
c_j X_j^{4\frac{\nu_o}{\upsilon_j}}
\quad\left(\mbox{with}\ c_j>0\right),
\end{equation}
are positive Rockland operators of degree $2\nu_o$ and $4\nu_o$ respectively.
By the celebrated result of Helffer and Nourrigat \cite{helffer+nourrigat-79},
any Rockland operator is hypoelliptic and satisfies subelliptic estimates.
Furthermore, by \cite[ch. 4.B]{folland+stein_bk82},
any positive Rockland operator $\mathcal R$,
as a differential operator defined on $\mathcal D(G)$,
admits an essentially self-adjoint extension on $L^2(G)$
for which we keep the same notation $\mathcal R$.
Let $E$ denote its spectral measure.
For any measurable function $\phi$ on $[0,\infty)$,
we define the operator
$$
\phi(\mathcal R) := \int_0^\infty \phi(\lambda) dE_\lambda
,
$$
which is invariant under left-translation.
If it maps continuously $\mathcal S(G)\rightarrow \mathcal S'(G)$ (for example if $\phi$ is bounded),
by the Schwartz kernel theorem, it is a convolution operator
with kernel $\phi(\mathcal R)\delta_o \in \mathcal S'(G)$, that is,
$$
\phi(\mathcal R) f = f \ * \ \phi(\mathcal R)\delta_o ,\quad f\in \mathcal S(G)
.
$$
Recall that the group convolution is defined via
$$
f_1*f_2(g) =\int_G f_1(g') f_2({g'}^{-1} g) dg'
, \quad f_1,f_2\in \mathcal S(G)
.
$$
The above hypotheses ensure the following Marcinkiewicz-type properties
proved by A. Hulanicki \cite{hula-1984}.
\begin{proposition}[Hulanicki]
\label{prop_hula}
For any $\alpha,\beta \in \mathbb N_0^n$,
there exists $k=k_{\alpha,\beta}\in \mathbb N_0$ and $C=C_{\alpha,\beta}>0$
such that for any $\phi\in C^\infty([0,\infty))$ we have
$$
\|x^\alpha X^\beta \phi(\mathcal R)\delta_o\|_{L^1(G)} \leq C
\sup_{\lambda\geq 0,\, k_1\leq k}
(1+\lambda)^k \partial_\lambda^{k_1} \left|\phi(\lambda)\right|
.
$$
By this we mean that if the supremum in the right hand side is finite, then
the distribution $x^\alpha X^\beta \phi(\mathcal R)\delta_o$ coincides with an integrable function and
the inequality holds.
\end{proposition}
\begin{remark}\label{rem_cq_hula}
Consequently if $\phi\in \mathcal S(\mathbb R)$ is Schwartz, then the kernel $\phi(\mathcal R)\delta_o$ is also Schwartz on $G$, i.e. $\phi(\mathcal R)\delta_o\in \mathcal S(G)$.
\end{remark}
The proof of Proposition \ref{prop_hula}
relies on using the $\mathcal R$-\emph{heat kernel} $h_t$,
defined as the kernel of
$\exp (-t\mathcal R)$ for each $t>0$.
In \cite{folland+stein_bk82}, it is proved that
the function $h=h_1$ is Schwartz and that
$$
h_t(x) =t^{-\frac Q \nu} h(t^{-\frac 1\nu} x)
.
$$
As in the Euclidean or stratified cases (see \cite{folland-1975}),
we can define \emph{Bessel potentials} associated with a positive Rockland operator $\mathcal R$ of degree $\nu$
via the integral
$$
\mathcal B_a (x)= \frac 1{\Gamma(\frac a \nu)}
\int_0^\infty t^{\frac a \nu -1 } e^{-t} h_t(x) dt
.
$$
Indeed for $a\in \mathbb C$ with $\text{\rm Re}\ a>0$,
this integral converges absolutely for $x\not=0$ and defines the Bessel potential
$\mathcal B_a\in C^\infty(G\backslash \{0\})$ which satisfies:
$$
\|\mathcal B_a\|_{L^1(G)} \leq \frac{\Gamma (\text{\rm Re}\ \frac a \nu)}{|\Gamma (\frac a \nu)|} \|h\|_{L^1}<\infty
, \quad a\in \mathbb C, \ \text{\rm Re}\ a>0
.
$$
Using the properties of semigroup of $e^{-t\mathcal R}$,
one obtains that
$\|\mathcal B_a\|_{L^2(G)}$ is square integrable if $\text{\rm Re}\ a >Q/2$.
The Bessel potential is the convolution kernel
of the $L^2(G)$-bounded left-invariant operator $(\text{\rm Id} +\mathcal R)^{-a/\nu}$
and
of the $L^2(G)$-bounded right-invariant operator $(\text{\rm Id} +\tilde\mathcal R)^{-a/\nu}$, so that we have
$$
(\text{\rm Id} +\mathcal R)^{-a/\nu} f= f* \mathcal B_a ,\quad
(\text{\rm Id} +\tilde\mathcal R)^{-a/\nu} f= \mathcal B_a*f
,\quad f\in L^2(G) .
$$
\subsection{Sobolev spaces}
\label{subsec_sobolev_spaces}
For $a\geq 0$ and $\mathcal R$ a positive Rockland operator,
we define the $\mathcal R$-Sobolev spaces as the domain of $(\text{\rm Id} +\mathcal R)^{\frac a\nu}$ , that is,
$$
L^2_a(G) = \{ f\in L^2(G) , \ (\text{\rm Id} +\mathcal R)^{\frac a\nu} f\in L^2(G) \}
.
$$
For $a<0$, $L^2_a(G)$ is the completion of $L^2(G)$ for the norm
$f\mapsto \|(\text{\rm Id} +\mathcal R)^{\frac a\nu} f\|_{L^2(G)}$.
It is easy to see that for any $a\in \mathbb R$, the Sobolev space $L^2_a(G)$ is a Hilbert space for the norm
$$
\|f\|_{L^2_a(G)}:= \|(\text{\rm Id} +\mathcal R)^{\frac a\nu} f\|_{L^2(G)}
.
$$
Adapting the stratified case \cite{folland-1975}
(see \cite{FR}),
one obtains:
\begin{proposition}[Sobolev spaces]
\label{prop_sobolev_spaces}
Let $\mathcal R$ be a positive Rockland operator of homogeneous degree $\nu_\mathcal R$.
\begin{enumerate}
\item
If $a\leq b$, then
$\mathcal S(G)\subset L^2_b (G)\subset L^2_a(G) \subset \mathcal S'(G)$
and an equivalent norm for $L^2_b(G)$ is
$f\mapsto \|f\|_{L^2_a(G)} + \|\mathcal R ^{\frac {b-a} \nu} f\|_{L^2_a(G)}$.
\item
If $a\in \nu_\mathcal R\mathbb N_0$, then an equivalent norm is given by
$f\mapsto\sum_{[\alpha]\leq a} \|X^\alpha f\|_{L^2(G)}$.
\item
The dual space of $L^2_a(G)$ is isomorphic to $L^2_{-a}(G)$ via
the bilinear form $(f_1,f_2) \mapsto \int_G f_1 f_2 dg$.
\item
We have the usual property of interpolation for Sobolev spaces:
let $T$ be a linear mapping from $L^2_{a_0}(G) + L^2_{a_1}(G)$
to locally integrable functions on $G$;
we assume that $T$ maps $L^2_{a_0}(G)$ and $L^2_{a_1}(G)$
boundedly into $L^2_{b_0}(G)$ and $L^2_{b_1}(G)$, respectively.
Then $T$ extends uniquely to a bounded mapping from
$L^2_{a_t}(G)$ to $L^2_{b_t}(G)$
with $(a_t,b_t) = t(a_0,b_0)+(1-t) (a_1,b_1)$.
\end{enumerate}
\end{proposition}
Consequently, the Sobolev spaces do not depend on the choice of operators $\mathcal R$ as in the statement above. Such operators always exist (see \eqref{ex_RO}) and we fix one of them until the end of the paper.
From the interpolation property of Sobolev spaces
(cf Proposition \ref{prop_sobolev_spaces}), we have:
\begin{lemma}
\label{lem_cq_interpolation}
Let $\kappa \in \mathcal S'(G)$ and $a\in \mathbb R$.
Let $\{\gamma_n,\, n\in \mathbb Z\}$
be a sequence of real numbers
which tends to $\pm \infty$ as $n\to\pm \infty$.
Assume that for any $n\in \mathbb Z$,
the operator $T_\kappa$ extends continuously
to a bounded operator $L^2_{\gamma_n} (G)\rightarrow L^2_{a+\gamma_n}(G)$.
Then the operator $T_\kappa$ extends continuously
to a bounded operator $L^2_\gamma (G)\rightarrow L^2_{a+\gamma}(G)$
for any $\gamma\in \mathbb R$.
\end{lemma}
As in the Euclidean and stratified cases \cite{folland-1975}, we can prove the following Sobolev inequalities:
\begin{lemma}[Sobolev inequality]
\label{lem_sob_ineq}
If $a>Q/2$ then any function $f\in L^2_a(G)$ admits a continuous bounded representative which
satisfies
$$
\|f\|_{L^\infty(G)} \leq C_a \|f\|_{L^2_a(G)}
,
$$
with $C_a=\|\mathcal B_a\|_{L^2(G)}$ independent of $f$.
\end{lemma}
\begin{proof}[Sketch of the proof]
It suffices to write
$$
f=(\text{\rm Id} +\mathcal R)^{-\frac a\nu} (\text{\rm Id} +\mathcal R)^{\frac a\nu} f = \{(\text{\rm Id} +\mathcal R)^{\frac a\nu} f\} * \mathcal B_a
.
$$
\end{proof}
\subsection{The Plancherel Theorem and the von Neumann algebras
$\mathscr L_L(L^2(G))$, $\mathcal K(G)$ and $L^\infty(\widehat G)$}
About representation theory and the Plancherel theorem,
we refer the reader to Dixmier's standard textbook \cite{dixmier_bk1969}, especially \S 18.8.
Recall that a bounded operator $A$ on a Hilbert space $\mathcal H$ is in the Hilbert-Schmidt class whenever $\| A\|_{HS} =\sqrt{\text{\rm Tr} \left(A^* A\right)}$ is finite.
If $f\in L^2(G) \cap L^1(G)$ then $\widehat{f}(\pi)$ is a Hilbert-Schmidt operator,
and the \emph{Plancherel formula} holds,
$$
\int_G |f(g)|^2 dg = \int_{\widehat{G}} \| \widehat f(\pi)\|_{HS}^2 d\mu(\pi)
,
$$
where $\mu$ is the Plancherel measure on $\widehat{G}$.
The group Fourier transform extends unitarily to $L^2(G)$
and
a square integrable function $f\in L^2(G)$ gives rise to a $\mu$-square-integrable field of Hilbert-Schmidt operators $\{\widehat{f}(\pi)\}$.
Conversely, a $\mu$-square-integrable field of Hilbert-Schmidt operators
$\{\sigma_\pi\}$ defines a square integrable function $f$ with
$$
(f,f_1)_{L^2(G)}=\int_{\widehat{G}} \text{\rm Tr} \left( \sigma_\pi \ \widehat{f_1}(\pi)^*\right) d\mu(\pi),\quad f_1\in L^2(G)
.
$$
Let $\mathscr L(L^2(G))$ denote the set of
bounded linear operators $L^2(G)\rightarrow L^2(G)$,
and let $\mathscr L_L(L^2(G))$
be the subset formed by the operators in $\mathscr L(L^2(G))$
which commute with the left regular representation $L(g): f\in L^2(G) \mapsto f(g^{-1} \cdot)$, $g\in G$.
Endowed with the operator norm and composition of operators,
$\mathscr L_L(L^2(G))$
is a von Neumann algebra.
If $T \in \mathscr L_L(L^2(G))$,
then there exists a $\mu$-measurable field
of uniformly bounded operators $\{\sigma^{(T)}_\pi\}$
such that for any $f\in L^2(G)$
the Hilbert-Schmidt operators $\widehat{Tf}(\pi)$ and $\sigma^{(T)}_\pi \widehat f(\pi)$
are equal $\mu$-almost everywhere;
the field $\{\sigma^{(T)}_\pi\}$ is unique up to a $\mu$-negligible set.
Let $L^\infty(\widehat G)$ denote the space of $\mu$-measurable fields of uniformly bounded operators on $\widehat G$, modulo equivalence with respect to the Plancherel formula $\mu$. As is usual, we will identify such fields with their classes in $L^\infty(\widehat G)$.
We have that if $T\in \mathscr L_L(L^2(G))$,
then there exists a unique $\sigma\in L^\infty(\widehat G)$ as above.
Note that
by the Schwartz kernel theorem,
the operator $T$ is of convolution type with kernel $\kappa\in \mathcal D'(G)$,
$$
Tf = f* \kappa ,\quad f\in \mathcal D(G)
.
$$
Conversely given a field $\{\sigma_\pi\}\in L^\infty(\widehat G)$
there exists a unique bounded linear operator $T\in \mathscr L_L(L^2(G))$
satisfying $\widehat{Tf}(\pi)=\sigma_\pi \widehat f(\pi)$ $\mu$-almost everywhere
for any $f\in L^2(G)$.
If $\kappa\in \mathcal D'(G)$ is such that the corresponding convolution operator
$f\in \mathcal D(G)\mapsto f*\kappa $ extends to a bounded operator
$T \in \mathscr L(L^2(G))$
then
$T \in \mathscr L_L(L^2(G))$
and
we abuse the notation by setting
$\sigma^{(T)}_\pi:=\pi(\kappa)\equiv\widehat\kappa(\pi)$.
We denote by $\mathcal K(G)$ the set of such distributions $\kappa$.
It is a von Neumann algebra isomorphic to $\mathscr L_L(L^2(G))$
when equipped with the $*$-product $\kappa\mapsto \kappa^*$ where $\kappa^*(x)=\bar \kappa(x^{-1})$,
and the operator norm
$$
\|\kappa\|_* :=
\| f\mapsto f*\kappa\|_{\mathscr L(L^2(G))}.
$$
Note that when we equip $L^\infty(\widehat G)$
with the operation
$\sigma\mapsto \sigma^*$
and the norm
$$
\|\sigma\|_* = \sup_{\pi\in \widehat{G}} \|\widehat\sigma_\pi\|_{op},
$$
where $\|\cdot\|_{op}$ denotes the operator norm
and the supremum is in fact the essential supremum with respect to the Plancherel measure $\mu$,
$L^\infty(\widehat G)$ becomes a von Neumann algebra
isomorphic with $\mathscr L_L(L^2(G))$ and $\mathcal K(G)$.
More precisely the group Fourier transform defined on $\mathcal K(G)$
gives the isomorphism between $\mathcal K(G)$ and $L^\infty(\widehat G)$.
\medskip
Throughout this paper, if $\kappa\in \mathcal D'(G)$,
$T_\kappa$ denotes the convolution operator
$$
T_\kappa: \mathcal D(G)\ni f\mapsto f*\kappa
,
$$
and we keep the same notation for any of its continuous extensions
$L^2_b(G)\rightarrow L^2_a(G)$ when they exist.
With norms possibly infinite,
$\|\kappa\|_*$ is equal to the operator norm
of $T_\kappa:L^2(G)\to L^2(G)$
by the Plancherel theorem,
and is less than $\|\kappa\|_{L^1(G)}$.
For any $a,b\in \mathbb R$, it is easy to see
that $T_\kappa$ admits a continuous extension
$L^2_b(G)\rightarrow L^2_a(G)$
if and only if
$(I+\tilde \mathcal R)^{-\frac b \nu}(\text{\rm Id} +\mathcal R)^{\frac a \nu} \kappa \in \mathcal K(G)$,
with equality between the $L^2_b(G)\rightarrow L^2_a(G)$-operator norm and the $\mathcal K(G)$-norm.
In this case we may abuse the notation and write
$$
\pi(\text{\rm Id} +\mathcal R)^{\frac a\nu} \pi(\kappa) \pi(\text{\rm Id} +\mathcal R)^{-\frac b\nu}
\quad\mbox{instead of}\quad
\pi\left((\text{\rm Id} +\mathcal R)^{\frac a\nu} (I+\tilde \mathcal R)^{-\frac b\nu} \kappa \right)
.
$$
\section{Quantization and symbols classes}
\label{sec_quantization_S}
As recalled in Introduction,
there exists a natural quantization
which is valid on any Lie group of type~I.
We will present it in this section
after defining symbols
for which this quantization makes sense
and produces operators $\mathcal D(G)\to\mathcal D'(G)$
with $G$ graded Lie groups.
Moreover, the resulting operators admit
integral representations with right convolution kernels
and these kernels play a major role in every subsequent proof.
We will also define symbol classes and give some examples of symbols.
\subsection{The symbols and their kernels}
\label{subsec_presymbol}
A \emph{symbol} is a family of operators
$\sigma=\{\sigma(x,\pi), \ x\in G, \ \pi\in \widehat{G}\}$
satisfying:
\begin{enumerate}
\item for each $x\in G$,
$\{\sigma(x,\pi), \ \pi\in \widehat{G}\}$
is a $\mu$-measurable field of operators $\mathcal H_\pi^\infty\rightarrow \mathcal H_\pi$,
\item
there exist $\gamma_1,\gamma_2\in \mathbb R$ such that
for any $x\in G$,
\begin{equation}
\label{field_sigma_gamma12}
\{\pi(\text{\rm Id} +\mathcal R)^{\gamma_1} \sigma(x,\pi) \pi(\text{\rm Id} +\mathcal R)^{\gamma_2}, \ \pi \in \widehat{G}\} \in L^\infty(\widehat G)
,
\end{equation}
\item
for any $\pi\in \widehat{G}$ and any $u,v\in \mathcal H_{\pi}$,
the scalar function $x\mapsto (\sigma(x,\pi) u,v)_{\mathcal H_\pi}$
is smooth over $G$.
\end{enumerate}
\medskip
Consequently at each $x\in G$ and $\pi\in \widehat{G}$,
the operator $\sigma(x,\pi)$ is densely defined on $\mathcal H_\pi$;
it is also the case for $X^\beta_x\sigma(x,\pi)$
for any $\beta\in \mathbb N_0^n$.
The second condition implies that for each $x\in G$, the $\mu$-measurable field
\eqref{field_sigma_gamma12} correspond to a distribution $\kappa_{x,\gamma_1,\gamma_2}\in \mathcal K(G)$ which depends smoothly on $x$;
hence $\sigma$ corresponds to a distribution
$$
\kappa(x, \cdot)=\kappa_x:=(\text{\rm Id} +\mathcal R)^{-\gamma_1} (I+\tilde \mathcal R)^{-\gamma_2} \kappa_{x,\gamma_1,\gamma_2}
,
$$
which we call its \emph{kernel}.
By injectivity of $\pi$ on $\mathcal K(G)$,
$\pi(X_x^\beta\kappa_x)=X_x^\beta \sigma(x,\pi)$.
Examples of symbols are the symbols within the classes $S^m_{\rho,\delta}$ defined later on.
More concrete examples of symbols which do not depend on $x\in G$
are $\pi(X)^\alpha$, $\alpha\in \mathbb N_0^n$
or the multipliers in $\pi(\mathcal R)$, that is,
$\phi(\pi(\mathcal R))$ with $\phi\in L^\infty(\mathbb R)$ (for example).
Indeed for any $\pi\in \widehat{G}$
the operator $\pi(\mathcal R)$ is essentially self-adjoint
\cite{hula+jenkins+ludwig-1985}
and we denote by $E_\pi$ its spectral projection,
hence giving a meaning to $\phi(\pi(\mathcal R))$.
The relation between the spectral projections $E$ and $E_\pi$ of $\mathcal R$ and $\pi(\mathcal R)$ is
$$
\pi(\phi(\mathcal R)f)=\phi(\pi(\mathcal R)) \pi(f)
, \quad \phi\in L^\infty(\mathbb R) , \ f\in L^2(G) .
$$
It is known \cite{hula+jenkins+ludwig-1985}
that the spectrum of $\pi(\mathcal R)$ consists of discrete eigenvalues in $(0,\infty)$.
This may add a further justification to using the word \emph{quantization}.
\subsection{The quantization mapping $\sigma\mapsto Op(\sigma)$}
\label{subsec_quantization}
Our quantization is analogous to the usual Kohn-Nirenberg quantization in the Euclidean setting, and has already been noticed by Taylor \cite{Tnma},
used indirectly on the Heisenberg group \cite{Tnma,bahouri+FK+gallagher_bk2012}
and explicitly on compact Lie groups \cite{ruzh+turunen_bk2010}.
It associates an operator
$T=Op(\sigma)$ to a symbol $\sigma$ in the following way
(with the same notation as in Subsection \ref{subsec_presymbol}).
For any $f\in \mathcal D(G)$ and $x\in G$,
\begin{eqnarray*}
\int_{\widehat{G}} \text{\rm Tr} \left| \sigma(x,\pi)\widehat f(\pi)\right| d\mu(\pi)
\leq
\sup_{\pi\in \widehat{G}} \| \pi(\text{\rm Id} +\mathcal R)^{\gamma_1} \sigma(x,\pi) \ \pi(\text{\rm Id} +\mathcal R)^{\gamma_2} \|_{op}\\
\int_{\widehat{G}} \text{\rm Tr} \left| \pi\left((\text{\rm Id} +\mathcal R)^{-\gamma_1}(I+\tilde \mathcal R)^{-\gamma_2} f\right)\right| d\mu(\pi)
,
\end{eqnarray*}
is finite and we can set
\begin{equation}\label{EQ:quant}
Tf(x):= \int_{\widehat{G}} \text{\rm Tr} \left(\pi(x)\sigma(x,\pi){\wh f(\pi)}\right) d\mu(\pi)
.
\end{equation}
We have obtained a continuous linear operator $T:\mathcal D(G)\to \mathcal D'(G)$.
By the Schwartz kernel theorem, $T=Op(\sigma)$ has an integral kernel in the distributional sense.
However since $\sigma$ is a symbol,
we obtain directly, still in the distributional sense,
the following integral representation in terms of the kernel $\kappa$ defined in Subsection \ref{subsec_presymbol},
$$
Tf(x) =f*\kappa_x(x)= \int_G f(y) \kappa(x, y^{-1}x) dy
.
$$
For example, the symbol $\sigma$ given by the identity operator on each space $\mathcal H_\pi$ is associated with the identity operator on $G$; its kernel is the Dirac measure at 0 denoted by $\delta_0$ (independent of the point $x\in G$).
More generally, for any $\alpha\in \mathbb N_0^n$,
the symbol $\pi(X)^\alpha$ is associated with the operator $X^\alpha$
with kernel $(-1)^{|\alpha|}X^\alpha \delta_0$ defined in the sense of distributions via
$$
\int_G f(g) (-1)^{|\alpha|} X^\alpha \delta_0(g) dg =
\int_G X^\alpha f(g) \delta_0(g) dg =
X^\alpha f(0) .
$$
It is easy to see that the quantization mapping $\sigma\mapsto T=Op(\sigma)$
is 1-1 and linear.
Before defining symbol classes,
we need to define difference operators.
\subsection{Difference operators}
Difference operators were defined on compact Lie groups in
\cite{ruzh+turunen_bk2010}, as acting on Fourier coefficients.
Its adaptation to our setting leads us to define difference operators on $L^\infty(\widehat G)$
viewed as fields.
More precisely for any $q\in C^\infty(G)$, we set
$$
\Delta_q\widehat f(\pi):=\widehat{qf}(\pi) =\pi(qf) .
$$
This defines an operator $\Delta_q$
with domain
$\text{\rm Dom}\ (\Delta_q):=\mathcal F_G \{f\in \mathcal K(G), \ qf\in \mathcal K(G)\}$,
and more generally $\pi(\text{\rm Id} +\mathcal R)^{-\gamma_1}\pi(I+\tilde \mathcal R)^{-\gamma_2}\text{\rm Dom}\ (\Delta_q)$ for any $\gamma_1,\gamma_2\in \mathbb R$.
Note that in general, it is not possible to define an operator $\Delta_q$ on each $\mathcal H_\pi$;
this can be seen quite easily by considering
the multiplication by the central variable
on the Heisenberg group for example.
The \emph{difference operators} are
$$
\Delta^\alpha :=\Delta_{\tilde q_{\alpha}}, \quad \alpha\in \mathbb N_0^n,
$$
where $\tilde q_{\alpha}(x)=q_\alpha(x^{-1})$
and the $q_\alpha$'s were defined via \eqref{def_qalpha}.
Lemma \ref{lem_property_qalpha} implies that
$\Delta^{\alpha_1}\Delta^{\alpha_2}$
is a linear combination
of $\Delta^{\alpha}$ with $[\alpha]=[\alpha_1]+[\alpha_2]$.
Furthermore, we have
\begin{eqnarray*}
&&\tilde q_\alpha(x) \ f_2*f_1(x)
=
\int_G q_\alpha(x^{-1} y \ y^{-1}) \ f_2(y)\ f_1(y^{-1} x) \ dy
\\
&&\qquad=
\!\!\!\!\!\!\sum_{[\alpha_1] + [\alpha_2] =[\alpha] }\!\!\!\!\!\!
c_{\alpha_1,\alpha_2}
\int_G
q_{\alpha_2}(y^{-1})
f_2(y)\ q_{\alpha_1}(x^{-1} y) f_1(y^{-1} x) \ dy
\\
&&\qquad=
\!\!\!\!\!\!\sum_{[\alpha_1] + [\alpha_2] =[\alpha] }\!\!\!\!\!\!
c_{\alpha_1,\alpha_2} \
(\tilde q_{\alpha_2}f_2)*(\tilde q_{\alpha_1}f_1) ,
\end{eqnarray*}
and
we get the \emph{Leibniz formula}:
\begin{equation}
\label{formula_leibniz}
\Delta^\alpha \left(\widehat{f_1}(\pi)\widehat{f_2}(\pi)\right)
= \sum_{[\alpha_1] + [\alpha_2] =[\alpha] } c_{\alpha_1,\alpha_2}
\ \Delta^{\alpha_1}\widehat{f_1}(\pi)\ \Delta^{\alpha_2} \widehat{f_2}(\pi).
\end{equation}
The idea of difference operators appear naturally
when considering operators on the torus $\mathbb T^{n}$.
In this case one recovers forward and backward difference
operators on the lattice $\mathbb Z^{n}$.
Difference operators were systematically defined and studied
on compact Lie groups in \cite{ruzh+turunen_bk2010}.
On the Heisenberg group, expressions of a related nature
were used to describe the Schwartz space in \cite{geller}
and with a hypothesis of unitary invariance in \cite{BJR}.
\subsection{The symbol classes $S^m_{\rho,\delta}$}
\begin{definition}
Let $m,\rho,\delta\in \mathbb R$ with $1\geq \rho\geq \delta\geq 0$ and $\delta\not=1$.
A symbol $\sigma$ is a {\em symbol of order $m$ and of type~$(\rho,\delta)$}
whenever,
for each $\alpha,\beta\in \mathbb N_0^n$ and $\gamma\in \mathbb R$,
the field
$$
\{\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho [\alpha]-m -\delta[\beta] +\gamma}\nu }
X_x^\beta\Delta^\alpha \sigma(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\gamma}\nu }, \ \pi\in \widehat{G}\}
,
$$
is in $L^\infty(\widehat G)$ uniformly in $x\in G$;
this means that we have
$$
\sup_{\pi\in \widehat{G}, \, x\in G}
\|\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho [\alpha]-m -\delta[\beta] +\gamma}\nu }
X_x^\beta\Delta^\alpha \sigma(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\gamma}\nu }\|_{op}
=C_{\alpha,\beta,\gamma} <\infty
.
$$
(The supremum over $\pi$ is in fact the essential supremum with respect to the Plancherel measure $\mu$.)
The \emph{symbol class} $S^m_{\rho,\delta}$ is the set of symbol of order $m$ and of type~$(\rho,\delta)$.
We also define $S^{-\infty}_{\rho,\delta}=\cap_{m\in \mathbb R}S^m_{\rho,\delta}$ the class of smoothing symbols.
\end{definition}
Let us make some comments on this definition:
\begin{enumerate}
\item
In the abelian case,
that is, $\mathbb R^n$ endowed with the addition law
and $\mathcal R=-\mathcal L$, $\mathcal L$ being the Laplace operator,
$S^m_{\rho,\delta}$ boils down to the usual H\"ormander class.
In the case of compact Lie groups
with $\mathcal R=-\mathcal L$, $\mathcal L$ being the Laplace-Beltrami operator,
a similar definition leads to the one considered in \cite{ruzh+turunen_bk2010}
since the operator $\pi(\text{\rm Id} +\mathcal R)$ is scalar.
However here, in the case of non-abelian graded groups,
the operator $\mathcal R$ can not have a scalar
Fourier transform.
\item
The presence of the parameter $\gamma$ is required to prove that the space of symbols $\cup_{m\in \mathbb R} S^m_{\rho,\delta}$ form an algebra of operators later on.
\item\label{rem_gamma_countable} The conditions on $\alpha$ and $\beta$ are of countable nature and it is also the case for $\gamma$.
Indeed,
by Lemma \ref{lem_cq_interpolation}, it suffices to prove the property above for one sequence $\{\gamma_n,\, n\in \mathbb Z\}$
with
$\gamma_n\underset {n\rightarrow \pm\infty}\longrightarrow\pm\infty$.
\item A symbol class $S^m_{\rho,\delta}$ is a vector space.
And we have the inclusions
$$
m_1\leq m_2,\quad \delta_1\leq\delta_2,\quad
\rho_1\geq\rho_2
\quad \Longrightarrow \quad
S^{m_1}_{\rho_1,\delta_1}\subset S^{m_2}_{\rho_2,\delta_2}
.
$$
\item If $\rho\not=0$,
we will show in Subsections \ref{subsec_1prop_kernels}
and \ref{subsec_composition}
that we obtain an algebra of operators
with smooth kernels $\kappa_x$ away from the origin.
\end{enumerate}
If $\sigma$ is a symbol and $a,b,c\in [0,\infty)$, we set
$$
\|\sigma(x,\pi)\|_{S^m_{\rho,\delta},a,b,c}: =
\!\!\!\!\!\! \sup_{\substack{|\gamma|\leq c \\ [\alpha]\leq a,\, [\beta]\leq b}}\!\!\!\!\!\!
\|\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho [\alpha]-m -\delta[\beta] +\gamma}\nu }
X_x^\beta\Delta^\alpha \sigma(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\gamma}\nu }\|_{op}
\, ,
$$
and
$$
\|\sigma\|_{S^m_{\rho,\delta},a,b,c} := \sup_{x\in G, \, \pi\in \widehat{G}} \|\sigma(x,\pi)\|_{S^m_{\rho,\delta},a,b,c}
.
$$
It is a routine exercise to check that for
any $m\in \mathbb R$, $\rho,\delta\geq 0$,
the functions
$\|\cdot\|_{S^m_{\rho,\delta},a,b,c}$,
$a,b,c\in [0,\infty) $, are semi-norms over the vector space $S^m_{\rho,\delta}$.
Furthermore,
with Comment \ref{rem_gamma_countable} above,
taking $a,b,c$ as non-negative integers, they endow
$S^m_{\rho,\delta}$ of a structure of Fr\'echet space.
The class of smoothing symbols is then equipped with the topology of projective limit.
The pseudo-differential operators of order $m\in \mathbb R\cup\{-\infty\}$ and type~$(\rho,\delta)$ are obtained by quantization from the symbols of the same order and type, that is,
$$
\Psi^m_{\rho,\delta}:=Op(S^m_{\rho,\delta})
,
$$
with the quantization given by \eqref{EQ:quant}.
They inherit a structure of topological vector space from the classes of symbols,
$$
\|Op(\sigma)\|_{\Psi^m_{\rho,\delta},a,b,c}:=
\|\sigma\|_{S^m_{\rho,\delta},a,b,c}
.
$$
It is not difficult to see from the computations in Subsection \ref{subsec_quantization}
that any operator $Op(\sigma)$ is a continuous operator $\mathcal D(G)\rightarrow \mathcal D'(G)$;
in fact, we can show
that $T$ is continuous $\mathcal S(G)\rightarrow \mathcal S(G)$
but the complete proof which uses Theorem \ref{thm_composition} and Proposition \ref{prop_multipliers}
can be found in \cite{FR}.
The type~$(1,0)$ is thought of as the basic class of symbols and the
types~$(\rho,\delta)$ as their generalisations,
the limitation on the parameters $(\rho,\delta)$
coming from reasons similar to the ones in the Euclidean settings.
For type~$(1,0)$, we set $S^m:=S^m_{1,0}$, $\Psi^m:=\Psi^m_{1,0}$
and,
$$
\|\sigma(x,\pi)\|_{S^m_{1,0},a,b,c}=\|\sigma(x,\pi)\|_{a,b,c}
, \
\|\sigma\|_{S^m_{1,0},a,b,c}=\|\sigma\|_{a,b,c} , \ \mbox{etc}\ldots
$$
Before proving that $\cup_{m\in \mathbb R} S^m_{\rho,\delta}$ and
$\cup_{m\in \mathbb R} \Psi^m_{\rho,\delta}$ are stable by composition,
let us give some examples.
\subsection{First examples}
\label{subsec_1ex}
As it should be, $\cup_{m\in \mathbb R} \Psi^m$
contains the calculus of left invariant differential operators.
More precisely the following lemma implies that
$\sum_{[\beta]\leq m} c_\beta X^\beta \in \Psi^m$.
The coefficients $c_\alpha$ here are constant
and it is easy to relax this condition
with each function $c_\alpha$ being smooth and bounded as well as all its derivatives.
\begin{lemma}
\label{lem_Xbeta_PsiDO}
For any $\beta_o\in \mathbb N_0^n$,
the operator $X^{\beta_o}=Op(\pi(X)^{\beta_{0}})$ is in $\Psi^{[\beta_o]}$.
\end{lemma}
\begin{proof}
For any $\alpha\in \mathbb N_0^n$,
in the sense of distributions,
$$
\int_G f(g) (\tilde q_\alpha (-1)^{|\beta_o|}X^{\beta_{0}} \delta_0)(g) dg
=
\int_G X^{\beta_o} \left\{\tilde q_\alpha(g) f(g) \right\} \delta_0(g) dg
,
$$
is always zero if $[\alpha]<[\beta_o]$
or $[\alpha]=[\beta_o]$ with $\alpha\not=\beta_o$.
If $[\alpha]>[\beta_o]$ or $[\alpha]=[\beta_o]$ with $\alpha=\beta_o$,
then it is equal to $X^{\beta_o-\alpha}f$ up to some constant $c_{\alpha,\beta_o}\in \mathbb R$.
Moreover, in the latter case, we get
\begin{eqnarray*}
\| f* (\tilde q_\alpha (-1)^{|\beta_o|}X^{\beta_{0}} \delta_0)\|_{L^2_{[\alpha]-[\beta_o] +\gamma}}
&=&
|c_{\alpha,\beta_{0}}|
\|X^{\beta_{0}-\alpha} f\|_{L^2_{[\alpha]-[\beta_o] +\gamma}(G)}
\\
&\leq& C_{\alpha,\beta_o} \|f\|_{L^2_\gamma(G)} .
\end{eqnarray*}
This shows
$\|\pi(\text{\rm Id} +\mathcal R)^{\frac{[\alpha]-[\beta_o] +\gamma}\nu }
\Delta^\alpha \pi(X)^{\beta_o}
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\gamma}\nu }\|_{op}\leq C_{\alpha,\beta_o}$.
\end{proof}
An example of smoothing operator is given by convolution with a Schwartz function:
\begin{lemma}
\label{lem_ex_smoothing_op}
If $\kappa\in \mathcal S(G)$ then $T_\kappa \in \Psi^{-\infty}$.
Furthermore, the mapping $\mathcal S(G)\ni \kappa\mapsto T_\kappa \in \Psi^{-\infty}$ is continuous.
\end{lemma}
\begin{proof}
For any $\kappa\in \mathcal S(G)$ and $a\geq 0$,
we have $(\text{\rm Id} +\mathcal R)^a\kappa \in L^1(G)$.
Indeed, it is true if $a\in \mathbb N_0$;
if $a\not\in \mathbb N_0$,
then writing
$$
(\text{\rm Id} +\mathcal R)^a\kappa = \left\{(\text{\rm Id} +\mathcal R)^{\lceil a\rceil }\kappa \right\}
* \mathcal B_{a-\lceil a\rceil}
,
$$
we get
$$
\|(\text{\rm Id} +\mathcal R)^a\kappa\|_{L^1(G)}
\leq
\| (\text{\rm Id} +\mathcal R)^{\lceil a\rceil }\kappa\|_{L^1(G)}
\| \mathcal B_{a-\lceil a\rceil}\|_{L^1(G)}
.
$$
We have also the same property for $\tilde \mathcal R$ by adapting the proof above.
Let $m\in \mathbb R$.
For any $\gamma\in \mathbb R$ and $\alpha\in \mathbb N_0^n$
such that $\gamma$ and $[\alpha]-m+\gamma$ are of the same sign, we have
\begin{eqnarray*}
&&\sup_{\pi\in \widehat{G}}
\|\pi(\text{\rm Id} +\mathcal R)^{\frac{[\alpha]-m +\gamma}\nu }
\Delta^\alpha \pi(\kappa)
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\gamma}\nu }\|_{op}\leq
\\&&
\left\{\begin{array}{ll}
\|\mathcal B_\gamma\|_{L^1(G)}
\|(\text{\rm Id} +\mathcal R)^{\frac{[\alpha]-m+\gamma} \nu}\tilde q_\alpha \kappa\|_{L^1(G)}
& \mbox{if} \ \gamma, [\alpha]-m+\gamma\geq 0 ,\\
\|(\text{\rm Id} +\tilde \mathcal R)^{-\frac{\gamma} \nu}\tilde q_\alpha\kappa\|_{L^1(G)}
\|\mathcal B_{-([\alpha] -m+\gamma)}\|_{L^1(G)}
& \mbox{if} \ \gamma,[\alpha]-m+\gamma\leq 0 .\\
\end{array}\right.
\end{eqnarray*}
It is now clear that $T_\kappa\in \Psi^m$ and that any semi-norm
$\|T\|_{\Psi^m,a,b,c}$ is controlled by some Schwartz semi-norm of $\kappa$.
\end{proof}
By Lemma \ref{lem_ex_smoothing_op} and Remark \ref{rem_cq_hula},
if $\phi\in \mathcal S(\mathbb R)$ then $\phi(\mathcal R)\in \Psi^{-\infty}$.
This last consequence could also be obtained via the next example.
\medskip
The $\mathcal R$-multipliers in the following class of functions yields operators in the calculus.
We consider the space $\mathcal M_m$ of smooth functions $\phi$ on $[0,\infty)$
satisfying for every $k\in\mathbb N_0$:
$$
\| \phi\|_{\mathcal M_m , k} := \sup_{\lambda \geq 0,\, k_1\leq k}
\left| (1+\lambda)^{-m+k_1} \partial_\lambda^{k_1} \phi(\lambda)\right| \
<\infty
.
$$
An important example is $\phi(\lambda)=(1+\lambda)^m$, $m\in \mathbb R$.
\begin{proposition}
\label{prop_multipliers}
Let $m\in \mathbb R$ and $\phi\in \mathcal M_{\frac m \nu}$.
Then $\phi(\mathcal R)$ is in $\Psi^m$ and its symbol satisfies
$$
\forall a,b,c \in \mathbb N\qquad
\exists k\in \mathbb N, \ C>0\; : \qquad
\|\phi(\pi(\mathcal R))\|_{a,b,c} \leq C \| \phi\|_{\mathcal M_{\frac m \nu} , k}
,
$$
with $k$ and $C$ independent of $\phi$.
\end{proposition}
The proof of Proposition \ref{prop_multipliers}
can be found in \cite{FR}.
It is based on Proposition \ref{prop_hula} and the Cotlar-Stein Lemma.
\section{Some properties of symbols, kernels and operators}
\label{sec_prop_symbol_kernel_op}
In this section, we give more explicitly the properties (R1), (R2) and (R4)
given in Introduction.
\subsection{First properties of the symbols}
\label{subsec_1prop_symbols}
The following properties of the symbol $\sigma\in S^m_{\rho,\delta}$ of an
operator with kernel $\kappa_x$
are not difficult to obtain.
\begin{enumerate}
\item If $\beta_o\in \mathbb N_0^n$ then the symbol $X^{\beta_o}_x \sigma(x,\pi)$ is in $S^{m+\delta[\beta_o]}_{\rho,\delta}$ with kernel $X_x^{\beta_o} \kappa_x$
and,
$$
\|X^{\beta_o}_x \sigma(x,\pi)\|_{S^m_{\rho,\delta}, a,b,c} \leq C_{b,\beta_o}
\| \sigma(x,\pi)\|_{S^m_{\rho,\delta}, a,b + [\beta_o],c}
.
$$
\item If $\alpha_o\in \mathbb N_0^n$ then the symbol $\Delta^{\alpha_o} \sigma(x,\pi)$ is in $S^{m-\rho[\alpha_o]} _{\rho,\delta}$ with kernel $\tilde q_{\alpha_o} \kappa_x$
and,
$$
\|\Delta^{\alpha_o}\sigma(x,\pi)\|_{S^m_{\rho,\delta}, a,b,c} \leq C_{b,\beta_o}
\| \sigma(x,\pi)\|_{S^m_{\rho,\delta}, a+[\alpha_o],b,c}
.
$$
\item The symbol $\sigma(x,\pi)^*$ is in $S^m_{\rho,\delta}$ with kernel
$\kappa_x^*:y\mapsto \bar\kappa_x(y^{-1})$ and,
$$
\|\sigma(x,\pi)^*\|_{S^m_{\rho,\delta},a,b,c}=
\!\!\!\!\!\! \sup_{\substack{|\gamma|\leq c \\ [\alpha]\leq a,\, [\beta]\leq b}}\!\!\!\!\!\!
\|
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\gamma}\nu }
X_x^\beta\Delta^\alpha \sigma(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho [\alpha]-m -\delta[\beta] +\gamma}\nu }\|_{op}
\, .
$$
\item Let
$\sigma_1\in S^{m_1}_{\rho,\delta}$ and $\sigma_2\in S^{m_2}_{\rho,\delta}$ with
respective kernels $\kappa_{1x}$ and $\kappa_{2x}$.
Then $\sigma(x,\pi):=\sigma_1(x,\pi)\sigma_2(x,\pi)$ defines the symbol
$\sigma$ in $S^{m}_{\rho,\delta}$, $m=m_1+m_2$,
with kernel
$\kappa_{2x}*\kappa_{1x}$; furthermore
$$
\|\sigma(x,\pi) \|_{S^m_{\rho,\delta},a,b,c}
\leq C
\|\sigma_1(x,\pi) \|_{S^{m_1}_{\rho,\delta},a,b,c+\rho a+|m_2|+\delta b}
\|\sigma_2(x,\pi) \|_{S^{m_2}_{\rho,\delta},a,b,c}
.
$$
where the constant $C=C_{a,b,c} >0$ does not depend on $\sigma$.
Indeed from the Leibniz rule for $\Delta^\alpha$ and $X^\beta$,
the operator
$$
\pi(\text{\rm Id} +\mathcal R)^{\frac{[\alpha]-m+\gamma}\nu} X^\beta_x \Delta^\alpha \sigma(x,\pi)\pi(\text{\rm Id} +\mathcal R)^{-\frac \gamma \nu}
,
$$
is a linear combination over $\beta_1,\beta_2,\alpha_1,\alpha_2\in \mathbb N_0^n$ satisfying $[\beta_1]+[\beta_2]=[\beta]$, $[\alpha_1]+[\alpha_2]=[\alpha]$, of terms
$$
\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho[\alpha]-m -\delta [\beta]+\gamma}\nu}
X^{\beta_1}_x \Delta^{\alpha_1} \sigma_1(x,\pi)
X^{\beta_2}_x \Delta^{\alpha_2} \sigma_2(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{-\frac \gamma \nu}
,
$$
whose operator norm is less than
\begin{eqnarray*}
\|\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho[\alpha]-m-\delta [\beta]+\gamma}\nu}
X^{\beta_1}_x \Delta^{\alpha_1} \sigma_1(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{-\frac{\rho[\alpha_2]-m_2-\delta[\beta_2]+\gamma}\nu}
\|_{op}\\
\|\pi(\text{\rm Id} +\mathcal R)^{\frac{\rho[\alpha_2]-m_2 -\delta[\beta_2] +\gamma}\nu}
X^{\beta_2}_x \Delta^{\alpha_2} \sigma_2(x,\pi)
\pi(\text{\rm Id} +\mathcal R)^{-\frac \gamma \nu}\|_{op}.
\end{eqnarray*}
Consequently, the collection of symbols $\cup_{m\in \mathbb R}S^m_{\rho,\delta}$ forms an algebra.
\item
Using the previous point and the left calculus
(see Lemma \ref{lem_Xbeta_PsiDO}),
if $\sigma\in S^m_{\rho,\delta}$ with kernel $\kappa_x$,
then $\pi(X)^{\beta} \sigma \, \pi(X)^{\tilde \beta}$ is in $S^{m+[\beta]+[\tilde \beta]}$ with kernel $X^\beta_y\tilde X^{\tilde \beta}_y \kappa_x(y)$.
\end{enumerate}
\subsection{First properties of the kernels}
\label{subsec_1prop_kernels}
As expected from pseudo-differential calculi on manifolds
such as homogeneous Lie groups,
the kernels of the operators of order 0 are of Calderon-Zygmund type
in the sense of Coifman-Weiss \cite[ch. III]{coifman+weiss-LNM71}.
This claim is a consequence of the following proposition together with the properties of the symbols.
\begin{proposition}
\label{prop_kernel}
Assume $\rho\in (0,1]$ and let us fix a homogeneous norm $|\cdot|$ on $G$.
Let $\sigma \in S^m_{\rho,\delta}$ and $\kappa_x$ the associated kernel.
Then for each $x\in G$, the distribution $\kappa_x$ coincides with a smooth function in $G\backslash \{0\}$.
Furthermore, $(x,y)\mapsto \kappa(x,y)$ is a smooth function on $G\times (G\backslash \{0\})$, and we have:
\begin{enumerate}
\item
There exists $C>0$ and $a,b,c\in \mathbb N_0$ such that
for any $y\in G$ with $|y|<1$,
$$
|\kappa_x(y)| \leq C
\sup_{\pi\in \widehat{G}} \|\sigma(x,\pi)\|_{S^m_{\rho,\delta},a,b,c}
\left\{\begin{array}{ll}
|y|^{-\frac{Q+m} \rho} & \mbox{if}\ Q+m>0\\
1&\mbox{if}\ Q+m<0\\
\ln |y|&\mbox{if}\ Q+m=0\\
\end{array}\right.
.
$$
\item
For any $M\in \mathbb N_0$,
there exists $C>0$ and $a,b,c\in \mathbb N_0$ such that
for any $y\in G$ with $|y|\geq 1$,
$$
|\kappa_x(y)| \leq C
\sup_{\pi\in \widehat{G}} \|\sigma(x,\pi)\|_{S^m_{\rho,\delta},a,b,c} |y|^{-M}
.
$$
\end{enumerate}
\end{proposition}
For example our operators of order 0 have singularities of the type $|y|^{-Q}$
with $Q$ homogeneous dimension strictly greater than the topological dimension.
Hence the calculus developed here can not coincide with the H\"ormander calculus on $\mathbb R^n$ (abelian).
This contrasts with the compact case: it was shown in \cite{RTW}
that the calculus developed in \cite{ruzh+turunen_bk2010} on compact Lie groups
leads to the usual H\"ormander operator classes on $\mathbb R^{n}$ extended
to compact connected manifolds.
\medskip
Before discussing the proof of Proposition \ref{prop_kernel},
let us prove the following couple of easy lemmata.
\begin{lemma}
\label{lem_square_integrable_kernel}
If $\sigma \in S^m_{\rho,\delta}$ and $a\in \mathbb R$ with $m+a<-Q/2$,
then the distribution $(\text{\rm Id} +\mathcal R)^{\frac a\nu} \kappa_x$
coincides with a square integrable function for each fixed $x$;
Furthermore, there exists a constant $C=C_{a,m}>0$,
independent of $\sigma$, such that we have
$$
\forall x\in G\qquad
\|(\text{\rm Id} +\mathcal R)^{\frac a\nu}\kappa_x\|_{L^2(G)}\leq C
\sup_{\pi\in\widehat{G}} \|\sigma(x,\pi)\|_{S^m_{\rho,\delta},0,0,0}
.
$$
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem_square_integrable_kernel}]
By the Plancherel formula and properties of Hilbert-Schmidt operators,
\begin{eqnarray*}
&&\|(\text{\rm Id} +\mathcal R)^{\frac a\nu}\kappa_x\|_{L^2(G)}^2
= \int_{\widehat{G}} \| (\text{\rm Id} +\mathcal R)^{\frac a\nu} \sigma(x,\pi) \|_{HS}^2 d\mu(\pi)
\\
&&\qquad\leq \sup_{\pi\in \widehat{G}} \| \pi(\text{\rm Id} +\mathcal R)^{\frac{-m}\nu} \sigma(x,\pi)\|_{op}^2
\int_{\widehat{G}} \| \pi(\text{\rm Id} +\mathcal R)^{\frac{m +a}\nu}\|_{HS}^2 d\mu(\pi)
\\
&&\qquad\leq
\sup_{\pi\in \widehat{G}} \|\sigma(x,\pi)\|_{S^m_{\rho,\delta}, 0,0,0}^2 \
\| \mathcal B_{-(m+a)}\|_{L^2(G)} ^2
,
\end{eqnarray*}
which is finite by the properties of Bessel potentials (see Subsection
\ref{SEC:RO}).
\end{proof}
\begin{lemma}
\label{lem_kernel_continuous}
If $\sigma \in S^m_{\rho,\delta}$ with $m<-Q$,
then the associated kernel $\kappa_x(y)$
coincides with a continuous function in $y$ for each $x\in G$.
Furthermore, there exists a constant $C=C_{a,m}>0$, independent of $\sigma$,
such that we have
$$
\forall x,y\in G\qquad
|\kappa_x(y)| \leq C
\sup_{\pi\in\widehat{G}} \|\sigma(x,\pi)\|_{S^m_{\rho,\delta},0,0,0}
,
$$
\end{lemma}
\begin{proof}
By Lemma \ref{lem_sob_ineq},
$$
\|\kappa_x\|_{L^\infty(G)}
\leq C_a
\|(\text{\rm Id} +\mathcal R)^{\frac a \nu} \kappa_x\|_{L^2(G)}
\quad\mbox{where}\quad
a= \frac {Q+m}2<-\frac Q2
,
$$
and we conclude by
Lemma \ref{lem_square_integrable_kernel}.
\end{proof}
Consequently,
for any $\sigma\in S^m_{\rho,\delta}$ with kernel $\kappa_x$,
we can apply Lemma \ref{lem_kernel_continuous}
to the symbol
$$
X^{\beta_o}_x \Delta^{\alpha_o} \pi(X)^\beta \sigma \pi(X)^{\tilde \beta}
\in S^{m+[\beta]+[\tilde\beta] +\delta[\beta_o]-\rho[\alpha_o]}_{\rho,\delta}
.
$$
The kernel is given by
$\tilde X^{\tilde \beta} X^\beta X^{\beta_o}_x \tilde q_{\alpha_o}\kappa_x$ (see Subsection \ref{subsec_1prop_symbols})
and, if $m+[\beta]+[\tilde\beta] +\delta[\beta_o]-\rho[\alpha_o] <- Q$,
it coincides with a continuous and bounded function,
$$
\forall x,y\in G\quad
|\tilde X^{\tilde \beta}_y X^\beta_y X^{\beta_o}_x \tilde q_{\alpha_o}\kappa_x(y)|
\leq
C \sup_{\pi\in \widehat{G}}\|\sigma(x,\pi)\|_{S^m_{\rho,\delta}, a,b,c}
$$
with $a=[\alpha_o]$,
$b=[\beta_o]$,
$c=\rho [\alpha_o]+ \delta[\beta_o]+ [\tilde \beta]$.
Hence if $\rho>0$ then the kernel of a symbol in $S^m_{\rho,\delta}$
is smooth on $G\times (G\backslash \{0\})$.
Let $|\cdot|_{\nu_o}$ be the homogeneous norm
defined in
\eqref{eq_norm_||nuo}
for $\nu_o$ the smallest common multiple of the $\upsilon_j$.
Clearly, for any $p\in \mathbb N$,
$|\cdot|_{\nu_o}^{2\nu_op}$ is a homogeneous polynomial of degree $p2\nu_o$
and thus can be written as a linear combination of $q_\alpha$ with $[\alpha]=2\nu_op$
(see Lemma \ref{lem_property_qalpha}).
Thus
\begin{eqnarray}
|y|_{\nu_o}^{2\nu_op} |\kappa_x(y)|
&=&
|y^{-1}|_{\nu_o}^{2\nu_op} |\kappa_x(y)|
\nonumber
\\
&\leq &C \sup_{[\alpha ]=2\nu_op}
\sup_{z\in G} |q_\alpha(z) |_{\nu_o}^{2\nu_o} |\kappa_x(g)|
\nonumber
\\
&\leq&
C \sup_{\pi\in \widehat{G}}\|\sigma(x,\pi)\|_{S^m_{\rho,\delta}, a,b,c}
\label{weak_control_kernel}
\end{eqnarray}
as long as $m-\rho 2\nu_o p>Q$.
Here $C=C_{m,p}$ is independent of $\sigma$.
The estimate \eqref{weak_control_kernel} proves the second point in Proposition \ref{prop_kernel} and is a weaker version of the first.
We will not show this first point
because of its argument's length.
The complete proof
can be found in \cite{FR}.
\subsection{A pseudo-differential operator as a limit of `nice' operators}
The definition of symbols presented above leads to kernels $\kappa_x$
in the distributional sense
and it is often needed to assume that the kernels are `nice' functions.
In this subsection we explain how we proceed to do so.
We fix a non-negative function
$\chi_o\in \mathcal D(\mathbb R)$
supported in $[1/4,4]$ such that $\chi_o\equiv 1$ on $[1/2,2]$.
For any $\epsilon>0$,
we write $\chi_\epsilon(x)=\chi_o(\epsilon |x|_{\nu_o}^{2\nu_o})$
where $|\cdot|_{\nu_o}$ is the homogeneous norm
defined in \eqref{eq_norm_||nuo}
for $\nu_o$ the smallest common multiple of the $\upsilon_j$.
Cleary $\chi_\epsilon\in \mathcal D(G)$.
We denote by $|\pi|$ a `norm' on $\widehat{G}$,
for example the distance between the co-adjoint orbits of $\pi$ and 1.
By definition, the $\mathcal H_\pi$'s, $\pi\in \widehat{G}$,
form a field of Hilbert spaces for the Plancherel measure.
So we can choose a generating sequence of vectors on $\mathcal H_\pi$
depending measurably on $\pi$.
We denote by $\text{\rm proj}_\epsilon$ the orthogonal projection on the $\lceil \epsilon^{-1}\rceil$-th first vectors of this sequence.
Let $\sigma\in S^m_{\rho,\delta}$.
We consider for any $\epsilon\in (0,1)$, the operator
$$
\sigma_\epsilon(x,\pi) :=\chi_\epsilon(x) 1_{|\pi|\leq \epsilon}
\sigma(x,\pi) \text{\rm proj}_\epsilon
.
$$
Clearly $\sigma_\epsilon\in S^m_{\rho,\delta}$
and for any $a,b,c\in \mathbb N_0$ there exists $C=C_{m,a,b,c}>0$ such that,
$$
\|\sigma_\epsilon\|_{S^m_{\rho,\delta},a,b,c}\leq C
\|\sigma\|_{S^m_{\rho,\delta},a,b,c}.
$$
The corresponding kernel is
$$
\kappa_\epsilon(x,y) =\chi_\epsilon (x) \int_{|\pi|\leq \epsilon}
\text{\rm Tr} \left( \sigma(x,\pi) \text{\rm proj}_\epsilon \right) d\mu(\pi)
,
$$
which is smooth in $x$ and $y$ and compactly supported in $x$.
From Proposition \ref{prop_kernel}, $\kappa_{\epsilon,x}$ decays rapidly at infinity in $y$ uniformly in $x$. Furthermore,
point-wise for $ x\in G$ and $y\in G\backslash\{0\}$,
or in the sense of $\mathcal S'(G)$-distribution for each $x\in G$,
we have the convergence
$\kappa_{\epsilon,x}(y) \underset {\epsilon\rightarrow 0} \longrightarrow
\kappa_x(y)$.
Let $T_\epsilon =Op(\sigma_\epsilon)$ be the corresponding operators.
For any $f\in \mathcal S(G)$, $T_\epsilon f\in \mathcal D(G)$ and
$$
Tf(x) = \lim_{\epsilon\rightarrow 0}T_\epsilon f(x)
\quad\mbox{where}\quad T=Op(\sigma)
.
$$
\medskip
In the proofs of the rest of the paper,
we will assume that the kernels of the operators are sufficiently regular
and compactly supported in $y$
so that all the performed operations,
e.g. composition of operators, convolution of kernels,
group Fourier transform of kernels etc... make sense.
This can be made rigorous via the procedure described above
since we will always obtain controls in $S^m_{\rho,\delta}$-semi-norms.
\subsection{Composition}
\label{subsec_composition}
We want to prove
\begin{theorem}
\label{thm_composition}
Let $1\geq \rho\geq \delta \geq 0$ with $\rho\not=0$ and $\delta\not=1$.
If $T_1\in \Psi^{m_1}_{\rho,\delta}$ and $T_2\in \Psi^{m_2}_{\rho,\delta}$
then $T_1T_2\in \Psi^{m}_{\rho,\delta}$ with $m=m_1+m_2$.
\end{theorem}
Let us start with some formal considerations.
Denoting $\sigma_j$ and $\kappa_j$ the symbol and kernel of $T_j$ for $j=1,2$,
it is not difficult to compute the following expression for the composition $T=T_1T_2$,
$$
Tf(x) = \int_G\int_G f(z) \kappa_2(y,z^{-1}y)\kappa_1(x,y^{-1} x) dy dz
.
$$
Thus the kernel of $T$ is
$$
\kappa_x(w)
=\int_G\kappa_2(xz^{-1},wz^{-1}) \kappa_1(x,z) dz
.
$$
Using the Taylor expansion for $\kappa_2$ in its first variable,
we have
$$
\kappa_2(xz^{-1},\cdot ) \approx \sum_\alpha \tilde q_\alpha (z) X^\alpha_x
\kappa_{2x} (\cdot)
\quad\mbox{thus}\quad
\kappa_x(w) \approx \sum_\alpha X^\alpha_x\kappa_{2x} * \tilde q_\alpha\kappa_1(w)
.
$$
Denoting $\sigma$ the group Fourier transform of $\kappa$,
we have
$$
\sigma(x,\pi):=\pi(\kappa_x)\approx
\sum_\alpha \Delta^\alpha \sigma_1(x,\pi) \
X^\alpha_x \sigma_2(x,\pi) .
$$
From Subsection \ref{subsec_1prop_symbols},
we know that
$$
\sum_{[\alpha]\leq M}
\Delta^\alpha \sigma_1(x,\pi) \
X^\alpha_x \sigma_2(x,\pi) \in S^{m-(\rho-\delta)M}_{\rho,\delta}
.
$$
Hence the main problem is to control the remainder coming from the use of the Taylor expansion;
this is the object of the following lemma.
\begin{lemma}
\label{lem_thm_composition}
We keep the notation defined just above and set
$$
\tau_M(x,\pi):=\sigma(x,\pi) -\sum_{[\alpha]\leq M}
\Delta^\alpha \sigma_1(x,\pi) \
X^\alpha_x \sigma_2(x,\pi)
.
$$
Let $\beta,\tilde\beta,\beta_o,\alpha_o\in \mathbb N_0^n$.
Then there exists $M_o\in \mathbb N_0$ such that for any integer $M>M_o$,
there exist $C>0$ and computable integers $a_1,b_1,c_1,a_2,b_2,c_2$
(independent of $\sigma_1$ and $\sigma_2$) such that we have
$$
\|\pi(X)^{\beta} \left\{ X_x^{\beta_o} \Delta^{\alpha_o}
\tau_M
\right\} \pi(X)^{\tilde \beta} \|_{op}
\leq
C
\|\sigma_1\|_{S^{m_1}_{\rho,\delta},a_1,b_1,c_1}
\|\sigma_2\|_{S^{m_2}_{\rho,\delta},a_2,b_2,c_2}
.
$$
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem_thm_composition}]
Proceeding as in the proof of \eqref{formula_leibniz}, we have:
\begin{eqnarray*}
&&
\tilde q_{\alpha_o}(y)\left(\kappa_x -\sum_{[\alpha]\leq M}
(\tilde q_\alpha \kappa_{1,x})*
(X_x^\alpha \kappa_{2,x})\right)
\\&&\quad=
\sum^{--}_{[\alpha_{o1}]+[\alpha_{o2}]=[\alpha_o]}
\int_G
\tilde q_{\alpha_{o1}}(z) \kappa_{1,x}(z)
\ R^{(\tilde q_{\alpha_{o2}}\kappa_{2,x\cdot})(yz^{-1})}_{0,M}(z^{-1})
\ dz.
\end{eqnarray*}
where the sign $\overset{--}\sum$ means linear combination,
here over $[\alpha_{o1}]+[\alpha_{o2}]=[\alpha_o]$,
Consequently,
$\pi(X)^{\beta_1} X_x^{\beta_o} \Delta^{\alpha_o}
\tau_M(x,\pi)
\pi(X)^{\beta_2}$
is the group Fourier transform of the function of $y$ given by
\begin{eqnarray}
&&\tilde X^{\beta_2}_y
X^{\beta_1}_y
X^{\beta_o}_x
\!\!\!\!\!\!\!\!\!\!\!\!
\sum^{--}_{[\alpha_{o1}]+[\alpha_{o2}]=[\alpha_o]}
\!\!
\int_G
\tilde q_{\alpha_{o1}}(z) \kappa_{1,x}(z) \
R^{(\tilde q_{\alpha_{o2}}\kappa_{2,x\cdot})(yz^{-1})}_{0,M}(z^{-1})
\ dz\nonumber
\\&&\quad =
\!\!\!\!\!\!\!\!\!
\sum^{--}_{\substack{
[\alpha_{o1}]+[\alpha_{o2}]=[\alpha_o]\\
[\beta_{o1}]+[\beta_{o2}]=[\beta_o]}}
\int_G
\tilde q_{\alpha_{o1}}(z) X^{\beta_{o2}}_x\kappa_{1,x}(z)\times
\nonumber
\\&&\qquad\qquad\qquad\times
R^{\tilde X^{\beta_2}_y
X^{\beta_1}_y (\tilde q_{\alpha_{o2}}X^{\beta_{o1}}_x\kappa_{2,x\cdot})(yz^{-1})}_{0,M}(z^{-1})
\ dz.
\label{eq_pf_lem2_thm_product_1}
\end{eqnarray}
After some manipulations involving integration by parts and Leibniz rules,
we obtain that
$\pi(X)^{\beta_1} X_x^{\beta_o} \Delta^{\alpha_o}
\tau_M(x,\pi)
\pi(X)^{\beta_2}$ is a linear combination
over
$[\alpha_{o1}]+[\alpha_{o2}]=[\alpha_o]$,
$[\beta_{o1}]+[\beta_{o2}]=[\beta_o]$ and
$[\beta_{11}]+[\beta_{12}]=[\beta_1]$ of
\begin{equation}
\int_G
X^{\beta_{12}}_{z_2=z}\tilde q_{\alpha_{o1}}(z_2) X^{\beta_{o2}}_x\kappa_{1,x}(z_2) \pi(z)^*
R^{(
(X^{\beta_{o1}}_x X_{x'}^{\beta_{11}} \Delta^{\alpha_{o2}}\sigma_2(xx',\pi)
\pi(X)^{\beta_2})}_{x'=0,M-[\beta_{11}]}(z^{-1})
dz,
\label{eq_pf_lem2_thm_product_mainexpression1}
\end{equation}
where we have extended the notation of the Taylor remainder to accept vector valued functions.
The adapted statement of Taylor's estimates given in Proposition \ref{prop_Taylor}
remains valid.
We consider each integral \eqref{eq_pf_lem2_thm_product_mainexpression1}
and insert
$\pi(\text{\rm Id} +\mathcal R)^{\frac{e_o}\nu}$
and its inverse
with the exponent $e_o$ to be determined in terms of $\beta_{1j},\beta_{oj}, \alpha_{o,j}$, $j=1,2$.
We decompose
$-\frac{e_o}\nu =e_1 +e_2$ with $e_1\in \mathbb N_0$.
Therefore, each integral in \eqref{eq_pf_lem2_thm_product_mainexpression1}
is equal to
\begin{eqnarray}
\int_G
X^{\beta_{12}}_{z_2=z}\tilde q_{\alpha_{o1}}(z_2) X^{\beta_{o2}}_x\kappa_{1,x}(z_2) \pi(z)^* \pi(\text{\rm Id} +\mathcal R)^{e_1}
\ \pi(\text{\rm Id} +\mathcal R)^{e_2} \nonumber\\
\{R^{(
(\pi(\text{\rm Id} +\mathcal R)^{\frac{e_o}\nu} X^{\beta_{o1}}_x X_{x'}^{\beta_{11}} \Delta^{\alpha_{o2}}\sigma_2(xx',\pi)
\pi(X)^{\beta_2})}_{x'=0,M-[\beta_{11}]}\}(z^{-1})dz
.\nonumber
\end{eqnarray}
We can always write
$$
\pi(z)^* \pi(\text{\rm Id} +\mathcal R)^{e_1}
= \left( \sum^{--}_{[\beta']=e_1\nu} X_z^{\beta'}\pi(z) \right)^*,
$$
and, therefore, after integrating by parts,
\eqref{eq_pf_lem2_thm_product_mainexpression1} is equal to
\begin{eqnarray*}
\sum^{--}_{[\beta'_1]+[\beta'_2]=[\beta']=e_1\nu}
\int_G
X^{\beta'_1}_{z_2=z} X^{\beta_{12}}_{z_2=z}\tilde q_{\alpha_{o1}}(z_2) X^{\beta_{o2}}_x\kappa_{1,x}(z_2) \pi(z)^*
\pi(\text{\rm Id} +\mathcal R)^{e_2} \\
X^{\beta'_2}_{z_1=z} \{R^{(
(\pi(\text{\rm Id} +\mathcal R)^{\frac{e_o}\nu} X^{\beta_{o1}}_x X_{x'}^{\beta_{11}} \Delta^{\alpha_{o2}}\sigma_2(xx',\pi)
\pi(X)^{\beta_2})}_{x'=0,M-[\beta_{11}]}\}(z_1^{-1})dz
.
\end{eqnarray*}
We fix a pseudo-norm $|\cdot|$ on $G$.
By Taylor's estimates and easy manipulations,
we obtain:
\begin{eqnarray*}
&&\|\{ X^{\beta'_2}_{z_1=z} \{R^{(
(\pi(\text{\rm Id} +\mathcal R)^{\frac{e_o}\nu} X^{\beta_{o1}}_x X_{x'}^{\beta_{11}} \Delta^{\alpha_{o2}}\sigma_2(xx',\pi)
\pi(X)^{\beta_2})}_{x'=0,M-[\beta_{11}]}\}(z_1^{-1})\|
\\ &&\leq C \sup_{x_1\in G}
\|\sigma_2(x,\pi)\|_{S^{m_2}_{\rho,\delta}, [\alpha_{o2}], b, [\beta_2]}
\!\!\!\!\!\!
\!\!\!\!\!\!
\!\!\!\!\!\!
\sum_{\substack{
[\gamma]>M-[\beta_{11}]-[\beta'_2]\\
|\gamma|<\lceil M-[\beta_{11}]-[\beta'_2]\rfloor +1}}
\sum_{\substack{[\beta_{o1}']\geq [\beta_{o1}]\\ |\beta_{o1}'|\leq |\beta_{o1}|}}
\!\!\!\!\!\!
|z|^{[\gamma]+[\beta'_{o1}]-[\beta_{o1}]},
\end{eqnarray*}
for $b:=
\max_{|\beta'_{o1}|\leq |\beta_o|} [\beta'_{o1}]+
\max_{|\gamma| <M+1}[\gamma]+e_1\nu +[\beta_1]$.
The last inequality is valid
if we have chosen $e_o,e_1,e_2$ satisfying for all $\beta_{o1}'$ and $\beta'_2$
as in the sums above, the condition
$$
e_o\leq \rho[\alpha_{o2}]-m_2-\delta(
[\beta_{o1}']+[\gamma]+[\beta'_2]+[\beta_{11}])
-[\beta_2],
$$
which is implied, in particular, by
$$
e_o\leq \rho[\alpha_{o2}]-m_2-\delta([\beta_{o1}]+M)
-[\beta_2].
$$
It is now time to choose:
\begin{itemize}
\item $e_o=e_1=e_2=0$
if $\rho[\alpha_{o2}]-m_2-\delta([\beta_{o1}]+M)
-[\beta_2]\geq 0$,
\item but if $\rho[\alpha_{o2}]-m_2-\delta([\beta_{o1}]+M)
-[\beta_2]< 0$,
\end{itemize}
$$
e_o:=\rho[\alpha_{o2}]-m_2-\delta([\beta_{o1}]+M)
-[\beta_2],\quad
e_1=\lceil -\frac{e_o}\nu\rceil\in \mathbb N,\quad
e_2=-\frac{e_o}\nu -e_1<0.
$$
We can now go back to
\eqref{eq_pf_lem2_thm_product_mainexpression1}
and obtain that
its operator norm is
\begin{eqnarray*}
\leq C
\sup_{x\in G}
\|\sigma_2(x,\pi)\|_{S^m_{\rho,\delta}, [\alpha_{o2}], b, [\beta_2]}
\sum_{[\beta'_1]+[\beta'_2]=e_1\nu}
\int_G |X^{\beta_{12}}(\tilde q_{\alpha_{o1}} X^{\beta_{o2}}_x\kappa_{1,x})(z)|
\\
\sum_{\substack{
[\gamma]>M-[\beta_{11}]-[\beta'_2]\\
|\gamma|<\lceil M-[\beta_{11}]-[\beta'_2]\rfloor +1}}
\sum_{\substack{[\beta_{o1}']\geq [\beta_{o1}]\\ |\beta_{o1}'|\leq |\beta_{o1}|}}
\!\!\!\!\!\!
|z|^{[\gamma]+[\beta'_{o1}]-[\beta_{o1}]}
dz
.
\end{eqnarray*}
Since $X^{\beta_{12}}(\tilde q_{\alpha_{o1}} X^{\beta_{o2}}_x\kappa_{1,x})$
behaves like a Schwartz function away from the origin
and like $\leq C|z|^{-p_o}$
with $p_o$ depending on $m_1,\beta_{12},\alpha_{o1},\beta_{o2}$
near the origin (see Proposition \ref{prop_kernel} and Section \ref{subsec_1prop_symbols}),
we see that all these integrals converge
when $-p_o+[\gamma]+[\beta'_{o1}]-[\beta_{o1}]>-Q$
for all indices as above,
and for this to be true it suffices that
$-p_o+M-[\beta_{11}]+e_o-\nu >-Q$.
We can always find $M$ such that this is satisfied
and in this case the operator norm of
\eqref{eq_pf_lem2_thm_product_mainexpression1}
is
$$
\leq C
\sup_{x\in G}
\|\sigma_2(x,\pi)\|_{S^{m_2}_{\rho,\delta}, [\alpha_{o2}], b, [\beta_2]}
\|\sigma_1\|_{S^{m_1}{\rho,\delta},a_1,b_1,c_1}
$$
for some (computable) integers $a_1,b_1,c_1$.
We choose $M$ the smallest integer which is a linear combination over $\mathbb N_0$ of the weights $\upsilon_1,\ldots,\upsilon_n$
such that all the operator norm of \eqref{eq_pf_lem2_thm_product_mainexpression1}
over
$[\alpha_{o1}]+[\alpha_{o2}]=[\alpha_o]$,
$[\beta_{o1}]+[\beta_{o2}]=[\beta_o]$,
$[\beta_{11}]+[\beta_{12}]=[\beta_1]$
are finite as above.
This proves that
the operator norm of
$\pi(X)^{\beta_1} X_x^{\beta_o} \Delta^{\alpha_o}
\tau_M(x,\pi)
\pi(X)^{\beta_2}$
is estimated as stated.
\end{proof}
Hence Lemma \ref{lem_thm_composition} yields
the following more precise version of Theorem \ref{thm_composition}.
\begin{corollary}
\label{cor_thm_composition}
Under the hypotheses of Theorem \ref{thm_composition},
writing $T_1=Op(\sigma_1)$ and $T_2=Op(\sigma_2)$,
there exists a unique symbol $\sigma\in S^m_{\rho,\delta}$ such that $T_1T_2=Op(\sigma)$.
Furthermore
$$
\sigma - \sum_{[\alpha]\leq M} \Delta^\alpha \sigma_1 \ X_x^\alpha \sigma_2
\ \in S^{m-(\rho-\delta)M}_{\rho,\delta}
,
$$
and the mapping
$$
\left\{\begin{array}{rcl}
S^{m_1}_{\rho,\delta} \times S^{m_1}_{\rho,\delta}
&\longrightarrow& S^{m-(\rho-\delta)M}_{\rho,\delta} \\
(\sigma_1,\sigma_2)
&\longmapsto&
\sigma - \sum_{[\alpha]\leq M} \Delta^\alpha \sigma_1 \ X_x^\alpha \sigma_2
\end{array}\right.
,
$$
is continuous.
\end{corollary}
With similar methods, we can prove that $\Psi^m_{\rho,\delta}$ is stable
by taking the formal adjoint of an operator, that is,
if $T\in\Psi^m_{\rho,\delta}$ then $T^*$ defined via
$(Tf_1,f_2)_{L^2} = (f_1,T^*f_2)_{L^2}$ is also in
$\Psi^m_{\rho,\delta}$:
\begin{theorem}
Let $1\geq \rho\geq \delta \geq 0$ with $\rho\not=0$ and $\delta\not=1$.
If $T\in \Psi^m_{\rho,\delta}$, then its formal adjoint $T^*$
is also in $\Psi^m_{\rho,\delta}$.
More precisely,
writing $T=Op(\sigma)$
there exists a unique symbol $\sigma^{(*)}\in S^m_{\rho,\delta}$ such that $T^*=Op(\sigma^{(*)})$.
Furthermore
$$
\sigma^{(*)} - \sum_{[\alpha]\leq M}
\Delta^\alpha X_x^\alpha \sigma^*
\ \in S^{m-(\rho-\delta)M}_{\rho,\delta}
,
$$
and the mapping
$$
\left\{\begin{array}{rcl}
S^{m}_{\rho,\delta}
&\longrightarrow& S^{m-(\rho-\delta)M}_{\rho,\delta} \\
\sigma
&\longmapsto&
\sigma^{(*)} - \sum_{[\alpha]\leq M}
\Delta^\alpha X_x^\alpha \sigma^*
\end{array}\right.
,
$$
is continuous.
\end{theorem}
Indeed let us perform formal considerations analogous to the ones for the composition.
Let $T=Op(\sigma)\in \Psi^m_{\rho,\delta}$ with kernel $\kappa_x$.
It is not difficult to compute that the kernel of $T^*$ is $\kappa_x^{(*)}$
given by
$$
\kappa_x^{(*)}(y)=\kappa^*_{xy^{-1}}(y)=\bar \kappa_{xy^{-1}}(y^{-1})
.
$$
Using the Taylor expansion for $\kappa^*_x$ in $x$,
we obtain
$$
\kappa_x^{(*)}(y) \approx \sum_\alpha
\tilde q_\alpha (y) X^\alpha_x\kappa_x^*(y)
.
$$
Denoting $\sigma^{(*)}$ the group Fourier transform of $\kappa^{(*)}$,
we have
$$
\sigma^{(*)}(x,\pi):=\pi(\kappa_x^{(*)})\approx
\sum_\alpha
\Delta^\alpha X^\alpha_x \sigma(x,\pi)^* .
$$
From Subsection \ref{subsec_1prop_symbols},
we know that
$$
\sum_{[\alpha]\leq M}
\Delta^\alpha X^\alpha_x \sigma(x,\pi)^*
\in S^{m-(\rho-\delta)M}_{\rho,\delta}
.
$$
Hence the main problem is as above
to control the remainder coming from the use of the Taylor expansion.
The proof proceeds in a similar way and is left to the reader.
Finally, we note that the proof of Theorem \ref{thm_composition}
can be adapted to provide the treatment of the remainder also
for composition of operators on compact Lie groups in
\cite[Theorem 10.7.8]{ruzh+turunen_bk2010}.
|
1,116,691,500,445 | arxiv | \section{Introduction}
The long standing problem of finding an exact description of the CFT dual to
M-theory on $AdS_4\times S^7$ (and orbifolds thereof), or the low energy limit of the world volume theory
of $N$ coinciding M2-branes, was solved beautifully in a recent paper of
Aharony, Bergman, Jafferis and Maldacena \cite{Aharony:2008ug}.
The dual gauge theory is a special case of the ${\cal N}=3$ superconformal Chern-Simons-matter (CSM) theories
studied in \cite{Gaiotto:2007qi} (see \cite{Schwarz:2004yj,Kapustin:1994mt,Chen:1992ee,Avdeev:1992jt,Avdeev:1991za}
for earlier works), which has quiver type matter content
and enhanced ${\cal N}=6$ supersymmetry.
In particular, the 't Hooft limit of the ${\cal N}=6$ CSM theory is argued to be dual to type IIA
string theory on $AdS_4\times \mathbb{CP}^3$. See also \cite{Benna:2008zy,
Minahan:2008hf,Imamura:2008nn,Honma:2008jd,Nishioka:2008gz,Bhattacharya:2008bj} for subsequent works on this
theory, and \cite{Bagger:2006sk,Gustavsson:2007vu,VanRaamsdonk:2008ft,
Distler:2008mk,Gomis:2008be,Ho:2008ei,Gustavsson:2008bf}
for recent works on M2-brane world volume theories.
In this paper we make a step toward understanding the details of the duality
between the ${\cal N}=6$ CSM theory and type IIA string on $AdS_4\times \mathbb{CP}^3$
in non(near)-BPS sectors, by exploring both spin chain operators in the superconformal gauge theory
(continuing on \cite{Gaiotto:2007qi}) and the Penrose limit of the string theory dual.
We study the two-loop dilatation operators in subsectors of the spin chain,
as well as the dispersion relation and scattering of impurities in an infinite chain that preserves
a centrally extended $SU(2|2)$ superconformal algebra.
The central charge of the $SU(2|2)$ algebra plays a key role in determining the exact dispersion
relations of the impurities. It is related to the momentum $P$ along the spin chain in the form
$$
Z = f(\lambda) (1-e^{2\pi i P})
$$
where $f(\lambda)$ is a nontrivial function of the 't Hooft coupling $\lambda=N/k$. We find that
$f(\lambda)$ scales differently with $\lambda$ at weak coupling (from perturbative gauge theory)
and at strong coupling (from the Penrose limit).
We discuss operator mixing and match multiplets in the weak coupling regime
with those in the pp-wave limit. We also present some preliminary discussions on the
giant magnons in $AdS_4\times\mathbb{CP}^3$.
Note added in proof: Upon completion of the bulk of this work, we received
\cite{Nishioka:2008gz} and \cite{Minahan:2008hf}, which contain results
that overlap with different parts of this paper.
\section{The ${\cal N}=6$ Chern-Simons-matter theory}
\subsection{Lagrangian}
It will be useful for us to formulate ${\cal N}=6$ Chern-Simons-matter theory in the ${\cal N}=2$ language.
The gauge group will be $U(N)\times U(N)$, with a pair of chiral fields $A_i$ ($i=1,2$) in the bifundamental
representation $({\bf N},{\bf \bar N})$, and $B_i$ in the conjugate representation
$({\bf \bar N},{\bf N})$. There is an ${\cal N}=2$ superpotential
\begin{equation}
W = {4\pi\over k} {\rm Tr} \left( A_1 B_1 A_2 B_2 - A_1 B_2 A_2 B_1 \right)
\end{equation}
This theory possesses ${\cal N}=6$ supersymmetry, and is exactly conformal, with superconformal group $OSp(6|4)$. The scalar
components of $(A_1, A_2, B_1^\dagger, B_2^\dagger)$ transform in the ${\bf 4}$ of $SU(4)_R$,
whereas $(B_1, B_2, A_1^\dagger, A_2^\dagger)$ transform in the ${\bf \bar4}$.
The scalar potential can be written as $V = V_D + V_F$, where $V_F=|\partial W/\partial A_i|^2
+|\partial W/\partial B_i|^2$, and $V_D$ comes the coupling of the scalar
fields to the auxiliary fields $\sigma$ and $\tilde\sigma$ (which lie in the ${\cal N}=2$ gauge multiplet
and take values in the adjoint of the two $U(N)$'s),
\begin{equation}
V_D = {\rm Tr} \left[(\sigma A_i-A_i\tilde\sigma) (A_i^\dagger\sigma -\tilde\sigma A_i^\dagger)\right]
+ {\rm Tr} \left[(\tilde\sigma B_i - B_i\sigma) (B_i^\dagger\tilde \sigma - \sigma B_i^\dagger)\right]
\end{equation}
where
\begin{equation}
\begin{aligned}
&\sigma = {2\pi\over k} (A_i A_i^\dagger - B_i^\dagger B_i),\\
&\tilde\sigma = -{2\pi\over k}(B_i B_i^\dagger - A_i^\dagger A_i).
\end{aligned}
\end{equation}
There are quartic boson-fermion coupling of the form
\begin{equation}
\begin{aligned}
{\cal L}_{F}&= {\cal L}_Y-{\rm Tr} \left(\psi_{A_i}^\dagger \sigma \psi_{A_i}- \psi_{A_i}^\dagger \psi_{A_i}\tilde\sigma\right)
-{\rm Tr}\left(\psi_{B_i}^\dagger \tilde\sigma \psi_{B_i}- \psi_{B_i}^\dagger \psi_{B_i}\sigma\right)
\\
&-{\rm Tr}\left(A_i^\dagger \chi^\dagger \psi_{A_i} - \tilde\chi^\dagger A_i^\dagger \psi_{A_i}+c.c.\right)
-{\rm Tr}\left(B_i^\dagger \tilde\chi^\dagger \psi_{B_i} - \chi^\dagger B_i^\dagger \psi_{B_i}+c.c.\right)
\end{aligned}
\end{equation}
where $\chi$ and $\tilde\chi$ are fermionic auxiliary fields in the ${\cal N}=2$ gauge multiplet,
\begin{equation}
\begin{aligned}
&\chi = {2\pi\over k} (\psi_{A_i} A_i^\dagger - B_i^\dagger \psi_{B_i}),\\
&\tilde\chi = -{2\pi\over k}(\psi_{B_i} B_i^\dagger - A_i^\dagger \psi_{A_i}).
\end{aligned}
\end{equation}
and ${\cal L}_Y$ is the Yukawa coupling,
\begin{equation}
\begin{aligned}
{\cal L}_Y &= {\partial^2 W\over \partial \phi_i \partial \phi_j } \psi_i \psi_j + c.c.\\
&= {4\pi\over k} {\rm Tr} (A_1 B_1 \psi_{A_2}\psi_{B_2} + \cdots)
\end{aligned}
\end{equation}
\subsection{Supersymmetry transformations}
\label{susy}
In manifestly ${\cal N}=6$ supersymmetric notation, we can write the supercharges as $Q_{IJ}=(Q^{IJ})^\dagger={1\over 2}\epsilon_{IJKL}
\bar Q^{KL}$, where $I,J,K,L=1,\cdots,4$. The scalars and fermions are denoted by $\phi_I$, $\bar\phi^I$,
$(\psi_I)_\alpha$ and $\bar\psi^I_\alpha$. One can explicitly identify them with the components fields
of ${\cal N}=2$ chiral multiplets as
\begin{equation}
\begin{aligned}
&\phi_1 = A_1,~~~\phi_2 = A_2,~~~\phi_3=B_1^\dagger, ~~~\phi_4=B_2^\dagger,\\
&\psi_1 = -\psi_{A_2}^\dagger,~~~\psi_2 = \psi_{A_1}^\dagger, ~~~
\psi_3 = -\psi_{B_2},~~~\psi_4 = \psi_{B_1}.
\end{aligned}
\end{equation}
The action of the supercharges on the fields is as follows
\begin{equation}\label{susygen}
\begin{aligned}
& Q_{IJ} \phi_K = \epsilon_{IJKL} \bar\psi^L,\\
& Q_{IJ} \bar\phi^K = \delta_I^K\psi_J - \delta_J^K \psi_I,\\
& (Q_{IJ})_\alpha (\psi_K)_\beta = \epsilon_{IJKL} i\sigma^\mu_{\alpha\beta} D_\mu \bar\phi^L
+{2\pi i\over k}\epsilon_{\alpha\beta} \epsilon_{IJKL}( \bar\phi^L \phi_M \bar\phi^M-\bar\phi^M \phi_M \bar\phi^L)
+{4\pi i\over k} \epsilon_{\alpha\beta}\epsilon_{IJLM} \bar\phi^L \phi_K \bar\phi^M,\\
& (Q_{IJ})_\alpha (\bar\psi^K)_\beta = \delta_I^K \left[i\sigma^\mu_{\alpha\beta} D_\mu \phi_J
-{2\pi i\over k}\epsilon_{\alpha\beta} ( \phi_J \bar\phi^M \phi_M-\phi_M \bar\phi^M \phi_J)\right]\\
& ~~~~~~ - \delta_J^K \left[i\sigma^\mu_{\alpha\beta} D_\mu \phi_I
-{2\pi i\over k}\epsilon_{\alpha\beta} ( \phi_I \bar\phi^M \phi_M-\phi_M \bar\phi^M \phi_I)\right]
-{4\pi i\over k} \epsilon_{\alpha\beta}(\phi_I \bar\phi^K \phi_J-\phi_J \bar\phi^K \phi_I),\\
& Q_{IJ} A_\mu = i\sigma_\mu \chi_{IJ}= {2\pi i\over k}\sigma_\mu(\phi_{[I} \psi_{J]} + {1\over 2} \epsilon_{IJKL} \bar\psi^{K}\bar\phi^{L}),\\
& Q_{IJ} \tilde A_\mu = i\sigma_\mu \tilde\chi_{IJ}= {2\pi i\over k}\sigma_\mu(\psi_{[J}\phi_{I]} + {1\over 2} \epsilon_{IJKL} \bar\phi^{L}\bar\psi^{K}).
\end{aligned}
\end{equation}
\section{Spin chains in ${\cal N}=6$ CS}
\subsection{$SU(2)_A\times SU(2)_B$ sector}
Let us focus on the $SU(2)_A\times SU(2)_B\times U(1)$ subgroup of $SU(4)_R$, where $A_i$ transform in the representation $({\bf 2},{\bf 1},+1)$, and $B_i$ in the representation $({\bf 1},{\bf 2},-1)$. Consider spin chains
of the form
\begin{equation}
{\rm Tr} (A_{i_1} B_1 A_{i_2} B_1 A_{i_3} B_1\cdots)
\end{equation}
These are chiral operators, but in general not primaries due to the superpotential. At two-loop,
the sextic scalar potential coming from the superpotential contributes to the anomalous dimension of
the above operator. The relevant potential term is
\begin{equation}
{16\pi^2\over k^2}{\rm Tr} \left[(A_1 B_1 A_2-A_2 B_1 A_1) (A_1 B_1 A_2-A_2 B_1 A_1)^\dagger\right]
\end{equation}
The potential terms in $V_D$ does not contribute at two-loop.
Similarly, the terms coupling the scalars to fermions in ${\cal L}_F$ do not have the right structure to
contribute to the two-loop anomalous dimension of the chiral operator either (other than an overall
shift which is fixed by the BPS bound for the chiral primaries, i.e. the operators with all the $A_i$'s
symmetrized).
\bigskip
\bigskip
\centerline{\begin{fmffile}{diag1}
\begin{tabular}{c}
\begin{fmfgraph*}(35,30)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3,i4,i5}
\fmflabel{$B_1$}{i1}
\fmflabel{$A_1$}{i2}
\fmflabel{$B_1$}{i3}
\fmflabel{$A_2$}{i4}
\fmflabel{$B_1$}{i5}
\fmflabel{$B_1$}{o1}
\fmflabel{$A_2$}{o2}
\fmflabel{$B_1$}{o3}
\fmflabel{$A_1$}{o4}
\fmflabel{$B_1$}{o5}
\fmfright{o1,o2,o3,o4,o5}
\fmf{fermion}{i1,o1}
\fmf{fermion}{i2,v1}
\fmf{fermion}{v1,o2}
\fmf{fermion}{i3,v1}
\fmf{fermion}{v1,o3}
\fmf{fermion}{i4,v1}
\fmf{fermion}{v1,o4}
\fmf{fermion}{i5,o5}
\fmfdot{v1}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}
}
\bigskip
\noindent
The two-loop integral in the above diagram is
\begin{equation}
\int {d^3y\over (4\pi)^6} {1\over |y|^3 |x-y|^3} \sim {1\over 8\pi^2}{\ln\Lambda\over (4\pi|x|)^3}
\end{equation}
where $1/(4\pi |x|)$ is the scalar propagator in position space. There is also a factor of $16\pi^2\lambda^2$
from the vertices and contraction of color indices, and a factor of $1/2$ since we were calculating the
two point function of the spin chain operator as opposed to the anomalous dimension. Putting these together,
we then find the two-loop
spin chain Hamiltonian
\begin{equation}\label{XXXH}
H = -\lambda^2 \sum_i(P_{i,i+1}-1)
\end{equation}
This is the Hamiltonian of the Heisenberg XXX spin-$1/2$ chain. The dispersion relation of an impurity
in this $SU(2)$ sector moving with momentum $p$ is
\begin{equation}\label{disptwo}
E=4\lambda^2 \sin^2(\pi p) + {\cal O}(\lambda^3)
\end{equation}
There may be a regularization scheme dependent order $\lambda^3$ term, but its structure is the same as the
$\lambda^2$, since the corresponding three loop diagrams are obtained by attaching gauge propagators to
the two-loop diagrams.
Now let us allow the $B_1$'s to change into $B_2$ as well, so that the spin chain takes the form
\begin{equation}
{\rm Tr} (A_{i_1} B_{j_1} A_{i_2} B_{j_2} A_{i_3} B_{j_3}\cdots)
\end{equation}
Once again, at two-loop the only potential term that contributes to the anomalous dimension
are the (${\cal N}=2$) F-terms. Furthermore, the exchanges of $A_1$ and $A_2$ across $B_1$ or $B_2$ have
the same amplitude, and similarly for the exchange of $B_1$ and $B_2$ across $A_i$. Therefore, we find that
at two-loop the $SU(2)_A\times SU(2)_B$ spin chain is two decoupled XXX spin-$1/2$ chains (of $A$'s and $B$'s respectively).
\subsection{The $SU(2|2)$ infinite chain}
To gain further insight we shall consider the infinite chain (the ``vacuum")
\begin{equation}
{\rm Tr} (A_1 B_1 A_1 B_1 A_1 B_1\cdots)
\label{vacuum}
\end{equation}
It preserves an $SU(2|2)$ subgroup of $OSp(6|4)$. The bosonic part of $SU(2|2)$ is $SU(2)_G\times
SU(2)_r\times U(1)_D$, where $SU(2)_G$ rotates $A_2, B_2^\dagger$ as a doublet, $SU(2)_r$ is
the rotation group in spacetime, and $U(1)_D$ is generated by $D$, defined to be
the anomalous dimension. More precisely, $D=\Delta -J$, where $\Delta$ is the conformal dimension and $J$ is the eigenvalue of the Cartan generator of $SU(2)_{G'}$, which is the group rotating $A_1, B_1^\dagger$ (and similarly $\psi_1,\psi_3$) as a doublet
\footnote{Choosing the ``vacuum'' (\ref{vacuum}), one considers the breaking
$SU(4)_R \rightarrow SU(2)_{G'} \times SU(2)_{G} \times U(1)$, and the vacuum preserves
$SU(2)_G\times U(1)$.
The extra $U(1)$, which assigns charge $+1$ to $A_1, B_1^\dagger$ and charge $-1$ to
$A_2, B_2^\dagger$, commutes with the generators of $SU(2|2)$.}.
Therefore one has $J(A_1)=J(B_1)=\frac{1}{2}$,$\, J(A_2)=J(B_2)=0$,
and similarly for the fermions. The odd generators of $SU(2|2)$ are denoted by $Q_{A\alpha}, \bar S_{A\alpha}$,
where $A$ is an $SU(2)_G$ doublet index, and $\alpha$ is the spacetime spinor index. The
superalgebra is
\begin{equation}\label{sutt}
\begin{aligned}
&\{Q_{A\alpha}, Q_{B\beta}\}= \epsilon_{AB} \epsilon_{\alpha\beta}
Z,~~~~\{\bar S_{A\alpha}, \bar S_{B\beta}\}= \epsilon_{AB} \epsilon_{\alpha\beta} \bar Z,\\
&\{Q_{A\alpha}, \bar S_{B\beta}\} = \epsilon_{AB} \epsilon_{\alpha\beta} D
+\epsilon_{AB} J_{\alpha\beta} + \epsilon_{\alpha\beta} T_{AB}.
\end{aligned}
\end{equation}
where $Z$ is a central charge, related to the momentum of the impurities in the infinite chain,
to be determined later.
Comparing with the supersymmetry transformations (\ref{susygen}), the pair of supercharges that preserve
the vacuum spin chain is
$(Q_{12},-Q_{14})\sim Q_A$. In particular,
$J(Q_A)=\frac{1}{2}$, and $D=\Delta-J$ commutes with the supercharges as required by the $SU(2|2)$ algebra.
The basic impurities are $A_2, B_2^\dagger, (\psi_{B_2}^\dagger)_\alpha$ in place of $A_1$, and similarly
$A_2^\dagger, B_2, (\psi_{A_2}^\dagger)_\alpha$ in place of $B_1$. At zero momentum they
transform in the minimal short representation of $SU(2|2)$. We will write $\phi_A = (A_2, B_2^\dagger)
=(\phi_2,\phi_4)$,
and $\chi_\alpha = (\psi_{B_2}^\dagger)_\alpha$. From (\ref{susygen}) we have the supersymmetry transformations
on $(\phi_A,\chi_\alpha)$
\begin{equation}\label{susya}
\begin{aligned}
&Q_{A\alpha} \phi_B \sim \epsilon_{AB} \chi_\alpha,\\
&Q_{A\alpha} \chi_\beta \sim \epsilon_{\alpha\beta} {2\pi i\over k}(\phi_A B_1 A_1 - A_1 B_1\phi_A).
\end{aligned}
\end{equation}
In terms of impurities with momentum $p$,
we have
\begin{equation}
\begin{aligned}
&Q_{A\alpha} |\phi_B(p)\rangle \sim \epsilon_{AB} |\chi_\alpha(p)\rangle,\\
&Q_{A\alpha} |\chi_\beta(p)\rangle \sim \epsilon_{\alpha\beta}{2\pi i\over k}(1-e^{2\pi ip}) |\phi_A(p)\rangle.
\end{aligned}
\end{equation}
In (\ref{sutt}) we have normalized $Q_A$ and $\bar S_A$ to be complex conjugates of one another in radial
quantization. In general they are related to the supercharges in (\ref{susygen})
by a rescaling, which a priori may depend on the coupling $\lambda$ due to quantum corrections
to $\bar S_A$. The central charge of the $SU(2|2)$ algebra takes the form $Z=f(\lambda)(1-e^{2\pi ip})$,
where $f(\lambda)$ is an undetermined function of $\lambda$.
The basic impurities (4 bosonic and 4 fermionic) fall into two short representations:
\begin{equation}
\begin{aligned}
\{[1,0]|[0,1]\} \oplus \{[1,0]|[0,1]\}
\end{aligned}
\end{equation}
We will call them $(2|2)_A$ and $(2|2)_B$ impurities for short.
The short multiplet saturates
the BPS bound \cite{Beisert:2005tm,Beisert:2006qh},
\begin{equation}\label{bound}
\Delta-J = D = \sqrt{{1\over 4}+4f(\lambda)^2 \sin^2(\pi p)}
\end{equation}
By comparison with the two-loop spin chain Hamiltonian (\ref{XXXH}), we determine that $f(\lambda)\simeq \lambda$
in the weak 't Hooft coupling limit.
Let us check this relation for the fermionic impurity $\psi_{B_2}^\dagger$. There is in fact only one
diagram allowed by the index structure that contributes to the exchange of $A_1$ with $\psi_{B_2}^\dagger$
across a $B_1$ along the chain, as follows.
\bigskip
\bigskip
\centerline{ \begin{fmffile}{diag2}
\begin{tabular}{c}
\begin{fmfgraph*}(35,30)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3,i4,i5}
\fmflabel{$B_1$}{i1}
\fmflabel{$A_1$}{i2}
\fmflabel{$B_1$}{i3}
\fmflabel{$\psi_{B_2}^\dagger$}{i4}
\fmflabel{$B_1$}{i5}
\fmflabel{$B_1$}{o1}
\fmflabel{$\psi_{B_2}^\dagger$}{o2}
\fmflabel{$B_1$}{o3}
\fmflabel{$A_1$}{o4}
\fmflabel{$B_1$}{o5}
\fmfright{o1,o2,o3,o4,o5}
\fmf{fermion}{i1,o1}
\fmf{fermion}{i2,v1}
\fmf{fermion}{i3,v1}
\fmf{fermion}{v2,o3}
\fmf{fermion}{v2,o4}
\fmf{fermion}{i5,o5}
\fmf{dbl_plain_arrow,tension=2}{v2,i4}
\fmf{dbl_plain_arrow,tension=0,lab.side=left,lab=$\psi_{A_2}$}{v2,v1}
\fmf{dbl_plain_arrow,tension=2}{o2,v1}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile} }
\bigskip
There is a factor of $16\pi^2\lambda^2$ coming from the F-term vertices,
and a factor $1/2$ to convert to the anomalous dimension. The fermion propagator in position space is
$i{\slash \!\!\!x}/(4\pi |x|^3)$.
The loop integral involved is
\begin{equation}
-i\int {d^3y d^3z\over (4\pi)^7} {(\slash \!\!\!x-\slash \!\!\!z)
(\slash \!\!\!z-\slash \!\!\!y)\slash \!\!\!y\over z^2 (x-y)^2 |y|^3 |x-z|^3 |y-z|^3}
\end{equation}
whose logarithmically divergent part is
\begin{equation}
\begin{aligned}
&-2i{\slash \!\!\!x\over |x|^5} \int {d^3y d^3z \over (4\pi)^7} {(z-y)\cdot y\over z^2 |y|^3 |y-z|^3}
= -2i{\slash \!\!\!x\over |x|^5} \int {d^3y \over (4\pi)^7|y|^3} y^\mu {\partial\over \partial y^\mu}\int d^3z {1\over z^2 |y-z|}
\\
&={1\over 8\pi^2} {i\slash \!\!\!x \ln\Lambda\over (4\pi)^3|x|^5}
\end{aligned}
\end{equation}
The resulting anomalous dimension is identical to that of the $A_2$ (or $B_2^\dagger$) impurity,
which is expected since they are in the same short multiplet.
\subsection{Scattering and bound states}
\subsubsection{$(2|2)_A\otimes (2|2)_A$ sector}
Now let us consider the scattering of a pair of basic $(2|2)$ impurities, working
perturbatively at two-loop. First consider a pair of impurities both in the $(2|2)_A$ multiplet
(or similarly, both in the $(2|2)_B$ multiplet), consisting of the fields
$(A_2, B_2^\dagger; \psi_{B_2}^\dagger)$. In particular, two $A_2$ impurities
with momenta $p_1$ and $p_2$ scatter according to the Hamiltonian (\ref{XXXH}),
and can form a bound state with dispersion relation \cite{Faddeev:1996iy}
\begin{equation}
\Delta-J -1= 2\lambda^2 \sin^2(\pi p).
\end{equation}
This saturates the BPS bound for the $(4|4)$ short multiplet of spin content
$\{[2,0],[0,0]|[1,1]\}$ under $SU(2)_G\times SU(2)_r\subset SU(2|2)$. The bosonic part of this short multiplet consists
of the bound states of the pairs
\begin{equation}
A_2 A_2, ~B_2^\dagger B_2^\dagger,~A_2 B_2^\dagger,~\epsilon^{\alpha\beta}
(\psi_{B_2}^\dagger)_\alpha(\psi_{B_2}^\dagger)_\beta
\end{equation}
moving with momentum $p$. The wave function decays exponentially as the pair is separated along the chain.
There is another $(4|4)$ ``multiplet" of asymptotic scattering states of
two $(2|2)_A$ multiplets, of spin content $\{[0,2],[0,0]|[1,1]\}$, whose bosonic part consists of
\begin{equation}\label{scs}
|A_2(p_1)B_2(p_2)^\dagger-B_2(p_1)^\dagger A_2(p_2)^\dagger\rangle,~~~
\sigma_\mu^{\alpha\beta}
|(\psi_{B_2}^\dagger)_\alpha(p_1)(\psi_{B_2}^\dagger)_\beta(p_2)\rangle
\end{equation}
However, they cannot form bound states at two loop. This is easiest to see from the scattering
of a bosonic $(2|2)_A$ impurity, say $A_2$, with a fermionic $(2|2)_A$ impurity $\psi_{B_2}^\dagger$.
There is an exchange amplitude between $A_2$ and $\psi_{B_2}^\dagger$ (or $B_2$ and $\psi_{B_2}^\dagger$), as in the diagrams below,
which allows only one bound state between them of given total momentum $p$. This bound state is
already included in the $\{[2,0],[0,0]|[1,1]\}$ multiplet, and hence there are no fermionic bound states
to pair up with potential bound states coming from (\ref{scs}).
\bigskip
\bigskip
\centerline{ \begin{fmffile}{diag3}
\begin{tabular}{c}
\begin{fmfgraph*}(35,30)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3,i4,i5}
\fmflabel{$B_1$}{i1}
\fmflabel{$A_2$}{i2}
\fmflabel{$B_1$}{i3}
\fmflabel{$\psi_{B_2}^\dagger$}{i4}
\fmflabel{$B_1$}{i5}
\fmflabel{$B_1$}{o1}
\fmflabel{$\psi_{B_2}^\dagger$}{o2}
\fmflabel{$B_1$}{o3}
\fmflabel{$A_2$}{o4}
\fmflabel{$B_1$}{o5}
\fmfright{o1,o2,o3,o4,o5}
\fmf{fermion}{i1,o1}
\fmf{fermion}{i2,v1}
\fmf{fermion}{i3,v1}
\fmf{fermion}{v2,o3}
\fmf{fermion}{v2,o4}
\fmf{fermion}{i5,o5}
\fmf{dbl_plain_arrow,tension=2}{v2,i4}
\fmf{dbl_plain_arrow,tension=0,lab.side=left,lab=$\psi_{A_1}$}{v2,v1}
\fmf{dbl_plain_arrow,tension=2}{o2,v1}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}
~~~~~~~~~~~~~~~~~~~~~~~\begin{fmffile}{diag4}
\begin{tabular}{c}
\begin{fmfgraph*}(35,30)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3,i4,i5}
\fmflabel{$B_1$}{i1}
\fmflabel{$B_2^\dagger$}{i2}
\fmflabel{$B_1$}{i3}
\fmflabel{$\psi_{B_2}^\dagger$}{i4}
\fmflabel{$B_1$}{i5}
\fmflabel{$B_1$}{o1}
\fmflabel{$\psi_{B_2}^\dagger$}{o2}
\fmflabel{$B_1$}{o3}
\fmflabel{$B_2^\dagger$}{o4}
\fmflabel{$B_1$}{o5}
\fmfright{o1,o2,o3,o4,o5}
\fmf{fermion}{i1,o1}
\fmf{fermion}{v1,i2}
\fmf{fermion}{i3,v1}
\fmf{fermion}{v2,o3}
\fmf{fermion}{o4,v2}
\fmf{fermion}{i5,o5}
\fmf{dbl_plain_arrow,tension=2}{v2,i4}
\fmf{dbl_plain_arrow,tension=0,lab.side=right,lab=$\psi_{B_1}$}{v1,v2}
\fmf{dbl_plain_arrow,tension=2}{o2,v1}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}}
\bigskip
\noindent
It is plausible that the $(4|4)$ bound state of a pair of $(2|2)_A$ impurities, which we denote by
$(4|4)_A$, remains a short multiplet at strong coupling.
\subsubsection{$(2|2)_A\otimes (2|2)_B$ sector}
From the earlier discussion on $SU(2)_A\times SU(2)_B$ sector of the spin chain we know that
the $A_2$ and $B_2$ impurities from $(2|2)_A$ and $(2|2)_B$ do not interact at two-loop (but
this is not necessarily the case for other pairs of impurities in $(2|2)_A\otimes (2|2)_B$).
In particular we have two-impurity states, with $A_2$ of momentum $p_1$ and $B_2$ of momentum $p_2$,
denoted by $|A_2(p_1)B_2(p_2)\rangle$, which are eigenstates of the two-loop dilatation operator.
By $SU(2|2)$ symmetry, there must be a 16-dimensional long multiplet of threshold (non-)scattering
states, of spin content $\{[2,0],[0,2],[0,0],[0,0]|[1,1],[1,1]\}$. The $[2,0]$ part consists of
the scalar triplet
\begin{equation}\label{trip}
|A_2(p_1)B_2(p_2)\rangle,~~|B_2^\dagger(p_1) A_2^\dagger(p_2)\rangle,~~
|A_2(p_1) A_2^\dagger(p_2) - B_2^\dagger(p_1) B_2(p_2)\rangle.
\end{equation}
The product representation
$(2|2)_A\otimes (2|2)_B$ also consists of the $SU(2)_r$ triplet of fermion bilinear
$$
\sigma_\mu^{(\alpha\beta)}|(\psi_{B_2})_\alpha^\dagger(p_1) (\psi_{A_2})_\beta^\dagger(p_2)\rangle
$$
Naively one may expect this to be the $[0,2]$ part of the long multiplet.
However, these pairs of basic impurities are interacting at two-loop, and the corresponding
states are not eigenstates of the Hamiltonian. In particular, the exchange amplitude
$$\sigma_\mu^{(\alpha\beta)}|\cdots(\psi_{B_2})_\alpha^\dagger (\psi_{A_2})_\beta^\dagger A_1\cdots\rangle
\to \sigma_\mu^{(\alpha\beta)}|\cdots A_1(\psi_{A_2})_\beta^\dagger(\psi_{B_2})_\alpha^\dagger\cdots\rangle$$
vanishes at two-loop, as the above diagram vanishes when the spinor indices
$\alpha,\beta$ are symmetrized. This effect
leads to a repulsive contact (i.e. nearest neighbor) interaction between the impurities
$(\psi_{B_2})_\alpha^\dagger(p_1)$ and $(\psi_{A_2})_\beta^\dagger(p_2)$ in the $SU(2)_r$ triplet
sector.
The resolution to this seeming puzzle is due to operator mixing, between say $\sigma_\mu^{(\alpha\beta)}|(\psi_{B_2})_\alpha^\dagger(p_1) (\psi_{A_2})_\beta^\dagger(p_2)\rangle$ and $|D_\mu(p)\rangle$, the state of an impurity $D_\mu A_1$
or $D_\mu B_1$ moving at momentum $p=p_1+p_2$. At two-loop this can be computed from the
amplitude
$$\sigma_\mu^{(\alpha\beta)}|\cdots(\psi_{B_2})_\alpha^\dagger (\psi_{A_2})_\beta^\dagger \cdots\rangle
\to |\cdots A_1 D_\mu B_1\cdots\rangle$$
via the following Feynman diagrams
\bigskip
\bigskip
\centerline{\begin{fmffile}{diag6}
\begin{tabular}{c}
\begin{fmfgraph*}(35,20)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2}
\fmfright{o1,o2}
\fmflabel{$\psi$}{i1}
\fmflabel{$\psi$}{i2}
\fmflabel{$D_\mu$}{o1}
\fmf{plain}{v1,o1}
\fmf{plain}{v2,o2}
\fmf{dbl_plain}{v1,i1}
\fmf{dbl_plain}{v2,i2}
\fmf{plain,right=.5,tension=0}{v1,v2}
\fmf{dbl_plain,left=.5,tension=0}{v1,v2}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}~~~~~~~
\begin{fmffile}{diag7}
\begin{tabular}{c}
\begin{fmfgraph*}(35,20)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmffixed{(0.85h,-.23h)}{i2,v1}
\fmffixed{(0.85h,-.23h)}{i3,v2}
\fmflabel{$\psi$}{i2}
\fmflabel{$\psi$}{i3}
\fmflabel{$D_\mu$}{o1}
\fmf{plain,tension=1}{v1,o1}
\fmf{plain,tension=0}{v2,o2}
\fmf{plain}{v2,o3}
\fmf{plain,tension=0.3}{v1,i1}
\fmf{dbl_plain}{v1,i2}
\fmf{dbl_plain}{v2,i3}
\fmf{dbl_plain,tension=.6}{v1,v2}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}~~~~~~or~~~~~
\begin{fmffile}{diag8}
\begin{tabular}{c}
\begin{fmfgraph*}(35,25)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfright{i1,i3,i4,u1}
\fmfleft{o1,o3,o4,w1}
\fmffixed{(0,0.2h)}{i2,i3}
\fmffixed{(0,0.2h)}{o2,o3}
\fmflabel{$\psi$}{i3}
\fmflabel{$\psi$}{i4}
\fmflabel{$A_\mu$}{o2}
\fmf{plain,tension=1}{v1,o3}
\fmf{plain,tension=1}{v1,o4}
\fmf{dbl_plain,tension=.7}{v2,i3}
\fmf{dbl_plain,tension=.7}{v1,i4}
\fmf{dbl_plain,tension=1}{v1,v2}
\fmf{wiggly,tension=0}{o2,v2}
\fmf{plain}{o1,i1}
\fmf{plain}{u1,w1}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}
}
\bigskip
\noindent It will turn out that we can determine the coefficients of these amplitudes simply based on the
consistency requirement that there are threshold non-scattering states of such mixed operators.
A simple example of such mixing at zero momentum (more precisely, at momentum $p=1$) is the following protected
operator obtained by acting on the vacuum chain with supercharges,
\begin{equation}\label{prot}
\begin{aligned}
&(Q_{13})_\alpha(Q_{24})_\beta |A_1B_1A_1B_1\cdots\rangle = -\sum_{n=even,\;m=odd} |
(\psi_{B_2})_\beta^\dagger(n) (\psi_{A_2})_\alpha^\dagger(m)\rangle\\
&~~~~~+i\sigma^\mu_{\alpha\beta}\sum_{n~even} |D_\mu A_1(n)\rangle
-i\epsilon_{\alpha\beta} \left(\sum_{n~odd}|\sigma_{24}(n)\rangle
-\sum_{n~even}|\tilde\sigma_{24}(n)\rangle\right) \\
&(Q_{24})_\beta(Q_{13})_\alpha |A_1B_1A_1B_1\cdots\rangle = \sum_{n=even,\;m=odd} |
(\psi_{B_2})_\beta^\dagger(n) (\psi_{A_2})_\alpha^\dagger(m)\rangle\\
&~~~~~+i\sigma^\mu_{\alpha\beta}\sum_{n~odd} |D_\mu B_1(n)\rangle
+i\epsilon_{\alpha\beta} \left(\sum_{n~odd}|\sigma_{24}(n)\rangle-\sum_{n~even}|\tilde\sigma_{24}(n)\rangle
\right) \\
\end{aligned}
\end{equation}
where $\sigma_{24}$ and $\tilde\sigma_{24}$ are defined as
\begin{equation}
\begin{aligned}
& \sigma_{24} = {2\pi \over k}(-\bar\phi^1\phi_1+\bar\phi^2\phi_2-\bar\phi^3\phi_3+\bar\phi^4\phi_4),\\
& \tilde\sigma_{24} = {2\pi \over k}(-\phi_1\bar\phi^1+\phi_2\bar\phi^2-\phi_3\bar\phi^3+\phi_4\bar\phi^4).
\end{aligned}
\end{equation}
The sum of the two lines in (\ref{prot}) gives the total derivative of the vacuum chain, whereas the difference
gives another protected operator (in both the $SU(2)_r$ triplet and singlet sector). A special case is when the length of the chain is 2,
and we obtain a component of the $SU(4)_R$ current
\begin{equation}
{(J_\mu)_1}^3={\rm Tr}\left[A_1 D_\mu B_1-(D_\mu A_1) B_1 -i (\psi_{B_2})_\alpha^\dagger(p_1)
\sigma_\mu^{\alpha\beta}(\psi_{A_2})_\beta^\dagger(p_2)\right]
\end{equation}
Let us now compute the operator mixing in the sector of a $(\psi_{B_2})_\alpha^\dagger$ and a
$(\psi_{A_2})_\beta^\dagger$ impurity in the triplet of $SU(2)_r$, or a single impurity $D_\mu$.
Denote by $|n,m\rangle$ the state with $(\psi_{B_2})_\alpha^\dagger$ at position $2n$
and $(\psi_{A_2})_\beta^\dagger$ at position $2m+1$, with the spinor indices contracted
by $\sigma_\mu^{\alpha\beta}$. Denote by $|D(n)\rangle$ the state of a $D_\mu$ acting on the
site $2n$, and by $|D'(n)\rangle$ the state of $D_\mu$ acting on the site $2n+1$. The two-loop dilatation operator then acts on these states as
\begin{equation}
\begin{aligned}
&H |n,m\rangle = -\lambda^2 (|n-1,m\rangle + |n+1,m\rangle + |n,m-1\rangle+|n,m+1\rangle -
4|n,m\rangle),\\
&~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
(|n-m-{1\over 2}|>{1\over 2}) \\
&H |n,n\rangle = -\lambda^2 (|n-1,n\rangle + |n,n+1\rangle -
(4-\delta)|n,n\rangle)\\
&~~~~~~~~~~~~~~~ - c_1 \lambda^2 (|D(n)\rangle - |D'(n)\rangle )
-c_2\lambda^2 (|D'(n-1)\rangle - |D(n+1)\rangle ), \\
&H |n,n-1\rangle = -\lambda^2 (|n+1,n-1\rangle + |n,n-2\rangle -
(4-\delta)|n,n-1\rangle)\\
&~~~~~~~~~~~~~~~- c_1 \lambda^2 (|D'(n-1)\rangle - |D(n)\rangle )
-c_2\lambda^2 (|D(n-1)\rangle - |D'(n)\rangle ), \\
& H|D(n)\rangle = -c_1\lambda^2 ( |n,n\rangle-|n,n-1\rangle)
-c_2\lambda^2 (|n+1,n\rangle-|n-1,n-1\rangle)
\\
&~~~~~~~~~~~~~~~ - c_3\lambda^2 (|D'(n-1)\rangle+|D'(n)\rangle-2|D(n)\rangle),\\
&~~~~~~~~~~~~~~~ - c_4\lambda^2 (|D(n-1)\rangle+|D(n+1)\rangle-2|D(n)\rangle),\\
& H|D'(n)\rangle = -c_1\lambda^2 ( |n+1,n\rangle-|n,n\rangle)
-c_2\lambda^2 (|n+1,n+1\rangle-|n,n-1\rangle)\\
&~~~~~~~~~~~~~~~- c_3\lambda^2 (|D(n)\rangle+|D(n+1)\rangle-2|D'(n)\rangle)\\
&~~~~~~~~~~~~~~~ - c_4\lambda^2 (|D'(n-1)\rangle+|D'(n+1)\rangle-2|D'(n)\rangle).
\end{aligned}
\end{equation}
where $\delta, c_1, c_2, c_3, c_4$ are constants that can be computed from the two-loop diagrams.
In particular, $\delta$ is a potentially nonzero correction to the anomalous dimension when
$(\psi_{B_2})_\alpha^\dagger$ and $(\psi_{A_2})_\beta^\dagger$ are next to each other (although
it will turn out to be zero in this case). The
coefficients $c_1$ and $c_2$ are due to the mixing between adjacent fermion pair $(\psi_{B_2})_\alpha^\dagger(\psi_{A_2})_\beta^\dagger$ and a $D_\mu$ on the nearest and
next-to-nearest neigboring sites, respectively. $c_3$ is due to the mixing of a $D_\mu$ impurity
with another $D_\mu$ on nearest
neighboring sites, according to the diagrams
\bigskip
\bigskip
\centerline{\begin{fmffile}{diag9}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmffixed{(0.2h,-0.4h)}{v2,v1}
\fmfleft{i3,i2,i1}
\fmfright{o3,o2,o1}
\fmflabel{$D$}{i3}
\fmflabel{$D$}{o1}
\fmf{plain}{i1,v2}
\fmf{plain}{v2,v3}
\fmf{plain}{v3,o1}
\fmf{plain}{i3,o3}
\fmf{photon,tension=0}{i2,v1}
\fmf{photon,tension=0}{v2,v1}
\fmf{photon,tension=0}{v3,v1}
\fmfdot{v1,v2,v3}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}~~~~~
\begin{fmffile}{diag10}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmffixed{(0,0.5h)}{v2,v1}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmflabel{$D$}{i1}
\fmflabel{$D$}{o3}
\fmf{plain}{i1,v2}
\fmf{plain}{v2,o1}
\fmf{plain}{i3,v3}
\fmf{plain}{v3,o3}
\fmf{photon,tension=0}{i2,v1}
\fmf{photon,tension=0}{v2,v1}
\fmf{photon,tension=0}{v3,v1}
\fmfdot{v1,v2,v3}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}~~~~~
\begin{fmffile}{diag11}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmflabel{$D$}{i1}
\fmflabel{$D$}{o3}
\fmf{plain}{i1,v1}
\fmf{plain}{v1,o1}
\fmf{plain}{i3,v2}
\fmf{plain}{v2,o3}
\fmf{photon,tension=0}{i2,v1}
\fmf{photon,tension=0}{v2,v1}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}
~~~~~
\begin{fmffile}{diag12}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmffixed{(0.35h,0.5h)}{v2,v1}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmflabel{$D$}{i1}
\fmflabel{$D$}{o3}
\fmf{plain}{i3,v1}
\fmf{plain,tension=2}{v1,o3}
\fmf{plain}{i1,o1}
\fmfv{d.sh=circle,d.fi=shaded,d.si=.3h}{v2}
\fmf{photon,tension=0}{i2,v2}
\fmf{photon,tension=0}{v2,v1}
\fmfdot{v1}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}
}
\bigskip
\noindent whereas $c_4$ is due to mixing of $D_\mu$'s on next-to-nearest neighboring sites, from the following diagrams
\bigskip
\bigskip
\centerline{\begin{fmffile}{diag13}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmffixed{(.8h,.3h)}{v3,v1}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmflabel{$D$}{i1}
\fmflabel{$D$}{o3}
\fmf{plain}{i1,o1}
\fmf{plain}{i2,v1,o2}
\fmf{plain}{i3,v2,o3}
\fmf{photon,tension=0}{v3,v1,v2}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}~~~~~
\begin{fmffile}{diag14}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmffixed{(.5h,.3h)}{v3,v1}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmflabel{$D$}{i1}
\fmflabel{$D$}{o3}
\fmf{plain}{i1,o1}
\fmf{plain}{i2,v1,v1a,o2}
\fmf{plain}{i3,v2}
\fmf{plain,tension=2}{v2,o3}
\fmf{photon,tension=0}{v3,v1}
\fmf{photon,tension=0}{v1a,v2}
\fmfdot{v1,v2,v1a}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}~~~~~
\begin{fmffile}{diag15}
\begin{tabular}{c}
\begin{fmfgraph*}(25,15)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmffixed{(.5h,.3h)}{v3,v1}
\fmfleft{i1,i2,i3}
\fmfright{o1,o2,o3}
\fmflabel{$D$}{i1}
\fmflabel{$D$}{o3}
\fmf{plain}{i1,o1}
\fmf{plain}{i2,v1,v1a,o2}
\fmf{plain}{i3,v2}
\fmf{plain,tension=2}{v2,o3}
\fmf{photon,tension=0}{v3,v1a}
\fmf{photon,tension=0}{v1,v2}
\fmfdot{v1,v2,v1a}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}
}
\bigskip
\noindent We will not compute these diagrams directly, but simply determine them from the existence of the
threshold states at general momenta below. The result is
\begin{equation}
\delta=0,~~~ c_1= {1\over \sqrt{2}},~~~c_2= -{1\over \sqrt{2}},~~~c_3=1,~~~c_4={1\over 2}.
\end{equation}
A general state of total momentum $p$ takes the form
\begin{equation}
|\Psi\rangle=\sum_{n,m} e^{\pi ip(n+m+{1\over 2})} f(n-m) |n,m\rangle
+ g\sum_n e^{2\pi ipn} |D(n)\rangle + g'\sum_n e^{2\pi ip(n+{1\over 2})} |D'(n)\rangle
\end{equation}
Suppose $|\Psi\rangle$ is an energy eigenstate $H|\Psi\rangle = \lambda^2 E|\Psi\rangle$.
This is equivalent to the equations
\begin{equation}\label{threeq}
\begin{aligned}
&2 \cos(\pi p) (f(n-1)+f(n+1))-4 f(n) = -Ef(n),~~~~~n\geq 2{~\rm or ~}n\leq -1 \\
&2\cos(\pi p) f(2)-4f(1)+\sqrt{2}\cos(\pi p) (-e^{-\pi i p/2}g+e^{\pi i p/2} g') = -E f(1),\\
&2\cos(\pi p) f(-1)-4f(0)+\sqrt{2}\cos(\pi p) (-e^{-\pi i p/2}g'+e^{\pi i p/2} g) = -E f(0),\\
& (\cos(2\pi p)-1)g+2(\cos(\pi p)g'-g)+\sqrt{2}\cos(\pi p) (-e^{\pi i p/2} f(1)+e^{-\pi i p/2} f(0))=-Eg,\\
& (\cos(2\pi p)-1)g'+2(\cos(\pi p)g'-g)+\sqrt{2}\cos(\pi p) (-e^{\pi i p/2} f(0)+e^{-\pi i p/2} f(1))=-Eg',\\
\end{aligned}
\end{equation}
The threshold state is given by
\begin{equation}
\begin{aligned}
&f(n) = -{i\over \sqrt{2}}e^{-i(n-1/2) \alpha} (\cos(\pi p)-e^{-i\alpha}),~~~~(n\geq 1)\\
&f(n) = -{i\over \sqrt{2}}e^{-i(n-1/2) \alpha} (\cos(\pi p)-e^{i\alpha}),~~~~(n\leq 0)\\
&g=\sin({\pi p-\alpha\over 2}),~~~~g'=\sin({\pi p+\alpha\over 2}),\\
&E = 4\sin^2({\pi p-\alpha\over 2})+4\sin^2({\pi p+\alpha\over 2}).
\end{aligned}
\end{equation}
where $\alpha=\pi(p_1-p_2)$ is the difference between the momenta of the two $\psi$ impurities.
In particular the protected operators in the triplet sector of (\ref{prot}) are given by the special case
$p=1$, $\alpha=0$. One can also check that there are no bound states at two loop.\footnote{An attempt
to find such bound states is to set say $e^{i\alpha}=\cos(\pi p)$ with purely imaginary $\alpha$
in (\ref{threeq}), but this state is growing as opposed to decaying, exponentially, in
the separation between $\psi_{B_2}^\dagger$ and $\psi_{A_2}^\dagger$.
} A priori these threshold states may not survive at higher loops, but they may survive in
the pp-wave limit as unbound $(2|2)_A$ and $(2|2)_B$ impurities.
There is also operator mixing in the $[0,0]$ part of the long multiplet. For instance, the fermion bilinear singlet
\begin{equation}\label{sing}
|\psi_{B_2}^\dagger(p_1) \psi_{A_2}^\dagger(p_2)\rangle=\epsilon^{\alpha\beta}|(\psi_{B_2})_\alpha^\dagger(p_1) (\psi_{A_2})_\beta^\dagger(p_2)\rangle
\end{equation}
can mix with four bosons, via diagrams such as the following
\bigskip
\bigskip
\centerline{\begin{fmffile}{diag5}
\begin{tabular}{c}
\begin{fmfgraph*}(35,30)
\fmfstraight
\fmfset{arrow_len}{.3cm}\fmfset{arrow_ang}{12}
\fmfleft{i1,i2,i3,i4,i5,i6}
\fmflabel{$B_1$}{i1}
\fmffixed{(0h,0.4h)}{v1,v2}
\fmffixed{(0h,-0.1h)}{s2,i2}
\fmffixed{(0h,0.1h)}{s5,i5}
\fmffixed{(0.5h,0h)}{s2,v1}
\fmffixed{(0.5h,0h)}{s5,v2}
\fmflabel{$\psi_{B_2}^\dagger$}{s2}
\fmflabel{$\psi_{A_2}^\dagger$}{s5}
\fmflabel{$B_1$}{i6}
\fmflabel{$B_1$}{o1}
\fmflabel{$A_1$}{o2}
\fmflabel{$B_1$}{o3}
\fmflabel{$A_i$}{o4}
\fmflabel{$A_i^\dagger$}{o5}
\fmflabel{$A_1$}{o6}
\fmfright{o1,o2,o3,o4,o5,o6}
\fmf{fermion}{i1,o1}
\fmf{fermion}{i6,o6}
\fmf{dbl_plain_arrow}{v1,s2}
\fmf{dbl_plain_arrow,lab=$\psi_{A_2}$}{v1,v2}
\fmf{dbl_plain_arrow}{v2,s5}
\fmf{fermion,tesion=.2}{v1,o2}
\fmf{fermion,tesion=.2}{v1,o3}
\fmf{fermion,tesion=.2}{o5,v2,o4}
\fmfdot{v1,v2}
\end{fmfgraph*}
\end{tabular}
\end{fmffile}}
\bigskip
\noindent We have seen this mixing at zero momentum already in (\ref{prot}).
\section{Penrose limit of type IIA string theory on $AdS_4\times {\mathbb{CP}}^3$}
The 't Hooft limit of the ${\cal N}=6$ superconformal Chern-Simons-matter theory is dual
to type IIA string theory on $AdS_4\times {\mathbb{CP}}^3$ \cite{Aharony:2008ug}.
The metric on $AdS_4\times {\mathbb{CP}}^3$ can be written as \cite{Pope:1984bd}
\begin{equation}\label{cpmetric}
\begin{aligned}
ds^2 &= R^2\left\{-\cosh^2\rho dt^2 + d\rho^2 + \sinh^2\rho d\Omega_2^2 \right. \\ &\left.+4 d\mu^2+ 4\sin^2\mu \left[d\alpha^2 +
{1\over 4}\sin^2\alpha(\sigma_1^2+\sigma_2^2+\cos^2\alpha\sigma_3^2)+{1\over 4}\cos^2\mu (d\chi+\sin^2\alpha \sigma_3)^2 \right]\right\}
\end{aligned}
\end{equation}
Here $R$ is the radius of the $AdS_4$, and $\sigma_{1,2,3}$ are left-invariant 1-forms on an $S^3$, parameterized by $(\theta,\phi,\psi)$,
\begin{equation}
\begin{aligned}
& \sigma_1 = \cos\psi d\theta + \sin \psi \sin\theta d\phi,\\
& \sigma_2 = \sin\psi d\theta - \cos\psi \sin\theta d\phi,\\
& \sigma_3 = d\psi+\cos\theta d\phi.
\end{aligned}
\end{equation}
The range of the coordinates is $0\le \mu,\alpha \le \pi/2\,,0\le \theta \le \pi\,,0\le\phi\le 2\pi\,,0\le \chi,\psi\le 4\pi$. The Penrose limit is defined by focusing on the geodesic along $\chi$, with $\mu=\pi/4$,
$\alpha=0$, $\rho=0$. To do this we can define the new variables
\begin{equation}
\rho={\tilde \rho\over R},~~~\mu={\pi\over 4} + {u\over 2R},~~~ \alpha = {r\over\sqrt{2} R},~~~
dx^+={dt+d\chi/2\over 2},~~~~
dx^-=R^2{dt-d\chi/2\over 2},
\end{equation}
and scale $R\to \infty$. The metric then reduces to
\begin{equation}
\begin{aligned}
ds^2 &= -4dx^+dx^- + du^2 + d\tilde\rho^2 + \tilde\rho^2 d\Omega_2^2 + dr^2 + {r^2\over 4}\sum_{i=1}^3\sigma_i^2
-(u^2+\tilde\rho^2)(dx^+)^2 +{1\over 2}r^2\sigma_3 dx^+ \\
&= -4dx^+dx^- + du^2 + \sum_{i=1}^3 dy_i^2 + \sum_{j=1}^2 dz_j d\bar z_j
-(u^2+\sum_{i=1}^3 y_i^2)(dx^+)^2 - {i\over 2}\sum_{j=1}^2 (\bar z_j dz_j - z_j d\bar z_j) dx^+
\end{aligned}
\end{equation}
where $z_1$, $z_2$ are standard complex coordintes on the ${\mathbb C}^2$ with radial coordinates
$(r,\theta,\phi,\psi)$.
To put the metric in standard pp-wave form, we make a further coordinate change
\begin{equation}
z_j = e^{-{i}x^+/2} w_j,~~~~\bar z_j = e^{{i}x^+/2} \bar w_j,
\end{equation}
and the metric becomes
\begin{equation}\label{ppmetric}
\begin{aligned}
ds^2 &= -4dx^+dx^- + du^2 + \sum_{i=1}^3 dy_i^2 + \sum_{j=1}^2 dw_j d\bar w_j
-(u^2+\sum_{i=1}^3 y_i^2 + {1\over 4}\sum_{j=1}^2 |w_j|^2)(dx^+)^2
\end{aligned}
\end{equation}
There are also fluxes in the $AdS_4\times {\mathbb CP}^3$ background, reducing to
\begin{equation}
\begin{aligned}
& F_2 = - dx^+\wedge du, \\
& F_4 = -3 dx^+ \wedge dy_1 \wedge dy_2 \wedge dy_3,
\end{aligned}
\end{equation}
in the Penrose limit.
This pp-wave solution was found in \cite{Bena:2002kq} (see also \cite{Lin:2005nh}),
and preserves 24 supersymmetries
as the $AdS_4 \times {\mathbb CP}^3$ background does.
We shall organize the coordinates $(u,y_i)$ as $(X^1,X^2,X^3,X^4)$,
and $w_i, \bar w_i$ as $(X^5,X^6,X^7,X^8)$.
In the light cone gauge $X^+=\tau$, $\Gamma^+\Theta=0$, the Green-Schwarz action
for the type IIA string is (we follow the conventions of \cite{Michelson:2002ps}; for earlier studies of the GS string in this pp-wave background see \cite{Sugiyama:2002tf},\cite{Hyun:2002wu})
\begin{equation}
\begin{aligned}
S &= {1\over 2\pi\alpha'}\int dt\int_0^{2\pi\alpha' p^+} d\sigma \left\{ {1\over 2}\sum_{i=1}^8 \left[ (\dot X^i)^2 -({X^i}')^2 \right]
-{1\over 2}\sum_{i=1}^4 (X^i)^2 - {1\over 8}\sum_{j=5}^8 (X_j)^2\right. \\
&\left.~~~~~~
-i \bar \Theta\Gamma^-\left[\partial_\tau+\Gamma^{11}\partial_\sigma
-{1\over 4} \Gamma^1\Gamma^{11} -{3\over 4}\Gamma^{234} \right]\Theta
\right\}
\end{aligned}
\end{equation}
The bosonic excitations of the type IIA string in this pp-wave background have light cone
spectrum
\begin{equation}\label{bosexc}
H = \sum_{i=1}^4\sum_{n=-\infty}^\infty N_n^{(i)}\sqrt{1+{n^2\over(\alpha'p^+)^2}}+\sum_{j=5}^8
\sum_{n=-\infty}^\infty N_n^{(j)}\sqrt{{1\over 4}+{n^2\over (\alpha'p^+)^2}}
\end{equation}
In terms of the gauge theory spin chain variables,
$p^+=J/R^2$,\footnote{To see this, note that
$p^+=-{1\over 2}p_-={i\over 2 R^2}{\partial\over\partial x^-} = {i\over R^2}({1\over 2}\partial_t - \partial_\chi)$.
Since $\chi\sim \chi+4\pi$, the charge quantization is such that $-i\partial_\chi=J/2$, and
$i\partial_t = \Delta$. For the chiral primary with $\Delta=J$ ($J$ is the length of the alternating $A_1B_1$ chain
divided by 2), we have $p^+=J/R^2$.
} $p=n/J$, $R^2/\alpha' = \pi \sqrt{2\lambda}$ \cite{Aharony:2008ug}, we find the dispersion relations
\begin{equation}\label{disp}
\begin{aligned}
&E^{(i)} = \sqrt{1+2\lambda(\pi p)^2},~~~~~i=1,\cdots,4,\\
& E^{(j)}=\sqrt{{1\over 4}+
2\lambda(\pi p)^2},~~~~~j=5,\cdots,8.
\end{aligned}
\end{equation}
It follows from the fermion equation of motion that
\begin{equation}
\begin{aligned}
(\partial_\tau^2-\partial_\sigma^2)\Theta&=\left(\partial_\tau-\Gamma^{11}\partial_\sigma\right)
\left(\partial_\tau+\Gamma^{11}\partial_\sigma\right) \Theta
\\
&=
-\left({1\over 4} \Gamma^1\Gamma^{11} +{3\over 4}\Gamma^{234} \right)^2\Theta\\
&=\left( {5\over 8}+{3\over 8}\Gamma^{1234}\Gamma^{11}\right)\Theta
\end{aligned}
\end{equation}
Hence there are four fermions of mass $1$, satisfying $\Gamma^{1234}\Gamma^{11}\Theta=\Gamma^{5678}\Theta=\Theta$,
and four fermions of mass $1/2$, satisfying $\Gamma^{1234}\Gamma^{11}\Theta=\Gamma^{5678}\Theta=-\Theta$. Consequently the fermion
spectrum takes the same form as the bosonic one (\ref{bosexc}). Note that the Green-Schwarz action has symmetry group
$SU(2)'\times SU(2|2)$, which contains bosonic subgroup $SU(2)'\times SU(2)_G\times SU(2)_r\times U(1)_D$. Here
$SU(2)'\times SU(2)_G\simeq SO(4)$ is the rotation group on $(X^5, X^6, X^7, X^8)$, whereas
$SU(2)_r$ rotates $(X^2, X^3, X^4)$. The supersymmetry transformations of the $X^i$'s take the form
\begin{equation}
\begin{aligned}
& \delta_{A\alpha} u \sim \Theta_{A\alpha},\\
& \delta_{A\alpha} y^i \sim {(\sigma^i)_\alpha}^\beta\Theta_{A\beta},\\
& \delta_{A\alpha} X^{B\dot C} \sim \delta_A^B \Theta^{\dot C}_{\alpha},
\end{aligned}
\end{equation}
where $X^{A\dot B}$ stand for $(X^5, X^6, X^7, X^8)$ in $SU(2)_G\times SU(2)'$ bispinor notation.
This is consistent with the fact that $\Theta_{A\alpha}$ (satisfying $\Gamma^{5678}\Theta=\Theta$)
have the same mass as $(u,y^i)$, and $\Theta_{\dot A\alpha}$ (satisfying $\Gamma^{5678}\Theta=-\Theta$) have the same mass as $X^{A\dot B}$. The
$SU(2)'$ symmetry appears to be an accidental symmetry in the pp-wave limit, and
reduces to a $U(1)$ away from the Penrose limit.
We shall note an important difference
of this pp-wave limit from say the BMN scaling of ${\cal N}=4$ SYM \cite{Berenstein:2002jq}: the limit here is defined by
taking $\lambda, J\to \infty$, while keeping $\lambda/J^2$ fixed.
This may appear surprising from perturbative gauge theory, as we might have expected from the two-loop dispersion relation
(\ref{disptwo}) that the BMN scaling might be defined by $\lambda/J$ kept fixed.
On other hand, in general the $\ell$-loop corrections may contribute to the
dispersion relation in the form
\begin{equation}
E^{(\ell)} = \lambda^{\ell} \sum_{n=1}^{\lfloor\ell/2\rfloor} c_{\ell,n}\sin^{2n}(\pi p)
\end{equation}
where $c_{\ell, n}$ are generically nonzero (say for $n=1$ and $\ell>2$), and hence alters the form of the BMN scaling
at strong coupling. This indeed seems to happen in ${\cal N}=6$ CSM theory.
At classical dimension $1/2$, there are 4 bosonic and 4 fermionic excitations. They are the
modes of $(X^{A\dot B},\Theta^{\dot B}_\alpha)$,
where $\dot B$ is an $SU(2)'$ spinor index, and lie
in the two short multiplets $(2|2)_A$ and $(2|2)_B$ with respect to $SU(2|2)$.
Their exact dispersion relation at general 't Hooft coupling is given by
the BPS bound (\ref{bound}), where the function $f(\lambda)$ scales differently with $\lambda$
in the weak and strong coupling limits, see (\ref{disptwo}) and (\ref{disp})
\begin{equation}
\begin{aligned}
&f(\lambda) \sim \lambda,~~~~\lambda\ll 1,\\
&f(\lambda) \sim \sqrt{\lambda/2},~~~~\lambda\gg 1.
\end{aligned}
\end{equation}
A similar phenomenon was observed in \cite{Lin:2005nh}. This is in contrast with ${\cal N}=4$ SYM, where the central charge of the extended superconformal algebra
of the infinite spin chain is proportional to $\sqrt{\lambda}$ in both the weak and strong coupling limits
(although there is no reason why this should be true at general finite 't Hooft coupling, as pointed out in \cite{Hofman:2006xt}).
At classical dimension $1$, we have pairs of free excitations in
$(2|2)_A\otimes (2|2)_A$, $(2|2)_B\otimes (2|2)_B$,
$(2|2)_A\otimes (2|2)_B$, as well as an additional $(4|4)$ multiplet of spin content
$\{[0,2],[0,0]|[1,1]\}$ under $SU(2)_G\times SU(2)_r$, which are the modes of $(y^i,u,\Theta_{A\alpha})$.
Note that the dispersion relation of the $(4|4)$ multiplet is consistent with the form of the BPS bound
for $(4|4)$ short multiplets at generic coupling,
\begin{equation}
D = \sqrt{1+4f(\lambda)^2 \sin^2(\pi p)}
\end{equation}
It is plausible that this $(4|4)$ multiplet survives as a short multiplet away from the pp-wave limit.
Naively, we may expect
this multiplet to include the $D_\mu$ impurities. But as we have seen
at two-loop, the $D_\mu$ impurities mix with the $(2|2)_A\otimes (2|2)_B$
sector to form threshold scattering states, and there are no bound states
(at least not at two loop). It is a puzzle to us how to describe this
$(4|4)$ multiplet perturbatively in the gauge theory, if it exists.
In the $(2|2)_A\otimes (2|2)_A$ (or
$(2|2)_B\otimes (2|2)_B$) sector, at two-loop we have found bound states
that saturate the BPS bound; they may survive as short multiplets at finite coupling,
and may become free pairs of $(2|2)_A$ (or $(2|2)_B$) excitations in the pp-wave limit.
In $(2|2)_A\otimes (2|2)_B$ sector, we have found an $8+8$ long multiplet of
threshold (non-)scattering states at two-loop. It is unclear whether these survive at finite coupling,
and match onto the free pairs of $(2|2)_A$ and $(2|2)_B$ excitations in the pp-wave limit.
\section{Giant magnons}
It is easy to find giant magnon
solutions to the Nambu action in $AdS_4\times {\mathbb{CP}}^3$, following \cite{Hofman:2006xt}.
Corresponding to our vacuum spin chain is a string moving along a geodesic in the ${\mathbb{CP}}^3$,
with $\mu=\pi/4$ and $\alpha=0$, parameterized by $\chi$, in the coordinate
system of (\ref{cpmetric}). Alternatively, we can work with projective coordinates
$[z_1,z_2,z_3,z_4]$, and consider the geodesic given by $|z_1|=|z_3|$, $z_2=z_4=0$. The first type of giant
magnons move on the $S^2$ parameterized by $\chi$ and $\mu$, at $\alpha=0$. In projective coordinates,
this is the ${\mathbb{CP}}^1$ defined by $z_2=z_4=0$. Note that this sphere preserves the $SU(2)_G$ which rotates
$z_2$ and $z_4$. In particular it is consistent to restrict the giant magnon solution
to this $S^2$. The $S^2$, or $\mathbb{CP}^1$, has its radius equal to the $AdS_4$ radius $R=\sqrt{\pi\alpha'}(2\lambda)^{1\over 4}$.
The giant magnon solution takes the identical form as the one in \cite{Hofman:2006xt}, with dispersion relation
\begin{equation}
E-J = \sqrt{2\lambda}|\sin({\Delta\chi\over 4})|=\sqrt{2\lambda}|\sin(\pi p)|
\end{equation}
where the angular difference between two ends of the magnon, $\Delta\chi/2$, is identified with $2\pi p$
($\chi$ has periodicity $4\pi$). This is consistent with (and saturates) the large $\lambda$ limit of the BPS bound due to the centrally
extended $SU(2|2)$ algebra of the infinite chain.
Interestingly, there is a second class of giant magnons, which lie in an $\mathbb{RP}^2\subset \mathbb{CP}^3$,
rather than the ${\mathbb{CP}}^1$. The $\mathbb{RP}^2$'s that contain the geodesic $|z_1|=|z_3|$, $z_2=z_4=0$
are defined by
\begin{equation}
|z_1|=|z_3|,~~~~{z_2\over z_1-z_3}=\alpha x,~~~~{z_4\over z_1-z_3}=\beta x,~~~~x\in\mathbb{R}.
\end{equation}
This is seen more explicitly in rotated coordinates $(\tilde z_1,\tilde z_2,\tilde z_3,\tilde z_4) =
({z_1-z_3\over\sqrt{2}},z_2, i{z_1+z_3\over \sqrt{2}},z_4)$, where
the original geodesic is $\tilde z_3/\tilde z_1\in \mathbb{R}$, $\tilde z_2=\tilde z_4=0$,
and the defining equation of the ${\mathbb{RP}}^2$ becomes
\begin{equation}
{\tilde z_3\over \tilde z_1}\in {\mathbb R},~~~~{\tilde z_2\over \tilde z_1}=\alpha x,~~~~{\tilde z_4\over \tilde z_1}=\beta x,~~~~x\in\mathbb{R}.
\end{equation}
where $\alpha$ and $\beta$ are complex constants, and we have a family of $\mathbb{RP}^2$'s parameterized by $(\alpha,\beta)$, which
transform as a doublet under $SU(2)_G$. In particular, a given $\mathbb{RP}^2$ lies in a $\mathbb{CP}^2\subset \mathbb{CP}^3$
which is fixed by a $U(1)$ symmetry, and it is the fixed locus of an involution of the $\mathbb{CP}^2$. It is therefore consistent
to restrict the giant magnon solution to this $\mathbb{RP}^2$.
We can describe the $\mathbb{RP}^2$ as the quotient of an auxiliary sphere $\tilde S^2$ by
the antipodal map. Note however that this $\tilde S^2$ has radius $2R$. Hence the giant magnon solutions
that move along the geodesic, which is half the equator of $\tilde S^2$ (with ends identified by the antipodal map),
has dispersion relation
\begin{equation}
E-J = 2\sqrt{2\lambda}|\sin({\Delta\varphi\over 2})|
\end{equation}
where $\varphi$ is the angular variable on the equator of $\tilde S^2$, ranging from 0 to $2\pi$, and
$\Delta\varphi$ is the difference between the two ends of the giant magnon. On the $\mathbb{RP}^2$, however, $\varphi$
is identified with periodicity $\pi$, and it is natural to propose the identification with spin chain momentum
$\Delta\varphi = \pi p$. So we obtain the dispersion relation
\begin{equation}
E-J = 2\sqrt{2\lambda}|\sin(\pi p/2)|
\end{equation}
Note that with given $0<p<1$, there is another giant magnon with $\Delta\varphi = \pi(1-p)$ with the same ends
as the one with $\Delta\varphi = \pi p$, and has dispersion relation $E-J=2\sqrt{2\lambda}|\cos(\pi p/2)|$. The minimal energy
configuration carrying momentum $p$ should then be
\begin{equation}
E-J = 2\sqrt{2\lambda} \min\{|\sin(\pi p/2)|,|\cos(\pi p/2)|\}
\end{equation}
Note that this obeys the large $\lambda$ limit of the BPS bound (\ref{bound}), but does not saturate it.
Naively, based on the transformation under $SU(2)_G$, one may want to identify the first type of giant magnons
with the $(4|4)$ multiplet in the pp-wave limit, since it involves excitations in the $u$-direction (see
(\ref{ppmetric})), and to identify
the second type of giant magnons with the $(2|2)$ multiplets. However, the second type of giant magnons does not saturate
the BPS bound of the $SU(2|2)$ algebra, and should correspond to long multiplets. A potential resolution to this puzzle
is that there are fermion zero modes of the giant magnons, which carry additional representations of the $SU(2)_G$. It is therefore
not clear to us how to identify these giant magnons with the excitations in the pp-wave limit or in perturbative gauge theory.
\subsection*{Acknowledgments}
We are grateful to H. Lin, J. Maldacena,
A. Tomasiello and E. Witten for useful discussions, and especially to D. Jafferis for sharing with us an early draft.
The work of D.G. is supported in part by DOE grant DE-FG02-90ER40542.
The work of S.G. is supported in part by the Center for the
Fundamental Laws of Nature at Harvard University and by NSF grants PHY-024482 and DMS-0244464.
The work of X.Y. is supported by a Junior Fellowship from the Harvard Society of Fellows. S.G. thanks the 6th Simons Workshop in Mathematics and Physics at Stony Brook for hospitality during completion of this work.
|
1,116,691,500,446 | arxiv | \section{KENN\ architecture}
\label{sec:architecture}
As seen in Section~\ref{sec:kenn}, the core of KENN\ method is in the TBF . In KENN\ the TBF is applied inside a \emph{Clause Enhancer} (CE), which is a module instantiated for each grounded clause $c$ that calculates function $\delta_c$ as defined in Equation~\ref{eq:deltac}. Figure~\ref{fig:CE} shows in details the architecture of the CE.
\begin{figure}[H]
\centering
\input{tikz/CE}
\caption{Clause Enhancer for $A \lor \lnot B$, where $A$ and $B$ are grounded atoms}
\label{fig:CE}
\end{figure}
The CE receives as input a vector containing all the grounded atoms and it first applies a pre-elaboration step ($\phi$) on the preactivations $\mathbf{z}$ to obtain $\mathbf{z}_c$, the preactivations of literals of clause $c$. Then, Equation \ref{eq:delta_wc_s} of $\delta_s^w$ is applied.
The term $w$ is a positive weight associated to clause $c$ that represents the strength of $c$: the higher $w$ is, the bigger the contribution of the clause in the final predictions. As opposed to methods based on the regularization of the Loss, $w$ is not a hyper parameter. On the contrary, it is a parameter of the model that is learned during training. This means that KENN\ can discover the importance of each clause based on the data. As a special case of this behavior, it is possible for KENN\ to learn to ignore clauses by setting $w$ to zero, making clause $c$ irrelevant for the final predictions.
The final step of the CE is a post elaboration ($\phi'$) that converts the results of the TBF \ into the changes to be applied on $\mathbf{z}$.
The outputs of the CEs are combined inside the \emph{Knowledge Enhancer} (KE), which is the final layer of KENN\ model. Figure~\ref{fig:KE} shows the architecture of KE, which calculates the final interpretation as in Equation~\ref{eq:final_intepretation}.
\begin{figure}[H]
\centering
\input{tikz/KE}
\caption{Knowledge Enhancer architecture. It implements Equation~\ref{eq:final_intepretation} by summing the deltas values of each clause enhancer (one per grounded clause) with the initial preactivations $\mathbf{z}$ and applying the logistic function }
\label{fig:KE}
\end{figure}
\subsection{Working with batches of data}
In the architecture defined so far, since all the grounded atoms are inside the vector $\mathbf{z}$, there are multiple instantiation of the same CE for each clause $c \in \mathcal{K}$, one for each grounding of $c$. For instance, in Figure~\ref{fig:KE_matrix}(a), the CE for clause $c: \lnot S(x) \lor C(x)$ is instantiated two times, one for object $a$, another for object $b$.
In many scenarios we want to use batches of data, which would require to modify the structure of the model graph for each batch, in particular after each training step, since the groundings are constantly changing and for each grounding a different CE is instantiated. Notice that, even if the CEs are different, the internal operations are the same and the only change is the provided input. For these reasons, we want to instantiate a single CE for each clause $c$ which works in parallel on all possible groundings of $c$.
If the groundings involve a single object, a simple solution is to define $\mathbf{z}$ as a matrix instead as a vector, where columns represent predicates and rows different groundings. More in details, $\mathbf{z}$ is now defined as a matrix such that the element $z_{ij}$ contains the preactivation of $P_j(o_i)$, with $P_j$ the $j^{th}$ predicate and $o_i$ the $i^{th}$ object.
Notice that this kind of representation is pretty common when working with neural networks since the columns (predicates) could correspond to the labels and the rows (groundings) to the samples.
Figure~\ref{fig:KE_matrix}(b) shows this new approach: CE is defined as before, but now it takes as input a matrix and it acts on each row separately. This can be done only if the matrix does not contain duplicate atoms, since the changes are applied independently to each row and are not aggregated together. This property always holds with clauses that
involve a single variable.
On the contrary, in a clause with multiple variables, there could be a problem because different groundings of such clause
could contain a common grounded atom. For instance, consider the clause
$$c: \lnot Smoke(x) \lor \lnot Friends(x,y) \lor Smoke(y)$$
The two groundings $c[x/Alice,y/Bob]$ and $c[x/Bob,y/Carl]$ share a common grounded atom: $Smoke(Bob)$. For this reason, using a unique CE for $c$ is problematic since the values returned by the same CE are not aggregated together and we would end up with multiple changes proposed for the same atom.
In the next sections, we will present the architecture of KENN\ that can be used to instantiate a single CE even in relational domains, i.e. when dealing with predicates with arity greater than one.
\begin{figure}[H]
\centering
\input{tikz/KE_matrix}
\caption{KE applied on a Prior Knowledge consisting of a single clause $c$: $\lnot S(x) \lor C(x)$ applied on a domain of two objects ($a$ and $b$): (a) using a vector with all the grounded atoms, each grounding of $c$ is managed by a different CE. Notice that the changes proposed by the first CE are zero for $C(b)$ and $S(b)$ while the second CE returns the value zero for the other two atoms: the two grounded clauses are independent; (b) using a matrix with a row for each grounding: all the grounding of $c$ are managed by the same CE, which operates in parallel on each row of the matrix. This is more convenient when dealing with mini-batches and results in a more efficient implementation.}
\label{fig:KE_matrix}
\end{figure}
\subsection{Unary and binary clauses}
\label{sec:SRL_representation}
Maintaining a single vector with all the grounded atoms as in Section~\ref{sec:architecture} could lead to an inefficient implementation since it forces to create multiple CEs for each clause. This means that it requires to reset the computational graph every time new data is provided. On the other hand, in the approach of Fig.~\ref{fig:KE_matrix}(b), for each clause only a CE is instantiated and it can work in parallel on all the groundings of the clause. This has two advantages: it simplifies the usage of batches during training and CEs' internal calculations are implemented as matrix operations, which are particularly efficient when working with a GPU.
However, such an approach can be applied with clauses that involve a single variable: we call this type of clauses \emph{unary}. In contrast, a clause that contains two variables is referred as a \emph{binary} clause. For simplicity, we don't take into account clauses with higher arity. Notice however that the proposed approach can be used with predicates and clauses with any number of variables.
Let $\mathcal{K}_U$ be the set of unary clauses and $\mathcal{K}_B$ the set of binary clauses. The Prior Knowledge is now defined as $\mathcal{K} = \mathcal{K}_U \cup \mathcal{K}_B$.
The idea now is to apply KE to the two sets separately: Equation~\ref{eq:final_intepretation} can be decomposed using the new defined partition of the knowledge:
\begin{equation}
\begin{split}
\mathcal{I}_E(A) &= \sigma \left( z_A + \sum_{c \in \mathcal{G}(\mathcal{K})} \delta_c(A) \right) \\
&= \sigma \left( z_A + \sum_{c \in \mathcal{G}(\mathcal{K}_U)} \delta_c(A) + \sum_{c \in \mathcal{G}(\mathcal{K}_B)} \delta_c(A) \right)
\end{split}
\label{eq:final_intepretation_binary}
\end{equation}
Notice that the approach defined in the previous section with a single CE for each clause can be directly applied to the unary knowledge $\mathcal{K}_U$. We need to define a strategy to deal with binary clauses.
\subsection{Relational data - Representation of a graph}
\label{sec:relational_representation}
Before introducing the approach for relation data, we need to define the representation to be used to store such data.
Fig.~\ref{fig:representation} shows this representation using the classical Smoker-Friends-Cancer example: there is a domain composed of three objects (persons) $O = \{o_0, o_1, o_2\}$, two unary predicates ($S$ and $C$, which stand for $Smoking$ and $Cancer$) and one binary predicate $F$ ($Friends$)). Fig.~\ref{fig:representation}(a) shows the graph representation with nodes and edges labeled with unary and binary predicates respectively. In this particular example, the graph has no self-loops, meaning that $Friends(x,x)$ is not taken into account. Please notice that this limitation is specific for this example since everyone is assumed to be friends of himself a priori. However, in general, KENN\ can work with self-loops.
Fig.~\ref{fig:representation}(b) shows the data structure used to encode the graph of Fig.~\ref{fig:representation}(a): the idea is to represent the relational data in two matrices that can be interpreted as two tables of a relational database. Matrix $\mathbf{U}$ contains the unary predicates. Like $\mathbf{z}$, it contains all the groundings of unary predicates (each column represents a predicate, each row a grounding). Index vector $i$ contains unique indexes for rows of $\mathbf{U}$ and can be interpreted as a primary key for the corresponding table. Matrix $\mathbf{B}$ contains the binary predicates. Finally, $s^x$ and $s^y$ contain the indexes of the pairs of objects in $\mathbf{B}$ and they can be interpreted as foreign keys to table $\mathbf{U}$.
More precisely, if we consider a row $i$, $\mathbf{B}_{i,j}$ will correspond to the value of predicate $B_j$ for the substitution $(x / o_{s^x_i}, y / o_{s^y_i})$:
$$\mathbf{B}_{i,j} = B_j(o_{s^x_i},o_{s^y_i})$$
For instance, in Fig.~\ref{fig:representation}(b): $s^x_0 = 0$ and $s^y_0 = 1$, which means that row $0$ of matrix $\mathbf{B}$ is referring to the pair of objects $(o_0,o_1)$.
\begin{figure}
\centering
\input{tikz/rel_representation}
\caption{
Relational data representation: in this example, three objects ($o_0$, $o_1$ and $o_2$), two unary predicates ($S$ and $C$) and only one binary predicate ($F$). (a): the graph representation; (b): the representation used by KENN: matrixes $\mathbf{U}$ and $\mathbf{B}$ can be interpreted as tables of a relational database: $\mathbf{U}$ contains all the groundings of unary predicates and an index column $i$ which correspond to the `primary key' of the table; matrix $\mathbf{B}$ contains the groundings of binary predicates together with columns $s^x$ and $s^y$ which corresponds to ``foreign keys'' to table $\mathbf{U}$.
Values refer to preactivations of the atoms.}
\label{fig:representation}
\end{figure}
The idea is to define \emph{differentiable queries} on the database that can be interleaved with other layers in the network: a JOIN operation can combine $\mathbf{U}$ and $\mathbf{B}$ in a unique matrix $\mathbf{M}$, then the KE can be applied on $\mathbf{M}$. The resulting matrix $\delta \mathbf{M}$ is again interpreted as a database table and three additional queries are used to combine properly the contributions of each CE to obtain the final predictions.
The next sections will dive deeper into this idea.
\subsection{Binary extensions of unary predicates}
\label{sec:join}
Lets now describe which predicates should be considered inside $\mathbf{M}$, the matrix used by the KE. In addition to binary predicates, we have that binary clauses can contain also unary predicates: we include in the columns of $\mathbf{M}$ also a binarized version of the unary predicates.
More formally, given a unary predicate $U_i$, we define its binary extensions as two binary predicates $U_i^x$ and $U_i^y$ such that:
$$
\forall x \ \forall y \ U_i^x(x,y) \leftrightarrow U_i(x)
$$
and
$$
\forall y \ \forall x \ U_i^y(x,y) \leftrightarrow U_i(y)
$$
Intuitively, $U_i^x$ and $U_i^y$ are binary predicates that ignore one of the two inputs. For instance, let $Smoker(x)$ be the unary predicate that is true if person $x$ smokes. The $Smoker^x(x,y)$ is the binary predicate corresponding to the sentence: ``The first object of the pair $(x,y)$ is a smoker''.
For each binary clause, we can substitute the original unary predicates with their binary extensions. For instance, the clause
$$c: \lnot Smoker(x) \lor \lnot Friends(x,y) \lor Smoker(y)$$
is converted to
$$c': \lnot Smoker^x(x,y) \lor \lnot Friends(x,y) \lor Smoker^y(x,y)$$
It is easy to prove that the truth value of $c'$ is the same of $c$:
$$\forall x \ \forall y \ \Big( Smoker^x(x,y) \leftrightarrow Smoker(x) \Big) \land \Big(Smoker^y(x,y) \leftrightarrow Smoker(y)\Big) \models \forall x \ \forall y \ c \leftrightarrow c'$$
Notice that while in $c$ there were both unary and binary predicates, in $c'$ all the predicates are binary. This is true for the original domain $O$, but they can be seen as unary predicates if we consider the domain of pairs in $O \times O$. The new solution for the binary knowledge comes exactly from this new perspective: if we extend matrix $\mathbf{B}$ to contain also the new binary predicates $U_i^x$ and $U_i^y$, then we can apply directly the KE on the resulting matrix using $\mathcal{K}_B$.
In summary, binary clauses can be seen as unary clauses in the domain of pairs of objects. Since KE can deal with unary clauses, it can be used to enhance binary clauses as well.
In the relational database analogy, this approach is equivalent to perform a JOIN query on $\mathbf{U}$ and $\mathbf{B}$ and apply the KE on the resulting matrix:
\begin{lstlisting}[language=SQL, escapeinside={*}{*}]
*$\mathbf{M}$* = SELECT *$s^x$*, *$s^y$*, *$U_i^x$*, *$U_i^y$*, *$B_j$*
FROM U AS Ux, U AS Uy, B
WHERE Ux.*$i$* = B.*$s^x$* AND Uy.*$i$* = B.*$s^y$*
\end{lstlisting}
where $U_i^x$ is a shortcut for selecting all the unary predicates in Ux
(Ux.$U_i$ AS $U_i^x$).
\begin{figure}[t]
\centering
\input{tikz/JOIN}
\caption{The JOIN query: all the binary predicates (including the binary extensions) are collected in a unique matrix. The red numbers are all referring to the same grounded atom $C(o_0)$.}
\label{fig:join}
\end{figure}
A practical example can be seen in Fig.~\ref{fig:join}. Notice that all grounded atoms values are repeated multiple times inside matrix $\mathbf{M}$. An example is $C(o_0)$ which is marked in red inside the figure. The same is true for $\delta \mathbf{M}$, where we have multiple delta values for each grounded unary predicate. This time, the values are different from each other since they all come from different groundings of the binary clauses. Accordingly to Equation~\ref{eq:final_intepretation_binary}, these values must be summed up. This will be achieved using GROUP BY queries.
\subsection{Relational Data - KENN\ architecture}
\label{sec:groupby}
In the previous section, we saw that matrix $\mathbf{M}$ defined by the JOIN query is used by the KE with $\mathcal{K}_B$. The KE returns $\delta \mathbf{M}$, a matrix of the same shape of $\mathbf{M}$ which contains the changes on the initial predictions induced by binary clauses. Finally, the different delta values associated with each grounded atom are aggregated together. This can be achieved by the following three queries:
\begin{lstlisting}[language=SQL, escapeinside={*}{*}]
*$\delta B$* = SELECT *$\delta B_i$*
FROM *$\delta M$*
*$\delta U_x$* = SELECT SUM(*$\delta U_i^x$*)
FROM *$\delta M$*
GROUP BY *$s^x$*
*$\delta U_y$* = SELECT SUM(*$\delta U_i^y$*)
FROM *$\delta M$*
GROUP BY *$s^y$*
\end{lstlisting}
\begin{figure}[H]
\centering
\input{tikz/GROUP_BY}
\caption{An example ofa GROUP BY query: for each unary predicate ($S$ and $C$), the values of their first binary extension ($S^x$ and $C^x$) are summed up based on the index
}
\label{fig:join}
\end{figure}
\noindent The final preactivations $\mathbf{U}'$ and $\mathbf{B}'$ are
$$\mathbf{U}' = \mathbf{U} + \delta \mathbf{U}_u + \delta \mathbf{U}_x + \delta \mathbf{U}_y$$
and
$$\mathbf{B}' = \mathbf{B} + \delta \mathbf{B}$$
\noindent Fig.~\ref{fig:unary_binary} shows a high-level overview of the entire model.
\begin{figure}[H]
\centering
\input{tikz/rel_overview}
\caption{
To obtain $\delta \mathbf{U}^u$, unary clauses $\mathcal{K}_U$ are used by KE directly on $\mathbf{U}$. The JOIN query is used to find matrix $\mathbf{M}$. Then, the KE is applied to $\mathbf{M}$ using binary clauses $\mathcal{K}_B$. Finally, to obtain $\delta \mathbf{U}^b$ and $\delta \mathbf{B}$ other three queries are used.}
\label{fig:unary_binary}
\end{figure}
\subsection{Knowledge for inputs/outputs relationships}
\label{sec:KENN_inputs}
Until now, we defined KENN\ as a method that injects logical knowledge which represents relationships between different predictions of a Neural Network. However, it is even possible to provide knowledge which expresses relationships between inputs and outputs values. In other words, if some of the inputs are truth values of some atomic formulas, we can add clauses that contain them, providing in this way information on the relations between inputs and outputs.
Figure~\ref{fig:KENN_inputs} introduces this idea: the solution is straightforward, we just need to concatenate the inputs of the base NN with its outputs before applying the KE. The KE will provide an updated version of both inputs ($\mathbf{x}'$) and outputs ($\mathbf{y}'$). Since we are interested only in the outputs, the last step discards the changed inputs $\mathbf{x}'$.
\begin{figure}[H]
\centering
\input{tikz/KENN_inputs}
\caption{KENN used with knowledge on both inputs and predictions: the ``preactivations'' $\mathbf{z}_x$ of inputs $\mathbf{x}$ are calculated with the logit function and concatenated to the preactivations of the outputs $\mathbf{y}$. The result can be used by KE to force the satisfaction of the knowledge.}
\label{fig:KENN_inputs}
\end{figure}
Notice that, since the KE works with preactivations, the logit function (the inverse of logistic function) must be applied to the inputs before the concatenation. Moreover, the logit is defined in the range $(0,1)$, which means that if the truth values in $\mathbf{x}$ are zeros and ones, logit can not be applied directly. For this reason, a small number $\epsilon$ is added or subtracted to $\mathbf{x}$ before applying the logit.
\section{Experiments on relational data}\label{sec:collective}
In this section, we focus on experiments with relational data. More precisely, KENN\ was tested on the context of Collective Classification: given a graph,
we are interested in finding a classification for its nodes using both features of the nodes (the objects) and the information coming from the edges of the graph (relations between objects)~\cite{collective_classification}.
In Collective Classification, there are two different learning paradigms: inductive and transductive learning. In inductive learning there are two separates graphs, one for training and the other for testing. On the contrary, in transductive learning there is only one graph that contains nodes both for training and testing. In other words, in inductive learning there are no edges between nodes for training and testing, while in transductive learning there are. Figure~\ref{fig:collective} shows the difference between the two paradigms.
\begin{figure}
\centering
\input{tikz/inductive_transductive.tex}
\caption{The two learning paradigms in Collective Classification. The colors represent the classes of the nodes. White nodes are nodes of the Test Set. In inductive learning, Training and Test sets are two distinct graphs: the network has to learn only from labeled data; in transductive learning, there is a unique graph with both training and test nodes: the network can make use of information coming from the Test Set at training time by considering the additional relations.}
\label{fig:collective}
\end{figure}
The tests were performed with the goal of obtaining a comparison with other important methods of Neural-Symbolic integration. For this reason, we followed the evaluation methodology of~\cite{RNM}, where the experiments have been carried out on Citeseer dataset~\cite{citeseer} using SBR and RNM. As in~\cite{RNM}, the experiments have been performed on both inductive and transductive paradigms.
\subsection{Citeseer Dataset}
The Citeseer dataset~\cite{citeseer} used in the evaluation is a citation network: the graph's nodes represent documents and the edges represent citations. The nodes' features are bag-of-words vectors, where an entry is zero if the corresponding word of the dictionary is absent in the document, and one if it is present. The classes to be predicted represent possible topics for a document.
The dataset contains 4732 nodes that must be classified in 6 different classes: AG, AI, DB, IR, ML and HCI.
The classification is obtained from the 3703 features of the nodes, with the addition of the information coming from the citations.
\subsection{The Prior Knowledge}
The Prior Knowledge codifies the idea that papers cite works that are related to them. This implies that the topic of a paper is often the same as the paper it cites. The clause
$$
\forall x \ \forall y \ T(x) \land Cite(x,y) \to T(y)
$$
is instantiated multiple times by substituting the topic $T$ with all the six classes.
\subsection{Experimental setup}
We used as base NN a dense network with three hidden layers, each with 50 hidden nodes and RELU activation function. This setting is the same as~\cite{RNM}, meaning that the results of KENN\ are directly comparable with the results of SBR and RNM provided by such work since also the used Knowledge is the same.
Notice that the base NN takes as input only the features of the nodes and it does not take into account relations. For this reason, by adding the knowledge, we could expect great improvements as compared to the base NN. Notice also that for the base NN model the two paradigms (inductive and transductive) are equivalent: the only
difference between the two is the presence or absence of relations between nodes in the training and test sets, and such relations are not taken into account by the NN.
Indeed, the relations are considered only on the level of the KE. This is true also for RNM and SBR. In the case of SBR, the difference is that it does not change the basic neural network, meaning that even at inference time the learned model does not take into account the citations. To obviate these problems, SBR optimizes the satisfaction of the constraints defined by the knowledge even at test time. This is a strong point in favor of KENN\ since among the three methods is the only one that does not require to solve an optimization problem during inference. Indeed, one of the main advantages of KENN\ is scalability.
Notice also that the citations are known a priori (the edges of the graph are given as inputs). Therefore, when applying KE, it is possible to focus only on pairs of documents for which the $Cite(x,y)$ predicate is known to be true. Indeed, all the grounded clauses in the Prior Knowledge are automatically satisfied if $Cite(x,y)$ is false for the specific grounding. This means that clauses are always satisfied when the pair of documents do not cite one another and that KE would not apply any changes for such pairs of objects. For this reason, $s_i^x$, $s_i^y$ and $\mathbf{B}$ were generated using only pairs of objects for which $Cite(x,y)$ is true, reducing both train and test time.
Finally, since each paper can not be classified with multiple classes, the activation used was the $\softmax$ function. Notice that the results obtained in Section~\ref{preact} were proved with the assumption that the activation is a logistic function. Therefore, there is not theoretic evidence that this approach can work. However, the results obtained give us empirical evidence that KENN\ can improve the base NN predictions even with $\softmax$ activation.
The training set dimension is changed multiple times to evaluate the efficacy of the three methods on the varying of training data. More precisely, tests have been conducted by selecting 10\%, 25\%, 50\%, 75\% and 90\% of the dataset for training. Like in~\cite{RNM}, for each of these values the training and evaluation is performed ten times, each of which with a different split of the dataset in training and test data. The final results are obtained as the mean of accuracy in the ten runs. This is justified by the fact that the choice of nodes in the Training Set has a strong effect on the learned model, and as a consequence on the test accuracy.
\subsection{Results}
\label{sec:results_collective}
Table~\ref{inductive_results} shows the results of KENN\ on the inductive version of the training.
\begin{table}[H]
\caption{Results in terms of accuracy on the inductive variant of the task ordered by the amount of data used for training. The first three columns contains the results reported in~\cite{RNM}.}
\centering
\begin{tabular}{c|lll|lll}
\% training & NN & SBR & RNM & NN & KENN$_g$ & KENN$_e$ \\
\hline
\rule{0pt}{3ex}\hspace{-0.12cm}
10 & 0.645 & 0.650 & {\bfseries 0.685} & 0.591 & 0.603 & 0.591 \\
&& (+0.005) & {\bfseries (+0.040)} && (+0.012) & (-0.000) \\
\rule{0pt}{3ex}\hspace{-0.12cm}
25 & 0.674 & 0.682 & {\bfseries 0.709} & 0.644 & 0.661 & 0.659 \\
&& (+0.008) & {\bfseries (+0.035)} && (+0.017) & (+0.015)\\
\rule{0pt}{3ex}\hspace{-0.12cm}
50 & 0.707 & 0.712 & {\bfseries 0.726} & 0.685 & 0.691 & 0.701\\
&& (+0.005) & {\bfseries (+0.019)} && (+0.006) & (+0.016)\\
\rule{0pt}{3ex}\hspace{-0.12cm}
75 & 0.717 & 0.719 & 0.726 & 0.723 & 0.725 & {\bfseries 0.742} \\
&& (+0.002) & (+0.009) && (+0.002) & {\bfseries (+0.019)}\\
\rule{0pt}{3ex}\hspace{-0.12cm}
90 & 0.723 & 0.726 & 0.732 & 0.722 & 0.725 & {\bfseries 0.740} \\
&& (+0.003) & (+0.009) && (+0.003) & {\bfseries(+0.018)}\\
\hline
\hline
\end{tabular}
\label{inductive_results}
\end{table}
Notice that, although the final results are calculated as the mean of ten runs, there are still some fluctuations and for this reason it is not possible to perfectly replicate the results obtained by~\cite{RNM} for the NN. For this reason, in the table below the NN has two columns, one with the results reported in~\cite{RNM}, another with the results obtained our my experiments. Since the results of the NN are slightly different in these two cases, the considered metric is the improvement over the base NN.
The behavior of RNM is consistent with the simple intuition that when the training data is scarce the usage of knowledge should bring higher benefits. On the other hand, KENN\ has an opposite behavior with respect to RNM, with lower improvements when the data is scarce and higher when it is abundant. As we have seen in Section~\ref{sec:multilabel}, KENN\ trained in the end-to-end fashion could suffer of overfitting when the training set is too small, and this explains why it performs poorly with 10\% of the dataset for training. In such a case, it reduces the accuracy with respect to the base NN (the improvement was of $-3.36 \cdot 10^{-5}$, which is approximated to zero in the table).
For this reason, as in the experiments with Yeast and Emotions, the usage of the greedy approach is beneficial for the smaller values of the training set dimension, since the greedy approach is less prone to overfitting. Indeed, KENN\ trained with the greedy approach gives better accuracy than the end-to-end one for the smallest training sets, with results that are always at least as good as SBR but worse than RNM.
\begin{table}[H]
\caption{Results in terms of accuracy on the transductive variant of the task ordered by the amount of data used for training. The first three columns contains the results reported in~\cite{RNM}.}
\centering
\begin{tabular}{c|lll|lll}
\% training & NN & SBR & RNM & NN & KENN$_g$ & KENN$_e$ \\
\hline
\rule{0pt}{3ex}\hspace{-0.12cm}
10 & 0.640 & 0.703 & {\bfseries 0.708} & 0.599 & 0.611 & 0.649 \\
&& (+0.063) & {\bfseries (+0.068)} && (+0.013) & (+0.050)\\
\rule{0pt}{3ex}\hspace{-0.12cm}
25 & 0.667 & 0.729 & {\bfseries 0.735} & 0.649 & 0.662 & 0.681 \\
&& (+0.062) & {\bfseries (+0.068)} && (+0.013) & (+0.032)\\
\rule{0pt}{3ex}\hspace{-0.12cm}
50 & 0.695 & 0.747 & {\bfseries 0.753} & 0.682 & 0.692 & 0.726\\
&& (+0.052) & {\bfseries (+0.058)} && (+0.010) & (+0.045)\\
\rule{0pt}{3ex}\hspace{-0.12cm}
75 & 0.708 & 0.764 & {\bfseries 0.766} & 0.705 & 0.709 & 0.755 \\
&& (+0.056) & {\bfseries (+0.058)} && (+0.004) & (+0.050)\\
\rule{0pt}{3ex}\hspace{-0.12cm}
90 & 0.726 & 0.780 & 0.780 & 0.728 & 0.747 & {\bfseries 0.785} \\
&& (+0.054) & (+0.054) && (+0.019) & {\bfseries(+0.057)}\\
\hline
\hline
\end{tabular}
\label{transductive_results}
\end{table}
Table~\ref{transductive_results} shows the results of the various models on the transductive learning task. Note that, as mentioned previously, the results of the base NN are similar to the case of inductive learning. In contrast, the other models, which take into account the citations, perform much better on this task since they have more information at their disposal during training. The only exception is KENN\ when trained with the greedy approach, which performs more or less the same in the two tasks. Notice that in this case the NN is trained separately from the KE and that, as already discussed, the NN does not benefit from the additional edges provided in transductive learning.
By looking at Table~\ref{transductive_results} we can notice that, like in the case of inductive learning, KENN\ works better than the other models when the percentage of training data is high. However, in this case, it seems to not suffer of overfitting as in the inductive learning, since the improvement over the base NN is pretty high even with a lower percentage of training nodes.
This can be explained by noticing that in transductive learning KENN\ can make use of the information carried out by the relations between training and test nodes. Like for inductive learning, the loss is calculated from the predictions on the training set nodes. However, if clause weights are high enough, the loss depends indirectly also on the values of the test nodes.
To explain this property, let's look at Figure~\ref{fig:bias_kenn}. As before, white nodes represent nodes of the test set. Suppose color red represents the class AI and blue ML. The value of the loss function does not depend directly on the value of node 9, since it is not in the training set. However, if the clause weight $w_{AI}$ of clause $c_{AI}: \forall x \ \forall y \ AI(x) \land Cite(x,y) \to AI(y)$ is high, then setting node 9 to class AI would force node 3 to be classified as AI as well. On the contrary, if we classify node 9 with the class ML, then node 3 would also be classified as ML (supposing high value also for $w_{ML}$).
Summarizing, in the simple scenario of Figure \ref{fig:bias_kenn}, it is convenient for the model to learn a high value for $w_{AI}$ and $w_{ML}$. By predicting nodes 9 and 8 as AI and ML respectively, the model will force also the training nodes to be correctly classified. As a consequence, the loss would be reduced and this kind of solution will be selected by the training process. In this case, the KE produces a sort of regularization similar to the one of LTN and SBR, since it rewards solutions that make the constraints satisfied.
\begin{figure}
\centering
\input{tikz/transductive_propagation.tex}
\caption{An extreme example of a graph in a transductive learning scenario: the white nodes are available during training, but not their classes. In this simple scenario, it is easy for the learning algorithm to make KENN\ fit the data by forcing the satisfaction of the constraints and by classifying the test set nodes accordingly.}
\label{fig:bias_kenn}
\end{figure}
\section{Conclusions and future work}\label{sec:conclusions}
We proposed KENN, a new method for injecting Prior Knowledge expressed as logical clauses into a neural network model, merging in this way the learning capabilities of Neural Networks with the expressivity of First Order Logic. KENN\ can be used in combination with many different types of neural networks and differs from other methods on the way logic is injected, that is by adding a new differentiable layer (the Knowledge Enhancer) at the top of a NN which provides predictions for the atomic formulas of the logic clauses.
Moreover, it has been introduced the concept of t-conorm boost function (TBF), which is a function that modifies the truth values of the literals of a clause to increase its satisfaction in terms of a fuzzy logic t-conorm. Furthermore, we introduced the concept of minimality of a TBF and provided a formal proof of the minimality of a specific function $\delta^f$ in respect to the G\"{o}del t-conorm. A soft approximation of $\delta^f$ is used inside the Clause Enhancer, a submodule of the KE.
In addition, the KE contains clause weights, which are learnable parameters that represent the strength of the clauses. At the best of our knowledge, KENN\ and RNM are the only approaches that can merge Neural Networks models and logical Prior Knowledge while learning the clause weights. The two methods work similarly, in the sense that both the methods use a neural network to predict the initial classification which is then updated based on the knowledge. The difference between the two can be summarized in the different choices made for this second step: KENN\ integrate it directly into the base NN as a differentiable function but it considers each rule separately and integrates the results via a linear combination; RNM instead take into account the entire knowledge at the same time, but it requires to solve an optimization problem (MAP estimation) at inference time and at each training step.
The ability to learn clause weights makes KENN\ suitable for tasks where the provided knowledge is not totally reliable or when some of the clauses are not hard constraints and it is not known a priori the strength of the constraints encoded in the clauses. This has proven to be a major advantage in the experiments with VRD Dataset, where the results obtained by KENN\ outperforms LTN using the same type of knowledge. Indeed, multiple clause weights have been set to zero by the learning process, meaning that KENN\ learned to ignore the corresponding clauses. On the other hand, LTN forces the satisfaction of constraints, meaning that those rules are enforced by it even if they are not satisfied in the training data. Moreover, the experiments on Citeseer provide other evidence in favour of approaches that integrate the knowledge into the model, since both KENN\ and RNM obtained higher results than SBR. Finally, notice that one of the advantages of learning the clause weights is that in principle it should be possible to generate new knowledge by generating random clauses and let KENN\ validate them by learning the weights. This approach could be the focus of further experiments.
While KENN\ proved to be able to effectively add the knowledge into the base NN with good results in multiple datasets, the flexibility in terms of usable knowledge is lower as compared to the other methods, in particular because of the absence of the existential quantifier. This could be the goal of future developments. Notice that, if such a change is done under the Close World Assumption, this should be a quite trivial extension to develop. Another missing feature is the possibility to represents functions. In this case, however, it is not a straightforward task. Further investigation are needed in this direction.
Another important contribution is in the solution found to manage relational data: the KE has been developed to work on a matrix where each column represents a unary predicate and the rows the objects of the domain. This means that the KE does not work with relational data. On the contrary, it works under the i.i.d. assumption. Instead of developing a version of the KE for relational data, the graph structure is managed by a pre elaboration step which provides the KE with a matrix in the required format, and a post elaboration step that aggregates the outputs generated by the KE to obtain the final results which are compliant with the relational structure of the domain. The main contribution is on the way these pre and post elaborations steps work: they are represented as queries of a relational database. Indeed, the data structure used to store a graph is very similar to the one of a relational database. However, in this case, such database is implemented in TensorFlow and the queries are implemented as differentiable operators. This strategy was used only in the context of KENN, but it could be the focus of further investigation. Indeed the idea of a differentiable relational database could provide a very convenient way to create Neural Network architectures for relational domains.
\section{Introduction}
\label{sec:introduction}
In the last decade, Deep Learning approaches gained a lot of interest in the AI community, becoming the state of the art on many fields, such as Computer Vision~\cite{image_classification}, Machine Translation~\cite{machine_translation}, Speech Recognition~\cite{speech_recognition} and so forth. However, the main downside of such methods is that they are demanding in terms of training data.
On the other hand, human beings are capable of learning new concepts with few examples (Few Shot Learning) and even in some cases with zero experience (Zero Shot Learning). One of the main reasons is due to their ability to make use of previously acquired knowledge. In other words, the human brain is not just a randomly initialized model that learns from data, on the contrary, it often contains some sort of prior knowledge when approaching a new task.
Such knowledge could come for experience on different tasks. For instance, when learning to move around and avoid obstacles, humans can effectively learn the three-dimensional structure of the world by constructing an internal representation of it. When approaching a new task where visual data must be processed, they can make use of the previously learned representation, transferring the acquired knowledge into a new domain.
Another type of knowledge is the one provided by other human beings. For example, suppose we tell a kid: ``A unicorn is a horse with a horn in its forehead''. Although the kid has no previous experience of unicorns, he will be able to recognize them, for example when watching a movie. This ability to exploit provided knowledge is crucial for effectively learn new concepts when training data is scarce.
In this paper, we are going to focus on this second type of knowledge. While in the case of humans such knowledge is often in the form of natural language definitions, in the proposed method it is provided as a set of logical formulas.
This work extends our previous research on KENN (\emph{Knowledge Enhanced Neural Network}), a method to inject knowledge into models for multi-label classification \cite{kenn}. While multi-label classification is an important topic in machine learning, the usage of knowledge is particularly relevant in the context of relational domains.
For this reason, we propose an updated version of KENN\ which can deal with these kinds of domains.
Suppose we have a Neural Network for a classification task, which is called \emph{base NN}, that takes in input the feature vector $\mathbf{x} \in \mathbb{R}^n$ and returns an output $\mathbf{y} \in [0,1]^m$ which contains the predictions for $m$ classes. A background knowledge $\mathcal{K}$ is also provided. $\mathcal{K}$ is defined as a set of clauses, i.e. disjunctions of literals, that represent constraints on the $m$ classes to be predicted. For instance, in an image classification task, the clause $\forall x \ (Dog(x) \to Animal(x))$, stating that dogs are animals, can be used by KENN, in conjunction with the base NN, to predict the two labels $Dog$ and $Animal$.
Figure~\ref{fig:KENN_overview} shows a high-level overview of KENN.
\begin{figure}[h]
\centering
\input{tikz/KENN_overview}
\caption{KENN model: features are given as input to a neural network (NN) and predictions on predicates values are returned. Knowledge Enhancer modifies the predictions based on logical constraints ($\mathcal{K}$)}
\label{fig:KENN_overview}
\end{figure}
Predictions $\mathbf{y}$ are revised by a differentiable function, called \emph{Knowledge Enhancer} (KE), which corresponds to a new layer on top of the base NN. Its role is to update the predictions $\mathbf{y}$ of the base NN into $\mathbf{y}'$ to increase the truth value of each clause $c \in \mathcal{K}$. Since both base NN and KE are differentiable, the entire architecture is differentiable end-to-end, making it possible to apply back-propagation algorithm directly on the whole model.
KE increases the satisfaction of each clause $c$ separately obtaining $|\mathcal{K}|$ different vectors, each of which represents the changes to be applied on $\mathbf{y}$ to obtain the improvement of satisfaction of $c$. These changes are combined linearly to obtain the final change to be applied on the base NN's predictions.
An important question arises: what does it mean to increase the satisfaction of a clause? We use fuzzy semantic, where the satisfaction of a clause is given by applying a t-conorm function $\bot$ on the truth values of the literals of the clause. Therefore, the objective of the final layer should be increasing the value of the chosen t-conorm. However, we are not relying on an optimization process to add knowledge, so it is not possible to just optimize the value of the t-conorm at run time. On the contrary, we want to inject the knowledge as a differentiable layer on top of a neural network: the optimization must be done analytically!
To do so, we propose the concept of \emph{t-conorm boost functions} (TBFs). Intuitively, a TBF is a function \mbox{$\delta: [0,1]^n \to [0,1]^n$} that proposes the changes to be applied on a set of $n$ truth values to increase the value of the \mbox{t-conorm} applied on them: $\bot({\bf \lit} + \delta({\bf \lit})) \geq \bot({\bf \lit})$. KENN\ include a soft differentiable approximation of a TBF in its last layer and it uses it to enhance the prediction of the base NN.
Moreover, KE contains additional parameters that can be learned as well: for each clause $c \in \mathcal{K}$ there is an associated \emph{clause weight} $w_c$ which determines the strength of $c$, i.e., it defines the influence that the clause has on the final predictions.
Differently from other Neural-Symbolic integration approaches, clauses weights are not given, but they are learned.
Moreover, by changing the weight of clause, KENN\ can learn to ignore clauses in the Prior Knowledge that are not fully satisfied inside training data by assigning zero to the weight.
We evaluated KENN\ on multiple aspects. Experiments on Yeast dataset~\cite{Yeast} and Emotions dataset~\cite{Emotions} (two standard dataset for multi-label classification), we used clauses generated directly from the training data. Results showed that KENN\ can make efficient usage of this kind of rules, proving to be robust to the presence of imperfect background knowledge.
In a second experiment, we tested KENN\ on the Predicate Detection task of Visual Relationship Detection Dataset (VRD Dataset)~\cite{visual2} using a manually curated prior knowledge proposed by~\cite{IvanThesis}. The task consists in finding the relationships between pairs of objects inside an image and it contains a \emph{Zero Shot Learning} sub-tasks where the objective is to predict types of relationships not available at training time. In this dataset, KENN\ outperformed state of the art methods, with the best results on \emph{Zero Shot Learning}.
Moreover, KENN\ outperformed Logic Tensor Networks, one of the major competitors of KENN, using the same knowledge.
Finally, the latest experiments were conducted to test the efficacy of KENN\ in the context of relational domains. For this purpose, we applied KENN\ on Citeseer~\cite{citeseer}, a standard dataset for Collective Classification~\cite{collective_classification}. The experiments on this dataset are particularly relevant since they provide some insight on the usage of KENN\ on relational domains and they give us a comparison between KENN\ and two other approaches: Semantic Based Regularization (SBR)~\cite{SBR} and Relational Neural Machines (RNM)~\cite{RNM}. The evaluation was conducted with two different learning paradigms of Collective Classification, namely Inductive and Transductive Learning. Moreover, the experiments were applied to different splits of the dataset to evaluate the three methods performances at the varying of the number of training samples.
\section{Prior Knowledge Enhancement}
\label{sec:kenn}
\subsection{T-conorm functions}
\label{sec:fuzzy}
In fuzzy logic, the satisfaction of a disjunction of literals is represented with a t-conorm function, which maps the truth values (expressed in the range $[0,1]$) of two literals to the truth value of their disjunction.
\begin{definition}
A t-conorm $\bot: [0,1] \times [0,1] \to [0,1]$ is a binary function which satisfies the following properties:
\begin{enumerate}
\item $\bot(a,b) = \bot(b,a)$
\item $\bot(a,b) \leq \bot(c,d) \ \ if \ a \leq c \ and \ b \leq d $
\item $\bot(a, \bot(b,c)) = \bot(\bot(a,b),c$)
\item $\bot(a,0)=a$
\end{enumerate}
\end{definition}
\noindent We represent a t-conorm as a unary function over vectors (${\bf \lit} = \left< t_1, t_2 ... t_n \right>$):
$$
\bot({\bf \lit}) = \bot(t_1, \bot(t_2, \bot(t_3 ... \bot(t_{n-1},t_n))))
$$
Given a clause $c$ composed of $n$ literals, if the $i^{th}$ component of ${\bf \lit}$ is the truth value of the $i^{th}$ literal of $c$, then $\bot({\bf \lit})$ is the truth value of $c$. In the following, we will use the vector ${\bf \lit}$ to represent the predictions of the literals' truth values and vector $\mathbf{y}$ for the predictions of all the grounded atoms.
\subsection{Prior Knowledge}
\label{sec:prior_knowledge}
We define the Prior Knowledge in terms of formulas of a function-free first order language $\mathcal{L}$. Its signature is defined with a set of constants $\mathcal{C} \triangleq \{a_1, a_2, ... a_l \}$
and a set of predicates $\mathcal{P} \triangleq \{P_1, P_2 ... P_q \}$.
Unary predicates can be used to express properties of singular objects. For instance, to represents that a person $a \in \mathcal{C}$ is a smoker, we can use $Smoker(a)$, where $Smoker \in \mathcal{P}$ is a predicate with arity one. Predicates with higher arity can express relations among multiple objects in the domain, e.g. $Friends(a,b)$ states that person $a$ is a friend of $b$.
\noindent The Prior Knowledge is defined as a set of clauses: $\mathcal{K} \triangleq \{ c_1, c_2, ... c_r \} $. A clause is a disjunction of literals, each of which is a possibly negated atom:
$$
c \triangleq \bigvee\limits_{i=1}^{m} l_i
$$
\noindent where $m$ is the number of literals in $c$ and $l_i$ is the $i^{th}$ literal. We assume that there are no repeated literals.
As an example, the clause $\lnot Smoker(x) \lor \lnot Friends(x,y) \lor Smoker(y)$ states that if a person $x$ is a smoker and he is a friend of another person $y$, then $y$ is also a smoker.
Notice that the previous clause has no constants and the two variables $x$ and $y$ in it are assumed to be universally quantified. This is because we are interested in representing general knowledge, and in the following, we will always apply this assumption implicitly to each clause in our knowledge.
We define the grounding of a clause $c$, denoted by $c[x/a, y/b ...]$, as the clause obtained by substituting all of its free variables with the corresponding constant symbol. For instance, if we take into consideration two specific persons $a$ and $b$, then
$$
(\lnot Smoker(x) \lor \lnot Friends(x,y) \lor Smoker(y))[ x/a,y/b ]$$
is equivalent to
$$
\lnot Smoker(a) \lor \lnot Friends(a,b) \lor Smoker(b)
$$
We will denote with $\mathcal{G}(c, \mathcal{C})$ the set of all the groundings of a clause $c$ and with $\mathcal{G}(\mathcal{K}, \mathcal{C})$ the set of all the grounded clauses.
\subsection{Semantic of $\mathcal{L}$}
\label{sec:semantic}
The semantic of $\mathcal{L}$ is defined by the Neural Interpretation, which is an interpretation induced by a neural network $\mathcal{N}$, which is responsible for calculating the initial interpretation for the grounded atoms.
More precisely, constants symbol are interpreted as continuous features in $\mathbb{R}$ while predicate symbols interpretation is given by $\mathcal{N}$.
Since we are dealing with neural network predictions, which returns continuos values, we can not make use of classic logic. Indeed, the semantic of logical connectives can not be defined with the usual truth tables. We rely on fuzzy logic instead.
\subsection{Neural interpretation}
The neural interpretation is defined by function $\mathcal{I}_N$ which maps constant symbols $o \in \mathcal{C}$ of $\mathcal{L}$ to real-valued vectors that represent the features of the corresponding objects in the real world. On the other hand, the truth values of atoms are mapped to the predictions of the base NN $\mathcal{N}$.
In the most simple case, given a predicate $P \in \mathcal{P}$ with arity $n$, $\mathcal{N}$ takes as input the interpretations (i.e. features) of $n$ constants symbols and returns the truth value of $P$ for such values.
Formally, given a list of constants $o_i \in \mathcal{C}$, a predicate $P \in \mathcal{P}$ with ariety $n$, an atomic formula $A$ and a clause $c$:
\begin{itemize}
\item $\mathcal{I}_N(o_i) = \mathbf{x}_i$, with $\mathbf{x}_i \in \mathbb{R}^l$
\item $\mathcal{I}_N(P) = \mathcal{N}_P$, where $\mathcal{N}_P: \mathbb{R}^{nl} \to [0,1]$ is a neural network which takes as input the interpretation (features) $\mathcal{I}_N(o_1), \mathcal{I}_N(o_2) \dots \mathcal{I}_N(o_n)$ of $n$ constant symbols and predicts the truth value of $P(o_1,o_2 \dots o_n)$
\item $\mathcal{I}_N(\lnot A) = 1 - \mathcal{I}_N(A)$
\item $\mathcal{I}_N(c) = \bot(\mathcal{I}_N(l_1),\mathcal{I}_N(l_2),\dots,\mathcal{I}_N(l_m))$, where $\bot$ is the t-conorm function (see Section~\ref{sec:fuzzy})
\end{itemize}
The enhanced interpretation $\mathcal{I}_E$ is defined by a function, called \emph{Knowledge Enhancer}, which takes as input the Neural Interpretation and updates it to increase the satisfaction of the clauses in $\mathcal{K}$ (taking also in consideration the learned weight of each clause). The enhanced interpretation is equivalent to the neural one, except for the interpretation of a predicate, which is now changed by the KE.
\subsection{t-conorm boost functions}
\label{TBFS}
Intuitively, the KE should implement a function $\delta: [0,1]^n \to [0,1]^n$, called \emph{t-conorm boost function}, which increases the value of $\bot({\bf \lit})$.
\begin{definition}
A function $\delta: [0,1]^n \to [0,1]^n$ is a \emph{t-conorm boost function} (TBF) if:
$$\forall n \in \mathbb{N} \ \ \forall {\bf \lit} \in [0,1]^n \ \ 0 \leq t_i + \delta({\bf \lit})_i \leq 1$$
\end{definition}
\noindent Let $\Delta$ denote the set of all TBF s.
\begin{proposition}
For every t-conorm $\bot$ and every TBF\ $\delta$,
$
\bot({\bf \lit}) \leq \bot({\bf \lit} + \delta({\bf \lit}))
$
\end{proposition}
\begin{proof}
By definition of TBF,
\mbox{$\forall i \in [1,n]$, $t_i \leq t_i + \delta({\bf \lit})_i$};
the conclusion directly follows from Property 2. of t-conorms.
\end{proof}
\begin{figure}
\centering
\input{tikz/truth_chart}
\caption{Effect of TBF s on base NN predictions for the grounded clause $c(a,b): \lnot Smoker(a) \lor \lnot Friends(a,b) \lor Smoker(b)$, where predicate $Friends$ is true when the two persons are friends. The gray areas represent the original predictions of the base NN while the green ones the changes applied by the TBF . The last bar represents the truth value of $c$ under the G\"{o}del t-conorm, which is increased in both cases. Indeed, both graphs shows a TBF. The improvements on the left are not minimal, since it is possible to reach the same improvement with smaller changes. On the contrary, the TBF \ on the right is minimal for the G\"{o}del t-conorm.}
\label{fig:truth_chart}
\end{figure}
TBF s are used in the KE to update the initial predictions $\mathbf{y}$ done by the base NN. While there are infinite many TBF s, not all of them can be used for our purposes. Consider for example the function $\delta({\bf \lit}) = \mathbf{1} - {\bf \lit}$: this of course is a TBF \ and it makes the t-conorm completely true for every possible initial predictions:
$$
\bot({\bf \lit} + \delta({\bf \lit})) = \bot({\bf \lit} + \mathbf{1} - {\bf \lit}) = \bot(\mathbf{1})
$$
Although the constraint reaches its maximum satisfaction, such a function returns a constant value for each literal ($\forall i \ t_i = 1$) which is useless for our purposes since the NN's predictions are not taken into account. We want to find a balance between the NN's predictions and the satisfaction of the clause: the TBF \ should keep the change on the initial predictions as minimal as possible. Therefore we look at TBF's that improve the t-conorm value \emph{in a minimal way} so that it is not possible to obtain a higher improvement with smaller modifications on literals values. An example of this idea can be seen in Fig.~\ref{fig:truth_chart}.
We now define the concept of minimality for a TBF s.
\begin{definition}
A function $\delta \in \Delta$ is minimal with respect to a norm $\|\cdot\|$ and a t-conorm $\bot$ iff:
\begin{equation}
\begin{split}
&\forall \delta' \in \Delta \ \ \forall n \in \mathbb{N} \ \ \forall {\bf \lit} \in [0,1]^n \\
&\| \delta'({\bf \lit}) \| < \| \delta({\bf \lit}) \| \to \bot({\bf \lit} + \delta'({\bf \lit})) < \bot({\bf \lit} + \delta({\bf \lit}))
\end{split}
\end{equation}
\end{definition}
\noindent Minimal TBF is defined with respect to a given norm $\|\cdot\|$ and a t-conorm $\bot$. We are going to focus on G\"{o}del t-conorm which is defined as
$$
\bot({{\bf \lit}}) = \max_{i=1}^n(t_i)
$$
and $l_p$-norm:
$$
\|{\bf \lit} \|_p = \left( \sum_{k=1}^{n} | t_k |^p \right)^{1/p}
$$
\noindent
Notice that G\"{o}del t-conorm value depends only on the highest literal in the clause. Since we are interested in the minimal change in the predictions it seems reasonable to increase only the highest truth value.
For any function $f:{\mathbb R}^n\rightarrow {\mathbb R}$
we define $\delta^{f}: {\mathbb R}^n\rightarrow {\mathbb R}^n$ as
\begin{equation}
\delta^f({\bf \lit})_i =
\left\{
\begin{array}{ll}
\label{eq:delta-f}
f({\bf \lit}) \ \ \ & \mbox{if $i = \argmax_{j=1}^n t_j$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
\begin{theorem}
If we choose function $f$ such that \mbox{$0 \leq f({\bf \lit}) \leq 1 - \max_{j=1}^nt_j$}, then $\delta^f$ function is minimal TBF s for the G\"{o}del t-conorm and $l_p$-norm.
\label{th:minimal_TBF}
\end{theorem}
\begin{figure}
\centering
\input{tikz/geometric_proof}
\caption{Geometric intuition for minimality of $\delta^f$: an example with ${\bf \lit}=[0.3,0.5]$ (black line). The blue circle represents the set of points ${\bf \lit} + \mathbf{k}$ with $\| \mathbf{k} \|_2 < \| \delta^f({\bf \lit})\|_2$. The green area is the set of all point ${\bf \lit} + \mathbf{k}$ with $\bot({\bf \lit}+ \mathbf{k}) < \bot({\bf \lit}+\delta^f({\bf \lit}))$. The blue area is completely inside the green square, i.e. $\delta^f({\bf \lit})$ (blue line) is the minimal change we could apply for incrementing the t-conorm of 0.15.}
\label{fig:proof}
\end{figure}
Fig.~\ref{fig:proof} shows a geometric interpretation for Theorem~\ref{th:minimal_TBF}. In the example, it is shown a specific case: the initial truth values for the two literals $\lnot Smoker(x)$ and $Cancer(x)$ are $0.5$ and $0.3$ respectively. The G\"{o}del t-conorm value is $0.5$, since it is the highest value. To be minimal, a TBF \ must increase the value of the t-conorm with the smallest possible change, meaning that it is not possible to reach the same improvement on the t-conorm with smaller changes. The blue circle contains the set of points that apply a smaller change to ${\bf \lit}$ than $\delta^f$ (in this case, we consider the $l_2$-norm). The picture shows that for all those points the G\"{o}del t-conorm is smaller than $0.65$, the value obtained by using $\delta^f$.
Of course, this does not represent a proof for the minimality of $\delta^f$, since it is applied to a specific set of truth values and it considers only $l_2$-norm. We need to prove it formally.
\begin{proof}
It is easy to see that $\delta({\bf \lit})$ is a TBF: the function always returns values inside the range $[0,1]$ and even the final predictions are in this range. We need to prove that it is minimal. Suppose that $\delta\in \Delta$ is such that
$$
\| \delta({\bf \lit}) \|_p < \| \delta^f({\bf \lit}) \|_p
$$
If $j =
\argmax_{k=1}^n(t_k + \delta({\bf \lit})_k)$,
we can derive:
$$
\bot({\bf \lit} + \delta({\bf \lit})) = t_j + \delta({\bf \lit})_j
$$
and, if $i = \argmax_{k=1}^nt_k$, we have that
$$
\bot({\bf \lit} + \delta^f({\bf \lit})) = t_i + f({\bf \lit})
$$
\noindent
Since $t_i\geq t_j$, we just need to demonstrate that \mbox{$\delta({\bf \lit})_j < f({\bf \lit})$}. Notice that:
\begin{align*}
\delta({\bf \lit})_j &= {(|\delta({\bf \lit})_j|^p)}^{1/p} \leq \left( \sum_{k=1}^{n} | \delta({\bf \lit})_k |^p \right)^{1/p}
= \| \delta({\bf \lit}) \|_p < \| \delta^f({\bf \lit}) \|_p
\end{align*}
Since $\delta^f({\bf \lit})$ changes only the value of the $i^{th}$ component of ${\bf \lit}$, \mbox{$\|\delta^f({\bf \lit})\|_p = f({\bf \lit})$}.
\end{proof}
\subsection{Boosting preactivations}
\label{preact}
Applying directly a TBF\ to the final prediction of the base NN could be problematic since we have to respect the constraint that the improved values remain in $[0,1]$. This limits the possible functions $f$ that can be used. For instance, $f$ cannot be a linear function.
Notice that the outputs $\mathbf{y}$ of the NN are calculated by applying an activation function over the preactivations $\mathbf{z}$ generated in the last layer. We assume the activation function of the last layer of the base NN to be the logistic function, i.e.:
$$
y_i = \sigma(z_i) = \frac{1}{1 + e^{-z_i}}
$$
\noindent where $y_i$ is the activation of the $i^{th}$ predicate and $z_i$ the corresponding preactivation. As with the activations, we distinguish between preactivations of the grounded atoms and the ones of the literals of a grounded clause by using $\mathbf{z}$ and $\mathbf{v}$ respectively. More precisely:
$$\mathbf{v} = \sigma^{-1}({\bf \lit})$$
In section~\ref{TBFS} we showed that if we increase only the value of the highest literal (highest activation), such a change is minimal for G\"{o}del t-conorm. We can apply the same strategy to preactivations and still have a minimal change that increases the t-conorm. In other words, we can increase $v_i$ instead of $t_i$, when $i = \argmax_{j=1}^n v_j$ and the previously proven properties still hold.
The initial predictions are
$
{\bf \lit} = \sigma(\mathbf{v})
$
and the final ones are
$
{\bf \lit}' = \sigma(\mathbf{v} + \delta^f(\mathbf{v}))
$.
\begin{proposition}
For any function $f:\mathbb{R}^n\rightarrow\mathbb{R}^+$, the function
\begin{equation}
\delta^g({\bf \lit}) = \sigma(\mathbf{v} + \delta^f(\mathbf{v})) - \sigma(\mathbf{v})
\end{equation}
is a minimal TBF \ for G\"{o}del t-conorm and $l_p$-norm.
\end{proposition}
\begin{proof}
Notice that final predictions ${\bf \lit}' = \sigma(\mathbf{v} + \delta^f(\mathbf{v}))$ are in $[0,1]$ since this range is the image of logistic function. Furthermore, $\sigma$ is monotonic increasing, which means that the highest preactivation corresponds to the highest activation:
$$\argmax_{j=1}^n \sigma(v_j) = \argmax_{j=1}^n v_j$$
Another implication of the monotonicity of $\sigma$ is that increasing a preactivation produces also an increase of the corresponding activation:
$$f(\mathbf{v}) \geq 0 \to \sigma(v_i + f(\mathbf{v})) \geq \sigma(v_i)$$
\noindent
Putting the previous two together we obtain that increasing the highest preactivation does indeed imply that an increase of the highest activation, meaning that $\delta^g$ is a minimal TBF.
More formally, lets define function $g({\bf \lit}) = \sigma(v_i + f(\mathbf{v})) - \sigma(v_i)$.
We obtain:
\begin{equation}\begin{split}
\delta^g({\bf \lit})_i &= \sigma(\mathbf{v} + \delta^f(\mathbf{v}))_i - \sigma(\mathbf{v})_i \\
&=
\left\{
\begin{array}{ll}
g({\bf \lit}) \ \ \ & \mbox{if $i = \argmax_{j=1}^n t_j$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{split}
\end{equation}
Theorem~\ref{th:minimal_TBF} guarantees that $\delta^g$ is a minimal TBF \ for G\"{o}del t-conorm and $l_p$-norm
\end{proof}
The function $\delta^g$ is not directly used by KENN; it is implicitly induced by the application of $\delta^f$ on $\mathbf{v}$. Therefore, by showing that it is a minimal TBF, we prove that applying $\delta^f$ on $\mathbf{v}$ is indeed equivalent to apply a minimal TBF on the NN predictions.
Until now we have seen that we can modify the preactivations $\mathbf{v}$ instead of changing directly the final truth values (${\bf \lit}$), but we haven't justified why it is better to do this. The first reason is that applying changes on $\mathbf{v}$ has the advantage of guaranteeing the final predictions to be in $\left[0,1\right]$ which means that we don't have any constraint on the choice of function $f$. Moreover, we could interpret the results of the Neural Network as levels of confidence in its predictions. According to this view, an initial value close to one (or zero) of the activation means high confidence in the prediction and we would like $f$ to apply a small change in such cases. On the other hand, if the initial prediction is close to $0.5$ (preactivation close to $0$) then we have the maximum uncertainty. Fig.~\ref{fig:logistic} shows how the same change on two different preactivation values results in different changes on the activations: the closer the initial value is to zero (maximum uncertainty), the higher the change on final truth values.
\DONE{Summarizing, to increase $c$ satisfaction we increase the preactivation of its highest literal. The extent of the change depends on the function $f$. KENN\ uses a distinct $f_c$ function for every clause $c$.} This is motivated by the fact that we want $\delta^{f_c}$ to be proportional to \emph{clause weight} $w_c$ (learnable parameter) which expresses the strength of the clause. The simplest function that conforms to this property is the constant function $w_c$ (with $w_c \in [0,\infty]$). The function applied to $\mathbf{v}$ to increase $c$ satisfaction is therefore $\delta^{w_c}$ defined as in \ref{eq:delta-f}:
\begin{equation}
\delta^{w_c}(\mathbf{v})_i =
\left\{
\begin{array}{ll}
w_c \ \ \ & \mbox{if $i = \argmax_{j=1}^n v_j$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
\begin{figure}[t]
\centering
\input{tikz/logistic}
\vspace*{-0.5cm}
\caption{The same change $\delta_v$ applied on two different values of preactivations results in different changes on activations: the more close the preactivation to zero (maximum uncertainty) the highest the modification on final predictions}
\label{fig:logistic}
\end{figure}
\subsection{Soft approximation of $\delta^{w_c}$}
\label{soft}
Although $\delta^{w_c}$ respects our minimality property, there are two problems when using it inside a neural network: first, it is not differentiable; second, it is too strict when multiple literals have close values. In those cases, it increases just one of the values even if the difference is minimal.
To obviate these problems, in our implementation we substitute $\delta^{w_c}$ with the $\softmax$ function multiplied by $w_c$, that can be seen as a soft differentiable approximation of $\delta^{w_c}$:
\begin{equation}
\delta^{w_c}_s(\mathbf{v})_i = w_c \cdot \softmax(\mathbf{v})_i = w_c \cdot \frac{e^{v_i}}{\sum_{j=1}^{n}e^{v_j}}
\label{eq:delta_wc_s}
\end{equation}
\subsection{Increasing the satisfaction of the entire knowledge}
\label{sec:entire_knowledge}
Until now we considered only the changes to be applied to increase a single clause, while in the general case $\mathcal{K}$ will be composed of multiple clauses. We need a way to aggregate the multiple changes proposed by each clause to obtain the final changes to be applied on all the predictions based on the entire knowledge.
Notice that we always used vectors $\mathbf{v}$ and ${\bf \lit}$ which contain only the preactivations and activations of the literals of the considered clause. In practice, when dealing with specific real world tasks, the base NN produces the preactivations $\mathbf{z}$ of every atomic formula. For this reason, the inputs of a specific TBF must be created from $\mathbf{z}$.
Given a grounded clause $c = \bigvee\limits_{i=1}^{m} l_i$
we define $\mathbf{z}_c$ as the vector containing preactivations of its literals:
$$
\mathbf{z}_c = \left< z_{l_1}, z_{l_2}, \dots z_{l_m} \right>
$$
\noindent where $z_{l_i}$ is the preactivation of literal $l_i$.
Calculating $\mathbf{z}_c$ is a straightforward task since we need just to select the right predictions of the base NN and negate them when necessary. Indeed, at the level of the base NN, the predictions are always on atomic formulas (positive), which means that we need to define the preactivations of a negated atom.
As seen in Section~\ref{sec:semantic}, the negation is interpreted as:
$$\mathcal{I}_N(\lnot A) = 1 - \mathcal{I}_N(A)$$
\noindent Logistic function has the following property:
$$1 - \sigma(x) = \sigma(-x)$$
and given that we work with preactivations:
$$\mathcal{I}_N(\lnot A) = 1 - \mathcal{I}_N(A) = 1 - \sigma(z_A) = \sigma(-z_A)$$
\noindent where $z_A$ is the preactivation of grounded atom $A$.
Therefore, the preactivation of a negated atom is equal to the one of the not negated atom multiplied by -1:
$$
z_{\lnot A} = - z_A
$$
\subsection{Vectorial representation of a grounded clause}
From now on, we are going to use a vectorial representation for grounded clauses: a clause $c$ is defined by a pair of vectors $(\mathbf{p}_c, \mathbf{s}_c)$:
\begin{itemize}
\item $\mathbf{p}_c \in \mathbb{N}^m$ is a vector containing the positions inside $\mathbf{z}$ of the grounded atoms in $c$
\item $\mathbf{s}_c \in \{0,1\}^m$ is a polarity vector which contains the sign of the literals of $c$
\end{itemize}
As an example, consider the grounded clause $c: A_1 \lor \lnot A_3$, where $A_1$ and $A_2$ are grounded atoms that can be find in positions $1$ and $3$ inside vector $\mathbf{z}$. The two vectors are $\mathbf{p}_c = \left<1,3\right>$ and $\mathbf{s}_c = \left<1,-1\right>$.
At this point, we define function $\phi: \mathbb{R}^n \times \mathbb{N}^m \times \{0,1\}^m \to \mathbb{R}^m$ which calculates the vector $\mathbf{z}_c$ with preactivations of literals of $c=(\mathbf{p},\mathbf{s})$:
\begin{equation}
z_{c,i} = \phi(\mathbf{z}, \mathbf{p}, \mathbf{s})_i = s_i \cdot z_{p_i}
\label{eq:phi}
\end{equation}
\noindent Given the vector $\mathbf{z}_c$, we can apply the (soft approximation of the) TBF to obtain $\delta_c^l$:
$$
\delta_c^l = \delta_s^{w}(\mathbf{z}_c)
$$
where $w$ is the weight of clause $c$. Notice that $\delta_c^l$ contains the changes to be applied on literals preactivations, not on the original grounded atoms in $\mathbf{z}$. To obtain the changes for the atomic formulas we rely on function $\phi': \mathbb{R}^n \times \mathbb{N}^m \times \{0,1\}^m \to \mathbb{R}^m$ which works in reverse of $\phi$:
\begin{equation}
\phi'(\delta_c^l, \mathbf{p}, \mathbf{s})_i =
\left\{
\begin{array}{ll}
s_j \cdot \delta_{c,j}^l & \qquad \mbox{if \ $\exists \ j : \ p_j = i$} \\
0 & \qquad \mbox{otherwise}
\end{array}
\label{eq:phip}
\right.
\end{equation}
\noindent The final changes on $\mathbf{z}$ based on clause $c=(\mathbf{p},\mathbf{s})$ are:
\begin{equation}
\begin{split}
\delta_c(\mathbf{z}) &= \phi'(\delta_c^l,\mathbf{p},\mathbf{s}) \\
&= \phi'(\delta_s^w(\phi(\mathbf{z}, \mathbf{p}, \mathbf{s})),\mathbf{p},\mathbf{s})
\end{split}
\label{eq:deltac}
\end{equation}
At this point, we need to define how to aggregate the various contributions of each clause. We use a simple sum of such values to obtain the final change. The final predictions $\mathbf{y}'$ are calculated as:
\begin{equation}
\mathbf{y}' = \sigma(\mathbf{z} + \sum_{c \in \mathcal{G}(\mathcal{K}, \mathcal{C})} \delta_c(\mathbf{z}))
\label{eq:final_intepretation}
\end{equation}
The choice of the sum as the aggregator of the various contributions makes learning and inference fast. In this way, the scalability is increased but also the probability of inconsistency between the final predictions and the logical rules. This aspect will be further investigated in Section~\ref{sec:rel_work_2}, where the extent of the problem will be analyzed and a comparison with RNM from this point of view will be provided.
\section{Multi-label Classification with labels constraints}\label{sec:multilabel}
We need to evaluate two different aspects of KENN: its ability to make use of the knowledge and the applicability of the model to relational domains. In this section we focus on the first aspect: the first experiments were conducted on three datasets for multilabel classification with no binary predicates. The next section will focus on evaluations on a relational domain.
\subsection{Multilabel classification tasks}
Multi-label classification is a supervised learning task relevant in many disciplines, e.g., bioinformatics~\cite{Yeast}, scene classification~\cite{multilabel_scene} and text categorization~\cite{text}.
In multi-label classification we are interested in mapping specific observations to subsets of all the possible labels~\cite{multilabel,PF2008}. It differs from binary classification and, more in general, from multi-class classification, because the classes are not mutually exclusive, i.e. multiple labels can be associated to a single instance.
Formally, it is given a set of labels $\mathcal{L} = \left\{ \lambda_i | i=1...m \right\}$ and a training set $\mathcal{T} = \left\{ (x_i, y_i) | i=1...n \right\}$, where $x_i$ denotes features of the $i^{th}$ observation and the classification $y_i \in 2^{\mathcal{L}}$ is a subset of labels associated to such observation. We want to find a classifier $\phi: \mathcal{X} \to 2^\mathcal{L}$ which, given features $x$ of an observation, returns the associated set of labels.
KENN\ is extremely suited to formulate multi-label classification with constraints. Indeed KENN\ uses vectorial representation for both features and corresponding classification, with $y_{i,j}$ equal to one if $\lambda_j$ is associated to observation $i$, zero otherwise.
While it is possible to train many different binary classifiers for each label, such a method does not take into account relationships among them. Indeed, in real-world applications, labels are often not independent. For this reason, it could be useful to exploit knowledge about labels relationships provided by some human experts.
The prior knowledge $\mathcal{K}$ defined in~\ref{sec:prior_knowledge} represents the labels relationships that are represented in $\mathcal{L}$ as predicates. For instance, a possible clause could be $\lambda_1 \lor \lnot \lambda_3 \lor \lambda_4$. Such a clause tells the system that at least one of the specified literal should be true.
We tested KENN\ on three datasets: Yeast~\cite{Yeast_1}, Emotions~\cite{Emotions} and VRD Dataset~\cite{visual2}. Table~\ref{datasets} reports some figures of the three datasets. Please note that for VRD the features are two bounding boxes of an image together with the class of the contained object (there are 100 possible classes and we used as features the one-hot encoding of them plus the 4 coordinates of each bounding box).
\begin{table}
\caption{Datasets statistics}
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
Dataset & Features & Labels & Train & Test \\
\hline
\hline
Yeast & 103 & 14 & 1500 & 917 \\
\hline
Emotions & 72 & 6 & 392 & 202 \\
\hline
VRD & 208 & 70 & 4000 & 1000 \\
\hline
\end{tabular}
\label{datasets}
\end{table}
\subsection{KENN with automatically generated knowledge}
\label{sec:yeast_emotions}
{
\subsection{Yeast and Emotions datasets}
Yeast dataset is a multilabel classification dataset for Bioinformatics. It contains 1500 training samples, each of which is composed of a vector of 103 features~\cite{Yeast_1, Yeast}. The features contain information about yeast microarray expressions. The 14 labels to be predicted are functional categories of yeast genes.
The Emotions dataset contains features of songs of 7 types of different genres~\cite{Emotions}. Features were extracted from sounds clips of 30 seconds and they are divided into two categories: Rhythmic and Timbre. The labels are divided into 6 classes which represent different types of emotions.
\subsection{Association rules learning for generating the knowledge}
In our experiments on Yeast and Emotions datasets we used automatically extracted association rules.
Association rules learning is the task of extracting association rules from a database of transactions~\cite{rules_mining}. Given a set of items $\mathcal{I} = \mathcal{I}_1, \mathcal{I}_2 ... \mathcal{I}_n$, a transaction consists of an itemset $\mathcal{T} \subseteq \mathcal{I}$ and an association rule is an implication of the form ${\lambda_1 \land ... \land \lambda_n \to \lambda_j}$ that must hold in the dataset with a certain confidence.
The idea is to use as prior knowledge for KENN\ the association rules extracted from labels of the training set.
For extracting association rules we used Apriori algorithm~\cite{Apriori}. As in~\cite{PF2008}, we generated each transaction from a sample using both positive and negative labels. More precisely, given a sample $(x_i, y_i)$, the transaction is calculated as:
$$
\Big\{ \lambda_j | y_{i,j} = 1 \Big\} \bigcup \Big\{ \lnot \lambda_j | y_{i,j} = 0 \Big\}
$$
where $\lambda_j$ is the $j^{th}$ label.
\subsection{Experimental setup}
In the experiments on both datasets, we used RMSProp~\cite{RMSProp} as learning algorithm.
To select the best values for support and confidence of the Apriori algorithm we used a random search: we tried different combinations of values of the two hyperparameters to generate the rules. we used the generated rules to train KENN\ using 2/3 of the Training Set and evaluated the model on the remain samples. Finally, the values that led to the highest accuracy were selected. Then, during the final evaluations, we used such optimal values for training KENN\ using the entire Training Set.
Finally, we tried two learning strategies: in the \emph{end-to-end} strategy, we trained KENN\ (base NN~+~KE) end-to-end using the entire training set; with \emph{greedy} strategy we split the training set into two subsets. With the first one we trained the base NN, with the second the KE (freezing the NN parameters). When showing results we will use KENN$_e$ and KENN$_g$ to distinguish between the two.
It is worth noticing that, when using the greedy approach of learning, for every configuration tried for Apriori parameters the accuracy of KENN\ was greater or equal than the one of the corresponding neural network, confirming the robustness of our method to poorly written knowledge bases.
The found values for support and confidence are respectively 0.2 and 0.99 for Yeast Dataset and 0.2 and 0.7 for Emotions.
\subsection{Results and discussion}
\label{sec:yeast_results}
\begin{table}[t]
\caption{Results with and without Prior Knowledge on Yeast and Emotions Datasets. The table shows Hamming Loss and Accuracy.}
\centering
\begin{tabular}{lllllll}
&
\multicolumn{1}{p{1.5cm}}{} &
\multicolumn{2}{c}{Yeast} &
\multicolumn{1}{p{0.5cm}}{} & \multicolumn{2}{c}{Emotions} \\
\hline
\rule{0pt}{3ex}
& & HL & Acc & & HL & Acc \\
\hline
\rule{0pt}{3ex}\hspace{-0.12cm}
& LR & 22.38 & 37.52 & & 27.89 & 33.70 \\
& KENN$_e$ & 22.24 & 46.43 & & 34.57 & 25.74 \\
& KENN$_g$ & \textbf{20.86} & \textbf{48.56} & & \textbf{24.59} & \textbf{38.78} \\
\hline
\hline
\end{tabular}
\label{YeastEmotions}
\end{table}
In Table~\ref{YeastEmotions} the final comparison between the logistic regression and KENN: the usage of prior knowledge improved both HL and accuracy when using the greedy approach of learning, while the end-to-end one results in smaller improvements in Yeast dataset and a degradation in Emotions.
In principle, we could expect better results training jointly the NN and KE, because there could be some combinations of parameters for the entire model such that the NN performs poorly while the entire system gives good results.
This type of configurations cannot be learned by the greedy approach. In other words, when training with the greedy approach, some solutions in the hypothesis space can not be reached.
For this reason, the greedy approach has a smaller ability to fit the data. However, this could also imply an increased risk of overfitting.
Therefore, we can expect better results of the end-to-end version when the amount of data is big enough to overcome overfitting.
Indeed, the problem with the end-to-end version is visible in particular on the Emotions dataset, which has less than 400 samples for its training.
\subsection{Visual Relationship Detection}
Visual Relationship Detection (VRD) is the task of finding objects in an image and capture their interactions~\cite{IvanThesis,visual2,visual1}. It is composed of three subtasks: Relationship Detection, Phrase Detection and Predicate Detection~\cite{visual2}. The general goal is to find relationships that are defined as triplets of the form $(subject, relation, object)$\footnote{in the original paper, the triplet was $(subject, predicate, object)$. Here we substituted $predicate$ with $relation$ to avoid ambiguity with the term ``predicate'' already in use}, where subject and object are defined with their class and a container bounding box. \DONE{Given an image, a bounding box of an object is the smallest rectangle that completely encloses the object and it is represented by the coordinates of the top left and the bottom right vertexes of the rectangle.}
An example of a relationship is the triplet $(person, kick, ball)$ in Fig.~\ref{fig:VRD}.
KENN\ was evaluated only on the Predicate Detection task where subject and object are already given and the objective is to find only the relations. More in detail, in Predicate Detection two bounding boxes are given. The first one encloses the subject, while the second contains the object. Together with the bounding boxes, the classes of the two contained objects are also given. The goal is to find the set of relations between the subject and the object.
Notice that each couple of objects can have multiple relations and the goal is to find all of them.
Moreover, even though the goal is to predict binary relations, we treated them as unary predicates by using the same strategy proposed in Section~\ref{sec:join}: a pair of bounding boxes is treated as a unique object and the binary predicates as unary predicates over the domain of pairs of bounding boxes. Notice that in this case we don't have to predict unary predicates in the original domain (the classes of the bounding boxes are already given) and for this reason, we don't need to use the RelationalKenn class.
\subsubsection{Visual Relationship Dataset}
The Visual Relationship Dataset (VRD Dataset) is a dataset for Visual Relationship Detection. It contains 5000 images (4000 for training and 1000 for testing). Each image contains multiple objects, each of which is categorized based on its class. These classes include animals, vehicles, clothes, and other generic objects.
There are 100 classes that subject and object can take and 70 relations that have to be predicted which are divided into five types. Fig.~\ref{fig:VRD} shows the five types of relations in the VRD Dataset. The types are: Action (like Ride or Kick), Spatial (for example Below or On), Preposition (for example with or at), Comparative (like Taller) and Verb (for example Wear).
A triplet type is a triple $(subject class, relation class, object class)$ which corresponds to the classes of the two objects and the relation.
In total, the dataset contains 377993 relationship instances, with 6672 triplets types. Among the types of triplets, 1877 can be found only in the Test Set and predicting them is the goal of the Zero Shot Learning variant of the task.
\begin{figure*}
\makebox[\textwidth][c]{
\includegraphics[scale=0.55]{Figures/VRD.png}
}
\caption{The five types of relations in VRD Dataset}
\label{fig:VRD}
\end{figure*}
\subsubsection{The knowledge}
We evaluated KENN\ on the Predicate Detection task using the manually curated knowledge base described in \cite{IvanThesis}. This knowledge base contains 206 clauses divided in three groups:
\begin{description}
\item[domain and range clauses] they restrict the domain (resp. range) of relations.
\\Example of domain clause: $\lnot Wear(x,y) \lor Person(x)$
\\Example of range clause: $\lnot ParkOn(x,y) \lor Street(y) \lor Road(y) \lor Grass(y)$
\item[mutual exlusivity clauses] they represents mutual exsclusivity between to relations.\\
An example is $\lnot Behind(x,y) \lor \lnot SitOn(x,y)$
\item[sub-relation clauses] they state the containment between relations. An example:\\$\lnot Ride(x,y) \lor On(x,y)$)
\end{description}
\subsubsection{Experimental setup}
\label{sec:VRD_settings}
The experiments were carried out once again with Logistic Regression as Base NN. Notice that some of the clauses involve also unary predicates, which are given as input. For this reason, the architecture used is the one defined in Figure \ref{fig:KENN_inputs}, where the input predicates (classes of the bounding boxes) are concatenated with the predictions of the base NN before the application of the KE.
Although KENN\ can learn clause weights, the presence of such parameters could slow down the learning process or cause overfitting. The system has to both use the knowledge and validate it.
Notice however that in KENN\ clause weight can be added as constant values, in particular when the knowledge is reliable and the constraints are hard. For this reason, in VRD experiments, the domain and range clauses were treated as hard constraints: the clause weight was set to a high constant value (10.0) which was not learnable. This is because we know for sure that such rules are always satisfied.
We trained both LR and KENN\ using RMSProp~\cite{RMSProp} and cross-entropy as the loss function. For evaluating the results we used the $Recall@n$ ($n \in \{50, 100\}$) metric proposed by Lu et al.~\cite{visual2} that is the percentage of times a correct relationship is found on the $n$ predictions with the highest score.
\subsubsection{Results and discussion}
\noindent Results on Predicate Detection task are shown in Table~\ref{table1}.
\begin{table}[t]
\centering
\caption{Results on VRD Predicate Detection task}
\begin{tabular}{llcclcc}
&
\multicolumn{1}{p{3.0cm}}{}
& \multicolumn{2}{c}{Standard} &
\multicolumn{1}{p{0.5cm}}{} &
\multicolumn{2}{c}{Zero Shot} \\
\hline
\rule{0pt}{3ex}
& & R@50 & R@100 & & R@50 & R@100 \\
\hline
& \rule{0pt}{3ex}\hspace{-0.12cm} \cite{visual2} & 47.87 & 47.87 & & 8.45 & 8.45 \\
& \cite{dai2017detecting} & 80.78 & 81.90 & & - & - \\
& \cite{Yu2017} & 85.64 & \textbf{94.65} & & 54.20 & 74.65 \\
& \cite{IvanThesis} & 78.63 & 91.88 & & 46.28 & 70.15 \\
& LR & 54.58 & 60.55 & & 33.88 & 44.91 \\
& KENN$_g$ & 59.87 & 71.42 & & 43.88 & 63.99 \\
& KENN$_e$ & \textbf{86.02} & 91.91 & & \textbf{68.95} & \textbf{83.83} \\
\hline
\hline
\end{tabular}
\label{table1}
\end{table}
KENN\ outperformed other methods on all the metrics except for Recall@100 of the standard variant of the task where it is surpassed by \cite{Yu2017}. Moreover, the best results of KENN\ can be seen on Zero Shot Learning version where the difference between KENN\ and the second best system is more than 10\%.
In \emph{Zero Shot Learning} the aim is to predict previously unseen triplets, therefore it is rather difficult to learn to predict them from the Training Set. This confirms the ability of KENN\ to use the Knowledge Base.
Another interesting result is the value obtained by KENN\ compared to LTN~\cite{IvanThesis}. In particular considering that the two works used the same Prior Knowledge. A possible explanation is given by the ability of KENN\ to learn \emph{clause weights}. Indeed, many weights results to be zero after learning. An example of a zero weighted clause is:~$\lnot Ride(x,y) \lor On(x,y)$.
Although the rule seems correct it is not in general satisfied in the training and test sets. This is because labels have been added manually, therefore there are plenty of missing relations. Our hypothesis is that people have a tendency to add the most informative labels making some of the clauses unsatisfied. For instance, suppose we need to label an image with a man riding a horse and we are not going to add all the relations since their number is pretty big (there are other objects and relations to label in the image, and multiple images). For the person and horse objects, there are plenty of different relations that are true: $Ride(x,y)$, $On(x,y)$, $Over(x,y)$, $Near(x,y)$ and so forth. It seems reasonable to think that a person that labels the image will tend to choose $Ride(x,y)$ over the others since it is the most informative one. This is because from $Ride(x,y)$ we can derive all the other relations.
Finally, notice that a perfect comparison between KENN\ and LTN is difficult to obtain since the neural networks used in the two cases are different: KENN\ is applied on a Logistic Regression network while LTN uses a Neural Tensor Network~\cite{NTN}. While it is possible to use the same NN in both cases, the NN architectures which works fine with one method do not work well with the other. See Section~\ref{sec:kenn_vs_sbr} for more details.
Another important aspect of the results is that KENN\ increased the metrics values of LR. As in previous experiments on Yeast and Emotions (see Section~\ref{sec:yeast_emotions}), we tested KENN\ both with greedy and end-to-end approaches. This time, although the greedy version brought improvements over the logistic regression, the best results are achieved by the end-to-end approach. This provides additional support for the overfitting hypothesis made in Section~\ref{sec:yeast_results} since VRD has a much bigger training set than Emotions and Yeast. Moreover, in previous experiments, prior knowledge was automatically extracted from the training data. Clauses could be satisfied within the training set just by chance. In such cases, the Apriori algorithm extracts misleading rules that are not satisfied at test time. This implies a further increase in the chances of overfitting.
}
\section{Related Work}
\label{sec:rel_work_2}
In this section we will present three Neural-Symbolic systems and provide a comparison with KENN. More specifically, the three methods are LTN, SBR, and RNM. These approaches are the most relevant in the context of this paper since they can combine general neural networks models with FOL knowledge and they can be applied in similar contexts of KENN. For this reason, it is particularly relevant to provide a comparison of KENN\ with them. More specifically, the comparison between KENN, LTN and SBR is relevant since it can highlight the pros and cons of adding the knowledge trough the Loss function as opposed to injecting it into the model structure. On the other hand, RNM follows a similar philosophy to KENN, but with a different choice on the way the knowledge is implemented inside the model. For these reasons, the three methods will be further compared with KENN\ later, but in terms of empirical results.
\subsection{Injecting knowledge using the Loss function}
To combine logical knowledge with machine learning approaches there are two common strategies: incorporate the knowledge by including it in the Loss function or introducing it inside the model. In the first category, the two most prominent works are Logic Tensor Networks and Semantic Based Regularization.
Both LTN and SBR are based on the concept of constraint. The idea is that, given a set of logical rules, those rules induce constraints on the acceptable outputs of the model. The two methods integrate those rules into the learning framework by regularizing the Loss function with an additional term that penalizes solutions that do not satisfy the constraints. To do so, they rely on a continuous relaxation of the logical rules inducted by fuzzy operators.
\subsubsection{Semantic Based Regularization}
As already mentioned, SBR introduces the knowledge during learning through the usage of a regularization term which increases when the constraints are not satisfied. The satisfaction of a constraint is calculated using a fuzzy generalization of the logic operators which is continuous and differentiable. The regularization term has the following form:
$$
R(f) = \sum_{h=1}^{H}{\lambda_h (1 - \phi_h(f))}
$$
where $H$ is the number of constraints, $\lambda_h$ is the weight associated to the $h^{th}$ constraint, $f$ is the vector of functions that represent the predicates (these are learned) and $\phi_h(f)$ is the level of satisfaction of the $h^{th}$ constraint.
Notice that, since the weights $\lambda_h$ are part of the Loss function, there is no way to let the back-propagation algorithm learn them and for this reason, they are assumed to be known a priori.
This is one of the major drawbacks of these kinds of methods as opposed to strategies that, like KENN, directly encode the knowledge into the model. Indeed, not always the final user of the method knows in advance the importance of a specific logical rule, and in some cases, some rule could be not correct. Moreover, by allowing rules' weights to be learnable instead of being hyper-parameters, it is in theory possible to incorporate random rules and rely on the learning algorithm to select the correct one by reducing the corresponding weights. In this way, it would be possible to discover new symbolic knowledge from data.
Figure~\ref{fig:hs_sbr} shows a representation of the Hypothesis Space (HS), i.e., the set of all the possible functions representable by the model, and uses colors to represent the value of the Loss function and the regularization term on the different hypothesis. In the shown example, the HS is represented as a subset of $\mathbb{R}^2$ and there is only one logical rule. Of course, this is not intended as a realistic HS of a neural network and the goal here is just to provide intuition on the effect of the regularizer on the training process. Red color represents high values for the loss and regularization term, while green color corresponds to low loss. The goal of the training process is to find a minimum of such function. To train a Neural Network, the standard approach is to use back-propagation, an efficient implementation of gradient descent. The algorithm starts from a random solution (the black circle in the figure). At each training step, the forward pass calculates the predictions of the current hypothesis and uses these predictions to calculate the value of the loss function. Then, in the backward pass, the gradient of the loss function with respect to the parameters of the model is calculated and used to update the parameters. After multiple steps, a local minimum is typically reached (black cross in the figure). The top left image shows a possible evolution of this procedure when applied to the original Loss function. The top right image depicts the regularization term. Finally, the bottom image shows the values of the combination of Loss and regularizer. Here, the gradient descent, starting from the same initial hypothesis of the top left image, reaches a different hypothesis which satisfied more the given constraint on the Training Set.
Finally, notice that while the constraints are enforced at training time, there are no guarantees that they will be satisfied at inference time as well since the Loss function is calculated based on the predictions on the Training Set. To obviate this problem, at training time unlabelled data can be provided as well. In this way, the learning process has more examples to learn to enforce the knowledge even at test time. Another possibility is to enforce the constraints not only during training, but also at inference time. However, notice that in this case the time complexity of inference is increased, since the back-propagation needs to be used even in this case.
\begin{figure}
\makebox[\textwidth][c]{
\includegraphics[scale=0.35]{Figures/hs_sbr.png}
}
\caption{The interaction of loss and regularization on the Hypothesis Space for SBR and LTN. On the top left, to each point of the HS it is associated a loss value. Similarly, on the top right the regularization term associates a penalty to each solution that does not satisfy a constraint $c$. The total loss, calculated by summing the two, is on the bottom. The image shows also a possible execution of the gradient descend for both the original Loss function and the regularized version. The training algorithm starts from an initial random solution (black circle) and moves inside the hypothesis space reaching a local minimum that is the final trained model. After the regularization (bottom) it is easier to reach solutions that satisfy $c$ as compared with the original Loss function (top left).}
\label{fig:hs_sbr}
\end{figure}
\subsubsection{Logic Tensor Networks}
LTN is another method for integrating logic and learning.
It is a very similar approach to SBR. Like in SBR, the learning process maximizes the satisfaction of the constraints by including a penalty on the loss for solutions that do not satisfy them.
One of the main differences between SBR and LTN is in the way the existential quantifier is taken into account. SBR, relies on the close world assumption, and the level of satisfaction of an existentially quantified rule is obtained as the conjunction of all its possible groundings.
On the other hand, in LTN existential quantifiers are skolemized. To explain skolemization, let us consider a simple formula as an example:
$$\forall x \exists y Father(x,y)$$
The rule states that every person has a father. With skolemization, a function $F$ is introduced and the previous formula is changed in
$$\forall x Father(x, F(x))$$
Intuitively, $F$ represents the function that, given a person, return her/his father.
Another difference of LTN with respect to SBR is that it can be used as a ``generative model'', in the sense that functions can be used to generate new objects of the domain. Take for instance the previously mentioned ``father'' function $F$. One could apply such a function on the features of a given person to obtain the vector with plausible features of her/his father (estimated based on the examples of father and sons in the training data).
In the case of KENN, functions are not taken into account, nor the existential quantifiers. From this perspective, KENN\ is less flexible in comparison with LTN and SBR on the type of knowledge that it can use. Notice that existential quantifiers could be added by assuming a close world like in SBR. However, here we assume the knowledge to be general, consisting only on universally quantified rules.
\subsection{KENN\ vs methods based on Loss}
\label{sec:kenn_vs_sbr}
KENN\ englobes the knowledge into the network in a very different way than SBR and LTN. As already discussed, SBR and LTN act on the Hypothesis Space (HS) by changing the associated loss function according to the constraint.
The approach of LTN and SBR is equivalent to ``remove'' solutions from the HS that do not satisfies the constraints by penalizing them during training (they are not really removed, but difficult or impossible to reach because of the penalty given by the Loss function), while in KENN\ the approach is the opposite: new solutions are added to the Hypothesis Space (HS) by a new additional layer, called \emph{Knowledge Enhancer}.
For this reason, LTN and SBR need to use a model that is already capable of representing functions that satisfy the constraints and a bias towards their satisfaction is introduced by penalizing the other solutions. If the HS does not contain solutions that satisfy the constraints, LTN and SBR can not impose their satisfaction since they are limited by the set of hypotheses in the HS.
On the other hand, KENN\ starts from a NN with a lower capacity which is not capable to satisfy the constraints on its own and the knowledge is introduced by adding new solutions to the HS by modifying the existing ones. For this reason, KENN\ typically does not works well with NNs that are already capable of satisfying the clauses, since it does not introduce any bias towards their satisfaction.
Summarizing, to work properly, LTN and SBR need a model with high capacity able to express the required Knowledge, while for KENN\ it is the opposite. As an example, consider a Logistic Regression (LR), i.e. a neural network with no hidden layers. It is well known that with this kind of network it is not possible to represent the XOR operator $\oplus$~\cite{XOR1,XOR2}:
\begin{equation}
\oplus(x_1,x_2) =
\left\{
\begin{array}{rl}
0 & \qquad \mbox{if \ $x_1 = 0 \land x_2 = 0$} \\
1 & \qquad \mbox{if \ $x_1 = 0 \land x_2 = 1$} \\
1 & \qquad \mbox{if \ $x_1 = 1 \land x_2 = 0$} \\
0 & \qquad \mbox{if \ $x_1 = 1 \land x_2 = 1$}
\end{array}
\label{eq:XOR}
\right.
\end{equation}
Suppose we want to express with the knowledge that the target function is indeed the XOR function. Both SBR and LRN do not change the model structure and they impose the knowledge by acting on the weights of the model. Therefore, it is not possible for them to force the satisfaction of the rule with an LR model.
On the other hand, such a goal can be achieved by KENN\ using the architecture and knowledge shown in Figure~\ref{fig:XOR}.
The strategy is the same of Section~\ref{sec:KENN_inputs}: the input vector $\mathbf{x} \in \{0,1\}^2$ is passed to a logistic regression network to obtain initial preactivations $\mathbf{z}_y$. In parallel, the inverse of logistic function is applied to $\mathbf{x}$ to find its ``preactivations'' $\mathbf{z}_x$. By concatenating the $\mathbf{z}_x$ and $\mathbf{z}_y$, it is possible to use the KE to inject the clauses which represent the XOR operator's behavior.
\begin{figure}[H]
\centering
\input{tikz/XOR}
\caption{XOR clauses applied on a LR model.}
\label{fig:XOR}
\end{figure}
To see how is this working, suppose the inputs values are $x_1 = 1$ and $x_2 = 1$. In this case, the preactivations calculated by the logit function $\sigma^{-1}$ will be very high (supposing small values of $\epsilon$). This means that preactivations for $\lnot x_1$ and $\lnot x_2$ will be very small. For this reason, the CE that enforce the first clause will increase $\lnot y$, since only the highest literal is increased. Similarly, the other three clauses will not affect the predictions of $y$. Indeed, if the clause weights have high values, the model of Figure~\ref{fig:XOR} represents the XOR function. This can be done because KENN\ modifies the model, meaning that the limitations of LR do not apply anymore.
Notice that the weights of the clauses can be set to zero and, as a consequence, the functions representable by LR can still be reached by the training process. Indeed, KENN\ modifies the HS only by adding new hypotheses, not removing them. Figure~\ref{fig:hs_kenn} shows the effects of the KE layer on the HS.
\begin{figure}
\makebox[\textwidth][c]{
\includegraphics[scale=0.5]{Figures/hs_kenn.png}
}
\caption{The effect of KENN\ on the Hypothesis Space. The left image shows the original Hypothesis Space, the right one the Hypothesis Space represented by KENN\ model. In this picture, the original HS is represented as a portion of a plane. The effect of adding a clause is to increase the number of parameters of the model of one unit (the clause weight $w_c$). On the HS, this can be seen as adding a new dimension, where points with high values in that dimension have a higher ability to satisfy the constraints.}
\label{fig:hs_kenn}
\end{figure}
Given a clause, the model is extended with one parameter (the clause weight), which is shown in the Figure as a new dimension in the HS. Higher values of this parameter imply higher satisfaction of the corresponding clause. The approach relies on the idea that if the clauses are satisfied in the Training Set, then the new introduced hypotheses (with clause weights greater than zero) are more capable of fitting the data since there are no solutions in the original HS that can satisfy the clauses as well as the new ones. Indeed, any additional hypothesis introduced by KENN\ is obtained by adding a new layer that changes the output of the underlying network to increase the constraint.
\subsection{Relational Neural Machines}
RNM is a framework that integrates a neural network model with a FOL reasoner. This is done in two stages: in the first one a Neural Network $\mathbf{f}$ is used to calculate initial predictions for the atomic formulas; in the second stage a graphical model is used to represent a probability distribution over the set of atomic formulas.
The distribution is defined as follow:
$$
P(\mathbf{y}|\mathbf{f}, \lambda) = \frac{1}{Z} exp\Big( \sum_{x \in S} \phi_0(\mathbf{f}(x), \mathbf{y}) + \sum_c \lambda_c \phi_c(\mathbf{y}) \Big)
$$
where $Z$ is the partition function, $\mathbf{y}$ are the grounded atoms predictions, $\mathbf{f}$ is the function codified by the Neural Network, $\phi_0$ is a potential that enforces the consistency between the predictions of the NN and the final predictions, $\lambda_c$ is the weight of constraint $c$ and $\phi_c$ is a potential that enforces the satisfaction of the constraint (higher if the constraint is satisfied).
To obtain the final predictions a Maximum a Posteriori (MAP) estimation is performed, finding the most probable assignment to the grounded atoms given the output of the Neural Network and the set of constraints:
$$
\mathbf{y}^* = \argmax_{y} P(\mathbf{y}|\mathbf{f}, \lambda)
$$
\subsection{Comparison with KENN}
At a high-level RNM approach is similar to KENN, since in both cases a Neural Network makes initial predictions and a post elaboration step is applied on such predictions to provide the final classification. However, RNM requires to solve an optimization problem at inference time and after each training step. This has the advantage of considering all the logical rules together at the same time at the expense of an increased computational effort. Contrary, in KENN\ each rule is considered separately from the others, and the second stage is directly integrated inside the model as a differentiable function that can be trained end to end with the base Neural Network.
However, one could argue that with this strategy there could be some contradictory changes when combining multiple clauses with the same predicates. For instance, if the knowledge is composed by the two clauses $c_1: A \lor B$ and $c_2: \lnot B \lor C$, then the summation strategy introduced in Section~\ref{sec:entire_knowledge} would not force the satisfaction of $A \lor C$, which is a logical consequence of the two clauses. Indeed, the effect of the TBF \ could be to increase the value of $B$ when applied to $c_1$ and decrease it when applied on $c_2$. We call this type of situations as \emph{collisions}.
For simplicity, let us take the assumption that the improvements are provided by $\delta^{w_c}$ instead of $\delta_s^{w_c}$, i.e., they are calculated by using the $argmax$ operator instead of the $softmax$. In the case of a collision, the final change on $A$ and $C$ would be $0$, while the change on $B$ would be the difference of the two clause weights: $w_{c_1} - w_{c_2}$. Therefore, when a collision happens only the stronger clause is taken into consideration, and its effect is lowered by the presence of the other clause.
A question arises: are collisions likely to happen? For now, let us assume that the base NN is a random classifier which extracts the initial predictions from a uniform distribution. The probability of $B$ to be increased by the first clause is $1/2$, which is also the probability of $\lnot B$ to be increased based on $c_2$. However, the probability of a collision is lower than $1/4$, since the two events are not independent. We remind the reader that $B$ is chosen to be increased based on $c_1$ if its truth value $B^N$ is higher than $A^N$, while $\lnot B$ is increased if $1 - B^N$ is greater than $C^N$. Therefore, if $x$ is the value of $B^N$, then the probability of a collision could be expressed as:
$$
\int_{0}^{1} x (1 - x) dx = (\tfrac{1}{2} x^2 - \tfrac{1}{3} x^3) \Big|_0^1 = \tfrac{1}{6}
$$
More in general, let $c_1$ and $c_2$ be two clauses that share a common predicate with opposite sign. Lets $n$ and $m$ be the number of literals of $c_1$ and $c_2$ respectively. The probability of a collision is expressed by the beta function:
$$
B(n,m) = \int_{0}^{1} x^{1-n} (1 - x)^{1-m} dx
$$
In table~\ref{beta_function} we can see that the probability of a collision with different numbers of literals for the two clauses. Notice that the probability becomes quite small with an increasing number of literals. Moreover, these values are calculated from the assumption that the neural network returns random predictions.
If we assume that the base NN performs at least as good as a random classifier, then the values of Table~\ref{beta_function} represent an upper bound for the probability of having a collision.
\begin{table}
\caption{Beta function values}
\centering
\begin{tabular}{l|l|l}
n & m & B(n,m) \\
\hline
2 & 2 & 0.167 \\
2 & 3 & 0.083 \\
3 & 3 & 0.033 \\
3 & 4 & 0.017 \\
4 & 4 & 0.007 \\
\end{tabular}
\label{beta_function}
\end{table}
In general, we could expect better results from RNM in respect to KENN, but faster training and inference from KENN. However, as we will see in Section~\ref{sec:collective}, when the amount of training data increase, KENN's results are in line with RNM, and some cases even better. One possible explanation is that in RNM the model is not trained end to end, making more difficult the learning process as compared to KENN.
\section{Implementaion}
KENN\ has been implemented as a library for python 3. It is based on TensorFlow 2 and Keras and it is available as an open-source project on \href{https://github.com/DanieleAlessandro/KENN2}{github}.
\section{State of the Art}
\label{sec:rel_works}
Many previous works attempt to learn in the presence of logical knowledge. Among them, works in Statistical Relational Learning use the knowledge to generate probabilistic graphical models which define a probability distribution. Two prominent examples are (Hybrid) Markov Logic Networks~\cite{MLN, HMLN}, that uses weighted FOL rules as a template for building Markov Random Fields \cite{MRF},
and
\emph{Probabilistic Soft Logic} (PSL)~\cite{PSL}, which extends MLN by using a continuous relaxation of the variables to gain efficiency.
A different line of research, called Neural-Symbolic Integration, combines neural network architectures with logical knowledge~\cite{neural_symbolic}.
One of the main techniques used to include prior knowledge into a neural network consists on treating the logical rules as constraints to e maximized through the usage of a regularization term in the loss function.
Among the methods that use this technique there is \emph{Semantic Loss Function}~\cite{semantic_loss} which includes propositional knowledge into a neural network by maximizing the probability of the knowledge to be true.
\emph{Logic Tensor Network} (LTN)~\cite{LTN,DonadelloSG17,IvanThesis,Donadello2019CompensatingSI} and \emph{Semantic Based Regularization} (SBR)~\cite{SBR}do not restrict on propositional logic, instead, they can work with FOL knowledge. This time, a fuzzy logic semantic is used.
Another approach that adds the knowledge trough the loss function is DL2~\cite{dl2}, which is used in regression tasks. In this case, the predicates represent comparison constraints between different terms (outputs of model and constants).
Finally, in \cite{semi_supervised} the regularization term is applied on unlabelled samples in the context of Semi-Supervised Learning.
A different approach, proposed in \cite{harnessing}, is to use a distillation mechanism to inject FOL rules where the model is composed of two networks: the teacher and the student. The teacher copies the behavior of the student network, but at the same time optimizes the satisfaction of the rules. The student network imitates the teacher, but it also optimize the prediction on the training data.
A different strategy, used also by the proposed method (KENN), is to inject the knowledge directly into the structure of the neural network.
Being part of the model, the knowledge is automatically enforced even at inference time instead of being optimized only during training. Another advantage of this type of technique is that it is possible to include rules weights as learnable parameters instead of defining them as hyper-parameters. This makes methods like KENN\ suitable for scenarios where the given knowledge contains errors or when rules are softly satisfied in the real world but it is not known in advance the extent on which they are correct.
Another method, proposed by Li and Srikumar, that injects the knowledge directly into the structure of the model is restricted to implications rules with exactly one consequent~\cite{augmenting}. However, it does not learn the clause weights, which are hyper-parameters.
Finally, Relational Neural Networks (RNM) is also an architecture that adds the logic directly into the model~\cite{RNM}. As the best of our knowledge, it is the only method other than KENN\ which is capable of integrating logical knowledge with a neural network while learning the clause weights. However, the logic is not included inside the model as a function, but it is enforced by an optimization process that applies MAP inference. This reduces the scalability of RNM since the optimization must be performed at inference time, and also at each training step. On the other hand, KENN\ uses the logic to build a new layer of the network which enforces the satisfaction of the knowledge without the need to perform optimization. A more detailed comparison between KENN\ and RNM is in Section~\ref{sec:rel_work_2}.
|
1,116,691,500,447 | arxiv | \section{Introduction}
The invention of the Loop algorithm\cite{Evertz} was a major breakthrough for Monte Carlo
simulations of quantum spin systems. This algorithm which is a quantum version of the
Swendsen-Wang cluster algorithm\cite{Kawashima} has many desirable features which makes it possible to study large
systems at low temperatures\cite{Depleted}.
Among them is that the algorithm can be formulated directly in continuous imaginary time\cite{Beard}, thus
avoiding the need to extrapolate data obtained at finite imaginary time discretization.
Another is that updates are done in an extended configuration space of spins and new entities called loops.
This makes big changes in the spin configuration possible in a single Monte Carlo step, resulting in very small
autocorrelation times. Furthermore the nonlocal updating procedure allows all topological sectors to be sampled. For a quantum spin system this means in particular that sectors with different magnetization are sampled. This is in contrast to
most other algorithms which operate at fixed magnetization.
Although excellent for a wide class of models,
the Loop algorithm does not do well when the Hamiltonian is made asymmetric by a uniform magnetic field or a chemical potential. In these cases autocorrelation times become very long at low temperatures and the performance of the algorithm is lowered drastically. Here it is shown how this can be overcome
by generalizing the loop algorithm. The
generalization is obtained by relaxing the condition of non-interacting loops, and
by taking the magnetic field into account in the loop building process.
To be specific, we consider the nearest neighbor XXZ-model
on a bipartite lattice in a magnetic field along the Z-axis
\begin{equation}
{\cal H} = \sum_{<ij>} \left( J_x S^x_i S^x_j +J_x S^y_i S^y_j + J_z S^z_i S^z_j \right)-H \sum_i S^z_i.
\end{equation}
Despite this choice of model it is expected that the procedure employed here should apply to other quantum models as well, such as lattice fermions in the presence of a chemical potential.
In the next section the generalized loop algorithm is presented. Then it is explained how an algorithm that
performs well in a magnetic field can be chosen. The usefulness of this algorithm
is demonstrated by measuring magnetization curves for the Heisenberg antiferromagnet on a dimer, a chain, and on a plane. Finally it is shown that the algorithm can also be used to determine
the critical temperature for the Kosterlitz-Thouless transition occuring at finite magnetic fields in
the Heisenberg antiferromagnet.
\section{The Algorithm}
To explain the algorithm we begin by formulating the $d$ dimensional quantum system as a classical system in $d+1$ dimensions. This is done in the
standard way\cite{Evertz} of dividing the Hamiltonian into sums of commuting pieces. Then a Trotter-Suzuki breakup is performed, and complete
sets of states, which are labelled by their eigenvalues for $S^z_i$, are inserted at each time-slice between each
sum of commuting pieces.
The matrix elements are easily evaluated
and corresponds to interactions around shaded plaquettes in a generalized checkerboard pattern.
As is shown in fig. \ref{weights}
there are six allowed spin configurations around a shaded plaquette
for the XXZ-model in a magnetic field. Other configurations have zero weights as for those $S^z$ is not conserved along the time direction.
\begin{figure}
\begin{center}
\epsfig{file=R16002PRB1.eps,width=7.0cm}
\end{center}
\caption{The different plaquettes for the XXZ-model in a magnetic field. The vertical is the imaginary time direction. \label{weights}}
\end{figure}
Because the loop algorithm can be formulated in continuous imaginary time
it is sufficient to consider the limit where the imaginary time spacing $\Delta \tau$ goes to zero. In this limit the plaquette weights are
\begin{eqnarray}
w(a_+) & = & 1- \left( \frac{J_z}{4}-\frac{H}{z} \right) \Delta \tau, \nonumber \\
w(a_-) & = & 1- \left( \frac{J_z}{4}+\frac{H}{z} \right) \Delta \tau, \\
w(b) & = & \frac{|J_x|}{2} \Delta \tau, \nonumber \\
w(c) & = & 1+ \frac{J_z}{4} \Delta \tau, \nonumber
\end{eqnarray}
where $z$ is the lattice coordination number.
The loop algorithm consists of two main steps.
The first is to build
loops. Loop building is a probabilistic process where each shaded plaquette $p$ is broken up into loop segments $G_p$ with a probability $P(s_p \rightarrow s_p,G_p)$, dependent
on the spin configuration $s_p$.
Each loop segment connects two or four spins.
The different types of loop segments are shown in fig.~\ref{segments}.
\begin{figure}
\begin{center}
\epsfig{file=R16002PRB2.eps,width=6.5cm}
\end{center}
\caption{The different breakups $G$ into loop segments around a shaded plaquette.\label{segments}}
\end{figure}
When this is done for all shaded plaquettes, the entire space-time lattice will be filled with loops. The second step is to flip spins along one or more loops. Because of the way the break-ups are constructed, this always results in an allowed spin configuration provided all the spins around a loop are flipped. The process of flipping the spins along a loop is also probabilistic.
It is governed by the probability $P_G(s \rightarrow s\prime)$ for changing spin configurations given a particular break-up $G$ for the whole lattice.
After this second step a new spin configuration is generated and one does
the measurements and start over again.
For the whole procedure to satisfy detailed balance it is sufficient\cite{Kandel} that
the probabilities are chosen such that
\begin{eqnarray}
P(s_p \rightarrow s_p,G_p) & = & \frac{w(s_p,G_p)}{w(s_p)}, \label{rule1} \\
P_G(s \rightarrow s\prime) \prod_p w(s_p,G_p) & = & P_G(s\prime \rightarrow s )
\prod_p w(s_p\prime,G_p), \label{rule2}
\end{eqnarray}
where $G$ and $s$ are the full loop and spin configuration, pieced together by
the loop segments $G_p$ and plaquette spins $s_p$ respectively.
$w(s_p,G_p)$ is the plaquette weight of plaquette $p$ in the extended configuration space of
both spins and loops. The weights $w(s_p,G_p)$ must be positive definite and satisfy
\begin{equation} \label{rule3}
\sum_{G_p} w(s_p,G_p) = w(s_p).
\end{equation}
The different loop algorithms described here correspond to different choices of these weights.
Writing out Eq.~(\ref{rule3}) explicitly we find
\begin{eqnarray}
w(a_+) & = & w(a_+,G_{||}) + w(a_+,G_\times)+w(a_+,G_{\otimes}) ,\label{eq6} \\
w(a_-) & = & w(a_-,G_{||}) + w(a_-,G_\times)+w(a_-,G_{\otimes}), \\
w(b) & = & w(b,G_{=}) + w(b,G_\times) + w(b,G_{\otimes}), \\
w(c) & = & w(c,G_{||}) + w(c,G_{=}) + w(c,G_{\otimes}). \label{eq9}
\end{eqnarray}
We have set the weights $w(a_+,G_{=})$, $w(a_-,G_{=})$, $w(b,G_{||})$
and $w(c,G_\times)$ to zero as flipping the spins along one loop segment
for such configurations leads to a configuration with zero weight.
It is therefore clear that we have eight parameters at our disposal. Let
us parametrize the weights in the following way
\begin{eqnarray}
w(a_+,G_\times) & = & s \Delta \tau, \\
w(a_-,G_\times) & = & t \Delta \tau, \\
w(b,G_=) & = & u \Delta \tau, \\
w(c,G_=) & = & v \Delta \tau, \\
w(a_+,G_{\otimes}) & = & e \Delta \tau, \\
w(a_-,G_{\otimes}) & = & f \Delta \tau, \\
w(b,G_{\otimes}) & = & g \Delta \tau, \\
w(c,G_{\otimes}) & = & h \Delta \tau.
\end{eqnarray}
The remaining four weights are given by Eqs.~(\ref{eq6})-(\ref{eq9}).
Note that although this is a convenient way of
parametrizing the weights it is not the most general one.
In selecting the parametrization above we have chosen
which weights are of order $\Delta \tau$ or 1.
With this parametrization it is easy to obtain the loop building probabilities $P(s_p \rightarrow s_p,G_p)$
from Eqs.~(\ref{rule1}) and (\ref{rule3}).
To have non-negative weights, all the parameters must be greater than or equal to zero and $(u+g) \le |J_x|/2$.
To satisfy detailed balance in the extended con\-figuration space
Eq.~(\ref{rule2}) we need the ratios $w(s_p\prime,G_p)/w(s_p,G_p)$
which are
\begin{eqnarray}
\frac{w(c,G_{||})}{w(a_+,G_{||})} & = & 1+\Delta \tau (
\frac{J_z}{2}-\frac{H}{z} + s+e -v-h), \label{first-flip} \\
\frac{w(c,G_{||})}{w(a_-,G_{||})} & = & 1+\Delta \tau (
\frac{J_z}{2}+\frac{H}{z} + t+f -v-h), \label{second-flip} \\
\frac{w(a_-,G_{||})}{w(a_+,G_{||})} & = & 1-\Delta \tau (
\frac{2H}{z}-s-e+t+f), \label{third-flip} \\
\frac{w(b,G_\times)}{w(a_+,G_\times)} & = & (\frac{|J_x|}{2}-u-g)/s, \label{fourth-flip} \\
\frac{w(b,G_\times)}{w(a_-,G_\times)} & = & (\frac{|J_x|}{2}-u-g)/t, \\
\frac{w(a_-,G_\times)}{w(a_+,G_\times)} & = & \frac{t}{s}, \\
\frac{w(c,G_=)}{w(b,G_=)} & = & \frac{v}{u}, \\
\frac{w(a_-,G_{\otimes})}{w(a_+,G_{\otimes})} & = & \frac{e}{f}.
\label{last-flip}
\end{eqnarray}
Given these ratios the flipping probabilities can be gotten from
\begin{equation}
P_G(s \rightarrow s\prime) = \min \left[
\frac{\prod_p w(s_p\prime,G_p )}{\prod_p w(s_p,G_p)},1 \right].
\end{equation}
\section{Parameter Choices}
There are many possibilities for the choice of parameters, but
not all of them lead to efficient ergodic algorithms.
To minimize autocorrelation times one must in particular ensure that
the loops generated have a reasonable chance of being flipped. This means that
the ratios in Eqs.~(\ref{first-flip})-(\ref{last-flip}) should be
as close to unity as possible.
Let us first consider $H=0$. The standard loop algorithm is constructed
such that all the ratios Eqs.~(\ref{first-flip})-(\ref{last-flip}) are one,
and by minimizing the weights $w(x,G_{\otimes})$:
\begin{eqnarray}
u_0 & = & \theta (J_z-|J_x|) \frac{|J_x|- J_z }{4}
+\theta (J_z+|J_x|) \frac{|J_x|+ J_z }{4}, \\
v_0 & = & u_0 , \\
s_0 & = & \frac{|J_x|}{2}-u_0, \\
t_0 & = & s_0, \\
e_0 & = & -\left[1- \theta(J_z+|J_x|) \right] \frac{J_z+|J_x|}{2}, \\
f_0 & = & e_0, \\
g_0 & = & 0, \\
h_0 & = & \theta(J_z-|J_x|) \frac{J_z-|J_x|}{2}.
\end{eqnarray}
In particular the nonzero parameters for the
Heisenberg antiferromagnet corresponds to, $u_0=v_0=J/2$, and for the
XY-model $s_0=t_0=u_0=v_0=J/4$.
For Ising anisotropy, $|J_x|<|J_z|$, certain $G_{\otimes}$ break-ups must be included as
otherwise some weights will be negative. For extreme anisotropy
$|J_x|=0$ the model is the classical Ising model and the world-lines
are all straight, $s_0=t_0=v_0=0$. In this limit the standard loop algorithm
above is the Swendsen-Wang algorithm for the Ising model.
Now consider $H \neq 0$. In this case it is not possible to set all
of the ratios Eqs.~(\ref{first-flip})-(\ref{last-flip}) to unity for any
parameter choices. With the parameter choices for the standard
loop algorithm these ratios are only unity for loops which do not change the
magnetization when flipped. For loops that can change the magnetization
the total ratio of weights is $\exp(-\beta H \Delta M)$,
where $\Delta M$ is the change in magnetization caused by flipping the loop.
This leads to autocorrelation times that increase exponentially with
$\beta H$. The reason is that the magnetic field
is not taken into account in the loop building process.
The process of changing the magnetization is a competition
between loosing Zeeman energy and gaining exchange energy. As
the exchange energy is gained in the loop building process, it is inefficient
to build the loops as if the magnetic field was absent.
What happens in the standard loop algorithm is that the number of loops generated which can change
the magnetization is very small.
An interesting observation is that one can construct an algorithm
where only loops which can change the magnetization are generated.
This choice is
\begin{equation}
u=v=e=f=g=h = 0 ,\; s = t = \frac{|J_x|}{2}, \label{High-H}
\end{equation}
which means that the only break-ups allowed are of the diagonal type.
This algorithm is ergodic, and in the classical Ising limit, $|J_x|=0$,
it corresponds to the standard local Ising model algorithm
in a magnetic field.
One can now ask for which magnetic field this algorithm
minimizes the autocorrelation times.
Eq. (\ref{first-flip}) is unity for
\begin{equation}
\frac{H}{z} = \frac{|J_x|}{2}+\frac{J_z}{2}.
\end{equation}
The important observation is that this value of $H$ is the saturation field, where
almost all plaquettes are of type $a_+$. At this field it is therefore
not important for the performance of the algorithm that Eqs.~(\ref{second-flip}) and
(\ref{third-flip}) deviate from one.
It is then natural to choose an algorithm
valid for all $H$ which interpolates between the standard algorithm at $H=0$ and the above at the saturation field.
For the Heisenberg antiferromagnet in a magnetic field we thus propose the following algorithm
\begin{eqnarray}
s=t & = & \frac{H}{2z}, \nonumber \\
u=v & = & \frac{J}{2}-\frac{H}{2z}. \label{interpolation}
\end{eqnarray}
For $H > J z$ we use the same algorithm as for $H=J z$.
Eq.~(\ref{interpolation}) implies that Eqs.~(\ref{first-flip}),(\ref{fourth-flip})-(\ref{last-flip}) becomes unity, whereas Eqs.~(\ref{second-flip})-(\ref{third-flip}) do not.
\section{Numerical Results}
The algorithm was first tested by measuring magnetization of a dimer or two-site $S=1/2$ antiferromagnetic chain. As shown by Kashurnikov {\em et al.}\cite{Worm} the integrated autocorrelation time for the standard loop algorithm increases exponentially with $\beta H$. We have verified this measuring the integrated autocorrelation
time as described in Ref.1, appendix B.
For the new algorithm the dimer integrated autocorrelation time is very small ($<2$) down to the lowest temperatures measured (T=.005J) and the magnetization measured agrees excellently with exact results.
Fig.~\ref{chain-fig} shows the magnetization per spin of a 64-site antiferromagnetic Heisenberg chain, and illustrates the improvement over the standard Loop algorithm. The results of the modified and standard loop algorithm were obtained using the same number of equilibration and measurement steps ($10^6$), and the lines are exact results obtained using Bethe Ansatz\cite{Takahashi}.
It is clear that, in contrast to the modified algorithm, results obtained using the standard loop algorithm have not converged for high magnetic fields.
Close inspection of the data at the lowest temperature reveals that the results of the modified algorithm deviate slightly from the exact results at {\em intermediate} magnetic fields. This deviation which is statistical is caused by increased autocorrelation times which arises because Eqs.~(\ref{second-flip})-(\ref{third-flip}) deviate from unity concomitant with the presence of a significant fraction of $a_{-}$ plaquettes at these fields. For the 64-site chain we measured the integrated autocorrelation time to be a maximum $3\cdot 10^4$ steps at $H/J=1.3$, going down to about 60 steps at low and high fields. It is quite conceivable that a different interpolation scheme than the one chosen in Eq.~(\ref{interpolation}) can reduce these autocorrelation times at intermediate fields.
\begin{figure}
\begin{center}
\epsfig{file=R16002PRB3.eps,width=6.5cm}
\end{center}
\caption{Magnetization per spin for a spin chain with 64 sites. \label{chain-fig}}
\end{figure}
Fig.~\ref{plane-fig} shows the full magnetization curve of a 64x64 square lattice antiferromagnet. Typical runs involved $10^7$ steps for equilibration and measurements. The inset shows the high field behavior for very low temperatures on the same lattice. The statistical errors, taking into account the autocorrelation times, are smaller than the symbol size. For the lowest temperature the integrated autocorrelation times reached a maximum of $4 \cdot 10^4$ steps at H/J=2.4 going down to about $4\cdot 10^3$ steps at low and high magnetic fields.
\begin{figure}
\begin{center}
\epsfig{file=R16002PRB4.eps,width=6.5cm}
\end{center}
\caption{Magnetization per spin for a plane with $64 \times 64$ sites. \label{plane-fig}}
\end{figure}
The Heisenberg antiferromagnet in a magnetic field undergoes
a Kosterlitz-Thouless transition at finite temperatures.
The transition temperature has previously been obtained for weak magnetic fields; $H < .2J$ \cite{Troyer}.
Fig.~\ref{helicity-fig} shows the
helicity modulus $\Upsilon$, which is the normalized free energy change due to phase twists in the x-y-plane, and which is proportional to the squared spatial winding number of world-lines\cite{Ceperley}, as a function of temperature for four different system sizes at $H=3.95J$. From a finite size analysis\cite{Harada} the transition temperature is found to be $T_c=.020(5) J$. Here a
single-cluster\cite{Evertz} implementation of the algorithm is used. It is expected that more precise estimates for $T_c$ can be obtained using a multi-cluster implementation.
\begin{figure}
\begin{center}
\epsfig{file=R16002PRB5.eps,width=6.5cm}
\end{center}
\caption{Helicity modulus as function of temperature for two different system sizes. The line is the Kosterlitz-Thouless-Nelson critical line. \label{helicity-fig}}
\end{figure}
The author wishes to thank P.A. Lee for useful discussions and
V. Chudnovsky and U.-J. Wiese for providing source code for the
continuous time loop algorithm.
|
1,116,691,500,448 | arxiv | \section{Introduction and results}
\subsection{Context}
Consider a real separable Hilbert space $\mathcal{H}$ equipped with the Hilbertian inner product $\langle\cdot,\cdot\rangle$ and the associated norm $\|\cdot\|$. The set of bounded operators from $\mathcal{H}$ to $\mathcal{H}$ will be denoted by $\mathcal{L}(\mathcal{H})$ and the associated norm by $\|\cdot\|$. Let $(e_n)_{n\in\mathbb{N}}$ be a Hilbertian basis of $\mathcal{H}$ and denote by $\mathcal{D}(\Hil)$ the subset of bounded diagonal operators with respect to that basis:
\[
T\in\mathcal{D}(\Hil) \Leftrightarrow \forall n\in\mathbb{N},\ \exists \lambda_n\in\mathbb{R},\ Te_n = \lambda_n e_n.
\]
The purpose of this paper is to analyze the structural and convergence properties of some flows of Hilbert-Schmidt operators for which $\mathcal{D}(\Hil)$ turns out to be an attractive Hilbert-Schmidt subspace. We obtain an alternative convergence proof that applies for example to the flows (in infinite dimension) of Brockett, Toda and Wegner. The present paper is mainly motivated by the work by Bach and Bru \cite{BB10} where they tackle in particular the example of the Brockett flow in infinite dimension. We extend one of their results to a wider class of flows of Hilbert-Schmidt operators. We may notice here that such flows of operators have deep applications in mathematical physics. For instance, as recently proved by Bach and Bru in \cite{BB15}, they can be used to diagonalize unbounded operators in boson quantum field theory.
This paper also aims at being a modest review of the methods and insights appearing in finite dimension with the idea to extend them in infinite dimension. These methods are explained, for instance, in the seminal papers \cite{CN88} and \cite{Bro89} (see also \cite{Bro91}, and \cite{W94} for the Wegner flow), but also in the nice survey \cite{Chu08}. Their relations with numerical algorithms such as the QR or LU decompositions are very deep and the idea of interpreting discrete algorithms as samplings of continuous flows has now a long story (see, for instance, \cite{R54a, R54b} and \cite{WL90}). Commenting this idea would lead us far beyond the scope of this paper, but we will briefly discuss it when describing our numerical illustrations.
\subsection{Results}
Let us now describe the results of this paper. For that purpose, we introduce some general notations.
\begin{notation}
Below are the notations that we will constantly use in the paper.
\begin{enumerate}[\rm (a)]
\item We denote $\mathcal{L}_2(\Hil)$ the set of the Hilbert-Schmidt operators on $\mathcal{H}$ and the associated norm is $\|\cdot\|_{\mathsf{HS}}$. In other words, $H\in\mathcal{L}_2(\Hil)$ means that, for any Hilbertian basis $(e_{n})_{n\in\mathbb{N}}$, we have
\[\sum_{n\geq 0}\|He_{n}\|^2<+\infty\,,\]
and this last quantity is independent from the Hilbert basis and is denoted by $\|H\|^2_{\mathsf{HS}}$ and we have $\|H\|\leq\|H\|_{\mathsf{HS}}$. We also recall that the Hilbert-Schmidt operators are compact and that, if $(\lambda_{n}(H))_{n\in\mathbb{N}}$ is the non-increasing sequence of the eigenvalues of $H$, we have
\[\|H\|_{\mathsf{HS}}^2=\sum_{n=0}^{+\infty}|\lambda_{n}(H)|^2\,.\]
\item We will denote by $\mathcal{S}(\Hil)$, $\mathcal{A}(\Hil)$ and $\mathcal{U}(\mathcal{H})$ the sets of the bounded symmetric, bounded skew-symmetric and unitary operators respectively. We also introduce $\mathcal{S}_{2}(\mathcal{H})=\mathcal{S}(\mathcal{H})\cap \mathcal{L}_{2}(\mathcal{H})$ and $\mathcal{A}_{2}(\mathcal{H})=\mathcal{A}(\mathcal{H})\cap \mathcal{L}_{2}(\mathcal{H})$. They are Hilbert spaces equipped with the scalar product $(A,B)\mapsto\mathsf{Tr}(A^\star B)$.
\item In the sequel we will use a fixed Hilbert basis $(e_n)_{n\in\mathbb{N}}$ and for any $H\in\mathcal{L}_2(\Hil)$ and any integers $i,j\in\mathbb{N}$, we will denote $h_{i,j}(t)=\langle H(t)e_i,e_j \rangle$.
\end{enumerate}
\end{notation}
\subsubsection{An elementary convergence result}
We will consider infinite systems determined by matrices that are balanced in the sense of the following definition.
\begin{definition}
A family of real-valued functions $(g_{i,j})_{i,j\in\mathbb{N}}$ defined on $\mathbb{R}$ is said to be \emph{balanced} if
\[
\forall i,j\in\mathbb{N},\ \exists c_{i,j}\in L^\infty(\mathbb{R},\mathbb{R}_+),\ \forall t\in\mathbb{R},\ |g_{i,j}(t)| \leq c_{i,j}(t) |g_{j,i}(t)|\,.
\]
It is said to be \emph{pointwise bounded} if, for all $t\in\mathbb{R}$, $\displaystyle{\sup_{i,j} |g_{i,j}(t)|<+\infty}$.
\end{definition}
Let us now state one of the main results of this paper. It is a general convergence result for infinite systems of differential equations. As we will see, many different classical situations may be reduced to this result.
\begin{theorem}\label{gen-theo}
Consider $H\in\mathcal{C}^1(\mathbb{R},\mathcal{L}_2(\Hil))$ such that
\begin{enumerate}[\rm i)]
\item the function $t\mapsto \|H(t)\|_{\mathsf{HS}}$ is bounded,
\item the family $(h_{i,j})_{i,j\in\mathbb{N}}$ is balanced,
\item there exists a pointwise bounded and balanced family of measurable functions $(g_{i,j})_{i,j\in\mathbb{N}}$ defined on $\mathbb{R}$ such that
\begin{equation}
\label{eq:ODEdiago}
\forall t\in\mathbb{R},\ h_{i,i}'(t)=\sum_{j=0}^{+\infty}g_{i,j}(t)|h_{i,j}(t)|^2\,.
\end{equation}
\end{enumerate}
Suppose that there exists $T>0$ and a sign sequence $(\epsilon_\ell)_{\ell\in\mathbb{N}}\in\{-1,1\}^{\mathbb{N}}$, such that
\begin{equation}
\label{eq:assumpSign}
\forall t\geq T,\ \forall k,\ell\in\mathbb{N},\ k\geq \ell,\ \epsilon_\ell g_{\ell,k}(t)\geq 0\,.
\end{equation}
Then we have the integrability property
\begin{equation}
\forall \ell\in\mathbb{N},\ \sum_{j=0}^{+\infty}\int_T^{+\infty} |g_{\ell,j}(t)| |h_{\ell,j}(t)|^2\,\mathrm{d} t<+\infty\,,
\end{equation}
and any diagonal term $h_{\ell,\ell}(t)$ converges at infinity to a real value denoted by $h_{\ell,\ell}(\infty)$.
Under the additional semi-uniform lower bound condition
\begin{equation}
\label{eq:assumpBound}
\forall \ell\in\mathbb{N},\ \exists c_\ell>0,\ \forall j\neq \ell,\ \forall t\geq T,\ |g_{\ell,j}(t)|\geq c_\ell\,,
\end{equation}
we have
\begin{equation}\label{eq.integrability}
\sum_{j\neq \ell}\int_T^{+\infty} |h_{\ell,j}(t)|^2\,dt = \int_T^{+\infty} \|H(t)e_\ell - h_{\ell,\ell}(t)e_\ell\|^2\,\mathrm{d} t <+\infty\,.
\end{equation}
Suppose moreover that, for any $\ell\in\mathbb{N}$, the function $t\mapsto \|H'(t)e_\ell\|$ is bounded, then for any $\ell\in\mathbb{N}$, $H(t)e_\ell$ converges to $h_{\ell,\ell}(\infty) e_\ell$.
\end{theorem}
\begin{remark}
Theorem~\ref{gen-theo} remains true if, instead of the assumption \eqref{eq:assumpSign}, we only assume that there exists a one-to-one integer mapping function $\phi:\mathbb{N}\to\mathbb{N}$, a time $T>0$ and a sign sequence $\epsilon_\ell\in\{-1,1\}$ ($\ell\in\mathbb{N}$) such that
\[
\forall t\geq T,\ \forall k\in\mathbb{N},\ k\notin\{\phi(i),\ 0\leq i \leq \ell-1\},\ \epsilon_\ell g_{\phi(\ell),k}(t)\geq 0\,.
\]
\end{remark}
We will apply Theorem \ref{gen-theo} to \enquote{bracket} flows. Let us state a proposition that describes the fundamental property of such flows.
\begin{proposition}\label{prop.cor1}
Let $G:\mathcal{S}(\Hil)\to\mathcal{A}(\Hil)$ be a locally Lipschitzian function and let us consider the following Cauchy problem
\begin{equation}
\label{eq:ODE}
H' = [H,G(H)]\, ,\qquad t\geq 0\,, \qquad H(0)=H_{0}\in\mathcal{S}_{2}(\mathcal{H})\,.
\end{equation}
There exists a unique global solution $H\in\mathcal{C}^1([0,+\infty),\mathcal{S}_{2}(\mathcal{H}))$ unitarily equivalent to the initial data $H_{0}$. More precisely there exists $U\in\mathcal{C}^1([0,+\infty),\mathcal{U}(\Hil))$ such that for any $t\geq 0$ one has $H(t) = U(t)^\star H_{0} U(t)$.
\end{proposition}
Let us now provide two corollaries of Theorem \ref{gen-theo}.
First let us introduce two convenient definitions.
\begin{definition}
Given a Hilbertian basis $(e_{n})_{n\in\mathbb{N}}$, we introduce
\begin{enumerate}[i.]
\item the symmetric family of symmetric Hilbert-Schmidt operators $(E_{i,j})_{(i,j)\in\mathbb{N}^2}$ defined by
\[E_{i,j}(\cdot)=\begin{cases}
\frac{1}{\sqrt{2}}(e^*_{j}(\cdot) e_{i}+e^*_{i}(\cdot) e_{j})& \text{if $ i\neq j$}\, \\
e^*_{i}(\cdot) e_{i} & \text{if $i=j$ }
\end{cases}\,.
\]
\item the skew-symmetric family of skew-symmetric Hilbert-Schmidt operators $(E^{\pm}_{i,j})_{(i,j)\in\mathbb{N}^2}$ defined by
\[E^\pm_{i,j}(\cdot)=\frac{1}{\sqrt{2}}(e^*_{j}(\cdot) e_{i}-e^*_{i}(\cdot) e_{j})\,.\]
\end{enumerate}
\end{definition}
Since the families $(E_{i,j})_{i\leq j}$ and $(E_{i,j}^\pm)_{i< j}$ are Hilbertian bases of $\mathcal{S}_{2}(\mathcal{H})$ and $\mathcal{A}_{2}(\mathcal{H})$ respectively, this motivates the following definition.
\begin{definition}
We say that $G\in\mathcal{L}\left(\mathcal{S}_{2}(\mathcal{H}),\mathcal{A}_{2}(\mathcal{H})\right)$ is diagonalizable when there exists a Hilbertian basis $(e_{n})_{n\in\mathbb{N}}$ and a skew-symmetric family $(g_{i,j})$ such that
\[G(E_{i,j})=g_{i,j}E^\pm_{i,j}\,.\]
In this case, we say that $(g_{i,j})$ is the matrix-eigenvalue of $G$. Note that the matrix-eigenvalue of $G$ is balanced and pointwise bounded ($G$ is bounded).
\end{definition}
\begin{corollary}\label{cor.cor1'}
Let $G\in\mathcal{L}\left(\mathcal{S}_{2}(\mathcal{H}),\mathcal{A}_{2}(\mathcal{H})\right)$ be a diagonalizable map such that its matrix-eigenvalue satisfies \eqref{eq:assumpSign} and \eqref{eq:assumpBound}. Then the Cauchy problem \eqref{eq:ODE} admits a unique global solution $H\in\mathcal{C}^1([0,+\infty),\mathcal{S}_{2}(\mathcal{H}))$ which weakly converges in $\mathcal{S}_{2}(\mathcal{H})$ to a diagonal Hilbert-Schmidt operator $H_\infty$. Moreover, if $\alpha$ is a diagonal term of $H_{\infty}$ with multiplicity $m$, then $\alpha$ is an eigenvalue of $H_{0}$ of multiplicity at least $m$.
\end{corollary}
Then we can apply our theorem to the finite dimensional case and even get an explicit convergence rate. Note that this presentation does not involves a Lyapunov function and relies on the a priori convergence of $H(t)$. This is a way to avoid, as far as possible, the considerations related to the evolution of the corresponding unitary matrices $(U(t))_{t\geq 0}$ in $\mathcal{U}(\mathcal{H})$ (that is not compact in infinite dimension).
\begin{corollary}\label{cor.dim.finie}
Consider a finite dimensional Hilbert space $\mathcal{H}$ of dimension $d$. Under the assumptions of Corollary \ref{cor.cor1'}, the solution of \eqref{eq:ODE} converges to a diagonal operator $H_{\infty}=\mathsf{diag}\left(\alpha_{\ell}\right)_{0\leq\ell\leq d-1}$ in $\mathcal{L}(\mathcal{H})$ and the $\alpha_{\ell}$ are exactly the eigenvalues (with multiplicity) of $H_{0}$. If, in addition, the eigenvalues of $H_{0}$ are simple, then there exist $T, C>0$ such that, for all $t\geq T$, we have
\[\|H(t)-H_{\infty}\|\leq Ce^{-\gamma t}\,,\]
where $\displaystyle{\gamma=\inf_{\mathcal{T}^-}|g_{i,j}(\alpha_{i}-\alpha_{j})|>0}$ with
\[\mathcal{T}^-=\{(i,j)\in\{0,\ldots, d-1\}^2 : i< j\quad\mbox{ and }\quad g_{i,j}(\alpha_{i}-\alpha_{j})<0\}\,.\]
\end{corollary}
Note that, in Corollary \ref{cor.dim.finie}, we have not to exclude special initial conditions to get an exponential convergence (see \cite[Theorem 3]{Bro91}).
The following proposition states that, even in infinite dimension, we may find exponentially fast eigenvalues of a generic Hilbert-Schmidt operator thanks to a bracket flow. Note that, in this case, we get a stronger convergence as the one established by Bach and Bru in \cite[Theorem 7]{BB10} for the Brockett flow and that we may also consider non symmetric operators.
\begin{proposition}\label{prop.House}
Assume that $H_{0}\in\mathcal{L}_{2}(\mathcal{H})$ (not necessarily symmetric) is diagonalizable with eigenvalues $(\lambda_{j})_{j\geq 1}$ with decreasing real parts.
Let us also assume that we may find $P\in\mathcal{L}(\mathcal{H})$ such that $P H_{0} P^{-1}=\Lambda$ where $\Lambda=\mathsf{diag}(\lambda_{j})_{j\in\mathbb{N}}$ is a diagonal operator and such that the minors of $P$, $P_{J}=(\langle Pe_{i},e_{j}\rangle)_{0\leq i,j\leq J}$ are invertible for all $J\in\mathbb{N}$.
We use the choice
\[G(H)=H^- -(H^-)^\star\,.\]
Then, we have, for all $\ell\in\mathbb{N}$,
\[H(t)e_{\ell}-\sum_{j=\ell+1}^{+\infty} \langle H(t)e_{\ell}, e_{j}\rangle e_{j}=\lambda_{\ell}e_{\ell}+\mathcal{O}\left(e^{-t\delta_{\ell}}\right)\,,\quad \delta_{\ell}=\min_{0\leq j\leq \ell}\Re(\lambda_{j}-\lambda_{j+1})\,.\]
\end{proposition}
Note in particular that, in the symmetric case, $H(t)$ converges to a diagonal operator $H_{\infty}$ that has exactly the same eigenvalues as $H_{0}$ (even in infinite dimension). This property is slightly stronger than the last one stated in Corollary \ref{cor.cor1'}: for the Toda flow, in infinite dimension, no eigenvalue is lost in the limit.
\subsubsection{Organization of the paper}
Section \ref{sec.existence} is devoted to the proof of Theorem \ref{gen-theo}, Proposition \ref{prop.cor1} and Corollary \ref{cor.cor1'}. We also provide examples of applications of our results (to the Brockett, Toda and Wegner flows) in infinite dimension. In Section \ref{sec.finite.dim}, we discuss the case of finite dimension to the light of our results and discuss the exponential convergence of the flows (Corollary \ref{cor.dim.finie} and Proposition \ref{prop.House}).
\section{Existence and global convergence results}\label{sec.existence}
\subsection{Proof of the main theorem}\label{sec.proof.main}
This section is devoted to the proof of Theorem~\ref{gen-theo}. We start by stating two elementary lemmas.
\begin{lemma}\label{L1}
Consider $h\in\mathcal{C}^1(\mathbb{R}_+)$ a real-valued bounded function, $F\in L^1(\mathbb{R}_+)$ and $G$ a nonnegative measurable function defined on $\mathbb{R}_+$ such that
\[
\forall t\geq 0,\ h'(t) = F(t) + G(t).
\]
Then, the function $G$ is integrable and $h$ converges to a finite limit at infinity.
\end{lemma}
\begin{proof}
Observe that, due to the sign of $G$, one gets for all $x\geq 0$
\[
\int_0^x |G(t)|\,\mathrm{d} t = \int_0^x G(t)\, dt = h(x)-h(0) - \int_0^x F(t)\,\mathrm{d} t \leq 2 \sup_{\mathbb{R}_+}|h| + \int_0^{+\infty} |F(t)| \,\mathrm{d} t \,.
\]
Therefore $G$ is integrable and, for all $x\geq 0$
\[
h(x) = h(0) + \int_0^{x} (F(t) + G(t))\,\mathrm{d} t\, ,
\]
which converges as $x$ tend to infinity.
\end{proof}
\begin{lemma}\label{L1+Lip}
Let $w$ be a function defined on $\mathbb{R}_+$ with nonnegative values. Suppose that $w$ is Lipschitz continuous with Lipschitz constant $C>0$. Then, for any $x\geq 0$, one has
\[
w(x)^2 \leq 2C \int_x^{+\infty} w(y)\,\mathrm{d} y \leq +\infty.
\]
Suppose moreover that $w$ is integrable, then $\displaystyle{\lim_{x\to +\infty}w(x) = 0}$.
\end{lemma}
\begin{proof}
Since $w$ is Lipschitzian, we have, for all $x, y\in\mathbb{R}$,
\[w(x)\leq C|x-y|+w(y)\,,\]
so that, integrating with respect to $y$ between $x$ and $x+\frac{w(x)}{C}$, we get
\[\frac{w(x)^2}{C}\leq \frac{w(x)^2}{2C}+\int_{x}^{x+\frac{w(x)}{C}}w(y)\,\mathrm{d} y\,,\]
and the conclusion follows.
\end{proof}
We can now complete the proof of Theorem~\ref{gen-theo}.
\subsubsection{Convergence}
Consider first Equation \eqref{eq:ODEdiago} with $i=0$. Thanks to the sign property \eqref{eq:assumpSign} we have
\[
\epsilon_0 h'_{0,0}(t) = \sum_{j\in\mathbb{N}} |g_{0,j}(t)| |h_{0,j}(t)|^2\, .
\]
The boundedness of $t\mapsto\|H(t)\|_{\mathsf{HS}}$ implies the boundedness of $t\mapsto h_{0,0}(t)$ and thus Lemma~\ref{L1} implies the existence of a finite limit $h_{0,0}(\infty)$ and the integrability property
\[
\int_T^{+\infty} \sum_{j\in\mathbb{N}} |g_{0,j}(t)| |h_{0,j}(t)|^2\, dt < +\infty\, .
\]
Then, the rest of the proof is by induction on $\ell\in\mathbb{N}$. Let $\ell\geq 1$ be an integer and assume that, for any integers $0\leq k\leq \ell-1$, we have
\[
\int_T^{+\infty} \sum_{j\in\mathbb{N}} |g_{k,j}(t)| |h_{k,j}(t)|^2\, dt < +\infty\, .
\]
Equation \eqref{eq:ODEdiago} with $i=\ell$ and the sign assumption \eqref{eq:assumpSign} give
\[
\epsilon_\ell h'_{\ell,\ell}(t) = \sum_{k\leq \ell-1} \epsilon_\ell g_{\ell,k}(t) |h_{\ell,k}(t)|^2 + \sum_{j\geq \ell} |g_{\ell,j}(t)| |h_{\ell,j}(t)|^2\, .
\]
Again, $t\mapsto h_{\ell,\ell}(t)$ is bounded. Let us now prove the integrability of the first sum in the right hand side.
Since the families $(g_{i,j})$ and $(h_{i,j})$ are balanced, we get the following
\[
\exists C_\ell\geq 0,\ \forall t\geq T,\ \forall k\leq \ell-1,\ |g_{\ell,k}(t)||h_{\ell,k}(t)|^2 \leq C_\ell |g_{k,\ell}(t)||h_{k,\ell}(t)|^2.
\]
Therefore we have
\begin{equation}
\label{eq:partintegrability}
\begin{aligned}
\int_T^{+\infty} \sum_{k\leq \ell-1} |g_{\ell,k}(t)| |h_{\ell,k}(t)|^2\,\mathrm{d} t & \leq C_\ell \sum_{k\leq \ell-1} \int_T^{+\infty} |g_{k,\ell}(t)||h_{k,\ell}(t)|^2\,\mathrm{d} t\\
& \leq C_\ell \sum_{k\leq \ell-1} \int_T^{+\infty} \sum_{j\in\mathbb{N}} |g_{k,j}(t)| |h_{k,j}(t)|^2\,\mathrm{d} t < +\infty\, .
\end{aligned}
\end{equation}
Finally, Lemma~\ref{L1} applies and we get the existence of a finite limit $h_{\ell,\ell}(\infty)$ together with the integrability property:
\[
\int_T^{+\infty} \sum_{j\geq \ell} |g_{\ell,j}(t)| |h_{\ell,j}(t)|^2\,\mathrm{d} t < +\infty\, ,
\]
and then, thanks to \eqref{eq:partintegrability}, we find
\[
\int_T^{+\infty} \sum_{j\in\mathbb{N}} |g_{\ell,j}(t)| |h_{\ell,j}(t)|^2\,\mathrm{d} t < +\infty\, .
\]
This ends the proof of the integrability property and of the convergence of diagonal terms.
\subsubsection{Stronger convergence}
Consider now the additional assumption~\eqref{eq:assumpBound}. For any fixed $\ell\in\mathbb{N}$, we have
\[
\sum_{j\neq \ell}\int_T^{+\infty} |h_{\ell,j}(t)|^2\,\mathrm{d} t \leq \dfrac{1}{c_\ell}\sum_{j\neq \ell}\int_T^{+\infty} |g_{\ell,j}(t)| |h_{\ell,j}(t)|^2\,\mathrm{d} t<+\infty\,.
\]
For any $t\geq T$, we let
\[w_{\ell}(t)=\sum_{j\neq \ell} |h_{\ell,j}(t)|^2\,\mathrm{d} t =\|K_{\ell}(t)\|^2\qquad \mbox{ with }\qquad K_{\ell}(t)=H(t)e_\ell - h_{\ell,\ell}(t)e_\ell\,.\]
For $t\geq s\geq T$, the Cauchy-Schwarz inequality implies that
\[
|w_{\ell}(t)-w_{\ell}(s)|\leq \| K_{\ell}(t)-K_{\ell}(s) \| \|K_{\ell}(t)+K_{\ell}(s)\|\,.
\]
On one hand, we have
\[
\|K_{\ell}(t) + K_{\ell}(s)\| \leq 2\sup_{\sigma\geq T} \|H(\sigma)\|_{\mathsf{HS}}.
\]
On the other hand, we write
\[
\begin{aligned}
\| K_{\ell}(t)-K_{\ell}(s) \| &\leq \|H(t)e_\ell-H(s)e_\ell\| + |h_{\ell,\ell}(t)-h_{\ell,\ell}(s)| \\
& \leq \int_s^t \| H'(\sigma)e_\ell\|\, d\sigma + \int_s^t |h'_{\ell,\ell}(\sigma)|\,\mathrm{d}\sigma\\
& \leq 2 \int_s^t \| H'(\sigma)e_\ell\|\, d\sigma \leq 2 M |t-s|\,,
\end{aligned}
\]
where $M$ is a uniform bound of $t\mapsto \|H'(t)e_\ell\|$. Finally, $w_{\ell}$ is a nonnegative and Lipschitz continuous function, which is moreover integrable thanks to \eqref{eq.integrability}. Therefore Lemma~\ref{L1+Lip} implies that $w_{\ell}(t)$ goes to $0$ as $t$ goes to $+\infty$.
\subsection{Proof of Proposition \ref{prop.cor1}}\label{sec.proof.cor1}
\subsubsection{Local existence}
Let us introduce the function $F:\mathcal{S}(\mathcal{H})\to \mathcal{S}(\mathcal{H})$ defined by
\begin{equation}\label{eq.F}
\forall H\in\mathcal{S}(\mathcal{H})\,,\qquad F(H)=[H, G(H)]\,.
\end{equation}
Note indeed that, for all $H\in\mathcal{S}(\mathcal{H})$, by using that $G$ is valued in $\mathcal{A}(\mathcal{H})$, we have
\[F(H)^\star=(HG(H)-G(H)H)^{\star}=G(H)^\star H^\star-H^\star G(H)^\star=F(H)\,.\]
Moreover $F$ is locally Lipschitzian since $G$ is. By using the Cauchy-Lipschitz theorem in the Banach space $(\mathcal{S}(\mathcal{H}),\|\cdot\|)$ with the function $F$, we get the local existence of a solution of the Cauchy problem \eqref{eq:ODE}, on a maximal time interval $[0,T_{\max})$, with $T_{\max}>0$.
\subsubsection{Global existence}
Let us now show that $T_{\max}=+\infty$ and that, we have, for all $t\in[0,+\infty)$, $H(t)\in\mathcal{L}_2(\Hil)$ and $\|H(t)\|_{\mathsf{HS}}=\|H_{0}\|_{\mathsf{HS}}$. Let us consider the following Cauchy problem:
\begin{equation}\label{eq.U}
U'=UG(H)\,,\qquad U(0)=\mathrm{Id}\,,
\end{equation}
where $H$ is the local solution of \eqref{eq:ODE} on $[0, T_{\max})$. Since the equation is linear, we know that there exists a unique solution $U$ defined on $[0, T_{\max})$. By using that $G(H(t))\in\mathcal{A}(\mathcal{H})$, it is easy to check that, on $[0, T_{\max})$, we have $U^\star U=U U^\star=\mathrm{Id}$. Then an easy computation shows that the derivative of $t\mapsto U(t)H(t)U(t)^\star$ is zero and thus
\[\forall t\in[0, T_{\max})\,,\qquad H(t)=U(t)^\star H_{0} U(t)\,.\]
From this, we infer that, for all $t\in[0, T_{\max})$, $H(t)\in\mathcal{L}_2(\Hil)$ and $\|H(t)\|_{\mathsf{HS}}=\|H_{0}\|_{\mathsf{HS}}$. Therefore, on $[0, T_{\max})$, the function $H$ is bounded in $\mathcal{L}_{2}(\mathcal{H})\subset\mathcal{L}(\mathcal{H})$. Thus we get $T_{\max}=+\infty$.
\subsubsection{Linear case}
Let us now discuss the proof of Corollary \ref{cor.cor1'}. First we notice that, since $\mathcal{L}_{2}(\mathcal{H})$ is an ideal of $\mathcal{L}(\mathcal{H})$, the function $F$ defined in \eqref{eq.F} sends the Hilbert space $\mathcal{S}_{2}(\mathcal{H})$ into itself. Thus, we may apply the Cauchy-Lipschitz theorem in this space and the global existence is ensured by the investigation in the previous section. Let us reformulate \eqref{eq:ODE} in the Hilbert basis $(e_{n})_{n\in\mathbb{N}}$. We easily have
\[h'_{i,i}(t)=2\langle G(H(t))e_{i}, H(t)e_{i}\rangle\,.\]
Then we notice that
\[H(t)=\sum_{k\leq \ell} h_{k,\ell}(t) E_{k,\ell}\,,\qquad G(H(t))=\sum_{k< \ell} g_{k,\ell}h_{k,\ell}(t)E^\pm_{k,\ell} \,.\]
We get
\[H(t)e_{i}=\sum_{j=0}^{+\infty} h_{i,j}(t)e_{j}\,,\qquad G(H(t))e_{i}=-\sum_{j=0}^\infty g_{i,j}h_{i, j}(t) e_{j}\,,\]
and thus
\[h'_{i,i}(t)=-2\sum_{j=0}^{+\infty} g_{i,j}|h_{i,j}(t)|^2\,.\]
Moreover we notice that
\[\|H'(t)\|\leq 2\|H(t)\|\|G(H(t))\|\leq 2\|G\|\|H(t)\|^2_{\mathsf{HS}}=2\|G\|\|H_{0}\|^2_{\mathsf{HS}}\,.\]
Therefore, thanks to the assumptions on $(g_{i,j})$, we may apply Theorem \ref{gen-theo}.
Then we let $\alpha_{\ell}=h_{\ell, \ell}(+\infty)$ and we notice that
\[\sum_{\ell=0}^{+\infty}\|H(t)e_{\ell}\|^2=\|H(t)\|^2_{\mathsf{HS}}=\|H_{0}\|^2_{\mathsf{HS}}\,,\]
and thus, with the Fatou lemma, we get
\[\sup_{\ell\in\mathbb{N}}|\alpha_{\ell}|^2\leq\sum_{\ell=0}^{+\infty}\alpha^2_{\ell}\leq\liminf_{t\to+\infty}\|H(t)\|^2_{\mathsf{HS}}=\|H_{0}\|^2_{\mathsf{HS}}\,.\]
We introduce the Hilbert-Schmidt operator $H_{\infty}:=\mathsf{diag}\left(\alpha_{\ell}\right)$ defined by
\[\forall x\in\mathcal{H}\,,\qquad H_{\infty}(x)=\sum_{\ell=0}^{+\infty} \alpha_{\ell}x_{\ell} e_{\ell}\,,\quad\mbox{ where }\quad x=\sum_{\ell=0}^{+\infty} x_{\ell} e_{\ell}\,.\]
In particular, we get that, for all $\ell\in\mathbb{N}$, $H(t)e_{\ell}$ converges to $H_{\infty}e_{\ell}$. Let us now consider $x\in\mathcal{H}$ and $\varepsilon>0$. There exists $L\in\mathbb{N}$ such that $\displaystyle{\|x-\sum_{\ell=0}^Lx_{\ell} e_{\ell}\|\leq\varepsilon}$. Since $\|H(t)\|\leq \|H(t)\|_{\mathsf{HS}}=\|H_{0}\|_{\mathsf{HS}}$ and $\|H_{\infty}\|\leq\|H_{0}\|_{\mathsf{HS}}$, we get that, for all $t\geq 0$,
\[\|(H(t)-H_{\infty})x\|\leq \|(H(t)-H_{\infty})\sum_{\ell=0}^Lx_{\ell} e_{\ell}\|+2\varepsilon\|H_{0}\|_{\mathsf{HS}}\,,\]
and, for $t$ large enough, it follows that
\[\|(H(t)-H_{\infty})x\|\leq 2\varepsilon\|H_{0}\|_{\mathsf{HS}}+\varepsilon\,.\]
Therefore $H(t)$ strongly converges to $H_{\infty}$ and thus $H(t)$ weakly converges to $H_{\infty}$ in $\mathcal{S}_{2}(\mathcal{H})$.
Let us consider an eigenvalue $\alpha$ of $H_{\infty}$, with multiplicity $m$ and associated with the eigenvectors $e_{\ell_{1}},\ldots, e_{\ell_{m}}$. Then, for all $\varepsilon>0$, there exists $T>0$ such that, for all $j\in\{1,\ldots, m\}$,
\[\|H(T)e_{\ell_{j}}-\alpha e_{\ell_{j}}\|\leq\varepsilon\,.\]
or equivalently
\[\|H_{0}f_{j}-\alpha f_{j}\|\leq\varepsilon\,,\quad\mbox{ with }\quad f_{j}=U(T)e_{\ell_{j}}\,.\]
Therefore, the spectral theorem implies that
\[\forall\varepsilon>0\,,\quad \mathrm{range}\left(\mathds{1}_{[\alpha-\varepsilon,\alpha+\varepsilon]}(H_{0})\right)\geq m\,.\]
This proves that $\alpha$ is an eigenvalue of $H_{0}$ with multiplicity at least $m$.
\subsection{Examples}
Let us now provide examples of applications of our convergence results.
\subsubsection{Brockett's choice}
For a given $A=\mathsf{diag}(a_{\ell})\in\mathcal{D}(\mathcal{H})$ with a non-increasing sequence $(a_{\ell})_{\ell\in\mathbb{N}}\in\mathbb{R}^\mathbb{N}_{+}$, we take $G_{1}(H) = [H,A]$. Since $\mathcal{L}_{2}(\mathcal{H})$ is an ideal of $\mathcal{L}(\mathcal{H})$, we have $G_{1} : \mathcal{S}_{2}(\mathcal{H})\to\mathcal{A}_{2}(\mathcal{H})$ and $\|G_{1}\|\leq 2\|A\|$. Moreover $G_{1}$ is diagonalizable and $g_{i,j}=a_{j}-a_{i}$.
\subsubsection{Toda's choice}
If $H\in\mathcal{S}_{2}(\mathcal{H})$, we introduce
\[H^-=\sum_{1\leq i\leq j} h_{i,j} e^*_{i} e_{j}\,\qquad \mbox{ and }\qquad G_{2}(H) = H^- - (H^-)^{\star}\,.\]
By definition, we have $G_{2} : \mathcal{S}_{2}(\mathcal{H})\to\mathcal{A}_{2}(\mathcal{H})$ and $\|G\|\leq 2\|H^-\|\leq 2\|H\|_{\mathsf{HS}}$. In addition, $G_{2}$ is diagonalizable with $g_{i,j}=-1$ for $i<j$.
\subsubsection{Wegner's choice}
Let us finally discuss a non-linear example. We take $G_{3}(H) = [H,{\rm diag} (H)]$. Since $G_{3}$ sends the symmetric operators in the skew symmetric operators, we get that $G_{3}$ generates a flow that preserves the Hilbert-Schmidt norm. Thus the flow is global. Explicitly we have $H'=[H, [H,\mathsf{diag} H]]$. In particular, we get
\[h'_{i,i}(t)=2\sum_{j=0}^{+\infty} (h_{i,i}(t)-h_{j,j}(t))|h_{ij}(t)|^2\,.\]
If, for large times, the diagonal terms become distinct (in the sense of \eqref{eq:assumpSign} and \eqref{eq:assumpBound}), then, by Theorem \ref{gen-theo}, $H(t)$ converges to a diagonal matrix $H_{\infty}$. Actually, in this case, the convergence is exponential in finite dimension (see Section \ref{sec.conv.exp} where the linearization of the same kind of system is explained).
\section{Exponential convergence}\label{sec.finite.dim}
\subsection{A stronger convergence in finite dimension}
This section is devoted to the proof of Corollary \ref{cor.dim.finie}.
\subsubsection{Reminder of the stable manifold theorem}
Let us recall a classical theorem.
\begin{theorem}\label{thm:SM}
Consider $F\in\mathcal{C}^1(\mathbb{R}^d,\mathbb{R}^d)$ such that $F(0) = 0$ and the differential $\mathsf{d}F_0$ is diagonalizable with no eigenvalues with zero real parts. Denote $E_-$ and $E_+$ respectively the stable and unstable subspaces of $\mathsf{d}F_0$ and $k=\mathrm{dim}(E_-)$
so that $d-k=\mathrm{dim}(E_+)$.
Let us consider the following Cauchy problem
\begin{equation}\label{eq.Cauchy}
H'(t) = F(H(t))\,,\quad t\in\mathbb{R}\,,\quad H(0)=H_0\,,
\end{equation}
and suppose, for any $H_0\in\mathbb{R}^d$, there exists a unique global solution $H\in\mathcal{C}^1(\mathbb{R},\mathbb{R}^d)$.
Then there exists $\mathcal{V}$ a neighborhood of $0$ in $\mathbb{R}^d$ and two $\mathcal{C}^1$-manifolds in $\mathcal{V}$, say $\mathcal{W}_-$ and $\mathcal{W}_+$, with respective dimensions $k$ and $d-k$, tangent at $0$ to $E_-$ and $E_+$ respectively, and such that for all $H_0\in\mathcal{V}$, the solution of \eqref{eq.Cauchy} satisfies:
\begin{enumerate}[\rm (a)]
\item $\lim_{t\to+\infty} H(t) = 0 \Leftrightarrow H_0\in \mathcal{W}_-$,
\item $\lim_{t\to-\infty} H(t) = 0 \Leftrightarrow H_0\in \mathcal{W}_+$.
\end{enumerate}
More precisely, the manifold $\mathcal{W}_-$ (stable at $+\infty$) is the graph of a $\mathcal{C}^1$-map denoted $\varphi_-$ defined over $E_-\cap\mathcal{V}$ and satisfying $\varphi_-(0)=0$ et $\mathsf{d}\varphi_-(0)=0$. The same result holds for $\mathcal{W}_+$.
\end{theorem}
The above result is a classical consequence of a fixed point argument and the convergence to such hyperbolic equilibrium point is exponential in the stable manifold, with a convergence rate similar to the one of the linearized Cauchy problem in the tangent space $E_-$, i.e. of order at worst $\mathcal{O}(e^{-\gamma t})$ where
\[
-\gamma = \max\left(\{\Re(\mathsf{sp}(\mathsf{d}F_0))\}\cap\mathbb{R}_-\right)<0\, .\]
\subsubsection{Convergence to the diagonal matrices: a naive estimate}
From Corollary \ref{cor.cor1'}, we know that $H(t)$ converges to $H_{\infty}=\mathsf{diag}(\alpha_{\ell})$ where the $\alpha_{\ell}$ are \textit{exactly} the eigenvalues of $H_{0}$ since $\mathcal{H}$ is of finite dimension. This convergence analysis is then reduced to a small neighborhood of $H_{\infty}$. The differential equation reads
\[H'=F(H)\,,\qquad F(H)=[H, G(H)]\,,\]
and, since $G$ and $H_{\infty}$ are diagonal in the canonical bases, we have $G(H_{\infty})=0$. We deduce that the differential of $F$ at $H_{\infty}$ is given by
\[\forall H\in\mathcal{S}(\mathcal{H})\,,\qquad \mathsf{d}F_{H_{\infty}}(H)=[H_{\infty}, G(H)]\,.\]
Thus $\mathsf{d}F_{H_{\infty}}$ is an endomorphism of the space of symmetric matrices and it is diagonalized in the basis $E_{i,j}$ since a straightforward computation gives
\[\mathsf{d}F_{H_{\infty}}(E_{i,j})=g_{i,j}(\alpha_{i}-\alpha_{j})E_{i,j}\,.\]
Since $0$ is an eigenvalue of $\mathsf{d}F_{H_{\infty}}$ of multiplicity $d$, we cannot directly apply the stable manifold theorem. Nevertheless, we can elementarily get the following proposition.
\begin{proposition}
\label{prop.exp.rate}
Let us introduce
\[\mathcal{T}=\{(i,j)\in\{0,\ldots, d-1\}^2 : i< j\}\,,\]
and assume that
\[-\gamma:=\displaystyle{\max_{\mathcal{T}}g_{i,j}(\alpha_{i}-\alpha_{j})}<0\,.\]
Then, for all $\varepsilon>0$, there exist $T, C>0$ such that, for all $t\geq T$,
\[\|H(t)-\mathsf{diag}(H(t))\|_{\mathsf{HS}}\leq Ce^{-\gamma t}\,.\]
\end{proposition}
\begin{proof}
Let us define
\[R(H)=F(H)-F(H_{\infty})-\mathsf{d}F_{H_{\infty}}(H-H_{\infty})=[H-H_{\infty}, G(H-\mathsf{diag}(H))]\,,\]
and
\[\Delta(t)=H(t)-\mathsf{diag}(H(t))\,.\]
Let us now consider $\varepsilon>0$ and $T>0$ such that, for all $t\geq T$, we have
\[\|H(t)-H_{\infty}\|_{\mathsf{HS}}\leq \varepsilon\,,\qquad\|R(H(t))\|_{\mathsf{HS}}\leq 2\varepsilon\|G\|\|\Delta(t)\|_{\mathsf{HS}}\,.\]
For all $H\in\mathcal{S}(\mathcal{H})$, we notice that $\mathsf{diag}\left(\mathsf{d}F_{H_{\infty}}(H)\right)$ is zero. Therefore we have
\[\Delta'(t)=\mathsf{d}F_{H_{\infty}}(\Delta(t))+R(H(t))-\mathsf{diag}\left(R(H(t))\right)\,.\]
The Duhamel formula reads, for $t\geq T$,
\[\Delta(t)=e^{(t-T)\mathsf{d}F_{H_{\infty}}} \Delta(T)+\int_{T}^t e^{(t-s)\mathsf{d}F_{H_{\infty}}}\{R(H(s))-\mathsf{diag}\left(R(H(s))\right)\} \,\mathrm{d} s\,,\]
where $\Delta$ and $R(H)-\mathsf{diag}\left(R(H)\right)\in\textsf{Span}\,(E_{i,j})_{i<j}$.
Since $(E_{i,j})$ is orthonormalized for the $\mathsf{HS}$-norm, we deduce that
\[\|\Delta(t)\|_{\mathsf{HS}}\leq e^{-\gamma(t-T)}\|\Delta (T)\|_{\mathsf{HS}}+\int_{T}^t e^{-(t-s)\gamma}\|R(H(s))-\mathsf{diag}\left(R(H(s))\right)\|_{\mathsf{HS}} \,\mathrm{d} s\,,\]
so that
\[e^{\gamma t}\|\Delta(t)\|_{\mathsf{HS}}\leq e^{\gamma T}\|\Delta (T)\|_{\mathsf{HS}}+2\varepsilon\|G\|\int_{T}^t e^{s\gamma}\|\Delta(s)\|_{\mathsf{HS}}\,\mathrm{d} s\,.\]
From the Gronwall lemma, we get, for $t\geq T$,
\[e^{\gamma t}\|\Delta(t)\|_{\mathsf{HS}}\leq e^{\gamma T}\|\Delta (T)\|_{\mathsf{HS}}e^{2\varepsilon\|G\| (t-T)}\,,\]
and thus
\[\|\Delta(t)\|_{\mathsf{HS}}\leq e^{(-\gamma +2\varepsilon\|G\|)(t-T)}\|\Delta (T)\|_{\mathsf{HS}}\,.\]
We deduce that the same convergence holds for the diagonal terms.
Let us now eliminate the dependence in $\varepsilon$ in the exponential. We easily have
\[H'(t)=F(H(t))=[H(t),G(H(t))]=[H(t),G(\Delta(t))]\,,\]
so that for all $t\geq T$
\[\|H'(t)\|_{\mathsf{HS}} \leq 2\|G\|\|H_0\|_{\mathsf{HS}}\|\Delta(t)\|_{\mathsf{HS}} \leq 2\|G\|\|H_0\|_{\mathsf{HS}} C e^{(-\gamma+\epsilon)t}\,.\]
It follows that
\[\|H(t)-H_{\infty}\|_{\mathsf{HS}}\leq \int_t^{+\infty} \|H'(s)\|_{\mathsf{HS}}\,\mathrm{d} s\leq 2\|G\|\|H_0\|_{\mathsf{HS}} \dfrac{C}{-\gamma+\epsilon} e^{(-\gamma+\epsilon)t}\,.\]
We get therefore the existence of a constant $D$ that depends on $\epsilon$ and $\|H_0\|_{\mathsf{HS}}$ only such that for all $t\geq T$,
\[\|H(t)-H_{\infty}\|_{\mathsf{HS}}\leq D e^{(-\gamma+\epsilon)t}\,.\]
Coming back to the previous proof with $De^{(-\gamma+\epsilon)t}$ in place of $\epsilon$, we get then from the Gronwall lemma
\[e^{\gamma t}\|\Delta(t)\|_{\mathsf{HS}}\leq e^{\gamma T}\|\Delta (T)\|_{\mathsf{HS}}e^{2D\|G\|\int_T^te^{(-\gamma+\epsilon)s}\,\mathrm{d} s}\,,\]
that may be bounded independently from $t\geq T$. Therefore we may omit the $\epsilon$ term in exponential convergence rate.
\end{proof}
\begin{remark}
In particular, this proposition may be used as follows. Let us consider $\varepsilon$ such that $\displaystyle{0<\varepsilon<\inf_{\mathcal{T}}|\alpha_{i}-\alpha_{j}|}$ and assume that $\|H_{0}-H_{\infty}\|_{\mathsf{HS}}<\varepsilon$ which may be insured by using the flow up to the time $T$. Then, there exists a permutation $\Pi$ such that
\[\|\tilde H_{0}-\tilde H_{\infty}\|_{\mathsf{HS}}=\mathcal{O}(\varepsilon)\,,\quad \tilde H_{0}=\Pi^\star H_{0}\Pi\,,\quad \tilde H_{\infty}=\mathsf{diag}(\tilde \alpha_{\ell})\,,\]
where $(\tilde \alpha_{\ell})_{\ell\in\{0,\ldots, d\}}$ denotes the sequence of the eigenvalues of $H_{0}$ in the opposite order with respect to $g_{i,j}$. Then we have
\[\|\tilde H(t)-\mathsf{diag}(\tilde H(t))\|_{\mathsf{HS}}\leq Ce^{-\gamma t}\,.\]
In other words, up to a permutation and as soon as we are close enough to the limit, we can insure an exponential convergence (with rate $\gamma$) to the diagonal matrices.
\end{remark}
\subsubsection{Proof of Corollary \ref{cor.dim.finie}}\label{sec.conv.exp}
It is actually more convenient to work on the side of the unitary transform $U$. We recall that, from \eqref{eq.U}, $U$ satisfies
\[U'=UG(U^\star H_{0}U)=\mathcal{F}(U)\,.\]
Let us first notice that $U$ converges.
\begin{proposition}\label{eq.CVU}
The function $t\mapsto U(t)$ converges as $t$ tends to $+\infty$. Moreover, the limit $U_{\infty}$ belongs to the finite set
\[\left\{\Pi^\star\mathsf{diag}(\epsilon_{\ell}) \Pi\,,\quad \Pi\in\mathfrak{S}_{n}\,, \epsilon_{\ell}\in\{\pm 1\}\right\}\,.\]
\end{proposition}
\begin{proof}
The orthogonal group is compact since the dimension is finite. If $U$ converges to $U_{\infty}$, then $U_{\infty}$ is still unitary and $H_{\infty}=U_{\infty}^\star H_{0} U_{\infty}$. There exists an orthogonal transformation $Q_{0}$ such that $Q_{0} H_{0}Q^\star_{0}=\mathsf{diag}(\alpha_{\ell})$ where the $\alpha_{\ell}$ are in non-increasing order. Thus, there exists a permutation $\Pi$ such that
\[H_{\infty}=(U_{\infty}^\star Q^\star_{0}\Pi^\star) H_{\infty} (\Pi Q_{0} U_{\infty})\,,\]
Thus, since $H_{\infty}$ is diagonal with simple coefficients, $\Pi Q_{0} U_{\infty}$ is itself diagonal and thus, we get $(\Pi Q_{0} U_{\infty})^2=\mathrm{Id}$.
Now we know that $H(t)$ tends to $H_{\infty}$ so that $U'(t)$ goes to $0$ as $t$ goes to $+\infty$. Since the orthogonal group is compact and that $\mathcal{F}$ is continuous with a finite number of zeros, we deduce that $U$ must converge to a zero of $\mathcal{F}$.
\end{proof}
We can notice that the proof of Proposition \ref{eq.CVU} relies on the convergence of $H(t)$ when $t$ goes to $+\infty$. Without this a priori convergence, we would not get the convergence of $U(t)$ for any given $H_{0}$. As in \cite[Theorem 3]{Bro91}, one should exclude from the initial conditions the union of the stable manifolds of all the stationary points that are not attractive. With the compactness of $\mathcal{U}(\mathcal{H})$, this would imply the convergence of $U(t)$ towards the (unique) attractive point. Actually such restrictions on the initial conditions are not necessary to get the convergence as explained in the following lines.
Let us now parametrize, locally near $U_{\infty}$, the orthogonal group $\mathcal{U}(\mathcal{H})$ by its Lie algebra $\mathcal{A}(\mathcal{H})$. By the local inversion theorem, it is well known that $\exp : \mathcal{A}(\mathcal{H})\to \mathcal{U}(\mathcal{H})$ is a local smooth diffeomorphism between a neighborhood of $0$ and a neighborhood of $\mathrm{Id}$. We recall that
\[(\mathsf{d}\exp (A))^{-1}(H)=\exp(-A)\sum_{k=0}^{+\infty} \frac{(\mathsf{ad}A)^k}{(k+1)!} H\,.\]
For $t\geq T$, we may write $U(t)=U_{\infty}\exp(A(t))$. Thus we have
\[U'=U_{\infty}e^{A(t)}G(e^{-A(t)}H_{\infty}e^{A(t)})\,.\]
We get
\[A'(t)=e^{A(t)} G(e^{-A(t)}H_{\infty}e^{A(t)})e^{-A(t)}+R_{1}(t)\,,\]
with $R_{1}(t)=\mathcal{O}(\|A(t)\|^2)$. We infer that
\[A'(t)= G([H_{\infty},A])+R_{2}(t)\,,\]
with $R_{2}(t)=\mathcal{O}(\|A(t)\|^2)$. Now $A\mapsto G([H_{\infty},A])$ is an endomorphism of the space of skew-symmetric matrices $\mathcal{A}(\mathcal{H})$ and it is diagonalized in the basis $E^\pm_{i,j}$ with $i<j$ with eigenvalues $g_{i,j}(\alpha_{i}-\alpha_{j})$. By assumption these eigenvalues are not zero. We may apply the stable manifold theorem. Since $A(t)$ goes to $0$, this means, by definition, that $A(t)$ belongs to the stable manifold near $0$. Therefore it goes exponentially to zero, with a convergence rate a priori of order $\mathcal{O}(e^{-\gamma t})$, with $\displaystyle{\gamma=\inf_{\mathcal{T}^-}|g_{i,j}(\alpha_{i}-\alpha_{j})|>0}$. Of course, the convergence may be stronger, depending on the stable manifold on which $A(t)$ is evolving. Then $H(t)$ inherits this convergence rate.
\subsection{Abstract factorization of Hilbert-Schmidt operators}
Actually it is possible to get a quantitative convergence to the eigenvalues of non self-adjoint operators in infinite dimension. Note that, in Theorem \ref{gen-theo}, we do not have such a quantification, even for symmetric operators.
\subsubsection{Chu-Norris decomposition for Hilbert-Schmidt operators}
Let us consider an application $G : \mathcal{L}(\mathcal{H})\to\mathcal{A}(\mathcal{H})$ of class $\mathcal{C}^1$ and the following Cauchy problem:
\[H'(t)=[H(t), G(H(t))]\,,\qquad H(0)=H_{0}\in\mathcal{L}_{2}(\mathcal{H})\,.\]
We also introduce the equations
\begin{align*}
g_{1}'(t)&=g_{1}(t)G(H(t))\,, &g_{1}(0)=\mathrm{Id}\,,\\
g'_{2}(t)&=(H(t)-G(H(t)))g_{2}(t)\,,&g_{2}(0)=\mathrm{Id}\,.
\end{align*}
Then we have the following lemma (coming from a straightforward adaptation of the theorems in \cite{CN88} that are actually true for bounded operators).
\begin{lemma}\label{lemma.ChuNorris}
The solutions $H, g_{1}$ and $g_{2}$ are global and of class $\mathcal{C}^1$. Then, we have $g_{1}\in\mathcal{C}^1(\mathbb{R},\mathcal{U}(\mathcal{H}))$ and
\begin{equation}\label{eq.g1g2}
H(t)=g_{1}(t)^{-1}H_{0}g_{1}(t)=g_{2}(t)H_{0}g_{2}^{-1}(t)
\end{equation}
and thus we have $H\in\mathcal{C}^1(\mathbb{R},\mathcal{L}_{2}(\mathcal{H}))$.
Moreover we have the relations
\begin{equation}\label{eq.g1g2'}
e^{tH_{0}}=g_{1}(t)g_{2}(t)\,,\qquad e^{tH(t)}=g_{2}(t)g_{1}(t)\,.
\end{equation}
\end{lemma}
\begin{proof}
Let us just give some insights of the proof. By easy calculations, we get $(g_{1}g_{1}^\star)'=0$, $(g_{1}Hg_{1}^\star)'=0$, and $(g_{2}^{-1}Hg_{2})'=0$, which ensures both the first algebraic properties and the global character of the solutions in the considered spaces. The first exponential factorization follows from the uniqueness result for the following Cauchy problem
\[Z'(t) = H_0Z(t),\quad t\in\mathbb{R},\quad Z(0)=\mathrm{Id}\,,\]
of which both $e^{tH_0}$ and $g_1(t)g_2(t)$ are solutions. The second exponential factorization then easily follows as a consequence.
\end{proof}
\subsubsection{Proof of Proposition \ref{prop.House}}
It is easy to see that
\[g_{2}(t) e_{\ell}=\sum_{j=0}^{\ell} \alpha_{\ell, j}(t)e_{j}\,.\]
Thus we may write
\[e^{t H_{0}}e_{\ell}=\sum_{j=0}^\ell \alpha_{\ell, j}(t) g_{1}(t)e_{j}\,.\]
We get
\[P^{-1}e^{t \Lambda}f_{\ell}=\sum_{j=0}^\ell \alpha_{\ell, j}(t) g_{1}(t)e_{j}\,,\qquad f_{\ell}=P e_{\ell}\,.\]
We have
\begin{equation}\label{eq.alpha}
\sum_{j=0}^\ell \alpha_{\ell, j}(t) g_{1}(t)e_{j}=P^{-1}e^{t \Lambda}f_{\ell}=\sum_{j=0}^J p_{\ell, j} e^{t\lambda_{j}}v_{j}+\sum_{j=J+1}^{+\infty} p_{\ell, j}e^{t\lambda_{j}} v_{j}\,,
\end{equation}
where $v_{j}=P^{-1}e_{j}$. We may also write
\begin{equation}\label{eq.alpha'}
A_{J}\left[\begin{array}{c}e_{0}\\ \vdots\\ e_{J} \end{array}\right] = P_{J}e^{t\Lambda_{J}}\left[\begin{array}{c}g^\star_{1}v_{0}\\ \vdots\\ g^\star_{1}v_{J} \end{array}\right]+\mathcal{O}(e^{t\Re(\lambda_{J+1})})\,.
\end{equation}
and
\[ A_{J} = P_{J}e^{t\Lambda_{J}}\left[\begin{array}{c}g^\star_{1}v_{0}\\ \vdots\\ g^\star_{1}v_{J} \end{array}\right][e_{0}\ldots e_{J}]+\mathcal{O}(e^{t\Re(\lambda_{J+1})})\,.\]
By using the simplicity of the eigenvalues, we get, by using a Neumann series and the fact that $P_{J}$ is invertible, that $A_{J}$ is invertible when $t$ is large enough. In addition, we get
\begin{equation}\label{eq.AJ-1}
\|A_{J}^{-1}\|=\mathcal{O}\left(e^{-t\Re(\lambda_{J})}\right)\,.
\end{equation}
From \eqref{eq.alpha}, we get
\begin{equation*}
\sum_{j=0}^\ell \alpha_{\ell, j}(t) H_{0}g_{1}(t)e_{j}=\sum_{j=0}^J p_{\ell, j} \lambda_{j}e^{t\lambda_{j}}v_{j}+\mathcal{O}\left(e^{t\Re(\lambda_{J+1})}\right)\,,
\end{equation*}
so that, with \eqref{eq.g1g2},
\begin{equation*}
\sum_{j=0}^\ell \alpha_{\ell, j}(t) g_{1}(t)H(t)e_{j}=\sum_{j=0}^J p_{\ell, j} \lambda_{j}e^{t\lambda_{j}}v_{j}+\mathcal{O}\left(e^{t\Re(\lambda_{J+1})}\right)\,,
\end{equation*}
which may be written as
\[ A_{J}\left[\begin{array}{c}H(t)e_{0}\\ \vdots\\ H(t)e_{J} \end{array}\right] = P_{J}\Lambda_{J}e^{t\Lambda_{J}}\left[\begin{array}{c}g^\star_{1}v_{0}\\ \vdots\\ g^\star_{1}v_{J} \end{array}\right]+\mathcal{O}(e^{t\Re(\lambda_{J+1})})\,.\]
or, with \eqref{eq.AJ-1},
\[ \left[\begin{array}{c}H(t)e_{0}\\ \vdots\\ H(t)e_{J} \end{array}\right] = A^{-1}_{J}P_{J}\Lambda_{J}e^{t\Lambda_{J}}\left[\begin{array}{c}g^\star_{1}v_{0}\\ \vdots\\ g^\star_{1}v_{J} \end{array}\right]+\mathcal{O}(e^{t\Re(\lambda_{J+1}-\lambda_{J})})\,.\]
From \eqref{eq.alpha'}, we get
\begin{equation}\label{eq.alpha''}
\left[\begin{array}{c}g^\star_{1}v_{0}\\ \vdots\\ g^\star_{1}v_{J} \end{array}\right]=e^{-t\Lambda_{J}}P^{-1}_{J}A_{J}\left[\begin{array}{c}e_{0}\\ \vdots\\ e_{J} \end{array}\right]+\mathcal{O}(e^{t\Re(\lambda_{J+1}-\lambda_{J})})\,,
\end{equation}
and thus
\begin{equation}\label{eq.approx-inv}
\left[\begin{array}{c}H(t)e_{0}\\ \vdots\\ H(t)e_{J} \end{array}\right] = A^{-1}_{J}P_{J}\Lambda_{J}P^{-1}_{J}A_{J}\left[\begin{array}{c}e_{0}\\ \vdots\\ e_{J} \end{array}\right]+\mathcal{O}(e^{t\Re(\lambda_{J+1}-\lambda_{J})})\,.
\end{equation}
Therefore the space spanned by $e_{0},\ldots, e_{J}$ is invariant by $H(t)$ up to an error of order $\mathcal{O}(e^{t\Re(\lambda_{J+1}-\lambda_{J})})$. In view of \eqref{eq.approx-inv}, we get the conclusion by induction on $J\geq 0$. In other words, we approximate the diagonal terms of $H(t)$ one by one by moving $J$ from $0$ to $+\infty$.
\subsection{Numerics}
In the following numerical illustrations, we consider the finite dimensional example of flows evolving in matrices of $\mathcal{M}_{d}(\mathbb{R})$ with $d=5$. As an initial data, let fix $H_0=Q\,\mathsf{diag}(\ell^2)_{1\leq \ell\leq d}\,Q^\star$, where $Q=\exp(B_5)$ is a unitary matrix defined through the following choice of $B_d\in\mathcal{A}_{d}(\mathbb{R})$ skew-symmetric:
\[B_d = (b_{i,j}),\quad b_{i,j}=1,\textrm{ for }i>j\,.\]
The numerical solver to compute any of the below approximate solutions is an adaptive 4th-order Runge-Kutta scheme.
\subsubsection{Brockett's choice}
The considered Brockett's bracket flow is given by the choice $A = \mathsf{diag}(d-\ell)_{0\leq \ell\leq d-1}\in\mathcal{D}(\mathbb{R}^d)$ with non-increasing diagonal elements. With these data, we observe the convergence to the limit $H_{\infty} = \mathsf{diag}([25,16,9,4,1])$, see Figure~\ref{figure.brockett}. The diagonal terms are sorted in a descending order, according to the coefficients in $A$. The effective exponential rate of convergence $\gamma=3$ coincides, for sufficiently large times, with the worst expected one (Corollary~\ref{cor.dim.finie}).
The matrix of $g_{i,j}(\alpha_i-\alpha_j)$ is there:
\[\begin{pmatrix}
0 & - 9 & - 32& - 63 & - 96 \\
- 9 & 0 & - 7 & - 24 & - 45 \\
- 32 & - 7 & 0 & - 5 & - 16 \\
- 63 & - 24 & - 5 & 0 & - 3 \\
- 96 & - 45 & - 16 & - 3 & 0
\end{pmatrix}\,.\]
\begin{figure}
\includegraphics[scale=0.32]{Brockett_Diag.pdf}
\includegraphics[scale=0.32]{Brockett_Cvg.pdf}
\caption{Brockett flow -- Convergence of diagonal entries (left), exponential convergence of $H(t)$ (right).}
\label{figure.brockett}
\end{figure}
Let change the initial data to the following one $\tilde{H}_0=\tilde{Q}\,\mathsf{diag}(\ell^2)_{1\leq \ell\leq d}\,\tilde{Q}^\star$, where $\tilde{Q}=\begin{pmatrix}1 & (0)\\ (0) & \exp(B_4)\end{pmatrix}$. Then $\tilde{H}_0$ takes the above block form
\begin{equation}\label{form.sev}
\begin{pmatrix}1 & (0) \\ (0) & K\end{pmatrix}\,,
\end{equation}
where $K\in\mathcal{S}_{4}(\mathbb{R})$ has spectrum $\{4,9,16,25\}$. Then we observe the convergence to the limit $H_{\infty} = \mathsf{diag}([1,25,16,9,4])$, with an exponential rate $\gamma=5$. Namely, the matrix of $g_{i,j}(\alpha_i-\alpha_j)$ is then:
\[\begin{pmatrix}
0 & 24 & 30 & 24 & 12 \\
24 & 0 & - 9 & - 32& - 63 \\
30 & - 9 & 0 & - 7 & - 24 \\
24 & - 32 & - 7& 0 & - 5 \\
12 & - 63 & - 24 & - 5 & 0
\end{pmatrix}\,.\]
In fact, in that case the exact flow evolves in the linear subspace described in~\eqref{form.sev} and the positive coefficients of the above matrix concern precisely the evolution of the flow transversally to that subspace. It is only because of the very specific form of the initial data that the numerical flow evolves in the same way, elsewhere numerical error due to the finite precision would have induced a convergence of the numerical solution to the totally sorted limit with rate $\gamma=3$.
\subsubsection{Toda's choice}
For the same test case, the Toda flow gives similar results, see Figure~\ref{figure.toda}. Once again, the effective exponential rate of convergence is $\gamma=3$ and coincides, for sufficiently large times with the theoretical one.
Actually the matrix of $g_{i,j}(\alpha_i-\alpha_j)$ is there:
\[\begin{pmatrix}
0 & -9 & -16 & -21 & -24 \\
- 9 & 0 & -7 & -12 & -15 \\
- 16& - 7 & 0 & -5 & -8 \\
- 21& - 12& - 5 & 0 & -3 \\
- 24& - 15& - 8& - 3 & 0
\end{pmatrix}\,.\]
On Figure~\ref{figure.toda2}, we present the numerical counterpart of Proposition~\ref{prop.House}. For any $1\leq \ell\leq d-1$, we compute the norm of the residual column $\|(h_{j,\ell}-\alpha_\ell\delta_{j,\ell})_{\ell\leq j\leq d}\|$ along the time. The thick dashed curves corresponds to the numerical computations and the thin continuous ones corresponds to the expected rate of convergence, namely $\Re(\alpha_{\ell}-\alpha_{\ell+1})\in\{9,7,5,3\}$ successively, because of the precise distribution of the eigenvalues of $H_0$ in this example.
\begin{figure}
\includegraphics[scale=0.32]{Toda_Diag.pdf}
\includegraphics[scale=0.32]{Toda_Cvg.pdf}
\caption{Toda flow -- Convergence of diagonal entries (left), exponential convergence of $H(t)$ (right).}
\label{figure.toda}
\end{figure}
\begin{figure}
\includegraphics[scale=0.32]{Toda_Cols.pdf}
\caption{Toda flow -- Convergence of extradiagonal entries.}
\label{figure.toda2}
\end{figure}
\subsubsection{Relation with the QR algorithm}
The exponential relations in Lemma~\ref{lemma.ChuNorris} are the keystone for the connection of the Toda flow to the well-known QR algorithm. Under assumptions of Proposition~\ref{prop.House}, $\exp(H_0)$ is invertible and diagonalizable with eigenvalues $(e^{\lambda_j})_{j\geq 1}$ with decreasing moduli. This is a sufficient condition for the convergence of the QR algorithm applied to $e^{H_0}$. We already get $g_1(t)\in\mathcal{U}(\mathcal{H})$ and, for that flow, the Cauchy-Lipschitz theorem ensures that $g_2$ takes value in the subset of upper triangular matrices. At $t=1$, $e^{H_0}=g_1(1)g_2(1)$ is therefore nothing but a QR factorization of $e^{H_0}$. Thus $e^{H(1)}=g_2(1)g_1(1)$ corresponds to the first iteration of the discrete algorithm and, since the differential equation is autonomous, that handling iterates along integer times so that the sampling sequence $(e^{H(n)})$ mimics the QR algorithm applied to $e^{H_0}$. In fact, the diagonal coefficients of $g_2$ are not all positive (even if they would be, up to a product with a $\mathsf{diag}(\pm 1)$ matrix, that corresponds to a change in the initial data for $g_1$ and $g_2$ depending on $H_0$), and the two algorithms slightly differ.
\subsubsection{Wegner's choice}
Consider now the above initial data for the Cauchy problem associated to Wegner's flow. The solution converges to $H_{\infty}=\mathsf{diag}([4,9,16,25,1])$. The exponential rate to the limit is $\gamma=-9$ that matches the expected value. Indeed a simple calculation gives, for $E\in\mathcal{S}(\mathcal{H})$
\[\textsf{d}F_{H_{\infty}}(E) = [H_\infty,[E,H_{\infty}]]\,,\]
so that for $i<j$
\[\textsf{d}F_{H_{\infty}}(E_{i,j}) = -(\alpha_i-\alpha_j)^2E_{i,j}\,.\]
Therefore, at the limit $H_\infty$, one has the following "matrix-eigenvalue":
\[\begin{pmatrix}
0 & - 25 & - 144 &- 441 & - 9 \\
- 25 & 0 & - 49 & - 256 &- 64 \\
- 144 & - 49 & 0 & - 81 &- 225 \\
- 441 &- 256 & - 81 & 0 & - 576 \\
- 9 &- 64 &- 225 & - 576 & 0
\end{pmatrix}\,.\]
\begin{figure}
\includegraphics[scale=0.32]{Wegner_Diag.pdf}
\includegraphics[scale=0.32]{Wegner_Cvg.pdf}
\caption{Wegner flow -- Convergence of diagonal entries (left), exponential convergence of $H(t)$ (right).}
\label{figure.wegner}
\end{figure}
In that case, for any possible diagonal limit $H_\infty$, the corresponding eigenvalues of $\textsf{d}F_{H_{\infty}}$ are all negative, except 0 that is an eigenvalue of multiplicity $d$.
|
1,116,691,500,449 | arxiv | \section{Introduction}
It is well-known that physics beyond the Standard Model (SM) must exist in order to address a number of outstanding questions such as
the nature of dark matter, the generation of neutrino masses, the origin of the observed baryon asymmetry and
the solution to the hierarchy problem -- all of which remain unanswered. The nature of this new physics is presently
mysterious: Not only is its form unknown, so is the energy scale at which it will first be revealed. Although constrained by data from Run I
at the LHC, dark matter searches, and flavor physics observables, Supersymmetry (SUSY)\cite{SUSYrefs} remains the leading theoretical framework to address at least some of these important puzzles.
However, Supersymmetry has so far been frustratingly elusive at the LHC, with numerous searches setting strong constraints on the
simplest SUSY scenarios\cite{SUSYsearches}. Nonetheless, the continual exploration of the SUSY parameter space remains mandatory, with
missing transverse energy (MET) based searches at the LHC continuing to be the most promising avenue for discovery.
Along these lines, the ATLAS experiment recently announced\cite{ATLASsignal} the observation of a $3\sigma$ excess in one of their Run I SUSY search
channels, $Z+$MET with $\geq 2j$, while a similar analysis by CMS\cite{CMSnonsignal} observed a result consistent with the expected
SM background. Importantly, the detailed nature of the cuts employed by these two experiments in this channel
are sufficiently different, as we will discuss below, so that the apparent null result from CMS does not necessarily exclude the possibility of
a signal being observed by ATLAS. However, an explanation of this potential signal within Supersymmetry remains challenging,
since any proposed scenario must also satisfy the constraints imposed by the plethora of ATLAS and CMS
searches\cite{SUSYsearches}. Nonetheless, a few new physics scenarios have been proposed\cite{ZMETpapers} that could give rise to the observed ATLAS excess
with varying degrees of success. In this work, we suggest a natural Supersymmetric scenario, based on the pMSSM, which comfortably explains the ATLAS excess in the $Z+$MET channel while evading all other searches.
Given the simple nature of the search channel, and the apparent rate of the excess, several features are
clearly necessary for a Supersymmetric model to provide a successful description of the data.
Since the $Z$-boson is observed in the dilepton mode, the signal rate demands a strong production cross section, implying the
production of relatively light gluinos or squarks which then decay to an intermediate
state accompanied by jets. This intermediate, apparently neutral state, {\it e.g.}, a neutralino, then decays via the
emission of a $Z$ plus the lightest Supersymmetric particle (LSP), which produces the MET in an R-parity
conserving scenario{\footnote {In the analysis considered here, we will assume the LSP to be the lightest neutralino.}}.
However, it is likely that such a spectrum would be easily excluded by, {\it e.g.}, the 0l, $2-6$ jets+MET searches
if the jets from the hadronic decay of the $Z$ were sufficiently hard. Clearly the details of the SUSY spectrum in such a scenario,
in particular the relative masses and compositions of the sparticles,
are highly constrained by multiple requirements and finding the right `balance'
presents a significant model building challenge.
To perform this study, it is necessary to incorporate a detailed analysis of the available SUSY parameter space that remains viable after
the Run I data at 7 and 8 TeV; there is no better way to accomplish this than to employ the 19-parameter
p(henomenological)MSSM\cite{pMSSM} which we have already studied in detail elsewhere\cite{us}. In particular, this recent work contains a large sample of
pMSSM models that are presently allowed, providing a viable playground for exploration. As will be discussed in detail below, an
examination of these models reveals an intriguing scenario that describes the excess, while complying with all the constraints.
Specifically, the $1^{st}/2^{nd}$ squarks, $\tilde Q_L, \tilde u_R$ or $\tilde d_R$ are identified as the leading
candidates for the objects that initiate the ATLAS $Z+$MET signal via a cascade decay. Once the other search constraints are taken
into account, the primary production of gluinos in
this role are found to yield too small of an event rate to explain the signal.
The $1^{st}/2^{nd}$ generation squark scenario benefits from having both of the first two generations of squarks being
produced simultaneously, as they are assumed to be degenerate in the pMSSM framework, yielding a large enough production
rate. Whereas, within our pMSSM model sample, the $3^{rd}$ generation
squarks are also too highly constrained by specialized searches to play the role of
the strong initiator of this signal. Within our successful scenario, the $1^{st}/2^{nd}$ generation squarks decay to a mostly bino-like neutralino
which then subsequently decays to a Higgsino-like
LSP triplet by emitting a $Z$-boson. The masses and splittings dictated by this spectrum control the overall production rates for the different sparticles, the hardness
of the jets and leptons, and the branching fractions for the intermediate neutralino decaying to the three light Higgsino states.
We note that considering only a single set of squarks presents a somewhat simplified picture and that other states (such as $\tilde t$ and $\tilde b$) may also contribute to the total signal, albeit in a secondary capacity. In the analysis below we use the successful models contained in our existing pMSSM sample as seeds to generate a small sample of
simplified models that describe the excess while remaining consistent with the many other LHC searches. We then study the detailed properties of these simplified models and discuss the Run II analyses that can be used to elucidate this scenario more fully, or exclude it from further consideration.
\section{Analysis}
The pMSSM\cite{pMSSM} is the most general version of the R-parity conserving MSSM subjected to the guiding principles of CP-conservation, Minimal Flavor Violation, and degenerate 1$^{st}$ and 2$^{nd}$ generation squark masses. Imposing these criteria reduces the number of free parameters in the MSSM to 19 (assuming a neutralino LSP): $m_{\tilde Q_{L1,2}}$, $m_{\tilde Q_3}$, $m_{\tilde u_{R1,2}/\tilde d_{R1,2}/\tilde t_R/\tilde b_R}$, $m_{\tilde L_{L1,2}}$, $m_{\tilde L_3}$, $m_{\tilde e_{R1,2}/\tilde\tau_R}$, $M_{1,2,3}$, $\mu$, $A_{t,b,\tau}$,
$M_A$, and $\tan\beta$. In our previous work\cite{us}, we generated a large set of models (with a `model' describing a point in the 19-dimensional parameter space) by randomly scanning the parameter space, setting the upper limit on the scan at 4 TeV for the dimensionful parameters and taking $\tan\beta=1-60$. The 4 TeV upper bound was chosen to facilitate collider studies at the 7,8 and 13,14 TeV LHC. We subjected these models to a global data set of collider, flavor, precision, dark matter and theoretical constraints.
In particular, we have examined this model sample in light of the SUSY search results from Run I of the LHC\cite{us}, subjecting them to roughly 40 separate analyses performed by the experiments at 7 and 8 TeV. The result is a sample of approximately 125k models that remain viable at the end of Run I, including many models with light squarks and gluinos, providing an ample playground for further studies.
Here, we investigate our pMSSM model sample to determine whether a region of the parameter space could adequately describe the excess in the $Z+$MET channel observed by ATLAS. For each model, we created SUSY production samples using Pythia 6.4 \cite{Sjostrand:2006za} for event generation and PGS \cite{PGS} for detector simulation, normalizing to NLO cross sections from Prospino \cite{Beenakker:1996ch}. The details of this procedure are the same as for our previous pMSSM studies \cite{us}. We then applied the cuts for the on-$Z$ region of the ATLAS search for final states containing a pair of opposite-sign dileptons, jets and MET \cite{ATLASsignal} to these simulated SUSY samples. In particular, events were required to have at least two leptons with $p_T > 25, 10$ GeV respectively, two jets with $p_T > 35$ GeV, and MET $>$ 225 GeV. The two hardest leptons were required to form a same-flavor opposite sign pair with invariant mass $m_{\ell\ell} = m_Z\ \pm\ 10$ GeV, and the scalar sum $H_T$ of their transverse momenta and the transverse momenta of all jets with $p_T > 35$ GeV was required to be at least 600 GeV. We also imposed standard cuts on the rapidity and isolation of jets and leptons, and required an angular separation $\Delta \phi (j, \mathrm{MET}) > 0.4$ between each of the two leading jets and the missing transverse momentum, as described by ATLAS.
Several of our pMSSM models predict significant numbers of events passing the ATLAS $Z+$MET cuts. The sparticle spectrum for such a representative pMSSM model is shown in Fig.~\ref{fig:spspect}. We observe a common pattern in these models, with light-flavor squarks decaying through gaugino cascades producing $Z$-bosons. The direct decay of the squarks to the lightest neutralino is usually suppressed by weak couplings to a Higgsino-like LSP (or alternatively a wino-like LSP if the squark is right-handed). Additionally, we find that these models tend to predict an observable excess in jets + MET searches, creating some tension with the null results in other Run I LHC SUSY searches that we have considered previously \cite{us}. In particular, it is challenging to reproduce the ATLAS $Z+$MET excess in the leptonic channel while simultaneously satisfying bounds from the ATLAS jets + MET search \cite{TheATLAScollaboration:2013fha}. However, the pMSSM models from our sample that reproduce the $Z+$MET excess often predict
a jets + MET event rate that is near the boundary of the existing limits. In particular, the number of events in the jets + MET search signal regions is typically reduced because the squarks decay mainly to a heavier bino-like neutralino rather than directly to the LSP.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\textwidth]{1046808-spectrum}
\caption{Sparticle spectrum for a representative pMSSM model that reproduces the ATLAS $Z+$MET excess.}
\label{fig:spspect}
\end{figure}
Encouraged by these results, we are motivated to consider points with similar spectra to these promising pMSSM models, which may predict a significant number of $Z+$MET events while fully evading constraints from the other LHC SUSY searches. Given the results of our pMSSM analysis, we focus on simplified spectra with the dominant decay pattern $\tilde{q} \rightarrow \tilde{B} \rightarrow \tilde{h}$, where $\tilde{q}$ is a light-flavor squark $\tilde{Q}_L$, $\tilde{u}_R$ or $\tilde{d}_R$.
Starting with a seed point taken from one of our successful pMSSM models, we vary the 3 most relevant
Lagrangian parameters ($m_{\tilde Q_{L1,2}}$, $\mu$, and $M_1$) in a grid around the region of interest. This corresponds to adjusting the most relevant
physical sparticle masses, specifically $m_{\tilde\chi{_1^0}}$, $m_{\tilde\chi{_3^0}}$, and the squark masses
$m_{\tilde q}$. Note that neither the sign of $\mu$ or the value of $\tan\beta$ are varied, as this would modify the
details of the Higgsino spectrum yet leave the gaugino branching fractions mostly unaltered.\footnote{Except for cases where some of the Higgsinos are kinematically inaccessible to decays of the bino-like $\tilde{\chi}_3^0$.} We expect the remaining pMSSM parameters to have a negligible impact on the simplified model phenomenology, as long as they are sufficiently heavy. We thus leave their values as given in the pMSSM seed model, with the exception of $A_t$ which is adjusted to produce the observed value of the Higgs mass within theoretical and experimental uncertainties. In all cases, the squark masses not being scanned are set above 2 TeV.
To produce this grid, we scan $\mu$ between 100 GeV and 254 GeV (with the lower limit set by LEP constraints and upper limit set by kinematic considerations), with 22 GeV steps. Note that given this small value of $\mu$ we would
expect these models to exhibit low values of fine-tuning from this source.
We then scan the \textit{physical} squark mass $m_{\tilde Q_L} \simeq m(\tilde{u}_L)\simeq m(\tilde{d}_L)$ between 350 GeV or $\mu + 150$ GeV (whichever is larger) and 900 GeV in increments of 35 GeV \footnote{Since we don't know the physical squark mass before spectrum generation, we estimate the soft mass required to give the desired physical mass. While approximate, this estimation is easily accurate enough to ensure that our scan grid is covering the region of interest.}. Finally, we scan $M_1$ between $\mu + 100$ GeV and $m_{\tilde Q_L}$ with 25 GeV increments. We employ the same procedure (with slightly different scan ranges noted in Table~\ref{GridScan}) to construct two additional grids, one each for $\tilde u_R/\tilde c_R$ and $\tilde d_R/\tilde s_R$.
Figure~\ref{fig:decay_patterns} shows the relevant spectrum and branching fractions for one of our grid points (from the $\tilde Q_L$ grid), which predicts 21 events in the ATLAS Z + MET search and is consistent with all other searches; this model is illustrative of the typical decay patterns for scenarios that reproduce the excess. The key features to note are the large branching fractions for squarks decaying to $\tilde{\chi}_3^0$, and the multiple possible decays of $\tilde{\chi}_3^0$, about a quarter of which result in $Z$ boson production.
\begin{table}[htbp]
\centering
\begin{tabular}{|l|l|l|l|} \hline\hline
Grid & $\mu$ (22 GeV steps) & $M_1$ (25 GeV steps)& $m_{\tilde Q_L}$ (35 GeV steps) \\
\hline
$\tilde{Q}_L$ & 100 GeV - 254 GeV & $\mu$ + 100 GeV - $m_{\tilde Q_L}$ & MAX(350 GeV, $\mu$ + 150 GeV) - 900 GeV \\
$\tilde{u}_R$ & 100 GeV - 254 GeV & $\mu$ + 100 GeV - $m_{\tilde Q_L}$ & MAX(300 GeV, $\mu$ + 150 GeV) - 800 GeV \\
$\tilde{d}_R$ & 100 GeV - 254 GeV & $\mu$ + 100 GeV - $m_{\tilde Q_L}$ & MAX(250 GeV, $\mu$ + 150 GeV) - 700 GeV \\
\hline\hline
\end{tabular}
\caption{Scan ranges for the 3 variable parameters in each of the 3 grid scans described in the text.}
\label{GridScan}
\end{table}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.8\textwidth,height=0.9\textwidth]{Spectrum_Diagram2.pdf}
\caption{Spectrum and decay patterns for a model in the $\tilde Q_L$ grid, which predicts 21 events in the ATLAS 20 fb$^{-1}$
$Z$+MET analysis and is consistent with all other implemented searches. Numbers indicate the branching fraction in percent for each decay mode (only branching fractions larger than 5\% are shown for simplicity).}
\label{fig:decay_patterns}
\end{figure}
\section{Results}
We now examine the results of our scan over the simplified pMSSM spectra.
As noted above, the strongest restrictions on the parameter space arise from the null results of other LHC SUSY searches, which are generally in tension with our goal of producing a large signal rate in the ATLAS 20 fb$^{-1}$ $Z$+MET analysis. Clearly, we require that a successful model point produce $\sim 15-20$ signal events for the ATLAS 20 fb$^{-1}$
$Z$+MET analysis. In addition, we also require the point to simultaneously satisfy the limit from the corresponding CMS search with different
selection criteria. Finally, a successful model point must satisfy all of the null ATLAS search results in other channels. In particular, it is clear that both the ATLAS 1l+jets search (arising in the spectra we consider from the heavy bino decay producing a $W$ instead of a $Z$) and the 0l, 2-6 jets search (when $W\,,Z$, or the Higgs are produced and decay hadronically) will also be important in determining the detailed nature of a successful parameter space point. The impact of these other searches will be discussed in more
detail below.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.3\textwidth]{zjmet_events_ql500}
\includegraphics[width=0.3\textwidth]{zjmet_events_ql600}
\includegraphics[width=0.3\textwidth]{zjmet_events_ql700}
\includegraphics[width=0.3\textwidth]{zjmet_events_ur450}
\includegraphics[width=0.3\textwidth]{zjmet_events_ur500}
\includegraphics[width=0.3\textwidth]{zjmet_events_dr450}
\caption{Signal event rate contours for the ATLAS $Z+$MET analysis in the $\chi_3^0-\chi_1^0$ mass difference and $\chi_1^0$ mass plane.
The top three panels correspond to the case of $\tilde Q_L=500\,, 600\,, 700$ GeV from left to right, while the bottom panels are for
$\tilde u_R=450\,, 500$ GeV and $\tilde d_R=450$ GeV, left to right.}
\label{fig:zjmetevents}
\end{figure}
To get an initial handle on the preferred parameter regions, Fig.~\ref{fig:zjmetevents} shows the LSP mass versus the mass splitting between the intermediate bino-like $\tilde \chi_3^0$ and the LSP for fixed values of the squark masses in the 450-700 GeV range {\it {before}} imposing any additional constraints.\footnote{Note that we have interpolated between grid points, smoothing out modest fluctuations in event yields.} The various
colored regions show the anticipated ATLAS $Z$+MET analysis event yields and, as we would naively expect, we see that
lighter squarks will generally lead to larger signal rates due to their significantly larger production cross sections. Also, we
see that the Higgsino-like LSP prefers to be relatively light, below $\sim 180-190$ GeV. Perhaps, most interestingly,
we observe that the most favored range for the electroweakino mass splitting lies above $\sim 150$ GeV. This might be
counter-intuitive since we would naively expect that a mass splitting, $\Delta m_{31}$, in the range of $\sim 90-125$ GeV would be
most desirable, since decays to the Higgs in this region would be kinematically forbidden, thereby increasing the
branching fraction for decays through the $Z$. Clearly, in all cases we see that the largest signal rates are obtained when the
$\chi_3^0$ is kinematically allowed to decay through both the $Z$ and the Higgs, due to the increased visibility of the decay products. Of course the preferred range
of $\Delta m_{31}$ is somewhat sensitive to the nature of the parent squark. In the $\tilde Q_L$ case, a value of $\Delta m_{31} \sim 150-200$
GeV is preferred while for $\tilde u_R(\tilde d_R)$ this value is significantly larger $\Delta m_{31} \sim 200-300(250-350)$ GeV (as we will see more clearly below) in obvious
correlation with the production cross sections, {\it i.e.}, the parent squark with the largest (smallest) production cross section
prefers the smallest (largest) corresponding value of $\Delta m_{31}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{mn1_mql+dm_Nzjm}
\caption{Results from the simplified spectra scan for a parent $Q_L$-squark in the $\tilde Q_L$ and $\chi_1^0+ (1/50)\Delta m_{31}$ mass plane. The vertical bars represent the coarse grid in our scan, with the value of the mass splitting $\Delta m_{31}$ increasing for successively higher slices of the bar. The color code
indicates the predicted event rate for the ATLAS 20 fb$^{-1}$ $Z+$MET channel. Black slices in a vertical bar correspond to points excluded by any of the simulated searches.}
\label{fig:scans1}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{mn1_mur+dm_Nzjm}
\caption{Results from the simplified spectra scan for a parent $u_R$-squark in the $\tilde u_R$ and $\chi_1^0+ (1/50)\Delta m_{31}$ mass plane. The vertical bars represent the coarse grid in our scan, with the value of the mass splitting $\Delta m_{31}$ increasing for successively higher slices of the bar. The color code
indicates the predicted event rate for the ATLAS 20 fb$^{-1}$ $Z+$MET channel. Black slices in a vertical bar correspond to points excluded by any of the simulated searches.}
\label{fig:scans2}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=1.00\textwidth]{mn1_mdr+dm_Nzjm}
\caption{Results from the simplified spectra scan for a parent $d_R$-squark in the $\tilde d_R$ and $\chi_1^0+ (1/50)\Delta m_{31}$ mass plane. The vertical bars represent the coarse grid in our scan, with the value of the mass splitting $\Delta m_{31}$ increasing for successively higher slices of the bar. The color code
indicates the predicted event rate for the ATLAS 20 fb$^{-1}$ $Z+$MET channel. Black slices in a vertical bar correspond to points excluded by any of the simulated searches.}
\label{fig:scans3}
\end{figure}
Figures \ref{fig:scans1}, \ref{fig:scans2} and \ref{fig:scans3} present
the results from our scans of the simplified pMSSM spectra for each parent squark type,
where the vertical bars represent the scanned regions in our somewhat
coarsely spaced grid. The location of the vertical bars is set by the approximate mass of the
parent squark and the LSP mass. The lowest slice
of each of the vertical bars corresponds to the smallest value of the bino-LSP mass splitting, $\Delta m_{31}$, scaled by a factor of 0.02,
while each successive higher slice (going up the bar) corresponds to increasing this value. The color code
indicates the number of events predicted for the ATLAS 20 fb$^{-1}$ $Z$+MET analysis, with yellow to red tones indicating a higher
event rate in agreement with the observed rate. Black regions represent points which
are excluded by the corresponding CMS $Z$+MET search, or by any of the $\sim$ 40 null SUSY searches described in~\cite{us}. In the $\tilde Q_L$ grid, the
0l+jets channel results gives by far the most important constraints, while for $\tilde u_R$, and particularly $\tilde d_R$, other searches also play an important role.
The results are seen to be quite different for the three parent squark
cases we consider. In particular, we see that the $\tilde Q_L$ case provides the best fit to the excess. However, even in this case
the most successful points are close to the black excluded regions, indicating that the other LHC SUSY searches are providing important constraints on this scenario. We also find the strong constraints at larger values of $\Delta m_{31}$, due to larger contributions to the 0l+jets channel when this splitting is too large. In general, we expect the 0l+jets rate to place strong restrictions on the $\tilde Q_L$ scenario because of either a large production rate for relatively light squark masses or because the rate is still reasonably large for heavier masses where the jets from the decay of the $W\,, Z$ and Higgs bosons appearing in the bino to LSP transitions are becoming sufficiently hard to pass the 0l+jets cuts. For surviving points explaining the Z + MET excess, we find that low LSP masses and moderate values of $\Delta m_{31}$ are preferred, and that the production
rate falls off too quickly for squark masses above $\sim 800$ GeV to generate a sufficient number of events.
In the case of a $\tilde u_R$ parent, both the favored region and the region excluded by the other searches
are smaller (a simple consequence of the lower production cross section) and, overall, lower signal rates are obtained.
These same features are seen to be further emphasized for the case where the parent squark is a $\tilde d_R$. In all cases, we find that the
sweet spot for describing the excess is in the region where the parent squark is $500-700$ GeV with a LSP mass of $100-200$ GeV and a bino-LSP mass splitting of $100-250$ GeV.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{Nzjm_0l_8tev20_dm_ql}
\includegraphics[width=0.45\textwidth]{Nzjm_0l_8tev20_dm_ur}
\includegraphics[width=0.45\textwidth]{Nzjm_0l_8tev20_dm_dr}
\caption{Ratio of the predicted number of events for models in our simplified grid scan to the ATLAS 95\% C.L. event limit for the 0l+jets
channel, $R_{0l}$, as a function of the number of signal events for the ATLAS 20 fb$^{-1}$ $Z+$MET search. The color code corresponds to the value of the
$\chi_3^0-\chi_1^0$ mass splitting. The top-left, top-right, and bottom panels correspond to the three grid scans, with $\tilde Q_L$,
$\tilde u_R$, and $\tilde d_R$ parent squarks, respectively.}
\label{fig:0l}
\end{figure}
For each parent squark type, we next examine the impact of the 0l,1l+jets and CMS $Z+$MET channels. Here, we only study the
set of models from our simplified grid scan that are consistent with the constraints in these channels. For each of these analyses, we compute the expected number of events in each signal region, and show the ratio of the expected number of events to the 95\% C.L. event exclusion limit for the most
important signal region for that channel (\textit{i.e.}, the signal region with the largest value of this ratio). For example, a ratio of 0.5 indicates that the model predicts
1/2 as many events as are allowed by the relevant null search result at $95\%$ C.L. Figure \ref{fig:0l} displays these results for the
case of the 0l+jets ATLAS search, {\it i.e.}, the event rate ratio $R_{0l}$, as a function of the number of predicted events for the ATLAS 20 fb$^{-1}$ $Z$+MET analysis for all three parent squark types. The color code indicates the value of $\Delta m_{31}$. The top-left, top-right, and bottom panels correspond to the parent squarks $\tilde Q_L$,
$\tilde u_R$, and $\tilde d_R$, respectively. In the $\tilde Q_L$ parent case, we see
that many of the models lie close to the 0l+jets exclusion boundary. In particular,
we see that for model points with at least 15 ATLAS 20 fb$^{-1}$ $Z+$MET signal events, the values of $R_{0l}$ lie in the range 0.6-1. Generally
an increase in the number of ATLAS 20 fb$^{-1}$ $Z+$MET signal events corresponds to a larger value of $R_{0l}$, so that at some point consistency
with the 0l+jets search prevents larger signal rates from being obtained. Also, we see that as
$\Delta m_{31}$ decreases for a fixed signal rate, the points are farther away from the 0l+jets exclusion boundary since the jets produced by $W\,, Z$ and Higgs
decays are becoming correspondingly softer. Considering the $\tilde u_R$ parent case, we find that the model points are a
bit further away from the 0l+jets boundary (due to the smaller production cross section), but we also find, correspondingly,
fewer models that produce a significant signal in the $Z+$MET analysis. This trend continues for the case of the
$\tilde d_R$ parent.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{Nzjm_1l_8tev20_dm_ql}
\includegraphics[width=0.45\textwidth]{Nzjm_1l_8tev20_dm_ur}
\includegraphics[width=0.45\textwidth]{Nzjm_1l_8tev20_dm_dr}
\caption{Ratio of the predicted number of events for models in our simplified grid scan to the ATLAS 95\% C.L. event limit for the 1l+jets
channel, $R_{1l}$, as a function of the number of signal events for the ATLAS 20 fb$^{-1}$ $Z+$MET search. The color code corresponds to the value of the
$\chi_3^0-\chi_1^0$ mass splitting. The top-left, top-right, and bottom panels correspond to the parent squark cases of $\tilde Q_L$,
$\tilde u_R$, and $\tilde d_R$, respectively.}
\label{fig:1l}
\end{figure}
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{Nzjm_2lCMS_8tev20_dm_ql}
\includegraphics[width=0.45\textwidth]{Nzjm_2lCMS_8tev20_dm_ur}
\includegraphics[width=0.45\textwidth]{Nzjm_2lCMS_8tev20_dm_dr}
\caption{Ratio of the predicted number of events for models in our simplified grid scan to the CMS 95\% C.L. event limit for the $Z+$MET
channel, $R_{CMS}$, as a function of the number of signal events for the ATLAS 20 fb$^{-1}$ $Z+$MET search. The color code corresponds to the value of the
$\chi_3^0-\chi_1^0$ mass splitting. The top-left, top-right, and bottom panels correspond to the parent squark cases of $\tilde Q_L$,
$\tilde u_R$, and $\tilde d_R$, respectively.}
\label{fig:2lCMS}
\end{figure}
Figure \ref{fig:1l} displays our results for the case of the 1l+jets ATLAS search where
the y-axis now shows the event rate ratio $R_{1l}$. For all three squark parent cases we see that the models tend to mostly lie
reasonably far away from the exclusion boundary for this search, implying that it has little impact on
shaping the parameter region for successful models. In fact, we find that few models are excluded by the 1l+jets analysis after the other
null search results have been applied. Figure \ref{fig:2lCMS} shows the results for the case of the CMS $Z+$MET analysis,
expressed as the ratio $R_{CMS}$; clearly for all squark parents there is a rough linear correlation between the
value of $R_{CMS}$ and the number of predicted ATLAS 20 fb$^{-1}$ $Z$+MET signal events. From this one might expect that requiring
$R_{CMS} \leq 1$ cuts off the corresponding ATLAS signal. However, this region is
already restricted by the 0l+jets ATLAS search, with the result that the CMS $Z+$MET analysis has
only a small additional impact on our model selection beyond the effect of the 0l+jets ATLAS search.
In addition to the 0l,1l+jets and CMS $Z+$MET searches, other ATLAS searches which are less clearly targeted for these types of models can still have an important impact on the allowed parameter space, especially for the $\tilde u_R$ and $\tilde d_R$ grids, where the 0l search is less dominant. In particular, the ATLAS 3l gaugino search~\cite{ATLAS:2012uks}, 4l search~\cite{ATLAS:2012hmt}, and same-sign dilepton search~\cite{ATLAS:2012sna} all make unique contributions to the combined exclusion region. Since the 3l and 4l searches are targeted at electroweak production, it is unsurprising that they are particularly sensitive to the lower mass regions allowed for the right-handed squarks, particularly $\tilde d_R$.
It is worth a short discussion to compare the event selection between the ATLAS and CMS $Z+$MET analyses. While both searches select events with a leptonic Z, at least two jets, and missing energy, the 600 GeV $H_T$ cut of the ATLAS search is highly effective at reducing Drell-Yan background, leaving $t\bar{t}$ as the dominant background process. The CMS analysis considers multiple search regions with missing energy bins to gain increased sensitivity, but even in the highest bin, requiring MET $> 300$ GeV, Drell-Yan production is still the most significant background. For comparison, the ATLAS analysis imposes the tight $H_T$ cut stated above and simply cuts on missing energy, MET $> 225$ GeV. As a result of these cut choices, the overlap between the ATLAS and CMS $Z+$MET search regions is small \cite{haastalk}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{ql_n3bfs}
\includegraphics[width=0.45\textwidth]{ql_n3bfs_surviving_g12events}
\caption{Distributions of the branching fractions for the decay modes, as indicated, of the $\chi_3^0$ intermediate state for the full set of models from our $\tilde{Q}_L$ grid
scan (left panel) and for models predicting 12 or more events in the ATLAS 20 fb$^{-1}$ $Z+$MET search while remaining consistent with all simulated searches, including the ATLAS 0l,1l+jets and CMS $Z+$MET searches (right panel).}
\label{fig:bfs}
\end{figure}
As discussed above, our successful models kinematically allow the decay of the intermediate bino-like state into the lighter
Higgsinos by $W\, ,Z$ and Higgs boson emission. Apart from phase space considerations, these relative branching fractions are
controlled by the bino and Higgsino content of the gauginos.\footnote{We essentially treat the winos as being decoupled with a
correspondingly large value of $M_2$.} Since $\tan \beta$ is being held fixed in our grid scans, the bino and Higgsino content of
these states are only being regulated (at tree-level) by the values of $M_1$ and $\mu$. Clearly as the mass splitting between these states,
$\Delta m_{31}$, increases, the purity of each state increases. If the intermediate state were to be
{\it pure} bino, then its decay via either the $W$ or $Z$-boson would be forbidden (as these channels require both the initial and final
states to have a nonzero Higgsino content), while decays through the Higgs would remain allowed (as this is controlled
instead by the product of the bino and Higgsino content of both the initial and final states). Figure \ref{fig:bfs} shows the distributions
for the $\chi_3^0$ branching fractions for the $\tilde Q_L$ parent squark case for all the models in the initial grid,
as well as after applying constraints from the CMS $Z+$MET search and null results in other ATLAS channels and requiring the point to predict 12 or more events in the ATLAS 20 fb$^{-1}$ $Z+$MET search. Here we see several things: ($i$) the typical $W$-boson branching fraction is rather large, although models with the largest values for this branching fraction are unable to satisfy the constraints applied to the right panel.
($ii$) In both panels, the $Z$-boson branching fraction is more than twice as large for the decay into
$\tilde \chi_2^0$ than for decays to the LSP. The reverse is true for decays producing a Higgs boson. ($iii$) The $\chi_3^0$ decays mediated via the $Z$-boson and the Higgs, to either the LSP or to $\tilde \chi_2^0$, are seen
to have similar branching fractions. Clearly it is not advantageous to completely suppress the Higgs mode, which can only be accomplished by reducing $\Delta m_{31}$ to values below the Higgs mass. Interestingly, this scenario would then also
predict a signal in the $h+$MET channel. We obtain similar results for the other squark parent scenarios.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{ql_n3_n1_hist_Nzjmg5}
\includegraphics[width=0.45\textwidth]{ur_n3_n1_hist_Nzjmg5}
\includegraphics[width=0.45\textwidth]{dr_n3_n1_hist_Nzjmg5}
\caption{Mass distributions of the parent squark, $\chi_3^0$ and $\chi_1^0$ states for the models from our grid scan that agree with all null search results and yield at least
5 events in the ATLAS 20 fb$^{-1}$ $Z+$MET channel. The top-left, top-right, and bottom panels correspond to the parent squark cases of $\tilde Q_L$,
$\tilde u_R$, and $\tilde d_R$, respectively.}
\label{massdis_histos}
\end{figure}
Lastly, it is interesting to examine the mass distributions of the squarks, bino-like $\tilde \chi_3^0$ and Higgsino-like LSP states
in the models that successfully reproduce the ATLAS 20 fb$^{-1}$ $Z+$MET signal (here defined to be $N \geq 5$ $Z$+MET events). This is presented for the three parent squark types in
Fig.~\ref{massdis_histos}. Here we see that the overall spectra of these three sparticles gradually become lighter as we compare
the $\tilde Q_L$ parent to $\tilde u_R$ and then to $\tilde d_R$, reflecting the corresponding falling squark pair
production cross sections, with these distributions peaking at 750, 650, and 600 GeV, respectively. In all three cases the peak
of the $\chi_3^0$ distribution is near $\sim 350$ GeV resulting in a softening of the jets on average, for the $\tilde u_R/\tilde d_R$ cases compared
to the $\tilde Q_L$ parent squark, due to a
compression of the spectrum.
The peak of the LSP mass distribution lies roughly near $\sim 200$ GeV for the $\tilde Q_L$ parent
squark and near $\sim 150$ GeV for both the $\tilde u_R$ and $\tilde d_R$ cases, implying that $\Delta m_{31}$ prefers to lie
near $\sim 150-200$ GeV in all cases.
\section{Conclusions}
We have examined the $3\sigma$ $Z+$MET excess observed by the ATLAS collaboration in Run I of the LHC in the context of a
Supersymmetric framework. We have employed the freedom inherent in the pMSSM parameter space to explore whether Supersymmetric
models can be constructed that generate the observed excess, while simultaneously being consistent with the numerous other null SUSY searches
at the LHC. Using a large pMSSM model sample that we had previously generated, we found a handful of points that satisfied our critieria,
demonstrating the power of this approach.
These points shared charateristics for the sparticle spectrum that are crucial for describing the data, namely relatively light $1^{st}/2^{nd}$
generation squarks that decay into a bino-like neutralino, which in turn decays into a light Higgsino multiplet, {\it i.e.}, $\tilde q\rightarrow\tilde B\rightarrow\tilde h$.
Using these points as seeds, we performed three grid scans, corresponding to the possible types of the parent squark, $\tilde Q_L$,
$\tilde u_R$ and $\tilde d_R$. We scanned over the set of relevant parameters, $m_{\tilde q}$, $M_1$ and $\mu$,
and generated three sets of simplified pMSSM models within a limited kinematic range. All other strongly-interacting sparticles were
set to the same value as in the parent pMSSM model ($\sim 2$ TeV) and $A_t$ was varied to reproduce the observed Higgs boson mass.
In principle it is possible that light stop and sbottom squarks could also contribute to the signal, but we
limited our analysis here to the simplest scenario.
We then examined the properties of these simplified models in detail.
They predict a range of event rates, up to 21 events, for the ATLAS 20 fb$^{-1}$ $Z+$MET channel, in agreement with the ATLAS measurement. Several
hundred of our grid points were found to be consistent with the 95\% C.L. bounds from all simulated searches, including the ATLAS 0l,1l+jet and CMS $Z+$MET search
channels. The case of a left-handed doublet parent squark, $\tilde Q_L$, is found to yield the best fit to the data, with the other scenarios giving slightly smaller event rates. The sweet spot for the sparticle spectrum is found to have squark masses in the 500-750 GeV range, with
bino masses near 350 GeV with a mass splitting of $150-200$ GeV with the Higgsino LSP. The bino $\chi_3^0$ state has important decays
involving $W$ and Higgs bosons, as well as the $Z$-boson. The predicted event rates for these models are close to the 95\% C.L. limits
from the 0l+jets search and the CMS $Z+$MET analysis, but lie somewhat further from the 1l+jet search results.
In conclusion, we have constructed a simplified Supersymmetric model based on the pMSSM, with specific characteristic features
that successfully yields an
excess for the 20 fb$^{-1}$ $Z+$MET ATLAS analysis, while evading all other SUSY searches at the Run I LHC. The operations at the 13 TeV LHC
currently underway will be able to quickly discover, or exclude, this scenario. If the ATLAS $3\sigma$ excess is confirmed with the new data
set, it could very well be a signal for Supersymmetry.
\section{Acknowledgments}
The authors would like to thank Brian Petersen for communications.
This work was supported by the Department of Energy, Contracts DE-AC02-06CH11357, DE-AC02-76SF00515 and DE-FG02-12ER41811.
|
1,116,691,500,450 | arxiv | \section{Introduction}
\para{}
A topological superconductor (SC) has a superconducting gap in the bulk but protected Majorana fermions on the boundaries or in the cores of vortices in an externally applied magnetic field \cite{qi-rmp-2011,sato-rpp-2017}.
There has been considerable excitement about the search for topological superconductors in recent years.
\para{}
While signatures of topological superconductivity have been observed in one-dimensional chains with proximity-induced superconductivity~\cite{mourik-s-2012,nadj-perge-s-2014}, the experimental search for topological superconductivity in two dimensions is a promising~\cite{lian-pnas-2018} and relatively unexplored territory~\cite{he-science-2017,menard-nc-2017,palacio-morales-a-2018,yin-np-2015}
\para{}
The honeycomb lattice, with special features of Dirac dispersion and opposite Berry curvature around the two inequivalent valleys in the Brillouin zone, has emerged as a paradigmatic system for exploring topological states.
In this paper, we extend these investigations to include attractive interactions between electrons and outline a route to topological superconductivity, highlighting the crucial role played by the Berry phase and valley degree of freedom.
\para{}
Transition metal dichalcogenides (TMDs) with the valley degree of freedom are a viable family of materials in the search for topological superconductivity.
TMDs are layered materials containing a transition metal layer that form a triangular layer sandwiched between two chalcogen layers.
Based on density functional theory (DFT) calculations that indicate considerable $d$-$p$ mixing between the chalcogen and transition metal ions~\cite{fang-prb-2015},
we expect the effective Hamiltonian to reduce to a honeycomb model, similar to graphene, but with the richness of strong spin-orbit coupling and interactions between electrons.
\para{}
In TMD materials like MoS$_2$ and WS$_2$~\cite{xiao-prl-2012}, superconductivity is observed below ${\sim}$10\,K~\cite{ye-s-2012,lu-s-2015,lu-pnas-2018}, although these appear to be trivial SCs.
Other TMD materials like $1T'$-WTe$_2$ exhibit gapless edge states, suggesting that they are topological insulators~\cite{fei-np-2017}.
WTe$_2$ is reported to become superconducting under pressure~\cite{kang-nc-2015,pan-nc-2015} and gating~\cite{sajadi-s-2018,fatemi-s-2018}, though whether it is a topological superconductor is still unclear. Also, more recently, magic angle twisted bilayer graphene~\cite{cao-n-2018} has emerged as a model system for understanding superconductivity in the strongly correlated regime.
References~\onlinecite{po-prx-2018,yuan-prb-2018,kang-prx-2018} suggest that, despite the concentration of charge
density on a triangular lattice, the low-energy physics is that of a Dirac honeycomb system.
This is also true for the naturally occurring layered mineral jacutingaite, Pt$_2$HgSe$_3$ where the low-energy physics is dominated by the Hg atoms on a honeycomb lattice \cite{marrazzo-prl-2018}, resulting in a room-temperature quantum spin Hall insulator with a gap of 110\,meV \cite{kandra-a-2019}.
Preliminary theoretical investigations suggest the possibility of unconventional superconductivity when gated/doped to the van Hove singularities in the band structure~\cite{wu-a-2018}.
The question of intrinsic topological superconductivity in this system is as yet unexplored.
\para{}
Given these motivations, we examine the superconducting states that emerge in
the Kane-Mele model~\cite{kane-prl-2005} as a result of various interactions.
This is the archetypal model on a honeycomb lattice that exhibits
a transition from a topological to a trivial insulator as a function of spin-orbit coupling~(see Fig.~\ref{fig:lattice}).
What are the superconducting instabilities of this gapped Dirac system?
Under what conditions do we get topological superconducting states?
These are the primary questions we address in this paper.
\para{}
We use self-consistent Bogoliubov--de Gennes theory to map out the phase diagrams of the Kane-Mele model with three different types of interactions, and analyze the topological invariants associated with the resulting superconducting phases.
Throughout this paper, we will use the terms ``trivial,'' and ``topological'' to refer to zero and nonzero topological invariants of the corresponding symmetry class.
For the three types of interactions, we find the following:
\begin{enumerate}[label=(\roman*)]
\item We show that onsite attraction, irrespective of whether the parent insulator is topological or trivial, the resulting superconductor is non-topological~[see Fig.~\ref{fig:pd-os}].
\item For nearest-neighbor attraction, topological superconductivity can arise from both the trivial as well as the topological insulator, and is most prominent near the transition~[see Fig.~\ref{fig:pd-nn}].
\item With antiferromagnetic nearest-neighbor interaction, we find exotic singlet states with broken rotation, translation, and time-reversal symmetries; however, none of these states are topological (see Fig.~\ref{fig:pd-heis}).
\end{enumerate}
\para{}
Our most significant results on topological superconducting states pertain to
Fig.~\ref{fig:pd-nn}] where we
find that two of the four superconducting states are topological, a time-reversal-symmetric helical superconductor and a chiral superconductor with Chern number $\pm 1$ that breaks time-reversal.
These topological states involve pairing within the same Dirac cone, and are stabilized when the underlying band structure is close to the transition between the topological and the trivial insulating phases.
\para{}
The topological superconducting states we find are different from those discussed in the literature.
For example, unlike $^3$He-B, the helical superconductor we predict has a nonzero center-of-mass (c.m.) momentum due to the valley degree of freedom.
The chiral superconductor too is different from the proposed paired state for the spinless $\nu=\frac{5}{2}$ quantum Hall state with Chern number 1, or the $p\pm ip$ superconducting state in spinful Sr$_2$RuO$_4$ or in $^3$He-A that have a Chern number of $\pm$2.
The chiral SC we predict is composed of a condensate of equal-spin pairs with nonzero c.m. momentum, and another condensate of opposite-spin pairs with the c.m. momentum reversed.
\para{}
In the final section, we compare our results with previous theoretical works on superconductivity in TMDs, and also comment on the implications of our results for cold atom experiments.
\section{Kane-Mele Model with Interactions}
\label{sec:model}
\begin{figure}\centering
\subfigure[\label{fig:honeycomb}]{%
\includegraphics[width=1.25in]{fig/honeycomb-sqrt3xsqrt3.pdf}%
}\qquad%
\subfigure[\label{fig:brillouinzone}]{%
\includegraphics[width=1.25in]{fig/brillouinzone.pdf}%
}
\subfigure[\label{fig:dispersion-trivial}]{%
\frame{\includegraphics[height=0.82in]{fig/DispersionCut-Trivial.pdf}}%
}
\subfigure[\label{fig:dispersion-gapclosing}]{%
\frame{\includegraphics[height=0.82in]{fig/DispersionCut-GapClosing.pdf}}%
}
\subfigure[\label{fig:dispersion-topological}]{%
\frame{\includegraphics[height=0.82in]{fig/DispersionCut-Topological.pdf}}%
}
\includegraphics[height=0.82in]{fig/red_black_blue_colorbar.pdf}%
\caption{\label{fig:lattice}%
\subref{fig:honeycomb}
Honeycomb lattice on which the Hamiltonian in Eq.~\eqref{eq:kanemele} is defined.
The blue hexagon marks the $\sqrt{3}\times\sqrt{3}$ supercell used in our study, which allows pairing with nonzero center-of-mass (c.m.) crystal momentum $K$ and $K'$ of Cooper pairs in addition to $\Gamma$.
\subref{fig:brillouinzone}
Brillouin zone of the honeycomb lattice.
The inner blue hexagon represents the reduced Brillouin zone of the supercell;
both $K$ and $K'$ defined for the original Brillouin zone are folded to the $\Gamma$ point in the reduced Brillouin zone.
\subref{fig:dispersion-trivial}--\subref{fig:dispersion-topological}
Dispersions of the non-interacting Kane-Mele model defined in Eq.~\eqref{eq:kanemele}.
The solid (dashed) curves show the dispersion of electrons with spin up (down).
The parameter $x = 3\sqrt{3}\lambda_{\text{so}}/(m_{\text{AB}}+3\sqrt{3}\lambda_{\text{so}})$ that represents the relative strength of the Ising spin-orbit coupling is varied between \subref{fig:dispersion-trivial} $0 \le x<1/2$ in the trivial insulator phase, \subref{fig:dispersion-gapclosing} $x=\frac{1}{2}$ at the topological transition, and \subref{fig:dispersion-topological} $\frac{1}{2} < x \le 1$ in the topological insulator phase.
The color of the curves indicates the sign of the Berry curvature:
In each spin sector, the signs of the Berry curvature at $K$ and $K'$ are opposite in the trivial phase, and the same in the topological phase.
At the topological transition ($x=\frac{1}{2}$), there is a single Dirac cone in each spin sector in the corresponding valley.
}
\end{figure}
\begin{figure*}[htb!]
\centering
\subfigure[\label{fig:pd-os}]{\includegraphics[height=2.2in]{fig/pd-os.pdf}}
\qquad
\subfigure[\label{fig:pd-nn}]{\includegraphics[height=2.2in]{fig/pd-nn.pdf}}
\caption{\label{fig:pd}%
Phase diagrams of Kane-Mele model in Eq.~\eqref{eq:kanemele} as functions of the tuning parameter $x = 3\sqrt{3}\lambda_{\text{so}}/(m_{\text{AB}}+3\sqrt{3}\lambda_{\text{so}})$ which interpolates between the trivial and topological insulating band structures, with \subref{fig:pd-os} onsite attractive interaction $U$,
\subref{fig:pd-nn} nearest-neighbor attractive density-density interaction $V$.
Solid lines mark continuous (topological) phase transitions, and the dotted lines mark first order transitions.
\subref{fig:pd-os} With $U$, we find an $s$-wave pairing state that is topologically trivial.
\subref{fig:pd-nn} With $V$, we find more exotic pairing states, two of which are topological:
The topological helical triplet superconductor(SC) (in green) near $x = 1/2$ has equal-spin spin-triplet pairing ($\Delta_{\up\up}, \Delta_{\dn\dn} \neq 0$) and a $\calT$-invariant topological superconducting ground state with $\tilde{\nu}=1$.
The trivial $p$-Kekule triplet SC (in blue) near $x=1$ has spin-triplet pairing between opposite spins ($d^z \neq 0$), is $\calT$-invariant and is topologically trivial.
Both of these states have nonzero center-of-mass momentum pairs, with non-trivial real-space patterns in the pairing order parameters, shown in Fig.~\ref{fig:nn-pairing-pattern}.
The other two superconducting phases (shown in purple and in pink) have a mixture of both types of triplet pairing and are $\calT$-breaking.
The topological chiral triplet SC (in purple) is a topological state with Chern number $\tilde{\calC}=\pm 1$.
The trivial $\calT$-breaking triplet SC (in pink) on the other hand is topologically trivial with $\tilde{\calC}=0$.
}
\end{figure*}
\para{}
To study the pairing instability of a two-dimensional Dirac system across the topological phase transition between topological and trivial insulating phases, we take the Kane-Mele model defined on a honeycomb lattice [Fig.~\ref{fig:honeycomb}] as the underlying band structure~\cite{kane-prl-2005}:
\begin{align}
\label{eq:kanemele}
\calH_{\mathrm{KM}}
&=
- t
\sum_{\langle i, j \rangle}
\psi_{i}^{\dagger} \psi_{j}
- \mu \sum_{i} \psi_{i}^\dagger \psi_{i} \nonumber\\
&\quad
- i \lambda_{\text{so}}
\sum_{\llangle i, j \rrangle}
\nu_{ij} \psi_{i}^{\dagger} \sigma^{z} \psi_{j}
+ m_{\text{AB}} \sum_{i} \xi_{i} \psi_{i}^{\dagger} \psi_{i}
\end{align}
where $\psi_{i}^{\dagger} \equiv (c_{i\up}^{\dagger}, c_{i\dn}^{\dagger})$ is the electron creation operator at site $i$, and $\langle \cdot, \cdot \rangle$ and $\llangle \cdot, \cdot \rrangle$ represent nearest-neighbor and next-nearest-neighbor pairs of sites.
Here, $t$ is the nearest-neighbor hopping amplitude,
$\mu$ the chemical potential,
$\lambda_{\text{so}}$ the strength of Ising spin-orbit coupling, with $\nu_{ij} = \mathrm{sgn} ( \hat{z} \cdot (\bfv_1 \times \bfv_2) )$ where $\bfv_1$ and $\bfv_2$ are nearest-neighbor vectors that connect an electron hop from site $i$ to site $j$, and $m_{\text{AB}}$ the sublattice potential, with $\xi_{i}=1$ ($-1$) if the site $i$ belongs to the sublattice $A$ ($B$).
The sublattice potential breaks inversion symmetry and reduces the symmetry group of the Hamiltonian to $D_{3h}$.
For the sake of simplicity, we do not include the Rashba spin-orbit coupling in our analysis.
Our main results, nevertheless, remain the same for a small Rashba coupling, as we discuss later.
\para{Symmetry and topology}
The topology of a non-interacting (or mean-field) Hamiltonian is characterized by different topological indices depending on the dimensionality and the symmetry of the system~\cite{schnyder-prb-2008,kitaev-acp-2009}.
The band structure $H_{\mathrm{KM}}$ has time-reversal symmetry ($\calT$ symmetry) with $\calT^2=-1$, and thus belongs to the class AII~\cite{altland-prb-1997}.
In two dimensions, this class has two distinct topological phases characterized by a $\bbZ_2$ topological index $\nu=0$ or 1.
To take the system across the topological phase transition, we introduce a parameter $x$ between 0 and 1, which is related to the spin-orbit coupling and sublattice potential by $3 \sqrt{3} \lambda_{\text{so}} = E_g x$ and $m_{\text{AB}} = E_g (1-x)$.
$H_{\mathrm{KM}}$ has a topological (trivial) ground state for $x>\frac{1}{2}$ ($x<\frac{1}{2}$).
The low-energy degrees of freedom involve two massive spin-polarized Dirac cones at each ``valley'' centered at $K$ and $K'$ [Figs.~\ref{fig:brillouinzone}--\subref{fig:dispersion-topological}].
At $x=\frac{1}{2}$, the band structure is at a topological phase transition, with one of the Dirac cones in each valley being massless.
The mass of the other Dirac cones remains constant at $E_g$ throughout the transition for all values of $x$.
For the purpose of our calculation we have chosen $E_g = t/2$.
Adding a small Rashba spin-orbit coupling does not affect the topology of the system, as long as the bulk gap remains finite \cite{kane-prl-2005}.
\para{Interactions}
We study the pairing instability of the Hamiltonian $H_{\mathrm{KM}}$ with three different types of interactions:
(1) attractive onsite interaction $-U \sum_{i} n_{i\up} n_{i\dn}$,
(2) attractive nearest-neighbor density-density interaction $-V \sum_{\langle i j \rangle} n_{i} n_{j}$,
or (3) antiferromagnetic nearest-neighbor Heisenberg interaction $J \sum_{\langle i j \rangle} \boldsymbol{\sigma}_{i} \cdot \boldsymbol{\sigma}_{j}$,
where $n_{i\sigma} \equiv c_{i\sigma}^{\dagger} c_{i\sigma}$, $n_{i} \equiv \psi^{\dagger}_{i} \psi_{i} $, and ${\sigma}_{i}^{\mu} \equiv \psi_{i}^{\dagger} \sigma^{\mu} \psi_{i}$ for $\mu=x,y,z$.
In each case, we decouple the interaction in the pairing channel and find the Bogoliubov--de Gennes (BdG) ground states.
All the superconducting states that emerge self-consistently in this analysis are fully gapped.
This allows us to calculate the relevant topological index in each phase corresponding to its symmetry class (see Appendix~\ref{sec:computetopoindex}).
Once again, Rashba spin-orbit coupling does not qualitatively affect the results, as long as it is weak compared to the Bogoliubov quasiparticle gap.
\begin{table}[ht!]
\centering
\caption{Summary of the superconducting phases in Fig.~\ref{fig:pd-nn} found with attractive nearest-neighbor density-density interaction $V$.
$\Phi^{K}$ and $\Phi^{K'}$ are spatial form factors defined by
$\Phi^{\bfQ}_{ij} = e^{i \bfQ \cdot (\bfr_{i} + \bfr_{j})}$,
representing pairing of two electrons at $K$ and $K'$ valleys, respectively.
}
\label{tab:phases-nn}
\begin{ruledtabular}
\begin{tabular}{cllcl}
& Superconducting & Order parameter $\Delta$ & $\calT$-sym. & Topo. \\
& phase &
& & index
\\
\hline
\fcolorbox[rgb]{0,0,0}{0.79, 0.98, 0.74}{\rule{0ex}{0.5ex}\rule{0.5ex}{0ex}}
& Topo. helical triplet
& $\Delta_{\up\up} \sim \Phi^{K}$, $\Delta_{\dn\dn} \sim \Phi^{K'}$
& $\checkmark$
& $\tilde{\nu}=1$
\\
\fcolorbox[rgb]{0,0,0}{0.79, 0.91, 0.99}{\rule{0ex}{0.5ex}\rule{0.5ex}{0ex}}
& Triv. $p$-Kekule triplet
& $d^z \sim \Phi^{K} - \Phi^{K'}$
& $\checkmark$
& $\tilde{\nu} = 0$
\\
\fcolorbox[rgb]{0,0,0}{0.78, 0.58, 1.0}{\rule{0ex}{0.5ex}\rule{0.5ex}{0ex}}
& Topo. chiral triplet
& $\Delta_{\up\up} \sim \Phi^{K}$, $d^z \sim \Phi^{K'}$
& $\bigtimes$
& $\tilde{\calC} = \pm 1$
\\
&
& (or its $\calT$ partner)
&
\\
\fcolorbox[rgb]{0,0,0}{0.99, 0.71, 0.97}{\rule{0ex}{0.5ex}\rule{0.5ex}{0ex}}
& Triv. $\calT$-breaking
& $\Delta_{\up\up}, \Delta_{\dn\dn} \sim \Phi^{K}-\Phi^{K'}$,
& $\bigtimes$
& $\tilde{\calC} = 0 $
\\
& triplet
& $d^z \sim \Phi^{K} + \Phi^{K'}$
&
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[ht!]\centering%
\includegraphics[width=3in]{fig/DiracConePairing.pdf}
\caption{\label{fig:DiracConePairing}%
As described in Table \ref{tab:phases-nn}, the phase diagram for nearest neighbor attractive interaction $V$ is understood in terms of six order parameters, corresponding to triplet pairing between up-up, down-down, and up-down pairs of fermions on each valley. (See Appendix~\ref{sec:kspacedescription} for details.)
The real-space pattern $\Phi^K_{ij}$ corresponding to the condensate at the valley $K$ is shown in Fig.~\ref{fig:nn-pairing-pattern-planewave}.
The $p$-Kekule pair potential exhibits interference between the two condensates $\Phi^K_{ij}$ and $\Phi^{K'}_{ij}$.
}
\end{figure}
\para{Nonzero c.m. momentum pairs}
Since the low-energy electronic degrees of freedom lie at valleys near $K$ and $K'$ [see Figs.~\ref{fig:brillouinzone}--\subref{fig:dispersion-topological}], we also allow pairing of two electrons from the same valley.
To incorporate such pairing with Cooper pairs having nonzero center-of-mass momentum $2K\equiv K'$ or $2K' \equiv K$, we use a supercell with six sites [blue hexagon in Fig.~\ref{fig:honeycomb}], whose reduced Brillouin zone folds the $K$ and $K'$ to $\Gamma$ [blue hexagon in Fig.~\ref{fig:brillouinzone}].
This introduces 6 onsite pairing order parameters and 36 nearest-neighbor pairing order parameters.
We then minimize the ground-state energy within this exhaustive parameter space averaging over $24\times 24$ momentum grid.
Note that we are not imposing a particular structure of the pairing order parameter;
we are allowing the self-consistency loop to pick the lowest-energy configuration in the space of 42 complex pairing order parameters.
\section{Various Superconducting Phases and Their Topology}
\para{Onsite attraction $U$}
In the Kane-Mele model at $\mu=0$ with onsite attractive interaction $U$, we find three different phases as shown in Fig.~\ref{fig:pd-os}.
Away from $x=\frac{1}{2}$, the system is an insulator for weak interaction due to the nonzero band gap:
its topological property is completely determined by the underlying band structure parametrized by $x$.
For strong enough interaction, we find a continuous transition to a uniform $s$-wave spin-singlet superconducting phase.
Note that in the presence of spin-orbit coupling (at $x \neq 0$), spin-singlet and spin-triplet are not symmetry-distinct, and pair amplitudes $\langle c_{i \sigma} c_{j \sigma'} \rangle$ in both spin channels can be nonzero in general.
The onsite interaction, however, allows pair potential $\Delta$ only in the spin-singlet channel.
Throughout this paper, we use the terms spin-singlet and spin-triplet pairings to refer to the spin component of $\Delta$ and not necessarily the pair amplitude.
\para{}
Since the pairing leaves the $\calT$ symmetry intact, the Bogoliubov--de Gennes Hamiltonian is in the class DIII, with a $\bbZ_2$ topological index $\tilde{\nu}=0$ or 1, defined analogously to the $\bbZ_2$ topological index $\nu$ of class AII topological insulator, but in terms of the Bogoliubov quasiparticles in Nambu space.
The superconducting state that arises from either the topological insulator or the trivial insulator is a trivial superconductor with $\tilde{\nu}=0$.
This can be understood in the following way:
The insulating phase can be seen as a $\calT$-invariant superconductor with zero pair potential.
Such a ``superconducting state'' is trivial since $\tilde{\nu}=2\nu=0 \text{ (mod 2)}$ independent of $\nu$; (the factor of 2 is due to the particle-hole redundancy of Nambu spinors).
At a continuous transition to a superconducting state, $\tilde{\nu}$ cannot change since the single-particle gap does not close.
Thus, it is natural that the superconductor that emerges from a continuous transition from a trivial or topological time-reversal-invariant insulator, is topologically trivial.
Conversely, a topological superconductor must be separated from a time-reversal-invariant insulator either by a discontinuous transition, or an intervening state where the single-particle gap closes.
\begin{figure}\centering%
\subfigure[\label{fig:nn-pairing-pattern-planewave}%
]{\includegraphics[width=1.75in]{fig/pairingpattern/pairing-planewave.pdf}}%
\subfigure[\label{fig:nn-pairing-pattern-pkekule}%
]{\includegraphics[width=1.75in]{fig/pairingpattern/pairing-pkekule.pdf}}%
\caption{\label{fig:nn-pairing-pattern}%
Real-space patterns of the pairing order parameters that we find with nearest-neighbor attractive density-density interaction.
A bond between sites $i$ and $j$ represents pair potential
\subref{fig:nn-pairing-pattern-planewave}
$\Delta_{i\up; j\up}$ of the ``topological helical triplet SC,'' which is $\sim \Phi^K_{ij}$, and
\subref{fig:nn-pairing-pattern-pkekule}
$d^{z}_{ij}$ of the ``$p$-Kekule SC,'' which is $\sim \Phi^{K}_{ij} - \Phi^{K'}_{ij}$.
The color of a bond marks the phase of the order parameter, which is also indicated $1$, $-1$, $\omega$, and $\omega^2$ on the bonds ($\omega \equiv e^{2\pi i / 3}$).
Since both $\Delta_{i\up; j\up}$ and $d^{z}_{ij}$ are antisymmetric under $i \leftrightarrow j$,
we choose a convention for the phases:
$i$ is always chosen from the A sublattice, and $j$ from the B sublattice.
}
\end{figure}
\para{Nearest-neighbor density-density attraction $V$}
With attractive nearest-neighbor density-density interaction $V$, we find a much richer phase diagram shown in Fig.~\ref{fig:pd-nn}.
(We have implicitly assumed the presence of long-range Coulomb repulsion to prevent phase separation at stronger interaction.)
Unlike $U$ which only allows spin-singlet pairing, $V$ also allows spin-triplet pairing channels.
The pair potential $\Delta_{i\sigma;j\sigma'}$ between electrons at sites $i$ and $j$ with spins $\sigma$ and $\sigma'$ can thus be decomposed into spin-singlet and three spin-triplet channels as
\begin{align}
\Delta_{i\sigma; j\sigma'}
&=
\left[
( \psi_{ij} \sigma^0 + \bfd_{ij} \cdot \boldsymbol{\sigma} ) i \sigma^y
\right]_{\sigma\sigma'},
\end{align}
where $\sigma^\mu$ for $\mu=0,x,y,z$ are the identity and the Pauli matrices in spin space.
Since, however, the Hamiltonian $H_{\mathrm{KM}}$ only has a U(1) spin rotation symmetry related to the $S_z$ conservation rather than the full SU(2) spin rotation symmetry, it is more convenient to decompose the pairing channels into $\psi$ (Cooper pairs with spin $S=0$), $\Delta_{\up\up}$ ($S=1$, $S_z=1$), $d^z$ ($S=1$, $S_z=0$), and $\Delta_{\dn\dn}$ ($S=1$, $S_z=-1$).
We find four distinct superconducting phases, all of which have $\Delta$ purely in the spin-triplet channel (with $\psi_{ij} = 0$).
These phases and their order parameters are summarized in Table~\ref{tab:phases-nn},
and can be understood in terms of spin and valley degrees of freedom, as shown in Fig.~\ref{fig:DiracConePairing}.
(Appendix~\ref{sec:symop} discusses how these order parameters transform under symmetry operations.)
\para{Topological helical SC}
Around $x=\frac{1}{2}$ at weaker interaction strength, we find a helical spin-triplet superconductor, which is $\calT$ invariant and characterized by a non-trivial topological $\bbZ_2$ index $\tilde{\nu}=1$ [green region in Fig.~\ref{fig:pd-nn}].
The pairing in this state is in the equal-spin channel ($\Delta_{\up\up}, \Delta_{\dn\dn} \neq 0$), with nonzero momentum Cooper pairs, as indicated by the real-space pattern of $\Delta_{i\up;j\up}$ shown in Fig.~\ref{fig:nn-pairing-pattern-planewave}, which goes as $\Delta_{i\up;j\up} \sim \Phi^{K}$, where $\Phi^{\bfQ}_{ij} \equiv e^{i \bfQ \cdot (\bfr_{i} + \bfr_{j})}$, for $i$ in sublattice A and $j$ in sublattice B.
$\Phi^{\bfQ}$ represents pairing with center-of-mass momentum $2\bfQ$.
The magnitude of the pair potential is uniform across all unit cells and only the phase modulates.
\para{}
This $\calT$-invariant superconducting state, whose non-trivial topology is characterized by the $\bbZ_2$ topological index $\tilde{\nu}=1$, can be understood in terms of the Dirac dispersions at each valley.
When $x \approx \frac{1}{2}$, the low energy electronic degrees of freedom are spin-valley locked [see Fig.~\ref{fig:dispersion-gapclosing}].
The order parameters $\Delta_{i\up;j\up} \sim \Phi^{K}_{ij}$ and $\Delta_{i\dn;j\dn} \sim \Phi^{K'}_{ij}$, therefore represent pairing between two electrons of the same spin from the same valley, which can be written in momentum space as
\begin{align}
\sum_{\bfq}
\Delta_{K+\bfq} c_{K + \bfq,\up}^{\dagger} c_{K - \bfq, \up}^{\dagger} +
\Delta_{K'+\bfq} c_{K' + \bfq,\dn}^{\dagger} c_{K' - \bfq, \dn}^{\dagger}
+ \mathrm{H.c.}
\end{align}
For small $\bfq$, $\Delta_{K+\bfq} \approx \Delta_{K} + O(q^2)$ with $\Delta_{K} \neq 0$.
The nonzero momentum pair potential $\Delta_{K+\bfq}$ thus plays the role of ``uniform $s$-wave'' gap within the Dirac cone at the $K$ valley (and similarly $\Delta_{K'+\bfq}$ for the $K'$ valley), which effectively becomes $p_x \pm i p_y$ pairing in the band basis \cite{sato-plb-2003,fu-prb-2006}.
This results in a nonzero Chern number $\tilde{\calC}=\pm1$ in each spin sector, leading to a non-trivial $\bbZ_2$ index $\tilde{\nu}=1$.
\para{}
As we have argued previously for the onsite attraction, a transition from an insulator to a topological superconductor must either involve an intermediate trivial superconducting phase if it is continuous, or be first order.
Within our exploration of the phase diagram, we have not found any intermediate phase between the insulating phases, both trivial and topological, and the topological helical superconducting phase.
Is the transition first order, or have we simply missed the intermediate phase?
In Appendix~\ref{sec:firstorder} we present a more careful study of the nature of this transition, where we identify a jump in the order parameter, a clear sign of a first-order transition.
\para{$p$-Kekule SC}
At $x=1$ and nearby where the underlying band structure is in the topological insulator phase, we find a $\calT$-invariant triplet SC which is topologically trivial ($\tilde{\nu}=0$) [blue region in Fig.~\ref{fig:pd-nn}].
The pairing in this state is in the opposite-spin spin-triplet channel ($d^z \neq 0$), and also has nonzero momentum Cooper pairs, forming the ``$p$-Kekule'' pattern in real space [see Fig.~\ref{fig:nn-pairing-pattern-pkekule}], which was originally discussed in the context of graphene~\cite{roy-prb-2010}.
This phase was previously found by \textcite{tsuchiya-prb-2016} who studied the same Hamiltonian ($H_{\text{KM}}$ with $V$) in the $x=1$ limit.
\para{Topological chiral SC}
In a thin region between the topological helical SC and the $p$-Kekule SC, we also find a $\calT$-breaking topological triplet SC with nonzero Chern number $\tilde{\calC} = \pm 1$ [purple region in Fig.~\ref{fig:pd-nn}].
We refer to this state as topological chiral SC, following Ref.~\onlinecite{qi-prl-2009}.
In this state, one of the valleys develops equal-spin pairing gap within the same cone, while the other valley develops an opposite-spin spin-triplet pairing gap across the two Dirac cones in the same valley.
This results in a nonzero Chern number with unequal contribution from the two valleys.
\para{Trivial $\calT$-breaking SC}
At $x \approx 0$ and at larger interaction strength,
the system favors a pairing state which is $\calT$ breaking with a mixture of equal-spin and opposite-spin pairing channels in both valleys [pink region in Fig.~\ref{fig:pd-nn}].
This is distinct from the chiral SC in that it is topologically trivial ($\tilde{\calC} = 0$).
(See Appendix~\ref{sec:tbreakingphase} for discussions on the structure of the order parameter in this phase.)
\begin{figure*}
\centering
\subfigure[\label{fig:pd-os-mu}]{\includegraphics[height=2.2in]{fig/pd-os-mu.pdf}}
\qquad
\subfigure[\label{fig:pd-nn-mu}]{\includegraphics[height=2.2in]{fig/pd-nn-mu.pdf}}
\\
\subfigure[\label{fig:pd-mu-scan}]{\quad\includegraphics[height=0.90in]{fig/DispersionCut-Scan.pdf}}
\caption{%
\label{fig:pd-mu}%
\subref{fig:pd-os-mu}, \subref{fig:pd-nn-mu}
Phase diagrams at chemical potential $\mu=t/4=E_g/2$ away from half-filling, with
\subref{fig:pd-os-mu} attractive onsite interaction $U$ and
\subref{fig:pd-nn-mu} attractive nearest-neighbor density-density interaction $V$.
\subref{fig:pd-mu-scan} Dispersions of the non-interacting band structure at different values of $x$, with the chemical potential $\mu$ marked by the horizontal dashed lines.
Within the range $\frac{1}{4} < x < \frac{3}{4}$, the normal-state band structure contains a non-spin-degenerate Fermi surface in each valley.
With $U$, we find $s$-wave superconducting phase as in Fig.~\ref{fig:pd-os}.
When there are Fermi surfaces ($\frac{1}{4} < x < \frac{3}{4}$), pairing amplitude should develop with infinitesimal $U$.
With $V$, we find similar phases as to Fig.~\ref{fig:pd-nn}, in addition to the ``metal'' phase near $x=\frac{1}{2}$.
The ``metal'' phase is defined to be regions with a very small pair amplitude ($\langle c_{i\sigma} c_{j\sigma'} \rangle <10^{-6}$), which is numerically difficult to distinguish from zero.
Unlike the trivial $s$-wave superconductivity, the nonzero center-of-mass momentum pairing is not necessarily an infinitesimal instability even in the presence of Fermi surfaces, due to their trigonal warping.
}%
\end{figure*}
\begin{figure}
\centering%
\includegraphics[width=3in]{fig/VcVsMu.pdf}
\caption{\label{fig:doping}%
The critical interaction strength for the transition to the topological helical SC is lowered by increasing $\mu$.
We show a doping-driven transition at $x=0.6$ on a $90\times90$ lattice with a temperature of $T=t/100$.
}
\end{figure}
\para{Finite doping $\mu\neq 0$}
So far, we have considered the band structure at half filling with $\mu=0$, and found topological superconducting phases with $V$.
Do these topological phases exist even when the underlying band structure is metallic?
Figures~\ref{fig:pd-mu} and Fig.~\ref{fig:doping} summarize the phase diagrams at nonzero chemical potential $\mu=t/4$.
Note that $E_g=t/2$, and therefore the band structure is metallic with a single non-spin-degenerate Fermi surface in each valley within the range $\frac{1}{4} < x < \frac{3}{4}$ [see Fig.~\ref{fig:pd-mu-scan}].
As shown in Fig.~\ref{fig:pd-mu}, the $\mu \neq 0$ phase diagrams contain the same superconducting phases as the $\mu = 0$ ones, in both cases of $U$ and of $V$.
The topological indices of these phases remain identical to the $\mu=0$ counterparts.
Importantly, we find that the topological helical superconductor that we find with $V$ is accessible at lower interaction strength with increasing $\mu$, as shown in Fig.~\ref{fig:doping}.
\para{}
Within the range $\frac{1}{4} < x < \frac{3}{4}$, where the normal state band structure contains Fermi surfaces, the $s$-wave superconductivity with $U$ becomes an infinitesimal instability.
For the superconducting phases that we find with $V$, all of which have spatially modulating pair potential, the electrons that form a Cooper pair are not time-reversal partners:
They reside at momenta opposite of $K$ or $K'$ (e.g. $c_{K+\bfq,\sigma}$ and $c_{K-\bfq,\sigma'}$).
Because of the trigonal warping of the Fermi surfaces, these two electrons cannot both be at the Fermi level, except on a finite number of $k$-points.
Therefore, such nonzero momentum pairings are no longer infinitesimal instabilities, even in the presence of Fermi surfaces, and requires finite interaction strength.
Following this argument, we mark the region near $x=\frac{1}{2}$ in Fig.~\ref{fig:pd-nn-mu} with very small pair potential (numerically indistinguishable from zero) as ``metal.''
The warping is minimal near the metal-insulator transition in the underlying band structure, but in spite of the finite density of states in this limit, intravalley pairing is still not an infinitesimal instability because the low energy fermions exactly at $K$ and $K'$ are sublattice polarized, and the nearest-neighbor interaction pairs fermions from opposite sublattices.
Nevertheless, this does not rule out the possibility that the underlying metallic state is unstable to other pairing channels, such as spin-singlet extended $s$-wave.
\begin{figure}[t]\centering
\subfigure[\label{fig:pd-haldane}]{\includegraphics[height=1.6in]{fig/pd-haldane.pdf}}
\subfigure[\label{fig:nn-pairing-pattern-stripe}]{%
\parbox[b][1.63in][t]{1.48in}{
\includegraphics[height=1.45in]{fig/pairingpattern/pairing-stripe.pdf}%
}%
}%
\caption{
\subref{fig:pd-haldane}
Phase diagram of Haldane model with nearest-neighbor attractive interaction $V$.
\subref{fig:nn-pairing-pattern-stripe}
The real-space pattern of the pairing gap $\Delta_{ij}$ of the ``stripe SC'' phase.
For the same reason as in Fig.~\ref{fig:nn-pairing-pattern}, $i$ is always chosen from the A sublattice and $j$ from the B sublattice.
}
\end{figure}
\para{Haldane model}
A natural corollary of the topological helical SC is that if we were to consider only one spin species, as in the Haldane model~\cite{haldane-prl-1988}, we expect a chiral SC near the topological transition in the band structure at $\mu=0$.
This turns out to be true:
By solving the self-consistent Bogoliubov--de Gennes equation of the following Hamiltonian,
\begin{align}
H_{\mathrm{Haldane}-V}
&=
- t
\sum_{\langle i, j \rangle}
c_{i}^{\dagger} c_{j}
- i \lambda
\sum_{\llangle i, j \rrangle}
\nu_{ij} c_{i}^{\dagger} c_{j} \nonumber\\
&\quad
+ m_{\mathrm{AB}} \sum_{i} \xi_{i} c_{i}^{\dagger} c_{i}
- V \sum_{\langle i, j \rangle} n_{i} n_{j}
\end{align}
as a function of $V$ and $x$ defined analogously to that of the Kane-Mele model above, we get a phase diagram shown in Fig.~\ref{fig:pd-haldane}.
For smaller values of $V$ we find the topological ``plane-wave SC,'' whose $\Delta_{ij}$ is equivalent to the $\Delta_{i\up;j\up}$ of the helical SC in Fig.~\ref{fig:pd-nn} and thus has Chern number $\tilde{\calC}=1$.
The chirality is determined by the underlying band structure, since the time-reversal symmetry is explicitly broken at the band-structure level, even without the interaction.
Due to the reduced degrees of freedom and thus less number of competing orders, the topological plane-wave SC phase expands and spans the whole range of $x$.
\para{}
At stronger $V$, we find two more superconducting phases, which we refer to as the ``$p$-Kekule (II) SC'' and ``stripe SC,'' both of which have zero Chern number.
Note that the ``$p$-Kekule (II) SC'' phase in the Haldane model is different from the
$p$-Kekule triplet SC phase of the Kane-Mele model:
the spatial structure of $\Delta_{ij}$ of ``$p$-Kekule (II) SC'' is identical to that of $d_{ij}^{z}$ of ``$p$-Kekule SC.''
However, while $p$-Kekule triplet SC pairs two electrons from different Dirac cones in the same valley,
``$p$-Kekule (II) SC'' pairs two electrons from the same Dirac cone in the same valley, due to the lack of the other Dirac cone.
The ``stripe SC,'' whose spatial structure of this phase is shown in Fig.~\ref{fig:nn-pairing-pattern-stripe}, breaks the $C_3$ rotation symmetry, but preserves the original translation symmetry of the lattice.
This state pairs electrons from the opposite valleys.
\para{}
The Haldane model has been experimentally realized with ultracold atoms~\cite{jotzu-n-2014} and there are proposals to engineer near-neighbor interactions~\cite{anisimovas-pra-2016}.
Based on our calculation, we predict that the resulting superconductivity with attractive interactions should be topological with a Chern number of $\tilde{\calC}=\pm1$.
\begin{figure}
\centering
\includegraphics[height=2.2in]{fig/pd-heis.pdf}
\caption{\label{fig:pd-heis}%
Phase diagrams of Kane-Mele model in Eq.~\eqref{eq:kanemele} as functions $x$ with nearest-neighbor antiferromagnetic Heisenberg interaction $J$.
We find two distinct topologically trivial singlet pairing states.
Near $x=0$ we find a topologically trivial nematic singlet SC that is $\calT$ invariant, and breaks the $C_3$ rotation symmetry of the system.
Near $x=1$ we find a topologically trivial chiral singlet SC, which is $\calT$ breaking with pairing in the spin-singlet channel.
}
\end{figure}
\begin{figure}
\centering
\subfigure[\label{fig:heisenberg-pairing-pattern-nematicvbs}%
]{\includegraphics[width=1.75in]{fig/pairingpattern/pairing-nematicvbs.pdf}}%
\subfigure[\label{fig:heisenberg-pairing-pattern-chiralvbs}%
]{\includegraphics[width=1.75in]{fig/pairingpattern/pairing-chiralvbs.pdf}}%
\caption{Real-space patterns of the spin-singlet pair potential $\psi_{ij}$ of
\subref{fig:heisenberg-pairing-pattern-nematicvbs} the nematic singlet SC and
\subref{fig:heisenberg-pairing-pattern-chiralvbs} the chiral singlet SC phases that we find with nearest-neighbor antiferromagnetic Heisenberg exchange $J$.
}
\label{fig:heisenberg-pairing-pattern}
\end{figure}
\para{Antiferromagnetic Heisenberg exchange $J$}
With antiferromagnetic Heisenberg exchange $J$ between nearest-neighboring sites at $\mu=0$, we find two distinct superconducting states as shown in Fig.~\ref{fig:pd-heis}.
Both of these states are topologically trivial, but have exotic characteristics:
The pairing state for $x \lesssim \frac{1}{2}$ is a nematic singlet SC, which is $\calT$ invariant but breaks rotation symmetry.
The pairing state for $x \gtrsim \frac{1}{2}$, on the other hand, is a chiral singlet SC, which is in the spin-singlet channel, yet is $\calT$ breaking and also breaks translation symmetry.
The real-space patterns of the singlet order parameter $\psi_{ij}$ in these phases are shown in Fig.~\ref{fig:heisenberg-pairing-pattern}.
\section{Discussion and Outlook}
\para{Summary}
To summarize, we have derived the phase diagram of the Kane-Mele model across its trivial-insulator-to-topological-insulator transition, with various interactions using the Bogoliubov--de Gennes framework.
With attractive onsite interaction $U$, we find trivial $s$-wave superconductivity as expected.
With nearest-neighbor interactions, both the attractive density-density interaction $V$, and the antiferromagnetic Heisenberg exchange $J$, we find exotic superconducting phases with finite Cooper-pair momentum.
Especially with $V$, we find two distinct topological superconducting phases, one $\calT$ invariant and one $\calT$ breaking, near the trivial-insulator-to-topological-insulator transition, where one pair of the Dirac cones become gapless.
\para{New route to topological superconductivity}
While the models we have solved are specific, the broad lessons we have learned are applicable to a more general class of phenomena.
The central thrust of our work is to understand the conditions under which we get topological superconductivity in a Dirac system.
Through our study of the Kane-Mele model, we have identified two crucial ingredients for obtaining a topological superconductor.
First, there needs to be uniform pairing within a Dirac cone~\cite{sato-plb-2003,fu-prl-2008}.
Second, such pairing must manifest on a single time-reversed pair of non-degenerate Dirac cones for $\calT$-invariant helical SC.
This corresponds to ``topological helical triplet SC'' in Fig.~\ref{fig:pd-nn} that is characterized by a $\bbZ_2$ topological index $\tilde{\nu}=1$.
If the intra-cone pairing is nonzero only on one Dirac cone, we have a chiral superconductor characterized by a nonzero Chern number $\tilde{\calC}$.
This corresponds to the purple region in Fig.~\ref{fig:pd-nn}, which is $\calT$ breaking.
\para{}
A single time-reversed pair of spin-polarized Dirac cones appears naturally at the topological transition of the Kane-Mele model at $x=\frac{1}{2}$.
Pairing internal to each of these Dirac cones is necessarily between equal-spin electrons.
It is only with nearest-neighbor density-density attraction that the equal-spin pairing channel is allowed.
Both onsite attraction and antiferromagnetic Heisenberg exchange enable pairing in the singlet channel, we therefore find no topological superconductivity with these interactions.
\para{}
Thus far, the search for topological superconductivity has been driven largely by one theme:
break $\calT$ and get effectively spinless fermions, and then induce (effective) $p$-wave pairing between them.
This originates from work by Kitaev in 1D~\cite{kitaev-pu-2001} and $\calT$ breaking is central to this quest.
One of the strengths of the work presented here is a route to 2D topological superconductivity in presence of $\calT$ invariance and an explicit demonstration in the context of the Kane-Mele model.
\para{BCS-BEC crossover and connection with topology}
The intuition from the $p+ip$ superconductors is that the strong coupling BEC regime is trivial whereas topological superconductivity only arises in the weak coupling BCS regime. We note, based on our studies, that
such a demarcation does not apply to the honeycomb Dirac system.
The most obvious difference is that in our model, the Fermi energy is in the middle of the band gap so that we have both electron and hole bands, each with nontrivial Berry phase.
Unlike the $p+ip$ superconductors where the sense of ``winding'' is related to the winding of the order parameter along the Fermi surface, in a Dirac system the winding is related to the Berry phase of the underlying band structure.
This makes our normal state qualitatively different from a trivial vacuum.
Therefore, upon including interaction in an otherwise insulating state, the system can enter topological superconducting state even in the BEC regime.
\para{Comparison with previous theoretical studies}
In previous theoretical studies,
pairing in the TMD materials has hitherto been studied without incorporating the full effect of the honeycomb lattice~\cite{yuan-prl-2014,hsu-nc-2017}, ignoring the Dirac physics and the $\pi$ Berry phase around the valley.
\textcite{yuan-prl-2014} considered onsite and nearest-neighbor attraction on a \emph{triangular} lattice, and found $\calT$-breaking topological superconductivity only in the presence of Rashba spin-orbit coupling.
We note that the phases discussed there are, in principle, included in our mean-field study and turn out to be energetically less favored than the finite momentum paired states that we encounter.
\textcite{hsu-nc-2017} used renormalization group analysis to explore the leading instability of one spin-polarized circular Fermi surface at $K$ and $K'$ with onsite repulsive interactions.
They found several degenerate paired states: an interpocket chiral SC, an intrapocket chiral SC and an intrapocket helical SC similar to our topological helical triplet SC phase.
\para{Experimental probes}
We expect that the theoretical phase diagrams and general principles for topological superconductivity that we have unearthed from simple models are relevant for the low-energy physics of monolayer TMD materials, such as $\mathrm{MoS_2}$, $\mathrm{WS_2}$, $\mathrm{WTe_2}$.
\para{}
Recent experiments on monolayer WTe$_2$ \cite{sajadi-s-2018,fatemi-s-2018} have observed gating-driven transition from quantum spin Hall insulator to superconductor.
The type of superconductivity induced in this system, and its topological properties, are not yet known.
If the superconductivity is driven by electron-phonon interaction, where the attractive onsite $U$ is the most relevant effective interaction, we can place the system in Fig.~\ref{fig:pd-os-mu} across the topological insulator and trivial $s$-wave superconductor phases.
If, on the other hand, the superconductivity is driven by electron-electron interaction, where the onsite pairing is suppressed by strong short-range repulsion, phase diagrams with $V$ [Fig.~\ref{fig:pd-nn}] or with $J$ (Fig.~\ref{fig:pd-heis}) may be relevant to superconductivity in these systems.
\para{}
The phases we have described could be experimentally identified by establishing signatures of spin-triplet pairing, of spatially modulated superconductivity, and of the Majorana edge modes characteristic of the topological superconductors.
The spin susceptibility measured using Knight shift and relaxation rates may be used to identify triplet pairing and discern whether it is equal-spin or opposite-spin pairing.
The $p$-Kekule SC with $S_z=0$ would exhibit a suppression of spin susceptibility to zero, with out-of-plane fields, unlike the other phases.
The equal-spin paired helical superconductor would have spin-polarized Majorana modes counterpropagating along the edges of the sample, which would contribute to a finite quantized thermal Hall conductivity in the superconducting state.
Time-reversal breaking in the chiral superconductor states could be identified by polar Kerr effect~\cite{kapitulnik-njp-2009} or muon spin rotation spectroscopy.
\para{}
Detecting the spatial modulation of the phase in the helical superconductor is possible using the dc-SQUID setup outlined in Ref.~\onlinecite{hsu-nc-2017}.
In addition, in realistic samples we expect finite Rashba spin-orbit coupling to result in a singlet order parameter derived from both the up-spin condensate with momentum $2K$ and the down-spin condensate with momentum $2K'$.
The resulting pair density wave in the singlet channel would be observable by scanning Josephson tunneling microscopy (SJTM)~\cite{hamidian-n-2016} with a superconducting tip with singlet pairs.
\para{}
The pair density wave nature of the $p$-Kekule SC would be expected to show up both in STM and in SJTM experiments with a tip exfoliated from the substrate.
However, as we show in Appendix~\ref{sec:sVspKekule}, this might require going to extremely low temperatures to prevent tunneling between the three equivalent $p$-Kekule configurations related to each other by a lattice translation.
\para{}
Spatial modulation of the order parameter is a direct consequence of intravalley pairing.
In the TMDs, it is now well established that circularly polarized light can be used to selectively excite fermions from one valley.
An observable consequence of intravalley pairing would then be a suppression of the cooperon energy observed with circularly polarized light as we approach the superconducting transition by lowering temperature.
\begin{acknowledgments}
We thank P. Coleman and Y.-T. Hsu for useful discussions.
K. L. and N. T. acknowledge support from the National Science Foundation Grant No. DMR-1629382.
T. H. and M. R. are supported by National Science Foundation Grant No. DMR-1410364.
\end{acknowledgments}
|
1,116,691,500,451 | arxiv | \section{Introduction}
Given a multigraph $G=(V,E)$, the {\em edge-coloring problem} (ECP) is to color the edges of $G$ with the minimum
number of colors so that no two adjacent edges have the same color. Its optimal value is called the {\em chromatic
index} of $G$, denoted by $\chi'(G)$. In addition to its great theoretical interest, ECP arises in a variety
of applications, so it has attracted tremendous research efforts in several fields, such as combinatorial
optimization, theoretical computer science, and graph theory. Holyer \cite{H} proved that it is $NP$-hard in general
to determine $\chi'(G)$, even when restricted to a simple cubic graph, so there is no efficient algorithm for
solving ECP exactly unless $NP=P$, and hence the focus of extensive research has been on near-optimal solutions to
ECP or good estimates of $\chi'(G)$.
Let $\Delta(G)$ be the maximum degree of $G$. Clearly, $\chi'(G)\ge \Delta(G)$. There are two classical upper bounds
on the chromatic index: the first of these, $\chi'(G) \le \lfloor \frac{3 \Delta(G)}{2} \rfloor$, was established
by Shannon \cite{Sh} in 1949, and the second, $\chi'(G) \le \Delta(G) +\mu(G)$, where $\mu(G)$ is the maximum multiplicity
of edges in $G$, was proved independently by Vizing \cite{V} and Gupta \cite{G} in the 1960s. This second result is widely known as
Vizing's theorem, which is particularly appealing when applied to a simple graph $G$, because it reveals that $\chi'(G)$
can take only two possible values: $\Delta(G)$ and $\Delta(G)+1$. Nevertheless, in the presence of multiple edges,
the gap between $\chi'(G)$ and these three bounds can be arbitrarily large. Therefore it is desirable
to find some other graph theoretic parameters connected to the chromatic index.
Observe that each color class in an edge-coloring of $G$ is a matching, so it contains at most $(|U|-1)/2$ edges
in $E(U)$ for any $U \subseteq V$ with $|U|$ odd, where $E(U)$ is the set of all edges of $G$ with both ends in $U$.
Hence {\em the density of $G$}, defined by
\[\Gamma(G)=\max \Big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm
{\rm and \hskip 2mm odd} \Big\},\]
provides another lower bound for $\chi'(G)$. It follows that $\chi'(G) \ge \max\{\Delta(G), \, \Gamma(G)\}$.
To facilitate better understanding of the parameter $\max\{\Delta(G), \, \Gamma(G)\}$, let $A$ be the edge-matching
incidence matrix of $G$. Then ECP can be naturally formulated as an integer program, whose linear programming (LP)
relaxation is exactly as given below:
\begin{center}
\begin{tabular}{ll}
\hbox{Minimize} \ \ \ & $ {\bm 1}^T {\bm x}$ \\
\hbox{\hskip 0.2mm subject to} & $ A{\bm x} = {\bm 1}$ \\
& \hskip 3mm ${\bm x} \ge {\bm 0}.$
\end{tabular}
\end{center}
This linear program is called the {\em fractional edge-coloring problem} (FECP), and its optimal
value is called the {\em fractional chromatic index} of $G$, denoted by $\chi^*(G)$. As shown by Seymour \cite{Se}
using Edmonds' matching polytope theorem \cite{e65}, it is always true that $\chi^*(G)=\max\{\Delta(G), \,\Gamma(G) \}$.
Thus the preceding inequality actually states that $\chi'(G) \ge \chi^*(G)$.
As $\chi'(G)$ is integer-valued, we further obtain $\chi'(G) \ge \max\{\Delta(G), \, \lceil \Gamma(G) \rceil \}$.
How good is this bound? In the 1970s Goldberg \cite{G73} and Seymour \cite{Se} independently made the following
conjecture.
\begin{conjecture} \label{GS}
Every multigraph $G$ satisfies $\chi'(G)\le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$.
\end{conjecture}
Let $r$ be a positive integer. A multigraph $G=(V,E)$ is called an $r$-{\em graph} if $G$
is regular of degree $r$ and for every $X \subseteq V$ with $|X|$ odd, the number of edges
between $X$ and $V-X$ is at least $r$. Note that if $G$ is an $r$-graph, then $|V(G)|$ is even
and $\Gamma(G) \le r$. Seymour \cite{Se} also proposed the following weaker version of Conjecture
\ref{GS}, which amounts to saying that $\chi'(G)\le \max\{\Delta(G), \, \lceil \Gamma(G) \rceil\}+1$ for
any multigraph $G$.
\begin{conjecture} \label{Sey}
Every $r$-graph $G$ satisfies $\chi'(G)\le r+1$.
\end{conjecture}
The following conjecture was posed by Gupta \cite{G} in 1967 and can be deduced from Conjecture \ref{GS}, as
verified by Scheide \cite{S07}.
\begin{conjecture} \label{Gupta67}
Let $G$ be a multigraph with $\mu(G)=\mu$, such that $\Delta(G)$ cannot be expressed in the form $2p\mu-q$,
where $q \ge 0$ and $p > \lfloor (q+1)/2 \rfloor$. Then $\chi'(G)\le \Delta(G)+\mu(G)-1$.
\end{conjecture}
A multigraph $G$ is called {\em critical} if $\chi'(H)<\chi'(G)$ for any proper subgraph $H$ of $G$.
In edge-coloring theory, critical multigraphs are of special interest, because they have much more
structural properties than arbitrary multigraphs. The following two conjectures are due to Jakobsen \cite{J0,J}
and were proved by Andersen \cite{An77} to be weaker than Conjecture \ref{GS}.
\begin{conjecture} \label{And0}
Let $G$ be a critical multigraph with $\chi'(G) \ge \Delta(G)+2$. Then $G$ contains an odd number of
vertices.
\end{conjecture}
\begin{conjecture} \label{And}
Let $G$ be a critical multigraph with $\chi'(G)>\frac{m \Delta(G)+(m-3)}{m-1}$ for an odd
integer $m \ge 3$. Then $G$ has at most $m-2$ vertices.
\end{conjecture}
Motivated by Conjecture \ref{GS}, Hochbaum, Nishizeki, and Shmoys \cite{HNS86} formulated the following
conjecture concerning the approximability of ECP.
\begin{conjecture} \label{HNS}
There is a polynomial-time algorithm that colors the edges of any multigraph $G$ using at most $\chi'(G)+1$
colors.
\end{conjecture}
Over the past four decades Conjecture \ref{GS} has been a subject of extensive research, and has stimulated a
significant body of work, with contributions from many researchers; see McDonald \cite{Mc} for a survey
on this conjecture and Stiebitz {\em et al.} \cite{SSTF} for a comprehensive account of edge-colorings.
Several approximate results state that $\chi'(G)\le \max\{\Delta(G)+\rho(G), \, \lceil \Gamma(G) \rceil\}$, where
$\rho(G)$ is a positive number depending on $G$. Asymptotically, Kahn \cite{K} showed that $\rho(G)=o(\Delta(G))$.
Scheide \cite{S} and Chen, Yu, and Zang \cite{CYZ} independently proved that $\rho(G)\le \sqrt{\Delta(G)/2}$.
Chen {\em et al.} \cite{GT} improved this to $\rho(G)\le \sqrt[3]{\Delta(G)/2}$. Recently, Chen and Jing \cite{GG}
further strengthened this as $\rho(G)\le \sqrt[3]{\Delta(G)/4}$.
There is another family of results, asserting that $\chi'(G)\le \max\{\frac{m \Delta(G)+(m-3)}{m-1}, \, \lceil \Gamma(G)
\rceil\}$, for increasing values of $m$. Such results have been obtained by Andersen \cite{An77} and Goldberg \cite{G73}
for $m=5$, Andersen \cite{An77} for $m=7$, Goldberg \cite{G84} and Hochbaum, Nishizeki, and Shmoys \cite{HNS86} for $m=9$,
Nishizeki and Kashiwagi \cite{NK} and Tashkinov \cite{T} for $m=11$, Favrholdt, Stiebitz, and Toft \cite{FST} for $m=13$,
Scheide \cite{S} for $m=15$, Chen {\em et al.} \cite{GT} for $m=23$, and Chen and Jing \cite{GG} for $m=39$. It is
worthwhile pointing out that, when $\Delta(G) \le 39$, the validity of Conjecture \ref{GS} follows instantly from
Chen and Jing's result \cite{GG}, because $\frac{39 \Delta(G)+36}{38}<\Delta(G)+2$.
Haxell and McDonald \cite{HM} obtained a different sort of approximation to Conjecture \ref{GS}: $\chi'(G)\le \max\{\Delta(G)+
2 \sqrt{\mu(G) \log \Delta(G)}, \, \lceil \Gamma(G) \rceil\}$. Another way to obtain approximations for Conjecture \ref{GS}
is to incorporate the order $n$ of $G$ (that is, number of vertices) into a bound. In this direction, Plantholt \cite{P99}
proved that $\chi'(G)\le \max\{\Delta(G), \, \lceil \Gamma(G) \rceil+1+ \sqrt{n \log (n/6)}\}$ for any multigraph
$G$ with ever order $n\ge 572$. In \cite{P13}, he established an improved result that is applicable to all multigraphs.
Marcotte \cite{M} showed that $\chi'(G)=\max\{\Delta(G), \, \lceil \Gamma(G) \rceil\}$ for any multigraph $G$ with no
$K_5^-$-minor, thereby confirming Conjecture \ref{GS} for this multigraph class. Recently, Haxell, Krivelevich, and
Kronenberg \cite{HKK} established Conjecture \ref{GS} for random multigraphs.
\vskip 3mm
The purpose of this paper is to present a proof of the Goldberg-Seymour conjecture.
\begin{theorem} \label{ThmGS}
Every multigraph $G$ satisfies $\chi'(G)\le \max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$.
\end{theorem}
As stated before, Conjectures \ref{Sey}-\ref{And} are all weaker than the Goldberg-Seymour conjecture,
so the truth of them follows from Theorem \ref{ThmGS} as corollaries.
\begin{theorem} \label{ThmSey}
Every $r$-graph $G$ satisfies $\chi'(G)\le r+1$.
\end{theorem}
\begin{theorem} \label{ThmGupta67}
Let $G$ be a multigraph with $\mu(G)=\mu$, such that $\Delta(G)$ cannot be expressed in the form $2p\mu-q$,
where $q \ge 0$ and $p > \lfloor (q+1)/2 \rfloor$. Then $\chi'(G)\le \Delta(G)+\mu(G)-1$.
\end{theorem}
\begin{theorem} \label{ThmAnd0}
Let $G$ be a critical multigraph with $\chi'(G) \ge \Delta(G)+2$. Then $G$ contains an odd number of
vertices.
\end{theorem}
\begin{theorem} \label{ThmAnd}
Let $G$ be a critical multigraph with $\chi'(G)>\frac{m \Delta(G)+(m-3)}{m-1}$ for an odd
integer $m \ge 3$. Then $G$ has at most $m-2$ vertices.
\end{theorem}
We have seen that FECP is intimately tied to ECP. For any multigraph $G$, the fractional chromatic index
$\chi^*(G)=\max\{\Delta(G), \,\Gamma(G) \}$ can be determined in polynomial time
by combining the Padberg-Rao separation algorithm for $b$-matching polyhedra \cite{PR} (see also \cite{LRT, PW}) with
binary search. In \cite{CZZ}, Chen, Zang, and Zhao designed a combinatorial polynomial-time algorithm for finding
the density $\Gamma(G)$ of any multigraph $G$, thereby resolving a problem posed in both Stiebitz {\em et al.} \cite{SSTF}
and Jensen and Toft \cite{JT}. Nemhauser and Park \cite{NP} observed that FECP can be solved in polynomial
time by an ellipsoid algorithm, because the separation problem of its LP dual is exactly the maximum-weight
matching problem (see also Schrijver \cite{Sc}, Theorem 28.6 on page 477). In \cite{CZZ},
Chen, Zang, and Zhao devised a combinatorial polynomial-time algorithm for FECP as well.
We believe that our proof of Theorem \ref{ThmGS} can be adapted to yield a polynomial-time algorithm for finding an
edge-coloring of any multigraph $G$ with at most $\max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$ colors, and we
are working on the design of this algorithm. A successful implementation would lead to an affirmative
answer to Conjecture \ref{HNS} as well.
Some remarks may help to put Theorem \ref{ThmGS} in proper perspective.
First, by Theorem \ref{ThmGS}, there are only two possible values for the chromatic index of a multigraph $G$:
$\max\{\Delta(G), \, \lceil \Gamma(G) \rceil\}$ and $\max\{\Delta(G)+1, \, \lceil \Gamma(G) \rceil\}$. Thus an
analogue to Vizing's theorem on edge-colorings of simple graphs, a fundamental result in graph theory, holds for
multigraphs.
Second, Theorem \ref{ThmGS} exhibits a dichotomy on edge-coloring: While Holyer's theorem \cite{H} tells us that it is
$NP$-hard to determine $\chi'(G)$, we can approximate it within one of its true value, because $\max\{\Delta(G)+1, \,
\lceil \Gamma(G) \rceil\} - \chi'(G) \le 1$. Furthermore, if $\Gamma(G)>\Delta(G)$, then $\chi'(G)=\lceil \Gamma(G) \rceil$,
so it can be found in polynomial time \cite{CZZ,PR}.
Third, by Theorem \ref{ThmGS} and aforementioned Seymour's theorem, every multigraph $G=(V,E)$ satisfies
$\chi'(G)-\chi^*(G) \le 1$, which can be naturally extended to the weighted case. Let $w(e)$ be a
nonnegative integral weight on each edge $e\in E$ and let ${\bm w}=(w(e): e\in E)$. The {\em chromatic index}
of $(G,{\bfm w})$, denoted by $\chi'_w(G)$, is the minimum number of matchings in $G$ such that
each edge $e$ is covered exactly $w(e)$ times by these matchings, and the {\em fractional chromatic index} of
$(G, {\bfm w})$, denoted by $\chi^*_w(G)$, is the optimal value of the following linear program:
\begin{center}
\begin{tabular}{ll}
\hbox{Minimize} \ \ \ & $ {\bm 1}^T {\bm x}$ \\
\hbox{\hskip 0.2mm subject to} & $ A{\bm x} = {\bm w}$ \\
& \hskip 3mm ${\bm x} \ge {\bm 0},$
\end{tabular}
\end{center}
where $A$ is again the edge-matching incidence matrix of $G$. Clearly, $\chi'_w(G)$ is the optimal value of
the corresponding integer program. Let $G_w$ be obtained from $G$ by replacing each edge $e$ with $w(e)$
parallel edges between the same ends. It is then routine to check that $\chi'_w(G)=\chi'(G_w)$ and
$\chi^*_w(G)=\chi^*(G_w)$. So the inequality $\chi'_w(G)-\chi^*_w(G) \le 1$ holds for all nonnegative
integral weight functions ${\bm w}$, and hence FECP has a fascinating integer rounding property (see Schrijver
\cite{Sc86,Sc}).
So far the most powerful and sophisticated technique for multigraph edge-coloring is the method of Tashkinov trees
\cite{T}, which generalizes the earlier methods of Vizing fans \cite{V} and Kierstead paths \cite{Ki}. (These
methods are named after the authors who invented them, respectively.) Most approximate results described above
Theorem \ref{ThmGS} were obtained by using the method of Tashkinov trees. As remarked by McDonald \cite{Mc}, the
Goldberg-Seymour conjecture and ideas culminating in this method are two cornerstones in modern edge-coloring. Nevertheless,
this method suffers some theoretical limitation when applied to prove the conjecture; the reader is referred to
Asplund and McDonald \cite{AM} for detailed information. Despite various attempts to extend the Tashkinov trees (see, for
instance, \cite{GT,GG,CYZ,S,SSTF}), the difficulty encountered by the method remains unresolved. Even worse, new
problem emerges: it becomes very difficult to preserve the structure of an extended Tashkinov tree under Kempe changes
(the most useful tool in edge-coloring theory). In this paper we introduce a new type of extended Tashkinov trees and
develop an effective control mechanism over Kempe changes, which can overcome all the aforementioned difficulties.
\vskip 2mm
The remainder of this paper is organized as follows. In Section 2, we introduce some basic concepts and
techniques of edge-coloring theory, and exhibit some important properties of stable colorings.
In Section 3, we define the extended Tashkinov trees to be employed in subsequent proof, and give an outline
of our proof strategy. In Section 4, we establish some auxiliary results concerning the extended Tashkinov trees
and stable colorings, which ensure that this type of trees is preserved under some restricted Kempe changes. In Section 5,
we develop an effective control mechanism over Kempe changes, the so-called good hierarchy of an extended
Tashkinov tree; our proof relies heavily on this novel recoloring technique. In Section 6, we derive some properties
satisfied by the good hierarchies introduced in the preceding section. In Section 7, we present the last step of our
proof based on these good hierarchies.
\section{Preliminaries}
This section presents some basic definitions, terminology, and notations used in our paper, along with some important
properties and results.
\subsection{Terminology and Notations}
Let $G=(V,E)$ be a multigraph. For each $X \subseteq V$, let $G[X]$ denote the subgraph of $G$ induced by $X$, and let
$G-X$ denote $G[V-X]$; we write $G-x$ for $G-X$ if $X=\{x\}$. Moreover, we use $\partial(X)$ to denote the set of all
edges with precisely one end in $X$, and write $\partial(x)$ for $\partial(X)$ if $X=\{x\}$. For each pair $x, y
\in V$, let $E(x,y)$ denote the set of all edges between $x$ and $y$. As it is no longer appropriate to represent
an edge $f$ between $x$ and $y$ by $xy$ in a multigraph, we write $f \in E(x,y)$ instead. For each subgraph $H$ of $G$,
let $V(H)$ and $E(H)$ denote the vertex set and edge set of $H$, respectively, let $|H|=|V(H)|$, and let $G[H]=G[V(H)]$ and $\partial(H) =\partial(V(H))$.
Let $e$ be an edge of $G$. A {\em tree sequence} with respect to $G$ and $e$ is a sequence
$T=(y_0,e_1,y_1, \ldots, e_p, y_p)$ with $p\ge 1$, consisting of distinct edges $e_1,e_2, \ldots, e_p$ and
distinct vertices $y_0,y_1, \ldots, y_p$, such that $e_1=e$ and each edge $e_j$ with $1\le j \le p$ is between
$y_j$ and some $y_i$ with $0\le i <j$. Given a tree sequence $T=(y_0,e_1,y_1, \ldots, e_p, y_p)$, we can naturally
associate a linear order $\prec$ with its vertices, such that $y_i \prec y_j$ if $i<j$. We
write $y_i \preceq y_j$ if $i\le j$. This linear order will be used repeatedly in subsequent sections.
For each vertex $y_j$ of $T$ with $j \ge 1$, let $T(y_j)$ denote $(y_0,e_1,y_1, \ldots, e_j, y_j)$. Clearly,
$T(y_j)$ is also a tree sequence with respect to $G$ and $e$. We call $T(y_j)$ the {\em segment} of $T$ induced
by $y_j$. Let $T_1$ and $T_2$ be two tree sequences with respect to $G$ and $e$. We write $T_2-T_1$ for
$E[T_2]-E[T_1]$, write $T_1 \subseteq T_2$ if $T_1$ is a segment of $T_2$, and write $T_1 \subset T_2$ if $T_1$
is a proper segment of $T_2$; that is, $T_1 \subseteq T_2$ and $T_1 \ne T_2$.
A {\em $k$-edge-coloring} of $G$ is an assignment of $k$ colors, $1,2, \ldots, k$, to the edges of $G$
so that no two adjacent edges have the same color. By definition, the chromatic index $\chi'(G)$ of $G$ is the minimum $k$
for which $G$ has a $k$-edge-coloring. We use $[k]$ to denote the color set $\{1,2, \ldots, k\}$,
and use ${\cal C}^k(G)$ to denote the set of all $k$-edge-colorings of $G$. Note that every $k$-edge-coloring
of $G$ is a mapping from $E$ to $[k]$.
Let $\varphi$ be a $k$-edge-coloring of $G$. For each $\alpha \in [k]$, the edge set $E_{\varphi, \alpha}=\{e\in E:\,
\varphi(e)=\alpha\}$ is called a {\em color class}, which is a matching in $G$. For any two distinct colors
$\alpha$ and $\beta$ in $[k]$, let $H$ be the spanning subgraph of $G$ with $E(H)=E_{\varphi, \alpha}
\cup E_{\varphi, \beta}$. Then each component of $H$ is either a path or an even cycle; we refer to such a component
as an $(\alpha, \beta)$-{\em chain} with respect to $\varphi$, and also call it an $(\alpha, \beta)$-{\em path}
(resp. $(\alpha, \beta)$-{\em cycle}) if it is a path (resp. cycle). We use $P_v(\alpha, \beta, \varphi)$ to
denote the unique $(\alpha, \beta)$-chain containing each vertex $v$. Clearly,
for any two distinct vertices $u$ and $v$, either $P_u(\alpha, \beta, \varphi)$ and $P_v(\alpha, \beta, \varphi)$
are identical or are vertex-disjoint. Let $C$ be an $(\alpha, \beta)$-chain with respect to $\varphi$, and let
$\varphi'$ be the $k$-edge-coloring arising from $\varphi$ by interchanging $\alpha$ and $\beta$ on $C$. We say
that $\varphi'$ is obtained from $\varphi$ by {\em recoloring} $C$, and write $\varphi'=\varphi /C$. This
operation is called a {\em Kempe change}.
Let $F$ be an edge subset of $G$. As usual, $G-F$ stands for the multigraph obtained from $G$ by deleting all
edges in $F$; we write $G-f$ for $G-F$ if $F=\{f\}$. Let $\pi \in {\cal C}^k(G-F)$. For each $K \subseteq E$,
define $\pi(K)=\cup_{e\in K-F} \, \pi(e)$. For each $v \in V$, define
\[\pi(v)=\pi(\partial(v)) \hskip 2mm {\rm and} \hskip 2mm \overline{\pi}(v)=[k]-\pi(v).\]
We call $\pi(v)$ the set of colors {\em present} at $v$ and call $\overline{\pi}(v)$ the set of colors {\em missing}
at $v$. For each $X\subseteq V$, define
\[\overline{\pi}(X)= \cup_{v\in X} \, \overline{\pi}(v).\]
We call $X$ {\em elementary} with respect to $\pi$ if $\overline{\pi}(u) \cap \overline{\pi}(v)
=\emptyset$ for any two distinct vertices $u, v\in X$. We call $X$ {\em closed} with respect to $\pi$ if
$\pi(\partial(X))\cap \overline{\pi}(X)=\emptyset$; that is, no missing coloring of $X$ appears on the
edges in $\partial(X)$. Furthermore, we call $X$ {\em strongly closed} with respect to $\pi$ if $X$ is closed
with respect to $\pi$ and $\pi(e) \ne \pi(f)$ for any two distinct colored edges $e, f \in
\partial(X)$. For each subgraph $H$ of $G$, write $\overline{\pi}(H)$ for $\overline{\pi}(V(H))$,
and write ${\pi}\langle H \rangle$ for ${\pi}(E(H))$. Moreover, define
\[\partial_{\pi, \alpha}(H)=\{e\in \partial(H): \pi(e)=\alpha\},\]
and define
\[I[\partial_{\pi, \alpha}(H)]=\{v\in V(H): v \hskip 2mm \mbox{is incident with an edge in} \hskip 2mm
\partial_{\pi, \alpha}(H)\}.\]
For an edge $e\in \partial(H)$, we call its end in (resp. outside) $H$ the {\em in-end} (resp. {\em out-end})
relative to $H$. Thus $I[\partial_{\pi, \alpha}(H)]$ consists of all in-ends (relative to $H$) of edges in
$\partial_{\pi, \alpha}(H)$. A color $\alpha$ is called a {\em defective color} of $H$ with respect to $\pi$
if $|\partial_{\pi, \alpha}(H)| \ge 2$. A color $\alpha \in \overline{\pi}(H)$ is called {\em closed} in $H$ under $\pi$
if $\partial_{\pi, \alpha}(H)=\emptyset$. For convenience, we say that $H$ is {\em closed} (resp. {\em strongly closed})
with respect to $\pi$ if $V(H)$ is closed (resp. strongly closed) with respect to $\pi$. Let $\alpha$ and $\beta$ be
two colors that are not assigned to $\partial(H)$ under $\pi$. We use $\pi/(G-H, \alpha, \beta)$ to denote the
coloring $\pi'$ obtained from $\pi$ by interchanging $\alpha$ and $\beta$ in $G-V(H)$; that is, for any edge $f$
in $G-V(H)$, if $\pi(f)=\alpha$ then $\pi'(f)=\beta$, and if $\pi(f)=\beta$ then $\pi'(f)=\alpha$. Obviously,
$\pi' \in {\cal C}^k(G-F)$.
\subsection{Elementary Multigraphs}
Let $G=(V,E)$ be a multigraph. We call $G$ an {\em elementary multigraph} if $\chi'(G)=\lceil \Gamma(G) \rceil$.
With this notion, Conjecture \ref{GS} can be rephrased as follows.
\begin{conjecture} \label{GS2}
Every multigraph $G$ with $\chi'(G) \ge \Delta(G)+2$ is elementary.
\end{conjecture}
Recall that $G$ is critical if $\chi'(H)<\chi'(G)$ for any proper subgraph $H$ of $G$. As pointed out by Stiebitz
{\em et al.} \cite{SSTF} (see page 7), to prove Conjecture \ref{GS2}, it suffices to consider critical multigraphs. To
see this, let $G$ be an arbitrary multigraph with $\chi'(G) \ge \Delta(G)+2$. Then $G$ contains a critical multigraph
$H$ with $\chi'(H)=\chi'(G)$, which implies that $\chi'(H) \ge \Delta(H)+2$. Note that if $H$ is elementary,
then so is $G$, because $\lceil \Gamma(G) \rceil \le \chi'(G) = \chi'(H) = \lceil \Gamma(H) \rceil \le
\lceil \Gamma(G) \rceil$. Thus both inequalities hold with equalities, and hence $\chi'(G)=\lceil \Gamma(G) \rceil$.
To prove Conjecture \ref{GS}, we shall actually establish the following statement.
\begin{theorem} \label{ThmGS2}
Every critical multigraph $G$ with $\chi'(G) \ge \Delta(G)+2$ is elementary.
\end{theorem}
In our proof we shall appeal to the following theorem, which reveals some intimate connection between elementary
multigraphs and elementary sets. This result is implicitly contained in Andersen \cite{An77} and Goldberg \cite{G84},
and explicitly proved in Stiebitz {\em et al.} \cite{SSTF} (see Theorem 1.4 on page 8).
\begin{theorem} \label{egraph2}
Let $G=(V,E)$ be a multigraph with $\chi'(G)=k+1$ for an integer $k \ge \Delta(G)+1$. If $G$ is critical, then the
following conditions are equivalent:
\begin{itemize}
\vspace{-2mm}
\item[(i)] $G$ is an elementary multigraph.
\vspace{-2mm}
\item[(ii)] For each edge $e\in E$ and each coloring $\varphi \in {\cal C}^k(G-e)$, the vertex set $V$ is elementary
with respect to $\varphi$.
\vspace{-2mm}
\item[(iii)] There exists an edge $e\in E$ and a coloring $\varphi \in {\cal C}^k(G-e)$, such that the vertex set $V$
is elementary with respect to $\varphi$.
\vspace{-2mm}
\item[(iv)] There exists an edge $e\in E$, a coloring $\varphi \in {\cal C}^k(G-e)$, and a subset $X$ of $V$,
such that $X$ contains both ends of $e$, and $X$ is elementary as well as strongly closed with respect to $\varphi$.
\end{itemize}
\end{theorem}
\subsection{Stable Colorings}
\vskip 2mm
In this subsection, we assume that $T$ is a tree sequence with respect to a multigraph $G=(V,E)$ and an edge $e$,
$C$ is a subset of $[k]$, and $\varphi$ is a coloring in ${\cal C}^k(G-e)$, where $k \ge \Delta(G)+1$.
\vskip 1mm
A coloring $\pi \in {\cal C}^k(G-e)$ is called a $(T, C, \varphi)$-{\em stable coloring} if the following two
conditions are satisfied:
\begin{itemize}
\vspace{-2mm}
\item[$(i)$] $\pi(f) = \varphi(f)$ for any $f\in E$ incident to $T$ with $\varphi(f)\in \overline{\varphi}(T)\cup C$; and
\vspace{-2mm}
\item[$(ii)$] $\overline{\pi} (v) = \overline{\varphi}(v)$ for any $v\in V(T)$.
\end{itemize}
By convention, $\pi(e)=\varphi(e)=\emptyset$. From the definition we see that if ${\varphi}\langle T \rangle \subseteq \overline{\varphi}(T)\cup C$, then $\pi(f) = \varphi(f)$ for all edges $f$ on $T$; this special type of stable colorings
will be our major concern.
In our proof we shall perform a sequence of Kempe changes so that the resulting colorings are stable in some sense. The
following lemma gives an equivalent definition of stable colorings.
\begin{lemma} \label{sc1}
A coloring $\pi \in {\cal C}^k(G-e)$ is $(T, C, \varphi)$-stable iff the following two conditions are satisfied:
\begin{itemize}
\vspace{-2mm}
\item[(i')] $\pi(f)=\varphi(f)$ for any $f\in E$ incident to $T$ with $\varphi(f)\in \overline{\varphi}(T)\cup C$ or
$\pi(f)\in \overline{\varphi}(T)\cup C$; and
\vspace{-2mm}
\item[(ii)] $\overline{\pi} (v) = \overline{\varphi}(v)$ for any $v\in V(T)$.
\end{itemize}
\end{lemma}
{\bf Proof.} Note that condition $(ii)$ described here is exactly the same as given in the definition
and that $(i')$ implies $(i)$, so the ``if" part is trivial. To establish the ``only if" part,
let $f \in E$ be an arbitrary edge incident to $T$ with $\pi(f)\in \overline{\varphi}(T)\cup C$. We claim that
$\varphi(f) = \pi(f)$, for otherwise, let $v\in V(T)$ be an end of $f$. By $(ii)$, we have $\overline{\pi} (v) = \overline{\varphi}(v)$. So ${\pi} (v) = {\varphi}(v)$ and hence there exists an edge $g \in \partial(v)-\{f\}$
with $\varphi(g) = \pi(f)$. It follows that $\varphi(g) \in \overline{\varphi}(T)\cup C$.
By $(i)$, we obtain $\pi(g)=\varphi(g)$, which implies $\pi(f)=\pi(g)$, contradicting the hypothesis that
$\pi \in {\cal C}^k(G-e)$. Our claim asserts that $\varphi(f)=\pi(f)$ for any $f\in E$ incident to $T$ with $\pi(f)\in \overline{\varphi}(T)\cup C$. Combining this with $(i)$, we see that $(i')$ holds. \hfill \rule{4pt}{7pt}
\vskip 3mm
Let us derive some properties satisfied by stable colorings.
\begin{lemma} \label{sc2}
Being $(T, C, \cdot )$-stable is an equivalence relation on ${\cal C}^k(G-e)$.
\end{lemma}
{\bf Proof.} From Lemma \ref{sc1} it is clear that being $(T, C, \cdot )$-stable is reflexive,
symmetric, and transitive. So it defines an equivalence relation on ${\cal C}^k(G-e)$. \hfill \rule{4pt}{7pt}
\iffalse
\vskip 3mm
\begin{lemma} \label{guantao}
Let $\pi \in {\cal C}^k(G-e)$ be a $(T, C, \varphi)$-stable coloring, and let $T' \subseteq T$ and $C' \subseteq [k]$
such that $\overline{\varphi}(T') \cup C' \subseteq \overline{\varphi}(T) \cup C$. Then $\pi$ is also a
$(T', C', \varphi)$-stable coloring.
\end{lemma}
{\bf Proof.} For any edge $f$ incident to $T'$ with $\varphi(f)\in \overline{\varphi}(T')\cup C'$, we have
$\varphi(f) \in \overline{\varphi}(T) \cup C$ by hypothesis. Since $\pi$ is a $(T, C, \varphi)$-stable coloring,
$\pi(f) = \varphi(f)$. Furthermore, $\overline{\pi} (v) = \overline{\varphi}(v)$ for all $v\in V(T)$. It follows
that $\pi$ is a $(T', C', \varphi)$-stable coloring. \hfill \rule{4pt}{7pt}
\fi
\vskip 3mm
Let $P$ be a path in $G$ whose edges are colored alternately by $\alpha$ and $\beta$ in $\varphi$,
with $|P|\ge 2$, and let $u$ and $v$ be the ends of $P$ with $v \in V(T)$. We say that $P$ is a $T$-{\em exit path}
with respect to $\varphi$ if $V(T) \cap V(P)=\{v\}$ and $\overline{\varphi}(u) \cap \{\alpha,\beta\} \ne \emptyset$;
in this case, $v$ is called a $(T,\varphi,\{\alpha,\beta\})$-{\em exit} and $P$ is also called a
$(T,\varphi,\{\alpha,\beta\})$-{\em exit path}. Note that possibly $\overline{\varphi}(v) \cap \{\alpha,\beta\}=\emptyset$.
Let $f\in E(u,v)$ be an edge in $\partial (T)$ with $v \in V(T)$. We say that $f$ is {\em $T\vee C$-nonextendable}
with respect to $\varphi$ if there exists a $(T, C \cup \{\varphi(f)\}, \varphi)$-stable coloring $\pi$ and a color
$\alpha \in \overline{\pi} (v)$, such that $v$ is a $(T, \pi, \{\alpha, \varphi(f)\})$-exit. Otherwise, we say that $f$
is {\em $T\vee C$-extendable} with respect to $\varphi$.
\begin{lemma}\label{LEM:extable}
Suppose $T$ is closed with respect to $\varphi$, and $f\in E(u,v)$ is an edge in $\partial (T)$ with $v \in V(T)$. If there
exists a $(T, C\cup \{\varphi(f)\}, \varphi)$-stable coloring $\pi$, such that $\overline{\pi}(u) \cap \overline{\pi}(T)
\ne \emptyset$, then $f$ is $T\vee C$-nonextendable with respect to $\varphi$.
\end{lemma}
{\bf Proof.} Let $\alpha \in \overline{\pi}(u) \cap \overline{\pi}(T)$ and $\beta \in \overline{\pi}(v)$.
By the definition of stable colorings, we have $\alpha\in \overline{\varphi}(T)$ and $\beta \in \overline{\varphi}(v)$.
Since both $\alpha$ and $\beta$ are closed in $T$ under $\varphi$, they are also closed in $T$ under $\pi$ by Lemma \ref{sc1}.
Define $\pi'=\pi/ (G-T, \alpha,\beta)$. Clearly, $\pi'$ is a $(T, C\cup \{\varphi(f)\}, \pi)$-stable coloring. By Lemma \ref{sc2},
$\pi'$ is also a $(T, C\cup \{\varphi(f)\}, \varphi)$-stable coloring. Since $P_v(\beta, \varphi(f), \pi')$ consists of
a single edge $f$, it is a $T$-exit path with respect to $\pi'$. Hence $f$ is $T\vee C$-nonextendable with respect
to $\varphi$. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{lem:ExiMissing}
Suppose $T$ is closed with respect to $\varphi$, and $f\in E(u,v)$ is an edge in $\partial (T)$ with $v \in V(T)$. If
$f$ is $T\vee C$-nonextendable with respect to $\varphi$, then for any $ \alpha \in \overline{\varphi}(v)$ there exists a
$(T, C\cup \{\varphi(f)\}, \varphi)$-stable coloring $\pi$, such that $v$ is a $(T, \pi, \{\alpha, \varphi(f)\})$-exit.
\end{lemma}
{\bf Proof.} Since $f$ is $T\vee C$-nonextendable, by definition, there exist a $(T, C\cup \{\varphi(f)\}, \varphi)$-stable
coloring $\varphi'$ and a color $\beta\in \overline{\varphi}(v)$, such that $v$ is a $(T,\varphi',\{\beta, \varphi(f)\})$-exit.
Since both $\alpha$ and $\beta$ are closed in $T$ under $\varphi$, they are also closed in $T$ under $\varphi'$ by Lemma \ref{sc1}.
Define $\pi= \varphi'/(G-T, \alpha, \beta)$. Clearly, $\pi$ is a $(T, C\cup \{\varphi(f)\}, \varphi')$-stable coloring. By
Lemma \ref{sc2}, $\pi$ is also a $(T, C\cup \{\varphi(f)\}, \varphi)$-stable coloring. Note that $P_v(\alpha, \varphi(f),
\pi)= P_v(\beta, \varphi(f), \varphi')$, so $P_v(\alpha, \varphi(f), \pi)$ is a $T$-exit path with respect to $\pi$, and
hence $v$ is a $(T, \pi, \{\alpha, \varphi(f)\})$-exit. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{zang1}
Suppose $T$ is closed but not strongly closed with respect to $\varphi$, with $|V(T)|$ odd, and suppose $\pi$ is
a $(T, C, \varphi)$-stable coloring. Then $T$ is also closed but not strongly closed with respect to $\pi$.
\end{lemma}
{\bf Proof.} Let $X=V(T)$ and let $t$ be the size of the set $[k]-\overline{\varphi}(X)$. Since $\pi$ is
a $(T, C, \varphi)$-stable coloring, from Lemma \ref{sc1} we deduce that $T$ is closed with respect to $\pi$
and that $\overline{\pi}(X)=\overline{\varphi}(X)$ (so $[k]-\overline{\pi}(X)$ is also of size $t$). By hypotheses, $|V(T)|$
is odd and $T$ is not strongly closed with respect to $\varphi$. Thus under the coloring $\varphi$ each color in $[k]-\overline {\varphi}(X)$ is assigned to at least one edge in $\partial(T)$, and some color in $[k]-\overline{\varphi}(X)$ is assigned
to at least two edges in $\partial(T)$. It follows that $|\partial(T)|\ge t+1$. Note that under the coloring $\pi$ only
colors in $[k]-\overline{\pi}(X)$ can be assigned to edges in $\partial(T)$, so some of these colors is used at least
twice by the Pigeonhole Principle. Hence $T$ is not strongly closed with respect to $\pi$. \hfill \rule{4pt}{7pt}
\subsection{Tashkinov Trees}
A multigraph $G$ is called $k$-{\em critical} if it is critical and $\chi'(G)=k+1$. Throughout this paper, by
a $k$-{\em triple} we mean a $k$-critical multigraph $G=(V,E)$, where $k \ge \Delta(G)+1$, together with an
uncolored edge $e\in E$ and a coloring $\varphi \in {\cal C}^k(G-e)$; we denote it by $(G,e, \varphi)$.
Let $(G,e, \varphi)$ be a $k$-triple. A {\em Tashkinov tree} with respect to $e$ and $\varphi$ is a
tree sequence $T=(y_0,e_1,y_1, \ldots, e_p, y_p)$ with respect to $G$ and $e$, such that
for each edge $e_j$ with $2\le j \le p$, there is a vertex $y_i$ with $0 \le i <j$ satisfying
$\varphi(e_j) \in \overline{\varphi}(y_i)$.
\vskip 2mm
The following theorem is due to Tashkinov \cite{T}; its proof can also be found in Stiebitz {\em et al.}
\cite{SSTF} (see Theorem 5.1 on page 116).
\begin{theorem} \label{TashTree}
Let $(G,e, \varphi)$ be a $k$-triple and let $T$ be a Tashkinov tree with respect to $e$ and $\varphi$.
Then $V(T)$ is elementary with respect to $\varphi$.
\end{theorem}
Let $G=(V,E)$ be a critical multigraph $G$ with $\chi'(G) \ge \Delta(G)+2$. For each edge $e\in E$ and each
coloring $\varphi \in {\cal C}^k(G-e)$, there is a Tashkinov tree $T$ with respect to $e$ and $\varphi$.
The {\em Tashkinov order} of $G$, denoted by $t(G)$, is the largest number of vertices contained in
such a Tashkinov tree. Scheide \cite{S} (see Proposition 4.5) has established the following
result, which will be employed in our proof.
\begin{theorem} \label{ThmScheide}
Let $G$ be a critical multigraph $G$ with $\chi'(G) \ge \Delta(G)+2$. If $t(G)<11$, then $G$ is an elementary
multigraph.
\end{theorem}
The method of Tashkinov trees consists of modifying a given partial edge-coloring with sequences of Kempe changes
and resulting extensions (that is, coloring an edge $e$ with a color $\alpha$, which is missing at both ends of
$e$). When applied to prove Conjecture \ref{GS}, the crux of this method is to capture the density $\Gamma(G)$
by exploring a sufficiently large Tashkinov tree (see Theorem \ref{TashTree}). However, this target may become
unreachable when $\chi'(G)$ gets close to $\Delta(G)$, even if we allow for an unlimited number of Kempe changes;
such an example has been found by Asplund and McDonald \cite{AM}. To circumvent this difficulty and to make this
method work, we shall introduce a new type of extended Tashkinov trees in this paper by using the procedure described
below.
Given a $k$-triple $(G,e, \varphi)$ and a tree sequence $T$ with respect to $G$ and $e$, we may construct a tree
sequence $T'=(T, e_1,y_1, \ldots, e_p, y_p)$ from $T$ by recursively adding edges $e_1,e_2, \ldots, e_p$ and
vertices $y_1,y_2, \ldots, y_p$ outside $T$, such that
$\bullet$ $e_1$ is incident to $T$ and each edge $e_i$ with $1\le i \le p$ is between $y_i$ and $
V(T)\cup \{y_1, y_2, \ldots, y_{i-1}\}$;
$\bullet$ for each edge $e_i$ with $1\le i \le p$, there is a vertex $x_i$ in $V(T)\cup \{y_1, y_2, \ldots,
y_{i-1}\}$, satisfying $\varphi(e_i) \in \overline{\varphi}(x_i)$.
Such a procedure is referred to as {\em Tashkinov's augmentation algorithm} (TAA). We call $T'$ a {\em closure} of $T$
under $\varphi$ if it cannot grow further by using TAA (equivalently, $T'$ becomes closed). We point out that,
although there might be several ways to construct a closure of $T$ under $\varphi$, the vertex set of these
closures is unique.
\section{Extended Tashkinov Trees}
The purpose of this section is to present extended Tashkinov trees to be used in our proof and to give an outline of
our proof strategy.
Given a $k$-triple $(G, e, \varphi)$, we first propose an algorithm for constructing a {\em Tashkinov series}, which
is a series of tuples $(T_n, \varphi_{n-1}, S_{n-1}, F_{n-1}, \Theta_{n-1})$ for $n=1,2, \ldots $, where
$\bullet$ $\varphi_{n-1}$ is the $k$-edge-coloring of $G-e$ exhibited in iteration $n-1$,
$\bullet$ $T_n$ is the tree sequence with respect to $e$ and $\varphi_{n-1}$ constructed in iteration $n-1$,
$\bullet$ $S_{n-1}$ consists of the connecting colors used in iteration $n-1$ with $|S_{n-1}| \le 2$,
$\bullet$ $F_{n-1}$ consists of the connecting edge used in iteration $n-1$ if $n \ge 2$ and $F_0=\emptyset$, and
$\bullet$ $\Theta_{n-1}\in \{RE, SE, PE\}$ if $n\ge 2$, which stands for the extension type
used in iteration $n-1$; we set $\Theta_0=\emptyset$.
For ease of description, we make some preparations. Since each $T_n$ is a tree sequence with respect to $G$ and
$e$, the linear order $\prec$ defined in Subsection 2.1 is valid for $T_n$. By $T_n+f_n$ we mean the tree sequence
augmented from $T_n$ by adding an edge $f_n$. By a {\em segment} of a cycle we mean a path contained in it.
Let $D_{n-1}$ be a certain subset of $[k]$ and let $\pi$ be a $(T_n, D_{n-1}, \varphi_{n-1})$-stable coloring. We use
$v_{\pi, \alpha}$ to denote the maximum vertex in
$I[\partial_{\pi, \alpha}(T_n)]$ in the order $\prec$ for each defective color $\alpha$ of $T_n$ with respect to
$\pi$, and use $v_{\pi}$ to denote the maximum vertex in the order $\prec$ among all these vertices
$v_{\pi, \alpha}$. We reserve the symbol $v_n$ for the maximum vertex in the order $\prec$ among all these vertices
$v_{\pi}$, where $\pi$ ranges over all $(T_n, D_{n-1}, \varphi_{n-1})$-stable colorings. We also reserve the symbol $\pi_{n-1}$
for the corresponding $\pi$ (that is, $v_n=v_{\pi_{n-1}}$), and reserve $f_n\in E(u_n,v_n)$ for an edge in $\partial(T_n)$
such that $\pi_{n-1}(f_n)$ is a defective color with respect to $\pi_{n-1}$. We call $v_n$ the {\em maximum
defective vertex} with respect to $(T_n, D_{n-1}, \varphi_{n-1})$.
\vskip 2mm
{\bf (3.1)} In our algorithm, there are three types of augmentations: revisiting extension (RE), series extension
(SE), and parallel extension (PE). Each iteration $n \,(\ge 1)$ involves a special vertex $v_n$, which is
called an {\em extension} vertex if $\Theta_n=SE$ and a {\em supporting} vertex if $\Theta_n=PE$.
\vskip 4mm
\noindent {\bf Algorithm 3.1}
\vskip 2mm
\noindent {\bf Step 0.} Let $\varphi_0=\varphi$ and let $T_1$ be a closure of $e$ under $\varphi_0$, which is
a closed Tashkinov tree with respect to $e$ and $\varphi_0$. Set $S_0 = F_0 = \Theta_0=\emptyset$ and set $n=1$.
\vskip 2mm
\noindent {\bf Step 1.} If $T_n$ is strongly closed with respect to $\varphi_{n-1}$, stop. Else, if there exists
a subscript $h \le n-1$ with $\Theta_h =PE$ and $S_h =\{\delta_h, \gamma_h\}$, such that $\Theta_i=RE$ for all
$i$ with $h+1 \le i \le n-1$, if any, and such that some $(\gamma_h, \delta_h)$-cycle with respect to $\varphi_{n-1}$
contains both an edge $f_n \in \partial_{ \varphi_{n-1}, \gamma_h}(T_n)$ and a segment $L$ connecting $V(T_h)$ and
$v_n$ with $V(L) \subseteq V(T_n)$, where $v_n$ is the end of $f_n$ in $T_n$, go to Step 2. Else,
let $D_{n-1}=\cup_{i \le n-1}S_i - \overline{\varphi}_{n-1}(T_{n-1})$, where $T_0= \emptyset$. Let $v_n$, $\pi_{n-1}$,
and $f_n\in E(u_n,v_n)$ be as defined above the algorithm, and let $\delta_n=\pi_{n-1}(f_n)$. If for every
$(T_n, D_{n-1}\cup\{\delta_n\}, \pi_{n-1})$-stable coloring $\sigma_{n-1}$, we have $\overline{\sigma}_{n-1} (u_n)
\cap \overline{\sigma}_{n-1} (T_n)=\emptyset$, go to Step 3. Else, go to Step 4.
\vskip 2mm
\noindent {\bf Step 2.} Let $\varphi_n = \varphi_{n-1}$, let $T_{n+1}$ be a closure of $T_n+f_n$ under $\varphi_{n}$,
and let $\delta_n=\delta_h$, $\gamma_n=\gamma_h$, $S_n =\{\delta_n,\gamma_n\}$, $F_n = \{ f_n\}$, and $\Theta_n = RE$.
Set $n=n+1$, return to Step 1. (We call this augmentation a {\bf revisiting extension} (RE), call $f_n$ an {\em RE
connecting edge}, and call $\delta_n$ and $\gamma_n$ {\em connecting colors}. Note that $v_n$ is neither called an
extension vertex nor called a supporting vertex (see (3.1)).
\vskip 2mm
\noindent {\bf Step 3.} Let $\varphi_n = \pi_{n-1}$, let $T_{n+1}$ be a closure of $T_n+f_n$ under $\varphi_{n}$,
and let $S_n= \{ \delta_n \}$, $F_n=\{f_n\}$, and $\Theta_n = SE$. Set $n=n+1$, return to Step 1. (We call this
augmentation a {\bf series extension} (SE), call $f_n$ an {\em SE connecting edge}, call $\delta_n$ a {\em connecting
color}, and call $v_n$ an {\em extension vertex}.)
\vskip 2mm
\noindent {\bf Step 4.} Let $A_{n-1}$ be the set of all iterations $i$ with $1\le i \le n-1$ such that
$\Theta_i =PE$ and $v_i=v_n$. Let $\gamma_n$ be a color in $\overline{\pi}_{n-1}(v_n)
\cap (\cup_{i \in A_{n-1}} S_i)$ if $A_{n-1} \ne \emptyset$ (see (3.5) below), and a color in $\overline{\pi}_{n-1}(v_n)$ otherwise.
By Lemmas \ref{LEM:extable} and \ref{lem:ExiMissing}, there exists a $(T_n, D_{n-1}\cup\{\delta_n\},
\pi_{n-1})$-stable coloring $\pi_{n-1}'$, such that $v_n$ is a $(T_n, \pi_{n-1}', \{\gamma_n, \delta_n\})$-exit.
Let $\varphi_n = \pi_{n-1}'/ P_{v_n}(\gamma_n, \delta_n, \pi_{n-1}')$, $S_n = \{\delta_n, \gamma_n\}$,
$F_n=\{f_n\}$, and $\Theta_n =PE$. Let $T_{n+1}$ be a closure of $T_n$ under $\varphi_n$.
Set $n=n+1$, return to Step 1. (We call this augmentation a {\bf parallel extension} (PE), call $f_n$ a {\em PE
connecting edge}, call $\delta_n$ and $\gamma_n$ {\em connecting colors}, and call $v_n$ a {\em supporting vertex}. Note
that $f_n$ is not necessarily contained in $T_{n+1}$.)
\vskip 3mm
Throughout the remainder of this paper, we reserve all symbols used for the same usage as in this algorithm. In particular,
$D_n=\cup_{i\leq n}S_i-\overline{\varphi}_n(T_n)$ (see Step 1) for $n \geq 0$. So $D_0=\emptyset$.
To help understand the algorithm better, let us make a few remarks and offer some simple observations.
{\bf (3.2)} In our proof we shall restrict our attention to the case when $|T_n|$ is odd (as we shall see). Suppose
$T_n$ is not strongly closed with respect to $\varphi_{n-1}$ (see Step 1). Then, by Lemma \ref{zang1}, $T_n$ is closed
but not strongly closed with respect to any $(T_n, D_{n-1}, \varphi_{n-1})$-stable coloring. Thus $v_n$, $\pi_{n-1}$,
and $f_n$ involved in Step 1 are all well defined. It follows that at least one of RE, SE and PE applies to each
iteration, and hence the algorithm terminates only when $T_n$ is strongly closed with respect to $\varphi_{n-1}$,
which contains the case when $V(T_n)=V(G)$.
{\bf (3.3)} As described in the algorithm, revisiting extension (RE) has priority over both series and parallel extensions
(SE and PE). If $\Theta_n=RE$, then from Algorithm 3.1 we see that the $(\gamma_h, \delta_h)$-cycle with respect to
$\varphi_{n-1}$ displayed in Step 1 must contain at least one edge in $G[T_h]$, at least two boundary edges
of $T_h$ colored with $\gamma_h$, and at least two boundary edges of $T_n$ colored with $\gamma_h$,
because $\delta_h$ is a missing color in $T_h$ under both $\varphi_{h}$ and $\varphi_{n-1}$.
{\bf (3.4)} It is clear that $\delta_n$ is a defective color of $T_n$ with respect to $\varphi_n$ when $\Theta_n =SE$
or $PE$ (as $|\partial_{ \pi_{n-1}, \delta_n}(T_n)|\ge 3$ when $|T_n|$ is odd), while $\gamma_n$ is a defective color
of $T_n$ with respect to $\varphi_n$ when $\Theta_n =RE$. Moreover, $D_{n-1}$ is the set of all connecting colors in
$\cup_{h\le n-1}S_h$ that are not missing in $T_{n-1}$ with respect to $\varphi_{n-1}$.
{\bf (3.5)} As we shall prove in Lemma \ref{uniquezang}, if $A_{n-1} \ne \emptyset$ in Step 4, then $\overline{\pi}_{n-1}(v_n)
\cap (\cup_{i \in A_{n-1}} S_i)=\overline{\varphi}_{n-1}(v_n)\cap (\cup_{i \in A_{n-1}} S_i)$ contains precisely one color,
so $\gamma_n$ can be selected in a unique way. This property will play an important role in our proof.
\setcounter{theorem}{1}
\begin{lemma}\label{hku}
For $n \ge 1$, the following statements hold:
\begin{itemize}
\vspace{-1.5mm}
\item[(i)] $\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}\subseteq \overline{\varphi}_{n}(T_n)\cup D_n \subseteq \overline{\varphi}_{n}(T_{n+1})\cup D_n$.
\vspace{-1.5mm}
\item[(ii)] For any edge $f$ incident to $T_n$, if $\varphi_{n-1}(f) \in \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$,
then $\varphi_{n}(f)=\varphi_{n-1}(f)$, unless $\Theta_n=PE$ and $f=f_n$. So $\varphi_{n}(f) \in \overline{\varphi}_{n}(T_n)\cup D_n$
provided that $\varphi_{n-1}(f) \in \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$.
\vspace{-1.5mm}
\item[(iii)] $\varphi_{n-1} \langle T_n \rangle \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$
and $\varphi_n \langle T_n \rangle \subseteq \overline{\varphi}_n(T_n)\cup D_n$. So $\sigma_n(f)=\varphi_n(f)$ for
any $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$ and any edge $f$ on $T_n$.
\vspace{-1.5mm}
\item[(iv)] If $\Theta_n=PE$, then $\partial_{\varphi_n, \gamma_n}(T_n)=\{f_n\}$, and edges in
$\partial_{\varphi_n, \delta_n}(T_n)$ are all incident to $V(T_n(v_n)-v_n)$. Furthermore,
each color in $\overline{\varphi}_n(T_n)-\{\delta_n\}$ is closed in $T_n$ under $\varphi_n$.
\end{itemize}
\end{lemma}
{\bf Proof.} By definition, $D_{n-1}=\cup_{i\leq n-1}S_i-\overline{\varphi}_{n-1}(T_{n-1})$. So $\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}=\overline{\varphi}_{n-1}(T_n) \cup [\cup_{i\leq n-1}S_i-\overline{\varphi}_{n-1}(T_{n-1})]$. Since $\overline{\varphi}_{n-1}(T_{n-1}) \subseteq \overline{\varphi}_{n-1}(T_n)$, we obtain
(1) $\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}=\overline{\varphi}_{n-1}(T_n) \cup (\cup_{i\leq n-1}S_i)$.
\noindent Similarly, we can prove that
(2) $\overline{\varphi}_{n}(T_n)\cup D_n= \overline{\varphi}_{n}(T_n)\cup (\cup_{i\leq n} S_i)$.
(i) For any $\alpha \in \overline{\varphi}_{n-1}(T_n)$, from Algorithm 3.1 and definition of stable colorings we see that $\alpha \in \overline{\varphi}_{n}(T_n)$, unless $\Theta_n=PE$ and $\alpha=\gamma_n$; in this exceptional case, $\alpha \in S_n$.
So $\overline{\varphi}_{n-1}(T_n) \subseteq \overline{\varphi}_{n}(T_n) \cup S_n$ and hence $\overline{\varphi}_{n-1}(T_n) \cup (\cup_{i\leq n-1}S_i) \subseteq \overline{\varphi}_{n}(T_n)\cup (\cup_{i\leq n} S_i)$. It follows from (1) and (2) that $\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}\subseteq
\overline{\varphi}_{n}(T_n)\cup D_n$. Clearly, $\overline{\varphi}_{n}(T_n)\cup D_n \subseteq \overline{\varphi}_{n}(T_{n+1})\cup D_n$.
(ii) Let $f$ be an edge incident to $T_n$ with $\varphi_{n-1}(f) \in \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$.
If $\Theta_n=RE$, then $\varphi_n=\varphi_{n-1}$ by Step 1 of Algorithm 3.1, which implies $\varphi_n(f) =
\varphi_{n-1}(f)$. So we may assume that $\Theta_n\ne RE$. Let $\pi_{n-1}$ be the $(T_n, D_{n-1}, \varphi_{n-1})$-stable
coloring as specified in Step 1 of Algorithm 3.1. By the definition of stable colorings, we obtain $\pi_{n-1}(f) =
\varphi_{n-1}(f)$. If $\Theta_n=SE$, then $\varphi_n(f)= \pi_{n-1}(f)$ by Step 3 of Algorithm 3.1. Hence $\varphi_n(f)
= \varphi_{n-1}(f)$. It remains to consider the case when $\Theta_n=PE$. Let $\pi_{n-1}'$ be the $(T_n, D_{n-1}
\cup\{\delta_n\}, \pi_{n-1})$-stable coloring as specified in Step 4 of Algorithm 3.1. By Lemma \ref{sc2},
$\pi_{n-1}'$ is $(T_n, D_{n-1}, \varphi_{n-1})$-stable. Hence $\pi_{n-1}'(f) = \varphi_{n-1}(f)$. Since $\varphi_n = \pi'_{n-1}/P_{v_n}(\delta_n, \gamma_n, \pi_{n-1}')$ and $P_{v_n}(\delta_n, \gamma_n, \pi_{n-1}')$ contains only
one edge $f_n$ incident to $T_n$ (see Step 4 of Algorithm 3.1), we have $\varphi_n(f) = \pi_{n-1}'(f)$, unless $f=f_n$.
It follows that $\varphi_n(f) = \varphi_{n-1}(f)$, unless $f=f_n$; in this exceptional case, $\varphi_{n-1}(f)=\delta_n$ and $\varphi_{n}(f) = \gamma_n \in S_n$. Hence $\varphi_{n}(f) \in \overline{\varphi}_{n-1}(T_n)\cup D_{n-1} \cup S_n \subseteq
\overline{\varphi}_{n}(T_n)\cup D_n \cup S_n =\overline{\varphi}_{n}(T_n)\cup D_n$ by (i) and (2), as desired.
(iii) Let us first prove the statement $\varphi_{n-1} \langle T_n \rangle \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$
by induction on $n$. As the statement holds trivially when $n=1$, we proceed to the induction step and assume
that the statement has been established for $n-1$; that is,
(3) $\varphi_{n-2} \langle T_{n-1} \rangle \subseteq \overline{\varphi}_{n-2}(T_{n-1})\cup D_{n-2}$.
By (3) and (ii) (with $n-1$ in place of $n$), for each edge $f$ on $T_{n-1}$ we have $\varphi_{n-1}(f) \in
\overline{\varphi}_{n-1}(T_{n-1}) \cup D_{n-1} \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$. For each edge $f \in T_n-T_{n-1}$, from
Algorithm 3.1 and TAA we see that $\varphi_{n-1}(f) \in D_{n-1}$ if $f$ is a connecting edge and $\varphi_{n-1}(f) \in \overline{\varphi}_{n-1}(T_n)$ otherwise. Combining these observations, we obtain $\varphi_{n-1}(f) \in \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$.
Hence $\varphi_{n-1} \langle T_n \rangle \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$, which together
with (ii) implies $\varphi_n \langle T_n \rangle \subseteq \overline{\varphi}_n(T_n)\cup D_n$.
It follows that for any edge $f$ on $T_n$, we have $\varphi_n(f) \in \overline{\varphi}_n(T_n)\cup D_n$. Thus $\sigma_n(f)=\varphi_n(f)$
for any $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$.
(iv) From the definitions of $\pi_{n-1}$ and stable colorings, we see that edges in $\partial_{\pi_{n-1}, \delta_n}(T_n)$
are all incident to $V(T_n(v_n))$, and each color in $\overline{\pi}_{n-1}(T_n)$ is closed in $T_n$ under $\pi_{n-1}$.
So, by the definitions of $\pi_{n-1}'$ and stable colorings, edges in $\partial_{\pi_{n-1}', \delta_n}(T_n)$ are all
incident to $V(T_n(v_n))$, and each color in $\overline{\pi}_{n-1}'(T_n)$ is closed in $T_n$ under $\pi_{n-1}'$.
Thus the desired statements follow instantly from the definition of $\varphi_n$ in Step 4. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{uniquezang}
Let $u$ be a vertex of $T_n$ and let $B_{n-1}$ be the set of all iterations $j$ with $1\le j \le n-1$, such that $\Theta_j
=PE$ and $v_j=u$. Suppose $B_{n-1}=\{i_1,i_2, \ldots, i_p\}$, where $1\le i_1<i_2< \ldots < i_p \le n-1$.
Then the following statements hold:
\begin{itemize}
\vspace{-1mm}
\item[(i)] $\overline{\varphi}_{n-1}(u) \cap (\cup_{j \in B_{n-1}} S_j) =\overline{\varphi}_{i_p}(u) \cap (\cup_{j \in B_{n-1}} S_j) = \{\delta_{i_p}\}$;
\vspace{-1.5mm}
\item[(ii)] $\gamma_{i_2}=\delta_{i_1}, \, \gamma_{i_3}=\delta_{i_2},\, \ldots, \, \gamma_{i_p}=\delta_{i_{p-1}}$; and
\vspace{-1.5mm}
\item[(iii)] $\overline{\varphi}_{i_1-1}(u)-\{\gamma_{i_1}\}=\overline{\varphi}_{i_p}(u)-\{\delta_{i_p}\}$. So
$\overline{\varphi}_{i_1-1}(u)=(\overline{\varphi}_{i_p}(u)-\{\delta_{i_p}\}) \cup \{\gamma_{i_1}\}$ and
$\overline{\varphi}_{i_p}(u)=(\overline{\varphi}_{i_1-1}(u)- \{\gamma_{i_1}\}) \cup \{\delta_{i_p}\}$.
\end{itemize}
\end{lemma}
{\bf Proof.} By the definition of $B_{n-1}$, for any iteration $j$ with $i_p+1 \le j \le n-1$, if $v_j=u$, then $\Theta_j=RE$
or $SE$. So $\overline{\varphi}_{n-1}(u)=\overline{\varphi}_{i_p}(u)$ by Algorithm 3.1 and the definition of stable colorings. Thus,
to prove (i), it suffices to show that $\overline{\varphi}_{i_p}(u) \cap (\cup_{j \in B_{n-1}} S_j) = \{\delta_{i_p}\}$.
Set $C_h=\{i_1,i_2, \ldots, i_h\}$ for $1 \le h \le p$. Then $C_p=B_{n-1}$ and hence $(i)$ is equivalent
to saying that
$(i')$ $\overline{\varphi}_{i_p}(u) \cap (\cup_{j \in C_p} S_j)= \{\delta_{i_p}\}$.
Let us prove statements $(i')$, $(ii)$, and $(iii)$ simultaneously by induction on $p$.
From Step 4 and the definition of stable colorings, we see that $\gamma_{i_1} \in \overline{\pi}_{i_1-1}(u)=
\overline{\varphi}_{i_1-1}(u)$, $\delta_{i_1} \notin \overline{\pi}_{i_1-1}(u)=\overline{\varphi}_{i_1-1}(u)$, and $\overline{\varphi}_{i_1}(u)$ is
obtained from $\overline{\varphi}_{i_1-1}(u)$ by replacing $\gamma_{i_1}$ with $\delta_{i_1}$. So $\overline{\varphi}_{i_1}(u) \cap
(\cup_{j \in C_1} S_j) = \overline{\varphi}_{i_1}(u) \cap \{\gamma_{i_1}, \delta_{i_1}\}=\{\delta_{i_1}\}$ and $\overline{\varphi}_{i_1-1}(u)-
\{\gamma_{i_1}\}= \overline{\varphi}_{i_1}(u)-\{\delta_{i_1}\}$. Thus both $(i')$ and $(iii)$ hold for $p=1$. For $(ii)$, there
is nothing to prove now.
Suppose we have established these statements for $p-1$. Let us proceed to the induction step for $p$.
By the induction hypotheses on $(i')$ and $(iii)$, we obtain the following two equalities:
(1) $\overline{\varphi}_{i_{p-1}}(u) \cap (\cup_{j \in C_{p-1}} S_j)= \{\delta_{i_{p-1}}\}$ and
(2) $\overline{\varphi}_{i_1-1}(u)-\{\gamma_{i_1}\}=\overline{\varphi}_{i_{p-1}}(u)-\{\delta_{i_{p-1}}\}$.
From the definition of $B_{n-1}$, we see that for any iteration $j$ with $i_{p-1}+1 \le j \le i_p-1$, if $v_j=u$, then
$\Theta_j=RE$ or $SE$. Thus, by Algorithm 3.1 and the definition of stable colorings, we obtain $\overline{\varphi}_{i_{p-1}}(u)=\overline{\varphi}_{i_p-1}(u)=\overline{\pi}_{i_p-1}(u)$. According to Step 4 and using (1),
(3) $\gamma_{i_p}=\delta_{i_{p-1}} \in \overline{\varphi}_{i_{p-1}}(u)$, $\delta_{i_p} \notin \overline{\varphi}_{i_{p-1}}(u)$, and
$\overline{\varphi}_{i_p}(u)$ is obtained from $\overline{\varphi}_{i_{p-1}}(u)$ by replacing $\gamma_{i_p}$ with $\delta_{i_p}$.
Clearly, (3) implies $(ii)$ and the following equality:
(4) $\overline{\varphi}_{i_p}(u)-\{\delta_{i_p}\}=\overline{\varphi}_{i_{p-1}}(u)-\{\delta_{i_{p-1}}\}$.
\noindent It follows from (2) and (4) that $\overline{\varphi}_{i_1-1}(u)-\{\gamma_{i_1}\}=\overline{\varphi}_{i_p}(u)-\{\delta_{i_p}\}$,
thereby proving $(iii)$.
By (1), we have $(\overline{\varphi}_{i_{p-1}}(u)-\{\delta_{i_{p-1}}\}) \cap (\cup_{j \in C_{p-1}} S_j)=\emptyset$. So
$(\overline{\varphi}_{i_{p-1}}(u)-\{\delta_{i_{p-1}}\}) \cap (\cup_{j \in C_p} S_j)= (\overline{\varphi}_{i_{p-1}}(u)-\{\delta_{i_{p-1}}\})
\cap S_{i_p} =(\overline{\varphi}_{i_{p-1}}(u)-\{\gamma_{i_{p}}\}) \cap \{\gamma_{i_p}, \delta_{i_p}\}=\emptyset$, where last
two equalities follow from (3). Combining this observation with (4) yields $\overline{\varphi}_{i_p}(u) \cap (\cup_{j \in C_p} S_j)= [(\overline{\varphi}_{i_{p-1}}(u)-\{\delta_{i_{p-1}}\}) \cup \{\delta_{i_p}\}] \cap (\cup_{j \in C_p} S_j)= \{\delta_{i_p}\}
\cap (\cup_{j \in C_p} S_j) =\{\delta_{i_p}\}$. Hence $(i')$ is established.
Since $\gamma_{i_1} \in \overline{\varphi}_{i_1-1}(u)$ and $\delta_{i_p} \in \overline{\varphi}_{i_p}(u)$, from the equality
$\overline{\varphi}_{i_1-1}(u)-\{\gamma_{i_1}\}=\overline{\varphi}_{i_p}(u)-\{\delta_{i_p}\}$, we immediately deduce that
$\overline{\varphi}_{i_1-1}(u)=(\overline{\varphi}_{i_p}(u)-\{\delta_{i_p}\}) \cup \{\gamma_{i_1}\}$ and
$\overline{\varphi}_{i_p}(u)=(\overline{\varphi}_{i_1-1}(u)- \{\gamma_{i_1}\}) \cup \{\delta_{i_p}\}$. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{Dnzang}
$|D_n|\leq n$.
\end{lemma}
{\bf Proof.} Recall that $D_n=\cup_{i\leq n}S_i-\overline{\varphi}_n(T_n)$. For $1 \le i \le n$, by Algorithm 3.1, we
have $S_i=\{\delta_i\}$ if $\Theta_i=SE$ and $S_i=\{\delta_i, \gamma_i\}$ otherwise.
To establish the desired inequality, we apply induction on $n$. Trivially, the statement holds when $n=0,1$. So we proceed
to the induction step, and assume that $|D_{n-1}|\leq n-1$ for some $n\ge 2$.
If $\Theta_n=RE$, then $\varphi_n = \varphi_{n-1}$ and $S_n = S_{n-1}$ by Step 2 in Algorithm 3.1. So $D_n \subseteq D_{n-1}$ and hence $|D_n|\leq n-1$.
If $\Theta_n=SE$, then $S_n=\{\delta_n\}$ and $\overline{\varphi}_n(T_n)=\overline{\varphi}_{n-1}(T_n)$ by Step 3 in Algorithm 3.1 and the definition of
stable colorings. It follows that $D_n \subseteq D_{n-1} \cup \{\delta_n\}$. Hence $|D_n|\leq |D_{n-1}|+1 \le n$.
It remains to consider the case when $\Theta_n=PE$. By Step 4 in Algorithm 3.1 and the definition of stable colorings, we
obtain $\delta_n \notin \overline{\varphi}_{n-1}(T_n)$ and $(\overline{\varphi}_{n-1}(T_n)-\{\gamma_n\}) \cup \{\delta_n \} \subseteq
\overline{\varphi}_{n}(T_n)$. So
\begin{equation*}
\begin{aligned}
D_n & =\cup_{i\leq n}S_i-\overline{\varphi}_n(T_n)\\
& \subseteq \cup_{i\leq n-1} S_i \cup \{\delta_n,\gamma_n\} -[(\overline{\varphi}_{n-1}(T_n)-\{\gamma_n\}) \cup \{ \delta_n \}]\\
& \subseteq \cup_{i\leq n-1} S_i \cup \{\gamma_n\} -(\overline{\varphi}_{n-1}(T_n)-\{\gamma_n\})\\
& \subseteq [\cup_{i\leq n-1} S_i - \overline{\varphi}_{n-1}(T_n)] \cup \{\gamma_n\}\\
& \subseteq D_{n-1} \cup \{\gamma_n\}.
\end{aligned}
\end{equation*}
Hence $|D_n|\leq |D_{n-1}|+1 \le n$. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{stablezang}
Suppose $\Theta_n=PE$ (see Step 4). Let $\sigma_n$ be a $(T_n, D_n, \varphi_n)$-stable coloring and let
$\sigma_{n-1}=\sigma_n/P_{v_{n}} (\gamma_{n},\delta_{n}, \sigma_n)$. If $P_{v_{n}}(\gamma_{n},\delta_{n},\sigma_n)
\cap T_{n}=\{v_{n}\}$, then $\sigma_{n-1}$ is $(T_n, D_{n-1} \cup \{\delta_n\}, \pi_{n-1})$-stable
and hence is $(T_{n},D_{n-1}, \varphi_{n-1})$-stable.
\end{lemma}
{\bf Proof.} Let $\pi_{n-1}'$ be as specified in Step 4 of Algorithm 3.1. Recall that
(1) $\pi_{n-1}'$ is $(T_n, D_{n-1} \cup \{\delta_n\}, \pi_{n-1})$-stable.
\noindent By definition, $\varphi_n = \pi_{n-1}'/P_{v_n}(\gamma_n, \delta_n, \pi'_{n-1})$. So
(2) $\pi_{n-1}' = \varphi_n /P_{v_n}( \gamma_n, \delta_n, \varphi_n)$.
We propose to show that
(3) $\sigma_{n-1}$ is $(T_n, D_{n-1} \cup \{\delta_n\}, \pi_{n-1}')$-stable.
To justify this, note that $\overline{\sigma}_n(v)=\overline{\varphi}_n(v)$ for all $v \in V(T_n)$, because $\sigma_n$ is a $(T_n, D_n, \varphi_n)$-stable coloring. Thus, by the definition of $\sigma_{n-1}$ and (2), we obtain
(4) $\overline{\sigma}_{n-1}(v)=\overline{\pi}_{n-1}'(v)$ for all $v \in V(T_n)$.
Let $f$ be an edge incident to $T_n$ with $\pi_{n-1}'(f) \in \overline{\pi}_{n-1}'(T_n) \cup D_{n-1} \cup \{\delta_n\}$.
By (1), we have $\pi_{n-1}'(f) \in \overline{\pi}_{n-1}(T_n) \cup D_{n-1} \cup \{\delta_n\}$. Since $\pi_{n-1}$ is
$(T_{n},D_{n-1}, \varphi_{n-1})$-stable, we further obtain $\pi_{n-1}'(f) \in \overline{\varphi}_{n-1}(T_n)
\cup D_{n-1} \cup \{\delta_n\}$. So $\pi_{n-1}'(f) \in \overline{\varphi}_{n}(T_n)\cup D_n$ by Lemma \ref{hku}(i). From Step 4 we
see that
(5) $\pi_{n-1}'(f)=\varphi_n(f)$ if $f\ne f_n$, $\pi'_{n-1}(f_n)=\delta_n$, and $\varphi_n(f_n)=\gamma_n$.
\noindent So $\varphi_n(f)\in \overline{\varphi}_n(T_n) \cup D_n$. Hence $\sigma_n(f)=\varphi_n(f)$, because $\sigma_n$
is a $(T_n, D_n, \varphi_n)$-stable coloring. From the definition of $\sigma_{n-1}$, we deduce that $\sigma_{n-1}(f)=
\sigma_n(f)$ if $f\ne f_n$ and $\sigma_{n-1}(f_n) = \delta_n$. Combining these observations with (5) yields
(6) $\sigma_{n-1}(f)=\pi'_{n-1}(f)$ for any edge $f$ incident to $T_n$ with $\pi_{n-1}'(f) \in
\overline{\pi}_{n-1}'(T_n) \cup D_{n-1} \cup \{\delta_n\}$.
\noindent Thus (3) follows instantly from (4) and (6). Using (1), (3) and Lemma \ref{sc2}, we conclude that
$\sigma_{n-1}$ is $(T_{n},D_{n-1} \cup \{\delta_n\}, \pi_{n-1})$-stable. So $\sigma_{n-1}$ is $(T_{n},
D_{n-1}, \pi_{n-1})$-stable. Since $\pi_{n-1}$ is $(T_{n},D_{n-1}, \varphi_{n-1})$-stable, from Lemma \ref{sc2}
it follows that $\sigma_{n-1}$ is $(T_{n},D_{n-1}, \varphi_{n-1})$-stable. \hfill \rule{4pt}{7pt}
\vskip 3mm
Let us now present a generalized version of Tashkinov trees to be used in our proof.
\begin{definition} \label{wz1}
{\rm Let $(G,e, \varphi)$ be a $k$-triple. A tree sequence $T$ with respect to $G$ and $e$ is called an {\em extended Tashkinov
tree} (ETT) if there exists a Tashkinov series $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$
constructed from $(G,e, \varphi)$ by using Algorithm 3.1, such that $T_n\subset T\subseteq T_{n+1}$, where $T_0=\emptyset$.}
\end{definition}
As introduced in Subsection 2.1, by $T_n\subset T\subseteq T_{n+1}$ we mean that $T_n$ is a proper segment of $T$, and
$T$ is a segment of $T_{n+1}$.
Observe that the extended Tashkinov tree $T$ has a built-in ladder-like structure. So we propose to call the sequence
$T_1\subset T_2 \subset \ldots \subset T_n \subset T$ the {\it ladder} of $T$, and call $n$ the {\it rung number} of $T$ and
denote it by $r(T)$. Moreover, we call $(\varphi_0, \varphi_1, \dots, \varphi_n)$ the {\it coloring sequence} of $T$, call $\varphi_n$
the {\em generating coloring} of $T$, and call $\mathcal {T}$ the Tashkinov series {\em corresponding} to $T$.
In our proof we shall frequently work with stable colorings; the following concept will be used to keep track of
the structures of ETTs.
\begin{definition} \label{wz2}
{\rm Let $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$ be a Tashkinov series
constructed from a $k$-triple $(G,e, \varphi)$ by using Algorithm 3.1. A coloring $\sigma_n\in {\cal C}^k(G-e)$
is called $\varphi_n \bmod T_n$ if there exists an ETT $T^*$ with corresponding Tashkinov series
$\mathcal {T}^*=\{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, satisfying
$\sigma_0 \in {\cal C}^k(G-e)$ and the following conditions for all $i$ with $1\le i \le n$:
\vskip 1mm
$\bullet$ $T_i^*=T_i$ and
$\bullet$ $\sigma_i$ is a $(T_i, D_i, \varphi_i)$-stable coloring in ${\cal C}^k(G-e)$,
where $D_i=\cup_{h\le i} S_h - \overline{\varphi}_{i}(T_i)$.
\vskip 1mm
\noindent We call $T^*$ an ETT {\em corresponding} to $(\sigma_n, T_n)$ (or simply {\em corresponding} to
$\sigma_n$ if no ambiguity arises).}
\end{definition}
\noindent {\bf Remark.} Comparing $\mathcal {T}^*$ with $\mathcal {T}$, we see that $T_{i+1}^*$ in $\mathcal {T}^*$ is obtained from $T_i$ by
using the same connecting edge, connecting color, and extension type as $T_{i+1}$ in $\mathcal {T}$ for $1\le i \le n$.
Furthermore, $T_1\subset T_2 \subset \ldots \subset T_n \subset T^*$ is the ladder of $T^*$ and $r(T^*)=n$.
Since $\sigma_i$ is a $(T_i, D_i, \varphi_i)$-stable coloring, by Lemma \ref{hku}(iii), we have $\sigma_i(f)=\varphi_i(f)$
for any edge $f$ on $T_i$ and $1\le i \le n$; this fact will be used repeatedly in our paper.
To ensure that the structures of ETTs are preserved under taking stable colorings, we impose some restrictions
on such trees.
\begin{definition} \label{wz3}
{\rm Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. We say that $T$ has the {\em maximum property}
(MP) under $(\varphi_0, \varphi_1, \ldots, \varphi_n)$ (or simply under $\varphi_n$ if no ambiguity arises),
if $|T_1|$ is maximum over all Tashkinov trees $T_1'$ with respect to an edge $e'\in E$ and a coloring $\varphi'_0 \in {\cal C}^k(G-e')$, and $|T_{i+1}|$ is maximum over all $(T_{i}, D_i, \varphi_{i})$-stable colorings for any $i$ with
$1\le i \le n-1$; that is, $|T_{i+1}|$ is maximum over all tree sequences $T_{i+1}'$, which is a closure of $T_i+f_i$
(resp. $T_i$) under a $(T_{i}, D_i, \varphi_{i})$-stable coloring $\varphi_i'$ if $\Theta_i=RE$ or $SE$
(resp. if $\Theta_i=PE$), where $f_i$ is the connecting edge in $F_i$. }
\end{definition}
At this point a natural question is to ask whether an ETT with sufficiently large size and satisfying the maximum property
can be constructed to fulfill our needs. We shall demonstrate that it is indeed the case (see Lemma \ref{welldefined}).
The statement below follows instantly from the above two definitions and Lemma \ref{sc2} (the details can also be found in
the proof of Lemma \ref{welldefined}).
\begin{lemma}\label{MP}
Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series $\mathcal {T}=\{(T_i,
\varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, let $\sigma_n$ be a $\varphi_n \bmod T_n$ coloring,
and let $T^*$ be an ETT corresponding to $(\sigma_n, T_n)$ (see Definition \ref{wz2}). If $T$ satisfies MP under
$\varphi_n$, then $T^*$ satisfies MP under $\sigma_n$. \hfill \rule{4pt}{7pt}
\end{lemma}
The importance of the maximum property is revealed by the following statement to be established: If $T$ enjoys the
maximum property under $\varphi_n$, then $V(T)$ is elementary with respect to $\varphi_n$; Theorem \ref{ThmGS2} follows
from it and Theorem \ref{egraph2} as a corollary. We shall prove this statement by induction on the rung number
$r(T)$. To carry out the induction step, we need several auxiliary results concerning ETTs with the maximum
property. Thus what we are going to establish is a stronger version.
Let us define a few terms before presenting our theorem. For each $v\in V(T)$, we use $m(v)$ to denote the minimum
subscript $i$ such that $v \in V(T_i)$. Let $\alpha$ and $\beta$ be two colors in $[k]$. We say that $\alpha$ and $\beta$ are
$T$-{\em interchangeable} under $\varphi_n$ if there is at most one $(\alpha, \beta)$-path with respect to $\varphi_n$
intersecting $T$. When $T$ is closed (that is, $T=T_{n+1}$), we also say that $T$ has the {\em interchangeability
property} with respect to $\varphi_{n}$ if under any $(T, D_n, \varphi_{n})$-stable coloring $\sigma_n$, any two
colors $\alpha$ and $\beta$ are $T$-interchangeable, provided that $\overline{\sigma}_n(T) \cap \{\alpha, \beta\}
\ne \emptyset$ (equivalently $\overline{\varphi}_{n}(T) \cap \{\alpha, \beta\} \ne \emptyset$).
The undefined symbols and notations in the theorem below can all be found in Algorithm 3.1.
\begin{theorem} \label{thm:tech10}
Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. If $T$ has the
maximum property under $\varphi_n$, then the following statements hold:
\begin{itemize}
\vspace{-1.2mm}
\item[(i)] $V(T)$ is elementary with respect to $\varphi_n$.
\vspace{-2mm}
\item[(ii)] $T_{n+1}$ has the interchangeability property with respect to $\varphi_n$.
\vspace{-2mm}
\item[(iii)] For any $i \le n$, if $v_i$ is a supporting vertex and $m(v_i) =j$, then every
$(T_i, D_i, \varphi_i)$-stable coloring $\sigma_i$ is $(T(v_i)-v_i, D_{j-1},\varphi_{j-1})$-stable. In
particular, $\sigma_i$ is $(T_{j-1},D_{j-1},\varphi_{j-1})$-stable. Furthermore, for any two distinct supporting
vertices $v_i$ and $v_j$ with $i,j \le n$, if $m(v_i)=m(v_j)$, then $S_i \cap S_j=\emptyset$.
\vspace{-2mm}
\item[(iv)] If $\Theta_n = PE$, then $P_{v_n}(\gamma_n, \delta_n, \sigma_n)$ contains precisely one vertex,
$v_n$, from $T_n$ for any $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$.
\vspace{-2mm}
\item[(v)] For any $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$ and any defective color $\delta$ of $T_n$
with respect to $\sigma_n$, if $v$ is a vertex but not the smallest one (in the order $\prec$)
in $I[\partial_{\sigma_n, \delta}(T_n)]$, then $v \preceq v_i$ for any supporting or extension vertex
$v_i$ with $i \ge m(v)$.
\vspace{-2mm}
\item[(vi)] Every $(T_n,D_n,\varphi_n)$-stable coloring $\sigma_n$ is a $\varphi_n\bmod T_n$ coloring. (So
every ETT corresponding to $(\sigma_n, T_n)$ (see Definition \ref{wz2}) satisfies
MP under $\sigma_n$ by Lemma~\ref{MP}.)
\end{itemize}
\end{theorem}
\vskip 1mm
Recall that in the definition of maximum property (see Definition \ref{wz3}), $|T_{n+1}|$ is not required
to be maximum over all $(T_n, D_n, \varphi_n)$-stable colorings. This relaxation allows us to proceed by
induction in our proof of Theorem \ref{thm:tech10}. Now let us show that Theorem \ref{ThmGS2} can be deduced
easily from this theorem.
\begin{lemma}\label{welldefined}
Let $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$ be a Tashkinov series
constructed from a $k$-triple $(G,e, \varphi)$ by using Algorithm 3.1. Suppose $T_{n+1}$ has MP under $\varphi_n$.
Then there exists a Tashkinov series $\mathcal {T}^*=\{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$,
satisfying $\sigma_0 \in {\cal C}^k(G-e)$, $|T_{n+1}^*|\ge |T_{n+1}|$, and the following conditions for $1\le i \le n$:
\begin{itemize}
\vspace{-2mm}
\item[(i)] $T_i^*=T_i$;
\vspace{-2mm}
\item[(ii)] $\sigma_i$ is a $(T_i, D_i, \varphi_i)$-stable coloring in ${\cal C}^k(G-e)$; and
\vspace{-2mm}
\item[(iii)] $|T_{i+1}^*|$ is maximum over all $(T_i, D_i, \sigma_i)$-stable colorings (see Definition \ref{wz3}).
\vspace{-2mm}
\end{itemize}
Furthermore, if $T_{n+1}^*$ is not strongly closed with respect to $\sigma_n$, then there exists a Tashkinov series
$\{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+2\}$, such that $T_{n+1}^* \subset T_{n+2}^*$
and $T_{n+2}^*$ satisfies MP under $\sigma_{n+1}$.
\end{lemma}
{\bf Proof.} Let $\mu$ be an arbitrary $(T_n, D_n, \varphi_n)$-stable coloring. Then $\mu$ is a $\varphi_n\bmod T_n$ coloring
by Theorem \ref{thm:tech10}(vi) (with $T=T_{n+1}$). Thus Definition \ref{wz2} guarantees the existence of an ETT, denoted by
$T_{n+1}(\mu)$, corresponding to $(\mu, T_n)$, which is a closure of $T_n+f_n$ (resp. $T_n$) under $\mu$ if $\Theta_n=RE$ or $SE$ (resp. if $\Theta_n=PE$). Let us reserve $\sigma_n$ for a $(T_n, D_n, \varphi_n)$-stable coloring such that $T_{n+1}(\sigma_n)$
has the maximum number of vertices among all these $T_{n+1}(\mu)$'s, and let $T_{i+1}^*=T_{n+1}(\sigma_n)$.
Then $|T_{n+1}^*|\ge |T_{n+1}|$. By Lemma \ref{sc2}, every $(T_n, D_n, \sigma_n)$-stable coloring is a
$(T_n, D_n, \varphi_n)$-stable coloring. So $|T_{n+1}^*|$ is also maximum over all $(T_n, D_n, \sigma_n)$-stable colorings.
Since $\sigma_n$ is a $\varphi_n\bmod T_n$ coloring, by Definition \ref{wz2}, there exists a Tashkinov series $\mathcal {T}^*=\{(T_i^*,
\sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$ that satisfies conditions (i) and (ii) as described
in the lemma. Using the same argument as employed in the preceding paragraph, we see that $|T_{i+1}^*|$ is maximum over
all $(T_i, D_i, \sigma_i)$-stable colorings as well for $1\le i \le n-1$.
Suppose $T_{n+1}^*$ is not strongly closed with respect to $\sigma_n$. Then we can construct a new tuple
$(T_{n+2}^*, \sigma_{n+1}, S_{n+1}, F_{n+1}, \Theta_{n+1})$ by using Algorithm 3.1. Clearly, $T_{n+1}^* \subset T_{n+2}^*$
and $T_{n+2}^*$ satisfies MP under $\sigma_{n+1}$. \hfill \rule{4pt}{7pt}
\vskip 3mm
{\bf Proof of Theorem \ref{ThmGS2}.} Let $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$ be a
Tashkinov series constructed from a $k$-triple $(G,e, \varphi)$, such that
$(a)$ $T_{n+1}$ satisfies MP under $\varphi_n$;
$(b)$ subject to $(a)$, $|T_{n+1}|$ is maximum over all $(T_n, D_n, \varphi_i)$-stable colorings; and
$(c)$ subject to $(a)$ and $(b)$, $T_{n+1}$ contains as many vertices as possible.
\noindent By Lemma \ref{welldefined}, such a Tashkinov series $\mathcal {T}$ exists, and $T_{n+1}$ is strongly closed with respect
to $\varphi_n$. By Theorem \ref{thm:tech10}(i), $V(T_{n+1})$ is elementary with respect to $\varphi_n$. From Theorem
\ref{egraph2}(i) and (iv), we thus deduce that $G$ is an elementary multigraph. \hfill \rule{4pt}{7pt}
\vskip 3mm
The proof of Theorem \ref{thm:tech10} will take up the entire remainder of this paper.
\section{Auxiliary Results}
We prove Theorem~\ref{thm:tech10} by induction on the rung number $r(T)=n$. The present section is devoted to a proof
of statement (ii) in Theorem~\ref{thm:tech10} in the base case and proofs of statements (iii)-(vi) in the general case.
For $n=0$, statement (i) follows from Theorem \ref{TashTree}, statements (iii)-(vi) hold trivially, and statement (ii)
is a corollary of the following more general lemma.
\begin{lemma}\label{tchange}
Let $(G,e, \varphi)$ be a $k$-triple, let $T$ be a closed Tashkinov tree with respect to $e$ and $\varphi$,
and let $\alpha$ and $\beta$ be two colors in $[k]$ with $\overline{\varphi}(T) \cap \{\alpha,\beta\}\ne
\emptyset$. Then there is at most one $(\alpha,\beta)$-path with respect to $\varphi$ intersecting $T$.
\end{lemma}
{\bf Proof.} Assume the contrary: there are at least two $(\alpha,\beta)$-paths $Q_1$ and $Q_2$ with respect
to $\varphi$ intersecting $T$. By Theorem \ref{TashTree}, $V(T)$ is elementary with respect to $\varphi$. So
$T$ contains at most two vertices $v$ with $\overline{\varphi}(v) \cap \{\alpha, \beta\} \ne \emptyset$,
which in turn implies that at least two ends of $Q_1$ and $Q_2$ are outside $T$. By hypothesis, $T$ is closed
with respect to $\varphi$. Hence precisely one of $\alpha$ and $\beta$, say $\alpha$, is in $\overline{\varphi}(T)$.
Thus we further deduce that at least three ends of $Q_1$ and $Q_2$ are outside $T$. Traversing $Q_1$ and $Q_2$
from these ends respectively, we can find at least three $(T, \varphi, \{\alpha, \beta\})$-exit paths
$P_1,P_2,P_3$. We call the tuple $(\varphi,T, \alpha, \beta, P_1,P_2,P_3)$ a {\em counterexample} and use ${\cal K}$
to denote the set of all such counterexamples.
With a slight abuse of notation, let $(\varphi, T, \alpha, \beta, P_1,P_2,P_3)$ be a counterexample in ${\cal K}$ with
the minimum $|P_1|+|P_2|+|P_3|$. For $i=1,2,3$, let $a_i$ and $b_i$ be the ends of $P_i$ with $b_i \in V(T)$,
and $f_i$ be the edge of $P_i$ incident to $b_i$. Renaming subscripts if necessary, we may assume that $b_1\prec b_2
\prec b_3$. Let $\gamma\in\overline{\varphi}(b_3)$ and let $\sigma_1=\varphi/(G-T,\alpha,\gamma)$. Clearly, $\sigma_1 \in
{\cal C}^k(G-e)$ and $T$ is also a Tashkinov tree with respect to $e$ and $\sigma_1$. Furthermore, $f_i$ is colored
by $\beta$ under both $\varphi$ and $\sigma_1$ for $i=1,2,3$.
Consider $\sigma_2=\sigma_1/P_{b_3}(\gamma, \beta, \sigma_1)$. Note that $\beta \in \overline{\sigma}_2(b_3)$.
Let $T'$ be obtained from $T(b_3)$ by adding $f_1$ and $f_2$ and let $T''$ be a closure of $T'$ under $\sigma_2$.
Obviously, both $T'$ and $T''$ are Tashkinov trees with respect to $e$ and $\sigma_2$. By Theorem \ref{TashTree},
$V(T'')$ is elementary with respect to $\sigma_2$.
Observe that none of $a_1,a_2,a_3$ is contained in $T''$, for otherwise, let $a_i \in V(T'')$ for some $i$
with $1\le i \le 3$. Since $\{\beta,\gamma\}\cap \overline{\sigma}_2(a_i) \ne \emptyset$ and
$\beta \in \overline{\sigma}_2(b_3)$, we obtain $\gamma \in \overline{\sigma}_2(a_i)$.
Hence from TAA we see that $P_1,P_2,P_3$ are all entirely contained in $G[T'']$, which in turn implies $\gamma \in \overline{\sigma}_2(a_j)$ for $j=1,2,3$. So $V(T'')$ is not
elementary with respect to $\sigma_2$, a contradiction. Each $P_i$ contains a subpath $Q_i$, which is a $T_2$-exit path
with respect to $\sigma_2$. Since $f_1$ is not contained in $Q_1$, we obtain $|Q_1|+|Q_2|+|Q_3|<|P_1|+|P_2|+|P_3|$.
Thus the existence of the counterexample $(\sigma_2, T'', \gamma, \beta, Q_1,Q_2,Q_3)$ violates the minimality
assumption on $(\varphi, T, \alpha, \beta, P_1,P_2,P_3)$. \hfill \rule{4pt}{7pt}
\vskip 3mm
So Theorem~\ref{thm:tech10} is true in the base case. Suppose we have established that
{\bf (4.1)} Theorem~\ref{thm:tech10} holds for all ETTs with at most $n-1$ rungs and satisfying MP, for some $n\ge 1$.
Let us proceed to the induction step. We postpone the proof of Theorem~\ref{thm:tech10}(i) and (ii) to
Section 7, and present a proof of Theorem~\ref{thm:tech10}(iii)-(vi) in this section. In our proof of the $(i+2)$th
statement in Theorem~\ref{thm:tech10} for $2\le i \le 4$, we further assume that
{\bf (4.} \hskip -2.8mm ${\bm i}${\bf )} the $j$th statement in Theorem~\ref{thm:tech10} holds for all ETTs with at most $n$ rungs and
satisfying MP, for all $j$ with $3\le j \le i+1$.
We break the proof of the induction step into a series of lemmas. The following lemma generalizes Lemma \ref{hku}(ii),
and will be used in the proofs of Theorem~\ref{thm:tech10}(iii) and (iv).
\begin{lemma}\label{samecolor}
Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series $\mathcal {T}=\{(T_i,
\varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. For any $1\le s\le n$ and any edge $f$ incident to $T_s$, if
$\varphi_{s-1}(f)\in\overline{\varphi}_{s-1}(T_s)\cup D_{s-1}$, then $\varphi_t(f) = \varphi_{s-1}(f)$ for any $t$ with $s\le t \le n$, unless
$f=f_{p}\in F_{p}$ for some $p$ with $s\le p \le t$ and $\Theta_{p}=PE$. In particular, if $f$ is an edge in $G[T_s]$ with $\varphi_{s-1}(f)\in\overline{\varphi}_{s-1}(T_s)\cup D_{s-1}$, then $\varphi_t(f) = \varphi_{s-1}(f)$ for any $t$ with $s\le t \le n$.
\end{lemma}
{\bf Proof.} By Lemma \ref{hku}(i), we have $\overline{\varphi}_{i-1}(T_i)\cup D_{i-1} \subseteq \overline{\varphi}_{i}(T_{i+1})\cup D_{i}$ for
all $i \ge 1$. So to establish the first half, it suffices to prove the statement for $t=s$, which is exactly the same as
Lemma \ref{hku}(ii).
Note that if $f$ is an edge in $G[T_s]$, then $f\notin \partial(T_{p})$ for any $p$ with $s\le p\le t$. Hence
$f\neq f_{p}\in F_{p}$ for any $p$ with $s\le p\le t$ and $\Theta_{p}=PE$. Thus the second half also holds. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{extension rules}
(Assuming (4.1)) Theorem~\ref{thm:tech10}(iii) holds for all ETTs with $n$ rungs and satisfying MP; that is,
for any $i \le n$, if $v_i$ is a supporting vertex and $m(v_i) =j$, then every
$(T_i, D_i, \varphi_i)$-stable coloring $\sigma_i$ is $(T(v_i)-v_i, D_{j-1},\varphi_{j-1})$-stable. In
particular, $\sigma_i$ is $(T_{j-1},D_{j-1},\varphi_{j-1})$-stable. Furthermore, for any two distinct supporting
vertices $v_i$ and $v_j$ with $i,j \le n$, if $m(v_i)=m(v_j)$, then $S_i \cap S_j=\emptyset$.
\end{lemma}
{\bf Proof}. By hypothesis, $T$ is an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, and $T$ satisfies MP under $\varphi_n$.
We prove the first half by contradiction. Assume the contrary: there exists a subscript $i \le n$, such that $v_i$ is a
supporting vertex, $m(v_i) =j$, and some $(T_i, D_i, \varphi_i)$-stable coloring $\sigma_i$ is not $(T(v_i)-v_i, D_{j-1},\varphi_{j-1})$-stable. By definition, there exists an edge $f$ incident to $T(v_i)-v_i$, with
$\varphi_{j-1}(f) \in \overline{\varphi}_{j-1}(T(v_i)-v_i)\cup D_{j-1}$, such that $\sigma_i(f) \ne \varphi_{j-1}(f)$,
or there exists a vertex $v$ of $T(v_i)-v_i$ such that $\overline{\sigma}_i(v)\ne \overline{\varphi}_{j-1}(v)$.
In the former case, since $j\leq i$, repeated application of Lemma \ref{hku}(i) and (ii) yields $\overline{\varphi}_{j-1}(T(v_i)-v_i)
\cup D_{j-1}\subseteq \overline{\varphi}_{j-1}(T_j)\cup D_{j-1}\subseteq \overline{\varphi}_{i-1}(T_i)\cup D_{i-1} \subseteq \overline{\varphi}_i(T_i)
\cup D_i$ and $\varphi_i(f) \in \overline{\varphi}_i(T_i)\cup D_i$. Hence $\sigma_i(f)=\varphi_i(f)$, which implies $\varphi_i(f) \ne \varphi_{j-1}(f)$. In the latter case, since $\overline{\sigma}_i(v)=\overline{\varphi}_i(v)$,
we have $\overline{\varphi}_i(v)\ne \overline{\varphi}_{j-1}(v)$. From Lemma \ref{sc1} we deduce that $\varphi_i$ is not
$(T(v_i)-v_i, D_{j-1},\varphi_{j-1})$-stable in either case.
Set $V_i^- = V(T(v_i)-v_i)$. Then there exists an edge $f$ incident to $V_i^-$ with $\varphi_{j-1}(f) \in \overline{\varphi}_{j-1}(V_i^-)\cup D_{j-1}$ such that $\varphi_{j-1}(f) \ne \varphi_i(f)$, or there exist a vertex $v\in V_i^-$ such that $\overline{\varphi}_{j-1}(v)\neq \overline{\varphi}_{i}(v)$. In either case, by Lemma~\ref{samecolor} and Algorithm 3.1, there exists a supporting vertex
$v_k\in V_i^-$ with $j\le k < i$. Thus $j\le i-1$ and $v_k\prec v_i$.
Since $v_i \in V(T_j)$, we have $v_i\in V(T_{i-1})$. Let $\pi_{i-1}$ be the $(T_i,D_{i-1},\varphi_{i-1})$-stable
coloring as specified in Steps 1 and 4 of Algorithm 3.1. Recall that $\delta_{i}=\pi_{i-1}(f_i)$. Since $v_i$ is
the maximum vertex in $I[\partial_{\pi_{i-1}, \delta_{i}}(T_{i})]$, we see that $\delta_{i}$ is a defective color
of $T_{i-1}$ with respect to $\pi_{i-1}$, and $v_i$ is not the smallest vertex in $I[\partial_{\pi_{i-1},
\delta_{i}}(T_{i-1})]$. As $\pi_{i-1}$ is also $(T_{i-1},D_{i-1},\varphi_{i-1})$-stable, applying (4.1) and
Theorem~\ref{thm:tech10}(v) to $v=v_i$ and $\pi_{i-1}$, we obtain $v_i \preceq v_k$; this contradiction
establishes the first half of the assertion. Since $m(v_i) =j$, we have $v_i \notin V(T_{j-1})$. So $T_{j-1}$ is
entirely contained in $T(v_i)-v_i$, and hence $\sigma_i$ is $(T_{j-1},D_{j-1},\varphi_{j-1})$-stable.
To establish the second half, let $v_i$ and $v_j$ be two distinct supporting vertices with $i< j\le n$ and $m(v_i)=m(v_j)$.
We aim to show that $S_i \cap S_j = \emptyset$.
For $k=i,j$, let $\pi_{k-1}$ be the $(T_k,D_{k-1},\varphi_{k-1})$-stable coloring as specified in Steps 1 and 4 of
Algorithm 3.1. Recall that $\delta_{k}=\pi_{k-1}(f_k)$ is a defective color of $T_k$ with respect to $\pi_{k-1}$,
and $v_k$ is the maximum vertex in $I[\partial_{\pi_{k-1}, \delta_{k}}(T_{k})]$. Let $r=m(v_i)=m(v_j)$.
Since $r\le i<j$ and $v_j \in V(T_r)$, we have $v_j \in V(T_{j-1})$. As $\pi_{j-1}$ is also $(T_{j-1}, D_{j-1},
\varphi_{j-1})$-stable, applying Theorem~\ref{thm:tech10}(v) to $\pi_{j-1}$, $T_{j-1}$ and $v=v_j$, we obtain
$v_j \prec v_i$. By definition, $S_i=\{\delta_i, \gamma_i\}$. Observe that
(1) $\gamma_i \notin S_j$. Indeed, since $\gamma_i \in \overline{\varphi}_{i-1}(v_i)$ and $V(T_i)$ is elementary with
respect to $\varphi_{i-1}$ by (4.1) and Theorem~\ref{thm:tech10}(i), we have $\gamma_i \notin \overline{\varphi}_{i-1}(v_j)$.
Let $f$ be the edge incident to $v_j$ with $\varphi_{i-1}(f) =\gamma_i$. Then $f$ is an edge in $G[T_i]$, because
$T_i$ is closed with respect to $\varphi_{i-1}$. By Lemma~\ref{samecolor}, we have $\varphi_{j-1}(f) = \varphi_{i-1}(f)
=\gamma_i$. So $\gamma_i \notin \overline{\varphi}_{j-1}(v_j)$ and $f\notin \partial(T_{j-1})$. Let $\pi'_{j-1}$ be as specified
in Step 4 in Algorithm 3.1. Since $\pi'_{j-1}$ is $(T_j,D_{j-1},\varphi_{j-1})$-stable, we have $\gamma_i \notin \overline{\pi}'_{j-1}(v_j)$ and $\pi'_{j-1}(f)= \gamma_i$, which implies $\gamma_i \notin S_j$.
(2) $\delta_i \notin S_j$. To justify this, note that $V(T_{i+1})$ is elementary with respect to $\varphi_{i}$
by (4.1) and Theorem~\ref{thm:tech10}(i). Since $\delta_i\in \overline{\varphi}_i(v_i)$, we have $\delta_i\notin
\overline{\varphi}_i(v_j)$. Let $f$ be the edge incident to $v_j$ with $\varphi_i(f) =\delta_i$. Since $T_{i+1}$ is closed
with respect to $\varphi_i$, edge $f$ is contained in $G[T_{i+1}]$. Since $j> i$ and $\varphi_i(f) \in \overline{\varphi}_i(T_i)
\cup D_i$, we have $\varphi_{j-1} (f) = \varphi_i(f)=\delta_i$ and $f\notin \partial(T_{j-1})$ by Lemma~\ref{samecolor}.
Let $\pi'_{j-1}$ be as specified in Step 4 of Algorithm 3.1. Then $\pi'_{j-1}$ is $(T_j,D_{j-1},\varphi_{j-1})$-stable.
By definition, $\delta_i \notin \overline{\pi}'_{j-1}(v_j)$ and $\pi'_{j-1}(f)= \delta_i$. Hence $\delta_i\notin S_j$.
Combining (1) and (2), we conclude that $S_i \cap S_j = \emptyset$, as desired. \hfill \rule{4pt}{7pt}
\vskip 3mm
The following lemma asserts that parallel extensions used in Algorithm 3.1 are preserved under
taking certain stable colorings.
\begin{lemma}\label{extension base}
(Assuming (4.1) and (4.2)) Theorem~\ref{thm:tech10}(iv) holds for all ETTs with $n$ rungs and satisfying MP;
that is, if $\Theta_n = PE$, then $P_{v_n}(\gamma_n, \delta_n, \sigma_n)$ contains precisely one vertex,
$v_n$, from $T_n$ for any $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$.
\end{lemma}
{\bf Proof}. Assume the contrary: $P_{v_n}(\gamma_n, \delta_n, \sigma_n)$ contains at least two vertices
from $T_n$ for some $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$. Let $j=m(v_n)$. By applying a series
of Kempe changes to $\sigma_n$, we shall construct a $(T_j(v_n) - v_n, D_{j-1}, \varphi_{j-1})$-stable
coloring $\mu$ and an ETT $T_j^*$ corresponding to $(\mu, T_{j-1})$ with ladder $T_1\subset T_2\subset \ldots \subset T_{j-1}
\subset T_j^*$, such that either $|T_j^*| > |T_j|$ or $V(T_j^*)$ is not elementary with respect to $\mu$, which
contradicts either the maximum property satisfied by $T$ or the induction hypothesis (4.1) on Theorem~\ref{thm:tech10}(i).
Let $L$ denote the set of all subscripts $i$ with $j\le i\le n$, such that $\Theta_i=PE$ and $m(v_i) =j$, where
$v_i$ is the supporting vertex involved in iteration $i$. We partition $L$ into disjoint subsets $L_1, L_2, \dots,
L_{\kappa}$, such that two subscripts $s, t\in L$ are in the same subset if and only if $v_s = v_t$. For $1\le i \le \kappa$,
write $L_i=\{i_1,i_2, \ldots,i_{c(i)}\}$, where $i_1<i_2< \ldots <i_{c(i)}$, and let $w_i$ denote the common supporting vertex
corresponding to $L_i$. Renaming subscripts if necessary, we may assume that $w_1\prec w_2\prec \ldots \prec w_{\kappa}$.
For each $L_i$, define $P_i$ to be the graph with vertex set $V(P_i) = \cup_{t\in L_i} S_t=\cup_{t\in L_i} \{\delta_t, \gamma_t\}$ and edge set $E(P_i) = \{\delta_t\gamma_t\ :\ t\in L_i\}$.
For each $t \in L$, we have $v_t \notin V(T_{j-1})$ because $m(v_t)=j$. It follows that $w_i \notin V(T_{j-1})$ for $1\le i
\le \kappa$. So each $L_i$ consists of all subscripts $t$ with $1\le t \le n$, such
that $\Theta_t=PE$ and $v_t=w_i$. By Lemma \ref{uniquezang}(ii) (with $w_i$ and $L_i$ in place of $u$ and $B_{n-1}$,
respectively), we obtain
(1) $\gamma_{i_2}=\delta_{i_1}, \, \gamma_{i_3}=\delta_{i_2},\, \ldots, \, \gamma_{i_{c(i)}}=\delta_{i_{c(i)-1}}$.
So $P_i$ is the walk: $\gamma_{i_1} \rightarrow \delta_{i_1}=\gamma_{i_2} \rightarrow \delta_{i_2}=\gamma_{i_3} \rightarrow \dots \rightarrow \delta_{i_{c(i)-1}}=\gamma_{i_{c(i)}} \rightarrow \delta_{i_{c(i)}}$, where $\gamma_{i_1} \in
\overline{\varphi}_{i_1-1}(w_i)$ and $\delta_{i_{c(i)}} \in \overline{\varphi}_{i_{c(i)}}(w_i)$.
(2) $P_1, P_2, \ldots, P_{\kappa}$ are pairwise vertex-disjoint paths. In particular, for any $1\le i \le \kappa$ and any
$1\le s<t \le c(i)$, we have $\gamma_{i_s}\ne \delta_{i_t}$.
To justify this, note that $S_p \cap S_q =\emptyset$ whenever $p$ and $q$ are contained in different $L_i$'s
by (4.2) and Theorem~\ref{thm:tech10}(iii). So $P_1, P_2, \ldots, P_{\kappa}$ are pairwise vertex-disjoint. It remains to
prove that each $P_i$ is a path.
Assume on the contrary that $P_i$ contains a cycle. Then $\gamma_{i_s}=\delta_{i_t}$ for some subscripts $s$ and $t$ with $s < t$
by (1). Let $v\in V(T)$ be an arbitrary vertex with $v\prec w_i$. Since $\gamma_{i_s}\in \overline{\varphi}_{i_s-1}(w_i)$, we have $\gamma_{i_s}\notin \overline{\varphi}_{i_s-1}(v)$ by (4.1) and Theorem~\ref{thm:tech10}(i). Let $f$ be the edge incident with $v$ with
$\varphi_{i_s -1}(f) = \gamma_{i_s}$. Since $T_{i_s}$ is closed with respect to $\varphi_{i_s-1}$, edge $f$ is contained
$G[T_{i_s}]$. By Lemma~\ref{samecolor}, we have $\varphi_{i_t-1}(f) = \varphi_{i_s -1}(f)=\gamma_{i_s}$. From the
definitions of $\pi_{i_t-1}$ and $\pi_{i_t-1}'$ in Step 4 of Algorithm 3.1, it follows that $v \not\in I[\partial_{\varphi_{i_t-1}, \gamma_{i_s}}(T_{i_t})]=I[\partial_{\varphi_{i_t-1}, \delta_{i_t}}(T_{i_t})]$. Therefore $w_i$ cannot be the supporting vertex of $T_{i_t}$ with respect to $\varphi_{i_t}$ and connecting color $\delta_{i_t}$ (see Algorithm 3.1); this contradiction proves (2).
(3) $v_n=w_1$.
Assume the contrary: $v_n\ne w_1$. Then $w_1 \prec v_n$. By (2), $P_1$ is a path and $\gamma_{1_1}\ne \delta_{1_{c(1)}}$.
From Lemma \ref{uniquezang}(iii) (with $w_i$ in place of $u$) we thus deduce that $\overline{\varphi}_{1_1-1}(w_1)=(\overline{\varphi}_{1_{c(1)}}(w_1) -\{\delta_{1_{c(1)}}\})\cup \{\gamma_{1_1}\} \neq \overline{\varphi}_{1_{c(1)}}(w_1)$. Since $L_1$ consists of all subscripts $t$ with $1\le t \le n$, such that $v_t=w_1$ and
$\Theta_t=PE$, by Algorithm 3.1 and the definition of stable colorings, $\overline{\varphi}_{j-1}(w_1)=
\overline{\varphi}_{1_1-1}(w_1)$ and $\overline{\varphi}_n(w_1)=\overline{\varphi}_{1_{c(1)}}(w_1)$. Hence $\overline{\varphi}_{j-1}(w_1)
\neq \overline{\varphi}_n(w_1)$. On the other hand, by (4.2) and Theorem~\ref{thm:tech10}(iii),
$\varphi_n$ is a $(T(v_n)-v_n, D_{j-1},\varphi_{j-1})$-stable coloring, which implies $\overline{\varphi}_{j-1}(w_1)=\overline{\varphi}_n(w_1)$; this contradiction justifies (3).
\vskip 2mm
For each $t$ with $1\le t \le n-1$ and $\Theta_t = PE$, let $\epsilon(t)$ be the smallest subscript $s>t$ such that
$\Theta_s \ne RE$. This $\epsilon(t)$ is well defined and $\epsilon(t) \le n$, as $\Theta_n = PE \ne RE$.
Given a coloring $\varphi$ and two colors $\alpha$ and $\beta$, recall that $\alpha$ and $\beta$ are called $T_t$-interchangeable
under $\varphi$ if there is at most one $(\alpha,\beta)$-path with respect to $\varphi$ intersecting $T_t$; that is,
all $(\alpha, \beta)$-chains intersecting $T_t$ are $(\alpha, \beta)$-cycles except possibly one, which is an $(\alpha, \beta)$-path. We say that $\alpha$ and $\beta$ are $T_t$-{\em strongly interchangeable} ($T_t$-SI) under $\varphi$ if for each vertex
$v$ in $T_t-v_t$, the chain $P_v(\alpha, \beta, \varphi)$ is an $(\alpha, \beta)$-cycle which is fully contained in
$G[T_{\epsilon(t)}]$ (equivalently, $V(P_v(\alpha, \beta, \varphi)) \subseteq V(T_{\epsilon(t)})$). Observe that
if $\alpha$ and $\beta$ are $T_t$-SI under $\varphi$, then they are $T_t$-interchangeable under $\varphi$.
Furthermore, $P_{v_t}(\alpha, \beta, \varphi)$ contains only one vertex $v_t$ from $T_t$, if it is
a path.
\begin{claim}\label{sigama-n} The coloring $\sigma_n$ satisfies the following properties:
\begin{itemize}
\vspace{-1.5mm}
\item[(a1)] $\sigma_n$ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable;
\vspace{-2mm}
\item [(a2)] $\sigma_n(f) = \varphi_{j-1}(f)$ for all edges $f$ in $G[T_{j}]$ with $\varphi_{j-1}(f) \in \overline{\varphi}_{j-1}(T_j)
\cup D_{j-1}$; in particular, this equality holds for all edges on $T_j$;
\vspace{-2mm}
\item [(a3)] $\overline{\sigma}_n(v) = \overline{\varphi}_{j-1}(v)$ for all $v\in V(T_j) - \{w_1, w_2, \dots, w_{\kappa}\}$
and $\overline{\varphi}_{j-1}(w_i) = (\overline{\varphi}_n(w_i) - \{\delta_{i_{c(i)}}\} )\cup \{\gamma_{i_1}\}
= (\overline{\sigma}_n(w_i) - \{\delta_{i_{c(i)}} \} )\cup \{\gamma_{i_1}\} $ for each $i=1, 2, \dots, \kappa$;
and
\vspace{-2mm}
\item [(a4)] for any $t\in L-\{n\}$, the colors $\gamma_t$ and $\delta_t$ are $T_t$-SI under $\sigma_n$
\end{itemize}
\end{claim}
To justify this claim, note that $(a1)$ follows instantly from (4.2) and Theorem~\ref{thm:tech10}(iii).
$(a2)$ For each edge $f$ in $G[T_{j}]$ with $\varphi_{j-1}(f) \in \overline{\varphi}_{j-1}(T_j) \cup D_{j-1}$, by Lemma~\ref{samecolor},
we have $\varphi_n(f) = \varphi_{j-1}(f)$. Repeated application of Lemma \ref{hku}(i) and (ii) implies that
$\varphi_{n-1}(f) \in \overline{\varphi}_{n-1}(T_n) \cup D_{n-1}$. By Lemma \ref{hku}(ii), we further obtain
$\varphi_{n}(f) \in \overline{\varphi}_{n}(T_n) \cup D_{n}$. Since $\sigma_n$ is a $(T_n, D_n, \varphi_n)$-stable coloring,
$\sigma_n(f)=\varphi_n(f)$, which implies $\sigma_n(f) = \varphi_{j-1}(f)$. By Lemma \ref{hku}(iii), each edge $f$ on
$T_j$ satisfies $\varphi_{j-1}(f) \in \overline{\varphi}_{j-1}(T_j) \cup D_{j-1}$, so the equality $\sigma_n(f) = \varphi_{j-1}(f)$
holds for all edges $f$ on $T_j$.
$(a3)$ Let $v$ be a vertex in $V(T_j) - \{w_1, w_2, \dots, w_{\kappa}\}$. If $v$ is contained in $T_j(v_n)-v_n$, then
$\overline{\sigma}_n(v)=\overline{\varphi}_{j-1}(v)$ by $(a1)$. So we assume that $v$ is outside $T_j(v_n)-v_n$. Note that $v$ is
not a supporting vertex during any iteration $p$ with $j \le p \le n$ by the definition of $L$. Hence $\overline{\varphi}_n(v)=\overline{\varphi}_{j-1}(v)$ by Algorithm 3.1 and the definition of stable colorings. As $\sigma_n$
is a $(T_n,D_n,\varphi_n)$-stable coloring, $\overline{\sigma}_n(v)=\overline{\varphi}_n(v)=\overline{\varphi}_{j-1}(v)$.
Since $L_i$ consists of all subscripts $t$ with $1\le t \le n$, such that $v_t=w_i$ and
$\Theta_t=PE$, we have $\overline{\varphi}_{j-1}(w_i)=\overline{\varphi}_{i_1-1}(w_i)$ and $\overline{\varphi}_n(w_i)=\overline{\varphi}_{i_{c(i)}}
(w_i)$ by Algorithm 3.1 and the definition of stable colorings. Furthermore, $\overline{\varphi}_{i_1-1}(w_i)=(\overline{\varphi}_{i_{c(i)}}(w_i)-\{\delta_{i_{c(i)}}\}) \cup \{\gamma_{i_1}\}$ by Lemma \ref{uniquezang}(iii)
(with $w_i$ in place of $u$). So $\overline{\varphi}_{j-1}(w_i)=(\overline{\varphi}_n(w_i)-\{\delta_{i_{c(i)}}\}) \cup \{\gamma_{i_1}\}$ for
$1\le i \le \kappa$. Since $\sigma_n$ is a $(T_n, D_n,\varphi_n)$-stable coloring, $\overline{\sigma}_n(w_i) =
\overline{\varphi}_{n}(w_i)$. Hence $\overline{\varphi}_{j-1}(w_i)= (\overline{\sigma}_n(w_i)- \{\delta_{i_{c(i)}}\}) \cup \{\gamma_{i_1}\}$ for $1\le i \le \kappa$.
$(a4)$ By the induction hypothesis (4.1) on Theorem~\ref{thm:tech10}(ii), $\gamma_t$ and $\delta_t$ are
$T_t$-interchangeable under $\varphi_t$. Since $\Theta_t=PE$, $P_{v_t}(\gamma_t, \delta_t, \varphi_t)$ is a path containing
only one vertex $v_t$ from $T_t$ by Algorithm 3.1. For each vertex $v$ in $T_t-v_t$, observe that $P_v(\gamma_t, \delta_t, \varphi_t)$
is a $(\gamma_t, \delta_t)$-cycle, for otherwise, $P_{v_t}(\gamma_t, \delta_t, \varphi_t)$ and $P_v(\gamma_t, \delta_t, \varphi_t)$
are two distinct $(\gamma_t, \delta_t)$-paths intersecting $T_t$, a contradiction. Since $RE$ has
priority over $PE$ and $SE$ in Algorithm 3.1, the cycle $P_v(\gamma_t, \delta_t, \varphi_t)$ is fully contained in
$G[T_{\epsilon(t)}]$, for otherwise, $\Theta_{\epsilon(t)}=RE$, contradicting the definition of $\epsilon(t)$.
Hence $\gamma_t$ and $\delta_t$ are $T_t$-SI under $\varphi_t$. By Lemma~\ref{samecolor}, we have $\varphi_t (f) = \varphi_n(f)$
for each edge $f$ on $P_{v}(\gamma_t, \delta_t, \varphi_t)$, because $\gamma_t, \delta_t\in D_t$. It follows that $\gamma_t$
and $\delta_t$ are $T_t$-SI under $\varphi_n$. Since $\sigma_n$ is $(T_n, D_n, \varphi_n)$-stable, $\{\gamma_t, \delta_t\}=S_t \subseteq \overline{\varphi}_{n}(T_n) \cup D_n$, and $T_{\epsilon(t)} \subseteq T_n$, we deduce that $\gamma_t$ and $\delta_t$ are also $T_t$-SI under
$\sigma_n$. Hence Claim \ref{sigama-n} is established.
\vskip 2mm
Let us now apply the following algorithm to construct a new coloring from $\sigma_n$, which has the same missing color set
as $\varphi_{j-1}$ at each $w_i$ with $i \ge 2$.
\vskip 2mm
{\flushleft \bf (A)} \hskip 1mm Let $I=\emptyset$ and $\sigma= \sigma_n$. While $I \ne L-L_1$, do: let $i\ge 2$ be a subscript
with $L_i - I \ne \emptyset$ and let $t$ be the largest member of $L_i -I$. Set
\[
{\bf A}(i, t) : \qquad \sigma = \sigma/P_{w_i}(\gamma_t, \delta_t, \sigma) \quad \mbox{ and }\quad I = I\cup\{t\}.
\]
Let us make some observations about this algorithm.
\vskip 1mm
(4) Let $I, i, t, \sigma$ be as specified in Algorithm $(A)$ before performing the iteration
$A(i, t)$. Then $P_{w_i}(\gamma_t, \delta_t, \sigma)$ is a path containing precisely one vertex $w_i$ from $T_t$,
with $\delta_t \in \overline{\sigma}(w_i)$. Furthermore, let $\sigma'= \sigma / P_{w_i} (\gamma_t, \delta_t, \sigma)$
and $I'= I\cup \{t\}$ denote the objects generated in the iteration $A(i, t)$. Then for any $s\in L-\{n\}-I'$,
the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under the coloring $\sigma'$.
To justify this, we apply induction on $|I|$. Note that $w_i=v_t$ for each iteration $A(i,t)$ by the definitions of $L_i$
and $w_i$.
Let us first consider the case when $I = \emptyset$. Now $t$ is the largest subscript $i_{c(i)}$ in $L_i$. By Algorithm 3.1
and the definition of stable colorings, we have $\delta_t \in \overline{\varphi}_t(w_i)$ and $\overline{\sigma}_n(w_i)=\overline{\varphi}_n(w_i)=\overline{\varphi}_t(w_i)$. So $\delta_t \in \overline{\sigma}_n(w_i)$.
By $(a4)$ in Claim~\ref{sigama-n},
(5) for any $s\in L-\{n\}$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\sigma_n$.
In particular, (5) holds for $s=t$, so $P_{w_i}(\gamma_t, \delta_t, \sigma_n)$ is a path containing precisely one
vertex $w_i$ from $T_t$. For any $s\in L-\{n,t\}$, either $s\in L_h$ for some $h$ with $h \ne i$ or
$s\in L_i$ with $s<t$. In the former case, $S_s\cap S_t = \emptyset$ by (4.2) and Theorem~\ref{thm:tech10}(iii),
so $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\sigma'$ by (5). In the latter case, $v_s=v_t=w_i$ and
$\epsilon(s) \le t$. Furthermore, no color on any edge in $G[T_t]$ is changed under the Kempe change that transforms
$\sigma$ into $\sigma'$. So $\gamma_s$ and $\delta_s$ are still $T_s$-SI under $\sigma'$.
So we proceed to the induction step. Let us show that $(4)$ holds for a general $I$ with $I \ne L-L_1$.
Let $I, i, t, \sigma$ be as specified in Algorithm $(A)$ before performing the iteration $A(i, t)$.
By induction hypothesis,
(6) for any $s\in L-\{n\}-I$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under the coloring $\sigma$.
\noindent It follows from $(6)$ (with $s=t$) that $\gamma_t$ and $\delta_t$ are $T_t$-SI under the coloring
$\sigma$. So $P_{w_i}(\gamma_t, \delta_t, \sigma)$ contains precisely one vertex $v_t=w_i$ from $T_t$, if
it is a path.
To prove that $P_{w_i}(\gamma_t, \delta_t, \sigma)$ is a path with $\delta_t \in \overline{\sigma}(w_i)$,
we first assume that $t$ is the largest subscript in $L_i$. By (4.2) and Theorem \ref{thm:tech10}(iii), neither
$\gamma_t$ not $\delta_t$ has been used in any Kempe change before the iteration $A(i, t)$. By Algorithm 3.1, the
definition of stable colorings, and the induction hypothesis, we have $\delta_t \in \overline{\varphi}_t(w_i)$ and $\overline{\sigma}(w_i)=\overline{\sigma}_n(w_i)=\overline{\varphi}_n(w_i)=\overline{\varphi}_t(w_i)$. So
$\delta_t \in \overline{\sigma}(w_i)$. Next, we assume that $t$ is not the largest subscript in
$L_i$. Let $t=i_p$. Then $i_{p+1}$ is the smallest element of $L_i$ greater than $i_p$. So the last Kempe change
involving $w_i$ before iteration $A(i, t)$ was performed on a path of the form $P_{w_i}(\gamma_{i_{p+1}},
\delta_{i_{p+1}}, \cdot)$. By induction hypothesis, $\delta_{i_{p+1}}$ was a color missing at $w_i$ before this
Kempe change. Thus $\gamma_{i_{p+1}}$ becomes a missing color at $w_i$ after this operation; it remains to be
missing at $w_i$ until the iteration $A(i, t)$ by (4.2) and Theorem \ref{thm:tech10}(iii).
Hence $\gamma_{i_{p+1}} \in \overline{\sigma}(w_i)$. By (1), we have $\delta_t=\delta_{i_p}=\gamma_{i_{p+1}}$. It follows that $\delta_t \in \overline{\sigma}(w_i)$. Therefore, $P_{w_i}(\gamma_t, \delta_t, \sigma)$ is a path containing precisely one vertex
$w_i$ from $T_t$, with $\delta_t \in \overline{\sigma}(w_i)$.
Let $I'= I\cup \{t\}$ and $\sigma'= \sigma / P_{w_i}(\gamma_t, \delta_t, \sigma)$. For each $s\in L-\{n\}-I'$,
either $s\in L_h$ for some $h$ with $h \ne i$ or $s\in L_i$ with $s<t$. In the former case, $S_s\cap S_t =
\emptyset$ by (4.2) and Theorem~\ref{thm:tech10}(iii), so $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\sigma'$ by (6).
In the latter case, $v_s=v_t$ and $\epsilon(s) \le t$. Furthermore, no color on any edge in $G[T_t]$ is changed under
the Kempe change that transforms $\sigma$ into $\sigma'$. So $\gamma_s$ and $\delta_s$ are still $T_s$-SI under $\sigma'$
by (6). Hence (4) holds.
\vskip 3mm
\begin{claim}\label{phistar}
Let $\varrho_1$ denote the coloring $\sigma$ output by Algorithm $(A)$. Then the following statements hold:
\begin{itemize}
\vspace{-1mm}
\item[(b1)] $\varrho_1$ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable;
\vspace{-2mm}
\item[(b2)] $\overline{\varrho}_1 (v)=\overline{\varphi}_{j-1}(v)$ for all $v\in V(T_j-v_n)$,
$\overline{\varrho}_1(v_n)=\overline{\varphi}_n(v_n)$, and $\varrho_1(f)=\sigma_n(f) =\varphi_{j-1}(f)$ for all edges $f$ on $T_j$;
\vspace{-2mm}
\item[(b3)] for any edge $f\in E(G)$, if $\varrho_1(f)\ne \sigma_n(f)$, then $f$ is not contained in $G[T_j]$ and
$\sigma_n(f) \in \cup_{i\in L-L_1} S_i$; and
\vspace{-2mm}
\item[(b4)] for any $i\in L_1-\{n\}$ (so $v_i=v_n$), the colors $\gamma_i$ and $\delta_i$ are
$T_i$-SI under $\varrho_1$.
\end{itemize}
\end{claim}
To justify this claim, recall from (4) that
(7) at each iteration $A(i, t)$ of Algorithm $(A)$, the chain $P_{w_i}(\gamma_t, \delta_t, \sigma)$ is a path
containing precisely one vertex $w_i=v_t$ from $T_t$, with $\delta_t \in \overline{\sigma}(w_i)$ and $i\ge 2$.
By (3) and the definitions of $L$ and $w_i$'s, we have
(8) $v_n =w_1 \prec w_i$ for all $i\ge 2$, and $T_j \subset T_t$ for each iteration $A(i, t)$ of Algorithm $(A)$.
It follows from (7) and (8) that $\sigma(f) = \sigma_n(f)$ for all edges $f$ incident to $T_j(v_n) - v_n$. So $\sigma$
and hence $\varrho_1$ is a $(T_j(v_n) - v_n, D_{j-1}, \sigma_n)$-stable coloring. By (4.2) and Theorem \ref{thm:tech10}(iii), $\sigma_n$ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable. From Lemma \ref{sc2} we deduce that $\varrho_1$ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable. So $(b1)$ holds.
By $(a3)$ in Claim \ref{sigama-n}, we have
(9) $\overline{\varphi}_{j-1}(w_i)=(\overline{\varphi}_n(w_i)-\{\delta_{i_{c(i)}}\}) \cup \{\gamma_{i_1}\}= (\overline{\sigma}_n(w_i)-
\{\delta_{i_{c(i)}}\}) \cup \{\gamma_{i_1}\}$ for each vertex $w_i$ with $i\ge 2$.
Recall that $S_p \cap S_q =\emptyset$ whenever $p$ and $q$ are contained in different $L_i$'s by (4.2) and Theorem~\ref{thm:tech10}(iii), and that $P_1, P_2, \ldots, P_{\kappa}$ are pairwise vertex-disjoint paths by (2).
After executing Algorithm $(A)$, the direction of each $P_i$ gets reversed (see (1)). Using
Lemma \ref{uniquezang}(iii), we obtain $\overline{\varrho}_1(w_i)=(\overline{\sigma}_n(w_i)-\{\delta_{i_{c(i)}}\})
\cup \{\gamma_{i_1}\}$, so $\overline{\varrho}_1(w_i)=\overline{\varphi}_{j-1}(w_i)$ for $i\ge 2$ by (9). Combining this with $(a3)$ in
Claim \ref{sigama-n}, we see that $\overline{\varrho}_1 (v)=\overline{\varphi}_{j-1}(v)$ for all $v\in V(T_j-v_n)$. By (4), the
path $P_{w_i}(\gamma_t, \delta_t, \sigma)$ involved in each iteration $A(i,t)$ of Algorithm (A) is disjoint from
$v_n=w_1$. So $\overline{\varrho}_1(v_n)= \overline{\sigma}_n(v_n)=\overline{\varphi}_n(v_n)$. In view of (7) and (8), we get
$\sigma(f) = \sigma_n(f)$ for all edges $f$ on $T_j$ at each iteration $A(i, t)$ of Algorithm $(A)$. Hence $\varrho_1(f)
=\sigma_n(f) =\varphi_{j-1}(f)$ for all edges $f$ on $T_j$, where the second equality follows from $(a2)$ in Claim
\ref{sigama-n}. Thus $(b2)$ is established.
Since the Kempe changes performed in Algorithm $(A)$ only involve edges outside $G[T_j]$ and colors in $\cup_{h\in L - L_1} S_h$, we immediately get $(b3)$. Clearly, $(b4)$ follows from $(4)$. This proves Claim \ref{phistar}.
\vskip 2mm
Consider the coloring $\varrho_1 \in {\cal C}^k(G-e)$ described in Claim \ref{phistar}. Let $T_j'$ be a closure of
$T_{j}(v_n)$ under $\varrho_1$. By (4.1) and Theorem~\ref{thm:tech10}(i), $V(T_n)$ is elementary with respect to
$\varphi_{n-1}$, so $|V(T_n)|$ is odd. From Step 4 in Algorithm 3.1, we see that $|\partial_{\pi'_{n-1}, \delta_n}(T_n)|
\ge 3$. Hence $|\partial_{\varphi_{n}, \delta_n}(T_n)| \ge 2$. Since $\sigma_n$ is a $(T_n,D_n,\varphi_n)$-stable coloring,
we have $|\partial_{\sigma_{n}, \delta_n}(T_n)| \ge 2$. By Lemma \ref{hku}(iv), edges in $\partial_{\sigma_n, \delta_n}(T_n)$
are all incident to $V(T_n(v_n)-v_n)$. Furthermore, each color in $\overline{\sigma}_n(T_n)-\{\delta_n\}$ is closed in
$T_n$ under $\sigma_n$. It follows from $(b3)$ and TAA that $T_j'-T_n \ne \emptyset$ and at least one edge in $T_j'-T_n$
is colored by $\delta_n$ under $\varrho_1$. By $(b1)$, $\varrho_1$ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable,
so it is a $(T_{j-1},D_{j-1},\varphi_{j-1})$-stable coloring and hence is a $\varphi_{j-1}\bmod T_{j-1}$ coloring by
(4.1) and Theorem \ref{thm:tech10}(vi). By $(b2)$, we have $\varrho_1(f)=\varphi_{j-1}(f)$
for any edge $f$ on $T_j(v_n)$. Note that $T_j(v_n)$ under $\varrho_1$ is obtained from $T_{j-1}$ by using the same
connecting edge, connecting color, and extension type as $T_j$. By (4.1) and Theorem \ref{thm:tech10}(vi), we obtain
(10) $T_j'$ is an ETT corresponding to coloring $\varrho_1$ and satisfies MP under $\varrho_1$. So $V(T_j')$ is elementary
with respect to $\varrho_1$ by (4.1) and Theorem \ref{thm:tech10}(i), because $j \le n$.
\vskip 2mm
Depending on the intersection of $\overline{\varrho}_1(T_j'-v_n)$ and $\cup_{i\in L_1}S_i$, we consider two cases.
{\flushleft \bf Case 1.} $\overline{\varrho}_1(T_j'-v_n)\cap (\cup_{i\in L_1}S_i)\ne \emptyset$.
\vskip 1mm
Let $u$ be the minimum vertex (in the order $\prec$) in $T_j'-v_n$ such that $\overline{\varrho}_1(u)
\cap (\cup_{i\in L_1}S_i)\neq \emptyset$. Clearly, $u \ne v_n$. By (10), $V(T_j')$ is elementary with respect to
$\varrho_1$. Since $\delta_n\in \overline{\varphi}_n(v_n) =\overline{\varrho}_1(v_n)$ by $(b2)$, we have
$\delta_n\notin \overline{\varrho}_1(T_j'-v_n)$; in particular, $\delta_n\notin \overline{\varrho}_1(u)$.
Recall that $L_1=\{1_1,1_2, \ldots,1_{c(1)}\}$ and that $n= 1_{c(1)}$. Since $\delta_{1_{c(1)}}
\notin \overline{\varrho}_1(u)$ and $\delta_{1_s} = \gamma_{1_{s+1}}$ for any $1_s\in L_1$ with $1_s< n$ (see $(1)$),
the definition of $u$ guarantees the existence of a minimum member $r$ (as an integer) of $L_1$, such that
$\gamma_r \in \overline{\varrho}_1(u)$. Note that $\gamma_r \in \cup_{i\in L_1}S_i$. Since $m(v_r)=j$, we have
$r \geq j$. Let us show that
(11) $u\in V(T_j')-V(T_r)$.
Otherwise, $u\in V(T_r)$. Since $\gamma_r\in \overline{\varrho}_1(u)$, we obtain $\gamma_r\in \overline{\sigma}_n(u)$,
for otherwise, there exists an edge $f$ incident with $u$ such that $\varrho_1(f) \ne \sigma_n(f) =\gamma_r$. It follows
from $(b3)$ that $\gamma_r \in \cup_{i\in L-L_1}S_i$, so $(\cup_{i\in L_1}S_i) \cap (\cup_{i\in L-L_1}S_i) \ne \emptyset$,
contradicting Theorem~\ref{thm:tech10}(iii). Since $\sigma_n$ is $(T_n,D_n,\varphi_n)$-stable, $\gamma_{r}\in \overline{\varphi}_n(u)$.
On the other hand, by (4.1) and Theorem~\ref{thm:tech10}(i), $V(T_r)$ is elementary with respect to $\varphi_{r-1}$.
From Step 4 in Algorithm 4.1, we see that $\gamma_r\in \overline{\varphi}_{r-1}(v_n)$ (as $v_r=v_n$), so $G[T_r]$ contains an edge
$f$ incident to $u$ with $\varphi_{r-1}(f)=\gamma_{r}$. By Lemma \ref{samecolor}, we obtain $\varphi_n(f)=\gamma_r$. Hence $\gamma_{r}\in \varphi_n(u)$; this contradiction justifies (11).
(12) $\overline{\varrho}_1(T'_j(u)-u)\cap (\cup_{i \in L_1} S_i-\{\delta_n\}) = \emptyset$.
By the minimality assumption on $u$, we have $\overline{\varrho}_1(T_j'(u)-\{v_n,u\}) \cap (\cup_{i\in L_1}S_i)=\emptyset$.
Using Lemma \ref{uniquezang}(i), we obtain $\overline{\varphi}_n(v_n) \cap (\cup_{i \in L_1} S_i)=\{\delta_n\}$. It follows
from $(b2)$ in Claim \ref{phistar} that $\overline{\varrho}_1(v_n) \cap (\cup_{i \in L_1} S_i)=\{\delta_n\}$. Thus
(12) holds.
Let $r$ be the subscript as defined above (11). Then $r=1_p$ for some $1\le p \le c(1)$. By (1), we have
$\gamma_r=\gamma_{1_p}=\delta_{1_{p-1}}$. Let $L_1^* =\{1_1, 1_2, \ldots, 1_{p-1}\}$. Since $1_{p-1}<1_p=r\le n$,
we have $n \notin L_1^*$. Observe that
(13) $\overline{\varrho}_1(v_n) \cap (\cup_{i\in L_1^*} S_i) = \emptyset$.
Indeed, by $(b2)$ in Claim \ref{phistar} and Lemma \ref{uniquezang}(i), we obtain $\overline{\varrho}_1(v_n)=
\overline{\varphi}_n(v_n)$ and $\overline{\varphi}_n(v_n) \cap (\cup_{i \in L_1} S_i)=\{\delta_n\}$. As $n \notin L_1^*$, from (1) and (2)
we see that $\delta_n \notin \cup_{i\in L_1^*} S_i$. So $\overline{\varphi}_n(v_n) \cap (\cup_{i \in L_1^*} S_i)=\emptyset$.
Hence (13) follows.
\vskip 2mm
We construct a new coloring from $\varrho_1$ by using the following algorithm.
{\flushleft \bf (B)} Let $I = \emptyset$ and $\varrho= \varrho_1$. While $I \ne L_1^*$, do: let $t$ be the largest
member of $L_1^* -I$ and set
\[
{\bf B}(t): \qquad \varrho = \varrho/P_u(\gamma_t, \delta_t, \varrho) \quad \mbox{ and } \quad I = I\cup \{t\}.
\]
Let us exhibit some properties satisfied by this algorithm.
(14) Let $I, t, \varrho$ be as specified in Algorithm $(B)$ before performing the iteration $B(t)$. Then
$\delta_t \in \overline{\varrho}(u)$, and $P_u(\gamma_t, \delta_t, \varrho)$ is a path containing at most one vertex
$v_n$ from $T_t$, but $v_n$ is not an end of $P_u(\gamma_t, \delta_t, \varrho)$. Furthermore, let $\varrho' =
\varrho/P_u(\gamma_t, \delta_t, \varrho)$ and $I'= I\cup \{t\}$ denote the objects generated in the iteration
$B(t)$. Then for any $s\in L_1^*-I'$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under the
coloring $\varrho'$.
To justify this, we apply induction on $|I|$. Note that $v_n=v_t$ for each iteration $B(t)$ by the definition of $L_1$.
Let us first consider the case when $I = \emptyset$. Now $t$ is the largest member of $L_1^*$ (that is, $t=1_{p-1}$).
So $\delta_t = \delta_{1_{p-1}}=\gamma_{1_p}=\gamma_r \in \overline{\varrho}(u)$. By $(b4)$ in Claim~\ref{phistar},
(15) for any $s\in L_1-\{n\}$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\varrho_1$.
In particular, (15) holds for $s=t$, so $P_u(\gamma_t, \delta_t, \varrho)$ is a path containing at most one vertex
$v_n=v_t$ from $T_t$. From (13) we see that $v_n$ is not an end of $P_u(\gamma_t, \delta_t, \varrho)$.
For any $s\in L_1^*-\{t\}$, we have $s<t$. So $\epsilon(s) \le t$. Since no color on any edge in $G[T_t]$ is changed
under the Kempe change that transforms $\varrho$ into $\varrho'$, the colors $\gamma_s$ and $\delta_s$ are still $T_s$-SI
under $\varrho'$ by (15).
So we proceed to the induction step. Let us show that $(14)$ holds for a general $I$ with $I \ne L_1^*$.
Let $I, t, \varrho$ be as specified in Algorithm $(B)$ before performing the iteration $B(t)$.
Let $t=i_q$. Then $i_{q+1}$ is the smallest element of $L_1^*$ greater than $i_q$. So in the iteration
$B(i_{q+1})$, the Kempe change was performed on a path of the form $P_{u}(\gamma_{i_{q+1}},
\delta_{i_{q+1}}, \cdot)$. By induction hypothesis, $\delta_{i_{q+1}}$ was a color missing at $u$ before
the iteration $B(i_{q+1})$. So $\gamma_{i_{q+1}}$ becomes a missing color at $u$ after this operation.
Hence $\gamma_{i_{q+1}} \in \overline{\varrho}(u)$. By (1), we have $\delta_t=\delta_{i_q}=\gamma_{i_{q+1}}$.
Thus $\delta_t \in \overline{\varrho}(u)$. By induction hypothesis,
(16) for any $s\in L_1^*-I$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\varrho$.
In particular, (16) holds for $s=t$, so $P_u(\gamma_t, \delta_t, \varrho)$ is a path containing at most one vertex
$v_n=v_t$ from $T_t$. Since none of the path involved in previous Kempe changes terminates at $v_n$, by (13) we
have $\overline{\varrho}(v_n) \cap (\cup_{i\in L_1^*} S_i) = \emptyset$. It follows that $v_n$ is not an end of
$P_u(\gamma_t, \delta_t, \varrho)$.
Let $I'= I\cup \{t\}$ and $\varrho' = \varrho/P_u(\gamma_t, \delta_t, \varrho)$. For each $s\in L_1^*-I'$, we have
$s<t$. So $\epsilon(s) \le t$. Since no color on any edge in $G[T_t]$ is changed under the Kempe
change that transforms $\varrho$ into $\varrho'$, the colors $\gamma_s$ and $\delta_s$ are still $T_s$-SI under $\sigma'$.
Hence (14) holds.
\begin{claim}\label{phistar2}
Let $\varrho_2$ denote the coloring $\varrho$ output by Algorithm $(B)$. Then the following statements hold:
\begin{itemize}
\vspace{-1mm}
\item[(c1)] $\varrho_2 $ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable;
\vspace{-2mm}
\item[(c2)] $\overline{\varrho}_2 (v)=\overline{\varrho}_1(v)$ for all $v\in V(T_j\cup T_j'(u)-u)$ and $\varrho_2 (f)=\varrho_1(f)$ for all $f\in E(T_j\cup T_j'(u))$;
\vspace{-2mm}
\item[(c3)] $\gamma_{1_1}\in \overline{\varrho}_2 (u)$.
\end{itemize}
\end{claim}
To justify this claim, recall from (14) that
(17) at each iteration $B(t)$, the path $P_u(\gamma_t, \delta_t, \varrho)$ contains at most one vertex $v_n$ from
$T_t$, but $v_n$ is not an end of $P_u(\gamma_t, \delta_t, \varrho)$.
Since $T_j \subseteq T_t$, we have $\varrho(f) = \varrho_1(f)$ (and hence $\varrho_2(f) = \varrho_1(f)$) for each edge $f$
incident to $T_j(v_n) - v_n$ by (17). It follows that $\varrho_2$ is a $(T_j(v_n) - v_n, D_{j-1}, \varrho_1)$-stable coloring.
By $(b1)$ in Claim \ref{phistar}, $\varrho_1$ is a $(T_j(v_n) -v_n, D_{j-1}, \varphi_{j-1})$-stable coloring. From
Lemma \ref{sc2} we see that $(c1)$ holds.
Similarly, from (17) we deduce that $\overline{\varrho}_2(v) = \overline{\varrho}_1(v)$ for all $v\in V(T_j)$ and $\varrho_2(f) = \varrho_1(f)$
for all $f\in E(T_j)$. In view of (12), it is clear that $T_j'(u)$ does not contain the other end of $P_u(\gamma_t,
\delta_t, \varrho)$ at each iteration $B(t)$. So $\overline{\varrho}_2(v) = \overline{\varrho}_1(v)$ for each $v\in V(T_j'(u) -u)$.
By (1), (2) and (12), we also have $\overline{\varrho}_1(T'_j(u)-u)\cap (\cup_{i \in L_1^*} S_i) = \emptyset$. Since $T_{j}'$ is a
closure of $T_j(v_n)$ under $\varrho_1$, from TAA we deduce that $\varrho_1 \langle T_j'(u) - T_j(v_n) \rangle
\cap (\cup_{i\in L_1^*}S_i) = \emptyset$. It follows that $\varrho(f) = \varrho_1(f)$ for all edges $f$ in $T_j'(u) - T_j(v_n)$
at each iteration $B(t)$. So $\varrho_2(f) = \varrho_1(f)$ for all edges $f$ in $T_j'(u) - T_j(v_n)$
and hence $(c2)$ holds.
By (14), we have $\delta_t \in \overline{\varrho}(u)$ before each iteration $B(t)$. So $\gamma_t$ becomes a
missing color at $u$ after performing iteration $B(t)$. It follows that $\gamma_{1_1}\in \overline{\varrho}_2 (u)$.
Hence $(c3)$ and therefore Claim \ref{phistar2} is established.
\vskip 2mm
By $(c1)$ in Claim~\ref{phistar2}, $\varrho_2$ is $(T_j(v_n)-v_n,D_{j-1},
\varphi_{j-1})$-stable, so it is a $(T_{j-1},D_{j-1},\varphi_{j-1})$-stable coloring and hence is a
$\varphi_{j-1}\bmod T_{j-1}$ coloring by (4.1) and Theorem \ref{thm:tech10}(vi). By $(b2)$ and $(c2)$, we have
$\varrho_2(f)=\varphi_{j-1}(f)$ for each edge $f$ on $T_j(v_n)$. Let $T_j^1$ be a closure of $T_{j}(v_n)$ under
$\varrho_2$. Then $T_j^1$ is an ETT corresponding to the coloring $\varrho_2$ and satisfies the maximum property
under $\varrho_2$ by (4.1) and Theorem \ref{thm:tech10}(vi), because it is obtained from $T_{j-1}$ by using the
same connecting edge, connecting color, and extension type as $T_j$. In view of $(b2)$ and $(c2)$, we have
$\bullet$ $\overline{\varrho}_2(v)=\overline{\varphi}_{j-1}(v)$ for all $v\in V(T_j-v_n)$;
$\bullet$ $\varrho_2 (f)=\varphi_{j-1}(f)$ for all $f\in E(T_j)$;
$\bullet$ $\overline{\varrho}_2(v)=\overline{\varrho}_1(v)$ for all $v\in V(T_j'(u)-u)$;
$\bullet$ $\varrho_2 (f)=\varrho_1 (f)$ for all $f\in E(T_j'(u))$; and
$\bullet$ $\overline{\varrho}_2(v_n)=\overline{\varphi}_{n}(v_n)$.
\noindent Using $(c3)$ and Lemma \ref{uniquezang}(iii), we obtain $\gamma_{1_1}\in \overline{\varrho}_2(u)$ and
$\overline{\varphi}_{j-1}(v_n) =\overline{\varphi}_{1_1-1}(v_n) \subseteq \overline{\varphi}_{1_{c(1)}}(v_n)
\cup \{\gamma_{1_1}\} = \overline{\varphi}_{n}(v_n) \cup \{\gamma_{1_1}\}=\overline{\varrho}_2(v_n) \cup
\{\gamma_{1_1}\}$. Therefore, $V(T_j \cup T_j'(u))\subseteq V(T_j^1)$ by TAA, which contradicts the
maximum property satisfied by $T$ under $\varphi_n$, because $u \notin V(T_j)$.
\vskip 2mm
{\bf Case 2.} $\overline{\varrho}_1(T_j'-v_n)\cap (\cup_{i\in L_1}S_i)= \emptyset$.
\vskip 1mm
Recall that $L_1=\{1_1,1_2,...,1_{c(1)}\}$. Set $S'=\cup_{i\in L_1}S_i$. Let us make some simple observations about
$T_j$ and $T_j'$.
\vskip 1mm
(18) $\overline{\varrho}_1 (T_j') \cap S'=\overline{\varrho}_1(v_n) \cap S'=\{\delta_n\}$ and $\varrho_1 \langle T_j' -T_j(v_n) \rangle \cap S'=\{\delta_n\}$.
To justify this, note that $V(T_j')$ is elementary with respect to $\varrho_1$ by (10) and that
$\overline{\varrho}_1(v_n)=\overline{\varphi}_n(v_n)$ by $(b2)$. By Lemma \ref{uniquezang}(i), we have $\overline{\varphi}_n(v_n) \cap S'=
\{\delta_n\}$. So $\overline{\varrho}_1(v_n) \cap S'=\{\delta_n\}$ and hence $\delta_n \notin \overline{\varrho}_1
(T_j'-v_n)$. By the hypothesis of the present case, we obtain $\overline{\varrho}_1 (T_j') \cap S'=\overline{\varrho}_1(v_n) \cap S'=\{\delta_n\}$. Since $T_j'$ is a closure of $T_{j}(v_n)$ under $\varrho_1$, from TAA and the paragraph above
(10), we deduce that $\varrho_1 \langle T_j' -T_j(v_n) \rangle \cap S'=\{\delta_n\}$. Hence (18) holds.
(19) $\delta_n\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$ and $\delta_n\notin \varrho_1 \langle T_j-T_j(v_n) \rangle$.
Assume on the contrary that $\delta_n\in \overline{\varrho}_1(T_j-V(T_j(v_n)))$. Then $\delta_n\in\overline{\varphi}_{j-1}(T_j-V(T_j(v_n)))$
by $(b2)$ in Claim~\ref{phistar}. Since $V(T_j)$ is elementary with respect to $\varphi_{j-1}$ by (4.1) and Theorem
\ref{thm:tech10}(vi), we have $\delta_n\notin\overline{\varphi}_{j-1}(v_n)$. So $G[T_j]$ contains an edge $f$ incident to $v_n$
colored by $\delta_n$ under $\varphi_{j-1}$. By Lemma \ref{samecolor}, $\varphi_n(f)=\varphi_{j-1}(f)=\delta_n$. Hence $\delta_n\in\varphi_n(v_n)$; this contradiction proves that $\delta_n\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$.
By (3) and Lemma \ref{uniquezang}(iii), we have $\overline{\varphi}_{j-1}(v_n)=\overline{\varphi}_{1_1-1}(v_n)=(\overline{\varphi}_{1_{c(1)}}(v_n)-
\{\delta_{1_{c(1)}} \}) \cup \{\gamma_{1_1}\} = (\overline{\varphi}_n(v_n)-\{\delta_n\}) \cup \{\gamma_{1_1}\}$. So $\delta_n\notin\overline{\varphi}_{j-1}(v_n)$ (see (2)). By (18), we obtain $\delta_n\notin \overline{\varrho}_1
(T_j(v_n)-v_n)$, which together with $(b2)$ implies $\delta_n\notin\overline{\varphi}_{j-1}(T_j(v_n)-v_n)$. Hence
$\delta_n\notin\overline{\varphi}_{j-1}(T_j(v_n))$. As $\delta_n\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$, we further conclude that $\delta_n\notin\overline{\varphi}_{j-1}(T_j)$ by $(b2)$. Hence no edge in $T_j-T_j(v_n)$ is colored by $\delta_n$ under
$\varphi_{j-1}$, because $T_j$ is a closure of $T_{j}(v_n)$ under $\varphi_{j-1}$ by TAA. It follows from $(b2)$
that $\delta_n\notin \varrho_1 \langle T_j-T_j(v_n) \rangle$. So (19) is justified.
\vskip 3mm
By Lemma \ref{hku}(iv), $\partial_{\varphi_n, \gamma_n}(T_n)=\{f_n\}$, and edges in $\partial_{\varphi_n, \delta_n}(T_n)$
are all incident to $V(T_n(v_n)-v_n)$. These two properties remain valid if we replace $\varphi_n$ by $\sigma_n$, because
$\sigma_n$ is $(T_n, D_n, \varphi_n)$-stable. Thus, by $(b3)$ in Claim~\ref{phistar}, they also hold true if we replace $\varphi_n$ by
$\varrho_1$. Since $T_j'$ is a closure of $T_j(v_n)$ under $\varrho_1$ and $\delta_n \in \overline{\varphi}_n(v_n)=
\overline{\varrho}_1(v_n)$ by $(b2)$, from TAA we see that no boundary edge of $T_n \cup T_j'$ is colored by
$\delta_n$ under $\varrho_1$. So $\partial_{\varrho_1, \gamma_n}(T_n)=\{f_n\}$ and
$\partial_{\varrho_1, \delta_n}(T_n \cup T_j')=\emptyset$.
At the beginning of our proof, we assume that $P_{v_n}(\gamma_n, \delta_n, \sigma_n)$ contains at least two vertices
from $T_n$. Let $P$ denote $P_{v_n}(\gamma_n, \delta_n, \varrho_1)$. Then $P=P_{v_n}(\gamma_n, \delta_n, \sigma_n)$
by $(b3)$ and hence $P \cap T_n \ne \{v_n\}$. Since $\partial_{\varrho_1, \gamma_n}(T_n)=\{f_n\}$ and
$\partial_{\varrho_1, \delta_n}(T_n \cup T_j')=\emptyset$, from the hypothesis of the present case, we deduce that the
other end $x$ of $P$ is outside $T_n \cup T_j'$. Furthermore, $P$ contains a subpath $P[w,x]$, which is a
$T_n \cup T_j'$-exit path with respect to $\varrho_1$. Note that $w$ is contained in $T_j'-V(T_n)$, because the edge
incident with $w$ on $P[w,x]$ is colored by $\gamma_n$ and $\partial_{\varrho_1, \gamma_n}(T_n)=\{f_n\}$. Let $\beta \in \overline{\varrho}_1(w)$. By the hypothesis
of the present case, we have
(20) $\beta\notin S'$.
\noindent Possibly $\beta \in \overline{\varrho}_1(T_j-V(T_j(v_n)))$; in this situation, let $z$
be the first vertex in $T_j-V(T_j(v_n))$ in the order $\prec$ such that $\beta \in \overline{\varrho}_1(z)$.
\begin{claim}\label{phi3}
There exists a coloring $\varrho_3 \in {\cal C}^k(G-e)$ with the following properties:
\begin{itemize}
\vspace{-1mm}
\item[(d1)] $\varrho_3$ is $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable;
\vspace{-2mm}
\item[(d2)] if $\beta\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$, then $\overline{\varrho}_3(v)=\overline{\varrho}_1(v)$
for all $v\in V(T_j\cup T_j'(w)-w)$ and $\varrho_3 (f)={\varrho}_1(f)$ for all $f\in E(T_j\cup T_j'(w))$;
\vspace{-2mm}
\item[(d3)] if $\beta\in \overline{\varrho}_1(T_j-V(T_j(v_n)))$, then $\overline{\varrho}_3(v)=\overline{\varrho}_1(v)$
for all $v\in V(T_j(z)\cup T_j'(w))-\{w,z\}$ and $\varrho_3(f)={\varrho}_1(f)$ for all $f\in E(T_j(z)\cup T_j'(w))$. Furthermore,
$\delta_n\in \overline{\varrho}_3(z)$; and
\vspace{-2mm}
\item[(d4)] $\gamma_{1_1}\in \overline{\varrho}_3(w)$.
\end{itemize}
\end{claim}
{\bf (Assuming Claim \ref{phi3})} By $(d1)$ in Claim~\ref{phi3}, $\varrho_3$ is a $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable
coloring. So it is a $(T_{j-1},D_{j-1},\varphi_{j-1})$-stable coloring and hence is a $\varphi_{j-1}\bmod T_{j-1}$ coloring by
(4.1) and Theorem \ref{thm:tech10}(vi). By $(b2)$, $(d2)$ and $(d3)$, we have $\varrho_3(f)=\varrho_1(f)=\varphi_{j-1}(f)$
for each edge $f$ on $T_j(v_n)$. Let $T_j^2$ be a closure of $T_j'(w)$ under $\varrho_3$. Then $T_j^2$ is an ETT
corresponding to the coloring $\varrho_3$ and satisfies MP under $\varrho_3$ by (4.1) and Theorem \ref{thm:tech10}(vi),
because it is obtained from $T_{j-1}$ by using the same connecting edge, connecting color, and extension type as $T_j$.
By (4.1) and Theorem \ref{thm:tech10}(i), $V(T_j^2)$ is elementary with respect to $\varrho_3$. By $(d4)$, we have $\gamma_{1_1}\in \overline{\varrho}_3(w)$. By Lemma \ref{uniquezang}(iii), we obtain $\overline{\varphi}_{j-1}(v_n) =\overline{\varphi}_{1_1-1}(v_n) \subseteq \overline{\varphi}_{1_{c(1)}}(v_n) \cup \{\gamma_{1_1}\} = \overline{\varphi}_{n}(v_n) \cup \{\gamma_{1_1}\}$. So $\overline{\varphi}_{j-1}(v_n) \subseteq \overline{\varrho}_3(v_n) \cup \{\gamma_{1_1}\}$ by $(b2)$, $(d2)$ and $(d3)$.
When $\beta\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$, by $(b2)$ and $(d2)$ we have
$\bullet$ $\overline{\varrho}_3(v)=\overline{\varphi}_{j-1}(v)$ for all $v\in V(T_j-v_n)$;
$\bullet$ $\varrho_3 (f)=\varphi_{j-1}(f)$ for all $f\in E(T_j)$;
$\bullet$ $\overline{\varrho}_3(v)=\overline{\varrho}_1(v)$ for all $v\in V(T_j'(w)-w)$; and
$\bullet$ $\varrho_3 (f)=\varrho_1 (f)$ for all $f\in E(T_j'(w))$.
\noindent From TAA we see that $V(T_j \cup T_j'(w))\subseteq V(T_j^2)$, which contradicts the maximum property satisfied by $T$.
When $\beta\in \overline{\varrho}_1(T_j-V(T_j(v_n)))$, by $(b2)$ and $(d3)$ we get
$\bullet$ $\overline{\varrho}_3(v)=\overline{\varphi}_{j-1}(v)$ for all $v\in V(T_j(z)-\{z,v_n\})$;
$\bullet$ $\varrho_3 (f)=\varphi_{j-1}(f)$ for all $f\in E(T_j(z))$;
$\bullet$ $\overline{\varrho}_3(v)=\overline{\varrho}_1(v)$ for all $v\in V(T_j'(w)-w)$; and
$\bullet$ $\varrho_3 (f)=\varrho_1 (f)$ for all $f\in E(T_j'(w))$.
\noindent From TAA we conclude that $V(T_j(z) \cup T_j'(w))\subseteq V(T_j^2)$. As $\delta_n\in \overline{\varrho}_3(z)
\cap \overline{\varrho}_3(v_n)$, $V(T_j^2)$ is not elementary with respect to $\varrho_3$, a contradiction again.
\vskip 2mm
To prove Claim \ref{phi3}, we consider the coloring $\varrho_0=\varrho_1/(G-T_j',\beta,\delta_n)$. Since $T_j'$ is closed with
respect to $\varrho_1$ and $\{v_n, w\} \subseteq V(T_j')$, no boundary edge of $T_j'$ is colored by $\beta$ or $\delta_n$ under $\varrho_1$. So $\varrho_0$ is $(T_j', D_{j-1}, \varrho_1)$-stable and hence is $(T_j(v_n)-v_n, D_{j-1}, \varrho_1)$-stable. Clearly, $P_w(\gamma_n, \beta, \varrho_0) = P_w(\gamma_n, \delta_n, \varrho_1)$. Define $\mu_0 = \varrho_0/P_w(\gamma_n, \beta, \varrho_0)$.
\begin{claim}\label{4-4-case2a} The coloring $\mu_0$ satisfies the following properties:
\begin{itemize}
\vspace{-1.5mm}
\item[(e1)] $\mu_0$ is a $(T_j(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable coloring;
\vspace{-2mm}
\item[(e2)] if $\beta\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$, then $\overline{\mu}_0(v)=\overline{\varrho}_1(v)$
for all $v\in V(T_j\cup T_j'(w)-w)$ and $\mu_0 (f)={\varrho}_1(f)$ for all $f\in E(T_j\cup T_j'(w))$;
\vspace{-2mm}
\item[(e3)] if $\beta\in \overline{\varrho}_1(T_j-V(T_j(v_n)))$, then $\overline{\mu}_0(v)=\overline{\varrho}_1(v)$
for all $v\in V(T_j(z)\cup T_j'(w))-\{w,z\}$ and $\mu_0(f)={\varrho}_1(f)$ for all $f\in E(T_j(z)\cup T_j'(w))$. Furthermore,
$\delta_n\in \overline{\varrho}_3(z)$;
\vspace{-2mm}
\item[(e4)] $\gamma_n = \delta_{1_{c(1)-1}}\in \overline{\mu}_0(w)$ and $\beta\notin \overline{\mu}_0(w)$;
\vspace{-2mm}
\item[(e5)] for any $t\in L_1-\{n\}$, the colors $\gamma_t$ and $\delta_t$ are $T_t$-SI under $\mu_0$; and
\vspace{-2mm}
\item[(e6)] $\overline{\mu}_0 (T_j'-w) \cap S'= \overline{\mu}_0 (v_n) \cap S' = \{\delta_n\}$ and
$\mu_0 \langle T_j' -T_j(v_n) \rangle \cap S'=\{\delta_n\}$.
\end{itemize}
\end{claim}
To justify this, recall that $\varrho_1$ is $(T_{j}(v_n)-v_n,D_{j-1},\varphi_{j-1})$-stable by $(b1)$. By the definitions
of $\varrho_0$ and $\mu_0$, the transformation from $\varrho_1$ to $\mu_0$ only changes colors on some edges disjoint
from $V(T_j(v_n))$. So $(e1)$ holds. Statement $(e4)$ follows instantly from the definition of $\mu_0$. Note that
$\delta_n, \beta \notin \cup_{t\in L_1-\{n\}}S_t$ by (1), (2) and (20), and that $T_{\epsilon(t)} \subseteq T_n$ for each
$t\in L_1 - \{n\}$. Furthermore, $P_w(\gamma_n, \beta, \varrho_0)$ is disjoint from $V(T_n)$. So $(e5)$ can be deduced
from $(b4)$ immediately. Using (18) and the definitions of $\varrho_0$ and $\mu_0$, we obtain $(e6)$.
By (10), $V(T_j')$ is elementary with respect to $\varrho_1$. Since $\beta \in \overline{\varrho}_1(w)$, we have
$\beta\notin \overline{\varrho}_1(T_j'-w)$. By $(b2)$, we obtain $\beta\notin \overline{\varphi}_{j-1}(T_j(v_n)-v_n)$ and
$\beta\notin\overline{\varphi}_n(v_n)$. From Lemma \ref{uniquezang}(iii) we deduce that $\overline{\varphi}_{j-1}(v_n)= \overline{\varphi}_{1_1-1}(v_n)
\subseteq \overline{\varphi}_{1_{c(1)}}(v_n) \cup \{\gamma_{1_1}\} = \overline{\varphi}_n(v_n) \cup \{\gamma_{1_1}\}$.
Since $\beta\ne \gamma_{1_1}$ by (20), we get $\beta\notin\overline{\varphi}_{j-1}(v_n)$. Hence
(21) $\beta\notin \overline{\varphi}_{j-1}(T_j(v_n))$.
Suppose $\beta\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$. Then $\beta\notin\overline{\varphi}_{j-1}(T_j)$ by $(b2)$ and (21).
Thus $\beta\notin \varphi_{j-1} \langle T_j-T_j(v_n) \rangle$, because $T_j$ is obtained from $T_j(v_n)$ by TAA under
$\varphi_{j-1}$. By $(b2)$ and (19), we obtain $\beta\notin \varrho_1 \langle T_j-T_j(v_n) \rangle$,
$\delta_n\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$, and $\delta_n\notin \varrho_1 \langle T_j-T_j(v_n) \rangle$.
From the definitions of $\varrho_0$ and $\mu_0$, we see that $(e2)$ holds.
Suppose $\beta\in \overline{\varrho}_1(T_j-V(T_j(v_n)))$. Recall that $z$ is the first vertex in $T_j-V(T_j(v_n))$
in the order $\prec$ with $\beta \in \overline{\varrho}_1(z)$. By $(b2)$ and (21), we get $\beta \in
\overline{\varphi}_{j-1}(z)$ and $\beta \notin \overline{\varphi}_{j-1}(T_j(z)-z)$. Since $T_j$ is obtained from
$T_j(v_n)$ by TAA under $\varphi_{j-1}$, we have $\beta\notin\varphi_{j-1} \langle T_j(z)-T_j(v_n) \rangle $.
It follows from $(b2)$ that $\beta\notin \varrho_1 \langle T_j(z)-T_j(v_n) \rangle$. By (19), we obtain $\delta_n\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$ and $\delta_n\notin \varrho_1 \langle T_j-T_j(v_n) \rangle$.
From the definition of $\varrho_0$ and $\mu_0$, we see that $(e3)$ holds. So Claim \ref{4-4-case2a} is established.
\vskip 2mm
Let $L_1^*= L_1 -\{n\}$. We construct a new coloring from $\mu_0$ by using the following algorithm.
{\flushleft \bf (C)} Let $I= \emptyset$ and $\mu = \mu_0$. While $I \ne L_1^*$, do: let $t$ be the largest member in
$L_1^* -I$ and set
\[
\mbox{ {\bf C}(t):} \qquad \mu = \mu/P_w(\gamma_t, \delta_t, \mu) \quad \mbox{ and } \quad
I = I\cup \{t\}.
\]
Let $\varrho_3$ denote the coloring $\mu$ output by Algorithm $(C)$. We aim to show that $\varrho_3$ is as described in
Claim~\ref{phi3}; our proof is based on the following statement.
(22) Let $I, t, \mu$ be as specified in Algorithm $(C)$ before performing the iteration $C(t)$. Then
$\delta_t \in \overline{\mu}(w)$, and $P_w(\gamma_t, \delta_t, \mu)$ is a path containing at most one vertex
$v_n$ from $T_t$, but $v_n$ is not an end of $P_w(\gamma_t, \delta_t, \mu)$. Furthermore, let $\mu' =
\mu/P_w(\gamma_t, \delta_t, \varrho)$ and $I'= I\cup \{t\}$ denote the objects generated in the iteration
$C(t)$. Then for any $s\in L_1^*-I'$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under the
coloring $\mu'$.
To justify this, we apply induction on $|I|$. Let us first consider the case when $I = \emptyset$. Now $t$ is the
largest member of $L_1^*$ (that is, $t=1_{c(1)-1}$). By $(e4)$ in Claim~\ref{4-4-case2a}, we have $\delta_t = \delta_{1_{c(1)-1}}=\gamma_n \in \overline{\mu}_0(w)$. By $(e5)$, we obtain
(23) for any $s\in L_1^*$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\mu_0$.
In particular, (23) holds for $s=t$, so $P_w(\gamma_t, \delta_t, \mu_0)$ is a path containing at most one vertex
$v_n=v_t$ from $T_t$. By (1), (2) and $(e6)$, we obtain
(24) $\overline{\mu}_0 (v_n) \cap (\cup_{i\in L_1^*} S_i)= \emptyset$.
\noindent From (24) we deduce that $v_n$ is not an end of $P_w(\gamma_t, \delta_t, \mu_0)$. For any
$s\in L_1^*-\{t\}$, we have $s<t$. So $\epsilon(s) \le t$. Since no color on any edge in $G[T_t]$ is changed under
the Kempe change that transforms $\mu=\mu_0$ into $\mu'$, the colors $\gamma_s$ and $\delta_s$ are still $T_s$-SI
under $\mu'$ by (23).
So we proceed to the induction step. Let us show that $(22)$ holds for a general $I$ with $I \ne L_1^*$.
Let $I, t, \mu$ be as specified in Algorithm $(C)$ before performing the iteration $C(t)$.
Let $t=i_q$. Then $i_{q+1}$ is the smallest element of $L_1^*$ greater than $i_q$. So in the iteration
$C(i_{q+1})$, the Kempe change was performed on a path of the form $P_w(\gamma_{i_{q+1}},
\delta_{i_{q+1}}, \cdot)$. By induction hypothesis, $\delta_{i_{q+1}}$ was a color missing at $w$ before
the iteration $B(i_{q+1})$. So $\gamma_{i_{q+1}}$ becomes a missing color at $w$ after this operation.
Hence $\gamma_{i_{q+1}} \in \overline{\mu}(w)$. By (1), we have $\delta_t=\delta_{i_q}=\gamma_{i_{q+1}}$.
Thus $\delta_t \in \overline{\mu}(w)$. By induction hypothesis,
(25) for any $s\in L_1^*-I$, the colors $\gamma_s$ and $\delta_s$ are $T_s$-SI under $\mu$.
In particular, (25) holds for $s=t$, so $P_w(\gamma_t, \delta_t, \mu)$ is a path containing at most one vertex
$v_n=v_t$ from $T_t$. Since none of the path involved in previous Kempe changes terminates at $v_n$, by (24)
we have $\overline{\mu}(v_n) \cap (\cup_{i\in L_1^*} S_i) = \emptyset$. It follows that $v_n$ is not an end of
$P_w(\gamma_t, \delta_t, \mu)$.
Let $I'= I\cup \{t\}$ and $\mu' = \mu/P_w(\gamma_t, \delta_t, \mu)$. For each $s\in L_1^*-I'$, we have
$s<t$. So $\epsilon(s) \le t$. Since no color on any edge in $G[T_t]$ is changed under the Kempe
change that transforms $\mu$ into $\mu'$, the colors $\gamma_s$ and $\delta_s$ are still $T_s$-SI under $\mu'$ by (25).
Hence (22) holds.
\vskip 2mm
To justify Claim~\ref{phi3}, recall from (22) that
(26) at each iteration $C(t)$, the path $P_w(\gamma_t, \delta_t, \mu)$ contains at most one vertex $v_n=v_t$ from
$T_t$, but $v_n$ is not an end of $P_w(\gamma_t, \delta_t, \mu)$.
Since $T_j \subseteq T_t$, we have $\mu(f) = \mu_0(f)$ (and hence $\varrho_3(f) = \mu_0(f)$) for each edge $f$
incident to $T_j(v_n) - v_n$ by (26). It follows that $\varrho_3$ is a $(T_j(v_n) - v_n, D_{j-1}, \mu_0)$-stable coloring.
By $(e1)$ in Claim \ref{4-4-case2a}, $\mu_0$ is a $(T_j(v_n) -v_n, D_{j-1}, \varphi_{j-1})$-stable coloring. From
Lemma \ref{sc2} we see that $(d1)$ holds.
Let us first assume that $\beta\notin \overline{\varrho}_1(T_j-V(T_j(v_n)))$. Again, since $T_j \subseteq T_t$, from (26) we
deduce that $\overline{\varrho}_3(v) = \overline{\mu}_0(v)$ for all $v\in V(T_j)$ and $\varrho_3(f) = \mu_0(f)$
for all $f\in E(T_j)$. By $(e6)$, we have $\overline{\mu}_0 (T_j'-w) \cap S'= \overline{\mu}_0 (v_n) \cap S' =
\{\delta_n\}$ and $\mu_0 \langle T_j' -T_j(v_n) \rangle \cap S'=\{\delta_n\}$. By (1) and (2),
we obtain $\delta_n \notin \cup_{i \in L_1^*} S_i$. So at each iteration $C(t)$ the path
$P_w(\gamma_t, \delta_t, \mu)$ neither contains any edge from $T_j'(w)$ nor terminate at a
vertex in $T_j'(w)-w$. It follows that $\overline{\varrho}_3(v) = \overline{\mu}_0(v)$ for all $v\in V(T_j'(w)-w)$
and $\varrho_3(f) = \mu_0(f)$ for all edges $f$ in $T_j'(w) -T_j(v_n)$. Hence $\overline{\varrho}_3(v)=
\overline{\mu}_0(v)$ for all $v\in V(T_j\cup T_j'(w)-w)$ and $\varrho_3 (f)={\mu}_0(f)$ for all $f\in
E(T_j\cup T_j'(w))$. Combining this with $(e2)$, we see that $(d2)$ holds.
Similarly, we can prove that if $\beta\in \overline{\varrho}_1(T_j-V(T_j(v_n)))$, then $(d3)$ is true.
By (22), we have $\delta_t \in \overline{\mu}(w)$ before each iteration $C(t)$. So $\gamma_t$ becomes a
missing color at $w$ after performing iteration $C(t)$. It follows that $\gamma_{1_1}\in \overline{\varrho}_3 (w)$.
Hence $(d4)$ is established. This completes the proof of Claim \ref{phi3} and hence of
Lemma~\ref{extension base}. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{basic2}
(Assuming (4.1) and (4.3)) Theorem~\ref{thm:tech10}(v) holds for all ETTs with $n$ rungs and satisfying MP;
that is, for any $(T_n, D_n, \varphi_n)$-stable coloring $\sigma_n$ and any defective color $\delta$ of $T_n$
with respect to $\sigma_n$, if $v$ is a vertex but not the smallest one (in the order $\prec$)
in $I[\partial_{\sigma_n, \delta}(T_n)]$, then $v \preceq v_i$ for any supporting or extension vertex
$v_i$ with $i \ge m(v)$.
\end{lemma}
{\bf Proof.} By hypothesis, $T$ is an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, and $T$ satisfies MP under $\varphi_n$.
Depending on the extension type $\Theta_n$, we consider two cases.
{\bf Case 1.} $\Theta_n=PE$. In this case, according to Step 4 in Algorithm 3.1, $\pi_{n-1}'$ is a
$(T_n, D_{n-1}\cup\{\delta_n\}, \pi_{n-1})$-stable coloring, $v_n$ is a $(T_n, \pi_{n-1}',
\{\gamma_n, \delta_n\})$-exit and $\varphi_n = \pi_{n-1}'/ P_{v_n}(\gamma_n, \delta_n, \pi_{n-1}')$.
Since $\sigma_n$ is a $(T_n, D_n, \varphi_n)$-stable coloring, from Lemma \ref{hku}(iv) we deduce that
$\partial_{\sigma_n, \gamma_n}(T_n)=\{f_n\}$. So $\delta\ne \gamma_n$.
By Theorem~\ref{thm:tech10}(iv), $P_{v_{n}}(\gamma_{n},\delta_{n},\sigma_n)\cap T_{n}=\{v_{n}\}$. Define $\sigma_{n-1}=\sigma_n/P_{v_{n}} (\gamma_{n},\delta_{n}, \sigma_n)$. Then
(1) $\sigma_{n-1}$ is $(T_{n},D_{n-1},\varphi_{n-1})$-stable by Lemma \ref{stablezang} and hence it
is also $(T_{n-1},D_{n-1},\varphi_{n-1})$-stable. Furthermore, $\partial_{\sigma_n, \delta}(T_n) \subseteq
\partial_{\sigma_{n-1}, \delta}(T_n)$ (as $\delta\ne \gamma_n$).
If $i<n$, then $v\in T_{n-1}$ because $m(v)\leq i< n$. Since $v$ is not the smallest vertex in $I[\partial_{\sigma_n,
\delta}(T_n)]$, from (1) it can be seen that $\delta$ is a defective
color of $T_{n-1}$ with respect to $\sigma_{n-1}$. Applying (4.3) and Theorem~\ref{thm:tech10}(v) to $T_{n-1}$ and $\sigma_{n-1}$
(see (1)), we obtain $v\preceq v_i$. So we assume that $i=n$. Since $v_n$ the maximum defective vertex with respect to
$(T_n, D_{n-1}, \varphi_{n-1})$ (see the definition above (3.1)), by (1) we also have $v\preceq v_n$.
{\bf Case 2.} $\Theta_n=RE$ or $SE$. In this case, $\varphi_n$ is $(T_{n},D_{n-1},\varphi_{n-1})$-stable (see Algorithm 3.1).
Since $\sigma_n$ is $(T_n, D_n, \varphi_n)$-stable and $\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}\subseteq \overline{\varphi}_{n}(T_n)\cup D_n$
by Lemma \ref{hku}(i), we deduce that $\sigma_n$ is $(T_n,D_{n-1},\varphi_{n-1})$-stable and
hence is also $(T_{n-1},D_{n-1},\varphi_{n-1})$-stable. If $i<n$, then $m(v)<n$. Since $v\in T_{n-1}$, $\delta$ is a
defective color of $T_{n-1}$ with respect to $\sigma_n$. Thus $v\preceq v_i$ by (4.3) and Theorem~\ref{thm:tech10}(v).
So we assume that $i=n$. Since $v_n$ the maximum defective vertex with respect to $(T_n, D_{n-1},
\varphi_{n-1})$, we also have $v\preceq v_n$. \hfill \rule{4pt}{7pt}
\vskip 3mm
The proof of Theorem~\ref{thm:tech10}(vi) is based on the following technical lemma.
\begin{lemma} \label{basic1}
(Assuming (4.1) and (4.4)) Let $\mathcal {T}_1=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$
(resp. $\mathcal {T}_2=\{(T_i, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n\}$) be a Tashkinov series
constructed from a $k$-triple $(G,e, \varphi_0)$ (resp. $(G,e, \sigma_0))$ by using Algorithm 3.1. Suppose
$T_{n+1}$ satisfies MP under $\varphi_n$, and $\sigma_i$ is a $(T_i, D_i, \varphi_i)$-stable coloring in
${\cal C}^k(G-e)$ for $1 \le i \le n-1$. Furthermore, $\sigma_{n-1}$ is a $(T_n, D_{n-1}, \varphi_{n-1})$-stable
coloring. If $\Theta_n=RE$, then there exists a Tashkinov series $\mathcal {T}_3=\{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1},
\Theta_{i-1}): 1\le i \le n+1\}$, such that $\sigma_n=\sigma_{n-1}$ and $T_i^*=T_i$ for $1\le i \le n$.
\end{lemma}
{\bf Proof.} Since $\Theta_n=RE$, according to Step 1 in Algorithm 3.1, there exists
a subscript $h \le n-1$ with $\Theta_h =PE$ and $S_h =\{\delta_h, \gamma_h\}$, such that for all $i$ with $h+1
\le i \le n-1$, if any, we have $\Theta_i=RE$ and $S_i =\{\delta_i, \gamma_i\}=S_h$, where $\delta_i=\delta_h$ and
$\gamma_i=\gamma_h$, and such that some $(\gamma_h, \delta_h)$-cycle $C$ with respect to $\varphi_{n-1}$ contains both
an edge $f_n \in \partial_{ \varphi_{n-1}, \gamma_h}(T_n)$ and a segment $L$ connecting $V(T_h)$ and $v_n$ with
$V(L) \subseteq V(T_n)$, where $v_n$ is the end of $f_n$ in $T_n$. According to Step 2 in this algorithm,
$\varphi_n = \varphi_{n-1}$, $T_{n+1}$ is a closure of $T_n+f_n$ under $\varphi_{n}$, $\delta_n=\delta_h$,
$\gamma_n=\gamma_h$, $S_n =\{\delta_n,\gamma_n\}$, and $F_n = \{ f_n\}$. Since $\Theta_i=RE$ for $h+1 \le i \le n-1$,
from Algorithm 3.1 we further deduce that
(1) $\varphi_h=\varphi_{h+1}=\ldots =\varphi_n$ and $S_h=S_{h+1}=\ldots =S_n$.
\noindent Moreover,
(2) $\sigma_h=\sigma_{h+1}=\ldots = \sigma_{n-1}$.
\noindent Set $\sigma_n=\sigma_{n-1}$. As $\sigma_i$ is a $(T_i,D_i,\varphi_i)$-stable coloring for $h \le i \le n-1$, by
(2) we get
(3) $\sigma_n$ is $(T_h,D_h,\varphi_h)$-stable.
Let $f_n$, $L$ and $C$ be as specified in the first paragraph of our proof. By the definition of $D_{n-1}$, we have
$\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}=\overline{\varphi}_{n-1}(T_n) \cup (\cup_{i\leq n-1}S_i)$ (see (1) in the proof of Lemma \ref{hku}).
So $\{\delta_h, \gamma_h\} \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$. Since $\sigma_{n-1}$ is $(T_n,D_{n-1},\varphi_{n-1})$-stable,
$\sigma_{n-1}(f)=\varphi_{n-1}(f)$ for all $f\in E(L) \cup \{f_n\}$. Thus edges on $L$ are also colored alternately by
$\delta_h$ and $\gamma_h$ in $\sigma_n$. Let $C^*$ be the $(\delta_h, \gamma_h)$-chain with respect to $\sigma_n$
containing $L$. Then $f_n\in C^*$.
We claim that $C^*$ is a cycle. Assume the contrary: $C^*$ is a $(\delta_h, \gamma_h)$-path with respect to $\sigma_n$.
By Step 4 in Algorithm 3.1, we have $\delta_h \in \overline{\varphi}_h(v_h)$. Using (1), we obtain $\delta_h \in \overline{\varphi}_{n-1}(v_h)$, so $v_h$ is outside $C$. It follows that $L$ and hence $C^*$ contains a vertex different
from $v_h$ in $T_h$. By (3) and Theorem~\ref{thm:tech10}(iv), $P_{v_h}(\delta_h, \gamma_h, \sigma_n)$ contains only vertex
$v_h$ from $T_h$. Thus $C^*$ and $P_{v_h}(\delta_h, \gamma_h, \sigma_n)$ are two disjoint $(\delta_h, \gamma_h)$-paths with
respect to $\sigma_n$. Since $\sigma_h=\sigma_{h+1}=\ldots = \sigma_{n}$, we see that $C^*$ and $P_{v_h}(\delta_h, \gamma_h, \sigma_n)$ are two disjoint $(\delta_h, \gamma_h)$-paths with respect to $\sigma_h$ intersecting $T_{h+1}$; this contradiction to Theorem~\ref{thm:tech10}(ii) justifies the claim.
The above claim and Algorithm 3.1 guarantee the existence of a Tashkinov series
$\mathcal {T}_3= \{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, such that $\sigma_n=\sigma_{n-1}$
and $T_i^*=T_i$ for $1\le i \le n$. \hfill \rule{4pt}{7pt}
\vskip 3mm
The lemma below actually states that ETTs along with the maximum property are also maintained under taking some stable
colorings.
\begin{lemma}\label{td}
(Assuming (4.1) and (4.4)) Theorem~\ref{thm:tech10}(vi) holds for all ETTs with $n$ rungs and
satisfying MP; that is, every $(T_n,D_n,\varphi_n)$-stable coloring $\sigma_n$ is a $\varphi_n\bmod T_n$ coloring.
(So every ETT corresponding to $(\sigma_n, T_n)$ satisfies MP under $\sigma_n$ by Lemma~\ref{MP}.)
\end{lemma}
{\bf Proof.} By hypothesis, $T$ is an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, and $T$ satisfies MP under $\varphi_n$. We aim
to show (recall Definition \ref{wz2}), by induction on $r(T)$, the existence of an extended Tashkinov tree $T^*$ with
corresponding Tashkinov series $\mathcal {T}^*=\{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$,
satisfying $\sigma_0 \in {\cal C}^k(G-e)$ and the following conditions for all $i$ with $1\le i \le n$:
\vskip 1mm
(1) $T_i^*=T_i$ and
(2) $\sigma_i$ is a $(T_i, D_i, \varphi_i)$-stable coloring in ${\cal C}^k(G-e)$,
where $D_i=\cup_{h\le i} S_h - \overline{\varphi}_{i}(T_i)$.
\noindent For this purpose, we shall define a $(T_{n-1},D_{n-1},\varphi_{n-1})$-stable coloring $\sigma_{n-1}$
based on $\sigma_n$, and apply induction hypothesis to $\sigma_{n-1}$.
Since $T_n$ is an ETT constructed from the $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}_n=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n\}$, with $r(T_n)=n-1$, and
since $T_n$ satisfies MP under $\varphi_{n-1}$, by (4.4) and Theorem~\ref{thm:tech10}(vi), $\sigma_{n-1}$
is a $\varphi_{n-1}\bmod T_{n-1}$ coloring. So
(3) there exists a Tashkinov series $\mathcal {T}^*_n=\{(T_i^*, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n\}$,
satisfying $\sigma_0 \in {\cal C}^k(G-e)$ and (1) and (2) for all $i$ with $1\le i \le n-1$.
Our objective is to find $\sigma_{n-1}$, such that
(4) $T_n^*$ can be set to $T_n$, and an ETT $T_{n+1}^*$ with respect to $e$ and $\sigma_n$ can be obtained from
$T_n^*=T_n$ by using the same connecting edge, connecting color, and extension type $\Theta_n$ as $T_{n+1}$ in $\mathcal {T}$.
Combining (3) and (4), we see that $\sigma_n$ is indeed a $\varphi_n\bmod T_n$ coloring. To establish (4), we consider
three cases, according to the extension type $\Theta_n$.
{\bf Case 1.} $\Theta_n=RE$. In this case, define $\sigma_{n-1}=\sigma_n$. By hypothesis, $\sigma_n$ is a
$(T_n,D_n,\varphi_n)$-stable coloring. So $\sigma_{n-1}$ is also $(T_n,D_n,\varphi_n)$-stable. Since $\varphi_n=\varphi_{n-1}$
by Algorithm 3.1 and $\overline{\varphi}_{n-1}(T_n)\cup D_{n-1}\subseteq \overline{\varphi}_{n}(T_n)\cup D_n$ by Lemma \ref{hku}(i), we deduce
that $\sigma_{n-1}$ is $(T_n,D_{n-1},\varphi_{n-1})$-stable and hence is also $(T_{n-1},D_{n-1},\varphi_{n-1})$-stable.
By Lemma \ref{hku}(iii), we have $\sigma_n(f)=\varphi_n(f)$ for any edge $f$ on $T_n$. It follows that
$\sigma_{n-1}(f)=\varphi_{n-1}(f)$ for any edge $f$ on $T_n$. Thus we can set $T^*_n=T_n$. Therefore, by Lemma \ref{basic1},
an ETT $T_{n+1}^*$ with respect to $e$ and $\sigma_n$ can be obtained from $T_n$ by using the same connecting edge,
connecting color, and extension type RE as $T_{n+1}$ in $\mathcal {T}$.
{\bf Case 2.} $\Theta_n=SE$. In this case, according to Step 3 of Algorithm 3.1, $\varphi_n = \pi_{n-1}$,
$T_{n+1}$ is a closure of $T_n+f_n$ under $\varphi_{n}$, $S_n= \{ \delta_n \}$, and $F_n=\{f_n\}$,
where $\pi_{n-1}$ is a $(T_n, D_{n-1}, \varphi_{n-1})$-stable coloring so that $v_{\pi_{n-1}}=v_n$, which is
the maximum defective vertex with respect to $(T_n, D_{n-1}, \varphi_{n-1})$ (see the paragraph above
(3.1)). By the definition of $\varphi_n$, we have
(5) $\varphi_n$ is $(T_n, D_{n-1}, \varphi_{n-1})$-stable and hence is also $(T_{n-1}, D_{n-1}, \varphi_{n-1})$-stable. Moreover,
$\partial_{\varphi_n, \delta_n}(T_n)= \partial_{\pi_{n-1}, \delta_n}(T_n)$.
Define $\sigma_{n-1}=\sigma_n$. Since $\sigma_n$ is a $(T_n,D_n,\varphi_n)$-stable coloring, so is
$\sigma_{n-1}$. In view of (5) and Lemma \ref{hku}(i), we obtain
(6) $\sigma_{n-1}$ is $(T_n,D_{n-1},\varphi_{n-1})$-stable and hence is also $(T_{n-1},D_{n-1},\varphi_{n-1})$-stable.
Moreover, $\partial_{\sigma_{n-1}, \delta_n}(T_n)= \partial_{\sigma_{n}, \delta_n}(T_n)=\partial_{\varphi_n, \delta_n}(T_n)$.
By Lemma \ref{hku}(iii), we have $\varphi_{n-1} \langle T_n \rangle \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$.
It follows from (6) that $\sigma_{n-1}(f)=\varphi_{n-1}(f)$ for any edge $f$ on $T_n$. Thus we can set $T^*_n=T_n$.
Moreover, by (5), (6) and Lemma \ref{sc2}, $v_n$ is also the maximum defective vertex with respect to $(T_n, D_{n-1},
\sigma_{n-1})$ (see the definition above (3.1)). We claim that
(7) for any $(T_n, D_{n-1}\cup\{\delta_n\}, \sigma_{n-1})$-stable coloring $\mu_{n-1}$, there holds $\overline{\mu}_{n-1} (u_n)
\cap \overline{\mu}_{n-1} (T_n)=\emptyset$, where $u_n$ is the vertex of $f_n$ outside $T_n$.
To justify this, note that $\sigma_{n-1}=\sigma_n$ is $(T_n,D_n,\varphi_n)$-stable. Since $\overline{\varphi}_n(T_n)
\cup D_n= \overline{\varphi}_n(T_n) \cup D_{n-1} \cup \{\delta_n\}$, by the definition of stable colorings,
$\sigma_{n-1}$ is $(T_n, D_{n-1} \cup \{\delta_n\}, \varphi_n)$-stable and hence $(T_n, D_{n-1} \cup \{\delta_n\},
\pi_{n-1})$-stable. Therefore $\mu_{n-1}$ is $(T_n, D_{n-1} \cup \{\delta_n\}, \pi_{n-1})$-stable by Lemma \ref{sc2}.
From Step 1 in Algorithm 3.1 we see that $\overline{\mu}_{n-1} (u_n) \cap \overline{\mu}_{n-1} (T_n)=\emptyset$.
By (7), a tree sequence $T_{n+1}^*$ with respect to $e$ and $\sigma_n$ can thus be obtained from $T_n$ by
using Step 3 in Algorithm 3.1 (with $\sigma_{n-1}$ in place of both $\varphi_{n-1}$ and $\pi_{n-1}$) and using
the same connecting edge, connecting color, and extension type SE as $T_{n+1}$ in $\mathcal {T}$.
Recall that RE has priority over both SE and PE in the construction of a Tashkinov series using Algorithm 3.1.
To prove that $T_{n+1}^*$ constructed above is an ETT, we still need to check that no ETT with respect
to $e$ and $\sigma_n$ can be obtained from $T_n$ by using RE. Assume the contrary: $T_{n+1}^*$ (with
a slight abuse of notation) is such an ETT. Since $T$ satisfies MP, so does the ETT $T_{n+1}^*$. Let $\mathcal {T}_1$
be the Tashkinov series obtained from $\{(T_i, \sigma_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n\}$
by adding a tuple $(T_{n+1}^*, \sigma_{n}, S_n^*, F_n^*, \Theta_n^*)$
corresponding to $T_{n+1}^*$, where $\Theta_n^*=RE$, and let $\mathcal {T}_2=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1},
\Theta_{i-1}): 1\le i \le n\}$. Since $\sigma_{n-1}$ is a $(T_n,D_{n-1},\varphi_{n-1})$-stable coloring by (6),
it follows from Lemma \ref{sc2} that $\varphi_{n-1}$ is a $(T_n,D_{n-1}, \sigma_{n-1})$-stable
coloring. Similarly, $\varphi_i$ is a $(T_i, D_i, \sigma_i)$-stable coloring for $1 \le i \le n-1$,
because $\sigma_i$ is a $(T_i, D_i, \varphi_i)$-stable coloring by (2) and (3). Applying Lemma \ref{basic1} to
$\mathcal {T}_1$ and $\mathcal {T}_2$, we see that an ETT with respect to $e$ and the coloring $\varphi_{n-1}$ in ${\cal C}^k(G-e)$
can be obtained from $T_n$ by using RE, contradicting the hypothesis of the present case.
{\bf Case 3.} $\Theta_n=PE$. In this case, define $\sigma_{n-1}=\sigma_n/P_{v_{n}} (\gamma_{n},\delta_{n}, \sigma_n)$.
Since $\sigma_n$ is a $(T_n, D_n, \varphi_n)$-stable coloring, by (4.4) and Theorem~\ref{thm:tech10}(iv), we obtain
$P_{v_{n}}(\gamma_{n},\delta_{n},\sigma_n)\cap T_{n}=\{v_{n}\}$. Using Lemma \ref{stablezang}, we have
(8) $\sigma_{n-1}$ is $(T_{n},D_{n-1} \cup \{\delta_n\}, \pi_{n-1})$-stable and hence is $(T_n,D_{n-1},
\varphi_{n-1})$-stable.
\noindent By Lemma \ref{hku}(iii), we have $\varphi_{n-1} \langle T_n \rangle \subseteq \overline{\varphi}_{n-1}(T_n)\cup D_{n-1}$.
It follows from (8) that $\sigma_{n-1}(f)=\varphi_{n-1}(f)$ for any edge $f$ on $T_n$. Thus we can set $T^*_n=T_n$.
Moreover, by (8) and Lemma \ref{sc2}, $v_n$ is also the maximum defective vertex with respect to $(T_n, D_{n-1}, \sigma_{n-1})$.
We claim that
(9) for some $(T_n, D_{n-1}\cup\{\delta_n\}, \sigma_{n-1})$-stable coloring $\mu_{n-1}$, there holds $\overline{\mu}_{n-1} (u_n)
\cap \overline{\mu}_{n-1} (T_n) \ne \emptyset$, where $u_n$ is the vertex of $f_n$ outside $T_n$.
To justify this, let $\mu_{n-1}$ be a $(T_{n},D_{n-1} \cup \{\delta_n\}, \pi_{n-1})$-stable coloring for which
$\overline{\mu}_{n-1} (u_n) \cap \overline{\mu}_{n-1} (T_n) \ne \emptyset$; such a coloring exists by Steps 1
and 4 in Algorithm 3.1. From (8) and Lemma \ref{sc2} we deduce that $\mu_{n-1}$ is a $(T_n, D_{n-1}\cup\{\delta_n\}, \sigma_{n-1})$-stable coloring.
By (9), a tree sequence $T_{n+1}^*$ with respect to $e$ and $\sigma_n$ can thus be obtained from $T_n$ by
using Step 4 in Algorithm 3.1 (with $\sigma_{n-1}$ in place of both $\varphi_{n-1}$ and $\pi_{n-1}$) and using
the same connecting edge, connecting color, and extension type PE as $T_{n+1}$ in $\mathcal {T}$.
Using the same argument as in Case 2, we conclude that no ETT with respect to $e$ and $\sigma_n$ can be obtained
from $T_n$ by using RE. So $T_{n+1}^*$ constructed above is an ETT with respect to $e$ and $\sigma_n$. \hfill \rule{4pt}{7pt}
\section{Good Hierarchies}
As is well known, Kempe changes play a fundamental role in edge-coloring theory. To ensure that ETTs are
preserved under such operations, in this section we develop an effective control mechanism, the so-called
good hierarchy of an ETT, which will serve as a powerful tool in the proof of Theorem~\ref{thm:tech10}(i). Throughout
this section, we assume that
{\bf (5.1)} Theorem~\ref{thm:tech10}(i) and (ii) holds for all ETTs with at most $n-1$ rungs and satisfying MP, and
Theorem~\ref{thm:tech10}(iii)-(iv) hold for all ETTs with at most $n$ rungs and satisfying MP.
Let $J_n$ be a closure of $T_{n}(v_n)$ under a $(T_n,D_n,\varphi_n)$-stable coloring $\sigma_n$. We use
$T_n\vee J_n$ to denote the tree sequence obtained from $T_n$ by adding all vertices in $V(J_n)-V(T_n)$ to
$T_n$ one by one, following the linear order $\prec$ in $J_n$, and using edges in $J_n$.
\begin{lemma}\label{elementary}
(Assuming (5.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Suppose $\Theta_n=PE$ and $T$ enjoys
MP under $\varphi_n$. If $J_n$ is a closure of $T_{n}(v_n)$ under a $(T_n,D_n,\varphi_n)$-stable coloring
$\sigma_n$, then $V(T_n\vee J_n)$ is elementary with respect to $\sigma_n$.
\end{lemma}
{\bf Proof.} Clearly, $T_n$ is an ETT with corresponding Tashkinov series $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \\
\Theta_{i-1}): 1\le i \le n\}$, and satisfies MP under $\varphi_{n-1}$. Since $r(T_n)=n-1$, by (5.1) and Theorem~\ref{thm:tech10}(i),
$V(T_n)$ is elementary with respect to $\varphi_{n-1}$. Let $\pi_{n-1}$ and $\pi_{n-1}'$ be as specified in Step 4
of Algorithm 3.1. Since $\pi_{n-1}$ is a $(T_n,D_{n-1}, \varphi_{n-1})$-stable coloring and $\pi_{n-1}'$ is $(T_n,D_{n-1}
\cup \{\delta_n\}, \pi_{n-1})$-stable coloring, by definition $V(T_n)$ is also elementary with respect to $\pi'_{n-1}$. As $\varphi_n=\pi'_{n-1}/P_{v_n}(\delta_n,\gamma_n, \pi'_{n-1})$ and $\delta_n\notin\pi'_{n-1}(T_n)$, we further obtain
(1) $V(T_n)$ is elementary with respect to $\varphi_{n}$ and hence elementary with respect
to $\sigma_n$.
As $\sigma_n$ is a $(T_n,D_n,\varphi_n)$-stable coloring, it follows from (5.1) and Theorem~\ref{thm:tech10}(iii) that
$\sigma_n$ is $(T_{j}(v_n)-v_n, D_{j-1},\varphi_{j-1})$-stable, where $j=m(v_n)$. So $\sigma_n$
is a $(T_{j-1}, D_{j-1}, \varphi_{j-1})$-stable coloring and hence is a $\varphi_{j-1}\bmod T_{j-1}$ coloring by
Theorem \ref{thm:tech10}(vi). By Lemma \ref{hku}(iii) and Lemma \ref{samecolor}, we obtain $\sigma_n(f)=\varphi_n(f)=
\varphi_{j-1}(f)$ for each edge $f$ on $T_j$. By (5.1) and Theorem \ref{thm:tech10}(vi), $J_n$ is an ETT corresponding to
$\sigma_n$, because it is obtained from $T_{j-1}$ by using the same connecting edge, connecting color, and extension
type as $T_j$. Clearly, $J_n$ also satisfies the maximum property under $\sigma_n$. Since $J_n$ has $j-1$ rungs,
using (5.1), we obtain
(2) $V(J_n)$ is elementary with respect to $\sigma_{n}$.
Suppose on the contrary that $V(T_n\vee J_n)$ is not elementary with respect to $\sigma_n$. Then $T_n\vee J_n$ contains
two distinct vertices $u$ and $v$ such that $\overline{\sigma}_n(u) \cap \overline{\sigma}_n(v) \ne \emptyset$. By (1)
and (2), we may assume that $u \in V(T_n)-V(J_n)$ and $v \in V(J_n)-V(T_n)$. Let $\alpha\in \overline{\sigma}_n(u) \cap \overline{\sigma}_n(v)$. Then $\alpha\neq \delta_n$ by (2), because $\delta_n \in \overline{\varphi}_n(v_n)=
\overline{\sigma}_n(v_n)$. Moreover, since $\gamma_n\in\overline{\varphi}_{n-1}(v_n)$ and $V(T_n)$ is elementary with respect to
$\varphi_{n-1}$, from Step 4 of Algorithm 3.1 and the definition of stable colorings, we deduce that $\gamma_n\notin
\overline{\varphi}_n(T_n)$ and hence $\gamma_n\notin \overline{\sigma}_n(T_n)$. So $\alpha\neq\gamma_n$. Consequently,
(3) $\alpha\notin S_n$.
Since $T_{n}(v_n)$ contains the uncolored edge $e$, it contains a vertex $w \ne v_n$. Note that $w$ is contained in
both $T_n$ and $J_n$. Let $\beta \in \overline{\sigma}_n(w)$. Since $\delta_n \in \overline{\sigma}_n(v_n)$ and
$\gamma_n\notin \overline{\sigma}_n(T_n)$, we obtain
(4) $\beta \notin S_n$ (see (2)).
As $V(J_n)$ is closed and elementary with respect to $\sigma_{n}$ (see (2)), the other end of $P_v(\alpha, \beta, \sigma_n)$
is $w$. From (3), (4), and Step 4 of Algorithm 3.1, we see that $\partial(T_n)$ contains no edge colored by $\alpha$ or
$\beta$ under $\varphi_{n}$ and hence under $\sigma_n$, because $\sigma_n$ is $(T_n,D_n,\varphi_n)$-stable.
Combining this with (1), we conclude that the other end of $P_u(\alpha, \beta, \sigma_n)$ is
also $w$. Thus $P_w(\alpha, \beta, \sigma_n)$ terminates at both $u$ and $v$, a contradiction. \hfill \rule{4pt}{7pt}
\vskip 3mm
Let $T$ be an ETT as specified in Theorem \ref{thm:tech10}; that is, $T$ is constructed from a $k$-triple $(G,e, \varphi)$
by using the Tashkinov series $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. To prove that
$V(T)$ is elementary with respect to $\varphi_n$, we shall turn to considering a restricted ETT $T'$ with
ladder $T_1\subset T_2 \subset \ldots \subset T_n \subset T'$ and $V(T')=V(T_{n+1})$, and then show that
$V(T')$ is elementary with respect to $\varphi_n$. For convenience, we may simply view $T'$ as $T$.
In the remainder of this paper, we reserve the symbol $R_n$ for a fixed closure of $T_n(v_n)$ under $\varphi_n$, if
$\Theta_n=PE$. Let $T_n\vee R_n$ be the tree sequence as defined above Lemma \ref{elementary}. We assume hereafter that
{\bf (5.2)} $T_{n+1}$ is a closure of $T_n\vee R_n$ under $\varphi_n$, which is a special closure of $T_n$ under $\varphi_n$
(see Step 4 in Algorithm 3.1), when $\Theta_n=PE$.
By Lemma~\ref{elementary}, $V(T_n\vee R_n)$ is elementary with respect to $\varphi_n$, so we may further assume that
{\bf (5.3)} $T \ne T_n\vee R_n$ if $\Theta_n=PE$, which together with (5.2) implies that $T_n\vee R_n$ is not
closed with respect to $\varphi_n$.
{\bf (5.4)} If $\Theta_n = PE$, then each color in $\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)$ is
closed in $T_n \vee R_n$ with respect to $\varphi_n$.
To justify this, note that each color in $\overline{\varphi}_n(R_n)$ is closed in $R_n$ under $\varphi_n$ because
$R_n$ is a closure. By Lemma \ref{hku}(iv), each color in $\overline{\varphi}_n(T_n)-\{\delta_n\}$ is closed in
$T_n$ under $\varphi_n$. Hence each color in $\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)-\{\delta_n\}$
is closed in $T_n \vee R_n$ with respect to $\varphi_n$. Lemma \ref{hku}(iv) also asserts that edges in $\partial_{\varphi_n, \delta_n}(T_n)$ are all incident to $V(T_n(v_n)-v_n)$. So $\delta_n$ is closed in $T_n \vee R_n$ as well, because
it is closed in $R_n$. Hence (5.4) follows.
To prove Theorem \ref{thm:tech10}(i), we shall appeal to a {\em hierarchy} of $T$ of the form
{\bf (5.5)} $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$, such that $T_n\vee R_n
\subset T_{n,1}$ if $\Theta_n=PE$ and $T_{n,i}=T(u_i)$ for $1\le i\leq q$, where $u_1\prec u_2\prec \ldots
\prec u_{q}$ are some vertices in $T-V(T_n)$, called {\it dividers} of $T$. (So $T$ has $q$ dividers in total.)
As introduced in Algorithm 3.1, $D_n=\cup_{h\leq n}S_h-\overline{\varphi}_n(T_n)$, where $S_h=\{\delta_h\}$
if $\Theta_h=SE$ and $S_h=\{\delta_h,\gamma_h\}$ otherwise. By Lemma \ref{Dnzang}, we have
{\bf (5.6)} $|D_n|\leq n$.
\noindent Write $D_n=\{\eta_1, \eta_2, \ldots ,\eta_{n'}\}$. In Definition \ref{R2} given below and the remainder of
this paper,
$\bullet$ $T_{n,0}^*= T_{n}\vee R_n$ if $\Theta_n=PE$ and $T_{n,0}^*= T_{n}$ otherwise, and $T_{n,j}^*=T_{n,j}$ if $j \ge 1$;
$\bullet$ $D_{n,j}=\cup_{h\leq n}S_h-\overline{\varphi}_n(T_{n,j}^*)$ for $0 \le j \le q$;
$\bullet$ $v_{\eta_h}$, for $\eta_h \in D_n$, is defined to be the first vertex $u$ of $T$ in the order $\prec$ with $\eta_h \in
\overline{\varphi}_n (u)$, if
\hskip 3mm any, and defined to be the last vertex of $T$ in the order $\prec$ otherwise;
$\bullet$ $\Lambda^0_h=\overline{\varphi}_n(T_n)-\varphi_n \langle T_{n,1}(v_{\eta_h})-T_{n,0}^* \rangle$ for $\eta_h \in D_{n,0}$,
where $T_{n,1}(v_{\eta_h})=T_{n,1}$ if $v_{\eta_h}$ is outside
\hskip 3mm $T_{n,1}$;
$\bullet$ $\Lambda^j_h=\overline{\varphi}_n(T_{n,j})-\varphi_n \langle T_{n,j+1}(v_{\eta_h})-T_{n,j} \rangle$ for $1 \le j \le q$
and $\eta_h \in D_{n,j}$, where $T_{n,j+1}(v_{\eta_h})=$
\hskip 3mm $T_{n,j+1}$ if $v_{\eta_h}$ is outside $T_{n,j+1}$; and
$\bullet$ $\Gamma^{j}=\cup_{\eta_h\in D_{n,j}}\Gamma^{j}_h$ for $0 \le j \le q$.
\iffalse
$\bullet$ $D_{n,0}=\cup_{h\leq n}S_h-\overline{\varphi}_n(T_n\vee R_n)$ if $\Theta_n=PE$ and $D_{n,0}=D_n$ otherwise;
$\bullet$ $D_{n,j}=\cup_{h\leq n}S_h-\overline{\varphi}_n(T_{n,j})$ for $1 \le j \le q$;
$\bullet$ $T_{n,j+1}(v_{\eta_h})=T_{n,j+1}$ if $v_{\eta_h}$ is not contained in $T_{n,j+1}$ for $0 \le j \le q$;
$\bullet$ $\Lambda^0_h= \overline{\varphi}_n(T_n)- \varphi_n \langle T_{n,1}(v_{\eta_h})-T_n \vee R_n \rangle$ if $\Theta_n=PE$
and $\Lambda^0_h= \overline{\varphi}_n(T_n)- \varphi_n \langle T_{n,1}(v_{\eta_h})-T_n \rangle$
\hskip 3mm otherwise;
$\bullet$ $T_{n,0}^*= T_{n}\vee R_n$ if $\Theta_n=PE$ and $T_{n,0}^*= T_{n}$ otherwise, and $T_{n,j}^*=T_{n,j}$ if $j \ge 1$; and
$\bullet$ $\Gamma^{j}=\cup_{\eta_h\in D_{n,j}}\Gamma^{j}_h$ for $0 \le j \le q$.
\fi
\vskip 2mm
Let $H$ be a subgraph of $G$ and let $C$ be a subset of $[k]$. We say that $H$ is $C$-{\em closed} with respect to
$\varphi_n$ if $\partial_{\varphi_n, \alpha}(H)=\emptyset$ for any $\alpha \in C$, and say that $H$ is $C^-$-{\em closed}
with respect to $\varphi_n$ if it is $(\overline{\varphi}_n(H)-C)$-closed with respect to $\varphi_n$.
\begin{definition} \label{R2}
{\rm Hierarchy (5.5) of $T$ is called {\em good} with respect to $\varphi_n$ if for any $j$ with $0 \le j \le q$ and
any $\eta_h\in D_{n,j}$, there exists a $2$-color subset $\Gamma^{j}_h=\{\gamma^{j}_{h_1},\gamma^{j}_{h_2}\}\subseteq
[k]$, such that
\begin{itemize}
\vspace{-2mm}
\item[(i)] $\Gamma^{j}_h \subseteq \Lambda^j_h$ (so $\Gamma^{j}_h \subseteq \overline{\varphi}_n(T_n)$ if $j=0$ and
$\Gamma^{j}_h \subseteq \overline{\varphi}_n(T_{n,j})$ if $j\ge 1$);
\vspace{-2mm}
\item[(ii)] $\Gamma^{j}_{g} \cap \Gamma^{j}_{h}=\emptyset$ whenever $\eta_{g}$ and $\eta_{h}$ are two distinct colors in $D_{n,j}$;
\vspace{-2mm}
\item[(iii)] for any $j$ with $1\le j \le q$, there exists precisely one color $\eta_g\in D_{n,j}$, such that
$\Gamma^{j}_{g} \subseteq \overline{\varphi}_n(T_{n,j}-V(T_{n,j-1}^*))$ (so $\Gamma^{j}_{g} \cap \Gamma^{j-1}_{g} =\emptyset$)
and $\Gamma^{j}_{h}=\Gamma^{j-1}_{h}$ for all $\eta_h\in D_{n,j}-\{\eta_g\}$;
\vspace{-2mm}
\item[(iv)] if $\Theta_n=PE$, then $T_{n}\vee R_n$ is not $(\Gamma^0)^-$-closed with respect to $\varphi_n$ and, subject
to this, $|\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)-\Gamma^0|$ is maximized (this maximum value is at least $4$, as we shall see); and
\vspace{-2mm}
\item[(v)] $T_{n,j}$ is $(\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h)^-$-closed with respect to $\varphi_n$
for all $j$ with $1\le j \le q$.
\vspace{-2mm}
\end{itemize}
\noindent The sets $\Gamma^{j}_h$ are referred to as $\Gamma$-{\em sets} of the hierarchy (or of $T$) under $\varphi_n$.}
\end{definition}
Some remarks may help to understand the concept of good hierarchies.
{\bf (5.7)} From Condition (i) we see that neither the color $\gamma^{j}_{h_1}$ nor $\gamma^{j}_{h_2}$ can be used by
edges on $T_{n,j+1}$ until after $\eta_h$ becomes missing at the vertex $v_{\eta_h}$ in $T_{n,j+1}$.
{\bf (5.8)} Condition (iv) implies that $T_{n,1}\neq T_{n}\vee R_n$ if $\Theta_n=PE$.
{\bf (5.9)} For $1\le j\le q$, by definitions, $D_{n,j} \subseteq D_{n,j-1}$, so $\Gamma^{j-1}_h$ is well defined for any
$\eta_h\in D_{n,j}$ and $\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h \subseteq \Gamma^{j-1}$. In view of Condition
(v), the first edge added to $T_{n,j+1}- T_{n,j}$ is colored by a color $\alpha$ in $\Gamma^{j-1}_g$
for some $g$ with $\eta_g\in D_{n,j}$. From Condition (i) and (5.7) we see that $\alpha\notin \Gamma^{j}_g$. So
$ \Gamma^{j}_g \ne \Gamma^{j-1}_g$. According to Condition (iii), now $\Gamma^{j}_g$ consists of two colors in
$\overline{\varphi}_n(T_{n,j} -V(T_{n,j-1}^*))$. Thus $\Gamma^{j-1}_g \cap \Gamma^{j}_g=\emptyset$ and hence $\alpha \notin
\Gamma^j$.
{{\bf (5.10)} If a color $\alpha \in \overline{\varphi}_n(T_{n,j}-V(T_{n,j-1}^*))$ for some $j$ with $1 \le j \le q$, then
$\alpha \notin \Gamma^{j-1}$ by Condition (i), and hence $\alpha$ is closed in $T_{n,j}$ with respect to $\varphi_n$
by Condition (v). This simple observation will be used repeatedly in subsequent proofs.
{{\bf (5.11)} Note that not every ETT admits a good hierarchy. Suppose $T$ does have such a hierarchy. To prove that
$V(T)$ is elementary with respect to $\varphi_n$, as usual, we shall perform a sequence of Kempe changes. Since
interchanging with colors in $D_{n,j}$ often results in a coloring which is not stable, in our proof we shall use colors
in $\Gamma^{j}_h$ as stepping stones to switch with the color $\eta_h$ in $D_{n,j}$ while maintaining stable colorings
in subsequent proofs. So we may think of $\Gamma^{j}_h$ as a color set exclusively reserved for $\eta_h$ and think of
a good hierarchy as a control mechanism over Kempe changes.
\vskip 3mm
We break the proof of Theorem \ref{thm:tech10}(i) into the following two theorems. Although the first theorem appears to
be weaker than Theorem \ref{thm:tech10}(i), the second one implies that they are actually equivalent.
We only present a proof of the second theorem in this section, and will give a proof of the first one
in the next two sections.
\begin{theorem}\label{hierarchy}
(Assuming (5.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Suppose $T$ admits a good hierarchy and satisfies
MP with respect to $\varphi_n$. Then $V(T)$ is elementary with respect to $\varphi_n$.
\end{theorem}
\begin{theorem}\label{good}
(Assuming (5.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by
using the Tashkinov series $\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. If $T$
satisfies MP under $\varphi_n$, then there exists a closed ETT $T'$ constructed from $T_n$ under $\varphi_n$
by using the same connecting edge, connecting color, and extension type as $T$, with $r(T')=n$ and $V(T')=V(T_{n+1})$,
such that $T'$ admits a good hierarchy and satisfies MP with respect to $\varphi_n$.
\end{theorem}
\noindent {\bf Remark}. As we shall see, our proof of Theorem \ref{good} is based on Theorem \ref{hierarchy},
while the proof of Theorem \ref{hierarchy} is completely independent of Theorem \ref{good}.
\vskip 2mm
{\bf Proof of Theorem \ref{good}.} By (5.1) and Theorem~\ref{thm:tech10}(i), $V(T_i)$ is elementary with respect to
$\varphi_{i-1}$ for $1 \le i\leq n$. So each $|T_i|$ is an odd number. Thus $|T_{i}|-|T_{i-1}|\geq 2$ for each $1\le
i\leq n$. By Theorem \ref{ThmScheide}, if $|T_1|\le 10$, then $G$ is an elementary multigraph, thereby proving Theorem
\ref{ThmGS2} in this case. So we may assume that $|T_1|\ge 11$. Hence
(1) $|T_i| \ge 2i+9$ for $1\le i \le n$.
We shall actually construct an ETT $T'$ from $T_n$ by using the same connecting edge, connecting
color, and extension type as $T$, which has a good hierarchy:
(2) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q+1}=T'$, such that $T_n\vee R_n \subset T_{n,1}$
if $\Theta_n=PE$ and such that $V(T')=V(T_{n+1})$.
Since $V(T_n)$ is elementary with respect to $\varphi_{n-1}$, by (1) we have $|\overline{\varphi}_{n-1}(T_{n})|\ge 2n+11$
(as $e$ is uncolored). From Algorithm 3.1 we see that $|\overline{\varphi}_{n-1}(T_{n})|=|\overline{\varphi}_n(T_{n})|$. So
(3) $|\overline{\varphi}_n(T_{n})|\ge 2n+11$. Moreover, $|D_{n,0}|\le |D_n|\le n$ by (5.6).
(4) If $\Theta_n=PE$, then we can find a $2$-color set $\Gamma^{0}_h=\{\gamma^{0}_{h_1},\gamma^{0}_{h_2}\}\subseteq
\overline{\varphi}_n(T_{n})$ for each $\eta_h\in D_{n,0}=\cup_{h\leq n}S_h-\overline{\varphi}_n(T_n\vee R_n)$, such that $\Gamma^{0}_{g} \cap \Gamma^{0}_{h}=\emptyset$ whenever $\eta_{g}$ and $\eta_{h}$ are two distinct colors in $D_{n,0}$, and such that $T_{n}\vee R_n$
is not $(\Gamma^0)^-$-closed with respect to $\varphi_n$, where $\Gamma^{0}=\cup_{\eta_h\in D_{n,0}}\Gamma^{0}_h$.
To justify this, let $\alpha$ be a color in $\overline{\varphi}_n(T_n \vee R_n)$ that is not closed in $T_n \vee R_n$
under $\varphi_n$; such a color exists by (5.3). In view of (3), $\overline{\varphi}_n(T_n)-\{\alpha\}$
contains at least $2n+10$ colors. So (4) follows if we pick all colors in $\Gamma^{0}$ from $\overline{\varphi}_n(T_n)-
\{\alpha\}$.
(5) If $\Theta_n=PE$, then there exists a $2$-color set $\Gamma^{0}_h=\{\gamma^{0}_{h_1},\gamma^{0}_{h_2}\}\subseteq
\overline{\varphi}_n(T_{n})$ for each $\eta_h\in D_{n,0}$ as described in (4), such that $|\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)-\Gamma^0|$
is maximized, which is at least $4$.
To justify this, let $\alpha$ be as specified in the proof of (4). Then $\alpha \notin \overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)$ by (5.4). If we pick all colors in $\Gamma^{0}$ from $\overline{\varphi}_n(T_n)-\{\alpha\}$,
with priority given to those in $\overline{\varphi}_n(T_n)-\overline{\varphi}_n(R_n)$, then $|\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n) -\Gamma^0| \ge 4$ by (3), because the ends of the uncolored edge $e$ are contained in both
$T_n$ and $R_n$. So (5) is established.
Thus Definition \ref{R2}(iv) is satisfied by these sets $\Gamma^{0}_h$. Using (3), we can similarly get
the following statement.
(6) If $\Theta_n \ne PE$, then we can find a $2$-color set $\Gamma^{0}_h=\{\gamma^{0}_{h_1},\gamma^{0}_{h_2}\}\subseteq
\overline{\varphi}_n(T_{n})$ for each $\eta_h\in D_{n,0}=D_n$, such that $\Gamma^{0}_{g} \cap \Gamma^{0}_{h}=\emptyset$ whenever
$\eta_{g}$ and $\eta_{h}$ are two distinct colors in $D_{n,0}$.
Note that the ETT $T'$ to be constructed is not necessarily $T$, so $T_{n,j}$ may not be a segment of $T$ for $1 \le j \le q$.
Since $T'$ is a tree sequence, we can obviously associate a linear order $\prec'$ with its vertices, so
that $\prec'$ is identical with $\prec$ when restricted to $T_{n,0}^*$. Thus, in Algorithms 5.5 and 5.6,
$v_{\eta_h}$ is defined to be the first vertex of $T'$ in the order $\prec'$ for which $\eta_h \in
\overline{\varphi}_n (v_{\eta_h})$, if any, and defined to be the last vertex of $T'$ in the order $\prec'$ otherwise;
and $T_{n,j+1}(v_{\eta_h})=T_{n,j+1}$ if $v_{\eta_h}$ is not contained in $T_{n,j+1}$ for $0\le j \le q$.
Given $\{\Gamma^0_h:\, \eta_h\in D_{n,0}\}$, let us construct $T_{n,1}$ using the following procedure.
\newpage
\noindent {\bf Algorithm 5.5}
\vskip 2mm
\noindent {\bf Step 0.} Set $T_{n,1}= T_{n}\vee R_n$ if $\Theta_n=PE$ and $T_{n,1}=T_n+f_n$ otherwise,
where $f_n$ is the connecting edge used in Step 2 or 3 of Algorithm 3.1, depending on $\Theta_n$.
\vskip 2mm
\noindent {\bf Step 1.} While there exists $f\in\partial(T_{n,1})$ with $\varphi_n(f)\in\overline{\varphi}_n(T_{n,1})$,
do: set $T_{n, 1}=T_{n,1}+f$ if the resulting $T_{n, 1}$ satisfies $\Gamma^{0}_h \cap \varphi_n \langle T_{n,1}(v_{\eta_h})
-T_{n,0}^* \rangle =\emptyset$ for all $\eta_h\in D_{n,0}$, where $T_{n,0}^*= T_{n}\vee R_n$ if $\Theta_n=PE$ and
$T_{n,0}^*= T_{n}$ otherwise.
\vskip 2mm
\noindent {\bf Step 2.} Return $T_{n, 1}$.
\vskip 3mm
Note that if $\Theta_n=PE$, then $T_{n}\vee R_n$ is not $(\Gamma^0)^-$-closed with respect to $\varphi_n$ by (4) and (5).
So Step 1 is applicable to $T_{n}\vee R_n$, and hence $T_{n,1}\ne T_{n}\vee R_n$. If $\Theta_n=RE$ or $SE$, then
$T_{n,1}\ne T_{n}$ by the algorithm. For each $\eta_h\in D_{n,0}$, it follows from (5), (6), and Step 1 that $\Gamma^{0}_h
\subseteq \overline{\varphi}_n(T_n)- \varphi_n \langle T_{n,1}(v_{\eta_h})- T_{n,0}^* \rangle $.
So $\Gamma^{0}_h \subseteq \Lambda^0_h$. Moreover, $T_{n,1}$ is $(\cup_{\eta_h\in D_{n,1}}\Gamma^{0}_h)^-$-closed with
respect to $\varphi_n$. To justify this, assume the contrary: there exists $f\in\partial(T_{n,1})$ with $\varphi_n(f) \in \overline{\varphi}_n(T_{n,1})- (\cup_{\eta_h\in D_{n,1}} \Gamma^{0}_h)$. Then either $\varphi_n(f) \in \overline{\varphi}_n(T_{n,1})-
(\cup_{\eta_h\in D_{n,0}} \Gamma^{0}_h)$ or $\varphi_n(f) \in \Gamma^{0}_h$ for some $\eta_h\in D_{n,0}$ but
$\eta_h\notin D_{n,1}$; in the latter case, ${\eta_h}$ is a missing color at the vertex $v_{\eta_h}$ in $T_{n,1}$.
Thus we can further grow $T_{n,1}$ by using $f$ and Step 1 in either case, a contradiction. Therefore, $T_{n,1}$
and $\{\Gamma^{0}_h:\, \eta_h\in D_{n,0}\}$ satisfy all the conditions stated in Definition \ref{R2}.
\vskip 2mm
Suppose we have constructed $T_{n,i}$ and $\{\Gamma^{i-1}_h:\, \eta_h\in D_{n,i-1}\}$ for all $i$ with
$1 \le i \le j$, which are as described in Definition \ref{R2}. If $T_{n,j}$ is closed with respect to
$\varphi_n$ (equivalently $V(T_{n,j})=V(T_{n+1}))$, set $T'=T_{n,j}$. Otherwise, we proceed to the construction
of $T_{n,j+1}$ and $\{\Gamma^{j}_h:\, \eta_h\in D_{n,j}\}$ using the following procedure.
\vskip 3mm
\noindent {\bf Algorithm 5.6}
\vskip 2mm
\noindent {\bf Step 0.} Set $\Gamma^{j}_h =\Gamma^{j-1}_h$ for each $\eta_h\in D_{n, j}$.
\vskip 2mm
\noindent {\bf Step 1.} Let $f$ be an edge in $\partial(T_{n,j})$ with $\varphi_n(f) \in \Gamma^{j-1}_h$ for some
$\eta_h\in D_{n,j}$, let $T_{n,j+1}=T_{n,j}+f$, and let $\{\gamma^{j}_{h_1},\gamma^{j}_{h_2}\}$ be a 2-subset of $\overline{\varphi}_n(T_{n,j}-V(T_{n,j-1}))$. Replace $\Gamma^{j}_h$ by $\{\gamma^{j}_{h_1},\gamma^{j}_{h_2}\}$.
\vskip 2mm
\noindent {\bf Step 2.} While there exists $f\in\partial(T_{n,j+1})$ with $\varphi_n(f)\in\overline{\varphi}_n(T_{n,j+1})$,
do: set $T_{n,j+1}=T_{n,j+1}+f$ if the resulting $T_{n,j+1}$ satisfies $\Gamma^{j}_h \cap \varphi_n \langle T_{n,j+1}(v_{\eta_h})-T_{n,j} \rangle =\emptyset$ for all $\eta_h\in D_{n,j}$.
\vskip 2mm
\noindent {\bf Step 3.} Return $T_{n,j+1}$ and $\{\Gamma^{j}_h:\, \eta_h\in D_{n,j}\}$.
\vskip 2mm
Let us make some observations about this algorithm and its output.
As $T_{n,j}$ is not closed with respect to $\varphi_n$, $V(T_{n,j})$ is a proper subset of $V(T_{n+1})$. By Definition
\ref{R2}(v), $T_{n,j}$ is $(\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h)^-$-closed with respect to $\varphi_n$.
So there exists a color $\beta \in \cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h$, such that $\partial_{\varphi_n, \beta}(T_{n,j}) \ne \emptyset$. Hence the edge $f$ specified in Step 1 is available.
For $1\le i \le j$, we have $|\overline{\varphi}_n(T_{n,i})|\ge |\overline{\varphi}_n(T_{n})|\ge 2n+11$ and $|D_{n,i}|
\le |D_{n,0}|\le |D_n|\le n$ by (3). So $\overline{\varphi}_n(T_{n,i})-(\cup_{\eta_h\in D_{n,i}}\Gamma^{i-1}_h) \ne \emptyset$;
let $\alpha$ be a color in this set. By Theorem \ref{hierarchy} (see the remark right above the proof of this theorem),
$V(T_{n,i})$ is elementary with respect to $\varphi_n$, which implies that $|T_{n,i}|$ is odd, because $\alpha$ is closed
in $T_{n,j}$ under $\varphi_n$ by Definition \ref{R2}(v). It follows that $|T_{n,j}|- |T_{n,j-1}|\ge 2$. So $\overline{\varphi}_n(T_{n,j}-V(T_{n,j-1}))$ contains at least two distinct colors, and hence the $2$-subset $\{\gamma^{j}_{h_1},\gamma^{j}_{h_2}\}$ involved in Step 1 exists.
Note that each color in $\overline{\varphi}_n(T_{n,j+1})-(\cup_{\eta_h\in D_{n,j+1}}\Gamma^{j}_h)$ is closed
in $T_{n,j+1}$ with respect to $\varphi_n$, for otherwise, $T_{n,j+1}$ can be augmented further using Step 2
(see the paragraph succeeding Algorithm 5.5 for details). Thus $T_{n,j+1}$ is $(\cup_{\eta_h\in D_{n,j+1}}\Gamma^{j}_h)^-$-closed with respect to $\varphi_n$ for $1\le j \le q-1$. From the algorithm we see that $\Gamma^{j}_h \subseteq \overline{\varphi}(T_{n,j})-\varphi_n \langle T_{n,j+1}(v_{\eta_h})-T_{n,j}^* \rangle = \Lambda^j_h$ for all $\eta_h\in D_{n,j}$. So $T_{n,j+1}$ and $\{\Gamma^{j}_h:\, \eta_h\in D_{n,j}\}$ satisfy all the conditions in Definition \ref{R2} and hence are as desired.
Repeating the process, we can eventually get a closed ETT $T'$, with $V(T')=V(T_{n+1})$, that admits a good hierarchy
with respect to $\varphi_n$. Clearly, $T'$ also satisfies MP under $\varphi_n$. \hfill \rule{4pt}{7pt} \\
Consider the case when $\Theta_n=PE$. By the definition of hierarchy (see (5.5)), $T_n\vee R_n$ is fully contained
in $T_{n,1}$. To maintain the structure of $T_n\vee R_n$ under Kempe changes, we need the following concept in subsequent proofs.
A coloring $\sigma \in {\cal C}^k(G-e)$ is called a $(T_n \oplus R_n, D_n,\varphi_n)$-{\em stable coloring} if it is both $(T_n,D_n,\varphi_n)$-stable and $(R_n,\emptyset,\varphi_n)$-stable; that is, the following conditions are satisfied:
$\bullet$ $\sigma(f)=\varphi_n(f)$ for any edge $f$ incident to $T_n$ with $\varphi_n(f)\in \overline{\varphi}_n(T_n)\cup D_n$;
$\bullet$ $\sigma(f)=\varphi_n(f)$ for any edge $f$ incident to $R_n$ with $\varphi_n(f)\in\overline{\varphi}_n(R_n)$; and
$\bullet$ $\overline{\sigma}(v)=\overline{\varphi}_n(v)$ for any $v\in V(T_n\cup R_n)$.
\vskip 2mm
{{\bf (5.12)} If $\sigma$ is a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring, then $\sigma(f)=\varphi_n(f)$ for any
edge $f$ on $T_n \cup R_n$. To justify this, note that, for any edge $f$ on $T_n$, this equality holds by Lemma \ref{hku}(iii).
For any edge $f$ in $R_n -T_n$, we have $\varphi_n(f)\in\overline{\varphi}_n(R_n)$ by the definition of $R_n$ and TAA. It follows from
the above definition that $\sigma(f)=\varphi_n(f)$.
From Lemma \ref{sc2} it is clear that being $(T_n \oplus R_n, D_n,\cdot)$-stable is also an equivalence relation
on ${\cal C}^k(G-e)$. Moreover, every $(T_n\vee R_n, D_n,\varphi_n)$-stable coloring is $(T_n\oplus R_n,D_n,\varphi_n)$-stable,
but the converse need not hold.
\setcounter{theorem}{6}
\begin{lemma}\label{splitter}
Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Suppose $\Theta_n=PE$ and
$T$ enjoys MP under $\varphi_n$. Let $T_n=T_{n,0}\subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ be a
hierarchy of $T$, and let $\sigma_n$ be a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring. If $T$ can be built from
$T_n \vee R_n$ by using TAA under $\sigma_n$, then $T$ is also an ETT satisfying MP with respect to $\sigma_n$,
and $T_n=T_{n,0} \subset T_{n,1}\subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains to be a hierarchy of $T$
under $\sigma_n$.
\end{lemma}
{\bf Proof.} Since $\sigma_n$ is a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring, we have $\sigma(f)=\varphi_n(f)$
for any edge $f$ on $T_n \vee R_n$ by (5.12). By definition, $\sigma_n$ is a $(T_n,D_n,\varphi_n)$-stable coloring, so
it is a $\varphi_n\bmod T_n$ coloring by (5.1) and Theorem~\ref{thm:tech10}(vi). Thus $T_n$ is an ETT corresponding to
$\sigma_n$. As $R_n$ is a closure of $T_n(v_n)$ under $\varphi_n$ and $\sigma_n$ is $(R_n,\emptyset,\varphi_n)$-stable,
$R_n$ is also a closure of $T_n(v_n)$ under $\sigma_n$. By hypothesis, $T$ can be built from $T_n \vee R_n$ by using TAA under $\sigma_n$. So $T$ is an ETT corresponding to the coloring $\sigma_n$ and satisfies MP under $\sigma_n$ by Theorem~\ref{thm:tech10}(vi). Obviously, $T_{n,0} \subset T_{n,1}\subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains
to be a hierarchy of $T$ under $\sigma_n$. \hfill \rule{4pt}{7pt}
\vskip 3mm
We define one more term before proceeding. Let $T$ be a tree sequence with respect to $G$ and $e$. A coloring
$\pi \in {\cal C}^k(G-e)$ is called $(T, \varphi_n)$-{\em invariant}
if $\pi(f) = \varphi_n(f)$ for any $f\in E(T-e)$ and $\overline{\pi} (v) = \overline{\varphi}_n(v)$ for any $v\in V(T)$.
Clearly, being $(T, \cdot)$-invariant is also an equivalence relation on ${\cal C}^k(G-e)$. Note that for any subset
$C$ of $[k]$, a $(T, C, \varphi_n)$-stable coloring $\pi$ is also $(T, \varphi_n)$-invariant, provided that ${\pi}\langle
T \rangle \subseteq \overline{\varphi}_n(T)\cup C$.
\vskip 3mm
\begin{lemma}\label{LEM:Stable}
(Assuming (5.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Suppose $T$ satisfies MP under
$\varphi_n$. Let $\sigma_n$ be obtained from $\varphi_n$ by recoloring some $(\alpha, \beta)$-chains fully contained in
$G-V(T)$. Then the following statements hold:
\begin{itemize}
\vspace{-1.5mm}
\item[(i)] $\sigma_n$ is $(T,D_n,\varphi_n)$-stable. In particular, $\sigma_n$ is $(T,\varphi_n)$-invariant.
Furthermore, if $\Theta_n=PE$, then $\sigma_n$ is $(T_n\oplus R_n,D_n,\varphi_n)$-stable.
\vspace{-2mm}
\item[(ii)] $T$ is an ETT satisfying MP with respect to $\sigma_n$.
\vspace{-2mm}
\item[(iii)] If $T$ admits a good hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q+1}=T$ under $\varphi_n$, then this hierarchy of $T$ remains good under $\sigma_n$, with the same $\Gamma$-sets (see Definition \ref{R2}). Furthermore, if $T$ is $(\cup_{\eta_h\in D_{n,q+1}}\Gamma^{q}_h)^-$-closed with respect to $\varphi_n$,
then $T$ is also $(\cup_{\eta_h\in D_{n,q+1}}\Gamma^{q}_h)^-$-closed with respect to $\sigma_n$.
\end{itemize}
\end{lemma}
{\bf Proof.} Since the recolored $(\alpha, \beta)$-chains are fully contained in $G-V(T)$, we have
(1) $\sigma_n(f) = \varphi_n(f)$ for each edge $f$ incident to $V(T)$ and $\overline{\varphi}_n(v) =
\overline{\sigma}_n(v)$ for each $v\in V(T)$.
\noindent Our proof relies heavily on this observation.
(i) By (1) and definitions, it is clear that $\sigma_n$ is a $(T,D_n,\varphi_n)$-stable. In particular,
$\sigma_n$ is $(T,\varphi_n)$-invariant. Furthermore, if $\Theta_n=PE$, then $\sigma_n$ is $(T_n\vee R_n,D_n,
\varphi_n)$-stable, which implies that $\sigma_n$ is $(T_n\oplus R_n,D_n,\varphi_n)$-stable.
(ii) In view of (1), we can construct $T$ from $T_n$ under $\sigma_n$ in exactly the same way as under $\varphi_n$. From
(1) we also deduce that $\sigma_n$ is a $(T_n,D_n,\varphi_n)$-stable coloring. Hence, by Theorem \ref{thm:tech10}(vi),
$T$ remains to be an ETT and satisfies MP under $\sigma_n$.
(iii) From (1), (5.5) and Lemma~\ref{splitter} (when $\Theta_n=PE$), we see that the given hierarchy $T_n=T_{n,0}
\subset T_{n,1} \subset \ldots \subset T_{n,q+1}=T$ is also a hierarchy of $T$ under $\sigma_n$. By hypothesis, this
hierarchy is good with respect to $\varphi_n$. Consider the $\Gamma$-sets specified in Definition \ref{R2} with respect
to $\varphi_n$. Using (1) it is routine to check that these $\Gamma$-sets satisfy all the conditions in Definition
\ref{R2} with respect to $\sigma_n$. So the given hierarchy of $T$ remains good under $\sigma_n$, with the same
$\Gamma$-sets. Furthermore, if $T$ is $(\cup_{\eta_h\in D_{n,q+1}}\Gamma^{q}_h)^-$-closed with respect to $\varphi_n$,
then $T$ is also $(\cup_{\eta_h\in D_{n,q+1}}\Gamma^{q}_h)^-$-closed with respect to $\sigma_n$. \hfill \rule{4pt}{7pt}
\section{Basic Properties}
As we have seen, Theorem \ref{thm:tech10}(i) follows from Theorems \ref{hierarchy} and \ref{good}.
In the preceding section we have proved Theorem \ref{good}. The remainder of this paper is devoted to a
proof of Theorem \ref{hierarchy}. In this section we make some technical preparations.
Let $T$ is an ETT that admits a good hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ and satisfies MP with respect to the generating coloring $\varphi_n$. To prove
Theorem \ref{hierarchy} (that is, $V(T)$ is elementary with respect to $\varphi_n$), we apply induction on
$q$, and the induction base is Theorem \ref{thm:tech10}(i) for $T_n$. For convenience, we view $T_{n,0}$ as
an ETT with $-1$ divider and $n$ rungs in the following assumption. Throughout this section we assume that
{\bf (6.1)} In addition to (5.1), Theorem \ref{hierarchy} holds for every ETT that admits a good hierarchy
and satisfies MP, with $n$ rungs and at most $q-1$ dividers, where $q\ge 0$.
Let us first prove two technical lemmas that will be used in the proof of Theorem \ref{hierarchy}.
\begin{lemma}\label{interchange}
(Assuming (5.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Suppose $\Theta_n=PE$ and $T$ enjoys
MP under $\varphi_n$. Let $\sigma_n$ be a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring and let $\alpha$ and $\beta$
be two colors in $[k]$. Then the following statements hold:
\begin{itemize}
\vspace{-2mm}
\item[(i)] $\alpha$ and $\beta$ are $R_n$-interchangeable under $\sigma_n$ if $\alpha\in \overline{\sigma}_n(R_n)$;
\vspace{-2mm}
\item[(ii)] $\alpha$ and $\beta$ are $T_n$-interchangeable under $\sigma_n$ if $\alpha\in \overline{\sigma}_n(T_n)$;
\vspace{-2mm}
\item[(iii)] $\alpha$ and $\beta$ are $T_n\vee R_n$-interchangeable under $\sigma_n$ if $\alpha \in \overline{\sigma}_n
(T_n \vee R_n)$ is closed in $T_n\vee R_n$ under $\sigma_n$; and
\vspace{-2mm}
\item[(iv)] $\alpha$ and $\beta$ are $T_n\vee R_n$-interchangeable under $\sigma_n$ if $\alpha\in
\overline{\sigma}_n(T_n)$ and $\beta\in \overline{\sigma}_n(R_n)$.
\end{itemize}
\end{lemma}
{\bf Proof.} Since $\sigma_n$ is a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring, it is $(T_n, D_{n},\varphi_{n})$-stable
by definition. Let $j=m(v_n)$. It follows from (5.1) and Theorem \ref{thm:tech10}(iii) that $\sigma_n$ is a
$(T_{j}(v_n)-v_n, D_{j-1},\varphi_{j-1})$-stable coloring. So $\sigma_n$ is $(T_{j-1}, D_{j-1}, \varphi_{j-1})$-stable and hence,
by (5.1) and Theorem \ref{thm:tech10}(vi), it is a $\varphi_{j-1}\bmod T_{j-1}$ coloring. Furthermore, $\sigma(f)=\varphi_n(f)$
for any edge $f$ in $T_n \cup R_n$ by (5.12) and $\overline{\sigma}_n(v)=\overline{\varphi}_n(v)$ for all $v\in V(T_n \cup R_n)$.
(i) Since $R_n$ is a closure of $T_n(v_n)$ under $\varphi_n$ and $\sigma_n$ is $(R_n, \emptyset, \varphi_n)$-stable,
$R_n$ is also a closure of $T_n(v_n)$ under $\sigma_n$. Since $R_n$ is obtained from $T_{j-1}$ by
using the same connecting edge, connecting color, and extension type as $T_j$, by (5.1) and Theorem \ref{thm:tech10}(vi),
$R_n$ is an ETT corresponding to $(\sigma_n,T_{j-1})$ and satisfies MP under $\sigma_n$. Let $\alpha$ and $\beta$ be as
specified in the lemma. As $r(R_n)=j-1$, by (5.1) and Theorem \ref{thm:tech10}(ii), there is at most one $(\alpha,
\beta)$-path with respect to $\sigma_n$ intersecting $R_n$. Hence $\alpha$ and $\beta$ are $R_n$-interchangeable under $\sigma_n$.
Let us make some observations before proving statements (ii) and (iii). By (5.4), each color in
$\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)$ is closed in $T_n\vee R_n$ with respect to $\varphi_n$.
Since $\sigma_n$ is a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring, by definition we obtain
(1) each color in $\overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(R_n)$ is closed in $T_n\vee R_n$ under $\sigma_n$.
(2) $\alpha$ and $\beta$ are $T_n$-interchangeable under $\sigma_n$ if $\alpha\in \overline{\sigma}_n(T_n)$,
$\alpha \ne \delta_n$, and $\beta \ne \delta_n$.
To justify this, note that $\alpha\neq \gamma_n$, because $\gamma_n\notin\overline{\varphi}_n(T_n)=\overline{\sigma}_n(T_n)$. So
$\alpha \notin S_n$. Nevertheless, there are two possibilities for $\beta$.
Let us first consider the case when $\beta \ne \gamma_n$. Since $\sigma_n$ is $(T_n, D_{n},\varphi_{n})$-stable, $P_{v_n}(\gamma_n,\delta_n,\sigma_n)\cap T_n=\{v_n\}$ by (5.1) and Theorem \ref{thm:tech10}(iv).
Define $\sigma_n'=\sigma_n/P_{v_n}(\gamma_n,\delta_n,\sigma_n)$. By Lemma \ref{stablezang},
$\sigma_n'$ is $(T_n,D_{n-1}, \varphi_{n-1})$-stable. From (5.1) and Theorem \ref{thm:tech10}(ii)
we deduce that $\alpha$ and $\beta$ are $T_n$-interchangeable under $\sigma_n'$. So they are $T_n$-interchangeable
under $\sigma_n$ because $\{\alpha,\beta\}\cap S_n=\emptyset$.
It remains to consider the case when $\beta=\gamma_n$. In this case, $f_n$ is the only edge in
$\partial_{\sigma_n, \gamma_n}(T_n)=\partial_{\varphi_n, \gamma_n}(T_n)$ by Lemma \ref{hku}(iv). Since $V(T_n)$ is elementary
with respect to $\varphi_n$, it is also elementary with respect to $\sigma_n$. As $\partial_{\sigma_n, \alpha}(T_n)
=\emptyset$, there is at most one $(\alpha,\gamma_n)$-path with respect to $\sigma_n$ intersecting $T_n$. So
$\alpha$ and $\beta$ are $T_n$-interchangeable under $\sigma_n$. Thus (2) is established.
By (1), $\delta_n$ is closed in $T_n\vee R_n$ with respect to $\sigma_n$. So statement (ii) follows instantly from
(2) and statement (iii).
(iii) Assume the contrary: there are at least two $(\alpha,\beta)$-paths $P_1$ and $P_2$ with respect to
$\sigma_n$ intersecting $T_n \vee R_n$. We may assume that
(3) $\alpha\in \overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(R_n)$.
To justify this, let $A$ be the set of four ends of $P_1$ and $P_2$. Then at least two vertices from $A$ are
outside $T_n\vee R_n$ because, by Lemma \ref{elementary}, $V(T_n\vee R_n)$ is elementary with respect to $\sigma_n$.
Using (i), it is then routine to check that $P_1 \cup P_2$ contains two vertex-disjoint subpaths $Q_1$ and $Q_2$, which are $T_n$-exit
paths with respect to $\sigma_n$. Let $u \in V(T_n) \cap V(R_n)$, let $\eta \in \overline{\sigma}_n(u)$, and let $\sigma_n'=
\sigma_n/(G-T_n\vee R_n,\alpha, \eta)$. By (1), $\eta$ is closed in $T_n\vee R_n$ with respect to $\sigma_n$; so is $\alpha$ by hypothesis. Hence $\sigma_n'$ is a $(T_n \oplus R_n, D_n,\varphi_n)$-stable coloring, and $Q_1$ and $Q_2$ are two $T_n$-exit paths with respect to $\sigma_n'$. Since $P_u(\eta, \beta, \sigma_n')$ contains at most one of $Q_1$ and $Q_2$, replacing
$\sigma_n$ and $\alpha$ by $\sigma_n'$ and $\eta$, respectively, we obtain (3).
Let $v$ be a vertex in $V(T_n)\cap V(R_n)$ with $\alpha\in \overline{\sigma}_n(v)$. Clearly, we may assume that $P_1=
P_v(\alpha,\beta,\sigma_n)$. By (i), we may further assume that $P_2$ is disjoint from $R_n$. So $P_2$ intersects $T_n$.
Therefore $\alpha$ and $\beta$ are not $T_n$-interchangeable under $\sigma_n$. Since $\gamma_n\notin\overline{\varphi}_n(T_n)=
\overline{\sigma}_n(T_n)$, we have $\alpha\neq\gamma_n$. By (2), we may assume that $\alpha = \delta_n$ or $\beta=\delta_n$.
Suppose $\beta=\delta_n$. By Lemma \ref{hku}(iv) and the definition of stable colorings, edges in $\partial_{\sigma_n, \delta_n}(T_n)$
are all incident to $V(T_n)\cap V(R_n)$. Thus both $P_1$ and $P_2$ intersect $V(T_n)\cap V(R_n)$, contradicting statement (i).
Suppose $\alpha=\delta_n$. By (1), $\delta_n$ is closed in $T_n\vee R_n$ under $\sigma_n$. Since $V(T_n) \cap V(R_n)$
contains both ends of the uncolored edge $e$, there exists a color $\theta \in \overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(R_n)-\{\delta_n\}$. Let $\sigma_n''= \sigma_n/(G-T_n\vee R_n,\delta_n, \theta)$. Then $\sigma_n''$ is also
$(T_n\oplus R_n, D_n,\varphi_n)$-stable. From the existence of $P_1$ and $P_2$, we see that $\theta$ and $\beta$ are not $T_n\vee R_n$-interchangeable under $\sigma_n''$, contradicting our observation (2) above the case when $\alpha \ne \delta_n$ and
$\beta \ne \delta_n$.
(iv) Assume the contrary: there are at least two $(\alpha,\beta)$-paths $P_1$ and $P_2$ with respect to $\sigma_n$
intersecting $T_n \vee R_n$. Let $u$ be a vertex in $T_n$ with $\alpha \in \overline{\sigma}_n (u)$ and let $v$ be
a vertex in $R_n$ with $\beta \in \overline{\sigma}_n(v)$. By (ii) (resp. (i)), $P_u(\alpha,\beta,\sigma_n)$
(resp. $P_v(\alpha,\beta,\sigma_n)$) is the only $(\alpha,\beta)$-path with respect to $\sigma_n$ intersecting $T_n$
(resp. $R_n$). Hence $P_1=P_u(\alpha,\beta,\sigma_n)$, $P_2=P_v(\alpha, \beta,\sigma_n)$ (rename subscripts if
necessary), and $P_u(\alpha,\beta,\sigma_n) \ne P_v(\alpha,\beta,\sigma_n)$. Moreover, neither $P_u(\alpha,\beta,\sigma_n)$
nor $P_v(\alpha, \beta,\sigma_n)$ has an end in $V(T_n) \cap V(R_n)$, which in turn implies that
(4) $u \in V(T_n)-V(R_n)$ and $v \in V(R_n)-V(T_n)$.
By (4) and statement (ii), $P_v(\alpha,\beta,\sigma_n)$ is disjoint from $T_n$. Let $\sigma_n'=\sigma_n/P_v(\alpha,
\beta,\sigma_n)$. By Lemma \ref{LEM:Stable}, $\sigma_n'$ is a $(T_n,D_n,\varphi_n)$-stable coloring. By Lemma~\ref{elementary},
$V(T_n\vee R_n)$ is elementary with respect to $\sigma_n$. Since $\alpha \in \overline{\sigma}_n(u)$ and $\beta\in \overline{\sigma}_n(v)$, from TAA we see that no edge in $R_n(v)-T_n(v_n)$ is colored by $\alpha$ or $\beta$
under both $\varphi_n$ and $\sigma_n$. Thus edges in $R_n(v)-T_n(v_n)$ are colored exactly the same under
$\sigma_n'$ as under $\sigma_n$ and $\overline{\sigma}_n(x)= \overline{\sigma}_n'(x)$ for any $x \in
V(R_n(v)-v)) \cup V(T_n)$. Let $R_n'$ be a closure of $T_n(v_n)$ under $\sigma_n'$. Then $v \in V(R_n')$. In view of
Lemma~\ref{elementary}, $V(T_n\vee R_n')$ is elementary with respect to $\sigma_n'$. However,
$\alpha \in \overline{\sigma}'_n(u) \cap \overline{\sigma}'_n(v)$, a contradiction. \hfill \rule{4pt}{7pt}
\vskip 3mm
As introduced in Section 5, $T_{n,0}^*= T_{n}\vee R_n$ if $\Theta_n=PE$ and $T_{n,0}^*= T_{n}$ otherwise. Throughout
a coloring $\sigma_n \in {\cal C}^k(G-e)$ is called a $(T_{n,0}^*, D_n,\varphi_n)$-{\em strongly stable coloring}
if it is a $(T_n\oplus R_n,D_n,\varphi_n)$-stable coloring when $\Theta_n=PE$ and is a $(T_n,D_n,\varphi_n)$-stable
coloring when $\Theta_n \ne PE$. By Lemma \ref{hku}(iii) and (5.12), every $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable
coloring is $(T_{n,0}^*, \varphi_n)$-invariant. It follows from Lemma \ref{sc2} that being $(T_{n,0}^*, D_n, \cdot )$-strongly
stable is an equivalence relation on ${\cal C}^k(G-e)$.
\begin{lemma}\label{step11}
(Assuming (6.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$, and let $\sigma_n$ be a
$(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring. Suppose $T'$ is an ETT
obtained from $T_{n,0}^*$ corresponding to $(\sigma_n,T_n)$ (see Definition \ref{wz2} and Theorem~\ref{thm:tech10}(vi))
that has a good hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,p} =T'$, where $1\le p \le q$
(see Definition \ref{R2} and (6.1)). Furthermore, $T'$ is $(\cup_{\eta_h\in D_{n,p}}\Gamma^{p-1}_h)^-$-closed
with respect to $\sigma_n$. Let $\alpha \in \overline{\sigma}_n(T')$ and $\beta \in [k]-\{\alpha\}$. If $\alpha$ is
closed in $T'$ under $\sigma_n$, then $\alpha$ and $\beta$ are $T'$-interchangeable under $\sigma_n$.
\end{lemma}
A very useful corollary of this lemma is given below.
\begin{corollary}\label{step1}
(Assuming (6.1)) Let $T$ be an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Suppose $T$ has a good hierarchy
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$. Let $p$ be a subscript with
$1\le p \le q$, and let $\alpha \in \overline{\varphi}_n(T_{n,p})$ and $\beta \in [k]-\{\alpha\}$. If $\alpha$ is
closed in $T_{n,p}$ under $\varphi_n$, then $\alpha$ and $\beta$ are $T_{n,p}$-interchangeable under $\varphi_n$.
\end{corollary}
{\bf Proof of Lemma \ref{step11}.} Assume the contrary: there are two $(\alpha,\beta)$-paths $Q_1$ and $Q_2$
with respect to $\sigma_n$ intersecting $T'=T_{n,p}$; subject to this, $p$ is minimum. Let us make some simple
observations about $T'$ before proceeding. Since $\sigma_n$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring,
by Theorem~\ref{thm:tech10}(vi) we have
(1) $T'$ satisfies MP under $\sigma_n$, and hence $V(T')$ is elementary with respect to $\sigma_n$
by (6.1) and Theorem \ref{hierarchy}.
By hypothesis, $\alpha$ is closed in $T'$ with respect to $\sigma_n$, which together with (1) implies that
(2) $|T'|$ is odd.
\noindent In our proof we shall repeatedly use the following hypothesis:
(3) $T'$ is $(\cup_{\eta_h\in D_{n,p}}\Gamma^{p-1}_h)^-$-closed with respect to $\sigma_n$.
Depending on whether $\beta$ is contained in $\overline{\sigma}_n(T')$, we consider two cases.
{\bf Case 1}. $\beta\in \overline{\sigma}_n(T')$.
In this case, $|\partial_{\sigma_n, \beta}(T')|$ is even by (1) and (2). From the existence of $Q_1$ and $Q_2$, we see that
$G$ contains two vertex-disjoint $(T', \sigma_n, \{\alpha,\beta\})$-exit paths $P_1$ and $P_2$.
For $i=1,2$, let $a_i$ and $b_i$ be the ends of $P_i$ with $b_i \in V(T')$. Renaming subscripts if necessary, we may assume
that $b_1 \prec b_2$. We distinguish between two subcases according to the location of $b_2$.
{\bf Subcase 1.1}. $b_2\in V(T')-V(T_{n,p-1}^*)$.
Since the edge on $P_1$ incident to $b_1$ is a boundary edge of $T'$ and is colored by $\beta$, we have
$\beta \in \Gamma^{p-1}_h$ for some $h$ with $\eta_h\in D_{n,p}$ by (3), which together with Definition \ref{R2}(i)
implies that $v_{\beta} \in V(T_{n,p-1})$, where $v_{\beta}$ is the vertex in $T'$ (see (1)) for which
$\beta\in \overline{\sigma}_n(v_{\beta})$. Let $\gamma\in \overline{\sigma}_n(b_2)$. By the assumption of
the present subcase and Definition \ref{R2}(i), we have $\gamma\notin \Gamma^{p-1}$. Hence $\gamma$ is closed
with respect to $\sigma_n$ in $T'$ by (3). So
(4) both $\alpha$ and $\gamma$ are closed in $T'$ under $\sigma_n$.
Let $\mu_1=\sigma_n/(G-T',\alpha,\gamma)$. Clearly, $\mu_1$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable
coloring. By Lemma~\ref{LEM:Stable},
(5) the given hierarchy of $T'$ remains good under $\mu_1$, with the same $\Gamma$-sets as those under $\sigma_n$
(see Definition \ref{R2}). Furthermore, $T'$ is $(\cup_{\eta_h\in D_{n,p}}\Gamma^{p-1}_h)^-$-closed under $\mu_1$.
Note that $P_1$ and $P_2$ are two $(T', \mu_1, \{\gamma, \beta\})$-exit paths.
Let $\mu_2=\mu_1/P_{b_2}(\gamma, \beta, \mu_1)$. Since $P_{b_2}(\gamma, \beta, \mu_1) \cap T'=\{b_2\}$, all edges incident to
$V(T'(b_2)-b_2)$ are colored the same under $\mu_2$ as under $\mu_1$. By (5) and Lemma~\ref{LEM:Stable},
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,p-1} \subset T'(b_2)-b_2$ is a good hierarchy
of $T'(b_2)-b_2$ under $\mu_2$, with the same $\Gamma$-sets as $T'$ under $\sigma_n$. So
(6) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,p-1} \subset T'(b_2)$ is a good hierarchy
of $T'(b_2)$ under $\mu_2$, with the same $\Gamma$-sets as $T'$ under $\sigma_n$.
Clearly, $\mu_2$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring.
By Theorem \ref{thm:tech10}(vi), $T'(b_2)$ satisfies MP under $\mu_2$. Thus from (6.1) we conclude that
$V(T'(b_2))$ is elementary with respect to $\mu_2$. However, $\beta\in \overline{\mu}_2(T_{n,p-1}) \cap
\overline{\mu}_2(b_2)$, a contradiction.
{\bf Subcase 1.2}. $b_2\in V(T_{n,p-1}^*)$.
We propose to show that
(7) there exists a color $\theta \in\overline{\sigma}_n(T_n)$ that is closed in both $T_{n,0}^*$ and $T_{n,1}$ under
$\sigma_n$ if $p=1$, and a color $\theta \in\overline{\sigma}_n(T_{n,p-1})$ that is closed in both $T_{n,p-1}$
and $T_{n,p}$ under $\sigma_n$ if $p\ge 2$.
Our proof is based on the following simple observation (see (3) in the proof of Theorem \ref{good}).
(8) $|\overline{\sigma}_n(T_n)|\ge 2n+11$ and $|D_{n,i}|\le |D_n|\le n$ for $0\le i \le q$.
Let us first assume that $p=1$. When $\Theta_n \ne PE$, let $\theta$ be a color in
$\overline{\sigma}_n(T_n)-(\cup_{\eta_h\in D_{n,1}}\Gamma^{0}_h)$; such a color exists by (8).
From Algorithm 3.1 we see that $T_n$ is closed under $\varphi_n$ and hence under $\sigma_n$.
By (3) and Definition \ref{R2}(v), $T_{n,1}$ is $(\cup_{\eta_h\in D_{n,1}}
\Gamma^{0}_h)^-$-closed under $\sigma_n$. So $\theta$ is as desired. When $\Theta_n=PE$, we have
$|\overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(R_n)-\Gamma^0|\ge 4$ by Definition \ref{R2}(iv).
Let $\theta \in \overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(R_n)-\Gamma^0-\{\delta_n\}$.
By the hypothesis of the present lemma, $\sigma_n$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring.
It follows from (5.4) that $\theta$ is closed in $T_n \vee R_n$ under $\sigma_n$. By Definition \ref{R2}(v),
$T_{n,1}$ is $(\cup_{\eta_h\in D_{n,1}} \Gamma^{0}_h)^-$-closed with respect to $\sigma_n$. So $\theta$ is also
as desired.
Next we assume that $p\geq 2$. By (8), we have $|\overline{\sigma}_n(T_{n,p-2})|\ge |\overline{\sigma}_n(T_{n})|
\ge 2n+11$ and $|D_{n,p-2}|\le |D_n|\le n$. So there exists a color $\theta$ in
$\overline{\sigma}_n(T_{n,p-2})-(\cup_{\eta_h\in D_{n,p-1}}\Gamma^{p-2}_h)$. Since $\overline{\sigma}_n(T_{n,p-2})
\subseteq \overline{\sigma}_n(T_{n,p-1})$, we have $\theta \in \overline{\sigma}_n(T_{n,p-1})-(\cup_{\eta_h\in D_{n,p-1}}\Gamma^{p-2}_h)$. By Definition \ref{R2}(v), $\theta$ is closed in $T_{n, p-1}$ under
$\sigma_n$. From the definition of $\theta$ and Definition \ref{R2}(iii), we also see that
$\theta \notin \Gamma^{p-1}$. So $\theta \in \overline{\sigma}_n(T_{n,p})- \Gamma^{p-1} \subseteq \overline{\sigma}_n(T_{n,p}) -
(\cup_{\eta_h\in D_{n,p}}\Gamma^{p-1}_h)$. By (3), $\theta$ is closed in $T_{n, p}$ under $\sigma_n$. Hence (7) is
established.
Let $\mu_3=\sigma_n/(G-T', \alpha,\theta)$. Since both $\alpha$ and $\theta$ are closed in $T'$ with
respect to $\sigma_n$, by Lemma~\ref{LEM:Stable}, $\mu_3$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable
coloring. Furthermore, $T_{n,p}$ admits a good hierarchy and satisfies
MP with respect to $\mu_3$. Thus $T_{n,p-1}$ also admits a good hierarchy and satisfies
MP with respect to $\mu_3$ if $p\ge 2$. By (7), $\theta$ is closed in $T_{n,0}^*$ if $p=1$ and closed in $T_{n,p-1}$
if $p \ge 2$ under $\mu_3$. Note that both $P_1$ and $P_2$ are $(T_{n,p-1}^*, \mu_3, \{\theta, \beta\})$-exit
paths. So $\theta$ and $\beta$ are not $T_{n,0}^*$-interchangeable under $\mu_3$ if $p=1$
and not $T_{n,p-1}$-interchangeable under $\mu_3$ if $p\ge 2$, which contradicts
Lemma \ref{interchange}(iii) or the interchangeability property of $T_n$ when $p=1$ and the minimality assumption on $p$ when $p\ge 2$.
\vskip 2mm
{\bf Case 2}. $\beta\notin \overline{\sigma}_n(T')$.
In this case, $|\partial_{\sigma_n, \beta}(T')|$ is odd and at least three by (1) and (2). From the existence of
$Q_1$ and $Q_2$, we see that $G$ contains at least three $(T, \sigma_n, \{\alpha, \beta\})$-exit paths $P_1,P_2,P_3$.
For $i=1,2,3$, let $a_i$ and $b_i$ be the ends of $P_i$ with $b_i \in V(T)$, and
$f_i$ be the edge of $P_i$ incident to $b_i$. Renaming subscripts if necessary, we may assume that $b_1\prec b_2 \prec b_3$.
{\bf Subcase 2.1}. $b_3\in V(T')-V(T_{n,p-1}^*)$.
For convenience, we call the tuple $(\sigma_n, T', \alpha, \beta, P_1,P_2,P_3)$ a {\em counterexample} and use ${\cal K}$
to denote the set of all such counterexamples. With a slight abuse of notation, we still use $(\sigma_n, T', \alpha, \beta, P_1,
P_2,P_3)$ to denote a counterexample in ${\cal K}$ with the minimum $|P_1|+|P_2|+|P_3|$. Let $\gamma\in\overline{\varphi}(b_3)$.
By the hypothesis of the present subcase and Definition \ref{R2}(i), we have $\gamma\notin \Gamma^{p-1}$. So $\gamma$
is closed in $T'$ under $\sigma_n$ by (3). Note that $\gamma$ might be some $\eta_h \in D_n$.
Let $\mu_4=\sigma_n/(G-T', \alpha,\gamma)$. By Lemma~\ref{LEM:Stable}, $\mu_4$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly
stable coloring. Furthermore, $T'$ admits a good hierarchy and satisfies MP under $\mu_4$.
Note that $P_1,P_2, P_3$ are three $(T', \mu_4, \{\gamma, \beta\})$-exit paths.
Consider $\mu_5=\mu_4/P_{b_3}(\gamma, \beta, \mu_4)$. Clearly, $\beta \in \overline{\mu}_5(b_3)$ and $\beta\notin
\Gamma^{p-1}$. Since $P_{b_3}(\gamma, \beta, \mu_4) \cap T'=\{b_3\}$, it is easy to see that $\mu_5$ is a
$(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring and all edges incident to
$V(T'(b_3)-b_3)$ are colored the same under $\mu_5$ as under $\mu_4$. By (5.1) and Theorem~\ref{thm:tech10}(vi), $T'(b_3)$ is an
ETT satisfying MP under $\mu_5$. By Lemma~\ref{splitter} and Lemma \ref{LEM:Stable}, $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,p-1} \subset T'(b_3)-b_3$ is a good hierarchy of $T'(b_3)-b_3$ under $\mu_5$, with
the same $\Gamma$-sets as $T'$ under $\sigma_n$. So
(9) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,p-1} \subset T'(b_3)$ is a good hierarchy
of $T'(b_3)$ under $\mu_5$, with the same $\Gamma$-sets as $T'$ under $\sigma_n$.
Let $H$ be obtained from $T'(b_3)$ by adding $f_1$ and $f_2$. Since $\beta\notin\Gamma^{p-1}$, it can be seen from (9) that
(10) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,p-1} \subset H$ is a good hierarchy of
$H$ under $\mu_5$, with the same $\Gamma$-sets as $T'$ under $\sigma_n$.
By (5.1) and Theorem~\ref{thm:tech10}(vi), $H$ satisfies MP under $\mu_5$. Set $T''=H$. Let us grow $T''$
by using the following algorithm:
(11) While there exists $f\in\partial(T'')$ with $\mu_5(f)\in\overline{\mu}_5(T'')$, do: set $T''=T''+f$ if
the resulting $T''$ satisfies $\Gamma^{p-1}_h \cap \mu_5 \langle T''(v_{\eta_h})-T_{n,p-1} \rangle =\emptyset$ for
all $\eta_h \in D_{n,p-1}$.
Note that this algorithm is exactly the same as Step 2 in Algorithm 5.6. From (11) we see that
(12) $T''$ is $(\cup_{\eta_h\in D''_{n,p}}\Gamma^{p-1}_h)^-$-closed with respect to $\mu_5$, where $D_{n,p}''=\cup_{h\leq n}S_h -\overline{\mu}_5(T'')$ (so $D_{n,p}'' \subseteq D_{n,p-1}$).
In view of (10) and (11), we conclude that
(13) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,p-1} \subset T''$ is a good hierarchy of $H$ under
$\mu_5$, with the same $\Gamma$-sets as $T'$ under $\sigma_n$.
Clearly, $T''$ satisfies MP under $\mu_5$. By (13), (6.1), and Theorem \ref{hierarchy}, $V(T'')$ is elementary with
respect to $\mu_5$. Observe that none of $a_1,a_2,a_3$ is contained in $T''$, for otherwise, let $a_i \in V(T_2)$ for
some $i$ with $1\le i \le 3$. Since $\{\beta,\gamma\}\cap \overline{\mu}_5(a_i) \ne \emptyset$ and
$\beta \in \overline{\mu}_5(b_3)$, we obtain $\gamma \in \overline{\sigma}_2(a_i)$. Hence from TAA we see that
$P_1,P_2,P_3$ are all entirely contained in $G[T'']$, which in turn implies $\gamma \in \overline{\sigma}_2(a_j)$
for $j=1,2,3$. So $V(T'')$ is not elementary with respect to $\mu_5$, a contradiction. Each $P_i$ contains a subpath
$L_i$, which is a $T''$-exit path with respect to $\mu_5$. Since $f_1$ is not contained in $L_1$, we obtain $|L_1|+|L_2|+|L_3|<|P_1|+|P_2|+|P_3|$. Thus, in view of (12), the existence of the counterexample $(\mu_5, T'', \gamma,
\beta, L_1, L_2,L_3)$ violates the minimality assumption on $(\sigma_n, T', \alpha, \beta, P_1,P_2,P_3)$.
{\bf Subcase 2.2}. $b_3\in V(T_{n,p-1}^*)$.
The proof in this subcase is essentially the same as that in Subcase 1.2. Let $\theta$ be a color as described in
(7). Consider $\mu_3=\sigma_n/(G-T', \alpha,\theta)$. Then we can verify that $\theta$ and $\beta$ are not $T_{n,0}^*$-interchangeable under $\mu_3$ if $p=1$ and not $T_{n,p-1}$-interchangeable under $\mu_3$ if $p\ge 2$, which contradicts
Lemma \ref{interchange}(iii) or the minimality assumption on $p$; for the omitted details, see the proof in Subcase 1.2. \hfill \rule{4pt}{7pt} \\
Let us make some further preparations before proving Theorem \ref{hierarchy}. Let $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q+1}=T$ be a good hierarchy of $T$ (see (5.5) and Definition \ref{R2}). Recall that $T_{n,0}^*=
T_{n}\vee R_n$ if $\Theta_n=PE$ and $T_{n,0}^*= T_{n}$ otherwise, $T_{n,0}^* \subset T_{n,1}$ by (5.5), and $T_{n,i}^*=T_{n,i}$
if $i \ge 1$. Let $T$ be constructed from $T_{n,q}^*$ using TAA by recursively adding edges
$e_1,e_2, \ldots, e_p$ and vertices $y_1,y_2, \ldots, y_p$, where $y_i$ is the end of $e_i$ outside
$T(y_{i-1})$ for $i\ge 1$, with $T(y_0)=T_{n,q}^*$. Write $T=T_{n,q}^* \cup\{e_1,y_1,e_2,...,e_p,y_p\}$. The {\em path number}
of $T$, denoted by $p(T)$, is defined to be the smallest subscript $i\in \{1,2,...,p\}$ such that the sequence $(y_i,e_{i+1},...,e_p,y_p)$ corresponds to a path in $G$.
A coloring $\sigma_n \in {\cal C}^k(G-e)$ is called a $(T_{n,i}^*, D_n, \varphi_n)$-{\em strongly stable coloring}, with
$1 \le i \le q$, if it is both a $(T_{n,0}^*, D_n, \varphi_n)$-strongly stable and a $(T_{n,i}^*,\varphi_n)$-invariant
coloring. Since every $(T_{n,0}^*, D_n, \varphi_n)$-strongly stable coloring is $(T_{n,0}^*,\varphi_n)$-invariant
by Lemma \ref{hku}(iii) and (5.12), this concept is a natural extension of $(T_{n,0}^*, D_n, \varphi_n)$-strongly
stable colorings. Let $v$ be a vertex of $G$. By $T \prec v$ we mean that $u \prec v$ for any $u \in V(T)$. Given a
color $\alpha \in [k]$, we use $v_{\alpha}$ to denote the first vertex $u$ of $T$ in the order $\prec$ for which $\alpha \in \overline{\varphi}_n(u)$, if any, and defined to be the last vertex of $T$ in the order $\prec$ otherwise.
\hskip 3mm
Recall that our proof of Theorem \ref{hierarchy} proceeds by induction on $q$ (see (6.1)). The induction step will be
carried out by contradiction. Throughout the remainder of this section and Subsection 7.1, $(T, \varphi_n)$ stands for
a minimum counterexample to Theorem \ref{hierarchy}; that is,
{\bf (6.2)} $T$ is an ETT that admits a good hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}$
\hskip 10mm $=T$ and satisfies MP with respect to the generating coloring $\varphi_n$;
{\bf (6.3)} subject to (6.2), $V(T)$ is not elementary with respect to $\varphi_n$;
{\bf (6.4)} subject to (6.2) and (6.3), $p(T)$ is minimum; and
{\bf (6.5)} subject to (6.2)-(6.4), $|T|-|T_{n,q}|$ is minimum.
\noindent Our objective is to find another counterexample $(T', \sigma_n)$ to Theorem \ref{hierarchy}, which violates
the minimality assumption (6.4) or (6.5) on $(T, \varphi_n)$.
The following fact will be used frequently in subsequent proof.
{\bf (6.6)} $V(T(y_{p-1}))$ is elementary with respect to $\varphi_n$.
\vskip 2mm
Let us exhibit some basic properties satisfied by the minimum counterexample $(T, \varphi_n)$ as specified above.
\vskip 3mm
\iffalse
\begin{lemma}\label{becare}
Suppose $\Theta_n=PE$. Then the following statements hold:
\begin{itemize}
\vspace{-2mm}
\item[(i)] $\varphi_n \langle R_n-T_n(v_n) \rangle \cap \overline{\varphi}_n (T_n-V(R_n))=\emptyset$;
\vspace{-2mm}
\item[(ii)] $\varphi_n \langle T_n \rangle-\overline{\varphi}_n(T_n)\subseteq D_n$; and
\vspace{-2mm}
\item[(iii)] $(\varphi_n \langle T_n \rangle-\overline{\varphi}_n(T_n))\cap \overline{\varphi}_n(R_n)\subseteq D_n\cap \overline{\varphi}_n(R_n-V(T_n))$.
\end{itemize}
\end{lemma}
{\bf Proof.} (i) By (6.6) or Lemma \ref{elementary}, $V(T_n\vee R_n)$ is elementary with respect to $\sigma_n$.
Since $R_n$ is obtained from $T_n(v_n)$ by TAA under $\varphi_n$ (see Algorithm 3.1), no edge in $R_n-T_n(v_n)$ is
colored by a color in $\overline{\varphi}_n(T_n-V(R_n))$; equivalently, $\varphi_n \langle R_n-T_n(v_n) \rangle \cap \overline{\varphi}_n (T_n-V(R_n))=\emptyset$.
(ii) According to Algorithm 3.1 and TAA, each edge in $T_n$ is colored by a color in $\overline{\varphi}_n(T_n) \cup D_n$.
So $\varphi_n \langle T_n \rangle \subseteq \overline{\varphi}_n(T_n) \cup D_n$, which yields
$\varphi_n \langle T_n \rangle-\overline{\varphi}_n(T_n)\subseteq D_n$.
(iii) Let $\alpha$ be an arbitrary color in $(\varphi_n \langle T_n \rangle-\overline{\varphi}_n(T_n))\cap \overline{\varphi}_n(R_n)$.
Then $\alpha \in D_n \cap \overline{\varphi}_n(R_n)$ by (ii) and $\alpha \notin \overline{\varphi}_n(T_n)$. So $\alpha \in D_n \cap \overline{\varphi}_n(R_n)
-\overline{\varphi}_n(T_n) \subseteq D_n\cap \overline{\varphi}_n(R_n-V(T_n))$. Hence the desired statement holds. \hfill \rule{4pt}{7pt}
\fi
\begin{lemma}\label{9n}
For $0 \le i \le p$, the inequality
$$|\overline{\varphi}_n(T(y_i))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle T(y_i) - T_{n,q}^* \rangle | \geq 2n+11$$
holds, where $T(y_0)=T_{n,q}^*$. Furthermore, if
$$|\overline{\varphi}_n(T(y_i))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle T(y_i) - T_{n,q}^* \rangle|
-|\Gamma^{q}\cup D_{n,q} |\le 4,$$
then there exist $7$ distinct colors $\eta_{h}\in D_{n,q}\cap \overline{\varphi}_n(T(y_i))$ such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \varphi_n \langle T(y_i) - T_{n,q}^* \rangle=\emptyset$, where $\Gamma^{q}$ and
$\Gamma^{q}_h$ are introduced in Definition \ref{R2}.
\end{lemma}
{\bf Proof.} Since the number of vertices in $T(y_i)-V(T_{n,q}^*)$ is $i$, and the number of edges in
$T(y_i) - T_{n,q}^*$ is also $i$, we obtain $|\overline{\varphi}_n (T(y_i) - V(T_{n,q}^*))| \ge | \varphi_n \langle
T(y_i) - T_{n,q}^* \rangle|$. Hence
\begin{equation*}
\begin{aligned}
& \hskip 2mm |\overline{\varphi}_n(T(y_i))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle T(y_i) - T_{n,q}^* \rangle |\\
\ge & \hskip 2mm |\overline{\varphi}_n(T(y_i))| - |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\overline{\varphi}_n (T(y_i) - V(T_{n,q}^*))| \\
= & \hskip 2mm |\overline{\varphi}_n(T_{n,q}^*)| - |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| \\
\ge & \hskip 2mm |\overline{\varphi}_n(T_{n,0}^*)| - |\overline{\varphi}_n(T_{n,0}^*)- \overline{\varphi}_n(T_n)| \\
= & \hskip 2mm |\overline{\varphi}_n(T_n)|\\
\ge & \hskip 2mm 2n+11,
\end{aligned}
\end{equation*}
where the last inequality can be found in the proof of Theorem \ref{good} (see (3) therein). So the first inequality is
established.
Suppose the second inequality also holds. Then these two inequalities guarantee the existence of at least $2n+7$ colors in
the intersection of $\overline{\varphi}_n(T(y_i))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n \langle T(y_i) - T_{n,q}^* \rangle
$ and $\Gamma^{q}\cup D_{n,q}$. Let $C$ denote this intersection. Then $|C|\ge 2n+7$. By (5.6), we have $|D_{n,q}|\le |D_{n}| \le n$
and $|\Gamma^{q}| \le 2 |D_{n,q}|\le 2n$. So $|\Gamma^{q}\cup D_{n,q}|\le 3n$. Since $|C|\le |\Gamma^{q}\cup D_{n,q}|$,
it follows that $2n+7 \le 3n$, which implies $n \ge 7$. Note that $C=\cup_{\eta_{h}\in D_{n,q}} (\Gamma^{q}_h \cup \{\eta_{h}\})
\cap C$ and $|(\Gamma^{q}_h \cup \{\eta_{h}\}) \cap C|\le 3$ for any $\eta_{h}$ in $D_{n,q}$. Since $|C|\ge 2n+7$ and $n\ge 7$,
by the Pigeonhole Principle, there exist at least $7$ distinct colors $\eta_{h}$ in $D_{n,q}$, such that $| (\Gamma^{q}_h \cup \{\eta_{h}\}) \cap C|=3$, or equivalently, $\Gamma^{q}_h \cup \{\eta_{h}\} \subseteq C$. For each of these $\eta_{h}$, clearly $\eta_{h}\in D_{n,q}\cap \overline{\varphi}_n(T(y_i))$ and $(\Gamma^{q}_h \cup \{\eta_{h}\}) \cap \varphi_n \langle T(y_i) - T_{n,q}^* \rangle=\emptyset$. \hfill \rule{4pt}{7pt}
\begin{lemma}\label{rutcor}
Suppose $q\geq 1$ and $\alpha\in \overline{\varphi}_n(T_{n,q})$. If there exists a subscript $i$ with $0\le i \le q$,
such that $\alpha$ is closed in $T^*_{n,i}$ with respect to $\varphi_n$, then $\alpha \notin \varphi_n \langle
T_{n,q}- T_{n,r}^* \rangle$, where $r$ is the largest such $i$. If there is no such subscript $i$, then
$\alpha\in \cup_{\eta_h\in D_{n,j}} \Gamma^{j-1}_h \subseteq \Gamma^{j-1}$ for $1 \le j \le q$,
$\Theta_n=PE$, $v_{\alpha}\in V(T_n)-V(R_n)$, and $\alpha \notin \varphi_n \langle T_{n,q}- T_n \rangle$.
\end{lemma}
{\bf Proof.} Let us first assume the existence of a subscript $i$ with $0\le i \le q$,
such that $\alpha$ is closed in $T^*_{n,i}$ with respect to $\varphi_n$. By definition, $r$ is the largest such $i$.
Suppose the contrary: $\alpha \in \varphi_n \langle T_{n,q}-T_{n,r}^* \rangle$. Then $r<q$ and there exists a
subscript $s$ with $r+1\le s \le q$, such that $\alpha \in \varphi_n \langle T_{n,s}-T_{n,s-1}^* \rangle$. From the definition
of $r$, we see that $\alpha$ is not closed in $T_{n,s}$ with respect to $\varphi_n$. It follows from Definition \ref{R2}(v)
that $\alpha\in\Gamma^{s-1}_h$ for some $\eta_h\in D_{n,s}$. By the definitions of $D_{n,s}$ and $D_{n,s-1}$,
we have $D_{n,s} \subseteq D_{n,s-1}$. So $\eta_h\in D_{n,s-1}$.
Since $\alpha$ in $\Gamma^{s-1}_h$ is used by at
least one edge in $T_{n,s}-T_{n,s-1}^*$, from Definition \ref{R2}(i) (with $j=s-1$) we deduce that $\eta_h$
is a color missing at some vertex in $T_{n,s}$ (see (5.7)). Thus $\eta_h\notin D_{n,s}$ by definition,
a contradiction.
Next we assume that there exists no subscript $i$ with $0\le i \le q$, such that $\alpha$ is closed in $T^*_{n,i}$ with
respect to $\varphi_n$. Since $\alpha\in \overline{\varphi}_n(T_{n,q})$, it follows from (5.10) that $\alpha \in \overline{\varphi}_n(T_{n,0}^*)$.
By Definition \ref{R2}(v), we obtain
(1) $\alpha\in \cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h \subseteq \Gamma^{j-1}$ for $1 \le j \le q$.
\noindent Hence $\alpha\in\Gamma^j$ for all $0 \le j \le q-1$. From the definition of $\Gamma^0$, we see that $v_{\alpha}
\in V(T_n)$. If $\Theta_n\neq PE$, then $\alpha$ would be closed in $T_{n}=T_{n,0}^*$ under $\varphi_n$, a contradiction.
So $\Theta_n=PE$. Moreover, by the assumption on $\alpha$, Algorithm 3.1 and (5.4), we have
$v_{\alpha} \in V(T_n)-V(R_n)$. Since $R_n$ is a closure of $T_n(v_n)$ under $\varphi_n$, using (6.6) and TAA we obtain
(2) $\alpha \notin \overline{\varphi}_n (R_n-V(T_n))$ and $\alpha \notin \varphi_n \langle R_n-T_n \rangle$.
(3) $\alpha \notin \varphi_n \langle T_{n,q}-T_{n,0}^* \rangle$.
Assume the contrary: $\alpha \in \varphi_n \langle T_{n,q}-T_{n,0}^* \rangle$. Then there exists a subscript $1\le s \le q$
such that $\alpha \in \varphi_n \langle T_{n,s}-T_{n,s-1}^* \rangle$. By (1), we have $\alpha\in\Gamma^{s-1}_h$ for some
$\eta_h \in D_{n, s}$. As $D_{n,s} \subseteq D_{n,s-1}$, we obtain $\eta_h \in D_{n, s-1}$. Since $\alpha$ is used by
at least one edge in $T_{n,s}-T_{n,s-1}^*$, from Definition \ref{R2}(i) (with $j=s-1$) we deduce that $\eta_h$ is a
color missing at some vertex in $T_{n,s}$ (see (5.7)). Thus $\eta_h\notin D_{n,s}$ by definition, a
contradiction.
Combining (2) and (3), we conclude that $\alpha \notin \varphi_n \langle T_{n,q}- T_n \rangle$. \hfill \rule{4pt}{7pt}
\vskip 3mm
Our proof of Theorem \ref{hierarchy} relies heavily on the following two technical lemmas.
\begin{lemma}\label{change}
Let $\alpha$ and $\beta$ be two colors in $\overline{\varphi}_n(T(y_{p-1}))$. Suppose $v_{\alpha} \prec v_{\beta}$ and $\alpha
\notin \varphi_n \langle T(v_{\beta}) - T_{n,q}^* \rangle$ if $\{\alpha, \beta\}-\overline{\varphi}_n(T_{n,q}^*)
\ne \emptyset$. Then $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)=P_{v_{\beta}}(\alpha,\beta,\varphi_n)$ if one of the
following cases occurs:
\begin{itemize}
\vspace{-1.5mm}
\item[(i)] $q\geq 1$, and $\alpha\in \overline{\varphi}_n(T_{n,q})$ or $\{\alpha, \beta\}\cap D_{n,q}=\emptyset$;
\vspace{-2mm}
\item[(ii)] $q=0$, and $\alpha\in \overline{\varphi}_n(T_{n})$ or $\{\alpha, \beta\} \cap D_{n}=\emptyset$; and
\vspace{-1.5mm}
\item[(iii)] $\alpha \in \overline{\varphi}_n(T_{n,q}^*)$ and is closed in $T_{n,q}^*$ with respect to $\varphi_n$.
\vspace{-1.5mm}
\end{itemize}
Furthermore, in Case (iii), $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)=P_{v_{\beta}}(\alpha,\beta,\varphi_n)$
is the only $(\alpha,\beta)$-path with respect to $\varphi_n$ intersecting $T_{n,q}^*$.
\end{lemma}
{\bf Proof.} Let $a=v_{\alpha}$ and $b=v_{\beta}$. We distinguish among three cases according to
the locations of $a$ and $b$.
{\bf Case 1}. $\{a,b\} \subseteq V(T_{n,q}^*)$.
By (6.6), $V(T_{n,q}^*)$ is elementary with respect to $\varphi_n$. So $a$ (resp. $b$) is the only vertex in $T_{n,q}^*$
missing $\alpha$ (resp. $\beta$). If both $\alpha$ and $\beta$ are closed in $T_{n,q}^*$ with respect to
$\varphi_n$, then no boundary edge of $T_{n,q}^*$ is colored by $\alpha$ or $\beta$. Hence $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,\beta,\varphi_n)$ is the only path intersecting $T_{n,q}^*$.
So we may assume that $\alpha$ or $\beta$ is not closed in $T_{n,q}^*$ with respect to $\varphi_n$.
It follows that if $q=0$, then $\Theta_n=PE$, for otherwise, Algorithm 3.1 would imply that
both $\alpha$ and $\beta$ are closed in $T_{n}=T_{n,0}^*$, a contradiction. Therefore
(1) $T_{n,0}^*= T_{n}\vee R_n$ if $q=0$.
Let us first assume that precisely one of $\alpha$ and $\beta$ is closed in $T_{n,q}^*$ with respect to $\varphi_n$. In
this subcase, by Corollary \ref{step1} if $q \ge 1$ and by (1) and Lemma~\ref{interchange}(iii) if $q=0$, colors
$\alpha$ and $\beta$ are $T_{n,q}^*$-interchangeable under $\varphi_n$, so $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,
\beta,\varphi_n)$ is the only path intersecting $T_{n,q}^*$.
Next we assume that neither $\alpha$ nor $\beta$ is closed in $T_{n,q}^*$ with respect to $\varphi_n$. In this subcase,
we only need to show that $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,\beta,\varphi_n)$. Symmetry allows us to assume
that $a \prec b$. Let $r$ be the subscript with $\beta\in\overline{\varphi}_n(T_{n,r}^*-V(T_{n,r-1}^*))$, where $0\leq r \le q$
and $T_{n,-1}^*=\emptyset$. Then $a, b \in V(T_{n,r}^*)$. By (6.2), $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ is a good hierarchy of $T$. If $r \ge 1$, then $\beta$ is closed in $T_{n,r}$ with respect
to $\varphi_n$ by Definition \ref{R2} (see (5.10)). From the above discussion about $T_{n,q}^*$ (with $r$ in place of $q$),
we similarly deduce that $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,\beta,\varphi_n)$. So we may assume that $r=0$. If
$\Theta_n\ne PE$, then both $\alpha$ and $\beta$ are closed in $T_n$ with respect to $\varphi_n$ (see Algorithm 3.1), so $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,\beta,\varphi_n)$ by (6.6). If $\Theta_n =PE$, then it follows from Lemma~\ref{interchange}(i), (ii) and (iv) that $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,\beta,\varphi_n)$.
{\bf Case 2}. $\{a,b\} \cap V(T_{n,q}^*)=\emptyset$.
By the hypotheses of the present case and the present lemma, we have $\{\alpha, \beta\}\cap D_{n,q}=\emptyset$ if
$q \ge 1$ and $\{\alpha, \beta\} \cap D_{n}=\emptyset$ if $q=0$. So
(2) $\alpha, \beta \notin D_{n,q}\cup \overline{\varphi}_n(T_{n,q}^*)$ if $q \ge 1$ and
$\alpha, \beta \notin D_{n}\cup \overline{\varphi}_n(T_{n,0}^*)$ if $q=0$.
\noindent By the definitions of $D_n$ and $D_{n,q}$, we have $ D_n \cup \overline{\varphi}_n(T_n) \subseteq D_{n,q}\cup
\overline{\varphi}_n(T_{n,q}^*)$. Since $\alpha \notin \varphi_n \langle T(b) - T_{n,q}^* \rangle$ by hypothesis, from (2),
Lemma~\ref{hku}(iii) and TAA we see that
(3) $\alpha,\beta\notin\varphi_n \langle T(b) \rangle$.
\noindent Suppose on the contrary that $P_a(\alpha,\beta,\varphi_n)\neq P_{b}(\alpha,\beta,\varphi_n)$.
Consider $\sigma_n=\varphi_n/P_{b}(\alpha,\beta,\varphi_n)$. Using (2), (3) and (6.6), it is routine to check
that $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring, and $T(b)$ is an ETT
satisfying MP with respect to $\sigma_n$. Moreover, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}
\subset T(b)$ is a good hierarchy of $T(b)$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$
(see Definition \ref{R2}). As $\alpha\in\overline{\sigma}_n(a)\cap \overline{\sigma}_n(b)$, the pair $(T(b), \sigma_n)$
is a counterexample to Theorem \ref{hierarchy}, which contradicts the minimality assumption (6.5) on $(T, \varphi_n)$.
{\bf Case 3}. $a \in V(T_{n,q}^*)$ and $b \notin V(T_{n,q}^*).$
By the hypotheses of the present case and the present lemma, (6.6) and TAA, we obtain
(4) $\alpha \notin \varphi_n \langle T(b) - T_{n,q}^* \rangle$ and $\beta \notin \overline{\varphi}_n (T(b)-b)$.
So $\beta$ is not used by any edge in $T(b) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$
(now $e_1=f_n$ in Algorithm 3.1 and $\beta \in D_n$).
Let us first assume that $\alpha$ is closed in $T_{n,q}^*$ with respect to $\varphi_n$. By Corollary \ref{step1} if $q \ge 1$
and by Lemma~\ref{interchange}(iii) or Theorem \ref{thm:tech10}(ii) (see (5.1)) if $q=0$, colors $\alpha$ and $\beta$ are $T_{n,q}^*$-interchangeable under $\varphi_n$. So $P_{a}(\alpha,\beta,\varphi_n)$ is the only $(\alpha,\beta)$-path
intersecting $T_{n,q}^*$. Suppose on the contrary that $P_{a}(\alpha,\beta,\varphi_n) \ne P_{b}(\alpha,\beta,\varphi_n)$.
Then $P_{b}(\alpha,\beta,\varphi_n)$ is vertex-disjoint from $T_{n,q}^*$ and hence contains no edge incident to $T_{n,q}^*$.
Consider $\sigma_n=\varphi_n/P_{b}(\alpha,\beta,\varphi_n)$. It is routine to check
that $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring, and $T(b)$ is an ETT
satisfying MP with respect to $\sigma_n$. Moreover, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}
\subset T(b)$ is a good hierarchy of $T(b)$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$,
by (4). As $\alpha\in\overline{\sigma}_n(a)\cap \overline{\sigma}_n(b)$, the pair $(T(b), \sigma_n)$
is a counterexample to Theorem \ref{hierarchy}, which contradicts the minimality assumption (6.5) on $(T, \varphi_n)$.
So we assume hereafter that
(5) $\alpha$ is not closed in $T_{n,q}^*$ with respect to $\varphi_n$.
Our objective is to show that $P_{a}(\alpha,\beta,\varphi_n)=P_{b}(\alpha,\beta,\varphi_n)$. Assume the contrary: $P_{a}(\alpha,\beta,\varphi_n)\\ \ne P_{b}(\alpha,\beta,\varphi_n)$. We distinguish between two subcases according to
the value of $q$.
{\bf Subcase 3.1.} $q=0$.
By the hypothesis of the present lemma, $\alpha\in \overline{\varphi}_n(T_{n})$ or $\{\alpha, \beta\} \cap D_{n}=\emptyset$.
So $\alpha\notin D_n$. From (5) and Algorithm 3.1 we deduce that $T_{n,0}^* \ne T_n$. Hence
(6) $\Theta_n=PE$, which together with (5) and (5.4) yields $a \notin V(T_n)\cap V(R_n)$.
Consider $\sigma_n=\varphi_n/P_{b}(\alpha,\beta,\varphi_n)$. We claim that
(7) $\sigma_n$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring.
To justify this, note that if $a\in V(T_n)-V(R_n)$, then $\alpha,\beta\notin \overline{\varphi}_n (R_n)$ by (6.6) and the
hypothesis of the present case. By definition, $\sigma_n$ is $(R_n,\emptyset,\varphi_n)$-stable. In view of Lemma~\ref{interchange}(ii), $P_{b}(\alpha,\beta,\varphi_n)$ is disjoint from $T_n$ and hence contains no edge
incident to $T_n$. So $\sigma_n$ is $(T_n,D_n,\varphi_n)$-stable. Hence (7) holds. Suppose $a\in V(R_n)-V(T_n)$.
By the hypothesis of the present lemma, $\{\alpha, \beta\} \cap D_{n}=\emptyset$. By (6.6), we also have
$\alpha,\beta\notin \overline{\varphi}_n(T_n)$. Thus $\alpha,\beta\notin \overline{\varphi}_n(T_n) \cup D_n$. By definition,
$\sigma_n$ is $(T_n,D_n,\varphi_n)$-stable. Using Lemma~\ref{interchange}(i), $P_{b}(\alpha,\beta,\varphi_n)$
is disjoint from $R_n$ and hence contains no edge incident to $R_n$. By definition, $\sigma_n$ is $(R_n,\emptyset,
\varphi_n)$-stable. Therefore (7) is true.
From (4), (7) and (6.6) we see that $\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T(b))$ (see the remark above
lemma \ref{step11}) and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(b)-b)$. Furthermore, $T(b)$ is an ETT
satisfying MP with respect to $\sigma_n$, and $T_n=T_{n,0} \subset T(b)$ is a good hierarchy of $T(b)$ under $\sigma_n$,
with the same $\Gamma$-sets as $T$ under $\varphi_n$. As $\alpha\in\overline{\sigma}_n(a)\cap \overline{\sigma}_n(b)$,
the pair $(T(b), \sigma_n)$ is a counterexample to Theorem \ref{hierarchy}, which contradicts the minimality assumption
(6.5) on $(T, \varphi_n)$.
{\bf Subcase 3.2.} $q\ge 1$.
Let us first assume that $\alpha$ is closed in $T^*_{n,i}$ with respect to $\varphi_n$ for some $i$ with
$0\le i \le q$. Let $r$ be the largest subscript $i$ with this property. Then $r\le q-1$ by (5).
By Lemma \ref{rutcor}, we have $\alpha \notin \varphi_n \langle T_{n,q}- T_{n,r}^* \rangle$, which together
with (4) yields
(8) $\alpha\notin \varphi_n \langle T(b)-T_{n,r}^* \rangle$.
By Corollary \ref{step1} if $r \ge 1$ and by Theorem \ref{thm:tech10}(ii) or Lemma~\ref{interchange}(iii) if $r=0$, colors
$\alpha$ and $\beta$ are $T_{n,r}^*$-interchangeable under $\varphi_n$. So $P_{a}(\alpha,\beta,\varphi_n)$ is the only $(\alpha,\beta)$-path with respect to $\varphi_n$ intersecting $T_{n,r}^*$. Hence $P_{b}(\alpha,\beta,\varphi_n)$ is
vertex-disjoint from $T_{n,r}^*$ and therefore contains no edge incident to $T_{n,r}^*$. Consider $\sigma_n=\varphi_n/P_{b}(\alpha,\beta,\varphi_n)$. By Lemma \ref{LEM:Stable}, $\sigma_n$ is a $(T_{n,r}^*, D_n,\varphi_n)$-strongly stable coloring, and $T_{n,r}^*$ is an ETT having a good hierarchy and satisfying MP with respect to $\sigma_n$. By (4)
and TAA, $\beta$ is not used by any edge in $T(b) - T_{n,r}^*$, except possibly $e_1$ when $r=0$ and $T_{n,0}^*=T_n$
(now $e_1=f_n$ in Algorithm 3.1 and $\beta \in D_n$). Since $\sigma_n$ is $(T_n, D_n,\varphi_n)$-stable,
it follows from (8) and (6.6) that $\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T(b))$ and $\overline{\sigma}_n(u)=
\overline{\varphi}_n(u)$ for each $u\in V(T(b)-b)$. So $T(b)$ is an ETT satisfying MP with respect to $\sigma_n$. Moreover,
(9) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T(b)$ is a good hierarchy of $T(b)$ under $\sigma_n$,
with the same $\Gamma$-sets as $T$ under $\varphi_n$.
To justify this, it suffices to verify that Definition \ref{R2}(v) is satisfied with respect to $\sigma_n$; that is,
$T_{n,j}$ is $(\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h)^-$-closed with respect to $\sigma_n$ for $1\le j \le q$.
As the statement holds trivially if $P_{b}(\alpha,\beta,\varphi_n)$ is vertex-disjoint from $T_{n,j}$, we may
assume that $P_b(\alpha, \beta, \varphi_n)$ intersects $T_{n,j}$. Thus $r+1\le j\le q$. Observe that $\alpha \in \cup_{\eta_h\in D_{n,j}} \Gamma^{j-1}_h$, for otherwise, $\alpha$ is closed in $T_{n,j}$ with respect to $\varphi_n$ by Definition \ref{R2}(v),
contradicting the definition of $r$. By (6.6), we also obtain $\beta \notin \overline{\varphi}_n(T_j)$. Consequently,
$T_{n,j}$ is $(\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h)^-$-closed with respect to $\sigma_n$. (Note that $\alpha$ may
become closed in $T_{n,j}$ with respect to $\sigma_n$. Yet, even in this situation the desired statement is true.)
This proves (9).
As $\alpha\in\overline{\sigma}_n(a)\cap \overline{\sigma}_n(b)$, the existence of $(T(b), \sigma_n)$ contradicts the minimality assumption (6.5) on $(T, \varphi_n)$.
Next we assume that $\alpha$ is not closed in $T^*_{n,i}$ with respect to $\varphi_n$ for any $i$ with $0\le i \le q$.
In view of Lemma \ref{rutcor}, we obtain
(10) $\alpha\in (\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h) \subseteq \Gamma^{j-1}$ for $1 \le j \le q$, $\Theta_n=PE$,
$a \in V(T_n)-V(R_n)$, and $\alpha \notin \varphi_n \langle T_{n,q}- T_n \rangle$.
It follows from (4), (10) and TAA that
(11) $\alpha\notin \varphi_n \langle T(b)-T_n \rangle$ and $\beta \notin \varphi_n \langle T(b)- T_{n,0}^* \rangle$.
\noindent Since $R_n$ is a closure of $T_n(v_n)$ under $\varphi_n$, using (10), (6.6) and TAA we obtain
(12) $\alpha, \beta \notin \overline{\varphi}_n (R_n)$ and $\beta \notin \varphi_n \langle R_n-T_n \rangle$.
By Lemma~\ref{interchange}(ii), colors $\alpha$ and $\beta$ are $T_n$-interchangeable under $\varphi_n$. So $P_{a}(\alpha,\beta,\varphi_n)$ is the only $(\alpha,\beta)$-path with respect to $\varphi_n$ intersecting
$T_n$. Hence $P_{b}(\alpha,\beta,\varphi_n)$ is vertex-disjoint from $T_n$ and therefore contains no edge incident
to $T_n$. Consider $\sigma_n=\varphi_n/P_{b}(\alpha,\beta,\varphi_n)$. By Lemma \ref{LEM:Stable},
$\sigma_n$ is a $(T_n, D_n,\varphi_n)$-stable coloring, and $T_n$ is an ETT satisfying MP with respect to $\sigma_n$.
From (11) and (12) we further deduce that $\sigma_n$ is a $(T_{n,0}^*, D_n, \varphi_n)$-strongly stable coloring,
$\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T(b))$, and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(b)-b)$.
So $T(b)$ is an ETT satisfying MP with respect to $\sigma_n$. Moreover, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots
\subset T_{n,q} \subset T(b)$ is a good hierarchy of $T(b)$ under $\sigma_n$, with the same $\Gamma$-sets as $T$
under $\varphi_n$ (see (10) and the proof of (9) for omitted details). As $\alpha\in\overline{\sigma}_n(a)\cap \overline{\sigma}_n(b)$, the existence of $(T(b), \sigma_n)$ contradicts the minimality assumption (6.5) on
$(T, \varphi_n)$. \hfill \rule{4pt}{7pt}
\vskip 3mm
\begin{lemma}\label{stablechange}
Let $\alpha$ and $\beta$ be two colors in $\overline{\varphi}_n(T(y_{p-1}))$, let $Q$ be an $(\alpha,\beta)$-chain with
respect to $\varphi_n$, and let $\sigma_n=\varphi_n/Q$. Suppose one of the following cases occurs:
\begin{itemize}
\vspace{-2mm}
\item[1)] $q\geq 1$, $\alpha\in\overline{\varphi}_n(T_{n,q})$, and $Q$ is an $(\alpha,\beta)$-path disjoint from $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)$;
\vspace{-2mm}
\item[2)] $q=0$, $\alpha\in\overline{\varphi}_n(T_{n})$, or $\alpha\in\overline{\varphi}_n(T_{n,0}^*)$ with $\alpha,\beta\notin D_n$, and
$Q$ is an $(\alpha,\beta)$-path disjoint from $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)$; and
\vspace{-2mm}
\item[3)] $T_{n,q}^*\prec v_{\alpha} \prec v_{\beta}$, $\alpha,\beta\notin D_{n,q}$, $\alpha\notin \varphi_n \langle T(v_{\beta})-T(v_{\alpha}) \rangle$, and $Q$ is an arbitrary $(\alpha,\beta)$-chain.
\vspace{-2mm}
\end{itemize}
Then the following statements hold:
\begin{itemize}
\vspace{-1.5mm}
\item[(i)] $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring;
\vspace{-2mm}
\item[(ii)] $T_{n,q}^*$ is an ETT satisfying MP with respect to $\sigma_n$; and
\vspace{-2mm}
\item[(iii)] if $q \ge 1$, then $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}$ is a good hierarchy
of $T_{n,q}$ under $\sigma_n$, with the same $\Gamma$-sets (see Definition \ref{R2}) as $T$ under $\varphi_n$,
and $T_{n,q}$ is $(\cup_{\eta_h\in D_{n,q}}\Gamma^{q-1}_h)^-$-closed with respect to $\sigma_n$.
\vspace{-2mm}
\end{itemize}
Furthermore, in Case 3, $T$ is also an ETT satisfying MP with respect to $\sigma_n$, and $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$, with the
same $\Gamma$-sets (see Definition \ref{R2}) as $T$ under $\varphi_n$.
\end{lemma}
\noindent {\bf Remark.} To prove Theorem \ref{hierarchy}, we shall perform a series of Kempe changes as described in
Lemma \ref{stablechange} starting from $\varphi_n$ and $T$. Let $\sigma'$ be a resulting coloring and let $T'$
be a resulting ETT. By the above statement (iii), to show that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T'$ is a good hierarchy of $T'$ under $\sigma'$, with the same $\Gamma$-sets as $T$ under $\varphi_n$,
it suffices to verify that Definition \ref{R2}(i) is satisfied, which is fairly straightforward in our proof, as we
shall see.
\vskip 3mm
{\bf Proof of Lemma \ref{stablechange}}. Write $a=v_{\alpha}$ and $b=v_{\beta}$. Let us consider the three cases
described in the lemma separately.
{\bf Case 1.} $q\geq 1$, $\alpha\in\overline{\varphi}_n(T_{n,q})$, and $Q$ is an $(\alpha,\beta)$-path disjoint from $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)$.
We distinguish between two subcases according to the location of $b$.
{\bf Subcase 1.1.} $b \in V(T_{n,q})$.
Let us first assume that there exists a subscript $i$ with $0\le i \le q$, such that $\alpha$ or $\beta$ is closed
in $T^*_{n,i}$ with respect to $\varphi_n$. Let $r$ be the largest such $i$. By (5.10) and Lemma \ref{rutcor}, we have
(1) $\{a, b\} \subseteq V(T_{n,r}^*)$ and $\alpha, \beta \notin \varphi_n \langle T_{n,q}- T_{n,r}^* \rangle$.
(2) $\alpha$ and $\beta$ are $T_{n,r}^*$-interchangeable under $\varphi_n$. So $P_{\alpha}(\alpha,\beta,
\varphi_n)=P_{\beta}(\alpha,\beta,\varphi_n)$.
To justify this, note that if $r \geq 1$, then (2) holds by Corollary~\ref{step1}. So we assume that $r=0$.
Then $\alpha$ or $\beta$ is closed in $T_{n,0}^*$ with respect to $\varphi_n$. Hence, by Lemma~\ref{interchange}(iii)
if $\Theta_n= PE$ and by (5.1) and Theorem \ref{thm:tech10}(ii) otherwise, $\alpha$ and $\beta$ are $T_{n,0}^*$-interchangeable
under $\varphi_n$. This proves (2).
It follows from (2) that $Q$ is vertex-disjoint from $T_{n,r}^*$ and hence contains no edge incident to $T_{n,r}^*$.
By Lemma \ref{LEM:Stable}, $\sigma_n=\varphi_n/Q$ is a $(T_{n,r}^*,D_n,\varphi_n)$-strongly stable coloring, and $T_{n,r}^*$
is an ETT satisfying MP with respect to $\sigma_n$. By (1) and (6.6), we obtain $\sigma_n(f)=\varphi_n(f)$ for each edge $f$ of $T_{n,q}$ and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each vertex $u$ of $T_{n,q}$. Therefore $\sigma_n$ is a $(T_{n,q}, D_n,\varphi_n)$-strongly stable coloring. By the definition of $r$, for any $r+1\le j \le q$ and $\theta \in \{\alpha,\beta\}$, we have $\partial_{\varphi_n, \theta}(T_{n,j})\ne \emptyset$, so $\theta \in \cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h$ by
Definition \ref{R2}(v). It is then routine to check that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}$ is a good
hierarchy of $T_{n,q}$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$\footnote{See the justification
of (9) in the proof of Lemma \ref{change} for omitted details. Note that $\alpha$ or $\beta$ may become closed in $T_{n,j}$
with respect to $\sigma_n$ for some $j$ with $r+1\le j \le q$. Yet, even in this situation Definition \ref{R2}(v) remains valid with respect to $\sigma_n$.}, and $T_{n,q}$ is $(\cup_{\eta_h\in D_{n,q}}\Gamma^{q-1}_h)^-$-closed with respect to $\sigma_n$.
Next we assume that there exists no subscript $i$ with $0\le i \le q$, such that $\alpha$ or $\beta$ is closed
in $T^*_{n,i}$ with respect to $\varphi_n$. By Lemma \ref{rutcor}, we have
(3) $\alpha, \beta\in (\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h) \subseteq \Gamma^{j-1}$ for $1 \le j \le q$,
$\Theta_n=PE$, $v_{\alpha}, v_{\beta}\in V(T_n)-V(R_n)$, and $\alpha, \beta \notin \varphi_n \langle T_{n,q}- T_n \rangle$.
Since $R_n$ is a closure of $T_n(v_n)$ under $\varphi_n$, using (6.6) and TAA we obtain
(4) $\alpha, \beta \notin \overline{\varphi}_n (R_n)$.
By Lemma~\ref{interchange}(ii), colors $\alpha$ and $\beta$ are $T_n$-interchangeable under $\varphi_n$. So $P_{a}(\alpha,\beta,\varphi_n)$ is the only $(\alpha,\beta)$-path with respect to $\varphi_n$ intersecting
$T_n$. Hence $Q$ is vertex-disjoint from $T_n$ and therefore contains no edge incident to $T_n$.
By Lemma \ref{LEM:Stable}, $\sigma_n=\varphi_n/Q$ is a $(T_n, D_n,\varphi_n)$-stable coloring, and
$T_n$ is an ETT satisfying MP with respect to $\sigma_n$. By (3), (4) and (6.6), we further deduce that
$\sigma_n$ is a $(T_{n,0}^*, D_n,\varphi_n)$-stable coloring, $\sigma_n(f)=\varphi_n(f)$ for each
edge $f$ of $T_{n,q}$, and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each vertex $u$ of $T_{n,q}$. It is then routine
to check that the desired statements hold.
{\bf Subcase 1.2.} $b \notin V(T_{n,q})$.
Let us first assume that there exists a subscript $i$ with $0\le i \le q$, such that $\alpha$ is closed
in $T^*_{n,i}$ with respect to $\varphi_n$. Let $r$ be the largest such $i$. By (5.10), Lemma \ref{rutcor} and TAA, we have
(5) $a \subseteq V(T_{n,r}^*)$ and $\alpha \notin \varphi_n \langle T_{n,q}- T_{n,r}^* \rangle$. Furthermore,
no edge in $T_{n,q}- T_{n,r}^*$ is colored by $\beta$, except possibly $e_1$ when $r=0$ and $T_{n,0}^*=T_n$
(now $e_1=f_n$ in Algorithm 3.1 and $\beta \in D_n$).
Using the same argument as that of (2), we obtain
(6) $\alpha$ and $\beta$ are $T_{n,r}^*$-interchangeable under $\varphi_n$.
It follows from (6) that $Q$ is vertex-disjoint from $T_{n,r}^*$ and hence contains no edge incident to $T_{n,r}^*$.
By Lemma \ref{LEM:Stable}, $\sigma_n=\varphi_n/Q$ is a $(T_{n,r}^*,D_n,\varphi_n)$-strongly stable coloring, and $T_{n,r}^*$
is an ETT satisfying MP with respect to $\sigma_n$. Using (5), we obtain $\sigma_n(f)=\varphi_n(f)$ for each edge $f$
of $T_{n,q}$ and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each vertex $u$ of $T_{n,q}$. Therefore $\sigma_n$ is a $(T_{n,q}, D_n,\varphi_n)$-strongly stable coloring, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}$ is a good
hierarchy of $T_{n,q}$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$, and $T_{n,q}$ is
$(\cup_{\eta_h\in D_{n,q}}\Gamma^{q-1}_h)^-$-closed with respect to $\sigma_n$ (see the justification
of (9) in the proof of Lemma \ref{change} for omitted details).
Next we assume that there exists no subscript $i$ with $0\le i \le q$, such that $\alpha$ is closed
in $T^*_{n,i}$ with respect to $\varphi_n$. By Lemma \ref{rutcor}, we have
(7) $\alpha \in (\cup_{\eta_h\in D_{n,j}}\Gamma^{j-1}_h) \subseteq \Gamma^{j-1}$ for $1 \le j \le q$,
$\Theta_n=PE$, $v_{\alpha} \in V(T_n)-V(R_n)$, and $\alpha \notin \varphi_n \langle T_{n,q}- T_n \rangle$.
It follows that (4) also holds. By Lemma~\ref{interchange}(ii), colors $\alpha$ and $\beta$ are $T_n$-interchangeable under $\varphi_n$. So $P_{a}(\alpha,\beta,\varphi_n)$ is the only $(\alpha,\beta)$-path with respect to $\varphi_n$ intersecting
$T_n$. Hence $Q$ is vertex-disjoint from $T_n$ and therefore contains no edge incident to $T_n$. By Lemma \ref{LEM:Stable}, $\sigma_n=\varphi_n/Q$ is a $(T_n, D_n,\varphi_n)$-stable coloring, and $T_n$ is an ETT satisfying MP with respect to
$\sigma_n$. Since $b \notin V(T_{n,q})$, no edge in $T_{n,q}- T_{n,0}^*$ is colored by $\beta$ by TAA, because
$T_{n,0}^*=T_n \vee R_n$ by (7). Using (4) and (7), it is routine to check that
the desired statements hold.
{\bf Case 2.} $q=0$, $\alpha\in\overline{\varphi}_n(T_{n})$, or $\alpha\in\overline{\varphi}_n(T_{n,0}^*)$ with $\alpha,\beta\notin D_n$, and
$Q$ is an $(\alpha,\beta)$-path disjoint from $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)$.
Let us first assume that $\alpha$ or $\beta$ is closed in $T_{n,0}^*$ with respect to $\varphi_n$. By Lemma~\ref{interchange}(iii)
or Theorem \ref{thm:tech10}(ii) (see (5.1)), colors $\alpha$ and $\beta$ are $T_{n,0}^*$-interchangeable under $\varphi_n$.
So $P_{a}(\alpha,\beta,\varphi_n)$ is the only $(\alpha,\beta)$-path intersecting $T_{n,0}^*$, and hence $Q$ is vertex-disjoint from
$T_{n,0}^*$. It is then routine to check that $\sigma_n=\varphi_n/Q$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable
coloring, and $T_{n,0}^*$ is an ETT satisfying MP with respect to $\sigma_n$ by Theorem \ref{thm:tech10}(vi). So we assume
hereafter that
(8) neither $\alpha$ nor $\beta$ is closed in $T_{n,0}^*$ with respect to $\varphi_n$.
By the hypothesis of the present case, $\alpha\in \overline{\varphi}_n(T_{n})$ or $\{\alpha, \beta\} \cap D_{n}=\emptyset$.
So $\alpha\notin D_n$. From (8) and Algorithm 3.1 we deduce that $T_{n,0}^* \ne T_n$. Hence
(9) $\Theta_n=PE$, which together with (5.4) yields $a, b \notin V(T_n)\cap V(R_n)$.
Let us show that
(10) $\sigma_n=\varphi_n/Q$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring.
To justify this, note that if one of $a$ and $b$ is contained in $V(T_n)-V(R_n)$ and the other is contained in
$V(R_n)-V(T_n)$, then $\alpha$ and $\beta$ are $T_{n,0}^*$-interchangeable under $\varphi_n$ by
Lemma~\ref{interchange}(iv). So $Q$ is vertex-disjoint from $T_{n,0}^*$ and hence (10) holds.
In view of (9), we may assume that
(11) if $a, b \in V(T_{n,0}^*)$, then either $a, b \in V(T_n)-V(R_n)$ or $a, b \in V(R_n)-V(T_n)$.
Let us first assume that $a\in V(T_n)-V(R_n)$. Then $\alpha \notin \overline{\varphi}_n (R_n)$ by (6.6) and
$b \in V(T_n)-V(R_n)$ if $b \in V(T_{n,0}^*)$ by (11). So $\alpha$ and $\beta$ are $T_n$-interchangeable under
$\varphi_n$ by Lemma~\ref{interchange}(ii) and $\beta \notin \overline{\varphi}_n (R_n)$ by (6.6). It follows that $Q$ is
vertex-disjoint from $T_{n}$ and that $\sigma_n(f)=\varphi_n(f)$ for any edge $f$ incident to $R_n$ with $\varphi_n(f)\in\overline{\varphi}_n(R_n)$. Hence (10) holds.
Next we assume that $a\in V(R_n)-V(T_n)$. Then $\alpha \notin \overline{\varphi}_n (T_n)$ by (6.6) and
$b \in V(R_n)-V(T_n)$ if $b \in V(T_{n,0}^*)$ by (11). So $\alpha$ and $\beta$ are $R_n$-interchangeable under
$\varphi_n$ by Lemma~\ref{interchange}(i) and $\beta \notin \overline{\varphi}_n (T_n)$ by (6.6). It follows that $Q$ is
vertex-disjoint from $R_{n}$. By the hypothesis of the present case, $\{\alpha, \beta\} \cap D_{n} =\emptyset$.
So $\alpha, \beta \notin \overline{\varphi}_n(T_n) \cup D_{n}$ and hence (10) holds.
From (10) we deduce that $T_{n,0}^*$ is an ETT satisfying MP with respect to $\sigma_n$.
{\bf Case 3}. $T_{n,q}^*\prec v_{\alpha} \prec v_{\beta}$, $\alpha,\beta\notin D_{n,q}$, $\alpha\notin \varphi_n \langle T(v_{\beta})-T(v_{\alpha}) \rangle$, and $Q$ is an arbitrary $(\alpha,\beta)$-chain.
By (6.6), $V(T(y_{p-1}))$ is elementary with respect to $\varphi_n$. So $\alpha,\beta\notin\overline{\varphi}_n(T_{n,q}^*)$.
By hypothesis, $\alpha,\beta\notin D_{n,q}$. Hence
(12) $\alpha,\beta\notin\overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$.
\noindent By the definitions of $D_n$ and $D_{n,q}$, we have $ D_n \cup \overline{\varphi}_n(T_n) \subseteq D_{n,q}\cup
\overline{\varphi}_n(T_{n,q}^*)$. So $\alpha,\beta \notin\overline{\varphi}_n(T_n)\cup D_n$. From Lemma \ref{hku}(iii), TAA and the
hypothesis of the present case, we further deduce that
(13) $\alpha, \beta \notin \varphi_n\langle T(b)\rangle$.
\noindent In view of Lemma~\ref{change}, we obtain
(14) $P_a(\alpha,\beta,\varphi)=P_b(\alpha,\beta,\varphi)$. (Possibly $Q$ is this path.)
Since $T_{n,q}^*\prec a \prec b$, using (12)-(14), it is straightforward to verify that $\sigma_n=\varphi_n/Q$ is a
$(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring.
From (12) and (13) we also see that $T(b)$ can be obtained from $T_{n,q}^*$ by using TAA, no matter whether $Q=
P_a(\alpha,\beta,\varphi)$. Thus $T$ is an ETT corresponding to $(\sigma_n,T_n)$. It is clear
that $T$ also satisfies MP under $\sigma_n$, and $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset
T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under
$\varphi_n$. \hfill \rule{4pt}{7pt}
\section{Elementariness and Interchangeability}
In Section 5 we have developed a control mechanism over Kempe changes; that is, a good hierarchies of an ETT. In Section 6
we have derived some properties satisfied by such hierarchies. Now we are ready to present a proof of Theorem \ref{hierarchy}
by using a novel recoloring technique based on these hierarchies.
\subsection{Proof of Theorem \ref{hierarchy}}
By hypothesis, $T$ is an ETT constructed from a $k$-triple $(G,e, \varphi)$ by using the Tashkinov series
$\mathcal {T}=\{(T_i, \varphi_{i-1}, S_{i-1}, F_{i-1}, \Theta_{i-1}): 1\le i \le n+1\}$. Furthermore, $T$ admits a good hierarchy
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q+1}=T$ and satisfies MP with respect to $\varphi_n$. Our
objective is to show that $V(T)$ is elementary with respect to $\varphi_n$.
As introduced in the preceding section, $T=T_{n,q}^* \cup\{e_1,y_1,e_2,...,e_p,y_p\}$, where $y_i$ is the end
of $e_i$ outside $T(y_{i-1})$ for $i\ge 1$, with $T(y_0)=T_{n,q}^*$. Suppose on the contrary that $V(T)$ is not
elementary with respect to $\varphi_n$. Then
{\bf (7.1)} $\overline{\varphi}_n(T(y_{p-1}))\cap\overline{\varphi}_n(y_p) \ne \emptyset$ by (6.6).
For ease of reference, recall that (see (3) in the proof of Theorem \ref{good})
{\bf (7.2)} $|\overline{\varphi}_n(T_{n})|\ge 2n+11$ and $|D_{n,j}|\le |D_n|\le n$ for $0 \le j \le q$.
In our proof we shall frequently make use of a coloring $\sigma_n \in {\cal C}^k(G-e)$ with properties
(i)-(iii) as described in Lemma \ref{stablechange}; that is,
{\bf (7.3)} $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring, and $T_{n,q}^*$ is an ETT
satisfying MP with respect to $\sigma_n$. Furthermore, if $q \ge 1$, then $T_{n,q}$ admits a good hierarchy
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}$ under $\sigma_n$, with the same $\Gamma$-sets
(see Definition \ref{R2}) as $T$ under $\varphi_n$, and $T_{n,q}$ is $(\cup_{\eta_h\in D_{n,q}}\Gamma^{q-1}_h)^-$-closed
with respect to $\sigma_n$ (see the remark succeeding Lemma \ref{stablechange}).
\begin{claim}\label{cla-p>0}
$p \ge 2.$
\end{claim}
\vspace{-1mm}
Assume the contrary: $p=1$; that is, $T=T_{n,q}^* \cup\{e_1,y_1\}$. Then
(1) there exists a color $\alpha$ in $\overline{\varphi}_n(T_{n,q}^*)\cap\overline{\varphi}_n(y_1)$ by (7.1).
\noindent We consider two cases according to the value of $q$.
{\bf Case 1.} $q=0$. In this case, from (1) and Algorithm 3.1 we see that $\Theta_n\ne SE$. Let us first assume
that $\Theta_n=RE$. Let $\delta_n, \gamma_n$ be as specified in Step 2 of Algorithm 3.1.
Since $\alpha, \delta_n \in \overline{\varphi}_n(T_n)$, both of them are closed in $T_n$ with
respect to $\varphi_n$. Hence $P_{y_1}(\alpha, \delta_n, \varphi_n)$ is vertex-disjoint from $T_n$. Let
$\sigma_n=\varphi_n/P_{y_1}(\alpha, \delta_n, \varphi_n)$. Then $\delta_n \in \overline{\sigma}_n(T_n) \cap
\overline{\sigma}_n(y_1)$. By Lemma \ref{LEM:Stable}, $\sigma_n$ is a $(T_n, D_n, \varphi_n)$-stable coloring.
It follows from Theorem \ref{thm:tech10}(vi) that $\sigma_n$ is a $\varphi_n\bmod T_n$ coloring.
From Definition \ref{wz2} and Step 1 of Algorithm 3.1, we see that $f_n=e_1$ is still an RE connecting edge
under $\sigma_n$ and is contained in a $(\delta_n,\gamma_n)$-cycle under $\sigma_n$, which is
impossible because $\delta_n \in \overline{\sigma}_n(y_1)$.
So we may assume that $\Theta_n=PE$. Let $\beta=\varphi_n(e_1)$. From TAA we see that $\beta\in\overline{\varphi}_n(T_{n,0}^*)$.
Let $\theta\in \overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)$. Then $\theta$ is closed in $T_{n,0}^*$ under $\varphi_n$ by (5.4).
In view of Lemma~\ref{interchange}(iii), $P_{v_{\theta}}(\alpha,\theta,\varphi_n)$
is the only $(\alpha,\theta)$-path intersecting $T_{n,0}^*$. Thus $P_{y_1}(\alpha,\theta,\varphi_n)\cap T_{n,0}^*=
\emptyset$. Let $\sigma_n=\varphi_n/P_{y_1}(\alpha,\theta,\varphi_n)$. By Lemma \ref{stablechange}
(the second case), $\sigma_n$ is a $(T_{n,0}^*, D_n, \varphi_n)$-strongly stable coloring, so
$\theta$ is also closed in $T_{n,0}^*$ with respect to $\sigma_n$. In view of
Lemma~\ref{interchange}(iii), $\beta$ and $\theta$ are $T_{n,0}^*$-interchangeable under $\sigma_n$.
As $P_{y_1}(\theta,\beta,\sigma_n)\cap T_{n,0}^*\neq\emptyset$, there are at least two $(\theta,\beta)$-paths with
respect to $\sigma_n$ intersecting $T_{n,0}^*$, a contradiction.
{\bf Case 2.} $q\ge 1$. In this case, by Definition \ref{R2}(v), we have
(2) $T_{n,q}$ is $(\cup_{\eta_h\in D_{n,q}}\Gamma^{q-1}_h)^-$-closed with respect to $\varphi_n$
\noindent So $e_1$ is colored by some color $\gamma_1$ in $\cup_{\eta_h\in D_{n,q}}\Gamma^{q-1}_h$. By Definition \ref{R2}(i)
and (5.9), we have $\gamma_1 \notin\Gamma^{q}$. Let $\theta\in\overline{\varphi}_n(T_{n,q})-\overline{\varphi}_n(T_{n,q-1}^*)$.
Then $\theta \notin \Gamma^{q-1}$ (so $\theta \neq\gamma_1$) by Definition \ref{R2}(i). Furthermore,
$\theta$ is closed in $T_{n,q}$ under $\varphi_n$ by (2). In view of Corollary \ref{step1}, $\alpha$ and $\theta$ are $T_{n,q}$-interchangeable under $\varphi_n$. So $P_{v_{\theta}}(\alpha,\theta,\varphi_n)=P_{v_{\alpha}}(\alpha,\theta,\varphi_n)$
is the unique $(\alpha,\theta)$-path intersecting $T_{n,q}$. Hence $P_{y_1}(\alpha,\theta,\varphi_n)\cap T_{n,q}=\emptyset$.
Let $\sigma_n=\varphi_n/P_{y_1}(\alpha,\theta,\varphi_n)$. Then $\sigma_n$ satisfies all the properties described in (7.3) by Lemma~\ref{stablechange}. Since $e_1$ is still colored by $\gamma_1\in\Gamma^{q-1}$ under $\sigma_n$ and
$\gamma_1\notin \Gamma^{q}$, we can obtain $T$ from $T_{n,q}$ by TAA under $\sigma_n$, so $T$ is an ETT satisfying MP under $\sigma_n$. Moreover, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$
under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Hence $(T, \sigma_n)$ is also a minimum counterexample
to Theorem \ref{hierarchy} (see (6.2)-(6.5)). As $P_{y_1}(\theta,\gamma_1,\sigma_n)\cap T_{n,q}\neq \emptyset$, there are
at least two $(\theta,\gamma_1)$-paths with respect to $\sigma_n$ intersecting $T_{n,q}$, contradicting Lemma~\ref{change}(iii)
(with $\sigma_n$ in place of $\varphi_n$), because $\theta, \gamma_1\in \overline{\sigma}_n(T_{n,q})$ and $\theta$ is also
closed in $T_{n,q}$ under $\sigma_n$ by (2). Hence Claim \ref{cla-p>0} is justified.
\vskip 3mm
Recall that the path number $p(T)$ of $T$ is the smallest subscript $i\in \{1,2,...,p\}$, such that the sequence $(y_i,e_{i+1},...,e_p,y_p)$ corresponds to a path in $G$, where $p\ge 2$ by Claim \ref{cla-p>0}. Depending on
the value of $p(T)$, we distinguish among three situations, labeled as Situation 7.1, Situation 7.2, and Situation 7.3. \\
\noindent {\bf Situation 7.1.} $p(T)=1$. Now $T - V(T_{n,q}^*)$ is a path obtained by using TAA under $\varphi_n$.
\begin{claim}\label{claim9}
We may assume that $\overline{\varphi}_n(y_i)\cap\overline{\varphi}_n(y_p) \ne \emptyset$ for some $i$ with $1\le i \le p-1$.
\end{claim}
\vspace{-1mm}
To justify this, let $\alpha \in \overline{\varphi}_n(T(y_{p-1})) \cap\overline{\varphi}_n(y_p)$ (see
(7.1)). If $\alpha\in \overline{\varphi}_n(y_i)\cap\overline{\varphi}_n(y_p)$ for some $i$ with $1\le i \le p-1$,
we are done. So we assume that
(1) $\alpha\in\overline{\varphi}_n(T_{n,q}^*) \cap\overline{\varphi}_n(y_p)$ and $\alpha \notin \overline{\varphi}_n(y_i)$
for all $1 \le i \le p-1$.
(2) If $\Theta_n=PE$ and $q=0$, then we may further assume that $\alpha\in \overline{\varphi}_n(T_n)$.
By (1), we have $\alpha\in \overline{\varphi}_n(T_{n,0}^*)$. Suppose $\alpha\in \overline{\varphi}_n(R_n-V(T_n))$. Then $\alpha\notin\Gamma^{0}$ by Definition \ref{R2}(i). In view of (7.2), we have $|\overline{\varphi}_n(T_n)|\geq 11+2n$ and $|\Gamma^{0}|\leq 2 |D_{n,0}| \le 2n$.
So there exists $\beta\in\overline{\varphi}_n(T_n)-\Gamma^{0}$. By Lemma \ref{interchange}(iv), $\alpha$ and $\beta$ are $T_{n,0}^*$-interchangeable under $\varphi_n$. Thus $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)=P_{v_{\beta}}(\alpha,\beta,\varphi_n)$
and $P_{y_p}(\alpha,\beta,\varphi_n)$ is disjoint from $T_{n,0}^*$. Let $\sigma_n=\varphi_n/P_{y_p}(\alpha,\beta,\varphi_n)$.
By Lemma~\ref{stablechange} (the second case), $\sigma_n$ is a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring,
and $T_{n,0}^*$ is an ETT satisfying MP with respect to $\sigma_n$. Note that $T$ can also be obtained from $T_{n,0}^*$ by
TAA under $\sigma_n$, because $\alpha,\beta \in \overline{\sigma}_n(T_{n,0}^*)$. Hence $T$ satisfies MP under $\sigma_n$ as
well. Since $\alpha,\beta\notin \Gamma^{0}$ and $\alpha,\beta\notin \overline{\varphi}_n(T(y_{p-1})-V(T_{n,0}^*))$, the hierarchy
$T_n=T_{n,0} \subset T_{n,1}=T$ remains to be good under $\sigma_n$, with the same $\Gamma$-sets
as those under $\varphi_n$. Therefore $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see
(6.2)-(6.5)). As $\beta\in \overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(y_p)$, replacing $\varphi_n$ by $\sigma_n$
and $\alpha$ by $\beta$ if necessary, we see that (2) holds.
Depending on whether $\alpha$ is used by edges in $T - T_{n,q}^*$, we consider two cases.
{\bf Case 1.} $\alpha\notin \varphi_n \langle T - T_{n,q}^* \rangle$. In this case, let $\beta\in\overline{\varphi}_n(y_{p-1})$.
Then $\beta$ is not used by any edge in $T - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$
in Algorithm 3.1 and $\varphi_n(e_1)=\beta \in D_n$). By (1) and (2), we have $\alpha\in\overline{\varphi}_n(T_{n,q})$ if $q \ge 1$ and $\alpha\in\overline{\varphi}_n(T_n)$ if $q=0$. It follows from Lemma~\ref{change} that $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)=P_{y_{p-1}}(\alpha,\beta,\varphi_n)$. So $P_{y_{p}}(\alpha,\beta,\varphi_n)$ is disjoint from $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)$. Let $\sigma_n = \varphi_n/P_{y_{p}}(\alpha,\beta,\varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3). In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\beta \in D_n$,
then $\sigma_n(e_1)=\varphi_n(e_1)$, which implies that $e_1$ is outside $P_{y_{p}}(\alpha,\beta,\varphi_n)$.
So $\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T)$ and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$.
Thus $T$ can be obtained from $T_{n,q}^*+e_1$ by TAA and satisfies MP under $\sigma_n$. Furthermore, $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy}
(see (6.2)-(6.5)). As $\beta\in \overline{\sigma}_n(y_{p-1}) \cap \overline{\sigma}_n(y_p)$, replacing $\varphi_n$ by
$\sigma_n$ if necessary, we see that Claim~\ref{claim9} is true.
{\bf Case 2.} $\alpha\in \varphi_n \langle T - T_{n,q}^* \rangle$. In this case, let $e_j$ be the edge with the smallest
subscript in $T -T_{n,q}^*$ such that $\varphi(e_j)=\alpha$. We distinguish between two subcases according to the value of
$j$.
{\bf Subcase 2.1.} $j\geq 2$. In this subcase, let $\beta \in \overline{\varphi}_n(y_{j-1})$. Then $\beta$ is not used
by any edge in $T(y_j) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1
and $\varphi_n(e_1)=\beta \in D_n$). By (1) and (2), we have $\alpha\in\overline{\varphi}_n(T_{n,q})$ if $q \ge 1$ and $\alpha\in\overline{\varphi}_n(T_n)$
if $q=0$. It follows from Lemma \ref{change} that $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)=P_{y_{j-1}}(\alpha,\beta,\varphi_n)$.
So $P_{y_{p}}(\alpha,\beta,\varphi)$ is disjoint from $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)$. Let $\sigma_n = \varphi_n/P_{y_{p}}(\alpha,\beta,\varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties
described in (7.3). In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\beta \in D_n$, then $\sigma_n(e_1)=\varphi_n(e_1)$,
which implies that $e_1$ is outside $P_{y_{p}}(\alpha,\beta,\varphi_n)$. So $T$ can be obtained from $T_{n,q}^*+e_1$ by TAA
under $\sigma_n$ and hence satisfies MP under $\sigma_n$.
Note that $\beta\notin\Gamma^{q}$ by Definition \ref{R2}(i) and that $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$ by (6.6). If $\alpha\notin \Gamma^{q}$, then clearly $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}
\subset T_{n,q+1}=T$ is a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$.
If $\alpha\in \Gamma^{q}$, say $\alpha\in \Gamma^{q}_{h}$ for some $\eta_h\in D_{n,q}$, then Definition \ref{R2}(i)
implies that $\eta_h \in\overline{\varphi}_n(w)$ for some $w\preceq y_{j-1}$. Since only edges outside $T(w)$ may
change colors between $\alpha$ and $\beta$ as we transform $\varphi_n$ into $\sigma_n$, it follows that $T_n=T_{n,0}
\subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$,
with the same $\Gamma$-sets as those under $\varphi_n$. Hence $(T, \sigma_n)$ is also a minimum counterexample to Theorem
\ref{hierarchy} (see (6.2)-(6.5)). Since $\beta\in \overline{\sigma}_n(y_{j-1}) \cap \overline{\sigma}_n(y_p)$, replacing
$\varphi_n$ by $\sigma_n$ if necessary, we see that Claim~\ref{claim9} holds.
{\bf Subcase 2.2.} $j=1$. In this subcase, $\alpha=\varphi(e_1)$. Note that $\alpha\notin\Gamma^{q}$ by Definition \ref{R2}(i)
and (5.9). We propose to show that
(3) there exists a color $\gamma$ in $\overline{\varphi}_n(T_{n,q})-\Gamma^{q}$ if $q\ge 1$ and in $\overline{\varphi}_n(T_{n})-\Gamma^{0}$
if $q=0$, such that $\gamma$ is closed in $T_{n,q}^*$ with respect to $\varphi_n$.
Let us first assume that $q\geq 1$. By (7.2), we obtain $|\overline{\varphi}_n(T_{n,q})|\ge |\overline{\varphi}_n(T_{n})|\ge 2n+11$ and
$|\Gamma^{q-1}|\le 2 |D_{n,q-1}|\le 2n$. So $|\overline{\varphi}_n(T_{n,q})-\Gamma^{q-1}|\ge 11$. By Definition \ref{R2}(iii), we
have $|\Gamma^{q}-\Gamma^{q-1}|= 2$. So $|\overline{\varphi}_n(T_{n,q})-(\Gamma^{q-1} \cup \Gamma^{q}) |\ge 9$. Let $\gamma$ be a
color in $\overline{\varphi}_n(T_{n,q})-(\Gamma^{q-1} \cup \Gamma^{q})$. By Definition \ref{R2}(v), $\gamma$ is closed in $T_{n,q}$
with respect to $\varphi_n$.
Next we assume that $q=0$. Again, by (7.2), we have $|\overline{\varphi}_n(T_{n})|\ge 2n+11$ and $|\Gamma^{0}|\le 2 |D_{n,0}|\le
2 |D_n| \le 2n$. Let $\gamma$ be a color in $\overline{\varphi}_n(T_{n})-\Gamma^{0}$ if $\Theta_n\ne PE$ and a color in
$\overline{\varphi}_n(T_n) \cap \overline{\varphi}_n(R_n)-\Gamma^0$ if $\Theta_n=PE$ (see Definition \ref{R2}(iv)). By Algorithm 3.1
and (5.4), $\gamma$ is closed in $T_{n,0}^*$ with respect to $\varphi_n$. So (3) holds.
By (3) and Lemma \ref{change}, $P_{v_{\alpha}}(\alpha,\gamma,\varphi_n)=P_{v_{\gamma}}(\alpha,\gamma,
\varphi_n)$ is the only $(\alpha,\gamma)$-path intersecting $T_{n,q}^*$. So $P_{y_{p}}(\alpha, \gamma,\varphi_n)$ is
disjoint from $T_{n,q}^*$ and hence it does not contain $e_1$.
Let $\sigma_n = \varphi_n/P_{y_{p}}(\alpha, \gamma,\varphi_n)$. Then $\sigma_n$ satisfies
all the properties described in (7.3) by Lemma~\ref{stablechange}. Moreover, $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$
for all $u\in V(T(y_{p-1}))$. Since $\alpha, \gamma\in \overline{\varphi}_n(T_{n,q}^*)$, we have $\alpha, \gamma\in \overline{\sigma}_n(T_{n,q}^*)$. Hence we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\sigma_n$, so $T$ satisfies MP under $\sigma_n$. Since $\alpha, \gamma \notin\Gamma^{q}$, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}
\subset T_{n,q+1}=T$ remains to be good under $\sigma_n$, with the same $\Gamma$-sets
as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy}
(see (6.2)-(6.5)). Since $e_1$ is outside $P_{y_p}(\alpha,\gamma,\varphi_n)$, we have $\sigma_n(e_1)=\alpha$. As $\gamma\in\overline{\sigma}_n(y_p)\cap \overline{\sigma}_n(v)$ for some $v\in V(T_{n,q})$ and $\alpha\neq\gamma$,
the present subcase reduces to Case 1 if $\gamma\notin \sigma_n \langle T-T_{n,q}^* \rangle$ or to Subcase 2.1 if
$\gamma\in \sigma_n \langle T-T_{n,q}^* \rangle$. This proves Claim \ref{claim9}.
\vskip 3mm
\begin{claim}\label{p-1}
We may assume that $\overline{\varphi}_n(y_{p-1})\cap\overline{\varphi}_n(y_p) \ne \emptyset$.
\end{claim}
\vspace{-1mm}
To justify this, let ${\cal K}$ be the set of all minimum counterexamples $(T, \varphi_n)$ to Theorem \ref{hierarchy}
(see (6.2)-(6.5)), and let $i$ be the largest subscript with $1 \le i \le p-1$, such that there exists a
member $(T, \mu_n)$ of ${\cal K}$ with $\overline{\mu}_n(y_{i})\cap\overline{\mu}_n(y_p) \ne \emptyset$;
this $i$ exists by Claim \ref{claim9}. We aim to show that $i=p-1$. Thus Claim \ref{p-1} follows by replacing
$\varphi_n$ with $\mu_n$, if necessary.
With a slight abuse of notation, we assume that $\overline{\varphi}_n(y_{i})\cap\overline{\varphi}_n(y_p) \ne
\emptyset$ and assume, on the contrary, that $i\le p-2$. Let $\alpha \in \overline{\varphi}_n(y_{i})\cap\overline{\varphi}_n(y_p)$.
Using (6.6) and TAA, we obtain
(1) $\alpha \notin \overline{\varphi}_n(T(y_{i-1}))$, where $T(y_0)=T_{n,q}^*$. So $\alpha$ is not used by any edge in
$T(y_{i+1}) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and
$\varphi_n(e_1)=\alpha \in D_n$).
Recall that Definition \ref{R2} involves $\Gamma^{q}_h=\{\gamma^{q}_{h_1}, \gamma^{q}_{h_2}\}$ for each $\eta_h\in
D_{n,q}$. Nevertheless, in our proof we only consider a fixed $\eta_h\in D_{n,q}$. For simplicity, we abbreviate
its corresponding $\gamma^{q}_{h_j}$ to $\gamma_{j}$ for $j=1,2$. By Definition \ref{R2}(i) and (5.9), we have
(2) $\gamma_{j}\in \overline{\varphi}_n (T_{n,q})$ if $q\geq 1$ and $\gamma_{j}\in \overline{\varphi}_n(T_{n})$ if $q=0$.
Moreover, if $\eta_h \in \overline{\varphi}_n(y_t)$ for some $t\ge 1$, then $\gamma_{j}\notin \varphi_n
\langle T(y_t)-T_{n,q}^* \rangle$ for $j=1,2$.
Depending on whether $\alpha \in D_{n,q}$, we consider two cases.
{\bf Case 1.} $\alpha\notin D_{n,q}$. In this case, let $\theta\in\overline{\varphi}_n(y_{i+1})$. From TAA and (6.6)
it follows that
(3) $\theta \notin \overline{\varphi}_n(T(y_i))$, so $\theta$ is not used by any edge in $T(y_{i+1}) - T_{n,q}^*$,
except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and $\varphi_n(e_1)=\theta \in D_n$).
If $\theta\notin D_{n,q}$, then $\{\alpha, \theta\}\cap D_{n,q} = \emptyset$. By the definitions of $D_n$ and
$D_{n,q}$, we have $\overline{\varphi}_n(T_n) \cup D_n \subseteq \overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$,
which together with (1) and (3) implies $\{\alpha, \theta\}\cap D_{n} = \emptyset$. Hence
$P_{y_{i}}(\alpha,\theta,\varphi_n)=P_{y_{i+1}}(\alpha,\theta,\varphi_n)$ by Lemma~\ref{change}.
Let $\sigma_n=\varphi_n/P_{y_{p}}(\alpha,\theta,\varphi_n)$. Since both $y_i$ and $y_{i+1}$ are contained in $T-V(T_{n,q}^*)$
and (1) holds, by Lemma~\ref{stablechange} (the third case), $\sigma_n$ satisfies all the properties described in
(7.3). Furthermore, $T$ is also an ETT satisfying MP with respect to $\sigma_n$, and $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$, with the
same $\Gamma$-sets as those under $\varphi_n$. Hence $(T, \sigma_n)$ is also a minimum counterexample to Theorem
\ref{hierarchy} (see (6.2)-(6.5)). Since $\theta\in\overline{\sigma}_n(y_p)\cap\overline{\sigma}_n(y_{i+1})$,
we reach a contradiction to the maximality assumption on $i$.
So we may assume that $\theta\in D_{n,q}$. Let $\theta=\eta_h\in D_{n,q}$. In view of (2) and Lemma \ref{change}, we obtain $P_{v_{\gamma_1}}(\alpha,\gamma_{1},\varphi_n)=P_{y_{i}}(\alpha,\gamma_{1},\varphi_n)$, which is disjoint from $P_{y_{p}}(\alpha,\gamma_{1},\varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_{p}}(\alpha,\gamma_{1},\varphi_n)$.
By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3). In particular,
if $e_1=f_n$ and $\varphi_n(e_1)=\alpha \in D_n$, then $\sigma_n(e_1)=\varphi_n(e_1)$, which implies that $e_1$ is outside
$P_{y_{p}}(\alpha,\gamma_{1},\varphi_n)$. By (6.6), (1) and (2), we have $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each
$u \in V(T(y_{p-1}))$ and $\sigma_n(f)=\varphi_n(f)$ for each edge $f$ in $T(y_{i+1})$. So $T$ can be obtained from
$T_{n,q}^*+e_1$ by TAA under $\sigma_n$, and hence satisfies MP under $\sigma_n$. Furthermore, $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$, with the same
$\Gamma$-sets as those under $\varphi_n$. Hence $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy}
(see (6.2)-(6.5)), with $\gamma_{1}\in \overline{\sigma}_n(y_p) \cap \overline{\sigma}_n(T_{n,q})$.
Using (2) and Lemma \ref{change}, we obtain $P_{v_{\gamma_{1}}}(\eta_h, \gamma_{1},\sigma_n) =P_{y_{i+1}}(\eta_h,\gamma_{1}, \sigma_n)$, which is disjoint from $P_{y_{p}}(\eta_h, \gamma_{1},\sigma_n)$. Let $\sigma_n'=\sigma_n/P_{y_{p}}(\eta_h,\gamma_{1},\sigma_n)$. By Lemma~\ref{stablechange}, $\sigma_n'$ satisfies all the
properties described in (7.3) (with $\sigma_n'$ in place of $\sigma_n$). In particular, if $e_1=f_n$ and $\sigma_n(e_1)
=\eta_h \in D_n$, then $\sigma_n'(e_1)=\sigma_n(e_1)$, which implies that $e_1$ is outside $P_{y_{p}}(\eta_h,\gamma_{1},\sigma_n)$.
By (6.6), (2) and (3), we have $\overline{\sigma}'_n(u)=\overline{\sigma}_n(u)$ for each $u \in V(T(y_{p-1}))$ and $\sigma_n'(f)=\sigma_n(f)$ for each edge $f$ in $T(y_{i+1})$. So $T$ can be obtained from $T_{n,q}^*+e_1$ by TAA under
$\sigma_n'$, and hence satisfies MP under $\sigma_n'$. Furthermore, since $\eta_h \in \overline{\sigma}'_n(y_{i+1})$,
the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains
to be good under $\sigma_n'$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore
$(T, \sigma_n')$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)). Since $\eta_h \in\overline{\sigma}_n'(y_p)\cap\overline{\sigma}_n'(y_{i+1})$, we reach a contradiction to the maximality assumption on $i$.
{\bf Case 2.} $\alpha\in D_{n,q}$. In this case, let $\alpha=\eta_h\in D_{n,q}$. Then $\Gamma^{q}_h=\{\gamma_{1},
\gamma_{2}\}$ (see the paragraph above (2)). Renaming subscript if necessary, we may assume that $\varphi_n(e_{i+1})
\neq \gamma_{1}$. By (1) and (2), we have
(4) $\gamma_{1}\notin \varphi_n \langle T(y_{i+1})-T_{n,q}^* \rangle$ and $\eta_h$ is not used by any edge in
$T(y_{i+1}) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1
and $\varphi_n(e_1)=\eta_h \in D_{n,q} \subseteq D_n$).
By (4) and Lemma~\ref{change}, we obtain $P_{v_{\gamma_1}}(\eta_h,\gamma_{1},\varphi_n)=P_{y_i}(\eta_h,\gamma_{1}, \varphi_n)$,
which is disjoint from the path $P_{y_p}(\eta_h,\gamma_{1},\varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p}(\eta_k,\gamma_{1},
\varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3).
In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\eta_h \in D_n$, then $\sigma_n(e_1)=\varphi_n(e_1)$, which implies that
$e_1$ is outside $P_{y_p}(\eta_k,\gamma_{1}, \varphi_n)$. By (6.6) and
(4), we have $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u \in V(T(y_{p-1}))$ and $\sigma_n(f)=\varphi_n(f)$ for
each edge $f$ in $T(y_{i+1})$. So $T$ can be obtained from $T_{n,q}^*+e_1$ by TAA under $\sigma_n$, and hence
satisfies MP under $\sigma_n$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$
remains to be a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore,
$(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), with $\gamma_{1}\in \overline{\sigma}_n(y_p) \cap \overline{\sigma}_n(T_{n,q})$. Let $\theta\in\overline{\sigma}_n(y_{i+1})$. From TAA
we see that
(5) $\theta$ is not used by any edge in $T(y_{i+1}) - T_{n,q}^*$ under $\sigma_n$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and $\sigma_n(e_1)=\theta \in D_n$).
By (6.6), we have $\theta\neq\gamma_1$. Using (4) and Lemma~\ref{change}, we get $P_{v_{\gamma_1}}(\theta, \gamma_{1},\sigma_n)=P_{y_{i+1}}(\theta, \gamma_{1}, \sigma_n)$. Let $\sigma_n'=\sigma_n/P_{y_p} (\theta,\gamma_{1},\sigma_n)$.
By Lemma~\ref{stablechange}, $\sigma_n'$ satisfies all the properties described in (7.3) (with $\sigma_n'$ in place of $\sigma_n$).
In particular, if $e_1=f_n$ and $\sigma_n(e_1)=\theta \in D_n$, then $\sigma_n'(e_1)=\sigma_n(e_1)$, which implies that
$e_1$ is outside $P_{y_p} (\theta,\gamma_{1},\sigma_n)$.
From (6.6) and (4) we deduce that $\overline{\sigma}'_n(u)=\overline{\sigma}_n(u)$ for each $u \in V(T(y_{p-1}))$,
and $\sigma_n'(f)=\sigma_n(f)$ for each edge $f$ in $T(y_{i+1})$. So $T$ can also be obtained from $T_{n,q}^*+e_1$
by TAA under $\sigma_n'$, and hence satisfies MP under $\sigma_n'$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains to be a good hierarchy of those under $\sigma_n'$, with the same
$\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n')$ is also a minimum counterexample to Theorem
\ref{hierarchy} (see (6.2)-(6.5)). Since $\theta \in\overline{\sigma}_n'(y_p)\cap\overline{\sigma}_n'(y_{i+1})$,
we reach a contradiction to the maximality assumption on $i$. Hence Claim \ref{p-1} is established.
\vskip 2mm
By Claim \ref{cla-p>0}, $p \ge 2$. By Claim \ref{p-1}, $\overline{\varphi}_n(y_{p-1})\cap\overline{\varphi}_n(y_p)
\ne \emptyset$. Let $\alpha \in \overline{\varphi}_n(y_{p-1})\cap\overline{\varphi}_n(y_p)$ and $\beta=\varphi_n(e_{p})$. Let
$\sigma_n$ be obtained from $\varphi_n$ by recoloring $e_p$ with $\alpha$ and let $T'=T(y_{p-1})$. Then
$\beta \in \overline{\sigma}_n(y_{p-1})\cap\overline{\sigma}_n(T'(y_{p-2}))$ and $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q}\subset T'$ is a good hierarchy of $T'$ under $\sigma_n$.
So $(T', \sigma_n)$ is a counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.4)), which violates the
minimality assumption (6.5) on $(T, \varphi_n)$. This completes our discussion about Situation 7.1.\\
\noindent {\bf Situation 7.2.} $p(T)=p$. Now $e_p$ is not incident to $y_{p-1}$.
\vskip 2mm
By (7.1), there exists a color $\alpha\in \overline{\varphi}_n(T(y_{p-1}))\cap\overline{\varphi}_n(y_p)$.
We divide this situation into $3$ cases and further into $6$ subcases (see figure below), depending on whether
$v_{\alpha}=y_{p-1}$ and $\alpha \in D_{n,q}$. Our proof of Subcase 1.1 is self-contained. Yet, in our discussion
Subcase 1.2 may be redirected to Subcase 1.1 and Subcase 2.1, and Subcase 2.1 may be redirected to Subcase 1.1, etc.
Figure 1 illustrates such redirections (note that no cycling occurs).
\begin{figure}[htpb]
\vspace{-1mm}
\centerline{\includegraphics[width=10cm]{egcolorfig2.eps}}
\vspace{-1mm}\caption{Redirections}
\end{figure}
Throughout this situation we reserve the symbol $\theta$ for $\varphi_n(e_p)$. Clearly, $\theta\neq \alpha$.
\vskip 2mm
{\bf Case 1.} $\alpha\in\overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(y_{p-1})$ and $\alpha \in D_{n,q}$.
\vskip 1mm
Let $\alpha=\eta_{m}\in D_{n,q}$. For simplicity, we abbreviate the two colors $\gamma^{q}_{m_1}$ and $\gamma^{q}_{m_2}$ in $\Gamma^{q}_m$ (see Definition \ref{R2}) to $\gamma_1$ and $\gamma_2$, respectively.
Since $\eta_m \in \overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(y_{p-1})$, from TAA and Definition \ref{R2}(i) we
see that
(1) $\gamma_{1},\gamma_{2}\notin \varphi_n \langle T(y_{p-1}) - T_{n,q}^* \rangle$ and $\eta_m$ is not used by
any edge in $T - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1
and $\varphi_n(e_1)=\eta_m \in D_{n,q} \subseteq D_n$).
By (1) and Lemma \ref{change} (with respect to $(T, \varphi_n)$), we have
(2) $P_{v_{\gamma_j}}(\eta_m, \gamma_{j},\varphi_n)=P_{y_{p-1}}(\eta_m,\gamma_{j},\varphi_n)$ for $j=1,2$.
\vskip 2mm
Let us consider two subcases according to whether $\theta\in\overline{\varphi}_n(y_{p-1})$.
{\bf Subcase 1.1.} $\theta\notin\overline{\varphi}_n(y_{p-1})$.
In our discussion about this subcase, we shall appeal to the following two tree sequences:
\vskip 1mm
$\bullet$ $T^-=(T_{n,q}^*, e_1, y_1, e_2, \ldots, e_{p-2}, y_{p-2}, e_p, y_p)$ and
\vskip 1mm
$\bullet$ $T^* \hskip 0.6mm =(T_{n,q}^*, e_1, y_1, e_2, \ldots ,y_{p-2}, e_p, y_p, e_{p-1}, y_{p-1})$.
\vskip 1mm
\noindent Note that $T^-$ is obtained from $T$ by deleting $y_{p-1}$ and $T^*$ arises from $T$ by interchanging
the order of $(e_{p-1}, y_{p-1})$ and $(e_p, y_p)$. We propose to show that both $T^-$ and $T^*$ are ETTs corresponding
to $\varphi_n$. Indeed, if $T(y_{p-2}) \ne T_n$, then both $T^-$ and $T^*$ can be obtained from $T(y_{p-2})$
by using TAA under $\varphi_n$. So we assume that $T(y_{p-2})= T_n$. By the hypothesis of the present subcase, $\varphi_n(e_p)=\theta \notin\overline{\varphi}_n(y_{p-1})$. From Algorithm 3.1 we deduce that now $\Theta_n=PE$.
Hence both $T^-$ and $T^*$ can be obtained from $T(y_{p-2})$ by using TAA under $\varphi_n$ as well. Therefore
both $T^-$ and $T^*$ are ETTs corresponding to $\varphi_n$. In view of the maximum property enjoyed by $T$, we
further conclude that both $T^-$ and $T^*$ are ETTs satisfying MP with respect to $\varphi_n$.
Let us first assume that $\theta\notin\Gamma^{q}$. Now it is easy to see that $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q} \subset T^-$ is a good hierarchy of $T^-$ under $\varphi_n$, with the same $\Gamma$-sets
(see Definition \ref{R2}) as $T$. (If $\theta\in\Gamma^{q}$, say $\theta \in \Gamma^{q}_h$, and
$\eta_h \in \overline{\varphi}_n(y_{p-1})$, then $T^-$ no longer satisfies Definition \ref{R2}(i).) Observe that $\gamma_{1} \notin
\overline{\varphi}_n(y_{p})$, for otherwise, $\gamma_{1}$ is missing at two vertices in $T^-$. Thus $(T^-, \varphi_n)$ is
a counterexample to Theorem \ref{hierarchy} (see (6.2) and (6.3)), which violates the minimality assumption
$(6.4)$ or $(6.5)$ on $(T, \varphi_n)$. Let us turn to considering $T^*$. Since $\theta \notin\overline{\varphi}_n(y_{p-1})$ and $\theta\notin\Gamma^{q}$, it is clear that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset
T^*$ is a good hierarchy of $T^*$ under $\varphi_n$, with the same $\Gamma$-sets as $T$.
Moreover, by (1), we have $\gamma_{1} \notin \varphi_n \langle T^*(y_{p}) - T_{n,q}^* \rangle$.
It follows from Lemma \ref{change} (with respect to $(T^*, \varphi_n)$) that
$P_{v_{\gamma_1}}(\eta_m, \gamma_{1},\varphi_n)=P_{y_{p}}(\eta_m,\gamma_{1},\varphi_n)$, contradicting (2).
Next we assume that $\theta\in\Gamma^{q}$. Then $\theta\in \Gamma^{q}_h$ for some $\eta_h \in D_{n,q}$.
If $\eta_h\notin \overline{\varphi}_n(y_{p-1})$, then $\eta_h \in\overline{\varphi}_n(T(y_{p-2}))$ by
Definition \ref{R2}(i). So we can still ensure that both $T^-$ and $T^*$ have good
hierarchies under $\varphi_n$. Thus, using the same argument as employed in the preceding paragraph, we can
reach a contradiction. Hence we may assume that $\eta_h\in \overline{\varphi}_n(y_{p-1})$.
Clearly, $\theta \ne \gamma_{1}$ or $\gamma_{2}$. Renaming subscripts if necessary, we may assume that
(3) $\theta \ne \gamma_2$.
\noindent Since $P_{v_{\gamma_2}}(\eta_m, \gamma_{2},\varphi_n)=P_{y_{p-1}}(\eta_m,\gamma_{2},\varphi_n)$ by (2), this
path is disjoint from $P_{y_{p}}(\eta_m,\gamma_{2},\varphi_n)$. Let $\mu_1 = \varphi_n/P_{y_{p}}(\eta_m,\gamma_2,\varphi_n)$.
By Lemma~\ref{stablechange}, $\mu_1$ satisfies all the properties described in (7.3) (with $\mu_1$ in place of
$\sigma_n$). In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\eta_m \in D_n$, then $\mu_1(e_1)=\varphi_n(e_1)$, which
implies that $e_1$ is outside $P_{y_{p}}(\eta_m,\gamma_2,\varphi_n)$.
By (1) and (3), we have $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T)$ and $\overline{\mu}_1(u)=
\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_1$; thereby
$T$ satisfies MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}
\subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_1$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\gamma_2$ is missing at two vertices.
By Lemma \ref{9n}, we have
$|\overline{\mu}_1(T(y_{p-2}))|- |\overline{\mu}_1(T_{n,0}^*-V(T_n))| - |\mu_1 \langle T(y_{p-2}) - T_{n,q}^* \rangle |
\geq 2n+11$, where $T(y_0)=T_{n,q}^*$. It follows that $|\overline{\mu}_1(T(y_{p-2}))|- |\overline{\mu}_1(T_{n,0}^*-V(T_n))|
- |\mu_1 \langle T - T_{n,q}^* \rangle | \geq 2n+9$. As $|\Gamma^{q}|\le 2 |D_{n,q}| \le 2|D_n| \le 2n$ by Lemma \ref{Dnzang},
using (6.6) we obtain
(4) there exists a color $\beta$ in $\overline{\mu}_1(T(y_{p-2}))- \overline{\mu}_1(T_{n,0}^*-V(T_n))-
\mu_1 \langle T - T_{n,q}^* \rangle- \Gamma^{q}$.
\noindent By Lemma \ref{change} (with $\gamma_2$ in place of $\alpha$), $P_{v_{\gamma_{2}}}(\beta, \gamma_{2},\mu_1)=P_{v_{\beta}}(\beta, \gamma_{2}, \mu_1)$, so it is disjoint from $P_{y_{p}}(\beta, \gamma_{2},\mu_1)$.
Let $\mu_2=\mu_1/P_{y_{p}}(\beta,\gamma_{2}, \mu_1)$. By Lemma~\ref{stablechange}, $\mu_2$ satisfies all the properties
described in (7.3) (with $\mu_2$ in place of $\sigma_n$). By (1), (3) and (4), we have $\beta, \gamma_2 \notin \mu_1 \langle T(y_{p})-T_{n,q}^* \rangle$. So $\mu_2(f)=\mu_1(f)$ for each $f\in E(T)$ and $\overline{\mu}_2(u)=\overline{\mu}_1(u)$
for each $u\in V(T(y_{p-1}))$. Hence we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_2$; thereby $T$ satisfies
MP under $\mu_2$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to
be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$.
Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem \ref{hierarchy}
(see (6.2)-(6.5)), in which $\beta$ is missing at two vertices. Since $\theta\in \Gamma^{q}_h$ and $\eta_h\in \overline{\varphi}_n(y_{p-1})=\overline{\mu}_1(y_{p-1})=\overline{\mu}_2(y_{p-1})$, we obtain
(5) $\theta \notin \mu_2 \langle T(y_{p-1}) - T_{n,q}^* \rangle$.
\noindent By (4), we also have
(6) $\beta \notin \mu_2 \langle T - T_{n,q}^* \rangle$.
\noindent It follows from (5) and Lemma~\ref{change} (with $\theta$ in place of $\alpha$) that $P_{v_{\theta}}(\beta,\theta,\mu_2)=P_{v_{\beta}}(\beta, \theta,\mu_2)$, so it is disjoint from
$P_{y_{p}}(\beta,\theta,\mu_2)$. Finally, set $\mu_3=\mu_2/P_{y_{p}}(\beta,\theta,
\mu_2)$. By Lemma~\ref{stablechange}, $\mu_3$ satisfies all the properties described in (7.3) (with $\mu_3$ in
place of $\sigma_n$). From (5) and (6) we see that $T$ can be obtained from $T_{n,q}^*+e_1$ by using TAA under $\mu_3$. Hence
$T$ satisfies MP under $\mu_3$. Note that $\mu_3(f)=\mu_2(f)$ for each $f\in E(T(y_{p-1}))$,
$\mu_3(e_p)=\beta$, and $\overline{\mu}_3(u)=\overline{\mu}_2(u)$ for each $u\in V(T(y_{p-1}))$. Moreover, $\beta \notin
\Gamma^q$ by (4). It is a routine matter to check that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_3$, with the same $\Gamma$-sets as those under $\mu_2$.
Since $\mu_3(e_p)=\beta \notin \Gamma^q$ and $v_{\beta} \prec y_{p-1}$, we see that
$T^-$ has a good hierarchy and satisfies MP with respect to $\mu_3$. As $\theta$ is missing at two vertices
in $T^-$, we conclude that $(T^-, \mu_3)$ is a counterexample to Theorem \ref{hierarchy} (see (6.2) and (6.3)),
which contradicts the minimality assumption $(6.4)$ or $(6.5)$ on $(T, \varphi_n)$.
{\bf Subcase 1.2.} $\theta \in\overline{\varphi}_n(y_{p-1})$.
In this subcase, from (6.6) and TAA we see that
(7) $\theta \notin \overline{\varphi}_n(T(y_{p-2}))$, so $\theta \notin \Gamma^q$ and hence $\theta\ne \gamma_1, \gamma_2$.
Furthermore, $\theta$ is not used by any edge in $T(y_{p-1}) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$
(now $e_1=f_n$ in Algorithm 3.1 and $\varphi_n(e_1)=\theta \in D_n$).
Since $P_{v_{\gamma_1}}(\eta_m, \gamma_{1},\varphi_n)=P_{y_{p-1}}(\eta_m,\gamma_{1},\varphi_n)$ by (2), this
path is disjoint from $P_{y_{p}}(\eta_m,\gamma_{1},\varphi_n)$. Let $\mu_1 = \varphi_n/P_{y_{p}}(\eta_m,\gamma_1,\varphi_n)$.
By Lemma~\ref{stablechange}, $\mu_1$ satisfies all the properties described in (7.3) (with $\mu_1$ in place of
$\sigma_n$). In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\eta_m \in D_n$, then $\mu_1(e_1)=\varphi_n(e_1)$, which
implies that $e_1$ is outside $P_{y_{p}}(\eta_m,\gamma_1,\varphi_n)$.
By (1) and (6.6), we have $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T)$ and $\overline{\mu}_1(u)=\overline{\varphi}_n(u)$
for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_1$, and hence $T$ satisfies
MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$
remains to be a good hierarchy of $T$ under $\mu_1$, with the same $\Gamma$-sets as those under $\varphi_n$.
Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\gamma_1$ is missing at two vertices.
From (1) and the definition of $\mu_1$, we see that
(8) $\gamma_1 \notin \mu_1 \langle T - T_{n,q}^* \rangle$.
\noindent From (8) and Lemma~\ref{change} (with $\gamma_1$ in place of $\alpha$), we deduce that $P_{v_{\gamma_{1}}}(\theta,\gamma_{1},\mu_1)=P_{y_{p-1}}(\theta,\gamma_{1},\mu_1)$, which is disjoint from
$P_{y_{p}}(\theta,\gamma_{1},\mu_1)$. Let $\mu_2=\mu_1/P_{y_{p}}(\theta,\gamma_{1},\mu_1)$. By Lemma~\ref{stablechange},
$\mu_2$ satisfies all the properties described in (7.3) (with $\mu_2$ in place of $\sigma_n$).
In particular, if $e_1=f_n$ and $\mu_1(e_1)=\theta \in D_n$, then $\mu_2(e_1)=\mu_1(e_1)$, which
implies that $e_1$ is outside $P_{y_{p}}(\theta,\gamma_{1},\mu_1)$. In view of (7), (8) and (6.6),
we have $\mu_2(f)=\mu_1(f)$ for each $f\in E(T(y_{p-1}))$, $\mu_2(e_p)=\gamma_1$, and $\overline{\mu}_2(u)=
\overline{\mu}_1(u)$ for each $u\in V(T(y_{p-1}))$. Moreover, $\theta \notin \Gamma^q$. So $T$ can be obtained
from $T_{n,q}^*+e_1$ by using TAA under $\mu_2$, and hence satisfies MP under $\mu_2$. It is a routine
matter to check that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to
be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$.
Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem
\ref{hierarchy} (see (6.2)-(6.5)). Since $\theta \in \overline{\mu}_2(y_p)\cap \overline{\mu}_2(y_{p-1})$
and $\mu_2(e_p)= \gamma_1 \notin \overline{\mu}_2(y_{p-1})$, the present subcase reduces to Subcase 1.1 if
$\theta\in D_{n,q}$ and reduces to Subcase 2.1 (to be discussed below) if $\theta\notin D_{n,q}$.
\vskip 2mm
{\bf Case 2.} $\alpha\in \overline{\varphi}_n(y_p)\cap \overline{\varphi}_n(y_{p-1})$ and $\alpha\notin D_{n,q}$.
\vskip 1mm
By the definitions of $D_n$ and $D_{n,q}$, we have $\overline{\varphi}_n(T_n) \cup D_n \subseteq
\overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$. Using (6.6) and this set inclusion, we obtain
(9) $\alpha \notin \overline{\varphi}_n(T(y_{p-2}))$ and $\alpha \notin D_n$. So $\alpha \notin\varphi_n \langle
T-T_{n,q}^* \rangle$ by TAA (see, for instance, (1)).
Recall that $T(y_0)=T_{n,q}^*$ and $\theta=\varphi_n(e_p)$. We consider two subcases according to whether $\theta \in \overline{\varphi}_n(y_{p-1})$.
{\bf Subcase 2.1.} $\theta \notin\overline{\varphi}_n(y_{p-1})$.
In our discussion about this subcase, we shall also appeal to the following two tree sequences:
\vskip 1mm
$\bullet$ $T^-=(T_{n,q}^*, e_1, y_1, e_2, \ldots, e_{p-2}, y_{p-2}, e_p, y_p)$ and
\vskip 1mm
$\bullet$ $T^* \hskip 0.6mm =(T_{n,q}^*, e_1, y_1, e_2, \ldots ,y_{p-2}, e_p, y_p, e_{p-1}, y_{p-1})$.
\vskip 1mm
\noindent As stated in Subcase 1.1, $T^-$ is obtained from $T$ by deleting $y_{p-1}$ and $T^*$ arises from $T$ by
interchanging the order of $(e_{p-1}, y_{p-1})$ and $(e_p, y_p)$. Furthermore, both $T^-$ and $T^*$
are ETTs satisfying MP with respect to $\varphi_n$. Observe that
(10) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T^*$ is a good hierarchy
of $T^*$ under $\varphi_n$, unless $\theta\in \Gamma^{q}_h$ for some $\eta_h\in D_{n,q}$ such that $\eta_h
\in\overline{\varphi}_n(y_{p-1})$.
Let us first assume that the exceptional case in (10) does not occur; that is, there exists no $\eta_h\in D_{n,q}$ such
that $\eta_h \in\overline{\varphi}_n(y_{p-1})$ and $\theta\in \Gamma^{q}_h$. It is easy to see that now
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T^-$ is a good hierarchy of $T^-$ under $\varphi_n$.
By Lemma \ref{9n}, we have
$|\overline{\varphi}_1(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_1 \langle T(y_{p-2}) - T_{n,q}^* \rangle |
\geq 2n+11$ holds, where $T(y_0)=T_{n,q}^*$. Since $|\Gamma^{q}|\le 2 |D_{n,q}| \le 2|D_n| \le 2n$ by
Lemma \ref{Dnzang}, using (6.6) we obtain
(11) there exists a color $\beta$ in $\overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n))-
\varphi_n \langle T - T_{n,q}^* \rangle-\Gamma^{q}$.
\noindent Note that $\beta \notin \overline{\varphi}_n(y_p)$, for otherwise, $(T^-, \varphi_n)$ would be
a counterexample to Theorem \ref{hierarchy} (see (6.2) and (6.3)), which violates the minimality assumption
$(6.4)$ or $(6.5)$ on $(T, \sigma_n)$. Since $\alpha, \beta \notin \varphi_n \langle T - T_{n,q}^* \rangle$ by
(9) and (11), applying Lemma \ref{change} to $(T, \varphi_n)$ and $(T^*, \varphi_n)$, respectively, we obtain
$P_{v_{\beta}}(\alpha,\beta,\varphi_n)=P_{y_{p-1}}(\alpha,\beta,\varphi_n)$
and $P_{v_{\beta}}(\alpha,\beta,\varphi_n)=P_{y_{p}}(\alpha,\beta,\varphi_n)$, a contradiction.
So we assume that the exceptional case in (10) occurs; that is, there exists $\eta_h\in D_{n,q}$ such
that $\eta_h \in\overline{\varphi}_n(y_{p-1})$ and $\theta\in \Gamma^{q}_h$. For simplicity, we abbreviate the two colors
$\gamma^{q}_{h_1}$ and $\gamma^{q}_{h_2}$ in $\Gamma^{q}_h$ (see Definition \ref{R2}) to $\gamma_1$ and $\gamma_2$,
respectively. Renaming subscripts if necessary, we may assume that $\theta=\gamma_1$. By Definition \ref{R2}(i) and
TAA, we have
(12) $\gamma_2 \notin\varphi_n \langle T-T_{n,q}^* \rangle$ and $\eta_h$ is not used by
any edge in $T - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1
and $\varphi_n(e_1)=\eta_h \in D_{n,q} \subseteq D_n$).
By (12) and Lemma \ref{change} (with $\alpha$ in place of $\beta$), we obtain $P_{v_{\gamma_2}}(\alpha,\gamma_{2},\varphi_n)=P_{y_{p-1}}(\alpha,\gamma_{2},
\varphi_n)$, which is disjoint from $P_{y_{p}}(\alpha,\gamma_{2},\varphi_n)$. Let $\mu_1=\varphi_n/P_{y_{p}}(\alpha,\gamma_{2},
\varphi_n)$. By Lemma~\ref{stablechange}, $\mu_1$ satisfies all the properties described in (7.3) (with $\mu_1$ in place of
$\sigma_n$). Since $\alpha, \gamma_2 \notin \varphi_n \langle T(y_{p})-T_{n,q}^* \rangle$ by (9) and (12), we have
$\mu_1(f)=\varphi_n(f)$ for each $f\in E(T)$ and $\overline{\mu}_1(u)= \overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$.
So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_1$, and hence $T$
satisfies MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$
remains to be a good hierarchy of $T$ under $\mu_1$, with the same $\Gamma$-sets as those under $\varphi_n$.
Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which
$\gamma_2$ is missing at two vertices.
If $\eta_h\in\overline{\mu}_1(y_{p})$, then $\eta_h \in\overline{\mu}_1(y_p)\cap\overline{\mu}_1(y_{p-1})$,
$\eta_h\in D_{n,q}$, and $\mu_1(e_p)=\gamma_{1} \notin \overline{\varphi}_n(y_{p-1})$. Thus the present subcase reduces
to Subcase 1.1. So we may assume that $\eta_h\notin\overline{\mu}_1(y_{p})$. By (12) and the definition of $\mu_1$, we have
(13) $\gamma_2 \notin\mu_1 \langle T-T_{n,q}^* \rangle$ and $\eta_h$ is not used by any edge in $T - T_{n,q}^*$
under $\mu_1$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and
$\mu_1(e_1)=\eta_h \in D_n$).
By (13) and Lemma \ref{change} (with $\gamma_2$ in place of $\alpha$), we obtain $P_{v_{\gamma_2}}(\eta_h,\gamma_{2},\mu_1)=P_{y_{p-1}}(\eta_h,\gamma_{2},
\mu_1)$, which is disjoint from $P_{y_{p}}(\eta_h,\gamma_{2},\mu_1)$. Let $\mu_2=\mu_1/P_{y_{p}}(\eta_h,\gamma_{2},
\mu_1)$. By Lemma~\ref{stablechange}, $\mu_2$ satisfies all the properties described in (7.3) (with $\mu_2$ in place of
$\sigma_n$). In particular, if $e_1=f_n$ and $\mu_1(e_1)=\eta_h \in D_n$, then $\mu_2(e_1)=\mu_1(e_1)$, which
implies that $e_1$ is outside $P_{y_{p}}(\eta_h,\gamma_{2},\mu_1)$. By (13), we have $\mu_2(f)=\mu_1(f)$ for each
$f\in E(T)$ and $\overline{\mu}_2(u)= \overline{\mu}_1(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from
$T_{n,q}^*+e_1$ by using TAA under $\mu_2$, and hence $T$ satisfies MP under $\mu_2$. Furthermore,
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$
remains to be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$.
Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\eta_h \in\overline{\mu}_2(y_p)\cap\overline{\mu}_2(y_{p-1})$, $\eta_h\in D_{n,q}$, and $\mu_2(e_p)=\gamma_1 \notin \overline{\mu}_2(y_{p-1})$. Thus the present subcase reduces to Subcase 1.1.
{\bf Subcase 2.2.} $\theta\in\overline{\varphi}_n(y_{p-1})$.
Let us first assume that $\theta \in D_{n,q}$; that is, $\theta=\eta_{m}$ for some $\eta_m\in D_{n,q}$. For
simplicity, we use $\varepsilon_1$ and $\varepsilon_2$ to denote the two colors $\gamma^{q}_{m_1}$ and
$\gamma^{q}_{m_2}$ in $\Gamma^{q}_m$ (see Definition \ref{R2}), respectively. By Definition \ref{R2}(i) and TAA, we have
(14) $\varepsilon_1, \varepsilon_2 \notin \varphi_n \langle T-T_{n,q}^* \rangle$ and $\eta_m$ is not used by
any edge in $T(y_{p-1}) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in
Algorithm 3.1 and $\varphi_n(e_1)=\eta_m \in D_n$).
By (14) and Lemma \ref{change}, we obtain $P_{v_{\varepsilon_1}}(\alpha, \varepsilon_1, \varphi_n)
=P_{y_{p-1}}(\alpha, \varepsilon_1, \varphi_n)$, which is disjoint from $P_{y_{p}}(\alpha, \varepsilon_1, \varphi_n)$.
Let $\mu_1=\varphi_n/P_{y_{p}}(\alpha, \varepsilon_1, \varphi_n)$. By Lemma~\ref{stablechange}, $\mu_1$ satisfies all
the properties described in (7.3) (with $\mu_1$ in place of $\sigma_n$). By (9) and (14), we have
(15) $\alpha, \varepsilon_1 \notin \mu_1 \langle T-T_{n,q}^* \rangle$ and $\eta_m$ is not used by any edge in
$T(y_{p-1}) - T_{n,q}^*$ under $\mu_1$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in
Algorithm 3.1 and $\mu_1(e_1)=\eta_m \in D_n$).
So $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T)$ and $\overline{\mu}_1(u)= \overline{\varphi}_n(u)$ for each
$u\in V(T(y_{p-1}))$. Thus $T$ can be obtained from $T_{n,q}^*+e_1$ by using TAA under $\mu_1$, and hence
satisfies MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_1$, with the same
$\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem
\ref{hierarchy} (see (6.2)-(6.5)), in which $\varepsilon_1$ is missing at two vertices.
By (15) and Lemma \ref{change} (with $\varepsilon_1$ in place of $\alpha$), we obtain $P_{v_{\varepsilon_1}}(\eta_m,
\varepsilon_1, \mu_1) =P_{y_{p-1}}(\eta_m, \varepsilon_1, \mu_1)$, which is disjoint from $P_{y_{p}}(\eta_m, \varepsilon_1,
\mu_1)$. Let $\mu_2=\mu_1/P_{y_{p}}(\eta_m, \varepsilon_1, \mu_1)$. By Lemma~\ref{stablechange}, $\mu_2$ satisfies all
the properties described in (7.3) (with $\mu_2$ in place of $\sigma_n$).
In particular, if $e_1=f_n$ and $\mu_1(e_1)=\eta_m \in D_n$, then $\mu_2(e_1)=\mu_1(e_1)$, which
implies that $e_1$ is outside $P_{y_{p}}(\eta_m, \varepsilon_1, \mu_1)$. In view of (15), we have
$\mu_2(f)=\mu_1(f)$ for each $f\in E(T(y_{p-1}))$, $\mu_2(e_p)=\varepsilon_1$, and $\overline{\mu}_2(u)= \overline{\mu}_1(u)$
for each $u\in V(T(y_{p-1}))$. So $T$ can be obtained from $T_{n,q}^*+e_1$ by using TAA under $\mu_2$, and hence satisfies
MP under $\mu_2$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$
remains to be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$.
Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\eta_m \in\overline{\mu}_2(y_p)\cap\overline{\mu}_2(y_{p-1})$, $\eta_m\in D_{n,q}$, and $\mu_2(e_p)=\varepsilon_1 \notin \overline{\mu}_2(y_{p-1})$. Thus the present subcase reduces to Subcase 1.1.
Next we assume that $\theta\notin D_{n,q}$. Set $T(y_0)=T_{n,q}^*$. We propose to show that
(16) there exists a color $\beta \in \overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) -
\varphi_n \langle T - T_{n,q}^* \rangle- D_{n,q}$, such that either $\beta \notin \Gamma^{q}$
or $\beta \in \Gamma^{q}_h$ for some $\eta_h \in D_{n,q}\cap \overline{\varphi}_n(T(y_{p-2}))$.
To justify this, note that if $|\overline{\varphi}_n(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T(y_{p-2}) - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\ge 5$, then
$|\overline{\varphi}_n(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\ge 3$, because $T-T(y_{p-2})$ contains precisely two edges.
Thus there exists a color $\beta \in \overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) -
\varphi_n \langle T - T_{n,q}^* \rangle- D_{n,q}$, such that $\beta \notin \Gamma^{q}$.
So we assume that $|\overline{\varphi}_n(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T(y_{p-2}) - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\le 4 $. By Lemma \ref{9n}, there exist $7$ distinct
colors $\eta_{h}\in D_{n,q}\cap \overline{\varphi}_n(T(y_{p-2}))$ such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \varphi_n \langle T(y_{p-2}) - T_{n,q}^* \rangle=\emptyset$. Let $\beta$ be an arbitrary color in such a
$\Gamma^{q}_h$. From Definition \ref{R2}, we see that $\Gamma^{q}_h \subseteq \overline{\varphi}_n(T_{n,q}^*)
\subseteq \overline{\varphi}_n(T(y_{p-2}))$, $\Gamma^{q}_h \cap \overline{\varphi}_n(T_{n,0}^*-V(T_n)) =\emptyset$,
and $\Gamma^{q}_h \cap D_{n,q} =\emptyset$. So $\beta \in \overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) -
\varphi_n \langle T(y_{p-2}) - T_{n,q}^* \rangle- D_{n,q}$. Since $T-T(y_{p-2})$ contains precisely two edges, there
exists $\beta \in \overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n \langle T - T_{n,q}^*
\rangle- D_{n,q}$, such that $\beta \in \Gamma^{q}_h$ for some $\eta_h \in D_{n,q}\cap \overline{\varphi}_n(T(y_{p-2}))$.
Hence (16) is established.
By the definitions of $D_n$ and $D_{n,q}$, we have $\overline{\varphi}_n(T_n) \cup D_n \subseteq
\overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$. By (16), $\beta \notin \overline{\varphi}_n(T_{n,0}^*-V(T_n)) \cup D_{n,q}$.
It follows from these two observations that
(17) if $q\ge 1$, then $\beta \in \overline{\varphi}_n(T_{n,q}^*)$ or $\beta \notin D_n$; if $q=0$, then
$\beta \in \overline{\varphi}_n(T_n)$ or $\beta \notin D_n$.
By (9), (17) and Lemma \ref{change}, we obtain $P_{v_{\beta}}(\alpha,\beta,\varphi_n)=P_{y_{p-1}}(\alpha,\beta,\varphi_n)$,
which is disjoint from $P_{y_{p}}(\alpha,\beta,\varphi_n)$. Let $\mu_3=\varphi_n/P_{y_{p}}(\alpha,\beta,\varphi_n)$.
By Lemma~\ref{stablechange}, $\mu_3$ satisfies all the properties described in (7.3) (with $\mu_3$ in place of
$\sigma_n$). By (9) and (16), we have $\alpha, \beta \notin \varphi_n \langle T-T_{n,q}^* \rangle$. So
(18) $\alpha, \beta \notin \mu_3 \langle T-T_{n,q}^* \rangle$,
\noindent $\mu_3(f)=\varphi_n(f)$ for each $f\in E(T)$, and $\overline{\mu}_3(u)= \overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$.
Thus we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_3$, and hence $T$ satisfies MP under $\mu_3$.
Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy
of $T$ under $\mu_3$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \mu_3)$ is also a minimum
counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\beta$ is missing at two vertices.
Since $\theta \in \overline{\varphi}_n(y_{p-1})$, it follows from (6.6) that $\theta\notin \overline{\varphi}_n(T_{n,q}^*)$.
By assumption, $\theta\notin D_{n,q}$. As $\overline{\varphi}_n(T_n) \cup D_n \subseteq \overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$, we obtain
(19) $\theta\notin D_n$ and hence $\theta \notin \mu_3 \langle T(y_{p-1})-T_{n,q}^* \rangle$ by TAA.
By (17)-(19) and Lemma \ref{change}, we obtain $P_{v_{\beta}}(\theta,\beta,\mu_3)=P_{y_{p-1}}(\theta,\beta,\mu_3)$,
which is disjoint from $P_{y_{p}}(\theta,\beta,\mu_3)$. Let $\mu_4=\mu_3/P_{y_{p}}(\theta,\beta,\mu_3)$.
By Lemma~\ref{stablechange}, $\mu_4$ satisfies all the properties described in (7.3) (with $\mu_4$ in place of
$\sigma_n$). By (18) and (19), we have $\mu_4(f)=\mu_3(f)$ for each $f\in E(T(y_{p-1}))$ and $\overline{\mu}_4(u)= \overline{\mu}_3(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_4$, and hence
$T$ satisfies MP under $\mu_4$. Since either $\beta \notin \Gamma^{q}$ or $\beta \in \Gamma^{q}_h$ for some $\eta_h
\in D_{n,q}\cap \overline{\mu}_3(T(y_{p-2}))$ by (16), it follows that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_4$, with the same $\Gamma$-sets as those under
$\mu_3$. Therefore, $(T, \mu_4)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which
$\theta \in\overline{\mu}_4(y_p)\cap\overline{\mu}_4(y_{p-1})$, $\theta \notin D_{n,q}$, and $\mu_4(e_p)=\beta \notin \overline{\mu}_4(y_{p-1})$. Thus the present subcase reduces to Subcase 2.1.
\vskip 2mm
{\bf Case 3.} $\alpha\in \overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(v)$ for some vertex $v\prec y_{p-1}$.
\vskip 1mm
Set $T(y_0)=T_{n,q}^*$. Let us first impose some restrictions on $\alpha$.
(20) We may assume that $\alpha\in\overline{\varphi}_n(T(y_{p-2}))-\varphi_n \langle T - T_{n,q}^* \rangle$, such that
either $\alpha\notin D_{n,q}\cup\Gamma^{q}$ if $q \ge 1$ and $\alpha\notin D_{n}\cup\Gamma^{0}$ if $q=0$,
or $\alpha$ is some $\eta_{h}\in D_{n,q}$ satisfying $\Gamma^{q}_h \cap \varphi_n \langle T - T_{n,q}^* \rangle=\emptyset$.
To justify this, note that if $|\overline{\varphi}_n(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T(y_{p-2}) - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\ge 5$, then
$|\overline{\varphi}_n(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\ge 3$, because $T-T(y_{p-2})$ contains precisely two edges.
Thus there exists a color $\beta \in \overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) -
\varphi_n \langle T - T_{n,q}^* \rangle- (\Gamma^{q}\cup D_{n,q})$. Clearly,
$\beta\in\overline{\varphi}_n(T(y_{p-2}))-\varphi_n \langle T - T_{n,q}^* \rangle$ and
$\beta\notin D_{n,q}\cup\Gamma^{q}$ if $q \ge 1$ and $\beta \notin D_{n}\cup\Gamma^{0}$ if $q=0$ (see the
definitions of $D_n$ and $D_{n,0}$).
If $|\overline{\varphi}_n(T(y_{p-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T(y_{p-2}) - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\le 4 $, then, by Lemma \ref{9n}, there exist $7$ distinct
colors $\eta_{h}\in D_{n,q}\cap \overline{\varphi}_n(T(y_{p-2}))$ such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \varphi_n \langle T(y_{p-2}) - T_{n,q}^* \rangle=\emptyset$. Since $T-T(y_{p-2})$ contains precisely two edges,
there exists one of these $\eta_{h}$, denoted by $\beta$, such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \varphi_n \langle T - T_{n,q}^* \rangle=\emptyset$.
Combining the above observations, we conclude that
(21) there exists $\beta \in\overline{\varphi}_n(T(y_{p-2}))-\varphi_n \langle T - T_{n,q}^* \rangle$, such that
either $\beta \notin D_{n,q}\cup\Gamma^{q}$ if $q \ge 1$ and $\beta \notin D_{n}\cup\Gamma^{0}$ if $q=0$,
or $\beta$ is some $\eta_{h}\in D_{n,q}$ satisfying $\Gamma^{q}_h \cap \varphi_n \langle T - T_{n,q}^* \rangle=\emptyset$.
If $\beta \in \overline{\varphi}_n(y_p)$, then (20) holds by replacing $\alpha$ with $\beta$. So we assume hereafter that
$\beta \notin \overline{\varphi}_n(y_p)$. Let $Q=P_{y_p}(\alpha, \beta, \varphi_n)$ and let $\sigma_n=\varphi_n/Q$.
We propose to show that one of the following statements (a) and (b) holds:
\begin{itemize}
\vspace{-2mm}
\item[(a)] $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring,
$T$ is also an ETT satisfying MP with respect to $\sigma_n$, and $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,q}\subset T_{n,q+1}=T$ remains to be a hierarchy of $T$ under $\sigma_n$, with the
same $\Gamma$-sets (see Definition \ref{R2}) as those under $\varphi_n$. Moreover, (20) holds with respect to
$(T, \sigma_n)$.
\vspace{-2mm}
\item[(b)] There exists an ETT $T'$ satisfying MP with respect to $\varphi_n$, such that $T_n=T_{n,0} \subset T_{n,1}
\subset \ldots \subset T_{n,q}\subset T'$ is a good hierarchy of $T'$ under $\varphi_n$, with the
same $\Gamma$-sets as $T$ under $\varphi_n$. Moreover, $V(T')$ is not elementary with respect to $\varphi_n$
and $p(T') < p(T)$.
\vspace{-2mm}
\end{itemize}
\noindent Note that if (b) holds, then $(T', \varphi_n)$ would be a counterexample to Theorem \ref{hierarchy} (see (6.2)
and (6.3)), which violates the minimality assumption $(6.4)$ on $(T, \varphi_n)$.
Let us first assume that $Q$ is vertex-disjoint from $T(y_{p-1})$. By Lemma \ref{LEM:Stable}, $\sigma_n$ is both $(T(y_{p-1}),D_n,\varphi_n)$-stable and $(T(y_{p-1}),\varphi_n)$-invariant. If $\Theta_n=PE$, then $\sigma_n$ is also
$(T_n\oplus R_n,D_n,\varphi_n)$-stable. Furthermore, $T(y_{p-1})$ is an ETT satisfying MP with respect to $\sigma_n$,
and $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T(y_{p-1})$ is a good hierarchy of
$T(y_{p-1})$, with the same $\Gamma$-sets as $T$ under $\sigma_n$. By definition, $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring. By the hypothesis of Case 3 and assumption on $\beta$, we have $\varphi_n(e_p) \ne \alpha,
\beta$. Thus it is clear that (a) is true, and (20) follows if we replace $\varphi_n$ by $\sigma_n$ and $\alpha$ by $\beta$.
Next we assume that $Q$ and $T(y_{p-1})$ have vertices in common. Let $u$ be the first vertex of $Q$ contained in
$T(y_{p-1})$ as we traverse $Q$ from $y_p$. Define $T'=T(y_{p-1}) \cup Q[u,y_p]$ if $u= y_{p-1}$
and $T'=T(y_{p-2})\cup Q[u,y_p]$ otherwise. By the hypothesis of Case 3 and (21), we have $\alpha,\beta \in
\overline{\varphi}_n(T(y_{p-2}))$. So $T'$ can be obtained from $T(y_{p-2})$ by using TAA under $\varphi_n$, with $p(T')<p(T)$. It
follows that $T'$ is an ETT satisfying MP with respect to $\varphi_n$.
By Definition \ref{R2}, we have $D_{n,q} \cap \Gamma^q=\emptyset$. Thus
(22) $\beta\notin \Gamma^{q}$ by (21).
Let us proceed by considering three possibilities for $\alpha$.
$\bullet$ $\alpha\notin \Gamma^{q}$. Since both $\alpha$ and $\beta$ are outside $\Gamma^{q}$ (see (22)),
it is easy to see that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T'$ is a good hierarchy of
$T'$ under $\varphi_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$. Hence (b) holds.
$\bullet$ $\alpha\in \Gamma^{q}\cap \varphi_n \langle T-T_{n,q}^* \rangle$. Let $\alpha \in \Gamma^{q}_h$ for some
$\eta_h \in D_{n,q}$. Since $\varphi(e_p)\neq\alpha$, we have $\alpha\in \varphi_n \langle T(y_{p-1})-T_{n,q}^* \rangle$.
Hence $\eta_h\in \overline{\varphi}_n(T(y_{p-2}))$ by Definition \ref{R2}(i). Furthermore, $\beta\in\overline{\varphi}_n(T(y_{p-2}))$
and $\beta\notin\Gamma^q$ by (21) and (22). Therefore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T'$
is a good hierarchy of $T'$ under $\varphi_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$. Hence (b) holds.
$\bullet$ $\alpha \in \Gamma^{q} - \varphi_n \langle T-T_{n,q}^* \rangle$. By the definition of $\Gamma^q$, we have
$\alpha \in \overline{\varphi}_n(T_{n,q})$ if $q \ge 1$ and $\alpha \in \overline{\varphi}_n(T_n)$ if $q=0$.
It follows from Lemma~\ref{change} that $P_{v_{\alpha}}(\alpha,\beta,\varphi_n)=P_{v_{\beta}}(\alpha,\beta,\varphi_n)$,
which is disjoint from $Q$. By Lemma~\ref{stablechange}, $\sigma_n=\varphi_n/Q$ satisfies all the properties described
in (7.3). Since $\alpha, \beta \notin \varphi_n \langle T-T_{n,q}^* \rangle$ by the assumption on $\alpha$ and (21),
we have $\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T)$ and $\overline{\sigma}_n(u)= \overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$.
So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\sigma_n$, and hence $T$ satisfies MP under $\sigma_n$.
Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$
under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\beta$ is missing at two vertices. So (a) holds and therefore (20) is established by replacing $\varphi_n$ with $\sigma_n$ and $\beta$ with $\alpha$.
\vskip 3mm
Let $\alpha$ be a color as specified in (20). Recall that $\theta=\varphi_n(e_p)$. We consider
two subcases according to whether $\theta\in \overline{\varphi}_n(y_{p-1})$.
{\bf Subcase 3.1.} $\theta \notin\overline{\varphi}_n(y_{p-1})$.
Consider the tree sequence $T^-=(T_{n,q}^*, e_1, y_1, e_2, \ldots, e_{p-2}, y_{p-2}, e_p, y_p)$. As stated in
Subcase 1.1, $T^-$ arises from $T$ by deleting $y_{p-1}$, and $T^-$ is an ETT satisfying MP with respect to $\varphi_n$.
Observe that
(23) $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T^-$ is a good hierarchy
of $T^-$ under $\varphi_n$, unless $\theta\in \Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$ such that $\eta_m
\in\overline{\varphi}_n(y_{p-1})$.
It follows that the exceptional case stated in (23) must occur, for otherwise, $(T^-, \varphi_n)$ would be a counterexample
to Theorem \ref{hierarchy} (see (6.2) and (6.3)), which violates the minimality assumption $(6.4)$ or $(6.5)$ on
$(T, \varphi_n)$. So $\theta\in \Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$ such that $\eta_m \in\overline{\varphi}_n(y_{p-1})$.
Since $\alpha\in\overline{\varphi}_n(T(y_{p-2}))$, we have $\alpha\neq \eta_m$ by (6.6). From Definition \ref{R2}(i),
we see that
(24) $\theta \notin \varphi_n \langle T(y_{p-1}) - T_{n,q}^* \rangle$.
\noindent By the definition of $\Gamma^q$, we have $\theta \in \overline{\varphi}_n(T_{n,q})$ if $q \ge 1$ and $\theta \in \overline{\varphi}_n(T_n)$ if $q=0$. Thus, by (20), (24) and Lemma~\ref{change}, we obtain $P_{v_{\alpha}} (\alpha, \theta,
\varphi_n) = P_{v_{\theta}}(\alpha, \theta, \varphi_n)$, which is disjoint from $P_{y_p}(\alpha, \theta, \varphi_n)$. Let $\mu_1=\varphi_n/P_{y_{p}}(\alpha,\theta, \varphi_n)$. By Lemma~\ref{stablechange}, $\mu_1$ satisfies all the
properties described in (7.3) (with $\mu_1$ in place of $\sigma_n$). Using (20) and (24), we get
(25) $\alpha, \theta \notin \mu_1 \langle T(y_{p-1})-T_{n,q}^* \rangle$,
\noindent $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T(y_{p-1}))$, $\mu_1(e_p)=\alpha \notin \Gamma^q$ (see (20)),
and $\overline{\mu}_1(u)= \overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by
using TAA under $\mu_1$ and hence $T$ satisfies MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_1$,
with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\theta$ is missing at two vertices.
By (25) and Lemma~\ref{change}, we obtain $P_{v_{\theta}} (\eta_m, \theta, \mu_1) = P_{y_{p-1}}(\eta_m, \theta,
\mu_1)$, which is disjoint from $P_{y_p}(\eta_m, \theta, \mu_1)$. Let $\mu_2=\mu_1/P_{y_{p}}(\eta_m,\theta,
\mu_1)$. By Lemma~\ref{stablechange}, $\mu_2$ satisfies all the properties described in (7.3) (with $\mu_2$ in
place of $\sigma_n$). Note that $\eta_m$ is not used by any edge in $T - T_{n,q}^*$ under $\mu_1$, except possibly
$e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and $\mu_1(e_1)=\eta_m \in D_n$). So
$e_1$ is outside $P_{y_p}(\eta_m, \theta, \mu_1)$. Hence
$\mu_2(f)=\mu_1(f)$ for each $f\in E(T)$, and $\overline{\mu}_2(u)= \overline{\mu}_1(u)$ for each $u\in V(T(y_{p-1}))$.
It follows that $T$ can be obtained from $T_{n,q}^*+e_1$ by using TAA and hence satisfies MP under $\mu_2$. Furthermore,
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$. Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)). Since $\eta_m \in \overline{\mu}_2(y_p) \cap \overline{\mu}_2(y_{p-1})$, $\eta_m\in D_{n,q}$, and $\mu_2(e_p)=\alpha \notin \overline{\mu}_2(y_{p-1})$, the present subcase reduces to Subcase 1.1.
{\bf Subcase 3.2.} $\theta \in\overline{\varphi}_n(y_{p-1})$.
We first assume that $\theta \in D_{n,q}$. Let $\theta=\eta_{m}\in D_{n,q}$. For simplicity, we abbreviate the
two colors $\gamma^{q}_{m_1}$ and $\gamma^{q}_{m_2}$ in $\Gamma^{q}_m$ (see Definition \ref{R2}) to $\gamma_1$ and
$\gamma_2$, respectively. By (20) and Definition \ref{R2}(i), we have
(26) $\{\alpha, \gamma_1, \gamma_2\}\cap \varphi_n \langle T - T_{n,q}^* \rangle=\emptyset$.
By (26) and Lemma~\ref{change}, we obtain $P_{v_{\alpha}} (\alpha, \gamma_1, \varphi_n) = P_{v_{\gamma_1}}(\alpha, \gamma_1, \varphi_n)$, which is disjoint from $P_{y_p}(\alpha, \gamma_1, \varphi_n)$. Let $\mu_1=\varphi_n/P_{y_{p}}(\alpha, \gamma_1,
\varphi_n)$. By Lemma~\ref{stablechange}, $\mu_1$ satisfies all the properties described in (7.3) (with $\mu_1$ in
place of $\sigma_n$). Since $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T)$, and $\overline{\mu}_1(u)=
\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$, we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under
$\mu_1$ and hence $T$ satisfies MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_1$, with the same $\Gamma$-sets as those under $\mu_1$. Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)),
in which $\gamma_1$ is missing at two vertices. In view of (26) and Definition \ref{R2}(i), we get
(27) $\{\alpha, \gamma_1, \gamma_2\}\cap \mu_1 \langle T - T_{n,q}^* \rangle=\emptyset$, and $\eta_m$ is not used by
any edge in $T - T_{n,q}^*$ under $\mu_1$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in
Algorithm 3.1 and $\mu_1(e_1)=\eta_m \in D_{n,q} \subseteq D_n$).
By (27) and Lemma~\ref{change}, we obtain $P_{v_{\gamma_1}} (\gamma_1, \eta_m, \mu_1) = P_{y_{p-1}}(\gamma_1,\eta_m,
\mu_1)$, which is disjoint from $P_{y_p}(\gamma_1,\eta_m, \mu_1)$. Let $\mu_2=\mu_1/P_{y_p}(\gamma_1,\eta_m, \mu_1)$.
By Lemma~\ref{stablechange}, $\mu_2$ satisfies all the properties described in (7.3) (with $\mu_2$ in
place of $\sigma_n$). In particular, if $e_1=f_n$ and $\mu_1(e_1)=\eta_m \in D_n$, then $\mu_2(e_1)=\mu_1(e_1)$, which
implies that $e_1$ is outside $P_{y_p}(\gamma_1,\eta_m, \mu_1)$.
Since $\mu_2(f)=\mu_1(f)$ for each $f\in E(T(y_{p-1}))$ by (27), and $\overline{\mu}_2(u)=
\overline{\mu}_1(u)$ for each $u\in V(T(y_{p-1}))$, we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under
$\mu_2$ and hence $T$ satisfies MP under $\mu_2$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$. Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)). Since $\eta_m \in \overline{\mu}_2(y_p) \cap \overline{\mu}_2(y_{p-1})$, $\eta_m\in D_{n,q}$, and $\mu_2(e_p)=\gamma_1 \notin \overline{\mu}_2(y_{p-1})$, the present subcase reduces to Subcase 1.1.
Next we assume that $\theta \notin D_{n,q}$. By (6.6) and the hypothesis of the present subcase, we have $\theta \notin \overline{\varphi}_n(T_{n,q}^*)$. So $\theta \notin \overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$, which implies
$\theta \notin \overline{\varphi}_n(T_n) \cup D_n$. In particular,
(28) $\theta \notin D_{n,q}\cup \Gamma^q$ if $q\ge 1$ and $\theta \notin D_n \cup \Gamma^0$ if $q=0$. Furthermore,
$\theta$ is not used by any edge in $T(y_{p-1}) - T_{n,q}^*$ by TAA (see, for instance, (1)).
We proceed by considering two possibilities for $\alpha$.
$\bullet$ $\alpha\notin D_{n,q}$. Now it follows from (20) that
(29) $\alpha \notin D_{n,q}\cup \Gamma^q$ if $q\ge 1$ and $\alpha \notin D_n \cup \Gamma^0$ if $q=0$.
By (20) and Lemma~\ref{change}, we obtain $P_{v_{\alpha}} (\alpha, \theta, \varphi_n) = P_{y_{p-1}}(\alpha, \theta, \varphi_n)$,
which is disjoint from $P_{y_p}(\alpha, \theta, \varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p}(\alpha, \theta, \varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3). Since $\sigma_n(f)=\varphi_n(f)$ for
each $f\in E(T(y_{p-1}))$ by (20) and (28), and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$,
we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\sigma_n$ and hence $T$ satisfies MP under $\sigma_n$. In view of (28) and (29), $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample
to Theorem \ref{hierarchy} (see (6.2)-(6.5)). Since $\theta \in \overline{\sigma}_n(y_p) \cap \overline{\sigma}_n(y_{p-1})$,
$\theta \notin D_{n,q}$, and $\sigma_n(e_p)=\alpha \notin \overline{\sigma}_n(y_{p-1})$, the present subcase reduces to Subcase 2.1.
$\bullet$ $\alpha\in D_{n,q}$. Let $\alpha=\eta_{h}\in D_{n,q}$. For simplicity, we use $\varepsilon_1$ and $\varepsilon_2$
to denote the two colors $\gamma^{q}_{h_1}$ and $\gamma^{q}_{h_2}$ in $\Gamma^{q}_h$ (see Definition \ref{R2}), respectively.
By (20), we have
(30) $\{\alpha, \varepsilon_1, \varepsilon_2\}\cap \varphi_n \langle T - T_{n,q}^* \rangle=\emptyset$.
By (30) and Lemma~\ref{change}, we obtain $P_{v_{\alpha}} (\alpha, \varepsilon_1, \varphi_n) = P_{v_{\varepsilon_1}}
(\alpha, \varepsilon_1, \varphi_n)$, which is disjoint from $P_{y_p}(\alpha, \varepsilon_1, \varphi_n)$.
Let $\mu_1=\varphi_n/P_{y_p}(\alpha, \varepsilon_1, \varphi_n)$. By Lemma~\ref{stablechange}, $\mu_1$ satisfies
all the properties described in (7.3) (with $\mu_1$ in place of $\sigma_n$). Since $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T)$
by (30), and $\overline{\mu}_1(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$, we can obtain $T$ from $T_{n,q}^*+e_1$ by
using TAA under $\mu_1$ and hence $T$ satisfies MP under $\mu_1$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_1$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)),
in which $\varepsilon_1$ is missing at two vertices. From (30) and Definition \ref{R2}(i) we see that
(31) $\varepsilon_1 \notin \mu_1 \langle T - T_{n,q}^* \rangle$.
By (31) and Lemma~\ref{change}, we obtain $P_{v_{\varepsilon_1}} (\theta, \varepsilon_1, \mu_1) = P_{y_{p-1}}
(\theta, \varepsilon_1, \mu_1)$, which is disjoint from $P_{y_p}(\theta, \varepsilon_1, \mu_1)$.
Let $\mu_2=\mu_1/P_{y_p}(\theta, \varepsilon_1, \mu_1)$. By Lemma~\ref{stablechange}, $\mu_2$ satisfies
all the properties described in (7.3) (with $\mu_2$ in place of $\sigma_n$). In view of (28) and (31),
we have $\mu_2(f)=\mu_1(f)$ for each $f\in E(T(y_{p-1}))$ and $\overline{\mu}_1(u)=\overline{\varphi}_n(u)$
for each $u\in V(T(y_{p-1}))$. So $T$ can be obtained from $T_{n,q}^*+e_1$ by using TAA and hence satisfies MP under
$\mu_2$. Furthermore, $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$. Therefore, $(T, \mu_2)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)). Since $\theta \in \overline{\mu}_2(y_p) \cap \overline{\mu}_2(y_{p-1})$,
$\theta \notin D_{n,q}$, and $\mu_2(e_p)= \varepsilon_1 \notin \overline{\mu}_2(y_{p-1})$, the present subcase reduces to
Subcase 2.1. This completes our discussion about Situation 7.2.\\
\noindent {\bf Situation 7.3.} $2\le p(T)\le p-1$.
\vskip 2mm
Recall that $T=T_{n,q}^* \cup\{e_1,y_1,e_2,...,e_p,y_p\}$, and the path number $p(T)$ of $T$ is the smallest subscript
$t \in \{1,2,...,p\}$ such that the sequence $(y_t,e_{t+1},...,e_p,y_p)$ corresponds to a path in $G$. Set $I_{\varphi_n}
=\{1\le t \le p-1: \, \overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(y_t)\ne \emptyset \}$. We use $\max(I_{\varphi_n})$
to denote the maximum element of $I_{\varphi_n}$ if $I_{\varphi_n} \ne \emptyset$. For convenience, set
$\max(I_{\varphi_n})=-1$ if $I_{\varphi_n}=\emptyset$.
If $\max(I_{\varphi_n}) \ge p(T)$, then we may assume that $\max(I_{\varphi_n})=p-1$ (the proof is exactly the same as
that of Claim \ref{claim9}). Let $\alpha \in \overline{\varphi}_n(y_{p-1})\cap\overline{\varphi}_n(y_p)$ and
$\beta=\varphi_n(e_{p})$. Let $\sigma_n$ be obtained from $\varphi_n$ by recoloring $e_p$ with $\alpha$ and let
$T'=T(y_{p-1})$. Then $\beta \in \overline{\sigma}_n(y_{p-1})\cap\overline{\sigma}_n(T')$ and
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T'$ is a good hierarchy of $T'$ under $\sigma_n$.
So $(T', \sigma_n)$ is a counterexample to Theorem \ref{hierarchy} (see (6.2) and (6.3)), which violates the minimality
assumption (6.4) or (6.5) on $(T, \varphi_n)$.
So we may assume hereafter that $\max(I_{\varphi_n})< p(T)$. Let $i=\max(I_{\varphi_n})$ if $I_{\varphi_n} \ne \emptyset$,
and let $j=p(T)$. Then $e_j$ is not incident to $y_{j-1}$. In our proof we reserve $y_0$ for the maximum vertex
(in the order $\prec$) in $T_{n,q}^*$.
\begin{claim}\label{j-1}
We may assume that there exists $\alpha\in\overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(T(y_{j-2}))$, such that
either $\alpha\notin\Gamma^q\cup \overline{\varphi}_n(T_{n,0}^*-V(T_n))$ or $\alpha \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$
with $v_{\eta_m} \preceq y_{j-2}$.
\end{claim}
To establish this statement, we consider two cases, depending on whether $I_{\varphi}$ is nonempty.
{\bf Case 1.} $I_{\varphi}\neq\emptyset$.
By assumption, $\max(I_{\varphi_n})< p(T)$. So $i \le j-1$. Let $\alpha\in\overline{\varphi}_n(y_p)\cap
\overline{\varphi}_n(y_i)$. By (6.6), we obtain
(1) $\alpha \notin \overline{\varphi}_n(T_{n,q}^*)$. So $\alpha \notin \Gamma^{q}\cup \overline{\varphi}_n(T_{n,0}^*-V(T_n))$.
\noindent If $i\le j-2$, then $\alpha\in\overline{\varphi}_n(T(y_{j-2}))$, as desired. Thus we may assume that $i=j-1$.
(2) There exists a color $\beta \in \overline{\varphi}_n(T(y_{j-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n \langle
T(y_{j-1}) - T_{n,q}^* \rangle -(\Gamma^{q}\cup D_{n,q})$ or a color $\beta \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$
with $v_{\eta_m} \preceq y_{j-2}$ and $(\Gamma^{q}_m \cup \{\eta_m\}) \cap \varphi_n \langle T(y_{j-1}) - T_{n,q}^*
\rangle=\emptyset$.
To justify this, note that if $|\overline{\varphi}_n(T(y_{j-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T(y_{j-2}) - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\ge 5$, then there exists a color $\beta$ in
$\overline{\varphi}_n(T(y_{j-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n \langle T(y_{j-1}) - T_{n,q}^* \rangle
-(\Gamma^{q}\cup D_{n,q})$, because $T(y_{j-1})-T(y_{j-2})$ contains only one edge.
If $|\overline{\varphi}_n(T(y_{j-2}))|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))| - |\varphi_n \langle
T(y_{j-2}) - T_{n,q}^* \rangle| -|\Gamma^{q}\cup D_{n,q} |\le 4 $, then, by Lemma \ref{9n}, there exist $7$ distinct
colors $\eta_{h}\in D_{n,q}\cap \overline{\varphi}_n(T(y_{j-2}))$ such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \varphi_n \langle T(y_{j-2}) - T_{n,q}^* \rangle=\emptyset$. Since $T(y_{j-1})-T(y_{j-2})$ contains only one edge,
there exists at least one of these $\eta_{h}$, say $\eta_m$, such that $(\Gamma^{q}_m \cup \{\eta_{m}\})
\cap \varphi_n \langle T(y_{j-1}) - T_{n,q}^* \rangle=\emptyset$. So (2) is true.
Depending on whether $\alpha$ is contained in $D_{n,q}$, we distinguish between two subcases.
{\bf Subcase 1.1.} $\alpha \in D_{n,q}$. In this subcase, let $\alpha=\eta_{h}\in D_{n,q}$. For simplicity, we abbreviate the two colors $\gamma^{q}_{h_1}$ and $\gamma^{q}_{h_2}$ in $\Gamma^{q}_h$ (see Definition \ref{R2}) to $\gamma_1$ and $\gamma_2$,
respectively. Since $\eta_{h} \in\overline{\varphi}_n(y_{j-1})$, by Definition \ref{R2}(i) and TAA, we have
(3) $\gamma_1, \gamma_2 \notin \varphi_n \langle T(y_{j-1}) - T_{n,q}^* \rangle$, and $\eta_h$ is not used by
any edge in $T(y_{j-1}) - T_{n,q}^*$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in
Algorithm 3.1 and $\varphi_n(e_1)=\eta_h \in D_{n,q} \subseteq D_n$).
By (3) and Lemma \ref{change}, we obtain $P_{v_{\gamma_1}}(\gamma_1,\eta_h,\varphi_n)=P_{y_{j-1}}(\gamma_1,\eta_h,
\varphi_n)$, which is disjoint from $P_{y_{p}}(\gamma_1,\eta_h,\varphi_n)$. Let $\mu_1=\varphi_n/P_{y_{p}}(\gamma_1,
\eta_h,\varphi_n)$. By Lemma~\ref{stablechange}, $\mu_1$ satisfies all the properties described in (7.3) (with $\mu_1$
in place of $\sigma_n$). In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\eta_h \in D_n$, then $\mu_1(e_1)=\varphi_n(e_1)$,
which implies that $e_1$ is outside $P_{y_{p}}(\gamma_1,\eta_h,\varphi_n)$.
Using (3) and (6.6), we get $\mu_1(f)=\varphi_n(f)$ for each $f\in E(T(y_{j-1}))$ and
$\overline{\mu}_1(u)= \overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by
using TAA under $\mu_1$, and hence $T$ satisfies MP under $\mu_1$. Furthermore, since $\eta_h \in \overline{\mu}_1(y_{p-1})$,
the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be good
under $\mu_1$, with the same $\Gamma$-sets as those under $\mu_1$. Therefore, $(T, \mu_1)$ is also a minimum counterexample
to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\gamma_1$ is missing at two vertices.
From (3) we see that
(4) $\gamma_1, \gamma_2 \notin \mu_1 \langle T(y_{j-1}) - T_{n,q}^* \rangle$, and $\eta_h$ is not used by
any edge in $T(y_{j-1}) - T_{n,q}^*$ under $\mu_1$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$
(now $e_1=f_n$ in Algorithm 3.1 and $\mu_1(e_1)=\eta_h \in D_{n,q} \subseteq D_n$).
Let $\beta$ be a color as specified in (2). Note that
(5) $\beta\notin \mu_1 \langle T(y_{j-1}) - T_{n,q}^* \rangle$, $\beta\notin D_{n,q}$, and $\beta\neq\eta_h=\alpha$.
Since $\gamma_1 \in \overline{\mu}_1(T_{n,q})$ if $q \ge 1$ and $\gamma_1 \in \overline{\mu}_1(T_n)$ if $q=0$,
from (4) and Lemma~\ref{change} we deduce that $P_{v_{\gamma_1}} (\gamma_1, \beta, \mu_1) = P_{v_{\beta}}
(\gamma_1, \beta, \mu_1)$, which is disjoint from $P_{y_p}(\gamma_1, \beta, \mu_1)$. Let $\mu_2=\mu_1/P_{y_p}(\gamma_1,
\beta, \mu_1)$. By Lemma~\ref{stablechange}, $\mu_2$ satisfies all the properties described in (7.3) (with $\mu_2$
in place of $\sigma_n$). By (4), (5) and (6.6), we have $\mu_2(f)=\mu_1(f)$ for each $f\in E(T(y_{j-1}))$, and $\overline{\mu}_2(u)=\overline{\mu}_1(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by
using TAA under $\mu_2$ and hence $T$ satisfies MP under $\mu_2$. If $\beta \notin \Gamma^q$, then clearly
$T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be a good hierarchy
of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$. So we assume that $\beta \in \Gamma^q$.
By (2), we have $\beta \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$ with $v_{\eta_m} \preceq y_{j-2}$ and
$(\Gamma^{q}_m \cup \{\eta_m\}) \cap \varphi_n \langle T(y_{j-1}) - T_{n,q}^* \rangle=\emptyset$. It follows
that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ is still a good hierarchy
of $T$ under $\mu_2$, with the same $\Gamma$-sets as those under $\mu_1$. Therefore, $(T, \mu_2)$ is also a minimum
counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\beta \in \overline{\mu}_2(y_p) \cap
\overline{\mu}_2(T(y_{j-2}))$. From (2) and the definitions of $\mu_1$ and $\mu_2$, we see that
either $\beta\notin\Gamma^q\cup \overline{\varphi}_n(T_{n,0}^*-V(T_n))$ or $\beta \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$
with $v_{\eta_m} \preceq y_{j-2}$. Thus Claim~\ref{j-1} holds by replacing $\varphi_n$ with $\mu_2$ and $\alpha$
with $\beta$.
{\bf Subcase 1.2.} $\alpha \notin D_{n,q}$. In this subcase, using (1) and the set inclusion $\overline{\varphi}_n(T_n) \cup D_n \subseteq \overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$, we get
(6) $\alpha\notin D_n$. So $\alpha$ is not used by any edge in $T(y_{j-1}) - T_{n,q}^*$ by TAA.
Let $\beta$ be a color as specified in (2). Then there are two possibilities for $\beta$.
$\bullet$ $\beta \in \overline{\varphi}_n(T(y_{j-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n \langle
T(y_{j-1}) - T_{n,q}^* \rangle -(\Gamma^{q}\cup D_{n,q})$. Now it follows from Lemma~\ref{change} that $P_{v_{\beta}}
(\alpha, \beta, \varphi_n) = P_{y_{j-1}}(\alpha, \beta, \varphi_n)$, so this path is disjoint from $P_{y_p}(\alpha, \beta, \varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p}(\alpha, \beta, \varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$
satisfies all the properties described in (7.3). By (6), the assumption on $\beta$ and (6.6), we have $\sigma_n(f)=
\varphi_n(f)$ for each $f\in E(T(y_{j-1}))$, and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each
$u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\sigma_n$ and hence $T$ satisfies
MP under $\sigma_n$. Since $\alpha, \beta \notin \Gamma^q$ (see (1)), the hierarchy $T_n=T_{n,0} \subset
T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be good under $\sigma_n$, with the
same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample to Theorem
\ref{hierarchy} (see (6.2)-(6.5)), in which $\beta \in \overline{\sigma}_n(y_p)\cap\overline{\sigma}_n(T(y_{j-2}))$.
Thus Claim~\ref{j-1} holds by replacing $\varphi_n$ with $\sigma_n$ and $\alpha$ with $\beta$.
$\bullet$ $\beta \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$ with $v_{\eta_m} \preceq y_{j-2}$ and $(\Gamma^{q}_m
\cup \{\eta_m\}) \cap \varphi_n \langle T(y_{j-1}) - T_{n,q}^* \rangle=\emptyset$. Note that $\eta_m \in \overline{\varphi}_n(T(y_{j-2}))$ and hence $\alpha\neq \eta_m$ by (6.6). In view of Lemma~\ref{change}, we obtain
$P_{v_{\beta}} (\alpha, \beta, \varphi_n) = P_{y_{j-1}} (\alpha, \beta, \varphi_n)$, which is disjoint from
$P_{y_p} (\alpha, \beta, \varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p} (\alpha, \beta, \varphi_n)$.
By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3). By (6), the assumption
on $\beta$ and (6.6), we have $\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T(y_{j-1}))$, and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$
by using TAA under $\sigma_n$ and hence $T$ satisfies MP under $\sigma_n$. Since $\alpha \notin \Gamma^q$
(see (1)) and $\eta_m \in \overline{\varphi}_n(T(y_{j-2}))$, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be good under $\sigma_n$, with the same $\Gamma$-sets as
those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy}
(see (6.2)-(6.5)), in which $\beta \in \overline{\sigma}_n(y_p)\cap\overline{\sigma}_n(T(y_{j-2}))$. Thus Claim~\ref{j-1} holds by replacing $\varphi_n$ with $\sigma_n$ and $\alpha$ with $\beta$.
{\bf Case 2.} $I_{\varphi}=\emptyset$.
Let $\alpha\in \overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(T(y_{p-1}))$. By the hypothesis of the present
case, we have $\alpha\in\overline{\varphi}_n(T_{n,q}^*)$. If $\alpha\notin\Gamma^q\cup \overline{\varphi}_n(T_{n,0}^*-V(T_n))$,
we are done. So we assume that $\alpha\in \Gamma^q\cup \overline{\varphi}_n(T_{n,0}^*-V(T_n))$.
{\bf Subcase 2.1.} $\alpha\in \overline{\varphi}_n(T_{n,0}^*-V(T_n))-\Gamma^q$. Let us first show that
(7) there exists a color $\beta\in\overline{\varphi}_n(T_{n,q}^*)-\overline{\varphi}_n(T_{n,0}^*-V(T_n))-\Gamma^q$.
Indeed, since $V(T_{n,q}^*)$ is elementary with respect to $\varphi_n$, we have $|\overline{\varphi}_n(T_{n,q}^*)|-|\overline{\varphi}_n(T_{n,0}^*-V(T_n))|-
|\Gamma^q| \ge |\overline{\varphi}_n(T_{n,0}^*)|- |\overline{\varphi}_n(T_{n,0}^*-V(T_n))|-|\Gamma^q| = |\overline{\varphi}_n(T_n)|- |\Gamma^q|$.
In view of (7.2), we obtain $|\overline{\varphi}_n(T_{n})|\ge 2n+11$ and $|\Gamma^q| \le 2 |D_{n,q}|\le 2n$. So
$|\overline{\varphi}_n(T_{n,q}^*)|-|\overline{\varphi}_n(T_{n,0}^*-V(T_n))|-|\Gamma^q| \ge 11$, which implies (7).
By (7) and Lemma~\ref{change}, we obtain $P_{v_{\alpha}} (\alpha, \beta, \varphi_n) = P_{v_{\beta}}(\alpha, \beta, \varphi_n)$,
which is disjoint from $P_{y_p}(\alpha, \beta, \varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p}(\alpha, \beta, \varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3). Since $\alpha, \beta\in
\overline{\varphi}_n(T_{n,q}^*)$, we have $\sigma_n(f)=\varphi_n(f)$ for each $f\in E(T_{n,q}^*)$, and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from
$T_{n,q}^*+e_1$ by using TAA under $\sigma_n$ and hence $T$ satisfies MP under $\sigma_n$. As $\alpha, \beta \notin \Gamma^q$,
the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to
be good under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is also a
minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\beta \in \overline{\sigma}_n(y_p)\cap\overline{\sigma}_n(T(y_{j-2}))$.
Thus Claim~\ref{j-1} holds by replacing $\varphi_n$ with $\sigma_n$ and $\alpha$ with $\beta$.
{\bf Subcase 2.2.} $\alpha\in \Gamma^q$. Let $\alpha \in\Gamma^{q}_m$ for some $\eta_{m}\in D_{n,q}$. Depending on
whether $\eta_m$ is contained in $\overline{\varphi}_n(T(y_{p-1}))$, we consider two possibilities.
$\bullet$ $\eta_m\notin\overline{\varphi}_n(T(y_{p-1}))$. By Definition \ref{R2}(i), we have $\alpha \notin\varphi_n
\langle T-T_{n,q}^* \rangle$. Since $T-T(y_{p-2})$ contains precisely two edges, Lemma~\ref{9n} guarantees
the existence of a color $\beta$ in $\overline{\varphi}_n(T(y_{p-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n
\langle T - T_{n,q}^* \rangle -(\Gamma^{q}\cup D_{n,q} )$ or a color $\beta=\eta_{h}\in D_{n,q}\cap
\overline{\varphi}_n(T(y_{p-2}))$ such that $(\Gamma^{q}_h \cup \{\eta_{h}\}) \cap \varphi_n \langle T - T_{n,q}^*
\rangle=\emptyset$. Note that $\beta \in \overline{\varphi}_n(T(y_{p-2}))-\varphi_n \langle T -
T_{n,q}^* \rangle$. By Lemma~\ref{change}, we obtain $P_{v_{\alpha}} (\alpha, \beta, \varphi_n) =
P_{v_{\beta}}(\alpha, \beta, \varphi_n)$, which is disjoint from $P_{y_p}(\alpha, \beta, \varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p}(\alpha, \beta, \varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all
the properties described in (7.3). Since $\alpha, \beta \notin\varphi_n \langle T-T_{n,q}^* \rangle$
and $\alpha, \beta\in \overline{\varphi}_n(T(y_{p-2}))$, we have $\sigma_n(f)=\varphi_n(f)$
for each $f\in E(T)$, and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$.
So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\sigma_n$ and hence $T$ satisfies MP under $\sigma_n$.
Furthermore, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains
to be good under $\sigma_n$, with the same $\Gamma$-sets as those under $\varphi_n$. Therefore, $(T, \sigma_n)$ is
also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\beta \in \overline{\sigma}_n(y_p)\cap
\overline{\sigma}_n(v_{\beta})$. Thus Claim~\ref{j-1} holds if $v_{\beta}\preceq y_{j-2}$, the present subcase
reduces to the case when $\max(I_{\sigma_n}) \ge p(T)$ if $y_j \preceq v_{\beta}$ (see the paragraphs above
Claim \ref{j-1}), and the present subcase reduces to Case 1 (where $I_{\sigma_n}\neq \emptyset$) if $y_{j-1}=v_{\beta}$.
$\bullet$ $\eta_m\in\overline{\varphi}_n(T(y_{p-1}))$. Note that $\eta_m\notin\overline{\varphi}_n(T_{n,q}^*)$ because $\eta_m\in D_{n,q}$.
So $\eta_{m}\in\overline{\varphi}_n(y_t)$ for some $1\leq t \leq p-1$. If $t\le j-2$, then Claim~\ref{j-1} holds. Thus we may assume
that $t \ge j-1$. Since $\eta_m \in \overline{\varphi}_n(y_t)$, it is not used by any edge in $T(y_t) - T_{n,q}^*$, except possibly
$e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and $\varphi_n(e_1)=\eta_m \in D_{n,q} \subseteq
D_n$). Since $\alpha \in\Gamma^{q}_m$, by Definition \ref{R2}(i), $\alpha$ is not used by any edge in $T(y_t) - T_{n,q}^*$.
It follows from Lemma~\ref{change} that $P_{v_{\alpha}} (\alpha, \eta_m, \varphi_n) = P_{y_t} (\alpha, \eta_m, \varphi_n)$,
which is disjoint from $P_{y_p} (\alpha,\eta_m, \varphi_n)$. Let $\sigma_n=\varphi_n/P_{y_p} (\alpha,\eta_m, \varphi_n)$. By Lemma~\ref{stablechange}, $\sigma_n$ satisfies all the properties described in (7.3).
In particular, if $e_1=f_n$ and $\varphi_n(e_1)=\eta_m \in D_n$, then $\sigma_n(e_1)=\varphi_n(e_1)$, which
implies that $e_1$ is outside $P_{y_p} (\alpha,\eta_m, \varphi_n)$. Since $\sigma_n(f)=\varphi_n(f)$
for each $f\in E(T(y_t))$ and $\overline{\sigma}_n(u)=\overline{\varphi}_n(u)$ for each $u\in V(T(y_{p-1}))$, we can
obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\sigma_n$, so $T$ satisfies MP under $\sigma_n$. Furthermore,
As $\alpha, \eta_m \in \overline{\sigma}_n(T(y_t))$, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots
\subset T_{n,q} \subset T_{n,q+1}=T$ remains to be good under $\sigma_n$, with the same $\Gamma$-sets as those under
$\varphi_n$. Therefore, $(T, \sigma_n)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)),
in which $\eta_m \in \overline{\sigma}_n(y_p)\cap\overline{\sigma}_n(y_t)$. Thus the present subcase
reduces to the case when $\max(I_{\sigma_n}) \ge p(T)$ if $j \preceq t$ (see the paragraphs above
Claim \ref{j-1}), and reduces to Case 1 (where $I_{\sigma_n}\neq \emptyset$) if $t=j-1$. This proves
Claim~\ref{j-1}.
\vskip 3mm
Let $\alpha$ be a color as specified in Claim \ref{j-1}; that is, $\alpha\in\overline{\varphi}_n(y_p)\cap\overline{\varphi}_n(T(y_{j-2}))$, such that either $\alpha\notin\Gamma^q\cup \overline{\varphi}_n(T_{n,0}^*-V(T_n))$ or $\alpha \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$ with $v_{\eta_m} \preceq y_{j-2}$.
Since $T(y_j)-T(y_{j-2})$ contains precisely two edges, Lemma~\ref{9n} guarantees the existence of a color $\beta$ in $\overline{\varphi}_n(T(y_{j-2}))- \overline{\varphi}_n(T_{n,0}^*-V(T_n)) - \varphi_n \langle T(y_j) - T_{n,q}^* \rangle -
(\Gamma^{q}\cup D_{n,q} )$ or a color $\beta=\eta_{h}\in D_{n,q}\cap \overline{\varphi}_n(T(y_{j-2}))$ such that $(\Gamma^{q}_h \cup \{\eta_{h}\}) \cap \varphi_n \langle T(y_j) - T_{n,q}^* \rangle=\emptyset$. Note that
(8) $\beta \notin \varphi_n \langle T(y_j) - T_{n,q}^* \rangle \cup \Gamma^{q}$.
\noindent Let $Q=P_{y_{p}}(\alpha,\beta,\varphi_n)$. We consider two cases, depending on whether $Q$ intersects $T(y_{j-1})$.
{\bf Case 1.} $Q$ and $T(y_{j-1})$ have vertices in common. Let $u$ be the first vertex of $Q$ contained in
$T(y_{j-1})$ as we traverse $Q$ from $y_p$. Define $T'=T(y_{j-1}) \cup Q[u,y_p]$ if $u= y_{j-1}$
and $T'=T(y_{j-2})\cup Q[u,y_p]$ otherwise. By the choices of $\alpha$ and $\beta$, we have $\alpha,\beta \in
\overline{\varphi}_n(T(y_{j-2}))$. So $T'$ can be obtained from $T(y_{j-2})$ by using TAA under $\varphi_n$. It follows
that $T'$ is an ETT satisfying MP with respect to $\varphi_n$, with $p(T')<p(T)$. If $\alpha\notin \Gamma^{q}$, then
both $\alpha$ and $\beta$ are outside $\Gamma^{q}$ (see (8)), so $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q}\subset T'$ is a good hierarchy of $T'$ under $\varphi_n$, with the same $\Gamma$-sets
as $T$ under $\varphi_n$. If $\alpha\in \Gamma^{q}$, then $\alpha \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$
with $v_{\eta_m} \preceq y_{j-2}$ by Claim \ref{j-1}. Since $\alpha,\eta_m \in \overline{\varphi}_n(T(y_{j-2}))$ and
$\beta \notin \Gamma^{q}$, it is clear that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T'$
is also a good hierarchy of $T'$ under $\varphi_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$.
So $(T', \varphi_n)$ is a counterexample to Theorem \ref{hierarchy} (see (6.2) and (6.3)), which violates the
minimality assumption $(6.4)$ on $(T, \varphi_n)$.
{\bf Case 2.} $Q$ is vertex-disjoint from $T(y_{j-1})$. Let $\sigma_n=\varphi_n/Q$. By Lemma \ref{LEM:Stable}, $\sigma_n$ is $(T(y_{j-1}),D_n,\varphi_n)$-stable. In particular, $\sigma_n$ is $(T(y_{j-1}),\varphi_n)$-invariant. If $\Theta_n=PE$,
then $\sigma_n$ is also $(T_n\oplus R_n,D_n,\varphi_n)$-stable. Furthermore, $T(y_{j-1})$ is an ETT satisfying MP with respect to $\sigma_n$, and $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T(y_{j-1})$ is a good hierarchy of
$T(y_{j-1})$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$. By definition, $\sigma_n$ is a $(T_{n,q}^*, D_n,\varphi_n)$-strongly stable coloring. If $\alpha\notin \Gamma^{q}$, then both $\alpha$
and $\beta$ are outside $\Gamma^{q}$ (see (8)), so $T_n=T_{n,0} \subset T_{n,1} \subset
\ldots \subset T_{n,q}\subset T$ is a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets
as $T$ under $\varphi_n$. If $\alpha\in \Gamma^{q}$, then $\alpha \in\Gamma^{q}_m$ for some $\eta_m\in D_{n,q}$
with $v_{\eta_m} \preceq y_{j-2}$ by Claim \ref{j-1}. Since $\alpha,\eta_m \in \overline{\varphi}_n(T(y_{j-2}))$ and
$\beta \notin \Gamma^{q}$, it is clear that $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q}\subset T$
is also a good hierarchy of $T$ under $\sigma_n$, with the same $\Gamma$-sets as $T$ under $\varphi_n$.
So $(T, \sigma_n)$ is a counterexample to Theorem \ref{hierarchy}, in which $\beta$ is missing at two vertices.
From the choice of $\beta$ above $(8)$ and the definition of $\sigma_n$, we see that
(9) either $\beta \notin \overline{\sigma}_n(T_{n,0}^*-V(T_n)) \cup \sigma_n \langle T(y_j) - T_{n,q}^* \rangle \cup
(\Gamma^{q}\cup D_{n,q} )$ or $\beta=\eta_{h}\in D_{n,q}\cap \overline{\sigma}_n(T(y_{j-2}))$, such that $(\Gamma^{q}_h
\cup \{\eta_{h}\}) \cap \sigma_n \langle T(y_j) - T_{n,q}^* \rangle=\emptyset$.
Let $\theta \in\overline{\sigma}_n(y_j)$. Then $\theta \notin\Gamma^{q}$. We proceed by considering two subcases.
{\bf Subcase 2.1.} $\theta \notin D_{n,q}$. In this subcase, using (6.6) and the set inclusion $\overline{\varphi}_n(T_n)
\cup D_n \subseteq \overline{\varphi}_n(T_{n,q}^*) \cup D_{n,q}$, we obtain
(10) $\theta \notin \overline{\sigma}_n(T(y_{j-1}))$ and $\theta \notin D_n$. So $\theta$ is not assigned to any
edge in $T(y_{j})-T_{n,q}^*$ by TAA.
As described in (9), there are two possibilities for $\beta$.
$\bullet$ $\beta \notin \overline{\sigma}_n(T_{n,0}^*-V(T_n)) \cup \sigma_n \langle T(y_j) - T_{n,q}^* \rangle \cup
(\Gamma^{q}\cup D_{n,q} )$. Observe that $\beta\notin D_{n}$ if $q=0$. By Lemma~\ref{change}, we obtain
$P_{v_{\beta}} (\beta, \theta, \sigma_n) = P_{y_j}(\beta, \theta, \sigma_n)$, which is disjoint from $P_{y_p}(\beta,
\theta, \sigma_n)$. Let $\mu_1=\sigma_n/P_{y_p}(\beta, \theta, \sigma_n)$. By Lemma~\ref{stablechange}, $\mu_1$
satisfies all the properties described in (7.3). By (10), the assumption on $\beta$ and (6.6), we have
$\mu_1(f)=\sigma_n(f)$ for each $f\in E(T(y_j))$ and $\overline{\mu}_1(u)=\overline{\sigma}_n(u)$ for each
$u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_1$ and hence $T$ satisfies MP under
$\mu_1$. As $\beta, \theta \notin \Gamma^q$, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ remains to be good under $\mu_1$, with the same $\Gamma$-sets as those under $\sigma_n$.
Therefore, $(T, \mu_1)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which
$\theta \in \overline{\mu}_1(y_p)\cap \overline{\mu}_1(y_j)$. Thus the present subcase reduces to the case when
$\max(I_{\mu_1}) \ge p(T)$ (see the paragraphs above Claim \ref{j-1}).
$\bullet$ $\beta=\eta_{h}\in D_{n,q}\cap \overline{\sigma}_n(T(y_{j-2}))$, such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \sigma_n \langle T(y_j) - T_{n,q}^* \rangle=\emptyset$. For simplicity, we abbreviate the two colors $\gamma^{q}_{h_1}$ and $\gamma^{q}_{h_2}$ in $\Gamma^{q}_h$ (see Definition \ref{R2}) to $\gamma_1$ and $\gamma_2$, respectively.
By Lemma~\ref{change}, we obtain $P_{v_{\beta}} (\beta, \gamma_1, \sigma_n) = P_{v_{\gamma_1}}(\beta, \gamma_1, \sigma_n)$,
which is disjoint from $P_{y_p}(\beta, \gamma_1, \sigma_n)$. Let $\mu_2=\sigma_n/P_{y_p}(\beta, \gamma_1, \sigma_n)$.
By Lemma~\ref{stablechange}, $\mu_2$ satisfies all the properties described in (7.3). By the assumption on $\beta$,
neither $\beta$ nor $\gamma_1$ is used by any edge in $T(y_j) - T_{n,q}^*$. So $\mu_2(f)=\sigma_n(f)$ for each $f\in E(T(y_j))$.
By (6.6), we get $\overline{\mu}_2(u)=\overline{\sigma}_n(u)$ for each $u\in V(T(y_{p-1}))$. It follows that $T$ can
be obtained from $T_{n,q}^*+e_1$ by using TAA under $\mu_2$ and hence $T$ satisfies MP under $\mu_2$. Furthermore, the
hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset T_{n,q+1}=T$ remains to be good under
$\mu_2$, with the same $\Gamma$-sets as those under $\sigma_n$. Therefore, $(T, \mu_2)$ is also a minimum counterexample
to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\gamma_1$ is missing at both $y_p$ and $v_{\gamma_1}$.
From the assumption on $\beta$ and the definition of $\mu_2$, we deduce that
(11) $\beta=\eta_{h}\in D_{n,q}\cap \overline{\mu}_2(T(y_{j-2}))$, such that $(\Gamma^{q}_h \cup \{\eta_{h}\})
\cap \mu_2 \langle T(y_j) - T_{n,q}^* \rangle=\emptyset$.
By (11) and Lemma~\ref{change}, we obtain $P_{v_{\gamma_1}} (\theta, \gamma_1, \mu_2) = P_{y_j}(\theta, \gamma_1,
\mu_2)$, which is disjoint from $P_{y_p}(\theta, \gamma_1, \mu_2)$. Let $\mu_3=\mu_2/P_{y_p}(\theta, \gamma_1, \mu_2)$.
By Lemma~\ref{stablechange}, $\mu_3$ satisfies all the properties described in (7.3). By (10), (11) and (6.6),
we have $\mu_3(f)=\mu_2(f)$ for each $f\in E(T(y_j))$ and $\overline{\mu}_3(u)=\overline{\mu}_2(u)$ for each
$u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_3$ and hence $T$ satisfies MP
under $\mu_3$. Furthermore, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset T_{n,q} \subset
T_{n,q+1}=T$ remains to be good under $\mu_3$, with the same $\Gamma$-sets as those under $\mu_2$. Therefore,
$(T, \mu_3)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which $\theta$ is
missing at both $y_p$ and $y_j$. Thus the present subcase reduces to the case when $\max(I_{\mu_3}) \ge p(T)$ (see
the paragraphs above Claim \ref{j-1}).
{\bf Subcase 2.2.} $\theta \in D_{n,q}$. Let $\theta=\eta_t \in D_{n,q}$. For simplicity, we use $\varepsilon_1$ and
$\varepsilon_2$ to denote the two colors $\gamma^{q}_{t_1}$ and $\gamma^{q}_{t_2}$ in $\Gamma^{q}_t$ (see Definition
\ref{R2}), respectively. Then
(12) $\varepsilon_1, \varepsilon_2 \notin \sigma_n \langle T(y_j) - T_{n,q}^* \rangle$ and $\eta_t$ is not used by
any edge in $T(y_j) - T_{n,q}^*$ under $\sigma_n$, except possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$
in Algorithm 3.1 and $\sigma_n(e_1)=\eta_t \in D_{n,q} \subseteq D_n$).
By (12) and Lemma~\ref{change} (with $\varepsilon_1$ in place of $\alpha$), we obtain $P_{v_{\varepsilon_1}} (\varepsilon_1,
\beta, \sigma_n) = P_{v_{\beta}}(\varepsilon_1, \beta, \sigma_n)$, which is disjoint from $P_{y_p}(\varepsilon_1, \beta,
\sigma_n)$. Let $\mu_4=\sigma_n/P_{y_p}(\varepsilon_1, \beta, \sigma_n)$. By Lemma~\ref{stablechange}, $\mu_4$ satisfies all
the properties described in (7.3). By (9), we have $\beta\notin \sigma_n \langle T(y_j) - T_{n,q}^* \rangle$,
which together with (12) and (6.6) implies $\mu_4(f)=\sigma_n(f)$ for each $f\in E(T(y_j))$ and $\overline{\mu}_4(u)=\overline{\sigma}_n(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain $T$ from $T_{n,q}^*+e_1$
by using TAA under $\mu_4$ and hence $T$ satisfies MP under $\mu_4$. Since $\beta \notin \Gamma^q$ by (9)
and $\eta_t \in\overline{\mu}_4(y_j)$, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ remains to be good under $\mu_4$, with the same $\Gamma$-sets as those under $\sigma_n$.
Therefore, $(T, \mu_4)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which
$\varepsilon_1$ is missing at both $y_p$ and $v_{\varepsilon_1}$.
From (12) and (6.6) it can be seen that
(13) $\varepsilon_1, \varepsilon_2 \notin \mu_4 \langle T(y_j) - T_{n,q}^* \rangle$ and $\eta_t \notin
\overline{\mu}_4(T(y_{j-1}))$. So $\eta_t$ is not used by any edge in $T(y_j)-T_{n,q}^*$ under $\mu_4$, except
possibly $e_1$ when $q=0$ and $T_{n,0}^*=T_n$ (now $e_1=f_n$ in Algorithm 3.1 and $\mu_4(e_1)=\eta_t \in D_{n,q} \subseteq D_n$).
By (13) and Lemma~\ref{change}, we obtain $P_{v_{\varepsilon_1}} (\varepsilon_1, \eta_t, \mu_4) =
P_{y_j}(\varepsilon_1, \eta_t, \mu_4)$, which is disjoint from $P_{y_p}(\varepsilon_1, \eta_t, \mu_4)$.
Let $\mu_5=\mu_4/P_{y_p}(\varepsilon_1, \eta_t, \mu_4)$. By Lemma~\ref{stablechange}, $\mu_5$ satisfies all
the properties described in (7.3). In particular, if $e_1=f_n$ and $\mu_4(e_1)=\eta_t \in D_n$, then $\mu_5(e_1)=\mu_4(e_1)$,
which implies that $e_1$ is outside $P_{y_p}(\varepsilon_1, \eta_t, \mu_4)$. By (13) and (6.6), we have $\mu_5(f)=\mu_4(f)$
for each $f\in E(T(y_j))$ and $\overline{\mu}_5(u)=\overline{\mu}_4(u)$ for each $u\in V(T(y_{p-1}))$. So we can obtain
$T$ from $T_{n,q}^*+e_1$ by using TAA under $\mu_5$ and hence $T$ satisfies MP under $\mu_5$. Since $\eta_t, \varepsilon_1 \in
\overline{\mu}_5(T(y_j))$, the hierarchy $T_n=T_{n,0} \subset T_{n,1} \subset \ldots \subset
T_{n,q} \subset T_{n,q+1}=T$ remains to be good under $\mu_5$, with the same $\Gamma$-sets as those under $\mu_4$.
Therefore, $(T, \mu_5)$ is also a minimum counterexample to Theorem \ref{hierarchy} (see (6.2)-(6.5)), in which
$\theta=\eta_t$ is missing at both $y_p$ and $y_j$. Thus the present subcase reduces to the case when $\max(I_{\mu_5})
\ge p(T)$ (see the paragraphs above Claim \ref{j-1}).
This completes our discussion about Situation 7.3 and hence our proof of Theorem \ref{hierarchy}}. \hfill \rule{4pt}{7pt}
\subsection{Proof of Theorem \ref{thm:tech10}(ii)}
In the preceding subsection we have proved Theorem \ref{hierarchy}} and hence Theorem \ref{thm:tech10}(i).
To complete the proof of Theorem \ref{thm:tech10}, we still need to establish the interchangeability property
as described in Theorem \ref{thm:tech10}(ii).
\begin{lemma}\label{rutgers}
Suppose Theorem~\ref{thm:tech10}(i), (iii)-(vi) hold for all ETTs with $n$ rungs and satisfying MP, and
suppose Theorem~\ref{thm:tech10}(ii) holds for all ETTs with $n-1$ rungs and satisfying MP. Then
Theorem~\ref{thm:tech10}(ii) holds for all ETTs with $n$ rungs and satisfying MP; that is,
$T_{n+1}$ has the interchangeability property with respect to $\varphi_n$.
\end{lemma}
{\bf Proof.} Let $T=T_{n+1}$, let $\sigma_n$ be a $(T, D_n, \varphi_{n})$-stable coloring, and let
$\alpha$ and $\beta$ be two colors in $[k]$ with $\alpha\in \overline{\sigma}_n(T)$ (equivalently
$\alpha\in \overline{\varphi}_{n}(T)$). We aim to prove that $\alpha$ and $\beta$ are $T$-interchangeable under
$\sigma_n$. Recalling (5.2), we may assume that $T_{n+1}$ is a closure of $T_n\vee R_n$ under $\varphi_n$,
which is a special closure of $T_n$ under $\varphi_n$, if $\Theta_n=PE$. As introduced in Section 5,
$T_{n,0}^*= T_{n}\vee R_n$ if $\Theta_n=PE$ and $T_{n,0}^*= T_{n}$ otherwise. From definitions we see that
$\sigma_n$ is also a $(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring (see the remark right above
Lemma \ref{step11}).
Assume the contrary: there are at least two $(\alpha,\beta)$-paths $Q_1$ and $Q_2$ with respect to $\sigma_n$ intersecting
$T$. By Theorem~\ref{thm:tech10}(i), $V(T)$ is elementary with respect to $\varphi_n$, so it is also elementary with
respect to $\sigma_n$. Since $T=T_{n+1}$ is closed with respect to $\varphi_n$, is it also closed with respect to $\sigma_n$.
As $\alpha\in \overline{\sigma}_n(T)$, it follows that $|V(T)|$ is odd and $\beta$ is outside $\overline{\sigma}_n(T)$.
From the existence of $Q_1$ and $Q_2$, we see that $|\partial_{\sigma_n, \beta}(T)|$ is odd and at least three. Thus
$G$ contains at least three $(T, \sigma_n, \{\alpha, \beta\})$-exit paths $P_1,P_2,P_3$.
We call the tuple $(\sigma_n, T, \alpha, \beta, P_1,P_2,P_3)$ a {\em counterexample} if $\sigma_n$ is a
$(T_{n,0}^*, D_n,\varphi_n)$-strongly stable coloring, and $T$ is a closed ETT corresponding to $(\sigma_n,T_n)$ (see Theorem~\ref{thm:tech10}(vi) and Definition \ref{wz2}) with $n$ rungs, with $T_{n,0}^* \subset T$, and satisfying MP
under $\sigma_n$. Moreover, $P_1,P_2,P_3$ are three $(T, \sigma_n, \{\alpha, \beta\})$-exit paths. We use
${\cal K}$ to denote the set of all such counterexamples. With a slight abuse of notation, let $(\sigma_n, T, \alpha, \beta, P_1,P_2,P_3)$ be a counterexample in ${\cal K}$ with the minimum $|P_1|+|P_2|+|P_3|$. For $i=1,2,3$, let $a_i$ and $b_i$ be the
ends of $P_i$ with $b_i \in V(T)$, and $f_i$ be the edge of $P_i$ incident to $b_i$. Renaming subscripts if necessary,
we may assume that $b_1\prec b_2 \prec b_3$. We propose to show that
(1) $b_3 \notin V(T_n)$ if $\Theta_n=SE$ or $RE$.
Otherwise, $b_3 \in V(T_n)$. Let $\gamma \in\overline{\sigma}_n(T_n)$. Since $T=T_{n+1}$ is closed with respect to
$\sigma_n$, both $\alpha$ and $\gamma$ are closed in $T$ with respect to $\sigma_n$. Let $\mu_1=\sigma_n/
(G-T,\alpha,\gamma)$. By Lemma \ref{LEM:Stable}, $\mu_1$ is a $(T,D_n,\varphi_n)$-stable
coloring. By definition, $\mu_1$ is a $(T_n,D_n,\varphi_n)$-stable coloring. Since $\Theta_n=SE$ or $RE$, by Algorithm 3.1,
$\mu_1$ is also a $(T_n,D_{n-1},\varphi_{n-1})$-stable coloring. By Theorem~\ref{thm:tech10}(vi), $T_n$
is an ETT corresponding to $\mu_1$ (see see Theorem~\ref{thm:tech10}(vi) and Definition \ref{wz2}) and satisfies MP
under $\mu_1$, with $n-1$ rungs. Since $P_1,P_2,P_3$ are three $(T_n, \mu_1, \{\gamma, \beta\})$-exit paths, there are
at least two $(\gamma, \beta)$-paths with respect to $\mu_1$ intersecting $T_n$. Hence $\gamma$ and
$\beta$ are not $T_n$-interchangeable under $\mu_1$, contradicting Theorem~\ref{thm:tech10}(ii) because $T_n$ has $n-1$ rungs.
So (1) is established.
(2) $b_3 \notin V(T_n \vee R_n)$ if $\Theta_n=PE$.
The proof is similar to that of (1). Assume the contrary: $b_3 \in V(T_n \vee R_n)$. Let $\gamma \in
\overline{\sigma}_n(T_n) \cap \overline{\sigma}_n(R_n)$ and let $\mu_1=\sigma_n/(G-T,\alpha,\gamma)$. By
Lemma \ref{LEM:Stable}, $\mu_1$ is a $(T_n \oplus R_n,D_n,\varphi_n)$-stable coloring.
Since $P_1,P_2,P_3$ are three $(T_n \vee R_n, \mu_1, \{\gamma, \beta\})$-exit paths, there are at least two
$(\gamma, \beta)$-paths with respect to $\mu_1$ intersecting $T_n \vee R_n$, contradicting Lemma~\ref{interchange}(iii).
So (2) holds.
Let $\gamma\in\overline{\sigma}_n(b_3)$ and let $\mu_2=\sigma_n/(G-T,\alpha,\gamma)$. By Lemma \ref{LEM:Stable},
$\mu_2$ is a $(T,D_n,\varphi_n)$-stable coloring. So
$\mu_2$ is a $(T_{n,0}^*,D_n,\varphi_n)$-strongly stable coloring. By Theorem~\ref{thm:tech10}(vi), $T$
is an ETT corresponding to $\mu_2$ (see Definition \ref{wz2}) and satisfies MP under $\mu_2$. Note that
$f_i$ is colored by $\beta$ under both $\mu_2$ and $\sigma_n$ for $i=1,2,3$.
Consider $\mu_3=\mu_2/P_{b_3}(\beta, \gamma, \mu_2)$. Clearly, $\beta \in \overline{\mu}_3(b_3)$. By (1), (2) and Lemma \ref{LEM:Stable}, $\mu_3$ is a $(T_{n,0}^*,D_n,\mu_2)$-strongly stable coloring. It follows from Lemma \ref{sc2} that
$\mu_3$ is a $(T_{n,0}^*,D_n,\varphi_n)$-strongly stable coloring. By Theorem~\ref{thm:tech10}(vi),
$T(b_3)$ is an ETT corresponding to $\mu_3$ (see Definition \ref{wz2}) and satisfies MP under $\mu_3$. Let $T'$ be obtained from $T(b_3)$ by adding $f_1$ and $f_2$ and let $T''$ be a closure of $T'$ under $\mu_3$. Obviously, both $T'$ and $T''$ are ETTs corresponding to $\mu_3$ and satisfies MP under $\mu_3$. By Theorem~\ref{thm:tech10}(i), $V(T'')$ is elementary with respect to $\mu_3$, because $T''$ has $n$ rungs.
Observe that none of $a_1,a_2,a_3$ is contained in $T''$, for otherwise, let $a_i \in V(T'')$ for some $i$
with $1\le i \le 3$. Since $\{\beta,\gamma\}\cap \overline{\mu}_3(a_i) \ne \emptyset$ and $\beta \in \overline{\mu}_3(b_3)$,
we obtain $\gamma \in \overline{\mu}_3(a_i)$. Hence from TAA we see that $P_1,P_2,P_3$ are all entirely
contained in $G[T'']$, which in turn implies $\gamma \in \overline{\mu}_3(a_j)$ for $j=1,2,3$. So $V(T'')$ is not
elementary with respect to $\mu_3$, a contradiction. Each $P_i$ contains a subpath $Q_i$, which is a $T''$-exit path
with respect to $\mu_3$. Since $f_1$ is not contained in $Q_1$, we obtain $|Q_1|+|Q_2|+|Q_3|<|P_1|+|P_2|+|P_3|$.
In view of (1) and (2), we have $T_n \subseteq T''$ if $\Theta_n \ne PE$ and $T_n \vee R_n \subseteq T''$ if $\Theta_n=PE$.
Thus the existence of the counterexample $(\mu_3, T'', \gamma, \beta, Q_1,Q_2,Q_3)$ violates the minimality assumption on
$(\sigma_n, T, \alpha, \beta, P_1,P_2,P_3)$.
This completes our proof of Lemma \ref{rutgers} and hence of Theorem \ref{thm:tech10}. \hfill \rule{4pt}{7pt}
|
1,116,691,500,452 | arxiv | \section{Introduction}\label{section1}
Traditionally, magnetic reconnection and MHD wave theory have been viewed as separate areas of solar physics (see, e.g., Priest \& Forbes \cite{magneticreconnection2000}; Roberts \cite{Bernie}; De Moortel \cite{DeMoortel2005}; Nakariakov \& Verwichte \cite{NV2005}; De Moortel \& Nakariakov \cite{DN2012}). However, this is a misconception: we know that (steady-state) reconnection models not only generate outflows/waves, but also require inflows/waves (e.g. Parker \cite{Parker}; Sweet \cite{Sweet}; Petschek \cite{Petschek}). Several authors have already challenged this point-of-view (e.g. Craig \& McClymont \cite{CraigMcClymont1991}; Longcope \& Priest \cite{LP2007}; Murray et al. \cite{Murray2009}; McLaughlin et al. \cite{McLaughlin2009}; \cite{McLaughlin2012}) and their investigations contribute to our understanding of dynamic or time-dependent models of magnetic reconnection. Of particular importance to this paper is the work of McLaughlin et al. (\cite{McLaughlin2009}) which is the first demonstration of reconnection {\emph{naturally}} driven by MHD wave propagation, via a process entitled {\emph{oscillatory reconnection}}.
MHD wave propagation in inhomogeneous media is a fundamental plasma process and the study of MHD waves in the neighbourhood of magnetic null points directly contributes to this area (see review by McLaughlin et al. \cite{McLaughlinREVIEW}). It is known that {\emph{null points}} - weaknesses in the magnetic field where the field strength, and hence the Alfv\'en speed, is zero - and {\emph{separatrices}} - topological features that separate regions of different magnetic flux connectivity - are an inevitable consequence of the distributed isolated magnetic flux sources at the photospheric surface, where the number of such null points will depend upon the magnetic complexity of the photospheric flux distribution (see, e.g., review by Longcope \cite{L2005}, and Close et al. \cite{Close2004}; R{\'e}gnier et al. \cite{RPH2008}; Longcope \& Parnell \cite{LP2009} for the statistics of coronal null points). It is also now known that MHD wave perturbations are omnipresent in the corona (e.g. Tomczyk et al. \cite{Tomczyk2007}). Thus, these two areas of scientific study; MHD waves and magnetic topology, {\emph{will}} encounter each other in the corona, i.e. MHD waves will propagate into the neighbourhood of coronal null points (e.g. blast waves from a flare will at some point encounter a null point).
McLaughlin \& Hood (\cite{MH2004}; \cite{MH2005}; \cite{MH2006a}; \cite{MH2006b}) investigated the behaviour of linear MHD waves (fast and slow magnetoacoustic waves and Alfv\'en waves) in the neighbourhood of a variety of 2D null points. It was found that the (linear) fast wave is focused towards the null point by a refraction effect and all the wave energy, and thus current density, accumulates close to the null, i.e. {\emph{null points will be locations for preferential heating by (linear) fast waves}}. The Alfv\'en wave propagates along magnetic fieldlines and so accumulates along the separatrices (in 2D) or along the spine or fan-plane (in 3D). Waves in the neighbourhood of a single 2D null point have also been investigated using cylindrical models, in which the generated waves encircled the null point (e.g. Bulanov \& Syrovatskii \cite{Bulanov1980}; Craig \& McClymont \cite{CraigMcClymont1991}; \cite{CraigMcClymont1993}; Craig \& Watson \cite{CraigWatson1992}; Hassam \cite{Hassam1992}) and it was found that the wave propagation leads to an exponentially-large increase in the current density (see also Ofman \cite{Ofman1992}; Ofman et al. \cite{Ofman1993}; Steinolfson et al. \cite{Steinolfson1995} and a comprehensive review by McLaughlin et al. \cite{McLaughlinREVIEW} for further details). 3D MHD wave activity about coronal null points has been investigated by various authors (e.g. Galsgaard et al. \cite{Galsgaard2003}; Pontin \& {Galsgaard} \cite{P1}; Pontin et al. \cite{P2}; McLaughlin et al. \cite{MFH2008}; Galsgaard \& Pontin \cite{klaus2011a}; \cite{klaus2011b}; Thurgood \& McLaughlin \cite{Thurgood2012}).
Reconnection can occur when strong currents cause the magnetic fieldlines to diffuse through the plasma and change their connectivity (Parker \cite{Parker}; Sweet \cite{Sweet}; Petschek \cite{Petschek}). However, these papers did not include the effect of gas pressure, which would act to limit the growth of the current density. In considering the relaxation of a 2D X-type neutral point disturbed from equilibrium, Craig \& McClymont (\cite{CraigMcClymont1991}) found that free magnetic energy is dissipated by a physical mechanism which couples resistive diffusion at the null to global advection of the outer field, which they called {\emph{oscillatory reconnection}}. An example of oscillatory reconnection generated by magnetic flux emerging into a coronal hole was reported by Murray et al. (\cite{Murray2009}) who found a series of ``reconnection reversals'' take places as the system searches for equilibrium, i.e. the system demonstrates oscillatory reconnection in a self-consistent manner. The physics behind oscillatory reconnection has been investigated by McLaughlin et al. (\cite{McLaughlin2009}), Murray et al. (\cite{Murray2009}) and Threlfall et al. (\cite{Threlfall2012}).
McLaughlin et al. (\cite{McLaughlin2012}) investigated the long-term evolution of an initially-buoyant magnetic flux tube emerging into a gravitationally-stratified coronal hole environment and reported on the resulting oscillations and outflows. They found that the physical mechanism of oscillatory reconnection {\emph{naturally}} generates quasi-periodic vertical outflows with a transverse/swaying aspect. There is currently a great deal of interest in observations of transverse motions in the solar atmosphere (e.g. Tomczyk et al. \cite{Tomczyk2007}; De Pontieu et al. \cite{Bart2007}; \cite{Bart2011}; Cirtain et al. \cite{Cirtain2007}; Erd\'elyi \& Taroyan \cite{Robertus2008}; Nishizuka et al. \cite{Nishizuka2008}; \cite{Nishizuka2011}; He et al. \cite{He2009a}; \cite{He2009b}; Liu et al. \cite{Liu2009}; \cite{Liu2011}; McIntosh et al. \cite{McIntosh2011}; Okamoto \& De Pontieu \cite{OBart2011}; Morton et al. \cite{Morton2012}; Yurchyshyn et al. \cite{Yurchyshyn2012}) and transverse/swaying motions have been observed over a range of wavelengths, speeds, temperatures and scales. However, the origin of these propagating, transverse oscillations remains a mystery, and these authors often note that the challenge remains to understand how and where these waves are generated in the solar atmosphere. Liu et al. (\cite{Liu2011}) summarises possible generation mechanisms for these transverse motions, including an oscillating wake from a coronal mass ejection or periodic reconnection (see, e.g., Chen \& Priest \cite{CP2006}; Sych et al. \cite{Sych2009}). The physical mechanism of oscillatory reconnection is another possible source of these transverse motions. As reported by McLaughlin et al. (\cite{McLaughlin2012}), the transverse behaviour seen in the periodic jets originating from the reconnection region of the inverted Y-shaped structure is specifically due to the oscillatory reconnection mechanism, and would be absent for a single, steady-state reconnection jet. The physical mechanism also naturally generates periodic outputs even though no periodic driver is imposed on the system.
Thus, there is a clear interest in furthering our understanding of the periodic nature of oscillatory reconnection. In this paper, we investigate the periodic signal generated by the mechanism, with a specific interest in measuring periods and decay rates as well as the robustness of our results, i.e. how do the results vary with the strength of the driver.
\subsection{Overview of McLaughlin et al. (2009)}\label{section1.1}
This paper will closely follow the work of McLaughlin et al. (\cite{McLaughlin2009}) as we investigate the periodic nature of oscillatory reconnection. These authors investigated the behaviour of nonlinear fast magnetoacoustic waves near a 2D X-type neutral point and found that the incoming wave deforms the null point into a cusp-like point which in turn collapses to a current sheet. The system then evolves periodically through a series of horizontal/vertical current sheets with associated changes in connectivity, i.e. the system demonstrates the mechanism of oscillatory reconnection.
More specifically, McLaughlin et al. (\cite{McLaughlin2009}) found that the incoming (fast) wave propagates across the magnetic fieldlines and the initial profile, an annulus, contracts as the wave approaches the null point. This is the refraction behaviour that {{is}} typical of fast wave behaviour around magnetic null points (see, e.g., McLaughlin et al. \cite{McLaughlinREVIEW}) and results from the spatially-varying (equilibrium) Alfv\'en-speed profile.
The incoming wave was observed to develop discontinuities (for a physical explanation, see Appendix B of McLaughlin et al. \cite{McLaughlin2009} or, alternatively, Gruszecki et al. \cite{Marcin2011}) and these discontinuities form fast oblique magnetic shock waves, where the shock makes ${\bf{B}}$ refract away from the normal. Interestingly, the shock locally heats the initially $\beta=0$ plasma, creating $\beta \neq 0$ at these locations.
At a later time, the shocks overlap, forming a shock-cusp, which leads to the development of hot jets and in turn these jets substantially heat the local plasma and significantly deform the local magnetic field. By the time the shock waves reach the null, the (initially X-point) magnetic field has been deformed such that the separatrices now touch one another rather than intersecting at a non-zero angle (Priest \& Cowley \cite{PC1975} call this \lq{cusp-like}\rq{}). The osculating field structure continues to collapse and forms a horizontal current sheet. However, the separatrices continue to evolve: the jets at the ends of the (horizontal) current sheet continue to heat the local plasma, which in turn expands. This expansion squashes and shortens the current sheet, forcing the separatrices apart. The (squashed) current sheet thus returns to a \lq{cusp-like}\rq{} null point that, due to the continuing expansion of the heated plasma, in turn forms a vertical current sheet. In effect, the (net) restoring force acts to return the (deformed) null point to its equilibrium state, but overshoots the equilibrium. The phenomenon then repeats itself: jets heat the plasma at the ends of this newly-formed (vertical) current sheet, the local plasma expands, the (vertical) current sheet is shortened, the system attempts to return itself to equilibrium, overshoots and forms a (second) horizontal current sheet. The evolution proceeds through a series of horizontal and vertical current sheets, and the system clearly displays oscillatory behaviour. It is also interesting to note that the final state is non-potential, where this is because the plasma to the left and right of the null is (locally) hotter than that above and below. Consequently, a thermal-pressure gradient exists and causes the X-point to be slightly closed up in the vertical direction (i.e. generating a small, positive current). It is important to note that the non-potential final state is still in force balance and will eventually return to a potential state, but on a far greater timescale than our simulations ($t_{{\rm{diffusion}}} \sim R_m = 10^4$ Alfv\'en times, where $R_m$ is the magnetic Reynolds number). We also note that there is nothing unique about the orientation of the first current sheet being horizontal followed by a vertical, this simply results from the particular choice of initial condition, and McLaughlin et al. (\cite{McLaughlin2012}) use the more general terminology {\emph{orientation 1}} and {\emph{orientation 2}}.
McLaughlin et al. (\cite{McLaughlin2009}) also present evidence of reconnection occurring in the system; reporting both a change in fieldline connectivity (qualitative evidence) and changes in the vector potential which directly showed a cyclic increase and decrease in magnetic flux on either side of the separatrices (see their Figures 12 and 13). Hence, since the system displayed both oscillatory behaviour {\emph{and}} reconnection, it was concluded that the system demonstrated the phenomenon of {{oscillatory reconnection}}.
Our paper has the following outline: the basic setup, equations and assumptions are described in \S\ref{section2}, the periodic nature of oscillatory reconnection is detailed in \S\ref{section4} and the conclusions are given in \S\ref{section:conclusions}.
\section{Basic Equations}\label{section2}
We consider the nonlinear, compressible, resistive MHD equations:
\begin{eqnarray}
\rho \left[ {\partial {\bf{v}}\over \partial t} + \left( {\bf{v}}\cdot\nabla \right) {\bf{v}} \right] &=&- \nabla p + \left( {{\frac{1}{\mu}}} \nabla \times {\bf{B}} \right)\times {\bf{B}} \; ,\nonumber \\
{\partial {\bf{B}}\over \partial t} &=& \nabla \times \left ({\bf{v}}\times {\bf{B}}\right ) + \eta \nabla ^2 {\bf{B}}\; ,\nonumber \\
\rho \left[{\partial {\epsilon}\over \partial t} + \left( {\bf{v}}\cdot\nabla \right) {\epsilon}\right] &=& - p \nabla \cdot {\bf{v}} + {{\frac{1}{\sigma}}} \left| {\bf{j}} \right| ^2 + Q_{\rm{shock}} \; \nonumber ,\\
{\partial \rho\over \partial t} + \nabla \cdot \left (\rho {\bf{v}}\right ) &=& 0\; , \label{MHDequations}
\end{eqnarray}
where $\rho$ is the mass density, ${\bf{v}}$ is the plasma velocity, ${\bf{B}}$ the magnetic induction (usually called the magnetic field), $p$ is the plasma pressure, $ \mu = 4 \pi \times 10^{-7} \/\mathrm{Hm^{-1}}$ is the magnetic permeability, $\sigma$ is the electrical conductivity, $\eta=1/ {\mu \sigma} $ is the magnetic diffusivity, $\epsilon= {p / \rho \left( \gamma -1 \right)}$ is the specific internal energy density, $\gamma={5 / 3}$ is the ratio of specific heats and ${\bf{j}} = {{\nabla \times {\bf{B}}} / \mu}$ is the electric current density.
We solve these governing equations numerically using a Lagrangian remap, shock-capturing code called {\emph{LARE2D}} (Arber et al. \cite{Arber2001}), which utilizes artificial shock viscosity to introduce dissipation at steep gradients. The details of this technique, often called Wilkins viscosity, can be found in Wilkins (\cite{Wilkins1980}). Thus, $Q_{\rm{shock}}$ represents the viscous heating at shocks.
We now introduce a change of scale to non-dimensionalise all variables. Letting ${\rm{\bf{v}}} = {\rm{v}}_0 {\mathbf{v}}^*$, ${\mathbf{B}} = B {\mathbf{B}}^*$, $x = L x^*$, $y=L y^*$, $z = L z^*$, $\rho={\rho}_0 \rho^*$, $p = p_0 p^*$, ${\bf{j}}=j_0\:{\bf{j}}^*$, $\nabla = \nabla^* / L$, $t={t}_0 t^*$ and $\eta = \eta_0$, where * denotes a dimensionless quantity and ${\rm{v}}_0$, $B$, $L$, ${\rho}_0$, $p_0$, $j_0$, ${t}_0$ and $\eta_0$ are constants with the dimensions of the variable they are scaling. We then set $ {B} / {\sqrt{\mu \rho _0 } } ={\rm{v}}_0$ and ${\rm{v}}_0 = {L} / {{t_0}}$ (this sets ${\rm{v}}_0$ as a constant background Alfv\'{e}n speed). We also set $j_0 = B/\mu L$ and ${\eta_0 {t}_0 } / {L^2} =R_m^{-1}$, where $R_m$ is the magnetic Reynolds number, and choose $R_m=10^4$. This process non-dimensionalises equations (\ref{MHDequations}) and under these scalings, $t^*=1$ (for example) refers to $t={t}_0= {L} / {{\rm{v}}_0}$; i.e. the time taken to travel a distance $L$ at the background Alfv\'en speed.
There is no fixed dimensional length scale to our X-point system (X-points are scale-free), and thus we have a great deal of freedom in choosing our dimensional constants. We choose $L=1\:$Mm and $B=1\:$G (for simplicity, and where these choices allow an intuitive understanding of equation \ref{Xpoint} below) and we chose a coronal density of ${\rho}_0= 5 \times 10^{-13}\:$kg/m$^3$ and coronal temperature of $T_0=10^6\:$K. This sets ${\rm{v}}_0={B}/\sqrt{\mu \rho_0} = 126.2\:$km/s, $t_0 = {L} / {{\rm{v}}_0} = 7.93 \:$seconds and $j_0 = B/\mu L = 8 \times 10^{-5}\:$A. The values returned from equations (\ref{MHDequations}) are made dimensional using these solar constants.
\subsection{Basic equilibrium and numerical set-up}\label{section:2.1}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=6.5in]{20234FIG1.eps}
\caption{Plot of time evolution of $j_z(0,0,t)$ (measured in milliAmps) for $0\le t\le 480\:$seconds. Red line indicates {\emph{impulsive phase}} and black line indicates {\emph{stationary}} phase. Insert shows the same time evolution over $158.9\le t\le 480\:$seconds (same horizontal axis, different vertical axis). The black dotted lines indicate the formation times of all the horizontal current sheets and the green line indicates $j_{{\rm{final}}} = 0.8165j_0= 6.5\times 10^{-5}\:$A. The blue dashed lines indicate an exponentially-damped envelope ${\rm{max}}(j_z)|_{\rm{stationary}}\times e^{-\lambda t} + j_{{\rm{final}}}$ and ${\rm{min}}(j_z)|_{\rm{stationary}}\times e^{-\lambda t} + j_{{\rm{final}}}$ where $\lambda = -0.015\:{\rm{s}}^{-1}$.}
\label{figure1}
\end{center}
\end{figure*}
To set-up our system, we follow the numerical framework of McLaughlin et al. (\cite{McLaughlin2009}). Thus, we consider a simple 2D X-type neutral point as our equilibrium magnetic field, where the initial field is taken as:
\begin{eqnarray}
{\bf{B}}_0 = \frac{B}{L} \left(y, x, 0\right) \;, \label{Xpoint}
\end{eqnarray}
where $B=1\:$G is a characteristic field strength and $L=1\:$Mm is the length scale for magnetic field variations. This magnetic field can be seen in McLaughlin et al. (\cite{McLaughlin2009}, their Figure 1).
Initially, we consider the equilibrium plasma to be cold: $T=0\:$K (i.e. $\beta (t=0)=0$) and, hence, ignore plasma pressure effects. However, McLaughlin et al. (\cite{McLaughlin2009}) showed that magnetic shocks will heat the plasma and so the plasma will not remain cold (see e.g. $\S 1.5$ in Priest \& Forbes \cite{magneticreconnection2000}).
In order to excite a pure fast magnetoacoustic wave, we consider an initial condition that perturbs velocity {\emph{purely across}} the equilibrium magnetic field. Using the terminology of McLaughlin et al. (\cite{McLaughlin2009}), there are three distinguishing velocity components that can be considered in our system:
\begin{itemize}
\item[$\bullet$]{${\rm{v}}_\perp= {\bf{v}}\times{\bf{B}}_0 \cdot {\hat{\bf{z}}}={\rm{v}}_x B_y - {\rm{v}}_y B_x$, which corresponds to the velocity across the equilibrium field, and hence corresponds to the fast magnetoacoustic wave (the only MHD wave that can cross fieldlines)},
\item[$\bullet$]{ ${\rm{v}}_\parallel= {\bf{v}}\cdot{\bf{B}}_0={\rm{v}}_x B_x + {\rm{v}}_y B_y$, which corresponds to the velocity along the equilibrium field and corresponds to the propagation of the slow magnetoacoustic wave},
\item[$\bullet$]{${\rm{v}}_z={\bf{v}}\cdot{\hat{\bf{z}}}$, which corresponds to the velocity in the invariant direction and hence corresponds to the Alfv\'en wave.}
\end{itemize}
Interestingly, these three velocity components - each isolating an individual MHD mode - are in good agreement with those reported by Thurgood \& McLaughlin (\cite{Thurgood2012}) who used the equilibrium magnetic field and the flux function (which is parallel to the invariant direction) to define an orthogonal coordinate system to isolate and identify the propagation of each of the MHD modes. Using their convention, ${\rm{v}}_\perp = {\bf{v}} \cdot { {\bf{B}}}_0 \times {\hat{{\bf{A}}}}$, where ${\bf{A}} = \frac{1}{2} \left(y^2-x^2\right) {\hat{\bf{z}}}$ is the flux function, and where perturbations in the ${\bf{B}}_0 \times \hat{ {\bf{A}}}$-direction were shown to correspond to those of the fast wave (for helicity-free systems, such as our 2D X-point).
To perturb our system, we consider an initial condition in velocity such that:
\begin{eqnarray}
{\rm{v}}_\perp \left(x,y,t=0 \right) &=& 2C \sin \left[ \pi \left(r - 4.5 \right) \right]{\rm{\;\; for\;\;\;}}4.5\le r \le 5.5 \nonumber\;,\\
{\rm{v}}_\parallel \left(x,y,t=0 \right) &=&{\rm{v}}_z \left(x,y,t=0 \right)= 0\;\label{ICs}
\end{eqnarray}
where $r^2=x^2+y^2$ and $2C$ is our initial amplitude and initial condition (\ref{ICs}) describes a circular, sinusoidal pulse in ${\rm{v}}_\perp$. Thus, as argued above, we initially generate (only) a pure fast wave in our system. {{Note that the velocity profile prescribed by equation (\ref{ICs}) appears as the symmetric $m = 0$ mode in ${{\rm{v}}_\perp}$, but corresponds to the asymmetric $m=2$ mode in Cartesian components. This is why the first current sheet has horizontal orientation, as per \S\ref{section1.1}.}}
This initial pulse will naturally split into two waves - an outgoing wave and an incoming wave - each of amplitude $C$. In this paper, we will focus on the incoming wave, i.e. the wave propagating towards the null point. In this paper, we will conduct a parameter study of initial wave amplitude $C$. Note that setting $C=1$ recovers the results of McLaughlin et al. (\cite{McLaughlin2009}) and choosing a small value for $C$, say $C=0.001$, recovers the the linear results from McLaughlin \& Hood ({\cite{MH2004}}; i.e. see Appendix A of McLaughlin et al. \cite{McLaughlin2009}). Under our dimensionalisation, a choice of $C=1$ corresponds to an incoming wave with maximum initial {{Cartesian}} velocity ${{\rm{v}}_\perp}/r = {\rm{v}}_0/5 = 25.2 \:$km/s at a distance of $r=5L= 5\:$Mm and an equilibrium magnetic field strength of $5\:$G.
The governing equations (\ref{MHDequations}) with initial conditions (\ref{ICs}) are solved computationally in a square domain $x,y \in [-20,20]$ with a numerical resolution of $6144 \times 6144$. Zero gradient boundary conditions are applied to the variables ${\bf{B}}$, $\rho$, $\epsilon$ at the four boundaries, and ${\bf{v}}$ is set to zero on all boundaries, i.e. reflective boundaries. A numerical damping region exists for $x^2+y^2 \ge 6$ {{which gradually removes kinetic energy from the outgoing waves}} and so all oscillations that enter this region are slowly damped away, and {{hence they do not influence the behaviour about the null}}. The (equilibrium) Alfv\'en speed increases with distance from the null point and, hence, waves accelerate as they propagate outwards. Since we do not want reflected waves to influence our null point, implementation of such a damping region is essential.
\section{Aperiodic driver leading to periodic behaviour}\label{section4}
We set $C=1$ in equations (\ref{ICs}) and, as expected, we recover the results of McLaughlin et al. (\cite{McLaughlin2009}) and {{readers are directed to that paper for full details (primarily their Figures 2, 5 \& 6) and also this paper's $\S\ref{section1.1}$.}} One of the key results from McLaughlin et al. (\cite{McLaughlin2009}) was the production of a {\emph{periodic}} response resulting from an {\emph{aperiodic}} input, i.e. the physical mechanism of oscillatory reconnection naturally gave rise to periodic behaviour. This periodic response can be quantitatively measured by analysing the time evolution of the (electric) current density, specifically ${\bf{j}}= (0,0,j_z)$, at the null point itself. This can be seen in McLaughlin et al. (\cite{McLaughlin2009}, their Figure 10) and the analysis of this time signal is the primary focus of this paper. In recreating the simulations of McLaughlin et al. (\cite{McLaughlin2009}), we recover this periodic time series, but at a higher numerical resolution, $6144\times6144$, and at a higher cadence, $dt=0.05\:t_0$. The time evolution of $j_z(0,0,t)$ for $0\le t \le 480\:$s can be seen in Figure \ref{figure1}. Note that due to the symmetry of our system, the null is {\emph{always}} located at the origin.
We identify two distinct regimes in Figure \ref{figure1}: $0\le t < 90\:$s (red line in Figure \ref{figure1}) which we refer to as the {\emph{impulsive phase}} and $t \ge 90\:$s (black line in Figure \ref{figure1}) which we refer to as the {\emph{stationary phase}}. The impulsive (or transient) phase is a spiky, irregular signal, with multiple local extrema indicating: the arrival time of the shock-cusp at the null point (at $t=20.6\:$s), the formation of the first horizontal current sheet (at $t=35.7\:$s, $j_z<0$), the formation of the first vertical current sheet (at $t=59.1\:$s, $j_z>0$) and the formation of the second horizontal current sheet (at $t=90.0\:$s). The (current-sheet) signal is further contaminated due to addition of small currents related to the propagation of shock waves across the null point. We define the end of the impulsive phase as the formation time of the second horizontal current sheet, which in our $C=1$ simulation occurs at $t=90.0\:$s.
After $t=90.0\:$s, the evolution of $j_z(0,0,t)$ is much cleaner and closer to (damped) sinusoidal. For $t \ge 90.0\:$s, the extrema exactly match the formation of the cyclic current sheets, with $j_z<0\:$/$\:>0$ indicating horizontal/vertical current sheets, respectively. We define this \lq{cleaner signal}\rq{ } regime as the {\emph{stationary phase}}, i.e. the regime characterised by a (damped) sinusoidal signal after the transients of the impulsive phase have leaked away, and this phase starts at the formation time of the second horizontal current sheet. The black dotted lines in Figure \ref{figure1} indicate the formation times of all the horizontal current sheets (for both phases).
Note that in labelling these two regimes, i.e the {{impulsive phase}} and the {{stationary phase}}, we have adopted the terminology usually associated with the excitation and damping of trapped and leaky modes in coronal loop oscillations (see, e.g., Terradas et al. \cite{Terradas2005}; \cite{Terradas2006}; Luna et al. \cite{Luna2008}; McLaughlin \& Ofman \cite{MO2008} and reference therein).
\begin{figure*}[t]
\begin{center}
\includegraphics[width=6.5in]{20234FIG2.eps}
\caption{Plot of $(a)$ formation time (measured in seconds) and $(b)$ length (measured in km and evaluated at the formation time) of first horizontal current sheet versus initial velocity amplitude (measured in km/s), where ${\rm{v}}_0=25.2\:$km/s corresponds to $C=1$ simulation.}
\label{figure2}
\end{center}
\end{figure*}
\subsection{Impulsive Phase}\label{section4.1}
Let us consider the impulsive phase of the evolution of $j_z(0,0,t)$, i.e. evolution over $0\le t < 90$ (indicated by red line in Figure \ref{figure1}). We see that $j_z(0,0,t)$ is zero (since the equilibrium is potential) until the arrival of the fast oblique magnetic shock (which has its own associated current density) at the null point, indicated by the first (negative) extrema at $t=20.6\:$s. This is followed by a second (local) minimum at $t=35.7\:$s, which is the formation time of the first horizontal current sheet (i.e. not at $t=20.6\:$s). The correspondence between extrema and current-sheet formation cannot be deduced from Figure \ref{figure1} alone, but can be determined by comparison with the evolution of the separatrices and contours of ${\bf{v}}$. The value of $j_z(0,0,t=35.7)=-41.25 j_0=-3.3\:$mA is proportional to the length of the first horizontal current sheet, which is measured directly from the numerical simulation as $0.1162\: L = 116.2\:$km (for the $C=1$ simulation).
The formation time and length (at formation time) of the first horizontal current sheet is of key interest here, since this directly reflects the driver (akin to a forcing term) of the system. We investigate how the formation time and length of the first horizontal current sheet vary as functions of the initial driving amplitude (from equation \ref{ICs}). Figure \ref{figure2}a shows the formation time (measured in seconds) of the first horizontal current sheet as a function of initial velocity amplitude (measured in km/s), where ${\rm{v}}_0=25.2\:$ corresponds to $C=1$ simulation. We see that the {\emph{greater}} the driving amplitude, the {\emph{earlier}} the first current sheet forms. This is intuitive as we expect fast oblique magnetic shocks with a larger amplitude to propagate faster and thus reach the null point more rapidly, compared to waves driven with a smaller amplitude.
Figure \ref{figure2}b shows how the current sheet length (measured in km) evaluated at the formation time of the first horizontal current sheet varies as a function of initial velocity amplitude. We see that the {\emph{greater}} the driving amplitude, the {\emph{longer}} the horizontal current sheet. The length of the current sheet is also directly proportional to the value of $j_z(0,0)$, and so we also conclude that the {\emph{greater}} the driving amplitude, the {\emph{stronger}} the value of $|j_z(0,0)|$ at the corresponding time. Again, this result is intuitive; it is the fast oblique magnetic shock that physically deforms the X-point into an osculating field structure (and ultimately into a horizontal current sheet) and thus we would expect {\emph{stronger}} fast oblique magnetic shocks (i.e. with a larger amplitude) to deform, i.e. refract ${\bf{B}}$ away from the normal, and \lq{squash}\rq{} the magnetic field to a greater extent, and thus to form {\emph{longer}} and {\emph{stronger}} current sheets.
\begin{figure}
\begin{center}
\includegraphics[width=3.75in]{20234FIG3.eps}
\caption{Plot of impulsive period, i.e. time taken to evolve from the first horizontal current sheet to the second horizontal current sheet (measured in seconds) versus initial velocity amplitude (measured in km/s), where ${\rm{v}}_0=25.2\:$km/s corresponds to $C=1$ simulation.}
\label{figure4}
\end{center}
\end{figure}
\begin{figure*}[t]
\begin{center}
\includegraphics[width=6.5in]{20234FIG4.eps}
\caption{Plot of $(a)$ stationary period, i.e. the time taken to evolve from the second horizontal current sheet to the third (measured in seconds) and $(b)$ $j_{{\rm{final}}}$, i.e. $j_z(0,0)$ evaluated at $t=480\:$s, (measured in milliAmps) versus initial velocity amplitude (measured in km/s), where ${\rm{v}}_0=25.2\:$km/s corresponds to $C=1$ simulation.}
\label{figure5}
\end{center}
\end{figure*}
Finally, let us investigate the time taken to evolve from the first horizontal current sheet (at $t=35.7\:$s) to the formation time of the second horizontal current sheet (at $t=90.0\:$s for the $C=1$ simulation), namely the time taken for one complete cycle (i.e. horizontal current sheet evolves to vertical, evolves back to horizontal). This time, which we refer to as the {\emph{impulsive period}}, is calculated to be $t=54.3\:$s (for initial amplitude ${\rm{v}}_0=25.2\:$km/s in the $C=1$ system). Again, we now investigate how this impulsive period varies with the initial driving amplitude ($\propto C {\rm{v}}_0$) and this can be seen in Figure \ref{figure4}. Here, we see that the {\emph{greater}} the initial driving amplitude, the {\emph{shorter}} the resulting impulsive period, i.e. the shorter the time taken to evolve from the first horizontal current sheet to the second. Recalling the dependency seen Figure \ref{figure2}b, this means that for {\emph{longer}} current sheets, the impulsive period is {\emph{shorter}}. This means that the restoring force must be stronger and so we conclude that {\emph{longer}} current sheets have a {\emph{stronger}} restoring force. In this way, the system acts as a {\emph{harmonic oscillator}}, i.e. the greater the displacement away from equilibrium, the stronger the restoring force.
\subsection{Stationary Phase}\label{section4.2}
Let us now investigate the {\emph{stationary phase}} of the oscillation seen in Figure \ref{figure1} (black line), i.e. the time evolution of $j_z(0,0,t)$ for $t_2 \le t\le 480\:$s, where $t_2$ is the formation time of the second horizontal current sheet ($t_2=90.0\:$s for the $C=1$ simulation). Figure \ref{figure1} also has an insert showing the (same) time evolution of $j_z(0,0,t)$ for $158.9\le t\le 480\:$s, i.e. same horizontal time axis, different vertical axis. Note that the insert does not show the start of the stationary phase, only a later part of it (starting at the time of the third horizontal current sheet). The green line indicates $j_{{\rm{final}}}$, i.e. the finite amount of current density left in the system at $t=480\:$s when the system has reached its final, non-potential state. For the $C=1$ system, this final state occurs at $t=480\:$s (8 mins) and $j_{{\rm{final}}} = 0.8165j_0=6.5\times 10^{-5}\:$A.
We see that there is clear oscillatory behaviour in the stationary phase and, moreover, the oscillation is {\emph{exponentially decaying}}. This can be seen in Figure \ref{figure1} and the blue dashed lines indicate an exponentially-damped envelope ${\rm{max}}(j_z)|_{\rm{stationary}}\times e^{-\lambda t} + j_{{\rm{final}}}$ and ${\rm{min}}(j_z)|_{\rm{stationary}}\times e^{-\lambda t} + j_{{\rm{final}}}$ where $\lambda = -0.015\:{\rm{s}}^{-1}$ is determined experimentally.
Let us now investigate the period associated with the stationary phase, which we define as the time taken to evolve between horizontal current sheets. Specifically, we define the {\emph{stationary period}} as the time taken to evolve from the second horizontal current sheet to the third, i.e. the first complete oscillation within the stationary phase. For the $C=1$ simulation, these formation times are $90.0\:$s and $158.9\:$s resulting in a stationary period of $69.0\:$s (note we present results here correct to 1 decimal places, but calculate periods to a greater degree of accuracy). Similar results are obtained for alternative definitions of the stationary periods, e.g. time taken to evolve from one vertical current sheet to the next.
We now investigate how the stationary period varies with the initial driving amplitude and this can be seen in Figure \ref{figure5}a. Here, we see that the {\emph{greater}} the initial driving amplitude, the {\emph{shorter}} the resulting stationary period. Thus, these results are in agreement with those in $\S \ref{section4.1}$, i.e. the {\emph{stronger}} the initial driving amplitude, the {\emph{longer}} the resulting current sheet, thus the {\emph{stronger}} the restoring force, thus the {\emph{shorter}} the resulting period. Coupled with the exponential decay, we see that in the stationary phase, the system acts akin to a {\emph{damped harmonic oscillator}}.
We also measure all the proceeding oscillations in the stationary phase, i.e. time taken to evolve from the second/third horizontal current sheet to the third/fourth horizontal current sheet, and obtain similar periods of $67.8\:$s and $66.6\:$s respectively. Interestingly, this means that the stationary period appears to be slightly decreasing by roughly $1.8\%$ per oscillation.
Finally, we investigate $j_{{\rm{final}}}$, the finite amount of current density left in the system when the system has reached its final, non-potential state and this can be seen in Figure \ref{figure5}b. For the $C=1$ system, $j_{{\rm{final}}} = 0.8165j_0=6.5\times 10^{-5}\:$A which is measured at $t=480\:$s. In all our simulations, $j_{{\rm{final}}}>0$ indicating that the (final) X-point is very slightly closed up in the vertical direction, i.e. $j_z>0$ is associated with vertical current sheets. This is because the (local) plasma to the left and right of the X-point is slightly hotter, since that is where the initial, strongest jet heating occurs. Thus, the existence of this thermal-pressure gradient coupled with force balance requires the final state to be non-potential.
From Figure \ref{figure5}b, we see that the {\emph{stronger}} the initial driving amplitude, the {\emph{greater}} the value of $j_{{\rm{final}}}$. Again, this is intuitive: fast oblique magnetic shocks with a {\emph{larger}} amplitude will intersect to form {\emph{stronger}}, {\emph{hotter}} jets to the left and right of the X-point. Thus, this local plasma will be hotter at the end of the simulation, indicating a stronger thermal-pressure gradient and thus, in order to achieve force balance, a greater absolutely value of the Lorentz force, i.e. a {\emph{greater}} value of $j_{{\rm{final}}}$.
\section{Conclusions}\label{section:conclusions}
This paper describes an investigation into the periodicity of oscillatory reconnection, specifically oscillatory reconnection initiated by a nonlinear fast magnetoacoustic wave deforming a 2D magnetic X-point. We have solved the compressible, resistive, nonlinear MHD equations using a Lagrangian remap, shock-capturing code ({\emph{LARE2D}}) and have followed the numerical set-up of McLaughlin et al. (\cite{McLaughlin2009}). As in that paper, we find that the fast magnetoacoustic wave develops into a fast oblique magnetic shock wave which significantly deforms the local magnetic fieldlines, to the extent that the incoming wave deforms the null point into a cusp-like point which in turn collapses to a current sheet. The system then evolves periodically through a series of horizontal and vertical current sheets with associated changes in connectivity, i.e. the system demonstrates the mechanism of oscillatory reconnection.
The main focus of this paper is on the periodic nature of this oscillatory cycle of horizontal and vertical current sheets. For the first time, we identify two distinct phases in the oscillation: a transient, {\emph{impulsive phase}}, encompassing the development of the first horizontal current sheet, the formation of the first vertical current sheet, and ending with the formation of the second horizontal current sheet. We define the {\emph{stationary phase}} to begin at the formation of the second horizontal current sheet and thus this phase includes all the proceeding cyclic behaviour.
In the impulsive phase, we find that {\emph{greater}} the driving amplitude ($C{\rm{v}}_0$) of the velocity initial condition (equation \ref{ICs}), [a] the {\emph{earlier}} the first horizontal current forms, [b] the {\emph{longer}} its maximum length and [c] the {\emph{greater}} its maximum current density. These results are intuitive since we would expect magnetic shocks with larger amplitudes to propagate faster and thus arrive at the null point more rapidly than those with smaller amplitudes. We would also expect magnetic shocks with larger amplitudes to deform the pre-existing magnetic field to a greater extent (specifically to refract ${\bf{B}}$ away from the normal to a greater extent) and thus, ultimately, to form {\emph{longer}} and {\emph{stronger}} current sheets.
We also investigate the time taken to evolve from the first horizontal current sheet to the second, which we labelled as the {\emph{impulsive period}}. We find that the {\emph{greater}} the initial velocity amplitude ($C{\rm{v}}_0$) the {\emph{shorter}} the resultant impulsive period. Coupled with the results on current sheet length, this means that {\emph{longer current sheets have shorter corresponding impulsive periods}}. In this way, the system is acts as a {\emph{harmonic oscillator}}, i.e. the greater the displacement away from equilibrium, the stronger the restoring force, and thus the shorter the impulsive period.
For a driving amplitude of $25.2\:$km/s (corresponding to $C=1$ simulation) we measure an impulsive period of $54.3\:$s. We also investigate the resultant impulsive periods for driving amplitudes $6.3 - 126.2\:$km/s and find associated impulsive periods in the range $46.9 - 65.4\:$s.
The stationary phase is found to be dominated by an exponentially-decaying oscillation, tending to a finite value, $j_{{\rm{final}}}$. We also investigate the {\emph{stationary period}}, namely the time taken to evolve from the second horizontal current sheet to the third, i.e. the first complete cycle of the stationary phase. As in the impulsive phase, we find that the {\emph{greater}} the initial velocity amplitude ($C{\rm{v}}_0$) the {\emph{shorter}} the resultant stationary period. This is explained just as before, the {\emph{greater}} the initial amplitude, the longer and stronger the current sheets at each stage, and thus the greater restoring force, leading to shorter periods (compared to smaller initial amplitude, shorter resultant current sheets, weaker restoring force and thus longer periods). Hence, again the system acts as a harmonic oscillator and, coupled with the exponential decay, we relate the oscillatory reconnection mechanism to that of a {{damped harmonic oscillator}} during the stationary phase.
For a driving amplitude of $25.2\:$km/s (corresponding to $C=1$ simulation) we measure a stationary period of $69.0\:$s. We also investigate the resultant stationary periods for driving amplitudes $6.3 - 126.2\:$km/s and find associated stationary periods in the range $56.3 - 78.9\:$s, i.e. these are high frequency ($0.0127 - 0.0178\:$Hz) oscillations.
{{
It is also prudent at this stage to ask what determines this stationary period and what determines the exponentially-decaying timescale. To this end, we can consider the work of Craig \& McClymont (\cite{CraigMcClymont1991}) who investigated the relaxation of a 2D X-point disturbed from equilibrium. By neglecting both nonlinear and thermal pressure effects, Craig \& McClymont (\cite{CraigMcClymont1991}) derived an analytical prediction for two timescales:
\begin{eqnarray}
\qquad t_{\rm{oscillation}} \approx 2 \ln {R_m}\;,\quad t_{\rm{decay}} \approx t_{\rm{oscillation}}^2/ 2 \pi^2 \nonumber\;
\end{eqnarray}
where we identify $t_{\rm{oscillation}}$ as our stationary period and $t_{\rm{decay}}$ as our decay time; $1/\lambda$. For a driving amplitude of $25.2\:$km/s, these correspond to $t_{\rm{oscillation}} \approx 109.6\:$s, compared to our measured stationary period of $69.0\:$s, and $t_{\rm{decay}} \approx 76.7\:$s compared to our measured decay time of $1/\lambda = 1/0.015 = 66.7\:$s, given that in our investigations $R_m= 10^4$ and time is made dimensional using $t_0=7.93\:$s. Thus, given the (relative) simplicity of the Craig \& McClymont (\cite{CraigMcClymont1991}) system, these estimates are in fair agreement with our results. Note, however, that these simple analytical formulae cannot predict the variation in period versus amplitude of the initial velocity driver. This suggests that nonlinear effects and thermal-pressure gradients play a crucial role, which seems reasonable given that the restoring force of oscillatory reconnection has been shown to be a dynamic competition between the thermal-pressure gradients and the Lorentz force (see $\S 3.2$ of Murray et al. \cite{Murray2009}; $\S 3.3$ of McLaughlin et al. \cite{McLaughlin2009}; Figure 7 of Threlfall et al. \cite{Threlfall2012}).
}}
It is also important to note that there is a significant difference between the impulsive period (e.g. $54.3\:$s for $C=1$ system) and the stationary period (e.g. $69.0\:$s for the $C=1$ system), and that in every numerical experiment we find that the stationary period is longer than the impulsive period. This indicates that different physical processes dominate in each phase (i.e. the deformation of the X-point by the shock and the importance of jet heating in the impulsive phase, and the elastic motion of the magnetic field trying to get back to equilibrium in the stationary phase) and validates our approach of dividing the whole evolution into two distinct phases.
This difference in impulsive and stationary periodicities actually has an intriguing caveat : it is important to note that when oscillatory reconnection is seen in, say, a numerical simulation, one must be careful to interpret which phase one is actually observing {\emph{and}} to observe several oscillations, i.e. if only two periods are seen, say the impulsive period followed by a single stationary period, then one would conclude that the period was actually {\emph{increasing}} between oscillations. A similar result would pertain in solar observations of oscillatory reconnection, i.e. the first period measured would be shorter than the proceding periods (assuming the first, i.e. impulsive, period is also observed). This is a clear prediction for the oscillatory reconnection mechanism.
It is also important to note that, as shown by McLaughlin et al. (\cite{McLaughlin2012}), the mechanism periodically generates ${\rm{v}}_x$ and ${\rm{v}}_y$ but that these are {\emph{generated}} exponentially damped. Thus, if such signals are detected, then they may be decaying not due to a particular damping mechanism, but {\emph{due to the generation mechanism itself}}.
In addition to the stationary period, we also measured all the proceeding periods in the stationary phase, e.g. time taken to evolve from third horizontal current sheet to fourth, etc. Interestingly, it was found that the period very slightly decreases by roughly $1.8\%$ per oscillation. The exact reason for this decrease in the period is uncertain and may be a numerical effect. This will be investigated further in future work.
As in McLaughlin et al. (\cite{McLaughlin2009}), it was found that the final state (i.e. velocity zero, oscillatory behaviour ceased) is in force balance but is non-potential and a small, finite amount of current density exists in the system, $j_{{\rm{final}}}$. The (final) X-point is very slightly closed up in the vertical direction, i.e. $j_z>0$ is associated with vertical current sheets. This is because the (local) plasma to the left and right of the X-point is slightly hotter, since that is where the initial, strongest, jet heating occurs. Thus, the existence of this thermal-pressure gradient in force balance requires the final state to be non-potential. We find that the {\emph{greater}} the initial velocity amplitude ($C{\rm{v}}_0$) the {\emph{larger}} the value of $j_{{\rm{final}}}$. Again, this is intuitive: fast oblique magnetic shocks with a {\emph{greater}} amplitude will overlap to form {\emph{stronger}}, {\emph{hotter}} jets to the left and right of the (equilibrium) X-point. Thus, this local plasma will be hotter at the end of the simulation, indicating a stronger thermal-pressure gradient and thus, in order to achieve force balance, a greater value of the Lorentz force, i.e. a {{larger}} value of $j_{{\rm{final}}}$.
We have presented an investigation into the periodic nature of oscillatory reconnection and have found that an aperiodic driver can {\emph{naturally}} generate a period signal via the physical mechanism of oscillatory reconnection. We have found that the system behaves akin to a {\bf{damped harmonic oscillator}}. Again, this is not surprising: effectively our velocity initial condition can be thought of as injecting a {\emph{finite amount of energy}} into the oscillatory reconnection mechanism and so intuitively the resultant periodic behaviour must be {\emph{finite in duration}}, i.e. this is a {\emph{dynamic}} reconnection phenomena as opposed to the classical steady-state, time-independent reconnection models.
Oscillatory Reconnection may also play a role in generating quasi-periodic pulsations (see, e.g., reviews by Aschwanden \cite{Aschwanden2003}; Nakariakov \& Melnikov \cite{Nakariakov2009}). Oscillatory behavior has been reported in a number of solar and stellar flare observations (e.g. Mathioudakis et al. \cite{Mathioudakis2003}; \cite{Mathioudakis2006}; McAteer et al. \cite{McAteer2005}; Inglis et al. \cite{Inglis2008}; Inglis \& Nakariakov \cite{Inglis2009}; Nakariakov et al. \cite{Nakariakov2010}; Nakariakov \& Zimovets \cite{Nakariakov2011}; Inglis \& Dennis \cite{Inglis2012}; Shen \& Liu \cite{Shen2012}) but the generation mechanism responsible remains an open question.
We believe the physical mechanism of oscillatory reconnection described in this paper is a robust, general phenomenon that will be observed in other systems that demonstrate finite-duration/non-steady-state reconnection (although we have only presented a specific example of oscillatory reconnection in this paper). For example, evidence of oscillatory reconnection in 3D flux emergence simulations has been reported by Archontis et al. (\cite{VasilisOSCILLATORYRECONNECTION}).
\begin{acknowledgements}
{The authors acknowledge IDL support provided by STFC. JAM and JOT acknowledge financial assistance from the Royal Astronomical Society. The computational work for this paper was carried out on the joint STFC and SFC (SRIF) funded cluster at the University of St Andrews (Scotland, UK).}
\end{acknowledgements}
|
1,116,691,500,453 | arxiv | \section{Introduction}
The convolutional codes are widely used in digital communication systems to correct the transmission errors, which have been adopted by almost all advanced wireless communication standards. There are strong requirements to develop high performance decoding components with good scalability and reconfigurability to support various standards, which can be applied to new radio communication systems such as the Software Defined Radio (SDR) and Cognitive Radio (CR). As the Viterbi algorithm \cite{Viterbi1967} is the most popular method for decoding convolutional codes, our discussion focuses on the techniques of Viterbi decoder implementations.
Traditional communication systems mainly use Field-Programmable Gate Array (FPGA) and Application Specific Integrated Circuit (ASIC) in hardware platforms. Enormous researches of the Viterbi decoder implementation are based on FPGA/ASIC and Gb/s throughput is achieved \cite{Fettweis1996} \cite{VLSI2010}. However, these high performances are always along with expensive cost and long development cycle, and these techniques can not provide the flexibility required by SDR or CR systems. Alternative microprocessors like Central Processor Unit (CPU) are more flexible than FPGAs/ASICs. Some works on CPU-based software decoding use single-instruction multiple-data (SIMD) instruction sets to achieve parallel decoding \cite{CPU2009} \cite{CPU2010}. But restricted by their computation resources, the data processing speed and decoding throughputs are much lower.
High performance computing (HPC) on GPUs is developed rapidly over the last decade.
Compared with FPGAs/ASICs, GPU-based implementations have very good flexibility and scalability using high-level programming language. Compared with CPUs, GPUs have more massive ALU cores to ensure large-scale parallel execution, which can gain higher throughput with appropriate optimization.
A lot more works have focused on GPU-based decoding in recent years. \cite{SDR2011} \cite{TVDA2011} \cite{TVDA_WCNC2013} \cite{SDR2010} \cite{TVDA_2014} use CUDA to design Viterbi decoders for SDR systems on NVIDIA GPUs. \cite{OPENCL2014} uses opening computing language (OpenCL) to achieve accelerating Viterbi decoding on an AMD GPU. However, most of these works just simply design a parallel decoder for block-coded convolutional codes, and basic level of optimizations are presented. Compared with these works, a better parallel Viterbi decoding algorithm with lower computational complexity is proposed in this paper. Fine-grained and coarse-grained parallelism optimizations are both presented, to maximize the execution efficiency of mathematical operations, memory transactions and data transfer between the host and the device. The good generality means our parallel block-based Viterbi decoder can work for most kinds of convolutional codes, and some optimizations can also be adopted to implementations of other GPU-based decoders.
\section{The Viterbi Decoding Algorithm}
The Viterbi algorithm (VA), a maximum likelihood sequence estimator, uses the trellis to exhaustively search the sequence that is closest to received bits from channel. It consists of two procedures in two directions: the forward procedure and the traceback procedure. Three kind of units will be calculated: the path metric (PM), the branch metric (BM) and the survivor path (SP). BM is calculated to measure Hamming/Euclidean distance from the received bits to the legal codewords at each stage. PM is the accumulated distance added by BMs. SP takes a record of the path with minimum distance to each state.
Forward computing starts at stage 0 with all metrics set to zero. For each state at current stage, an add-compare-select (ACS) operation is carried out to update their PM at next stage and rewrite their SP relatively. The ACS operation can be described by equation (\ref{Eq_ACS}). $PM_n^j$ denotes the path metric of state $j$ at stage $n$. $BM_n^{i,j}$ denotes the branch metric from state $i$ at stage $n-1$ to state $j$ at stage $n$. $PM_{n - 1}^i$ and $BM_n^{i,j}$ are added up for all state $i$ connected to state $j$ and a minimum result is chosen to update $PM_n^j$.
\begin{equation}\label{Eq_ACS}
PM_n^j = \mathop {\min }\limits_i \left( {PM_{n - 1}^i + BM_n^{i,j}} \right)
\end{equation}
While the forward ACS computing finishes at the end of the data stream, a state with minimum PM should be estimated as the beginning of traceback procedure. The selected state $S_E$ is believed to be the true encoding tail state with high probability. Therefore, the traceback process goes along the final survivor path $SP_T^E$ to obtain decoded bits.
In below sections, the $(R, 1, K)$ convolutional code with code rate $1/R$ and constraint length $K$ is concerned. The number of states is denoted by $N$.
\section{Proposed GPU-based Decoder and Methods for Efficient Decoding}
\subsection{Parallel Block-based Viterbi Decoder}
The original VA is not suitable to decode the convolutional codes encoded in a stream, as a huge amount of storage resource would be required and high decoding latency would not be acceptable. Thus, we propose a parallel block-based Viterbi decoder (PBVD) based on the GPU architecture.
Fig.\ref{Fig_SBVD} shows a schematic of the decoding procedures using PBVD. A real-time constraint is introduced into the decoding procedure. A data segment from stage $t-M$ to $t+D+L$ called a parallel block (PB) consists of a truncated block, a traceback block and a decoding block. Assuming that the block to be decoded starts at stage $t$, with the length of $D$, the PBVD should start at stage $t-M$. A forward ACS procedure is carried out with unknown initial state metrics (typically set to zero). The ACS operation goes through stage $t-M$ to $t+D+L$ and survivor paths with length of $M+D+L$ are estimated and stored, so as the PMs for each state. At the end of the interval, a traceback procedure starts from a random state (state $S_0$, for example). After $L$ times traceback along a randomly picked survivor path, state $S_E$ is reached and regarded as the authentic state at stage $t+D$. Afterwards, traceback procedure would continue and the data segment from stage $t$ to $t+D$ is decoded.
\begin{figure}[b]
\centering
\includegraphics[width=3.5in]{PBVD.pdf}
\caption{The diagram of decoding procedures inside a data segment.}
\label{Fig_SBVD}
\end{figure}
Unlike the original VA, there is no state estimation between forward and backward procedure in PBVD. That means the shortest path would not need to be picked out as the unique selection for backward decoding. This simplification benefits from the traceback block, which provides $L$ stage for all survivor paths merging to an authentic state at stage $t+D$. The length $L$ is called decoding depth and typically equal to $5K$ \cite{SBVD1997}. Similarly, by a number of $M$ iterations on the truncated block, the truncation error due to the unknown initial metrics is negligible. Thus, the strong probability of successful decoding for the mid block is guaranteed.
\begin{figure}[tb]
\centering
\includegraphics[width=3.2in]{PBVD_stream3.pdf}
\caption{The diagram of parallel decoding for data stream using two individual GPU kernels. (Note that the composition of VP in K1 and K2 are different.)}
\label{Fig_SBVDstream}
\end{figure}
To decode a stream of convolutional codes, the input data could be blocked to a series of segments of length $D$. Each segment extends a length of $L$ in both sides as the truncated block and traceback block, to form a parallel block ($M$ is set equal to $L$ in the following description), so the biting length for adjacent PB is $2L$.
To achieve high decoding throughput on a GPU, these $N_t$ PBs should be decoded concurrently by two individual GPU kernels (denoted by K1 and K2) with different parallelism to match the different computational complexity of procedures in two directions. Each PB should be successively handled by GPU thread cluster in K1 and K2, which are named virtual processors (VP). After synchronization, the outputs of all VPs in K2 are finally gathered to form the decoded stream. An example of the design for stream decoding using GPU-based PBVD with $N_t = 4$ is shown in Fig.\ref{Fig_SBVDstream}.
\subsection{Optimized Parallelism for Forward ACS Computation}
Typically, the commonly used schemes for the forward ACS operations are the state-based parallel execution \cite{TVDA_WCNC2013} and the butterfly-based parallel execution \cite{TVDA_2014}. In this paper, a group-based parallel scheme is proposed by exploiting the characteristics of the trellis, to reduce the amount of branch metric computation in the forward procedure.
For a $(R,1,K)$ convolutional code, the state in the trellis is defined by the contents of the $v$ binary memory cells $D_{v-1}{\sim}D_0$ in the encoder, which can be denoted by $S_d$ and $d=(D_{v-1}D_{v-2} \cdot \cdot \cdot D_1D_0)_2$. There are $R$ filters in the encoder, the $r$th of which has impulse response ${\textbf{\emph{g}}^{(r)}} = [ {g_{K - 1}^{(r)}g_{K - 2}^{(r)} \cdot \cdot \cdot g_1^{(r)}g_0^{(r)}} ]$, called the generator polynomials. $\textbf{\emph{c}}(S_d, x) = [ {{c^{(1)}}{c^{(2)}} \cdot \cdot \cdot {c^{(R)}}}]$ is used to express the encoder output corresponding to input bit $x$ at state $S_d$. $c^{(r)}$ is the output of the $r$th filter, 0 or 1, which can be calculated by:
\begin{equation}\label{Eq_cr}
{c^{(r)}} = (x \cdot g_{K - 1}^{(r)}) \oplus ({D_{K - 2}} \cdot g_{K - 2}^{(r)}) \oplus \cdot \cdot \cdot \oplus ({D_0} \cdot g_0^{(r)})\\
\end{equation}
All operations $\oplus$ are module-2 additions in field GF(2). Consider a butterfly structure from the trellis,
the contiguous states $S_{2j}$ and $S_{2j+1}$ in $j$th butterfly ($j = 0,1,2,...,N/2-1$) would like to shift to the states $S_j$ or $S_{j+2^{v-1}}$ for different input bits. $\bm{\alpha}$ and $\bm{\beta}$ denote the output of encoder at state $S_{2j}$ with input bit $x=0$ and 1 respectively. So as the $\bm{\gamma}$ and $\bm{\theta}$ for state $S_{2j+1}$. The $r$th bit ${\alpha}^{(r)}$ in $\bm{\alpha} = [ {{{\alpha}^{(1)}}{{\alpha}^{(2)}} \cdot \cdot \cdot {{\alpha}^{(R-1)}}{{\alpha}^{(R)}}}]$ can be obtained by:
\begin{align}\label{Eq_alpha}
{{\alpha}^{(r)}} &= c^{(r)}(S_{2j},0) \notag\\
&= (x \cdot g_{K - 1}^{(r)}) \oplus \cdot \cdot \cdot \oplus ({D_1} \cdot g_1^{(r)}) \oplus ({D_0} \cdot g_0^{(r)}) \notag\\
&= (0 \cdot g_{K - 1}^{(r)}) \oplus \cdot \cdot \cdot \oplus ({D_1} \cdot g_1^{(r)}) \oplus (0 \cdot g_0^{(r)}) \notag\\
&= ({D_{K - 2}} \cdot g_{K - 2}^{(r)}) \oplus \cdot \cdot \cdot \oplus ({D_1} \cdot g_1^{(r)})
\end{align}
Similarly, ${\beta}^{(r)}$, ${\gamma}^{(r)}$ and ${\theta}^{(r)}$ can be obtained as follows:
\begin{align}
\label{Eq_beta}
{{\beta}^{(r)}} &= c^{(r)}(S_{2j},1) = g_{K - 1}^{(r)} \oplus {\alpha}^{(r)} \\
\label{Eq_gamma}
{{\gamma}^{(r)}} &= c^{(r)}(S_{2j+1},0) = {\alpha}^{(r)} \oplus g_0^{(r)}\\
\label{Eq_theta}
{{\theta}^{(r)}} &= c^{(r)}(S_{2j+1},1) = g_{K - 1}^{(r)} \oplus {\alpha}^{(r)} \oplus g_0^{(r)}
\end{align}
From equation (\ref{Eq_alpha}) to (\ref{Eq_theta}) we can conclude that for given generator polynomials, once the $\bm{\alpha}$ is established, other outputs $\bm{\beta}$, $\bm{\gamma}$ and $\bm{\theta}$ in the butterfly would be uniquely derived. Therefore, all the $N/2$ butterflies in the $N$-state trellis can be classified to $2^R$ (denoted by $N_c$) groups. The groups are distinguished by $\bm{\alpha}$, which means that butterflies in the same group have the same branch metrics at one stage.
As a result, for the $N / N_c$ states in the same group, only four branch metrics need to be calculated, to update the $N / N_c$ path metrics. Thus, the total computation of branch metrics for all the ACS operations at one stage can be calculated as $2^{R+2}$. For the widely used convolutional codes which have $R=2$ and $K=5,7,9$, or $R=3$ and $K=7,9$, the forward ACS operations can be accelerated due to lower computation of branch metrics than state-based or butterfly-based parallelism scheme ($2^{R+2} < 2^K$).
\section{Framework of Kernels and Memory Organization on GPU}
\subsection{Kernel Execution and Thread Mapping Strategies}
\renewcommand\arraystretch{1.1}
\begin{table}[bp]
\centering
\caption{Thread Dimensions and Execution Parallelism of Two Kernels}
\label{Tab_Parallelism}
\begin{tabular}{ccccc}
\hline
\multirow{2}{*}{Kernel} & \multicolumn{2}{c}{Thread dimension} & \multicolumn{2}{c}{Parallelism}\\
\cline{2-5}
& BlockDim & \multicolumn{1}{c|}{ThreadDim} & Inter-frame & Intra-frame \\
\hline
K1 & $N_{bl}$ & \multicolumn{1}{c|}{$32N_c$} & $32N_{bl}$ & $N_c$ \\
K2 & $N_{bl}/N_c$ & \multicolumn{1}{c|}{$32N_c$} & $32N_{bl}$ & $1$ \\
\hline
\end{tabular}
\end{table}
In our GPU-based implementation, two individual kernels K1 and K2 with different thread dimensions are initiated. K1 finishes the forward computing, followed by K2 which carries out the traceback and decoding procedures. To describe the thread organizations in kernels, blockDim and threadDim are used to represent the number of threadblocks and the number of threads in each threadblock. In K1, the group-based parallel execution mode is employed. For the forward computing of a PB, all the $N$ states will be sorted to $N_c$ groups using the given criteria. Then for each group, a thread is dispatched to calculate four (or two in special) branch metrics to update all the path metrics and survivor paths at each stage. Thus, $N_c$ threads are required to build a virtual processor in K1. Considering that 32 CUDA threads are managed cooperatively in batches called a warp, a threadblock in K1 is regulated to accommodate 32 virtual processors. That means the threadDim of K1 is $N_c$ times the warp size.
\begin{algorithm}[tb]
\caption{Parallel block-based Viterbi decoding algorithm}
\label{Alg_PBVD}
\begin{algorithmic}[1]
\item[\algorithmickernelone]
\textbf{Forward procedure}
\FOR {thread block $b=0$ to $N_{bl} -1$, warp $w=0$ to $N_c-1$ and thread $t=0$ to $31$ \textbf{parallel}}
\FOR {stage $s=0$ to $D+2L-1$}
\STATE $sp = 0$, $tid= b \times 32 + t$;
\STATE Load input symbol and calculate four branch metrics;
\FOR {\textbf{all} $j \in Group(w)$}
\STATE Load: $pm_1 = {\rm PM}[2j][t]$, $pm_2 = {\rm PM}[2j+1][t]$;
\STATE $reg[j] = min( pm_{1} + BM_{\bm{\alpha}}, pm_{2} + BM_{\bm{\gamma}} )$;
\STATE take a bitwise record in $sp$ for state $j$;
\STATE $reg[j+2^{K-2}] = min( pm_{1} + BM_{\bm{\alpha}}, pm_{2} + BM_{\bm{\gamma}} )$;
\STATE take a bitwise record in $sp$ for state $j+2^{K-2}$;
\ENDFOR
\STATE Store: ${\rm PM[\ast][t]} = reg[\ast]$, ${\rm SP}[s][w][tid] = sp$;
\ENDFOR
\ENDFOR
\item[\algorithmickerneltwo]
\textbf{Backward procedure}
\FOR {thread block $b=0$ to $N_{bl} / N_c - 1$, warp $w=0$ to $N_c-1$ and thread $t=0$ to $31$ \textbf{parallel}}
\STATE $i=j=g=state=0$, $tid=b \times N_c \times 32 + w \times 32 + t$;
\FOR {stage $s=D+2L-1$ to $L$}
\STATE Obtain $i$ by $state$ from lookup tables;
\FOR {$g=0$ to $N_c-1$}
\STATE Load ${\rm SP}[s][g][tid]$ and store into $sp$;
\ENDFOR
\IF {$s \leq D+L-1$}
\STATE Output decoded bit: $(state>>(K-2)) \& 0x01$;
\ENDIF
\STATE $j = state \% 2^{K-2}$, $sp = (sp >> i) \& 0x01$;
\STATE $state = 2 \times j + sp$;
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
In the second kernel K2, as the backward procedure is a completely serial processing that can not be executed in parallel, only one thread is enough to constitute the virtual processor in K2. For convenience narration, we let the threadDim of K2 equal to K1, so that each threadblock in K2 contains $32 \times N_c$ virtual processors. If we allocate $N_{bl}$ threadblocks in K1, the total number of PBs $N_t$ should be equal to $32 \times N_{bl}$. Thus, to handle the $N_t$ PBs simultaneously, $N_{bl} / N_c$ threadblocks should be allocated in K2. Inter-frame parallelism and intra-frame parallelism are introduced to indicate the number of virtual processors in each kernel and the number of threads each virtual processor contains, respectively. Table \ref{Tab_Parallelism} gives a summary about the thread dimensions and execution parallelism of K1 and K2.
\begin{figure}[tb]
\centering
\includegraphics[width=3.4in]{Coalesced_access.pdf}
\caption{The diagram of coalesced memory accesses for survivor paths.}
\label{Fig_Coalesced}
\end{figure}
\subsection{Memory Organization for Various Information}
In the parallel block-based Viterbi decoder, there are several kinds of data information: (i) the input/output data streams, which can only be stored in the off-chip global memory as they need to be exchanged from the host machine; (ii) the cumulative path metrics and the branch metrics, which are only updated in the forward procedure, so on-chip register resources and shared memory can be used under the conditions of enough capacity; (iii) the survivor paths, which are generated in forward procedure and fetched in decoding phase, that they can only be placed in global memory and designed to meet the alignment requirement for coalesced memory access of two individual kernels.
It is a challenge to design a suitable data structure for the survivor paths due to the different intra-frame parallelism in K1 and K2. Once the coalesced memory access is satisfied in one of the two kernels, the memory transactions in the other kernel would face to horrible inefficiencies.
To solve this inconsistency, an optimized construction is exploited in Fig.\ref{Fig_Coalesced}. At the current stage, states from 32 PBs are gathered and reordered. All states would be collected to $N_c$ groups followed by the state classification criteria. These $N_c$ groups of states are mapped to different warps allocated in K1. Inside a group, $N/N_c$ states from the same PB are processed in order by the same thread, which means threads with the same threadIdx.x from these $N_c$ warps make up a virtual processor in K1. As the survivor path is a record of selected forward path which can be presented by bit data (for example, bit 0 denotes the upper branch, and bit 1 denotes the lower branch), these $N/N_c$ results can be stored by bit in a same unit as:
\begin{equation}\label{Eq_bit}
{\rm SP}[ x ][ y ][ z ] = \underbrace {1101 \cdot \cdot \cdot 01}_{(N/{N_c})bits} \notag
\end{equation}
As a result, the survivor paths should be allocated as ${\rm SP}[D+2L][N_c][N_t]$
to ensure coalesced access for contiguous PBs inside a warp. For each backward stage in K2, $N_c$ individual results are merged because only one warp is needed for the backward phase of these 32 PBs. For a single thread in this warp, all survivor path messages from a PB are loaded with $N_c$ memory requests, but all in the form of aligned transaction. After all, the memory requests in both K1 and K2 are managed without duplicate transactions and extra time overhead.
The shared memory are allocated based on thread blocks and threads with the same threadIdx.x in different warps need to swap data to jointly accomplish the forward phase for a PB. To avoid the bank conflict in shared memory transactions, the data structure should be devised as ${\rm PM}[N][32]$
to ensure that the accesses for path metrics with the same state id are aligned and fall into individual shared memory banks. As a result, for each shared memory store/load instruction, no transaction for the same request replays and maximum bandwidth utilization is reached.
Remarkably,
additional registers are necessary as the temporary places to store the updated results for path metrics, and shared memory store transactions would not be carried out until all the calculations at a stage are finished.
\subsection{Asynchronous Data Transfer and Throughput Analysis}
The time overhead of data transfer between host and device should be taken into account when evaluating a GPU-based decoder. CUDA supports asynchronous streams technique to achieve the overlap for data transfer tasks and kernel launches in different streams. The decoder should activate a suitable number of CUDA streams and arrange tasks to the idle streams consecutively to ensure the high occupancy of the GPU device.
For our GPU-based Viterbi decoder, the H2D messages are blocked input data streams and D2H messages are decoded bits. A kernel throughput $S_k$ is introduced to evaluate the kernel execution efficiency and it can be obtained by $\frac{D \times N_{t}}{T_k}$
where $T_k$ is kernel execution time. For the H2D data transfer, a parameter $U_1$ is defined to indicate the number of bytes for an input symbol storage. Similarly, a parameter $U_2$ is defined to indicate the number of bytes for the storage of a decoded bit in D2H data transfer. Thus, the time cost of H2D and D2H transfer can be calculated by: $T_{\rm H2D} = \frac{(D+2L) \times N_{t} \times U_1}{B}$ and $T_{\rm D2H} = \frac{D \times N_{t} \times U_2}{B}$,
where $B$ denotes the PCI-E bandwidth. To hide data transfer latency, $N_s$ CUDA streams can be allocated (in each stream, $N_t$ parallel blocks are arranged).
Ideally, all the data transfer batches can be completely hidden by the kernel executions, besides the first H2D batch and the last D2H batch. Thus, the decoding throughput can be approximately calculated by :
\begin{align}\label{Eq_TP}
{\rm T/P} &\approx \frac{D \times N_t \times N_s}{T_{\rm H2D} + \sum{T_k} + T_{\rm D2H}} \notag\\
&\approx \frac{B \times N_s}{(1+2L/D) \times U_1 + N_s/S_k + U_2}
\end{align}
Notice that the approximation $\sum{T_k} \approx N_s \times T_k$ can be used, though the concurrent kernel execution (CKE) technique or the Hyper-Q technique in CUDA may be applied.
\renewcommand\arraystretch{0.9}
\begin{table}[bp]
\centering
\caption{Classification of states for a (2, 1, 7) convolutional code}
\label{Tab_Group}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Group & $\bm{\alpha}$ & $\bm{\beta}$ & $\bm{\gamma}$ & $\bm{\theta}$ & Index of states\\%\tnote{*}\\
\hline
\multirow{2}{*}{0} & \multirow{2}{*}{00} & \multirow{2}{*}{11} & \multirow{2}{*}{11} & \multirow{2}{*}{00} & \multicolumn{1}{l|}{0, 1, 4, 5, 24, 25, 28, 29, 42, 43}\\
& & & & & \multicolumn{1}{l|}{46, 47, 50, 51, 54, 55}\\\hlin
\multirow{2}{*}{1} & \multirow{2}{*}{01} & \multirow{2}{*}{10} & \multirow{2}{*}{10} & \multirow{2}{*}{01} & \multicolumn{1}{l|}{2, 3, 6, 7, 26, 27, 30, 31, 40, 41}\\
& & & & & \multicolumn{1}{l|}{44, 45, 48, 49, 52, 53}\\\hlin
\multirow{2}{*}{2} & \multirow{2}{*}{11} & \multirow{2}{*}{00} & \multirow{2}{*}{00} & \multirow{2}{*}{11} & \multicolumn{1}{l|}{8, 9, 12, 13, 16, 17, 20, 21, 34}\\
& & & & & \multicolumn{1}{l|}{35, 38, 39, 58, 59, 62, 63}\\\hlin
\multirow{2}{*}{3} & \multirow{2}{*}{10} & \multirow{2}{*}{01} & \multirow{2}{*}{01} & \multirow{2}{*}{10} & \multicolumn{1}{l|}{10, 11, 14, 15, 18, 19, 22, 23, 32}\\
& & & & & \multicolumn{1}{l|}{33, 36, 37, 56, 57, 60, 61}\\%5, 7, 9, 11, 16, 18, 28, 30
\hline
\end{tabular}
\end{table}
To improve the decoding throughput, one way is to make the kernels operate efficiently by the approaches in above sections, to achieve a higher $S_k$. Another way is to develop suitable methods of the storage for input/output messages, to reduce $U_1$ and $U_2$. For a soft-decision decoding over the AWGN channel, received symbols should be converted to soft messages and stored by several bits.
In fact, a $q$-bit fixed-point quantization scheme can be designed and $\lfloor 32/q \rfloor$ messages can be packed and stored into a same integer unit. As a result, the value $U_1$ decreases from $4R$ to $4R/\lfloor 32/q \rfloor$. For the storage of decoded bits,
we can use a similar packing scheme to store each decoded bit by bitwise operations. In this case, a character type can store 8 individual decoded bits that reduce $U_2$ to $1/8$.
\renewcommand\arraystretch{1.0}
\begin{table*}[htbp]
\centering
\caption{Time consumption and throughput of original and optimized decoder under different devices and various Parallelism}
\label{Tab_R2}
\begin{tabular}{cccccccc|ccccccc}
\hline
\multirow{2}{*}{Device} & \multirow{2}{*}{$N_{bl}$} & \multirow{2}{*}{$N_t$} & \multicolumn{5}{c|}{Original results} & \multicolumn{7}{c}{Optimized results}\\
\cline{4-15}
& & & $T_k$ & $T_{\rm H2D}$ & $T_{\rm D2H}$ & $S_k$ & T/P(1S) & $T_{k1}$ & $T_{k2}$ & $T_{\rm H2D}$ & $T_{\rm D2H}$ & $S_k$ & T/P(1S) & T/P(3S)\\
\hline
\multirow{5}{*}{GTX580} & 64 & 2048 & 2.914 & 1.532 & 0.636 & 359.8 & 181.5 & 1.443 & 0.611 & 0.377 & 0.023 & 509.5 & 403.4 & 508.3\\
& 128 & 4096 & 5.811 & 2.968 & 1.280 & 362.9 & 185.4 & 3.046 & 0.859 & 0.747 & 0.043 & 571.4 & 446.4 & 547.7\\
& 192 & 6144 & 8.514 & 4.506 & 1.969 & 368.0 & 189.1 & 4.050 & 1.232 & 1.155 & 0.063 & 594.5 & 472.2 & 571.0\\
& 256 & 8192 & 11.361 & 5.986 & 2.556 & 368.2 & 189.3 & 5.250 & 1.456 & 1.571 & 0.082 & 628.7 & 498.4 & 590.0\\
& 320 & 10240 & 14.224 & 7.502 & 3.192 & 369.6 & 189.4 & 6.513 & 1.807 & 1.893 & 0.101 & 641.8 & 504.9 & 598.3\\ \hline
\multirow{5}{*}{GTX980} & 64 & 2048 & 1.681 & 0.865 & 0.325 & 620.6 & 294.7 & 0.591 & 0.377 & 0.261 & 0.012 & 1082.5 & 764.9 & 1243.5\\
& 128 & 4096 & 3.232 & 1.771 & 0.652 & 647.1 & 298.6 & 0.840 & 0.386 & 0.454 & 0.023 & 1575.4 & 1051.4 & 1623.7\\
& 192 & 6144 & 4.831 & 2.684 & 0.981 & 650.8 & 304.9 & 1.172 & 0.392 & 0.678 & 0.032 & 2005.2 & 1253.0 & 1767.5\\
& 256 & 8192 & 6.436 & 3.613 & 1.333 & 652.3 & 308.8 & 1.568 & 0.414 & 0.896 & 0.042 & 2116.8 & 1290.6 & 1785.2\\
& 320 & 10240 & 8.034 & 4.334 & 1.657 & 652.5 & 309.1 & 1.899 & 0.523 & 1.102 & 0.052 & 2122.7 & 1324.7 & 1802.5\\
\hline
\end{tabular}
\begin{tablenotes}
\item $T_k$, $T_{\rm H2D}$ and $T_{\rm D2H}$ are in ms. $S_k$ and T/P are in Mbps.
\end{tablenotes}
\end{table*}
\section{Experimental Results and Discussions}
The experimentations are carried out on Intel i7-4790k platform with NVIDIA GTX580 (1544MHz, 512 CUDA cores, and PCI-E 2.0 supported) and Nvidia GTX980 (1126MHz, 2048 CUDA cores, and PCI-E 3.0 supported). The program
are complied with GCC 4.8.2 and CUDA 6.5.
A (2,1,7) convolutional code with generator polynomials $\textbf{\emph{g}}^{(1)}=[1111001]$ and $\textbf{\emph{g}}^{(2)}=[1011011]$ is chosen from CCSDS standard \cite{CCSDS} for convenient comparison with other works. As the code rate is 1/2, the 64 states can be divided into $2^2=4$ groups using the given classification methods, and the result is shown in Table \ref{Tab_Group}. The BER performance under AWGN channel for various $L$ are presented in Fig.\ref{Fig_BER} ($D$ is fixed to 512, which is an less important factor). It is shown that as $L$ rises to 42, which is about 6 times the constraint length, the BER result is approximate to the theoretical performance. Actually, in the proposed decoder, larger $L$ results in better error correction performance, but too large $L$ can cut down the decoding throughput. Thus, $D=512$ and $L=42$ are selected for the parallel block in the following tests.
\begin{figure}[tb]
\centering
\includegraphics[width=3.0in]{BER_black.pdf}
\caption{BER performance of the (2,1,7) convolutional code. ($D=512$, 8-bit quantization)}
\label{Fig_BER}
\end{figure}
To demonstrate the improvements by using the proposed strategies and methods, experimental results of both the original decoder and the optimized decoder are given for comparison in Table \ref{Tab_R2}, including the kernel execution times, the data transfer times, the kernel throughput and the decoding throughput. The proposed decoder is operated on different GPU devices with various numbers of $N_{bl}$ and $N_t$. The original parallel block-based Viterbi decoder launches only one kernel to finish the whole decoding procedure. 32-bit float-point quantization is used for the input soft messages, and decoded bits are stored in integers. In the optimized decoder, two kernels with different number of threads are launched and execution times $T_{k1}$ and $T_{k2}$ are recorded individually. It can be seen that the total execution times are reduced significantly by at least 40\%, which results in an improvement of kernel throughput $S_k$. Input messages are quantized to 8-bit, which are stored using the packing scheme, and bitwise storage is designed for decoded bits. As a result, the H2D/D2H data transfer sizes are both cut down and $T_{\rm H2D}$/$T_{\rm H2D}$ are greatly shorted to improve the decoding throughput (T/P). To hide data transfer latency, asynchronous transfer technique is adopted and throughput results with three CUDA streams (3S) are presented. By comparing with the performance under the synchronous mode which only uses one CUDA stream (1S), it shows that the more powerful the GPU is, the more efficient overlap and more throughput improvement become. Futhermore, as the increase in the number of concurrently executed parallel blocks $N_t$, the GPU will finally run at full capacity and the decoder will reach the peak throughput.
Table \ref{Tab_R3} shows the decoding throughput comparison between our work and existing works on various GPU platforms, which are all for convolutional codes with code rate 1/2 and constraint length 7. A metric named TNDC (Throughput under Normalized Decoding Cost) introduced in \cite{TNDC} is provided in order to make fair comparison. As the normalized results show, the proposed decoder achieves about 1.5$\sim$9.2 times speedup compared with the existing GPU-based implementations.
In addition, compared with the existing fastest x86-CPU work \cite{CPU2010}, which runs a 64-state VA decoder on the Intel Core 2 Extreme X9650 (4 cores, 3.0GHz) at the speed of 60Mbps, our results show significant throughput advantages. Compared with the newest results on FPGA platforms, e.g., 865Mbps for a 64-state VA decoder on Stratix III 340 (216MHz) \cite{FPGA2014} and 10Gbs for a 32-state VA decoder on Xilinx Virtex 7 XC7VX690T-2 \cite{FPGA2015}, our results reach a comparable speed, and the good scalability and compatibility make it easy to transplant our decoder onto future powerful GPU devices to achieve higher performance.
\renewcommand\arraystretch{1.1}
\begin{table}[htb]
\centering
\caption{Decoding throughput comparison with existing works}
\label{Tab_R3}
\begin{tabular}{ccccc}
\hline
Work & Device & T/P(Mbps) & TNDC & Speedup\\
\hline
\cite{SDR2011} & GTX275 & 28.7 & $ \approx $0.085 & $\times$9.20\\%\hline
\cite{TVDA2011} & 8800GTX & 29.4 & $ \approx $0.170 & $\times$4.60\\%\hline
\cite{TVDA_WCNC2013} & GTX580 & 67.1 & $ \approx $0.085 & $\times$9.20\\%\hline
\cite{SDR2010} & 9800GTX & 90.8 & $ \approx $0.420 & $\times$1.86\\%\hline
\cite{OPENCL2014} & HD7970 & 391.5 & $ \approx $0.207 & $\times$3.78\\%\hline
\multirow{2}{*}{\cite{TVDA_2014}} & Tesla C2050 & 240.9 & $ \approx $0.468 & $\times$1.67\\
& GTX580 & 404.7 & $ \approx $0.512 & $\times$1.53\\\hline
\multirow{2}{*}{This work} & GTX580 & 598.3 & $ \approx $0.757 & $\times$1.03\\
& GTX980 & 1802.5 & $ \approx $0.782 & $\times$1.00\\
\hline
\end{tabular}
\end{table}
\section{Conclusion}
This paper introduces a parallel block-based Viterbi decoder. The data stream is divided to a series of parallel blocks for concurrently decoding. Implementation on GPU uses two individual kernels mapping to two decoding phases, and optimized parallelism inside kernels are presented, which are based on the proposed state classification criteria. Aiming to accelerate the decoding, appropriate GPU memory and data structure are developed for intermediate messages. Storage for input/output data are designed and multiple CUDA streams are used to reduce the overhead of data transfer. Experimental results show that proposed GPU-based decoder achieves about 1.5 times speedup than the existing fastest work on GPU. The proposed decoding architecture can be used in the software-defined radio systems, as a flexible Viterbi decoding unit with strong reconfigurable ability.
\section*{Acknowledgment}
This work was supported by the National Natural Science Foundation of China (91438116).
\bibliographystyle{IEEEtran}
|
1,116,691,500,454 | arxiv | \section{Introduction}
The study of metal poor, compact star forming galaxies was initiated
by \citet{sarsearl72}. Their importance was rapidly recognized because they were
considered to be ideal benchmarks in the study of the earliest stages of galaxy
evolution, and were studied in many works
(\citet{campbell88,pagel92,french80}, among many others).
While these nearby systems were originally thought to be genuinely young
systems, experiencing their very first star forming episodes, even the
most metal poor objects in the local universe have been found to posses
significant underlying evolved stellar populations,
e.g., \citep{papa96,taylor95}. In fact, very deep Hubble Space Telescope
images of the most metal-poor object known, I Zw 18, indicate that it has formed
the bulk of its present stellar population 0.5--1.0 Gyr ago \citep{aloisitosi99}.
Other works on the stellar populations of this and similar galaxies yield
similar results \citep{papa02,aloisi07}. A motivation to
examine samples at higher redshifts is to locate truly young systems
that could be experiencing their first major
star formation episodes. Some exploratory steps in this direction were taken by
\citet{hoyoskoo05}, who found that distant ($z \sim 0.7$), compact star
forming galaxies showing the [\textsc{OIII}]$\lambda$4363 line deviated from the
usual Luminosity-Metallicity relationships, implying that these distant, metal
poor systems might turn out to be truly young systems. One of the aims of this paper is
to further explore this hypothesis.
Knowledge of the metallicity of external galaxies is crucial for galaxy evolution theories
because it is a direct result of the integral history of the star formation
and mass assembly of galaxies. The metallicity of galaxies has been traditionally
used in conjunction with the luminosity to build the luminosity-metallicity relation (LZR)
see, e.g., \citet{lequeux79,rich_call95}. This relationship is thought to arise from
depth of the gravitational well of massive galaxies which does not
allow newly created metals being cast into the ISM amidst
hot supernovae ejecta, where it is assumed that the more luminous galaxies are also the most
massive, and have a higher star formation efficiency. In this model,
lower mass systems will not be able to keep their heavy elements and
will probably be
of lower metallicity. Galaxies in cluster environments undergo other processes, which greatly complicate the picture.
All these phenomena will naturally create a complex metallicity distribution for galaxies at various redshifts, of which only a few data points
at intermediate redshifts are precisely known.
This paper addresses the star formation history of galaxies by
examining the properties of a sample of star forming galaxies
showing the [\textsc{OIII}]$\lambda 4363$ line.
The [\textsc{OIII}]$\lambda$4363 auroral line is generated by the
collisionally excited transition (2p$^{2 1}$D to 2p$^{2 1}$S)
of doubly ionised oxygen atoms. This line is key in the study of the metallicity of the ionised phase
of the interstellar medium (ISM) since it allows determining the electron temperature of ionised regions
without any previous assumption on the metal content of the observed nebula \citep{ost89}.
Unfortunately, the [\textsc{OIII}]$\lambda$4363 line is difficult to
detect for several reasons:
(i) This line is stronger in low metallicity clouds. In higher metallicity environments, it is
exponentially depressed because the increased cooling leaves no energy to excite oxygen atoms to the upper level
of this transition. (ii) This line is also stronger in young and strong starbursts. Older starbursts (older than 5--6 Myr)
are unable to maintain a large fraction of oxygen atoms doubly ionised. (iii) This line is also stronger in systems in which the
contribution to the continuum from the underlying stellar population is less important relative to the contribution
from the newly created ionising stars. The latter case is the one where
this weak line is less likely to be obliterated by
the continuum noise. In \citet{hoyosdiaz06}, it was shown that all these issues affect the detection
of the [\textsc{OIII}]$\lambda$4363 \AA\ line for the case of local \textsc{HII} galaxies.
The discussion above explains why the majority of the most accurate metallicity determinations for intermediate redshift sources
have been obtained using measurements of the auroral oxygen lines in galaxies with strong starbursts. At the same time, it also explains
why these measurements may not be fully tracing the complete metallicity distribution at each redshift, since the upper end of this distribution is increasingly difficult
to trace accurately through the use of the oxygen lines. Furthermore,
even in the $12+\log O/H \sim 8.10-8.35$ regime, the oxygen-based
strong line calibrators such as $\mathrm{R}_{23}$ allow two solutions, a problem difficult to overcome.
An alternative is to use [\textsc{SIII}]$\lambda \lambda 9096, 9532$,
suggested for instance by \citet{2006A&A...449..193P}, which does
not exhibit the degeneracies that $\mathrm{R}_{23}$ has at intermediate metallicities. Finally, it is also possible
to determine electron temperatures through the detection of [\textsc{SIII}]$\lambda 6312$ line up to at least
solar metallicities and then derive good abundances for high-metallicity, vigorously star forming systems.
This has been shown in \citet{1994A&A...282L..37K}, \citet{2005A&A...441..981B}, \citet{2000MNRAS.318..462D}, and \citet{2002MNRAS.329..315C}.
One disadvantage in the use of sulfur lines is that they shift
into the NIR for objects at even moderate redshifts. However, the work
of \citet{loschinos2013} shows that this is being addressed with new
IR instrumentation. The latter work presents a study on the nature of
the [\textsc{SIII}]$\lambda \lambda 9096, 9532$ emitters,
showing that these objects are usually star forming systems where a Compton-thick AGN has little or no effect in exciting the sulfur lines.
The work presented in \cite{hoyoskoo05} also investigated the differences between star forming systems with and without the [\textsc{OIII}]$\lambda$4363 \AA\ line
using intermediate redshift ($z\simeq0.7$) galaxies observed by the DEEP2 survey. Compact, star forming
galaxies showing the [\textsc{OIII}]$\lambda 4363$ auroral line have lower metallicities and higher emission
line equivalent widths than objects without this feature, in spite of their
H$\beta$ line luminosities not being lower. This was also found by \citet{hoyosdiaz06}.
The underlying stellar populations of galaxies presenting the [\textsc{OIII}]$\lambda$4363 are
less luminous relative to their newly created ionising populations,
when compared to other star forming galaxies \citep{hoyosdiaz06}.
In this paper, \S \ref{ods} describes the observations and sample
selection, \S \ref{resultaos} shows our metallicity calculations based
on electron temperatures.We compare our results to other recent works dealing with both local and intermediate
redshift sources in \S \ref{discutir}. This section also
summarizes the paper. We use $(\Omega_{M},\Omega_{\Lambda},h_{70})=(0.3,0.7,1)$. Magnitudes are
given in the $AB$ system.
\section{Observations and Sample Selection.}
\label{ods}
The data used in this work were taken for the second phase of the Deep Extragalactic Evolutionary Probe survey
(DEEP2\footnote{See \texttt{http://deep.ucolick.org/}}, \citet{deep2}).
This survey uses the DEIMOS\footnote{See \texttt{http://www2.keck.hawaii.edu/inst/deimos/}} \citep{deimospaper} spectrograph on the W.M. Keck telescope.
DEEP2 is a densely sampled, high precision redshift survey which, thanks to a
$BRI$ colour pre-selection criterion, preferentially targets galaxies in
the $0.7<z<1.4$ redshift range in three of the four areas of the sky
it covers.
It has collected a grand total of 53000 spectra, measuring 38000 reliable redshifts.
The DEIMOS instrument was used with a 1200$\mathrm{mm}^{-1}$ grating
centered at 7800\AA, thus covering on average the 6500\AA-9100\AA \
wavelength range. The wavelength range shows slight changes that
depend on the slit position on the mask. The
resolving power ($R \simeq 5000$) allows separating the [\textsc{OII}]$\lambda\lambda3726,3729$ doublet.
Because of the relatively high resolution, the DEEP2 spectra also
yield accurate velocity dispersions for emission line objects.
We select galaxies showing the strong oxygen emission lines
[\textsc{OII}]$\lambda3727$, [\textsc{OIII}]$\lambda4959$ and that have
a reliable detection of the weak auroral line [\textsc{OIII}]$\lambda4363$
in their spectra. There is no need to include the [\textsc{OIII}]$\lambda5007$ in
the selection criteria since its intrinsic emission is linked to that of the [\textsc{OIII}]$\lambda4959$ line and
its reddening-corrected flux can be found as $I_{5007}=2.98 \times I_{4959}$ for normal star forming systems.
The combination of these requirements limit the redshift range to $0.69<z<0.88$, which includes
$\sim 11000$ potential candidates.
This initial culling was made possible by the Weiner, B. J. \textit{priv. comm.} redshifts and
equivalent width measurements.
The remaining spectra were visually inspected
for reliable detections of the auroral [\textsc{OIII}]$\lambda4363$ line, and to
ensure the [\textsc{OII}]$\lambda3727$ doublet is cleanly
separated. This allows measuring the electron density accurately, without any previous assumptions.
We measured the emission line parameters measured using the \textsc{IRAF} tasks \textsc{ngaussfit} and \textsc{splot}.
Our final sample consists of 22 sources, or about 0.2\% of the potential candidates in the relevant redshift window.
Figure \ref{fig1} shows two example spectra of the final sample.
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.50]{Plots/z74_3.eps} & \includegraphics[scale=0.50]{Plots/z79_3.eps} \\
\end{tabular}
\caption{Rest frame spectra of two metal-poor galaxies - DEEP2\_130016475 at $z=0.747$
(left panel) and DEEP2\_31046514 at $z=0.789$ (right panel). Both show the temperature sensitive
[\textsc{OIII}]$\lambda 4363$ line used to select the sample and the other oxygen
emission lines used to measure the ionic and total abundances. Because
of the slit location on the mask, the spectrum in the left panel is
cutoff at a rest-wavelength blueward of [\textsc{OIII}]$\lambda5007$.}
\label{fig1}
\end{center}
\end{figure*}
The \emph{number count}\footnote{This error does not take into account flux calibration issues. These other error sources will be dealt with separately.}
$S/N$ ratio of the $H{\gamma}$ line is in the $15<S/N_{\gamma}<100$ interval with a typical value of $S/N_{\gamma}=50-60$.
In the case of the [\textsc{OIII}]$\lambda 4363$, we have $4.1<S/N_{4363}\leq 11.5$ with an average value of $S/N_{4363}=10.0$.
We thus have a worst-case uncertainty in the [\textsc{OIII}]$\lambda 4363 / \mathrm{H}\gamma$ of 25\%, with a typical value of 10\%.
The relative error in the [\textsc{OIII}]$\lambda 4363 / \mathrm{H}\gamma$ ratio
is a major contributor to the error in the electron temperatures and
therefore oxygen abundances in the direct method we use to derive
metallicities. However, in this analysis, the determination of the
electron temperatures are robust relative to these uncertainties.
We stress here that, because of our visual sample selection, the
galaxies used here do not define a complete set in statistical terms.
However, they are representative of the intermediate redshift
population of emission line galaxies with good metallicity
determinations that show [\textsc{OIII}]$\lambda 4363$.
The DEEP2 parent sample was selected from photometry obtained at the
Canada-France-Hawaii Telescope (CFHT) in the $B$, $R$ and $I$ bands using the
CFHT 12k$\times$8k camera \citep{2004ApJ...617..765C}. The catalogues
were generated using the \textit{imcat} software
\citep{1995ApJ...449..460K} and the object magnitudes in $R$ are
measured within a circular aperture with a radius of 3 $r_g$ where $r_g$ is
the optimal Gaussian profile unless 3 $r_g$ were less than 1$"$ , in
which case the flux is measured inside a 1$''$ aperture. The $B-R$ and $R-I$ colours
are measured with apertures of 1$''$ in order to minimise the noise \citep{2004ApJ...617..765C}.
The rest-frame magnitudes were obtained following \citet{cnaw06} and
use as templates a set of 34 local galaxies observed by \citet{1996ApJ...467...38K}.
For each galaxy a parabolic fit between the synthetic $B-R$ and $R-I$ colours and $U-B$ measured at the galaxy redshift is used to estimate
the rest-frame B magnitude and $U-B$ colour. The \textit{rms} errors for the
K-corrections are usually smaller than 0.15 magnitudes
measured at the high redshift edge of DEEP2 ($z \sim 1.5$). The \textit{rms}
errors for the $U-B$ colours range from 0.12 mag at $z = 1.2$ (worst value)
to 0.03 mag at redshifts where the observed filters best overlap $U-B$
\citep{cnaw06}. For more detailed descriptions of both procedures we
refer the reader to \citet{2004ApJ...617..765C} and \citet{cnaw06}.
Table \ref{tab1} summarises the selected sample, where we present the
object identification from the DEEP2 catalogue, coordinates on the
sky, observed and rest-frame magnitudes, the rest-frame equivalent width of
H$\beta$ and the velocity dispersions. The rest-frame colours
are consistent with these sources containing vigorous star forming clusters.
The measured velocity dispersions indicate that none of the sources hosts Active Galactic
Nucleus (AGN) activity, as
defined by the criterion presented in \citet{ost89}, which sets the
limit between AGNs and normal star forming systems at around $180
\mathrm{km}\mathrm{s}^{-1}$.
\begin{table*}
\caption{Full DEEP2 IDs, redshifts, J2000 coordinates, AB apparent and absolute
B-band magnitudes, AB $UBV$ colours, $H\beta$ rest frame equivalent width and velocity dispersion.
Uncertainties in EW(H$\beta$) are a few \AA .}
\label{tab1}
\begin{tabular}{cccccccccc}
\hline
ID & $z$ & RA & DEC & $m_{B}$ & $M_{B}$ & $U-B$ & $B-V$ & EW(H$\beta$) & $\sigma$ \\
& & $(hh:mm:ss)$ & $(dd:mm:ss)$ & $(mag_{ob})$ & $(mag)$ & $(mag)$ & $(mag)$ & ${(\AA)}$ & (km/s) \\
\hline
\hline
21007232 & 0.71659 & 16:47:26.19 & 02:19:00.81 & 23.40 & -20.04 & 0.47 & 0.35 & 36 & 37$\pm$2 \\
41022570 & 0.72138 & 02:27:30.46 & 00:02:04.43 & 23.43 & -19.47 & 0.39 & 0.31 & 310 & 29$\pm$2 \\
42009827 & 0.72939 & 02:29:33.65 & 00:01:44.53 & 23.97 & -19.32 & 0.36 & 0.25 & 80 & 30$\pm$2 \\
42025672 & 0.73143 & 02:29:02.03 & 00:02:00.54 & 22.87 & -20.07 & 0.36 & 0.28 & 122 & 48$\pm$3 \\
31047738 & 0.73232 & 23:26:41.18 & 00:01:13.06 & 22.60 & -20.53 & 0.33 & 0.23 & 93 & 38$\pm$3 \\
22006008 & 0.73280 & 16:51:08.82 & 02:19:04.23 & 24.52 & -18.92 & 0.44 & 0.32 & 13 & 25$\pm$5 \\
22032374 & 0.73839 & 16:53:06.12 & 02:19:57.79 & 23.60 & -19.94 & 0.51 & 0.38 & 38 & 35$\pm$2 \\
32018903 & 0.73961 & 23:30:55.46 & 00:00:47.86 & 23.73 & -19.65 & 0.48 & 0.36 & 89 & 46$\pm$7\\
13016475 & 0.74684 & 14:20:57.85 & 52:56:41.81 & 22.97 & -20.16 & 0.49 & 0.38 & 162 & 47$\pm$6 \\
22032252 & 0.74872 & 16:53:03.49 & 34:58:48.95 & 24.21 & -19.30 & 0.49 & 0.36 & 78 & 37$\pm$3 \\
31019555 & 0.75523 & 23:27:20.37 & 00:05:54.76 & 23.56 & -19.27 & 0.53 & 0.43 & 165 & 52$\pm$2 \\
14018918 & 0.77091 & 14:21:45.41 & 53:23:52.70 & 23.14 & -20.23 & 0.44 & 0.32 & 124 & 42$\pm$2 \\
12012181 & 0.77166 & 14:17:54.62 & 52:30:58.42 & 23.37 & -19.77 & 0.37 & 0.27 & 42 & 41$\pm$3 \\
41059446 & 0.77439 & 02:26:21.48 & 00:48:06.81 & 22.68 & -20.92 & 0.35 & 0.24 & 35 & 45$\pm$3 \\
41006773 & 0.78384 & 02:27:48.87 & 00:24:40.08 & 23.97 & -19.27 & 0.33 & 0.23 & 36 & 30$\pm$2 \\
31046514 & 0.78856 & 23:27:07.50 & 00:17:41.50 & 23.84 & -20.11 & 0.46 & 0.33 & 47 & 48$\pm$2 \\
22020856 & 0.79448 & 16:51:31.47 & 34:53:15.96 & 23.50 & -20.13 & 0.45 & 0.32 & 68 & 42$\pm$4 \\
22020749 & 0.79679 & 16:51:35.22 & 34:53:39.48 & 23.53 & -20.16 & 0.42 & 0.30 & 102 & 49$\pm$4 \\
22021909 & 0.79799 & 16:50:55.34 & 34:53:29.88 & 24.06 & -19.35 & 0.24 & 0.16 & 25 & 32$\pm$7 \\
21027858 & 0.84107 & 16:46:29.01 & 02:19:41.33 & 23.56 & -20.14 & 0.29 & 0.20 & 93 & 56$\pm$1 \\
22022835 & 0.84223 & 16:50:34.61 & 02:19:31.49 & 23.47 & -19.85 & 0.32 & 0.23 & 250 & 50$\pm$4 \\
31047144 & 0.85623 & 23:26:55.43 & 00:01:11.53 & 23.33 & -20.27 & 0.25 & 0.18 & 87 & 55$\pm$2 \\
\hline
\end{tabular}
\end{table*}
In Figure \ref{fig2}, we compare our selected sample against other DEEP2 sources in the same
redshift interval. The comparison galaxies were selected
according to their $\mathrm{H}\beta$ line equivalent widths.
The purpose of this comparison is to highlight the nature of the [\textsc{OIII}]$\lambda 4363$ galaxies, showing that
their star formation episodes are both very intense and young.
The first comparison sample contains 4550 galaxies with $\mathrm{EW}_{\beta}>10\mathrm{\AA}$, which represents
the general population of DEEP2 emission-line galaxies at these redshifts.
The second one contains 218 galaxies with $\mathrm{EW}_{\beta}>50\mathrm{\AA}$, which represents
either galaxies with an AGN or systems with very young and vigorous star forming episodes.
It is seen that the [\textsc{OIII}]$\lambda 4363$ galaxies are very blue
and luminous. Some of the galaxies selected for this work are actually amongst
the bluest DEEP2 targets, even in the $U-B$ colour, hinting that our
spectroscopically selected sample must be comprised of galaxies with very intense starbursts.
\begin{figure*}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.38]{Plots/U-B_1.eps} & \includegraphics[scale=0.38]{Plots/B-V_2.eps} \\
\end{tabular}
\caption{$U-B$ \textsl{vs.} $B$ and $B-V$ \textsl{vs.} $V$ CMDs. The blue dots
depict 4550 objects with $\mathrm{EW}(H\beta) \geq 10 \mbox{\AA}$. The black
dots show the 218 galaxies with $\mathrm{EW}H\beta) \geq 50\mbox{\AA}$.
Our sample is presented as large red solid circles.}
\label{fig2}
\end{center}
\end{figure*}
\section{Results}
\label{resultaos}
\subsection{Relative flux calibration. Special case for the [\textsc{OII}]$\lambda 3727$ doublet.}
\label{rfc}
The DEEP2 spectra are not flux calibrated so that it is necessary
to make a relative flux calibration for each spectrum in order to
obtain the physical line ratios.
Here, we use the results provided by \citet{nkp_private}, who fits a smooth fourth order polynomial
to model the relative throughput with wavelength. This allows measuring meaningful observational line ratios, without the need to
perform a full flux-calibration of the spectra.
In our procedure, we first normalize the observed number counts of any given oxygen line to the
number counts of its nearest hydrogen Balmer line, obtaining an \emph{instrumental} line ratio.
Therefore, we first compute
$F(\mathrm{[OIII]}\lambda,\lambda 4959,5007)/F(\mathrm{H}\beta)$ and
$F(\mathrm{[OIII]}\lambda 4363)/F(\mathrm{H}\gamma)$
in instrumental units and then apply the relative calibration
explained in \citet{nkp_private}, for which a brief summary is
presented as an appendix.
In cases where two lines are close in
wavelength space, there is no need to apply the throughput correction,
since the latter is almost identical for both the numerator and
denominator in the two expressions.
However, for the
[\textsc{OII}]$\lambda3727$ doublet the \citet{nkp_private} calibration is used
as there are no H series lines close enough in wavelength, as the case
of the [\textsc{OIII}] lines. In this case the relative flux
calibration uses the
$F(\mathrm{[OII]}\lambda 3727) /F(\mathrm{H}\gamma)$ ratio.
The typical uncertainty in the empirical \citet{nkp_private}
calibration is of the order of 10\%, which must be propagated into the
aforementioned ratio. This is a major contributor to the error budget
in this specific line ratio, together with the uncertainty in fixing the
[\textsc{OII}]$\lambda 3727$ continuum level for vigorously star
forming objects.
An additional source of error is related to the possible presence
of an underlying stellar population contributing to the continuum, although
most of the objects studied here do not show strong absorption wings in the Balmer lines.
Typical values of the absorption $H\beta$ equivalent width found for
line-emiting star forming galaxies are in the range 0--3.5 \AA\ for
a spectral resolution of about 7 \AA\ \citep{1994ApJ...435..647I} .
For the higher spectral resolution of the data analysed here ($R \sim 5000$, at 7800\AA\ )
these contributions are considerably reduced, involving only the core of the line
\citep{1988MNRAS.231...57D}.
If we adopt a rest frame $\mathrm{H}\beta$ absorption equivalent width of the order of 1\AA\ ,
the underlying absorption correction is typically small, about 2\% in H$\beta$ and
3\% in H$\gamma$ due to the large Balmer emission equivalent widths.
However, given that we are measuring electron temperatures
via equation \ref{eq:ratio} below, the impact of this
uncertainty in our determination of the electron temperature
affects the $R_{O3}$ ratio defined below at the 1\% level at most.
\subsection{Temperatures, densities, and oxygen content calculations.}
\label{tdo}
Given the expected physical conditions of the line emitting regions in
the galaxies of this sample, it is safe to assume that all oxygen is either singly or
doubly ionised. The $O^{+}/O^{0}$ ratio is fixed to the
$H^{+}/H^{0}$ ratio via a charge exchange reaction, and is almost surely negligible
in this case. It would also require exceptionally hard
radiation in order to produce a significant amount of $O^{3+}$.
Therefore, we use a two phase model with a low ionisation zone which
depends on the emission of the [\textsc{OII}]$\lambda 3727$ doublet, and
a high ionisation zone in which the \textsc{OIII} lines are formed.
Table \ref{tab2} summarises the required line ratios to solve the two-phase
scenario, assuming the electron density of the higher ionization zone to be equal
to the electron density in the lower ionization zone. At any rate, the electron densities
found are well below the critical value for de-excitation. From our observations, we
can obtain $R_{O3}$ and $R_{ne}$. It is, however, not possible to calculate $R_{O2}$ as we cannot measure the
auroral [\textsc{OII}] lines as their observed wavelengths for this
sample of galaxies fall beyond the coverage of DEIMOS.
\begin{table}
\caption{Line ratios used to derive electron densities and temperatures. $I(nnnn)$ is the reddening-corrected intensity of each emission line.}
\label{tab2}
\begin{tabular}{ll}
Quantity. & Diagnostic. \\ \hline
$t_{e}$[\textsc{OIII}] & $R_{O3}=(I(4959)+I(5007))/I(4363)$ \\
$t_{e}$[\textsc{OII}] & $R_{O2}=I(3727)/(I(7319)+I(7330))$ \\
$n_{e}$[\textsc{OII}] & $R_{ne}=I(3726)/I(3729)$ \\ \hline
\end{tabular}
\end{table}
We measure the physical ratio via the closest H line using the
normalised flux measurements defined in \S \ref{rfc} and simultaneously
account for internal extinction, assuming the theoretical case B recombination
value for Balmer decrements I(H$\gamma$)/I(H$\beta$)=0.471, which
is the mean between the values corresponding to $T_{e}=10000K$ and
$T_{e}=20000K$ for $n_{e}=100 \mathrm{cm}^{-3}$, as is seen in \citet{ost89}.
\begin{equation} \label{eq:ratio}
R_{O3}=\frac{[f(4959)+f(5007)]/f(H\beta)}{f(4363)/f(H\gamma)}\frac{I(H\beta)}{I(H\gamma)}
\end{equation}
\noindent
where, in this case, $f(nnnn)$ represents the \emph{observed} number counts of each
emission line.
\indent
This expression is used because the $R_{O3}$ ratio involves lines with
different wavelengths and therefore we need to minimise errors arising from the relative flux calibration. This is accomplished by measuring the
line number counts in units of the nearest Balmer line.
On the other hand, the ratio between the two lines in the [\textsc{OII}]$\lambda3727$ doublet $R_{ne}$ bears the
lowest uncertainty, as it is not affected by flux calibration or reddening issues.
The error in our electron density determinations is thus dominated by the uncertainty in the electron
temperature of the O$^+$ zone.
We have derived the physical conditions of the $O^{++}$ dominated region
using the expressions given by \citet{guille08} for
the oxygen emission lines of \textsc{HII} galaxies. This formula is an approximation
for the statistical equilibrium model in a five level
atom. We present below the adequate fitting function they used from
the \textsc{temden} task of \textsc{iraf}, which is based on the program \textsc{fivel} (\citet{robertis87} and \citet{shaw95}):
\begin{equation}
t_{e}\mbox{[OIII]}=0.8254-0.0002415 \times R_{O3}+\frac{47.77}{R_{O3}}.
\end{equation}{\par}
The expected deviations in electron temperatures that could arise from the use of this expression are 5\% or lower.
In our case, the electron temperatures in the $O^{+}$ ionisation zone needs to be estimated from
$t_{e}\mathrm{[\textsc{OIII}]}$ because the [\textsc{OII}] auroral
lines are not included in the spectral range covered by the DEIMOS spectra.
Thus, the empirical expression given in \citet{pagel92}, which is based on a
fit to the theoretical models first presented in \citet{stasinska90} is used:
\begin{equation}
t_{e}^{-1}\mbox{[\textsc{OII}]}=0.5(t_{e}^{-1}\mbox{[\textsc{OIII}]}+0.8)
\end{equation}
The uncertainties in $t_{e}\mathrm{[\textsc{OII}]}$ are therefore a convolution between the $t_{e}\mathrm{[\textsc{OIII}]}$
errors and the variety of models\footnote{Chemical compositions, stellar atmospheres, ages, recombination case and geometry assumed, etc.} used to derive
the above expression for the electron temperature in the $O^{+}$ zone. These errors are bound to be higher
than the errors for $t_{e}\mathrm{[\textsc{OIII}]}$, and are estimated
to be about 40\%. The errors quoted in Table \ref{tab3} only
reflect the standard error propagation from the \citet{stasinska90} expressions and our $t_{e}\mathrm{[\textsc{OIII}]}$ uncertainties. Once the 40\% intrinsic scatter
of the $t_{e}\mathrm{[\textsc{OII}]}$ calibration is taken into account, this translates into a 50\% error in the $O^{+}/H^{+}$ ionic abundance, or 0.3 dex.
Because most of the oxygen in the ionised gas-phase will probably be in the form of $O^{2+}$,
the large uncertainties for the $O^{+}$ content will not
affect significantly the error budget for the final oxygen abundance.
Table \ref{tab3} shows the resulting electron temperatures and densities. It
is seen that the electron density is close to the expected value of
$100\mathrm{cm}^{-3}$. This is much lower than the critical density
$n_{\mathrm{crit}}$ where the higher energy level of the observed lines are as
likely to be de-excited by collisions as by radiative decay.
This indicates that our $t_{e}\mathrm{[\textsc{OII}]}$ calculations
are valid. For some galaxies, the resulting electron densities we
obtain are smaller than $50\mathrm{cm}^{-3}$, and for these only upper
limits are quoted.
Using the results above and the expressions of \citet{guille08} we
calculate the partial ionic abundances $12+\log (O^{+}/H^{+})$ and $12+\log (O^{++}/H^{+})$. These are then combined to obtain the
final oxygen abundance, where we assume the fractions of O$^0$ and O$^{3+}$ are negligible:
\begin{equation}
\frac{O}{H}=\left(\frac{O^{+}}{H^{+}}\right)+\left(\frac{O^{2+}}{H^{+}}\right)
\end{equation}
Table \ref{tab3} presents the resulting ionic and total abundances and the ratio between the two ionisation states $\log(O^{2+}/O^{+})$
in the gas. The errors are smaller than 0.1 dex for the oxygen
ionic and total abundances and between 0.1 and 0.15 dex for
the O$^{2+}$/O$^+$ ratio. The low oxygen abundances, corresponding to
metallicities from 1/3 to 1/10 of the solar value, combined
with the extreme colour-magnitude properties of the sample
show that these galaxies are metal-poor.
\begin{table*}
\caption{Results from nebular analysis. Abundances, electron densities and electron temperatures for the O{2+} zone.}
\label{tab3}
\begin{tabular}{ccccccccc}
\hline
ID & $z$ & $n_{e}$ & $T_{e} (O^{2+})$ & $12+\log(O^{+}/H^{+})$ & $12+\log (O^{2+}/H^{+}$) & $12+\log(O/H)$ & $\log(O^{2+}/O^{+})$ \\
& & $(cm^{-3})$ & $(10^{4} K)$ & & & & \\
\hline
21007232 & 0.71659 & 100 & 1.16$\pm$0.09 & 7.72$\pm$0.11 & 7.98$\pm$0.10 & 8.17$\pm$0.11 & 0.26$\pm$0.21 \\
41022570 & 0.72138 & 230 & 1.34$\pm$0.03 & 7.35$\pm$0.06 & 7.84$\pm$0.03 & 7.96$\pm$0.04 & 0.50$\pm$0.09 \\
42009827 & 0.72946 & 70 & 1.47$\pm$0.07 & 7.42$\pm$0.08 & 7.80$\pm$0.05 & 7.95$\pm$0.06 & 0.38$\pm$0.13 \\
42025672 & 0.73152 & $<50$ & 1.17$\pm$0.03 & 7.42$\pm$0.07 & 8.03$\pm$0.04 & 8.13$\pm$0.04 & 0.61$\pm$0.10 \\
31047738 & 0.73239 & 110 & 1.13$\pm$0.04 & 7.73$\pm$0.07 & 8.00$\pm$0.04 & 8.19$\pm$0.05 & 0.27$\pm$0.12 \\
22006008 & 0.73282 & 100 & 1.78$\pm$0.13 & 7.45$\pm$0.09 & 7.49$\pm$0.07 & 7.77$\pm$0.08 & 0.04$\pm$0.16 \\
22032374 & 0.73839 & $<50$ & 1.53$\pm$0.09 & 7.29$\pm$0.08 & 7.69$\pm$0.07 & 7.83$\pm$0.07 & 0.40$\pm$0.15 \\
32018903 & 0.73956 & 65 & 1.38$\pm$0.06 & 7.42$\pm$0.08 & 7.89$\pm$0.05 & 8.02$\pm$0.06 & 0.47$\pm$0.13 \\
13016475 & 0.74684 & 200 & 1.29$\pm$0.02 & 7.08$\pm$0.06 & 7.97$\pm$0.02 & 8.03$\pm$0.02 & 0.89$\pm$0.08 \\
22032252 & 0.74872 & $<50$ & 1.61$\pm$0.08 & 7.14$\pm$0.08 & 7.60$\pm$0.06 & 7.73$\pm$0.06 & 0.46$\pm$0.13 \\
31019555 & 0.75523 & 230 & 1.48$\pm$0.04 & 6.92$\pm$0.06 & 7.74$\pm$0.03 & 7.80$\pm$0.04 & 0.82$\pm$0.09 \\
14018918 & 0.77091 & 180 & 1.18$\pm$0.04 & 7.51$\pm$0.07 & 8.04$\pm$0.05 & 8.15$\pm$0.05 & 0.53$\pm$0.12 \\
12012181 & 0.77166 & 70 & 1.63$\pm$0.07 & 7.27$\pm$0.07 & 7.71$\pm$0.04 & 7.84$\pm$0.05 & 0.44$\pm$0.11 \\
41059446 & 0.77439 & 110 & 1.44$\pm$0.09 & 7.28$\pm$0.08 & 7.74$\pm$0.07 & 7.87$\pm$0.07 & 0.46$\pm$0.15 \\
41006773 & 0.78384 & 140 & 1.69$\pm$0.10 & 7.22$\pm$0.08 & 7.58$\pm$0.06 & 7.74$\pm$0.07 & 0.36$\pm$0.14 \\
31046514 & 0.78856 & 80 & 1.24$\pm$0.05 & 7.48$\pm$0.08 & 7.93$\pm$0.06 & 8.06$\pm$0.06 & 0.45$\pm$0.14 \\
22020856 & 0.79449 & 140 & 1.44$\pm$0.08 & 7.39$\pm$0.08 & 7.73$\pm$0.06 & 7.89$\pm$0.07 & 0.34$\pm$0.14 \\
22020749 & 0.79679 & 80 & 1.69$\pm$0.12 & 7.32$\pm$0.08 & 7.45$\pm$0.07 & 7.69$\pm$0.08 & 0.13$\pm$0.16 \\
22021909 & 0.79800 & $<50$ & 1.61$\pm$0.05 & 7.14$\pm$0.06 & 7.70$\pm$0.03 & 7.81$\pm$0.04 & 0.56$\pm$0.10 \\
21027858 & 0.84107 & 115 & 1.14$\pm$0.03 & 7.59$\pm$0.07 & 8.03$\pm$0.05 & 8.16$\pm$0.05 & 0.44$\pm$0.12 \\
22022835 & 0.84220 & 370 & 1.60$\pm$0.20 & 7.19$\pm$0.12 & 7.62$\pm$0.13 & 7.76$\pm$0.13 & 0.43$\pm$0.25 \\
31047144 & 0.85630 & 280 & 1.92$\pm$0.10 & 6.96$\pm$0.07 & 7.51$\pm$0.05 & 7.62$\pm$0.05 & 0.55$\pm$0.12 \\ \hline
\end{tabular}
\end{table*}
\section{Discussion and Conclusions}
\label{discutir}
Metallicity is one of the most important parameters to understand
galaxy evolution and the total oxygen abundance is a common way to
trace the metallicities of line-emitting galaxies.
The main result of this work is the direct metallicity determination using \textsc{[OIII]} electron temperatures, for a sample of 22
galaxies at intermediate redshifts ($0.69<z<0.88$). This is summarized in
the LZR diagram presented in Figure \ref{fig3}.
We measure total oxygen abundances between $1/10$ and $1/3$ of the solar value: 12+$\log$[O/H]$_{\odot}$
=8.69 \citep{asplund09} for approximately $-21<\mathrm{M}_{B}<-19$.
These results can be compared to \citet{hoyoskoo05} hereafter H05, \citet{kakazu07} hereafter K07,
\citet{salzer09} hereafter S09, \citet{A14} hereafter A14 and
\citet{Ly14} hereafter L14.
The sample from K07 was collected from spectroscopic observations of 161 Ultra
Strong Emission Line galaxies (USELs) using the DEIMOS spectrograph on the Keck II telescope. The galaxies are spread in a wide range of redshifts
($0.38<z<0.83$) and its selection criteria are geared towards the detection of extremely low metallicities.
The majority of the objects in this sample have low luminosities and metallicities.
While no diagnostic diagrams could be used to flag objects hosting AGN
or with the presence of shock heating, the high electron temperatures
measured for some galaxies ($20000<T(K)<30000$) suggest that some
objects have some contribution from these latter phenomena.
The sample from S09 presents some of the most luminous objects of this type
(-22 $<M_{B}<$ -20) at intermediate redshifts (0.35$<$z$<$0.41). The star-forming galaxies of
this sample are selected from a wide-field Schmidt survey that picks
emission line objects by the presence of H$\alpha$ emission in their objective-prism spectra. This naturally forces this sample to exclude low-luminosity objects.
The sample from A14 is composed by Extreme Emission Line Galaxies
(EELGs) selected from the 20k zCOSMOS Bright Survey by their unusually large
$\mathrm{[\textsc{OIII}]} \lambda5007$ equivalent widths. They are seven purely star-forming
galaxies with redshifts from 0.43 to 0.63 and intermediate luminosities.
The sample from L14 encompasses a wide luminosity range (from $M_{B}$=-21.1 to
-17.5) with abundances going from extreme metal poor galaxies
(12+$\log$(O/H)$<$7.65) to galaxies with about half solar abundances and a redshift range similar to that of K07.
The data were obtained using optical spectroscopy with DEIMOS and the MMT Hectospec spectrographs.
The sample from H05 and this work have many features in common since both of
them have been taken from the DEEP2 redshift survey with the DEIMOS spectrograph.
It covers the central region of the LZR diagram, though the presence of the [OII]$\lambda\lambda3726,3729$ doublet
within the wavalength range was not an explicit selection requirement
for H05. The redshift range of the H05 sample is $0.51<z<0.85$.
It is also possible to compare the sample presented here to
the ``green pea" population, first described by
\citet{2009MNRAS.399.1191C}. These systems were identified by
the Galaxy Zoo project because of
their peculiar bright green colour and small sizes, being unresolved
in the Sloan Digital Sky Survey imaging. These galaxies show very strong [\textsc{OIII}]$\lambda 5007$ emission lines and very large H$\alpha$ equivalent widths up to 1000\AA .
Here, we have chosen for comparison the 66 $0.112<z<0.360$ ``green pea" sub-sample studied by \citet{2011ApJ...728..161I} (hereafter I11), who collected a sample of 803 $0.02<z<0.63$ star-forming luminous compact galaxies. The global properties
of these star-forming luminous compact galaxies very closely resemble the properties of the ``green pea" population, but have been selected by both their spectroscopic and photometric signatures. These 66
``green peas" selected by \citet{2011ApJ...728..161I} were also
studied by \citet{2009MNRAS.399.1191C}, but the former metallicities
were obtained using direct ($T_ {e}$) methods, which allows comparing them
to our sample in an optimal way. The oxygen abundances of the
\citet{2011ApJ...728..161I} ``green peas'' do not differ from those of
nearby low-metallicity blue compact dwarf galaxies. We here note that, at the resolution of SDSS, the objects in the sample presented here
would be almost point-like.
\begin{figure}
\begin{center}
\includegraphics[scale=0.37]{Plots/MB_metal_JM2_3_figure3.eps}
\caption{Luminosity-Metallicity Diagram for intermediate star-forming galaxies for objects of this study and comparable samples from the literature. H05 includes 15 luminous star-forming galaxies from DEEP2. K07 includes 7 EELGs, 4 of them are extremely low metallicity galaxies. S09 includes 13 metal-poor galaxies. A14 includes 7 EELGs. L14 includes 16 metal-poor objects. I11 represents a sample of ``green peas" with good metallicity determinations. This work presents 22 new metal-poor galaxies. The dashed line represents the luminosity binned LZR by
\citet{zahid11} (Z11 in the diagram) with a $1-\sigma$ error area. The
dotted line on the upper zone of the diagram represents the Tremonti
luminosity-metallicity relation for local SDSS galaxies
(\citet{tremonti04}, T04 in the diagram) while the solid line
represents the best fit to the combined samples.}
\label{fig3}
\end{center}
\end{figure}
We find that our values for luminosities and metallicities are in good agreement with previous
determinations, with our error estimates being smaller. Our sample shares the
same L-Z locus as the 66 ``green peas'' from I11, though our galaxies tend
to be less luminous and more metal-poor than the ``green peas''.
Taken together, all these studies define a locus in the LZR diagram,
that is offset from the local LZR of the SDSS sample of
\citet{tremonti04} towards lower metallicities.
These two loci can be represented by linear fits to the data, which
are shown in Figure \ref{fig3}; the local fit is represented a dotted line, and the intermediate redshift LZR
as a solid line. The linear regressions are:
\begin{eqnarray*}
12+\log(O/H) &=&-0.185(\pm0.001)M_{B}+5.238(\pm0.018) \\
12+\log(O/H )&=&-0.135(\pm0.019)M_{B}+5.272(\pm0.38) \\
\end{eqnarray*}
\noindent
for the local SDSS and intermediate redshift samples respectively.
\indent
Both lines have similar slopes, but are offset by a factor of 10 to lower oxygen content for the high luminosity end
considered in this work. In principle, this could be due to (i) the different nature of the objects involved, (ii) selection effects regarding both luminosity and chemical
abundances, or (iii) a genuine evolutionary effect. However, we should
note that the abundances have been derived by two different methods for the local
and for the distant samples. The local sample uses empirical
calibrations while our work uses direct abundance determinations.
The comparison of our $T_{e}$ abundances with those of the
intermediate $z$ sample of \citet{zahid11}, who used empirical
calibrations of the oxygen line equivalent widths,
is a means of exploring the differences between both methods.
Here, it is critical that the parent samples in both works are essentially the same. The luminosity binned LZR, taking the median abundance value as metallicity, by
\citet{zahid11} is plotted in our Figure \ref{fig3}. It shows that, at the same luminosity, there is an $8-\sigma$
deviation between both abundance distributions. Given the numbers involved in the two samples (about 1700 \textsl{vs} 22), this large deviation
implies totally incompatible distributions. This huge discrepancy cannot be solved by taking into account the inherent uncertainties in both methods which amount to
0.2$dex$ at most. This could, in the best scenario, reduce the discrepancy to $4-\sigma$, which is still not plausible.
Our sample also shows similar luminosities, metallicities and optical appearances with the ``green peas" population of luminous star forming systems
found at lower redshift, although it tends to be fainter and less
metal rich than the ``green peas''.
The full description of the metallicity distribution
of star forming systems at intermediate redshift will require the use
of IR spectroscopy that allows detecting the [\textsc{SIII}] lines.
After carefully considering the systematics of the various metallicity determination methods, we conclude
that the metallicity distributions provided by other works like \cite{zahid11} and \cite{maier14}
and the one found in this work are not compatible.
\section*{Acknowledgments}
We are grateful to Ben Weiner, for providing us with the automated line
measurements.
Financial support has been provided by projects
AYA2010-21887-C04-03 (former Ministerio de Ciencia e In-
novaci\'on, Spain) and AYA2013-47742-C4-3-P (Ministerio
de Econom\'\i a y Competitividad), as well as the exchange
programme ‘Study of Emission-Line Galaxies with Integral-
Field Spectroscopy’ (SELGIFS, FP7-PEOPLE-2013-IRSES-
612701), funded by the EU through the IRSES scheme.
This work is based on observations taken at the W. M. Keck
Observatory, which is operated jointly by the National Aeronautics and
Space Administration (NASA), the University of California, and the
California Institute of Technology. Funding for the DEEP2 Galaxy
Redshift Survey has been provided by NSF grants AST-9509298,
AST-0071048, AST-0507428, and AST-0507483.
We recognize and acknowledge the highly significant cultural role and
reverence that the summit of Mauna Kea has always had within the
indigenous Hawaiian community; it has been a privilege to be given the
opportunity to conduct observations from this mountain.
\input{referencias.tex}
|
1,116,691,500,455 | arxiv |
\section{Introduction} \vspace{-0.6em}
\noindent
The identification of dynamical systems from data is a powerful tool in Robotics \cite{aastrom1971system}.
Learnt analytic models may be used for control synthesis and can be utilized for gravitational and inertial compensation \cite{spong2020robot}.
Moreover, when used as simulators, they can be used to reduce the sample complexity of data-driven control methods such as Reinforcement Learning \cite{deisenroth2011pilco, chua2018deep}.
For these control applications, where out-of-distribution prediction is typically required, the ability to generalize beyond the acquired data is critical.
Any modelling error may be exploited by a controller, and such exploitation may result in catastrophic system failure.
To ensure sufficient out-of-sample generalization, the model's hypothesis space is an important consideration.
Ideally, this space should be defined such that only plausible trajectories, that are physically consistent and
have bounded energy, are generated.
\medskip
\noindent Standard black-box models such as deep networks or Gaussian processes, which are a common choice for model learning \cite{nguyen2011survey, deisenroth2011pilco, chua2018deep}, have a broad hypothesis space and so can simulate dynamics with unbounded energy.
To overcome this shortcoming, `grey-box' models that combine deep networks with physical insights have been recently proposed, e.g., incorporating Lagrangian \cite{lutter2018deep, Lutter2019Energy, gupta2019general} and Hamiltonian Mechanics \cite{greydanus2019hamiltonian} for energy-conserving models.
These empirically show better generalization compared to black-box models, but the local representations of deep networks cannot guarantee out-of-sample generalization.
Only white-box models \cite{atkeson1986estimation, ting2006bayesian, traversaro2016identification, wensing2017linear, sutanto2020encoding}, which infer the physical parameters of the system given the analytic equations of motion, can guarantee out-of sample generalization as these models are valid globally.
While these combinations of physics with data-driven learning can obtain more robust representations, the usage of physics priors commonly reduces model accuracy compared to black-box methods as most physics priors cannot capture reality to a sufficient fidelity.
The priors of Newtonian-, Lagrangian- and Hamiltonian mechanics typically cannot describe the complex nonlinear phenomena of friction, hysteresis and contact. Therefore, the need of a compromise between physics-inspired models and black-box functions approximation is clear.
\medskip
\noindent In this work, we examine a spectrum of hybrid models for the domain of multibody robot dynamics.
We motivate a computation graph architecture that embodies the Newton Euler equations, emphasising the utility of the Lie Algebra form in translating the dynamical geometry into an efficient computational structure for learning \cite{handa2016gvnn}.
We describe the used actuator models (Section \ref{sec:actuators}) and the virtual parameters (Appendix).
In the experiments, we evaluate 26 Newton-Euler based system identification approaches and benchmark these models on the simulated and physical Furuta Pendulum and Cartpole.
The comparison shows that the kinematic parameters, required by previous Newton-Euler methods \cite{atkeson1986estimation, sutanto2020encoding, ledezma2017first}, can be accurately inferred from data.
Furthermore, we highlight that models with guaranteed bounded energy of the uncontrolled system generate non-divergent trajectories, while more general models have no such guarantee. Therefore, their performance strongly depends on the data distribution.
The main contributions of this work are the introduction of a white-box model that jointly learns dynamic and kinematics parameters and can be combined with black-box components.
We then provide an extensive empirical evaluation on challenging systems and different datasets that elucidates the comparative performance of our grey-box architecture with comparable white- and black-box models.
\section{Newton Euler Equations} \vspace{-0.6em}
\noindent Consider a set of coupled rigid bodies, such as a robotic manipulator.
While the global dynamics of such a system is complex, the behaviour of each individual body and its effect on its neighbours are easily understood.
This is the key insight of algorithms such as Recursive Newton Euler Algorithm (RNEA) and Articulated Body (ABA) \cite{Featherstone2007rigid}, which efficiently solve for the global dynamics by successively propagating the local solutions forward and backwards.
While alternative methods for modelling, such as Lagrangian and Hamiltonian Mechanics, may be appreciated for their mathematical elegance and convenience; the Newton Euler approach is based on global and interpretable physical quantities enabling robust out-of-sample generalization.
Moreover, the efficient algorithmic form is applicable for both forward and inverse dynamics
We describe the Newton Euler method in its Lie Algebra form \cite{kim2012lie}, which is not only compact, but easy to represent as a differentiable computation graph \cite{handa2016gvnn}.
\medskip
\noindent In this paper we focus on kinematic trees composed on $n$ components.
Given a base reference frame, the pose of each body can be described via successive application of an affine transform ${\bm{T}}$ containing the 3x3 rotation matrix ${\bm{R}}$ and translation ${\bm{p}}{\in}\mathbb{R}^3$.
To describe the dynamics of each body, we use a 'generalised' velocity $\bar{{\bm{v}}} {=}[{\bm{v}}, \bm{\omega}]^\intercal$, which is composed of linear (${\bm{v}}$) and rotational ($\bm{\omega}$) components. The generalised velocity $\bar{{\bm{v}}}_i$ expresses the link velocity in the inertial base frame but expressed in the $i$th link coordinate frame. The generalised notation is extended to acceleration $\bar{{\bm{a}}}$, force $\bar{{\bm{f}}}{=}[{\bm{f}}, \bm{\tau}]^\intercal$ (with moment $\bm{\tau}$), momentum $\bar{{\bm{l}}}$, and inertia
\begin{align}
\bar{{\bm{M}}} &=
\begin{bmatrix}
{\bm{J}} & m[{\bm{p}}_m] \\
m[{\bm{p}}_m]^\intercal & m{\bm{I}}
\end{bmatrix}
\end{align}
with link mass $m$, link inertia ${\bm{J}}$ and link center of mass (CoM) position ${\bm{p}}_m$ relative to the body frame, and $[{\bm{x}}]$ mapping the vector ${\bm{x}}$ to a skew-symmetric matrix such that ${\bm{x}}{\times}{\bm{y}}{=}[{\bm{x}}]{\bm{y}}$.
The benefit of the Lie Algebra formulation is apparent when considering the kinematics, dynamics and differential equations.
Transforming the kinematic terms (e.g. $\bar{{\bm{v}}}, \bar{{\bm{a}}}$) of the $i$th body in the $j$th frame is a compact linear operations through use of the adjoint transform of ${\bm{T}}$ ($\text{Ad}_{{\bm{T}}}$), i.e.,
\begin{align}
\bar{{\bm{v}}}_{j} &=
\text{Ad}_{{\bm{T}}_{j,i}}\bar{{\bm{v}}}_i,
\hspace{1.5em}
\bar{{\bm{a}}}_{j}=
\text{Ad}_{{\bm{T}}_{j,i}}\bar{{\bm{a}}}_i.
\end{align}
In the Lie algebra formulation, the kinematic terms can be described in the SE(3) manifold due to the properties of the affine transform ${\bm{T}}$.
Dynamic terms (e.g. $\bar{{\bm{l}}}, \bar{{\bm{f}}}$) can be shown to act in the \textit{dual space} for SE(3), DSE(3).
Hence, these terms can be transformed between frames using the coadjoint operator $\text{Ad}^*_{{\bm{T}}_{j,i}}{=}\text{Ad}^\intercal_{{\bm{T}}_{j,i}}$, i.e.,
\begin{align}
\bar{{\bm{l}}}_{j}= \text{Ad}^*_{{\bm{T}}_{j,i}}\bar{{\bm{l}}}_i,
\hspace{1.5em}
\bar{{\bm{f}}}_{j}= \text{Ad}^*_{{\bm{T}}_{j,i}}\bar{{\bm{f}}}_i.
\end{align}
Differential equations in the inertial base frame can also be expressed compactly, i.e., the Newton-Euler equation is described by
\begin{align*}
\bar{{\bm{f}}} = \frac{d}{dt}\bar{{\bm{l}}} &=
\bar{{\bm{M}}}\bar{{\bm{a}}} - \text{ad}^*_{\bar{{\bm{v}}}}\bar{{\bm{M}}}\bar{{\bm{v}}},
\hspace{5pt} \text{where} \hspace{5pt}
\text{ad}^*_{\bar{{\bm{v}}}} =
\begin{bmatrix}
[\bm{\omega}] & \mathbf{0} \\
[{\bm{v}}] & [\bm{\omega}]
\end{bmatrix}.
\end{align*}
These transformations and differential equations can be embedded in articulated body algorithm (ABA) and the Recursive Newton Euler Algorithm (RNEA). These different formulation yield more compact and intuitive description compared to the original formulation \cite{Featherstone2007rigid}, e.g., the ABA algorithm in Lie algebra is described in Algorithm \ref{alg:articulated-rigid-body}.
\section{Actuator Models} \label{sec:actuators} \vspace{-0.6em}
\noindent The models of the previous sections mostly focused on simulating rigid body dynamics. This representation does commonly not capture reality with sufficient fidelity for mechanical systems with actuation. Actuators exerting the control signal are affected by non-linear transformation of the set-points, hysteresis and friction. To be able to learn such models with non-ideal actuators, we augment the rigid-body dynamics model with an actuator model. This actuation model can either be white-box model relying on existing friction models or black-box models. We define six different joint independent actuator models, building from $\tau = \tau_d$,
\begin{align*}
\text{Viscous:}& &\tau &= \tau_d {-} \mu_v \dot{q}, \\
\text{Stribeck:}& &\tau &= \tau_d {-} \text{sign}(\dot{q}) \left(f_s {+} f_d \exp\left({-}\nu_s \dot{q}^2 \right)\right) {-} \mu_v \dot{q},\\
\text{NN Friction:}& &\tau &= \tau_d {-} \text{sign}(\dot{q}) \: \| f_{\text{NN}}(q,\dot{q}; \psi_F) \|_1 ,\\
\text{NN Residual:}& &\tau &= \tau_d {-} f_{\text{NN}}(q,\dot{q}; \psi_R), \\
\text{FF-NN:}& &\tau &= f_{\text{NN}}(\tau_d, q,\dot{q}; \psi_M),
\end{align*}
where $\tau_d$ is the desired torque. The Viscous, Stribeck and MLP friction actuator models are guaranteed to learn a stable uncontrolled system (i.e., $\tau_d{\coloneqq}0$) as all possible actuator parameters can only dissipate energy, i.e., $\dot{E} = \bm{\tau}^T \dot{{\bm{q}}} \leq 0 \hspace{4pt} \forall \hspace{4pt} \theta_F$.
The Residual and MLP actuator models are capable of generating energy and so the performance will depend on the training data.
\section{Experimental Setup} \vspace{-0.6em}
\noindent In the following the experimental setup containing the used systems, datasets and models is described.
\subsection{Systems} \vspace{-0.6em}
\noindent For the comparison we choose the Quanser Cartpole and the Quanser Furuta Pendulum as these systems are under-actuated and the physical systems have different peculiarities, which make system identification challenging. System identification for under-actuated systems is harder compared to fully-actuated systems as all data points lie on a low dimensional manifold and one cannot capture the whole space.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{figures/rollout/RSS_Rollouts.pdf}
\caption{Qualitative model comparison of the 26 different models performing forward roll-outs on the Cartpole and Furuta Pendulum. The roll-outs start from the starting state and are computed with 250Hz sampling frequency and integrated with RK4. The models are trained on three different datasets ranging from uniformly sampled and ideal observations (i.e., Simulated Data from Uniform Sampling) to trajectory data of noisy observation from the physical system.}
\label{fig:rollout}
\end{figure*}
\subsection{Datasets} \vspace{-0.6em}
\noindent To evaluate the impact of data quality, we are evaluating the performance on three different datasets with different levels of complexity. The simulated data from uniform sampling dataset is generated by sampling joint positions, velocities and torques from a uniform distribution spanning the complete state domain and computing the acceleration with the true analytic forward dynamics. The simulated data from trajectories dataset is generated by simulating the ideal system with viscous friction and small state and action noise. The real system data from trajectories dataset is generated on the physical system by applying an energy controller that repeatedly swings-up the pendulum and lets it fall down without actuation.
\subsection{Models} \vspace{-0.6em}
\noindent For the evaluation we compare three different instantiations of the previously described white-box model family with the 6 different actuation models, if applicable.
\textbf{No-Kin Differential Newton Euler Alg.:}
The no-Kin DiffNEA model assumes knowledge of the kinematic tree but no knowledge of the kinematics ${\bm{\theta}}_K$ or dynamics parameters ${\bm{\theta}}_I$. These link parameters are learned from data containing only generalized coordinates using ADAM to minimize the squared loss of the forward dynamics.
\textbf{Differential Newton Euler Alg.:}
The DiffNEA model assumes knowledge of the kinematic chain and the kinematics parameters ${\bm{\theta}}_K$ and only learns ${\bm{\theta}}_I$. These parameters are learnt by minimizing the squared loss of the forward dynamics using ADAM. This approach was introduced by \cite{sutanto2020encoding}.
\textbf{Newton Euler Alg.:}
The NEA model assumes knowledge of the kinematic chain and the kinematics parameters ${\bm{\theta}}_K$ and only learns ${\bm{\theta}}_I$. These parameters are learnt by linear regression. This model learning approach was initially introduced by \cite{atkeson1986estimation}. Due to the linear regression this model cannot be augmented with the different actuation models.
\textbf{Feed-Forward Neural Network (FF-NN)}
As black-box model learning baseline, we are using a feed-forward neural network trained via ADAM. This network is a continuous time model and predicts the joint acceleration.
\vspace{-0.3em}
\subsection{Model Initialization} \vspace{-0.6em}
For the white box models we are differentiating between two initialization strategies, without prior and with prior. Without prior means that the link parameters are initialized randomly. With prior means that the parameters are initialized with the known values given by the manufacturer. This differentiation enables us to evaluate the impact of good initialization for white box models.
\section{Experimental Results}\vspace{-0.6em}
\noindent The qualitative experimental results are shown in Figure \ref{fig:rollout}. The overall performance depends heavily on the system as well as the dataset.
The numerically sensitive conditioning of the Furuta Pendulum causes all models to be worse on all datasets and model classes.
Conversely, the magnitude of the physical parameters of the Cartpole make the identification and long-term prediction is simpler.
Regarding the datasets, the overall forward prediction performance, with some outliers, decreases with dataset complexity.
Regarding the different models, no clear best system identification representation and identification approach from those studied is apparent.
One interesting result is that the white-box model approach with \emph{unknown} kinematics (no-Kin DiffNEA) performs comparable to the white-box model with \emph{known} kinematics (DiffNEA, NEA), demonstrating the kinematics and dynamics can be learned jointly.
Furthermore, one can observe that the three different model classes, energy-conserving, energy-bounded and energy-unbounded models achieve very different long-term forward prediction.
\subsection{Energy Conserving Models} \vspace{-0.6em}
\noindent The energy-conserving models, i.e. no-Kin DiffNEA, DiffNEA, NEA and no-actuator model, can only represent energy conserving dynamics.
When this prior is correct, as in the simulated uniform dataset, these models perform well.
If the prior is not correct, e.g., the large viscous friction of the \emph{simulated} Cartpole, the forward prediction degrades significantly even for simulated data.
The identified parameters are physically plausible but unreasonable. While the mass of the cart is about $0.5$kg,
this model class sets the mass to $1$kg
for the simulation and $2$kg
for the physical system. The mass of the about $120$g
pendulum is set to $340$g
for the simulation and $3000$g
for the physical system.
\vspace{-0.3em}
\subsection{Energy Bounded Models}\vspace{-0.6em}
\noindent The energy-bounded models, i.e., NEA plus Viscous, Stribeck and NN-Friction actuator, guarantee that the energy does not increase without actuation and hence, guarantee to learn a global stable system. This model class contains the best performing models of this benchmark.
The learnt models always yield non-diverging trajectories and can model systems with and without friction.
The NN-Friction actuator achieves particularly good performance by exploiting its black-box flexibility within the inductive bias.
Despite its expressiveness, the model does not overfit and the obtained physical parameters are comparable to the white-box actuator models.
\vspace{-0.3em}
\subsection{Energy Unbounded Models} \vspace{-0.6em}
\noindent The energy-unbounded models, i.e., FF-NN, NEA plus FF-NN and NN-Residual actuator, can potentially learn to increase the system energy during simulation without actuation, which is physically implausible.
The benchmark of Figure \ref{fig:rollout} shows that models of this class learn to pump energy into the system even for perfect sensor measurements (i.e., FF-NN of the Cartpole with simulated uniformly sampled data).
For the more challenging trajectory datasets, all black-box and hybrid models learn models that increase in energy during simulation without actuation.
Many of these models also generate divergent trajectories during simulation, which leave the training domain.
\vspace{-0.3em}
\subsection{Model Initialization} \vspace{-0.6em}
\noindent For the the hybrid white-box models the different model initializations were compared.
In the simulation experiments no clear difference between models with and without prior initialization is visible. Evaluating the identified physical parameters also yields no clear improvement of the initialization with prior, e.g., even for unreasonably large physical parameters, the identified parameter with a prior was not necessarily smaller.
Therefore, we conclude that the initialization with the best known parameters does not necessarily improve model performance when using stochastic gradient descent with ADAM.
\vspace{-0.3em}
\section{Conclusion} \vspace{-0.6em}
\noindent In this paper, we have described the classical Newton-Euler system identification approach \cite{atkeson1986estimation} using the elegant Lie algebra formulation \cite{kim2012lie} and the differential programming paradigm. We combined this formulation with white- and black-box actuation models for end-to-end learning.
In a large-scale benchmark, we compared 26 different models on three datasets and two different systems.
The two main conclusions of this benchmark are, (1) the no-Kin DiffNEA model, which learns the kinematics, performs equally well compared to the same DiffNEA with the kinematics given, and (2) models with guaranteed global stability yield the best long term forward simulations.
\section*{Acknowledgment}
This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No \#640554 (SKILLS4ROBOTS). Furthermore, this research was also supported by grants from ABB, NVIDIA and the NVIDIA DGX Station.
|
1,116,691,500,456 | arxiv | \section{Two convex relaxations for sparse linear regression}
Given a collection of observed sample points $(x_i, y_i) \in \mathbb{R}^p \times \mathcal{Y}$, the goal
of a sparse learning task is to learn a linear function $x \mapsto \beta^T x$ that is then used
to predict an outcome of $y\in \mathcal{Y}$ for future/unseen data, where $\beta$
is restricted to have a small number of nonzero entries).
Such a task can be modeled as the following cardinality constrained optimization problem
\begin{equation}\label{SpML}
\min_{\substack{\beta \in \mathbb{R}^p, \\ \|\beta\|_0 \leq k} }
\frac{1}{n} \sum_{i=1}^n f \left(\beta^T x_i; y_i\right).
\end{equation}
With the cardinality constraint, (\ref{SpML}) is usually highly nonconvex and difficulty
to solve to global optimality. The authors in \cite{PilanciWainwrightGhaoui2015}
considered the following regularized version,
\begin{equation}\label{SpMLreg}
\min_{\substack{\beta \in \mathbb{R}^p, \\ \|\beta\|_0 \leq k} }
\frac{1}{n} \sum_{i=1}^n f \left(\beta^T x_i; y_i\right)+ \rho\|\beta\|_2^2.
\end{equation}
One of the key results in \cite{PilanciWainwrightGhaoui2015} shows that (\ref{SpMLreg})
can be equivalently formulated as minimizing a convex function over a subset of binary vectors,
\begin{equation}\label{SpMLbin}
\min_{\substack{z \in \{0,1\}^p, \\ \sum_{j} z_j \leq k} }
\ \ \ \underbrace{\max_{v \in \mathbb{R}^n} \left\{-\frac{1}{2\rho} v^T X \mathbf{D}(z) X^T v - \sum_{i=1}^n f^*(v_i; y_i) \right\}}_{G(z)},
\end{equation}
where $G(z)$ is convex because as it is the max function of infinite many linear functions,
and $f^*(v,y) := \sup_{t\in \mathbb{R}} \left\{st - f(t, y)\right\}$ is the conjugate function of $f(\cdot; y)$.
In this note we focus on the important special case of sparse linear regression, i.e., we consider
the following cardinality-constrained quadratic program,
\begin{equation}\label{l0_card}
\nu_{\ell0} := \min_{\beta} \ \ \frac{1}{2} \|X\beta - y\|_2^2 + \frac{1}{2}\rho\|\beta\|_2^2, \ \ s.t. \ \ \|\beta\|_0 \leq k,
\tag{$\ell_0:{card}$}
\end{equation}
The authors of \cite{PilanciWainwrightGhaoui2015} further proposed to relax the binary condition $z \in \{0,1\}^p$ in
(\ref{SpMLbin}) to $z \in [0,1]^p$, and studied the conditions under which such a relaxation is exact.
When specialized to the sparse linear regression problem, the continuous relaxation takes
the following form of a semidefinite program,
\begin{equation}\label{SDP_PWG}
\begin{aligned}
\nu_{PWG} := \min_{t \in \Re, z \in [0,1]^p} \ \ & 0.5 t \\
s.t. \ \ & \begin{bmatrix}t & y \\ y & I_n + \frac{1}{\rho} X \mathbf{D}(z) X^T \end{bmatrix} \succeq 0, \ \ \ e^T z \leq k,
\end{aligned}\tag{$SDP_{PWG}$}
\end{equation}
where $e$ is a vector with all entries 1 in proper dimension, and $\mathbf{D}(z)$ is a diagonal matrix whose entries are $z_i, i=1,...,p$.
It can also be equivalently written as the following compact form,
\[
\nu_{PWG} = \frac{1}{2} \min_{z \in [0,1]^p, e^T z \leq k} \ \ y^T \left(\frac{1}{\rho} X \mathbf{D}(z) X^T + I_n\right)^{-1} y.
\]
Following a different approach, authors of \cite{DongChenLinderoth2015} recently proposed another semidefinite relaxation
for sparse linear regression where the $\ell$-0 norm appears as a regularized term. When modified as a convex relaxation
for the cardinality constrained form (\ref{SpMLbin}), their proposed semidefinite relaxation is,
\begin{equation}\label{SDP_DCL}
\begin{aligned}
\nu_{DCL} := \min_{b \in \mathbb{R}^p, B \in \mathcal{S}^p} \ \ & \frac{1}{2}\left\langle \begin{bmatrix}y^T y & -y^T X \\ -X^T y & \rho I_p + X^T X \end{bmatrix},
\begin{bmatrix}1 & b^T \\ b & B\end{bmatrix}\right\rangle \\
s.t. \ \ & \begin{bmatrix}1 & b^T \\ b & B\end{bmatrix} \succeq 0 \\
& \begin{bmatrix}z_i & b_i \\ b_i & B_{ii}\end{bmatrix} \succeq 0, \forall i, \ \ \sum_{i=1}^p z_i \leq k.
\end{aligned}\tag{$SDP_{DCL}$}
\end{equation}
In this note we compare these two semidefinite relaxations. We show that the relaxation (\ref{SDP_DCL})
is no weaker than (\ref{SDP_PWG}) in this section. In section \ref{sec:cert} we establish
a result that characterizes a certificate of exactness for the convex relaxation (\ref{SDP_DCL}), hence extends
a key result in \cite{PilanciWainwrightGhaoui2015} to (\ref{SDP_DCL}). Section \ref{sec:empirical} concerns
the probability of exact recovery for the case of Gaussian ensemble, where we show empirically (\ref{SDP_DCL})
can recover the true support of with much less data points.
We first state a technical lemma that will be used soon.
\begin{lemma}\label{lem:ridge}
For any $X \in \Re^{n\times p}$ and $\rho > 0$, we have
\[
\min_{\beta \in \mathbb{R}^p} \left\{\frac{1}{2} \|X\beta - y\|_2^2 + \frac{1}{2}\rho\|\beta\|_2^2 \right\} \ = \
\frac{1}{2} y^T \left(\frac{1}{\rho} X X^T + I_n\right)^{-1} y
\]
\end{lemma}
\begin{proof}
Straightforward computation.
\end{proof}
By Lemma \ref{lem:ridge}, (\ref{SDP_PWG}) can be reformulated as
\begin{equation}\label{PWG:card:Poly}
\nu_{PWG} =
\min_{z \in [0,1]^p, e^T z \leq k} \ \
\min_{\beta \in \mathbb{R}^p} \ \ \frac{1}{2} \left\|X\sqrt{\mathbf{D}(z)}\beta - y\right\|_2^2 + \frac{1}{2}\rho
\left\|\beta\right\|_2^2,
\end{equation}
where $\sqrt{\mathbf{D}(z)}$ is a diagonal matrix with the i-th diagonal entry $\sqrt{z_i}$.
\begin{proposition}\label{prop:dominate} $\nu_{\ell 0} \geq \nu_{DCL} \geq \nu_{PWG}$.
\end{proposition}
\begin{proof}
Suppose that $(\tilde{b}, \tilde{B}, \tilde{z})$ is optimal in (\ref{SDP_PWG}). Without loss of generality we may assume that $\tilde{z}_i = \frac{\tilde{b}_i^2}{\tilde{B}_{ii}}$ for all $\tilde{B}_{ii} \neq 0$,
and $\tilde{z}_i = 0$ otherwise. Therefore $\tilde{z}_i \in [0,1], \forall i$.
Define $\tilde{\beta}$ as
\[
\tilde{\beta}_i = \begin{cases}\tilde{z}^{-\frac{1}{2}}_i \tilde{b}_i, & \ if \ \tilde{z}_i > 0\\
0 & \ if \ \tilde{z}_i = 0.\end{cases}
\]
Then obviously
$\tilde{b} = \sqrt{\mathbf{D}(\tilde{z})} \tilde{\beta}$ and $(\tilde{\beta}, \tilde{z})$ is feasible in (\ref{PWG:card:Poly}). We have
\begin{align*}
\frac{1}{2}\left\langle \begin{bmatrix}y^T y & -y^T X \\ -X^T y & \rho I_n +
X X^T \end{bmatrix},
\begin{bmatrix}1 & \tilde{b}^{T} \\ \tilde{b} & \tilde{B}\end{bmatrix}\right\rangle
\geq &
\frac{1}{2}\left\langle \begin{bmatrix}y^T y & -y^T X \\ -X^T y &
X X^T \end{bmatrix},
\begin{bmatrix}1 & \tilde{b}^{T} \\ \tilde{b} & \tilde{b}\tilde{b}^T\end{bmatrix}\right\rangle +
\frac{1}{2}\rho \mathbf{trace}(\tilde{B}) \\
\geq & \frac{1}{2} \left\|X \sqrt{\mathbf{D}(\tilde{z})} \tilde{\beta} - y\right\|_2^2 +
\frac{1}{2} \rho \left\|\tilde{\beta}\right\|_2^2 \geq \nu_{PWG}.
\end{align*}
The first inequality is because of $\tilde{B} \succeq \tilde{b}\tilde{b}^T$. The second inequality
is because of $\tilde{B}_{ii} \tilde{z}_i \geq \tilde{b}_i$, which implies
$\tilde{B}_{ii} \geq \tilde{\beta}_i^2$, and the final inequality is by the characterization (\ref{PWG:card:Poly}).
\end{proof}
As (\ref{SDP_DCL}) satisfies the Slater condition, strong duality holds and the dual of (\ref{SDP_DCL}) is
\begin{equation}\label{SDP_DCL:dual}
\begin{aligned}
\nu_{DCL} = \frac{1}{2}y^T y + \max_{\tau, \lambda, t, d}& \ \ - \frac{1}{2}\tau - \frac{1}{2}k \lambda\\
s.t. & \ \
\begin{bmatrix}
\tau & -y^T X - t^T \\
-X^T y - t & X^T X + \rho I - \mathbf{D}(d)
\end{bmatrix} \succeq 0 \\
& \ \ \begin{bmatrix}
\lambda & t_i \\
t_i & d_i
\end{bmatrix} \succeq 0 ,\forall i.
\end{aligned}\tag{$SDP_{DCL}: dual$}
\end{equation}
\section{Certificate of exactness}\label{sec:cert}
Proposition \ref{prop:dominate} implies that
if $\nu_{PWG} = \nu_{\ell 0}$, then $\nu_{DCL} = \nu_{\ell 0}$.
Therefore all sufficient conditions for the exactness of (\ref{SDP_PWG}) readily carry over to (\ref{SDP_DCL}).
Authors of \cite{PilanciWainwrightGhaoui2015} provided a characterization of a \textit{certificate of exactness} for the continuous relaxation
of (\ref{SpMLbin}), as well as a specialized result on (\ref{SDP_PWG}). We restate their characterization result in Theorem \ref{pwg-cert}, and provide
a parallel result in Theorem \ref{thm:dualcert} for (\ref{SDP_DCL}).
\begin{theorem}[Corollary 2 in \cite{PilanciWainwrightGhaoui2015}]\label{pwg-cert}
The convex relaxation (\ref{SDP_PWG}) is exact if and only if there is a subset $S \subseteq \{1,...,p\}$, where $|S| \leq k$, such that there exists $\lambda \in \mathbb{R}_+$,
\begin{align}
|X_j^T M y | > \lambda, \ \ & \forall j \in S, \mbox{ and } \label{pwg_1}\\
|X_j^T M y | \leq \lambda, \ \ & \forall j \notin S, \label{pwg_2}
\end{align}
where $X_j\in \mathbb{R}^n$ is the j-th column of $X$, and $M := \left(I_n + \rho^{-1} X_S X_S^T\right)^{-1}$.
\end{theorem}
Using this result, the authors were able to prove a high-probability exact recovery condition for the special case of Gaussian ensembles. We leave the discussion
of Gaussian ensemble in the next section. Here we provide a parallel characterization of certificates of exactness for (\ref{SDP_DCL}).
\begin{theorem}\label{thm:dualcert}
Let $S \subseteq \{1,...,n\}$, $|S| = k$ and $z^*$ be a binary vector such that $z^*_i = 1, \forall i \in S$ and $z^*_i = 0, \forall i
\notin S$. Further let $b^*$ be the optimal solution of the ridge regression in the restricted subspace, i.e.,
\[
b^* \in \arg\min_{\beta\in \mathbb{R}^p} \left\{ \|X\beta - y\|_2^2 + \rho\|\beta\|_2^2 \ \middle| \ \beta_j = 0, \forall j\notin S \right\}
\]
Then $(b^*, b^* b^{*T}, z^*)$ is optimal to (\ref{SDP_DCL})
if and only if there exists a vector $\tilde{d} \in \mathbb{R}^p_+$ and
scalar $\tilde\lambda \in \mathbb{R}_+$ such that
\begin{align}
\rho^{-1} X^T X + I_p - \mathbf{D}(\tilde{d}) \succeq 0, &\label{cond:1}\\
\tilde\lambda = \tilde{d}_i \left(X_i^T M y \right)^2, &\qquad \forall i \in S, \label{cond:2}\\
\tilde\lambda \tilde{d}_i \geq \left(X_i^T M y \right)^2, &\qquad \forall i\notin S \label{cond:3}
\end{align}
where $M := \left(I_n + \rho^{-1} X_S X_S^T\right)^{-1}$.
\end{theorem}
The proof of Theorem \ref{thm:dualcert} exploits the optimality conditions of (\ref{SDP_DCL}) and its dual, and is
given in detail in the appendix section. We remark that one can directly show that the conditions in Theorem \ref{thm:dualcert}
are no stronger than those in Theorem \ref{pwg-cert}.
\begin{remark}
Suppose that $\lambda$ is the scalar such that
(\ref{pwg_1}) and (\ref{pwg_2}) hold, then (\ref{cond:1}) -- (\ref{cond:3}) hold for $\tilde{\lambda}$ and $\tilde{d}$, where
\[
\tilde{\lambda} := \ \max \ \left\{ \left(X_j^T M y\right)^2 : i\in S\right\}, \tilde{d}_i = \tilde{\lambda} \left(X_j^T M y\right)^{-2}, \forall i \in S,
\mbox{ and } \tilde{d}_i = 1,\forall i \notin S.
\]
Note that $\tilde{d}_i \in [0,1]$ for all $i$ by construction. Therefore (\ref{cond:1}) holds. $(\ref{cond:2})$ and $(\ref{cond:3})$ are also valid by construction.
\end{remark}
\section{Empirical comparison on exact recovery rate for Gaussian ensemble}\label{sec:empirical}
In this section we consider the special case of Gaussian ensemble, where the design matrix $X \in \mathbb{R}^{n\times p}$ is generated with i.i.d. N(0,1) entries.
A ``true" signal $\beta^*$ is generated to be $k$-sparse, i.e., it has only $k$ number of nonzero entries, and each nonzero entry is of the order $1/\sqrt{k}$.
The response vector $y$ is generated by $y = X\beta^* + \epsilon$, where $\epsilon$ has i.i.d $N(0,\gamma^2)$ entries.
The following result is established in \cite{PilanciWainwrightGhaoui2015}, which characterizes the size of $n$ needed to guarantee the exact recovery of the
support of $\beta^*$ with high probability.
\begin{theorem}
There are constants $c_0$ and $c_1$, such that the following holds. Suppose that we are given a sample size
$n > c_0\frac{\gamma^2 + \|\beta_S^*\|_2^2}{\beta_{min}^{*2}} \log p$,
and that we solve (\ref{SDP_PWG}) with $\rho = \sqrt{n}$. Then with probability at least
$1-2 e^{-c_1n}$, the relaxation (\ref{SDP_PWG}) is exact, i.e., $\nu_{PWG} = \nu_{\ell 0}$.
\end{theorem}
Here $\beta_{min}^*$ is the minimal nonzero entry (in absolute value) of $\beta^*$. Note that Proposition \ref{prop:dominate} ensures that under the same
conditions, $\nu_{DCL} = \nu_{\ell 0}$ with (at least the same) high probability.
In the remaining part of this section we empirically evaluate the exact recovery for the Gaussian ensemble case. We compare the
probabilities of exact recovery by (\ref{SDP_PWG}) and (\ref{SDP_DCL}) for various $n$ and $p$. To avoid potential numerical issues in solution precision,
we exploit Theorem \ref{pwg-cert} and Theorem \ref{thm:dualcert} to directly search for the certificates. Such a strategy enables us to test whether
(\ref{SDP_PWG}) and (\ref{SDP_DCL}) provide the global solution to (\ref{l0_card}) on large number of simulated data sets without explicitly solving the
semidefinite relaxations many times.
Given simulated data $(X,y,\beta^*)$, let $S$ denote the support of $\beta^*$. Let $\rho > 0$ be fixed, it is straightforward to test whether conditions in
Theorem \ref{pwg-cert} are satisfied. If so, then (\ref{SDP_PWG}) recovers the true support of $\beta^*$. The situation is slightly more complicated
for (\ref{SDP_DCL}) and Theorem \ref{thm:dualcert}. Here we describe a bisection algorithm to search for the dual certificates $\tilde{\lambda}$ and
$\tilde{d}$, provided that the support of $\beta^*$ is used as the index set $S$.
\subsection{A bisection algorithm to search for the dual certificates for (\ref{SDP_DCL})}
Without loss of generality we assume that $S = \{1,...,|S|\}$.
Firstly if $X_i^T M y = 0$ for some $i \in S$, then the convex relaxation (\ref{SDP_DCL}) is not exact unless the trivial case where $X^T M y = 0$ for all $i$.
Then without loss of generality we can assume that
$\tilde{d}_i = \tilde\lambda \left(X_i^T M y\right)^{-2}$, for all $i\in S$, and
$\tilde{d}_i = \tilde\lambda^{-1} \left(X_i^T M y\right)^{2}$, for all $i\notin S$. Therefore the problem of testing (\ref{cond:1}) -- (\ref{cond:3})
is then equivalent to testing whether there exists $\tilde\lambda > 0$ such that the following function is nonpositive,
\begin{equation}\label{eq:f_lambda}
f(\tilde\lambda) := \lambda_{\max} \left\{\begin{bmatrix}D_S(\tilde\lambda) & 0 \\ 0 & D_{\bar{S}}(\tilde\lambda)\end{bmatrix} - \rho^{-1} X^T X - I_p\right\},
\end{equation}
where $\lambda\{\cdot\}$ is the largest eigenvalue function, $D_S(\tilde\lambda)$ is a $|S|\times|S|$ diagonal matrix with diagonal entries
$\tilde\lambda \left(X_i^T M y\right)^{-2}, i=1,...,|S|$,
and similarly $D_{\bar{S}}(\tilde\lambda)$ is a diagonal matrix with diagonal entries $\tilde\lambda^{-1} \left(X_i^T M y\right)^{2}, i =|S|+1,...,p$.
Note that $f(\tilde\lambda)$ is a convex function when $\tilde\lambda>0$. This is because $\lambda_{\max}\{\cdot\}$ is a convex function
and non-decreasing in terms of diagonal entries, and $\tilde\lambda^{-1}$ is convex when $\tilde\lambda > 0$. It is known that a subgradient of
$f(\tilde\lambda)$ can be computed from an eigenvector associated with the largest eigenvalue in (\ref{eq:f_lambda}).
Indeed, let $u$ be such an eigenvector, then
\[
h(\tilde\lambda) := \sum_{i\in S}\left(X_i^T M y\right)^{-2} u_i^2 - \lambda^{-2} \sum_{i \in \bar{S}} \left(X_i^T M y\right)^{2} u_i^2 \in \partial f(\tilde\lambda).
\]
In other words, given any $\hat{\lambda}>0$, $f(\lambda) \geq f(\hat\lambda) + h(\hat\lambda) (\lambda- \hat\lambda)$ for all $\lambda > 0$.
A final ingredient needed for a bisection algorithm is the initial interval. Consider the diagonal entries of the matrix in (\ref{eq:f_lambda}),
obviously if $\tilde\lambda \left(X_i M y\right)^{-2} - (\rho^{-1}X_i^T X_i + 1) \geq 0$ for some $i \in S$, or
$\lambda^{-1} \left(X_i M y\right)^{2} - (\rho^{-1}X_i^T X_i + 1) \geq 0$ for some $i \notin S$, then $f(\tilde\lambda) \geq 0$.
Therefore we can restrict ourself in a region such that $\tilde\lambda \left(X_i M y\right)^{-2} - (\rho^{-1}X_i^T X_i + 1) \leq 0$ for all $i \in S$,
and $\lambda^{-1} \left(X_i M y\right)^{2} - (\rho^{-1}X_i^T X_i + 1) \leq 0$ for all $i \notin S$.
This provides initial upper and lower bounds such that if there exists $\hat\lambda$ such that $f(\hat{\lambda}) < 0$, $\hat\lambda$ must be in
\[
\left[\max_{i \in \bar{S}} \left\{\left(X_i M y\right)^{2} (\rho^{-1}X_i^T X_i + 1)^{-1}\right\}, \min_{i\in S} \left\{\left(X_i M y\right)^{2} (\rho^{-1}X_i^T X_i + 1)\right\}\right].
\]
Then the problem of testing whether there is $\tilde{\lambda}$ such that $f(\tilde{\lambda})$ can be solved by the following bisection algorithm,
\begin{enumerate}
\item Start with $\ell = \max_{i \in \bar{S}} \left\{\left(X_i M y\right)^{2} (\rho^{-1}X_i^T X_i + 1)^{-1}\right\}$ and
$u = \min_{i\in S} \left\{\left(X_i M y\right)^{2} (\rho^{-1}X_i^T X_i + 1)\right\}$;
\item Let $\hat{\lambda} = \frac{\ell+u}{2}$ and evaluate $f(\hat\lambda)$;
\item If $f(\hat\lambda) \leq 0$, return YES; otherwise compute $h(\hat\lambda)$;
\item If $h(\hat\lambda) = 0$, return NO. If $h(\hat\lambda) > 0$, $u \leftarrow \hat{\lambda} -\frac{\hat\lambda}{h(\hat\lambda)}$; otherwise if $h(\hat\lambda) < 0$,
$\ell \leftarrow \hat{\lambda} -\frac{\hat\lambda}{h(\hat\lambda)}$. If $u-\ell > \epsilon$, where $\epsilon$ is a fixed precision tolerance, then go to step 2. Otherwise return NO.
\end{enumerate}
\subsection{Numerical simulations}
Using this bisection algorithm, we conduct similar experiments as shown in Figure 1 of \cite{PilanciWainwrightGhaoui2015}. For each value of $p$ (denoted as $d$ in all plots), the true
sparsity is set as $\left\lceil \sqrt{p} \right\rceil$, and the number of data points $n=\alpha k \log(p-k)$. The true signal $\beta^{*}_i, (i=1,...k)$ is generated to be 1 or -1 with same probability. The
Figures \ref{fig:1} through \ref{fig:6} show the exact support recovery rate for $\alpha \in [1,10]$, when $\rho$ are chosen to be $2 \sqrt{n}, 3\sqrt{n}, 4\sqrt{n}, 6\sqrt{n}, 8\sqrt{n}, 12\sqrt{n}$.
The numerical simulation illustrates that (\ref{SDP_DCL}) can recover the support of true signals with significant less data points than that of (\ref{SDP_PWG}). Also the exact
recovery rate of (\ref{SDP_DCL}) appears to be much less sensitive to the choice of $\rho$. This result further motivates us to study scalable approximate methods, such
as those based on low rank factorization of the matrix $B$, to solve (\ref{SDP_DCL}).
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{Rho2-Num100.eps}
\caption{Exact support recovery rate when $\rho = 2\sqrt{n}$.}
\label{fig:1}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{Rho3-Num100.eps}
\caption{Exact support recovery rate when $\rho = 3\sqrt{n}$}
\label{fig:2}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{Rho4-Num100.eps}
\caption{Exact support recovery rate when $\rho = 4\sqrt{n}$}
\label{fig:3}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{Rho6-Num100.eps}
\caption{Exact support recovery rate when $\rho = 6\sqrt{n}$}
\label{fig:4}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{Rho8-Num100.eps}
\caption{Exact support recovery rate when $\rho = 8\sqrt{n}$}
\label{fig:5}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{Rho12-Num100.eps}
\caption{Exact support recovery rate when $\rho = 12\sqrt{n}$}
\label{fig:6}
\end{center}
\end{figure}
\newpage
\bibliographystyle{plain}
|
1,116,691,500,457 | arxiv | \section{Introduction}
Dialog based interaction between a customer and a bot may become tedious and irrelevant if the system’s beliefs are not consistent with the set of beliefs of the customer. Beliefs are cognitive representational states that represent the presumed context of the conversation perceived by each agent. A key limitation in today's conversational interfaces is their lack of robustness when understanding the latent beliefs. The inherent difficulties of conversational systems (chatbots) are further increased by the conditions under which these systems typically operate: increasingly larger vocabularies, large and diverse user populations, spontaneous input, etc. Unless mediated by better belief identification and robust reasoning mechanisms, these errors propagate to subsequent stages of processing in a dialog system and exert a considerable negative impact on the quality and ultimately the success of the interaction.
Identifying latent beliefs is a challenging task since it depends on various factors like identifying and analyzing context. Other challenges that exist in identifying latent beliefs is the reference to multiple concepts and the need to extract a large amount of facts, commonsense knowledge, anaphora resolution, and logical reasoning. Consider the example, \textit{``I am looking for a heavy course"}, it is difficult to identify the latent belief if the student is looking for a course which is heavy in terms of the workload; class size and/or the nature/quality of assignments.
We propose a novel approach for identifying more accurate beliefs over concept values in conversational systems by integrating information across multiple turns in the conversation. Traditional machine learning approaches train a system with extremely large dialog corpus that covers a variety of scenarios. Another approach is to build a system with a complex set of hand-crafted rules that may address some specific instances. Both approaches may be impractical in many real-world domains. Our framework is based on a combination of machine-learning mechanism and logical reasoning to understand the latent beliefs. Our model then evaluates the beliefs to tailor the dialog and make it consistent with the set of beliefs of the user. This process then helps drive the conversation in a meaningful way.
\section{Related Work}
Deep learning based dialog systems~\cite{miller2016key} use memory networks to learn the underlying dialog structure and carry out goal-oriented dialog.
However, they do not factor in beliefs or trigger epistemic rules in modifying the conversation given the evolving context.
In~\cite{williams2016dialog} Williams et.al, describe the dialog state tracking challenge and mention ``how the task of correctly inferring the state of the conversation - such as the user's goal - given all of the dialog history up to that turn'' is important. It is in this overall context, we propose that it is important to evaluate the probable beliefs held by the human and tailor the dialog system suitably to be consistent with the beliefs in order to hold a relevant conversation.
Although a number of attempts have been made to build dialog systems~\cite{henderson2014word}, ~\cite{Weston16}, ~\cite{williams2016dialog}, the use of epistemic rules in driving the dialog in a consistent way with the beliefs has not yet been tackled. Various approaches to dialog management have been proposed and these can be broadly classified into finite-state methods, probabilistic methods and deep learned methods.
There are recent motivating examples of works that make use of machine-learning to build intelligent dialog systems.
Traditional dialog systems are specialized for a domain and rely on slot-filling driven by a knowledge base and a finite-state model~\cite{lemon2006hierarchical}. The finite-state model represents the dialog structure in the form of a state transition network in which the nodes represent the system's interactions and the transitions between the nodes determine all the possible paths through the network.
Uckelman~\cite{sara2010obligationes} describes how in a formal dialog system, dynamic epistemic logic can be used in an \textit{Obligatio}, where two agents, an \textit{Opponent} and a \textit{Respondent}, engage in an alternating-move dialog to establish the consistency of a proposition.
Sadek et al.~\cite{Sadek:1997:AND:1622270.1622305} proposes a reasoning engine to build effective and generic communicating agents.
Motivated by the above works, we propose to identify the beliefs, use these beliefs to trigger epistemic rules, and use the assertions of the rules to drive the conversation by tailoring the states in a finite-state machine dialog system.
Opinion mining methods have been in use for over long time. Recently, these methods have been applied to dialog systems. Roy et al.~\cite{aaai_Roy_16} propose a novel approach to consider customer satisfaction to tailor the dialog. While general sentiment analysis methods are useful to understand a customer's mental situation, they may need to be complemented with more domain specific information leading to richer fine grained classes of sentiments.
For this reason, it is necessary to develop methods which take into account domain specific sentiments information.
\section{Supervised Approach for Automatically Learning Latent Beliefs}
A key issue in dialogue management is the design of the dialogue policy incorporating latent beliefs. While it is possible to design this by hand, such an approach has several shortcomings. Development is manually intensive and expensive, the systems are difficult to maintain and performance is often sub-optimal. In our previous work, we highlighted some of the issues in hand crafting belief rules~\cite{DBLP:conf/aaai/SangroyaASR18}.
An alternative to hand-crafting belief rules is to automatically learn them given a large annotated corpus of utterances and corresponding labeled beliefs. However, it is challenging to design a machine learning model as the performance of the model would be degraded if it is based on simple linguistic features such as \textit{bag-of-words} (See Figure~\ref{word_cloud}). Sometimes, the available training data consist of large number of closely related classes that makes the problem of belief identification more difficult.
\begin{figure}
\centering
\includegraphics[scale=0.2]{images/word_cloud.png}
\caption{Word Cloud of Dialog Utterances of a \textit{Curious} Student}
\label{word_cloud}
\end{figure}
It is challenging task to build an automatic system that can understand the latent beliefs of humans (more specifically students). For example, there may be a student who is confused and directionless, might come up with something like \textit{``I have no inkling of where I want my life to go and am unable to determine what classes to take. I'm interested in machine learning. Can you help me decide what to do?”} . There may be another student who has done his homework on what he wants but needs only little direction from the advisor. These two different categories of students needs different conversational flow and a different set of dialog policy needs to be tailored for each case.
We take the initial input from the student to advisory bot and categorize it into one of the tree categories based on there emotional belief. We consider 3 categories ``\textbf{curious}", ``\textbf{neutral}", ``\textbf{confused}" and manually annotate them to student's utterances for each dialog. Most of the conversations seem to be similar in nature. Therefore, it becomes a challenging task to annotate the data correctly and pass it to machine learning model.
Figure~\ref{fig:senti_methodology} illustrates the proposed methodology for identifying and updating the latent beliefs. Its inputs include the utterances and a domain ontology.
\begin{figure}
\centering
\includegraphics[scale=0.25]{images/arch.png}
\caption{Domain Specific Latent Belief Extraction Process}
\label{fig:senti_methodology}
\end{figure}
\subsection{Step 1: Classifying Latent Beliefs}
If an utterance belongs to more critical categories such as \textit{confused}, it is assigned a higher weight as compared to category such as \textit{curious}. This is intuitive that student with a more confused state of mind would need more attention and a specific dialog plan. The latent belief classification is done with the help of machine learning in an automatic fashion.
\subsection{Step 2: Enriching Beliefs using Domain Ontology}
We make use of knowledge mining derived using the domain ontology to update the latent beliefs. Our ontology consists of a large knowledge graph expressing the information about a domain. For example, this includes the course information, their easiness ratings, workload ratings and class size (See Figure~\ref{fig:meta_info}). This information is available in a structured RDF graph. Use of ontology assists in enriching the beliefs further. It also helps to map the utterance which can be in any natural language form to a structured information that can be used to tailor the FSM based dialog manager.
\begin{figure}
\centering
\includegraphics[scale=0.3]{images/meta_info.png}
\caption{Domain Knowledge for Belief Enrichment}
\label{fig:meta_info}
\end{figure}
\subsection{Epistemic Reasoning over Latent Beliefs and Domain Knowledge}
The extracted latent beliefs and the domain knowledge trigger the epistemic rules. For example ``\texttt{Belief (Student is confused) and Course\_Load(High) => Knows-Agent (student is not confident), Knows-Agent (should advise light courses)}'' asserts facts about the current epistemic state of the agent. We use \texttt{Prolog} to define epistemic rules and carry out the reasoning against facts and domain knowledge.
\subsection{Step 3: Learning the Model for Latent Belief Extraction}
We now train a LSTM (Long Short Term Memory) network to build a model that can automatically categorize the beliefs based upon the information described in previous section (See Algorithm~\ref{alg1}). Like many other studies of LSTM on text, words are first converted to low-dimensional dense word vectors via a word embedding layer. The first layer is therefore the embedding layer that uses 32 length vectors to represent each word. The next layer is the LSTM layer with 100 units. Finally, we use a dense output layer with 5 neurons (5 classes/labels) and a softmax activation function to make the predictions. We used \textit{Categorical\_Cross\_Entropy} as the loss function (in \textit{Keras}) alongwith ADAM optimizer. For regularization, we employ dropout to prevent co-adaptation. We run the experiment for 20 epochs with a batch size of 64.
\begin{algorithm}
\caption{Algorithm for Updating Dialog States using Latent Belief Extraction}
\label{alg1}
\begin{algorithmic}
\REQUIRE FSM for dialog
\REQUIRE User Input as Dialog Utterance $T$
\ENSURE Fine Grain Set of Latent Beliefs
\ENSURE Meaningful dialog states
\FOR{all Utterances $r$ in $T$ }
\FOR{all states ${s_i}$ of FSM}
\STATE Remove Stopwords and Punctuations
\STATE Convert to Lowercase
\STATE Tokenize
\STATE Mark special name-entities
\FOR{all sentence $m$ in $r$ }
\STATE Extract Latent Beliefs {$Curious$, $Confused$ etc.}
\STATE Using Domain Ontology {$courses$, $workload$, $size$ etc., update latent beliefs $b$}
\STATE Using Information Extraction, update $b$
\STATE Using facts ${F_i}$ and EPISTEMIC\_rulebase, fill slots of Dialog manager Frames
\STATE update weight ${w_i}$ of state ${s_i}$ using Latent Beleifs and Epistemic Reasoning
\STATE Using threshold and weights, build meaningful states of FSM (ask\_states, skip\_states)
\ENDFOR
\ENDFOR
\ENDFOR
\end{algorithmic}
\end{algorithm}
In our experiments we consider dialog dataset student advisor conversations from \url{https://www.ibm.com/blogs/research/2018/07/sentence-selection-dstc7}. We manually annotate latent beliefs across three categories such as \emph{Curious, Confused and Neutral}. Total number of utterances after data processing were 3500 (Figure~\ref{fig:belief_classes}). The clean-up process involves converting text to lower case; tokenizing the sentences; and removal of punctuations and stopwords. Each input utterance as input is converted into a vector form. We identify the top 300 unique words and every word in this vocabulary is given a index. If the word is not present in vocabulary we consider it as 0. For Example: \textit{I am very disappointed today}. The vector representation of this sentence is [10, 100, 23, 467, 0]. Next, we need to truncate and pad the input sequences so that they are all the same length. We take the max length of utterance to be 50. We divide the data into training set (75\%) and test set (25\%). We ensure that we do not have data sparsity issue i.e. We keep approximate equal proportion of data for each class.
\subsection{Experimental Results}
\begin{figure}
\centering
\includegraphics[scale=0.45]{images/belief_classes_v2.png}
\caption{Number of Utterances in Training Data for each Category (Latent Beliefs)}
\label{fig:belief_classes}
\end{figure}
We evaluate the proposed method on the seventh Dialog State Tracking Challenge (DSTC-7) dataset (``Flex Data: Student – Advisor dialogue"~\cite{Flex}). We used 3500 conversations including list of paraphrases for each utterance as training data for belief classification. It becomes a little challenging to tag them to correct categories because of lexical similarity between these three classes. We got 84\% classification accuracy for latent belief extraction. We also tried bag-of-words and naive bayes models that performed badly for this experiment. As we discussed earlier, this is possible due to lot of common words between three classes.
\subsection{Example: Tailoring the Chatbot}
\label{chatbot}
We parse a dialog through Dependency parsers (such as Stanford-CoreNLP, GATE, MITIE etc.) and
extract triples from the description by focusing on the dependencies identified among nouns and verbs. For example, for a description \textit{''I prefer morning classes as I sleep early at night"}, triples such as (\textit{I, prefer, morning classes}) are extracted. Once the triples are extracted, we use a hand-crafted fact-assertion rulebase to assert facts implied by the triples. This is done by evaluating the triples in the context of a student-advisor ontology, synonym-and-slang dictionary, information-extraction patterns that are relevant for the category of the belief, and by triggering the fact-assertion rules. The latent belief identification process helps to understand the student's emotional beliefs and tailor the conversations accordingly.
We assume that we have a hand-crafted dialog-management finite-state-machine (FSM) to carry out the dialog with the student. The FSM operates on slots that are filled by extracting information from the input dialog and subsequent interaction. The probable beliefs of the student that were asserted as facts are then evaluated by the epistemic rules encoded in a knowledge base for the domain. The rules make assertions about the states in the FSM that need to be skipped and the states that need to be evaluated in order to be consistent with the beliefs of the student. The subsequent dialog is carried out and the next set of beliefs are then asserted. The cycle then continues. As shown in Table~\ref{Sample_Output}, we can observe that chatbot is able to skip some FSM states as a result of latent beliefs. The beliefs and epistemic rules helped tailor the dialog to the user's expectations. In this work, we demonstrate the overall architecture where we use the output of RNN based belief identification model as an input to the epistemic rule engine. Our approach is generic and can be applied easily in any other domains.
\rowcolors{2}{gray!25}{white}
\begin{table}
\footnotesize
\begin{center}
\caption{Sample Output of Dialog System}
\label{Sample_Output}
\begin{tabular}{|p{3.2cm}|p{4.1cm}|}
\rowcolor{gray!50}
\hline
Advisor & Hi! I am your advisor. You can ask any doubts in selection of your courses for next semester.
\\
\hline
\hline
Student & I am a junior year student with interest in statistics and data analysis
\\
\textcolor{blue}{Output ($ML$-$Model$)} & \texttt{\textcolor{blue}{Latent Belief: Student(Curious)}}\\
\hline
Advisor \textcolor{red}{(skipstate(ask\_interest), skipstate(ask\_semester))} & Do you have any specific requirement \textit{about the workload} of the course
\\
\hline
Student & I would prefer a \textit{class with lighter workload} and higher helpfulness rating
\\
\hline
Advisor & Do you have any timing preferences?
\\
\hline
Student & I \textit{prefer morning classes} as I sleep early at night.\\
\hline
Advisor \textcolor{red}{(SkipState(ask\_extra\_details))} & I would advise you STATS250 ``Statistics and Data Analysis" which is an easy course.\\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusion}
This paper presents a supervised approach to identify latent beliefs from dialog utterances and tailor the dialog. In future, we would like to investigate reinforcement learning based approaches for belief identification.
\bibliographystyle{aaai} |
1,116,691,500,458 | arxiv | \section{Introduction}
\label{sec:1}
microRNAs (miRNAs) --short, endogenous, noncoding RNAs that operate post-transcriptionally via sequence-specific binding to target RNAs-- are increasingly recognized as key actors in the regulation of eukaryotic gene expression \cite{bartel,flynt,cech,gurt,metaz}. Following transcription (from either introns of protein-coding genes or from miRNA-specific genes) and maturation, miRNAs get incorporated into specialized, multiprotein complexes known as RISCs (short for RNA-induced silencing complexes) \cite{risc}. Once within a RISC, the miRNA provides the pattern to bind specific sites called miRNA response elements (MREs) found on their target RNAs \cite{chan,why}. Effective base pairing typically requires 6- to 9-nucleotide complementarity, and leads to negative gene expression control through either mRNA destabilization or translational repression \cite{chek,jona,djur}. The fact that miRNA expression is significantly tissue-specific places miRNAs at the center of the regulatory layer that controls the composition of the protein repertoire and cell type specificity \cite{bart,liang,fran,eber}. Still, many aspects of miRNA biology suggest that this role might be exerted through a broader and more complex, yet possibly more subtle, class of mechanisms.
In first place, miRNAs appear to be highly conserved in vertebrates and invertebrates, and their mRNA target structure displays a significant degree of conservation in higher organisms \cite{bere,josh}. For instance, more than half of human genes are conserved miRNA targets, including a large number of weak-interacting sites that appear to be under selective pressure to be maintained \cite{frie}. Such a strong degree of conservation suggests that protein levels may need to be fine tuned within extremely precise ranges \cite{baek}. Quantitative studies together with the statistical overrepresentation of noise-buffering motifs within the miRNA-RNA network indeed supports this idea \cite{shim,tsang,reda}, and recent experiments have confirmed miRNA's ability to stabilize output levels for lowly expressed proteins \cite{sici}. Yet, the amount of noise reduction that can be achieved even in optimal conditions does not seem to justify a view of noise suppression as the key evolutionary driver for a significantly conserved miRNA targeting pattern \cite{wangs,das,ober,schm}.
Secondly, miRNA targets are known to include, together with messenger RNAs, a host of ncRNA species like lncRNAs as well as pseudogenes \cite{guil,hans,eber2}. On one hand, miRNA sponging by ncRNAs can clearly be critical in determining both miRNA levels and their potential for translational repression. On the other, it substantially increases the complexity of the network of miRNA-RNA interactions. It is now clear that each long RNA molecule can typically be targeted by multiple miRNAs, while every miRNA can interact with a very large number of distinct RNAs, generating an extended interaction network stretching across the entire transcriptome \cite{suma,helw,kimd,zavo}. Now the ability of miRNAs to regulate gene expression is ultimately linked to the overall target availability, and tends to get weaker as the number of targets (more precisely, of potential binding sites) increases, the so-called `dilution' effect \cite{arve}. This leaves room to search for alternative mechanisms through which miRNAs could exert a regulatory function, even at the non-local (up to system-scale) level.
The heterogeneity of the miRNA-RNA network and the fact that repression potential depends tightly on molecular levels suggest that competition to bind miRNAs might be a contributing factor in the establishment of robust protein profiles \cite{levine,fzor}. In rough terms, the essence of the so-called ceRNA hypothesis (whereby `ceRNA' stands for `competing endogenous RNA') is that, due to a cross-correlation of molecular levels, competition can induce an effective positive coupling between miRNA targets, such that a perturbation affecting the level of one target could be broadcast to its competitor via the subsequent shift in miRNA availability \cite{salm}. In this respect, one might say that RNAs form a sort of `molecular ecosystem', where mutual dependencies can be established post-transcriptionally via miRNA-mediated interactions driven by competition. The ceRNA scenario has received much attention since its formulation, both {\it ex vivo} and in synthetic systems (see e.g. \cite{tay,vano,karr,tayy,yuany,sgro}). Effective interactions coupling RNAs targeted by the same miRNAs (which can be probed e.g. by over-expressing miRNAs or targets) are now known to be implicated in a variety of processes, from development and differentiation \cite{fati}, to stress response \cite{stress} and disease \cite{alva,anas}, and have been investigated in connection to their perspective therapeutic usefulness \cite{sanc}.
Still, it has also become clear that the theoretical appeal of the ceRNA effect is not easily translated into quantitative understanding. A key issue is that of fine tuning. Several conditions clearly factor in the emergence of the ceRNA scenario. The possibility to turn competition between miRNA targets into an effective positive coupling between them presupposes for instance a cross-coordination of molecular levels, as a large excess (resp. scarcity) of miRNAs with respect to targets or binding sites will necessarily result into a completely repressed (resp. unrepressed) profile \cite{jens,denzler}. The ceRNA scenario would naturally become less realistic if kinetic parameters had to be tightly tuned in order to allow for ceRNA crosstalk conditions to arise. In addition, experiments suggest that a relatively small number of targets are usually sensitive to modulation in miRNA availability. Moreover, which targets are responsive depends on miRNA levels \cite{alau,boss,denz}. The emergent selectivity and adaptability of ceRNA interactions should be reconciled with the heterogeneity observed in the miRNA-RNA interaction network in which each miRNA can regulate up to hundreds of targets.
Mathematical and {\it in silico} models developed in recent years have shed light on several of these issues and revealed many unexpected traits \cite{wang,laix}. This chapter aims at reviewing the methods employed and the key features of the ceRNA scenario that such studies suggest.
Our starting point is a generic, minimal deterministic mathematical model of post-transcriptional regulation whose steady states can be fully characterized analytically and numerically. Despite its roughness, it allows to precisely quantify the sensitivity of a ceRNA to alterations in the level of one of its competitors, sufficing to capture many of the central characteristics of miRNA-based regulation from basic assumptions about the underlying processes. In particular, miRNA-ceRNA interaction strengths and silencing/sequestration mechanisms emerge, together with the relative abundance of regulators and targets, as key factors for the onset and character of ceRNA crosstalk, including its selectivity. Moreover, heterogeneities in kinetic parameters as well as in miRNA-ceRNA interaction topology are major drivers of ceRNA crosstalk in a broad range of parameter values. The picture obtained at stationarity can be extended to out-of-equilibrium regimes. In particular, one can characterize a `dynamical' ceRNA effect, which can be stronger than the equilibrium one, as well as the typical timescales required to reach stationary crosstalk.
Passing from a deterministic to a stochastic description, one can address the behaviour of fluctuations in molecular levels and evaluate the ability of miRNA-based regulatory elements to process noise. We will show in particular that the ceRNA mechanism can provide a generic pathway to the reduction of intrinsic noise both for individual proteins and for complexes formed by sub-units sharing a miRNA regulator (which might explain why interacting proteins are frequently regulated by miRNA clusters). The processing of extrinsic (transcriptional) noise is more involved. While ceRNA crosstalk is generically hampered by it, specific patterns of transcriptional correlations can actually result in enhanced noise buffering and in the emergence of complex (e.g. bistable) expression patterns. On the other hand, one can quantify the physical limits to crosstalk intensity by considering how different sources of noise affect it. It turns out that the size of target derepression upon the activation of its competitor is a crucial determinant of the strength of miRNA-mediated ceRNA regulation. When it is sufficiently large, post-transcriptional crosstalk can be as effective as direct transcriptional regulation in controlling expression levels. In specific cases, ceRNA crosstalk may even represent the most effective mechanism to tune gene expression.
An especially important question (and a difficult one, in view of the fact that the effect can be rather modest) concerns the quantification of ceRNA crosstalk intensity, and specifically the identification of unambiguous crosstalk markers that can be validated both experimentally and through the analysis of transcriptional data. We shall examine a few alternatives that have been employed, highlighting the different motivations underlying their use, their physical meaning and their respective limitations.
\section{Models and methods}
\label{sec:2}
\subsection{Deterministic model}
The simplest mathematical representation of the dynamics of $N$ ceRNA species and $M$ miRNA species interacting in a miRNA-ceRNA network is based on deterministic mass-action kinetics. We shall denote by $m_i$ the level of ceRNA species $i$ (with $i$ ranging from 1 to $N$), by $\mu_a$ the level of miRNA species $a$ (ranging from 1 to $M$), and by $c_{ia}$ the levels of miRNA-ceRNA complexes. Based on experimental evidence, one can assume that all miRNA molecules are `active', i.e. bound to an Argonaute protein and ready to attach to a target ceRNA. This allows to discard the kinetic steps leading to the formation of the RNA-induced silencing complex (RISC). In such conditions, concentration variables evolve in time due to
\begin{enumerate}
\item synthesis and degradation events,
\item complex binding and unbinding events,
\item the processing of complexes.
\end{enumerate}
The latter in turn can follow two distinct pathways: a catalytic one, leading to the degradation of the ceRNA with the re-cycling of the miRNA; and a stoichiometric one, where both molecules are degraded, possibly after sequestration into P-bodies \cite{vale,bacc}. The relevant processes (see Fig. 1A and B for a sketch) are therefore
\begin{equation}\label{processes}
\arraycolsep=1pt\def1.8{1.8}
\begin{array}{r@{}l}
&{}\emptyset \xrightleftharpoons[d_i]{b_i} m_i ~~~~~~~~~~~~
\emptyset \xrightleftharpoons[\delta_a]{\beta_a} \mu_a ~~~~~~~~~~~~
\mu_a+m_i \xrightleftharpoons[k_{ia}^-]{k_{ia}^+} c_{ia}\\
&{}c_{ia} \xrightharpoonup{\sigma_{ia}} \emptyset ~~~~~~~~~~~~
c_{ia} \xrightharpoonup{\kappa_{ia}} \mu_a
\end{array}~~~~~~~.
\end{equation}
Correspondingly, the mass action kinetic equations take the form (see e.g. \cite{figl,bosi,meht})
\begin{equation}\label{uno}
\arraycolsep=1pt\def1.8{2}
\begin{array}{r@{}l}\frac{d m_i}{dt}&{}=b_i-d_i m_i-\sum_a k_{ia}^+ m_i\mu_a +\sum_a k_{ia}^- c_{ia}~~, \\
\frac{d \mu_a}{dt}&{}=\beta_a-\delta_a \mu_a-\sum_i k_{ia}^+ m_i\mu_a + \sum_{i}(k_{ia}^-+\kappa_{ia})c_{ia}~~,\\
\frac{d c_{ia}}{dt}&{}= k_{ia}^+ m_i \mu_a-(\sigma_{ia}+\kappa_{ia}+k_{ia}^-)c_{ia}~~,
\end{array}
\end{equation}
where the physical meaning of parameters is summarized in Table \ref{tab1} and where the indices $i$ and $a$ range from $1$ to $N$ and from $1$ to $M$, respectively.
\begin{table}[b]
\caption{Variables and parameters appearing in the basic model, Eq. (\ref{uno}). Note that the levels of molecular species can be specified by copy numbers (as indicated below) as well as by (continuous) concentrations, depending on whether the modeling framework is stochastic (see Section \ref{stochastic}) or deterministic (as in Eq. (\ref{uno})), respectively. \label{tab1}}
\begin{tabular}{p{1.5cm}p{2.4cm}p{7.5cm}}
\hline\noalign{\smallskip}
Variable & Units & Description \\
\noalign{\smallskip}\svhline\noalign{\smallskip}
$m_i$ & molecules & Number of free copies of ceRNA species $i$\\
$\mu_a$ & molecules & Number of free copies of miRNA species $a$\\
$c_{ia}$ & molecules & Number of copies of $i-a$ complex\\ [0.2cm]
\hline\noalign{\smallskip}
Parameter & Units & Description \\
\noalign{\smallskip}\svhline\noalign{\smallskip}
$b_i$ & molecule min$^{-1}$ & Transcription rate of ceRNA species $i$\\
$d_i$ & min$^{-1}$ & Degradation rate of ceRNA species $i$\\
$\beta_a$ & molecule min$^{-1}$ & Transcription rate of miRNA species $a$\\
$\delta_a$ & min$^{-1}$ & Degradation rate of miRNA species $i$\\
$k_{ia}^+$ & molecule$^{-1}$ min$^{-1}$ & $i-a$ complex association rate\\
$k_{ia}^-$ & min$^{-1}$ & $i-a$ complex dissociation rate\\
$\kappa_{ia}$ & min$^{-1}$ & Catalytic decay rate (with miRNA re-cycling) of $i-a$ complex\\
$\sigma_{ia}$ & min$^{-1}$ & Stoichiometric decay rate (without miRNA re-cycling) of $i-a$ complex\\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig1}
\end{center}
\caption{{\bf (A)} Sketch of an interaction network formed by miRNAs and their targets (ceRNAs). The network is a weighted bipartite graph. Line thickness is proportional to the coupling strength (i.e. to the miRNA-ceRNA binding affinity). {\bf (B)} Sketch of the individual processes lumped in each interaction represented in (A). Details of reactions and rates are given in Eq. (\ref{processes}). {\bf (C)} Sketch of the behaviour of the level of free targets (ceRNA or miRNA) as a function of the level of free regulators (miRNA or ceRNA, respectively). {\bf (D)} Sketch of the ceRNA mechanism: competition to bind a miRNA can induce an effective positive coupling between its targets.}
\label{Fig1}
\end{figure}
For several purposes it is useful to introduce the ``stoichiometricity ratio''
\begin{equation}\label{sr}
\alpha_{ia}=\frac{\sigma_{ia}}{\sigma_{ia}+\kappa_{ia}}
\end{equation}
quantifying the probability that the $i-a$ complex is processed without miRNA re-cycling.
\paragraph{{\bf \textsf{Note}} }
The model just described, that is the one on which we will mostly focus, is limited to miRNAs and ceRNAs and excludes, for instance, upstream regulators (e.g. transcription factors, TFs) and downstream products (e.g. proteins). Integrating some of these ingredients is however straightforward and it has been done in the literature. For instance, upstream TFs independently regulating the synthesis of ceRNAs and miRNA can be accounted for by assuming that transcription requires the cooperative binding of $H$ TF molecules for each of the RNA species involved (labeled $\ell$, including both miRNAs and ceRNAs). Denoting by $k_{\textrm{on}}$ and $k_{\textrm{off}}$ the binding and unbinding rates of TFs to DNA, respectively, the fractional occupancies of TF binding sites on the DNA evolve as
\begin{equation}
\frac{d n_{\ell}}{dt} = k_{\textrm{on}}(1-n_{\ell}) f_\ell^H - k_{\textrm{off}} n_{\ell}~~,
\end{equation}
where $n_{\ell}$ ($0\leq n_{\ell}\leq 1$) stands for the probability that the binding site for the TF controlling the transcription of species $\ell$ is occupied and $f_\ell$ stands for the level of the TF controlling species $\ell$. In most cases, the variables $n_\ell$ will equilibrate on timescales much shorter than those characterizing the dynamics of molecular levels \cite{alon}. In such conditions, each $n_{\ell}$ can be thought to take on its stationary value, i.e.
\begin{equation}
\avg{n_\ell}=\frac{f_\ell^H } {f_\ell^H + K^H}~~~~~,~~~~~K=\left(\frac{k_{\textrm{off}}}{k_{\textrm{on}}}\right)^{1/H}~~.
\end{equation}
Such occupancies in turn modulate the transcription rates appearing in (\ref{uno}). In particular, the effective transcription rate of ceRNA (resp. miRNA) species $i$ (resp. $a$) becomes $b_{i,{\rm eff}}= b_i\,\avg{n_i}$ (resp. $\beta_{a,{\rm eff}}= \beta_a\,\avg{n_a}$) \cite{prob}.
An extension of (\ref{uno}) including downstream species (proteins) is briefly discussed in Sec. \ref{prots}.
\subsection{Analysis of the steady state: threshold behaviour and competition-induced responses}
At steady state, molecular populations evolving according to Eqs (\ref{uno}) are given by the solutions of
\begin{equation}\label{eq:steadystateM}
\arraycolsep=1pt\def1.8{2}
\begin{array}{r@{}l}
\avg{m_i} &{}= \frac{b_i + \sum_a k_{ia}^- \avg{c_{ia}} }{d_i + \sum_a k_{ia}^+ \avg{\mu_a} }~~ \\
\avg{\mu_a} &{}= \frac{\beta_a + \sum_i (k_{ia}^- + \kappa_{ia}) \avg{c_{ia}} }{ \delta_a + \sum_i k^+_{ia} \avg{m_i} }~~\\
\avg{c_{ia}} &{}= \frac{k^+_{ia} \avg{\mu_a} \, \avg{m_i} }{\sigma_{ia} + \kappa_{ia} + k^-_{ia}}~~
\end{array}~~.
\end{equation}
(We shall henceforth represent the steady state level of species $x$ by angular brackets, i.e. $\avg{x}$.) These conditions have been rigorously shown to describe the unique, asymptotically stable steady state of (\ref{uno}) \cite{flon}. Eqs (\ref{eq:steadystateM}) provide a full description of the molecular network in terms of the populations of all species at sufficiently long times, given all kinetic parameters, and are easily solved numerically for any $N$ and $M$. It is however possible to get a mathematical intuition about how miRNAs affect ceRNA levels at stationarity by eliminating complexes (i.e. $\avg{c_{ia}}$) from (\ref{eq:steadystateM}). This allows to re-cast the steady-state in terms of miRNA and ceRNA levels only. Specifically, one gets
\begin{equation}\label{ss}
\arraycolsep=1pt\def1.8{1.8}
\begin{array}{r@{}l}
\avg{m_i} &{}= \frac{m_i^\star}{1+\sum_a \mu_a/\mu_{0,ia}}~~, \\
\avg{\mu_a} &{}= \frac{\mu_a^\star}{1+\sum_i m_i/m_{0,ia}}~~,
\end{array}
\end{equation}
where $m_i^\star\equiv b_i/d_i$ and $\mu_a^\star=\beta_a/\delta_a$ stand for the maximum values achievable by ceRNA and miRNA levels at stationarity, while
\begin{equation}\label{thresholds}
\arraycolsep=1pt\def1.8{1.8}
\begin{array}{r@{}l}
m_{0,ia} &{}= \frac{\delta_a}{k_{ia}^+}\left(1+\frac{k_{ia}^-+\kappa_{ia}}{\sigma_{ia}}\right)~~, \\
\mu_{0,ia} &{}= \frac{d_i}{k_{ia}^+}\left(1+\frac{k_{ia}^-}{\sigma_{ia}+\kappa_{ia}}\right)~~
\end{array}
\end{equation}
represent `reference' concentrations that depend on the specific miRNA-ceRNA pair. For sakes of simplicity, we shall refer to these values as ``thresholds''. The gist of (\ref{ss}) is the following (see Fig. \ref{Fig1}C) \cite{figl}:
\begin{description}
\item[{\bf Free or unrepressed regime}~:]If the levels of all miRNA species interacting with ceRNA $i$ are sufficiently low (specifically, much lower than the respective thresholds $\mu_{0,ia}$, so that $\sum_a \mu_a/\mu_{0,ia}\ll 1$), then the steady-state level of ceRNA $i$ will be very close to the maximum possible, $m_i^\star$. In such conditions, ceRNA species $i$ will be roughly insensitive to changes in miRNA levels. We call this the `unrepressed' or `free' regime for ceRNA $i$.
\item[{\bf Susceptible regime}~:]As the quantity $\sum_a \mu_a/\mu_{0,ia}$ increases, e.g. following an increase in the level of one or more miRNA species, $\avg{m_i}$ deceases in a sigmoidal fashion. This occurs most notably when $\sum_a \mu_a/\mu_{0,ia}\simeq 1$ (corresponding, for $M=1$, to a miRNA level close to the threshold value $\mu_{0,ia}$). Here ceRNA $i$ is very sensitive to a change in miRNA levels. We shall therefore term this the `susceptible' regime for ceRNA $i$.
\item[{\bf Repressed regime}~:]When miRNA levels become sufficiently large, ceRNA $i$ will eventually become fully repressed. In order for this to occur, it suffices that $\sum_a \mu_a/\mu_{0,ia}\gg 1$ (which occurs e.g. when the level of at least one of the miRNA species targeting $i$ significantly exceeds its corresponding threshold $\mu_{0,ia}$). We shall call this the `repressed' regime for ceRNA $i$.
\end{description}
(Notice that, because the role of miRNAs and ceRNAs is fully interchangeable, similar regimes can be defined for miRNAs, with the reference concentrations $m_{0,ia}$ playing the role of the threshold ceRNA levels characterizing the distinct regimes.)
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig2}
\end{center}
\caption{Characterization of the steady state for a system with 2 ceRNA species competing for one miRNA species. {\bf (A)} Steady state molecular levels as a function of the miRNA transcription rate $\beta_1$. {\bf (B)} Fano Factor (FF) of each molecular species versus $\beta_1$. {\bf (C)} Coefficient of variation (CV) of each molecular species versus $\beta_1$. {\bf (D)} Steady state molecular levels as a function of the transcription rate of ceRNA 1, $b_1$. {\bf (E)} Fano Factor of each molecular species versus $b_1$. {\bf (F)} Coefficient of variation of each molecular species versus $b_1$. In panels (A) and (D), continuous lines describe analytical results (from Eq~(\ref{ss})) while markers denote mean values obtained from stochastic simulations performed using the Gillespie algorithm (see Sec. \ref{gillespie}). In panels (B), (C), (E) and (F), continuous lines describe analytical results obtained by the Linear Noise Approximation (see Sec. \ref{lna}) while markers represent numerical results derived from stochastic simulations. Parameter values are reported in Table \ref{pars}.}
\label{Fig2}
\end{figure}
Fig. \ref{Fig2}A and D report results obtained for the case $N=2$, $M=1$ (two ceRNA species competing for a single miRNA regulator). One sees that ceRNA levels get increasingly repressed as the miRNA transcription rate increases while all other parameters remain fixed (Fig. \ref{Fig2}A). The range of values of $\beta_1$ where ceRNA levels change most strongly corresponds to the susceptible regime. One also sees that ceRNAs 1 and 2 have slightly different thresholds ($\mu_{0,11}\simeq 2$ and $\mu_{0,21}\simeq 15$), as ceRNA 1 is clearly sensitive to variations in miRNA availability for smaller values of $\beta_1$ compared to ceRNA 2. Fig. \ref{Fig2}D shows instead how molecular levels change upon modulating the transcription rate of ceRNA species 1. As $b_1$ increases, $m_1$ grows as expected while concentration of free miRNAs decreases as they increasingly engage targets. This in turn derepresses the other ceRNA species, whose level also increases as the transcription rate of ceRNA 1 is upregulated. That the level of ceRNA 2 can increase upon changing $b_1$ is the key signature of the miRNA-mediated crosstalk that can be established between competing RNAs.
\begin{table}[b]
\caption{Values of kinetic parameters used in the different figures. \label{pars}}
\begin{tabular}{p{2.6cm} p{1.4cm} p{0.9cm} p{0.9cm} p{0.9cm} p{0.9cm} p{0.9cm} p{1.1cm} p{1.1cm} p{1.1cm}}
\hline\noalign{\smallskip}
Parameter & Fig. 2A--C & 2D--F & 3A & 3B & 3C & 3D & 5 & 6A,B & 6C,D\\
\noalign{\smallskip}\svhline\noalign{\smallskip}
$b_1$ [molec min$^{-1}$] & 10 & -- & 10 & 20 & 2 & 1 & 1 (mean) & -- & 10\\
$b_2$ [molec min$^{-1}$] & 15 & 10 & 15 & 10 & 10 & 10 & 1 (mean) & 0 & 0\\
$\beta_1$ [molec min$^{-1}$] & -- & 20 & 15 & -- & 15 & -- & -- & 15 & 15\\
$d_1$ [min$^{-1}$] & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.005 & 0.1 & 0.1\\
$d_2$ [min$^{-1}$] & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.005 & 0 & 0.1\\
$\delta_1$ [min$^{-1}$] & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.1 & 0.01 & 0.1 & 0.1 \\
$k_{11}^+$ [molec$^{-1}$ min$^{-1}$] & $e^{-3}$ & $e^{-2}$ & 1 & $e^{-2}$ & $e^{15}$ & $e^{15}$ & $e^{-2}$ & shown & caption\\
$k_{21}^+$ [molec$^{-1}$ min$^{-1}$] & $e^{-5}$ & $e^{-4}$ & $e^{-3}$ & $e^{-3}$ & $e^{-4}$ & $e^{-4}$ & $e^{-3}$ & 0 & caption\\
$k_{11}^-$ [min$^{-1}$] & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.1 & 0.001 & 0.001\\
$k_{21}^-$ [min$^{-1}$] & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.1 & 0 & 0.001\\
$\kappa_{11}$ [min$^{-1}$] & 0.001 & 0.001 & 0.001 & 0.001 & 0.1 & 0.1 & 0.05 & 0.001 & 0.001\\
$\kappa_{21}$ [min$^{-1}$] & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 & 0.05 & 0 & 0.001\\
$\sigma_{11}$ [min$^{-1}$] & 1 & 1 & 1 & 1 & 1 & 1 & 0.001 & 1 & 1\\
$\sigma_{21}$ [min$^{-1}$] & 1 & 1 & 1 & 1 & 1 & 1 & 0.001 & 0 & 1\\
\noalign{\smallskip}\hline\noalign{\smallskip}
\end{tabular}
\end{table}
\paragraph{{\bf \textsf{Note}} }
The reference levels (\ref{thresholds}) ultimately represent the combinations of parameters that are most relevant in order to elucidate many of the network's features. As one would expect, the leading behaviour for $\mu_{0,ia}$ is determined by the ratio $d_i/k_{ia}^+$: the threshold gets smaller as the miRNA-ceRNA interaction gets stronger (i.e., lower miRNA levels suffice to repress a target in presence of stronger coupling), whereas larger intrinsic ceRNA decay rates impose larger repression thresholds. Expectedly, catalytic decay rate affects the thresholds $\mu_{0,ia}$ and $m_{0,ia}$ differentially: while the former decreases as catalytic processing gets more efficient (i.e., miRNA recycling strengthens repression by effectively increasing miRNA availability), $m_{0,ia}$ increases as $\kappa_{ia}$ gets larger (i.e., higher ceRNA levels are required to repress miRNAs at high catalytic processing rates). Note however that $m_{0,ia}$ diverges as $\sigma_{ia}\to 0$, i.e. when all miRNAs are recycled after complex degradation. In other words, in absence of stoichiometric processing of the $i-a$ complex, miRNA $a$ can never be repressed by ceRNA $i$. This implies that, in order for the ceRNA scenario described above to take place, it is necessary that the stoichiometricity ratio $\alpha_{ia}$, Eq. (\ref{sr}), is strictly positive.
\subsection{Stochastic model}
\label{stochastic}
Like all regulatory processes \cite{kond}, the individual reactions reported in (\ref{processes}), i.e. transcription, degradation and titration events due to miRNA-ceRNA interactions, are intrinsically stochastic. This means in practice that molecular levels evolving in time according to (\ref{processes}) are bound to be subject to random fluctuations, with the strength of the noise affecting each molecular species roughly proportional to the square root of its mean. After a transient, concentrations will stabilize and fluctuate around the steady state of the deterministic model (\ref{uno}), described by (\ref{eq:steadystateM}). The deterministic model thereby yields a description of the miRNA-ceRNA network that is all the more accurate when the system is well mixed and concentrations are sufficiently large, making noise negligible. Besides giving a more realistic description of the dynamics of molecular populations, accounting for randomness is however crucial to characterize ceRNA crosstalk in detail, and particularly to disentangle competition-induced effects from fluctuation-induced ones. We shall now therefore briefly review some of the frameworks that have been employed to analyze the stochastic dynamics of (\ref{processes}).
\subsubsection{The master equation}
The direct mathematical route to account for stochasticity is based on the chemical Master Equation (ME) \cite{vkam}, which describes the time evolution of the probability $P(\pmb{\mu},{\bf m},{\bf c} ,t)$ to find the system with prescribed molecular levels ${\bf m}=\{m_i\}_{i\in\{1,\dots,N\}}$ for ceRNAs, $\pmb{\mu}=\{\mu_a\}_{a\in\{1,\dots,M\}}$ for miRNAs and ${\bf c}=\{c_{\ell}\}_{\ell\in{1,\dots,M\cdot N}}$ for the $N\cdot M$ species of miRNA-ceRNA complexes at time $t$. The ME reads
\begin{align}
\label{eq:masterCompleta}
&\frac{\partial P}{\partial t} =\sum_{a=1}^{M}\beta_{a}\left(P_{\mu_{a}-1}-P\right)& \, &\emptyset \xrightharpoonup[\beta_a]{}\mu_a&\nonumber\\
&+\sum_{i=1}^{N}b_{i}\left(P_{m_{i}-1}-P\right)& \, &\emptyset\xrightharpoonup[b_i]{}m_i&\nonumber\\
&+\sum_{a=1}^{M}\delta_{a}\left[(\mu_{a}+1)P_{\mu_{a}+1}-\mu_{a}P\right]&\, &\mu_a\xrightharpoonup[\delta_a]{}\emptyset&\nonumber\\
&+\sum_{i=1}^{N}d_{i}\left[(m_{i}+1)P_{m_{i}+1}-m_{i}P\right]&\, &m_i \xrightharpoonup[d_i]{}\emptyset&\nonumber\\
&+\sum_{i=1}^{N}\sum_{a=1}^{M}k_{ia}^{+}\left[(\mu_{a}+1)(m_{i}+1)P_{\mu_{a}+1,m_{i}+1,c_{ia}-1}-\mu_{a}m_{i}P\right]&\, &\mu_a+m_i\xrightharpoonup[k_{ia}^+]{}c_{ia}&\\
&+\sum_{i=1}^{N}\sum_{a=1}^{M}k_{ia}^{-}\left[(c_{ia}+1)P_{\mu_{a}-1,m_{i}-1,c_{ia}+1}-c_{ia}P\right]& \, &c_{ia}\xrightharpoonup[k_{ia}^-]{}\mu_a+m_i&\nonumber \nonumber\\
&+\sum_{i=1}^{N}\sum_{a=1}^{M}\sigma_{ia}\left[(c_{ia}+1)P_{c_{ia}+1}-c_{ia}P\right]& \, &c_{ia}\xrightharpoonup[\sigma_{ia}]{}\emptyset&\nonumber\\
&+\sum_{i=1}^{N}\sum_{a=1}^M\sum_{i=M+1}^{M+N}\kappa_{ia} \left[(c_{ia}+1)P_{\mu_{a}-1,c_{ia}+1}-c_{ia}P\right]& \, &c_{ia}\xrightharpoonup[\kappa_{ia}]{}\mu_a&\nonumber
\end{align}
where we adopted for simplicity the compact notation $P_{x_i\pm1}:=P(x_1,\dots,x_i\pm1,\dots,x_{N+M+NM})$. Eq~(\ref{eq:masterCompleta}) relies on the (unrealistic) hypothesis that chemical species live in a well mixed environment without compartments, so that they are all in principle capable of interacting. An interesting and fundamental connection between the mass action kinetics in Eq~(\ref{uno}) and the ME is provided by the so-called {\em mean field approximation}, which assumes a simplified factorized form for the joint probability distribution $P$:
\begin{equation}
\label{eq:meanfield}
P(\{\mu_a\},\{m_i\},\{c_{ia}\},t) =\prod_{i=1}^N P_i(m_i)\prod_{a=1}^M P_a(\mu_a) \prod_{\ell = 1}^{N \cdot M} P_\ell(c_{\ell})
\end{equation}
Plugging (\ref{eq:meanfield}) into (\ref{eq:masterCompleta}) and computing the mean value of all chemical species, one can see that the differential equation governing their the time evolution coincides with Eq.~(\ref{uno}). This point of view casts in a new perspective the deterministic mass action kinetics: as long as the correlations between the different variables can be neglected, the deterministic scheme is expected to provide an accurate description of the dynamics of the model. On the other hand, by construction, the deterministic mass action kinetic is blind to statistical correlations between variables. If one is interested in this aspect, Eq.~(\ref{eq:masterCompleta}) provides the correct theoretical framework.
Unfortunately, the ME is notoriously hard to handle analytically. Therefore, in the following, we will outline different approximation schemes that have been used to obtain useful indications about fluctuations and correlations between molecular levels.
\subsubsection{Gaussian Approximation}
The Gaussian approximation is probably the simplest one going beyond mean-field. The rationale of the method is rooted in Van Kampen's expansion \cite{vkam}, and specifically in the fact that, if molecules are assumed to be enclosed in a sufficiently large volume, the solution of the ME is Gaussian except for small corrections. Adopting the following vector notation already implicitly used in Eq.~(\ref{eq:meanfield}), i.e.
\begin{eqnarray}
\vec x &:=& \{x_1 ,\dots x_M ,x_{M+1}\dots x_{M+N},x_{M+N+1},\dots x_{M+N+MN}\} \nonumber\\
&=& \{\mu_1,\dots,\mu_M,m_1,\dots,m_N,c_{11},\dots,c_{NM}\}\,~~,
\end{eqnarray}
the Gaussian approximation assumes that $\vec x$ is distributed as a multivariate Gaussian, namely
\begin{equation}
\label{eq:gauss-multi}
P(\vec x) \simeq G(\vec x | \vec a, \Sigma^{-1}) = \frac{\exp\left[ -\frac12 \left(\vec x - \vec a \right)^T\Sigma^{-1} \left(\vec x - \vec a \right)\right]}{\sqrt{(2\pi)^{M+N+MN}\mathrm{det}(\Sigma)}}
\end{equation}
where the covariance matrix $\boldsymbol{\Sigma}$ has element $\Sigma_{ij}=E(x_i x_j)-E(x_i)E(x_j)$, the vector $\vec a$ has coordinates $a_i = E(x_i)$, and the expectation value $E(\cdot)$ is with respect to the Gaussian measure $G$ defined in Eq~(\ref{eq:gauss-multi}). One of the characteristics that make Gaussian distributions useful in this context lies the property that all moments of a Gaussian measure can be expressed in terms of the mean $\vec a$ and the covariance matrix $\boldsymbol{\Sigma}$, so that, for instance, the generic third and fourth order moments read $E(x_i x_j x_k) = \Sigma_{ij}a_k + \Sigma_{ik}a_j + \Sigma_{jk}a_i$ and $ E(x_i x_j x_k x_l)=\Sigma_{ij}\Sigma_{kl}+\Sigma_{ik}\Sigma_{jl}+\Sigma_{il}\Sigma_{jk}$ respectively. In analogy with the closure of the system of equations in the first moments that the factorization hypothesis in Eq~(\ref{eq:meanfield}) induces, a shrewd use of the moment generating function produces a closed system of equations for $\vec a$ and $\boldsymbol{\Sigma}$. The natural formalism to impose this moment closure is that of the {\em moment-generating function}, defined as
\begin{equation}
\label{eq:momgen}
F({\bf z},t) = \sum_{\bf x}\prod_{i=1}^{N+M+N\cdot M}z_i^{x_i}P({\bf x},t)~~.
\end{equation}
It is simple to show that the time evolution of $F({\bf z},t)$ is ruled the second-order partial differential equation
\begin{equation}
\label{eq:momgenevol}
\partial_t F({\bf z},t) = {\cal H}({\bf z})F({\bf z},t)\quad,
\end{equation}
where, for the miRNA-ceRNA network, the operator ${\cal H}$ is defined as
\begin{eqnarray}
{\cal H}({\bf z}) &=& \sum_{a=1}^M \beta_a (z_a-1) + \sum_{i=M+1}^{M+N}b_i(z_i-1) \nonumber\\
&+& \sum_{a=1}^M \delta_a (\partial_{z_a}-z_a\partial_{z_a}) + \sum_{i=M+1}^{M+N}d_i(\partial_{z_i}-z_i\partial_{z_i}) + \sum_{l=N+M+1}^{N+M+N\cdot M}\sigma_{l}(\partial_{z_l}-z_l\partial_{z_l})\nonumber\\
&+& \sum_{a=1}^M\sum_{i=M+1}^{M+N} k_{ia}^+(z_{ia}\partial^2_{z_i\,z_a} - z_i z_a \partial^2_{z_i\,z_a})
+ \sum_{a=1}^M\sum_{i=M+1}^{M+N} k_{ia}^-(z_i z_a \partial_{z_{ia}} - z_{ia} \partial_{z_{ia}})\nonumber\\
&+& \sum_{a=1}^M\sum_{i=M+1}^{M+N} \kappa_{ia}( z_i \partial_{z_{ia}}-z_{ia}\partial_{z_{ia}})~~.
\end{eqnarray}
The moment-generating function $F$ owes its name to the following
constitutive property:
\begin{equation}
\partial^{l_1+l_2+\dots+l_k}_{z^{l_1}_{i_1},z^{l_2}_{i_2},\dots,z^{l_k}_{i_k}}F({\bf z},t)|_{\bf z=1}=\langle x_{i_1}^{l_1}x_{i_2}^{l_2}\cdots x_{i_k}^{l_k}\rangle_{P({\bf x},t)}\quad.
\end{equation}
In other terms, consecutive derivatives of $F$ generate all moments of the distribution $P$. The ME Eq~(\ref{eq:masterCompleta}) allows us to write a hierarchy of equations for the moments. However, it turns out that moments of order $k$ are usually expressed in terms of moments of order $k+1$, not allowing to close the system of equations for the moments. The Gaussian approximation truncates the hierarchy of moment dependencies by expressing third-order cumulants in terms of second-order ones (an approximation that turns out to be correct for Gaussian distributions). Thanks to this moment-closure approximation one ends up with a complete system of $N+M+N\cdot M+ {{N+M+N\cdot M}\choose2}$ equations for the mean molecular levels and all covariances.
\subsubsection{The Langevin approach}
A possibly more intuitive description of the stochastic dynamics is obtained by noting that, under broad conditions \cite{vkam}, one can effectively represent molecular fluctuations by adding specific noise terms to each of the factors appearing in the kinetic Eqs (\ref{uno}). This leads to a Langevin dynamics given by
\begin{equation}\label{due}
\arraycolsep=1pt\def1.8{2}
\begin{array}{r@{}l}
\frac{d m_i}{dt}&=b_i-d_i m_i\,\uudl{+\xi_i}\,-\sum_a k_{ia}^+ m_i\mu_a \,\uudl{+\sum_a \xi_{ia}^+}\,+\sum_a k_{ia}^- c_{ia}\,\uudl{+\sum_a\xi_{ia}^-}~~, \\
\frac{d \mu_a}{dt}&=\beta_a-\delta_a \mu_a\,\uudl{+\xi_a}\,-\sum_i k_{ia}^+ m_i\mu_a\,\uudl{+\sum_i \xi_{ia}^+}\, + \sum_{i}(k_{ia}^-+\kappa_{ia})c_{ia}\,\uudl{+\sum_i(\xi_{ia}^-+\xi_{ia}^{\rm cat})}~~,\\
\frac{d c_{ia}}{dt}&= k_{ia}^+ m_i \mu_a\,\uudl{+\xi_{ia}^+}\,-(\sigma_{ia}+\kappa_{ia}+k_{ia}^-)c_{ia}\,\uudl{+(\xi_{ia}^{\rm st}+\xi_{ia}^{\rm cat}+\xi_{ia}^-)}~~,
\end{array}
\end{equation}
where the mutually independent stochastic `forces' associated to each process have been inserted after the corresponding term and underlined. In specific,
\begin{itemize}
\item $\xi_i$ and $\xi_a$ represent the intrinsic noise due to random synthesis and degradation events that affect $m_i$ and $\mu_a$, respectively;
\item $\xi_{ia}^+$ and $\xi_{ia}^-$ model the noise affecting the random association and dissociation of complexes, respectively;
\item $\xi_{ia}^{\rm cat}$ and $\xi_{ia}^{\rm st}$ represent the noise of catalytic and stoichiometric complex processing events, respectively.
\end{itemize}
Each of these noise terms has zero mean. Correlations are instead given by
\begin{equation}\label{lang_noise}
\arraycolsep=1pt\def1.8{1.5}
\begin{array}{r@{}l}
&{} \avg{\xi_{i}(t)\xi_{i}(t')} = (b_i+d_i \avg{m_i}) ~ \delta(t-t')~~,\\
&{}\avg{\xi_{a}(t)\xi_{a}(t')} = (\beta_a+\delta_a \avg{\mu_a}) ~ \delta(t-t')~~, \\
&{}\avg{\xi_{ia}^+(t)\xi_{ia}^+(t')} = k_{ia}^+ \avg{m_i} \avg{\mu_a} ~ \delta(t-t')~~,\\
&{}\avg{\xi_{ia}^-(t)\xi_{ia}^-(t')} = k_{ia}^- \avg{c_{ia}} ~ \delta(t-t')~~,\\
&{}\avg{\xi_{i}^{\rm cat}(t)\xi_i^{\rm cat}(t')} = \kappa_{ia} \avg{c_{ia}} ~ \delta(t-t')~~,\\
&{}\avg{\xi_{i}^{\rm st}(t)\xi_{i}^{\rm st}(t')} = \sigma_{ia} \avg{c_{ia}} ~ \delta(t-t')~~,
\end{array}
\end{equation}
where steady state abundances (in angular brackets) are given by the solutions of Eqs (\ref{eq:steadystateM}). The specific form (\ref{lang_noise}), involving steady state vaues, can be derived within the so-called Linear Noise Approximation (LNA, \cite{vkam}), assuming that stationary molecular levels are sufficiently large \cite{swain}. As we show next, the LNA also provides direct access to the covariances of molecular levels.
\subsubsection{Linear Noise Approximation}
\label{lna}
Denoting by $\mathbf{x}$ the vector of molecular levels of all species involved, i.e. $\mathbf{x}= (\{m_i\},\{\mu_a\},\{c_{ia}\})$, the stochastic dynamics (\ref{due}) can be written in vector notation as
\begin{equation}\label{dueprime}
\frac{d\mathbf{x}}{dt}=\mathbf{f(x)}+\boldsymbol{\xi}~~,
\end{equation}
where the vector function $\mathbf{f}$ accounts for the deterministic terms in (\ref{due}) while the vector noise $\boldsymbol{\xi}$ contains the overall noise affecting each component. The LNA is based on the assumption that, at stationarity, random fluctuations cause $\mathbf{x}$ to deviate from is steady state value $\avg{\mathbf{x}}$ by a quantity $\delta \mathbf{x} = \mathbf{x} - \avg{\mathbf{x}}$ that is small enough to allow for the linearization of Eq (\ref{dueprime}) around $\avg{\mathbf{x}}$. In such conditions, $\delta \mathbf{x}$ changes in time as \cite{vkam}
\begin{equation}
\label{SDE}
\frac{d }{d t} \delta \mathbf{x} = \mathbf{S} \delta \mathbf{x} + \boldsymbol{\xi}~~~~~,~~~~~\mathbf{S}=\frac{d\mathbf{f}}{d\mathbf{x}}\bigg|_{\mathbf{x=\avg{x}}}~~,
\end{equation}
where $\mathbf{S}$ is the stability matrix of first-order derivatives evaluated at the steady state. Assuming that $\boldsymbol{\xi}$ is a Gaussian noise with zero mean and cross-correlations described by a matrix $\boldsymbol{\Gamma}$, i.e. $\avg{\xi_s(t) \xi_{s'} (t')} = \Gamma_{ss'} \delta (t-t')$ (where the indices $s$ and $s'$ range over the components of $\mathbf{x}$), one can show that the covariances of molecular levels at steady state obey \cite{swain}
\begin{equation}
\avg{ \delta x_a \delta x_b } = - \sum_ {i, l, s, r } B_{as} B_{br} \frac{\Gamma_{il} } {\lambda_s + \lambda_r } (B^{-1})_{si} (B^{-1})_{rl} ~~,
\end{equation}
where $\boldsymbol{\lambda}$ denotes the vector of eigenvalues of the stability matrix, while $\mathbf{B}$ stands for its eigenvectors (i.e. $\sum_{b}S_{ab}B_{br}=\lambda_r B_{ar}$).
The above formula provides a way to estimate correlations (and hence Pearson coefficients) of all molecular species involved in the system. The continuous lines in Fig. \ref{Fig2}B, C, E and F have indeed been obtained by the LNA.
\subsubsection{The Gillespie algorithm}
\label{gillespie}
The standard numerical route to simulate systems like Eq (\ref{due}) relies on the Gillespie algorithm (GA), a classical stochastic simulation method that computes the dynamics of a well-mixed system of molecular species interacting through a set of possible processes \cite{gill}. The GA allows to simulate the dynamics of systems like (\ref{processes}) without solving the ME, i.e. without the full knowledge of the probability $P(\mathbf{x},t)$ of the system being in state vector $\mathbf{x}$ (encoding for the population of each molecular species) at time $t$. In short (see however \cite{gibs} for a more detailed presentation), one can say that the GA essentially relies on two assumptions: (i) each process occurs with a specific rate constant; and (ii) the current state of the system (in terms of the number of molecules of each species) determines which process is going to occur next, independently of the previous history. Under these conditions, one can simulate trajectories of a system described by a set of processes such as (\ref{processes}) simply from the knowledge of the probability density $P(k,\tau|\mathbf{x},t)$ that process $k$ takes place between time points $t+\tau$ and $t+\tau+d\tau$ given that the state of the system at time $t$ is $\mathbf{x}$ (with no other processes occurring between time $t$ and time $t+\tau$). Because the dynamics is memoryless, $P(k,\tau|\mathbf{x},t)$ factorizes as
\begin{equation}\label{gilles}
\arraycolsep=1pt\def1.8{1.5}
\begin{array}{r@{}l}
P(k,\tau|\mathbf{x},t)d\tau=&\,\,\text{Prob}\{\text{no process between time $t$ and time $t+\tau$}\}\times\\
& \,\,\times \,\,\text{Prob}\{\text{process $k$ between time $t+\tau$ and time $t+\tau+d\tau$}\}\\
\equiv& \,\, P_0\times P_k~~.
\end{array}
\end{equation}
The probability $P_k$ is given by the intrinsic rate of process $k$ ($c_k$) times a function of $\mathbf{x}$ ($g_k(\mathbf{x})$) that quantifies the number of different ways in which process $k$ might occur and which basically encodes for the law of mass action. We shall use the shorthand $c_k g_k(\mathbf{x})=f_k(\mathbf{x})$. Hence $P_k=f_k(\mathbf{x}(t+\tau))d\tau$.
$P_0$ can instead be evaluated by sub-dividing the interval $[t,t+\tau]$ in $K$ parts ($K\gg 1$), each of duration $\tau/K$. If $f_k$ denotes the rate of process $k$, then $P_0$ is just the probability that no process occurs in any of the $K$ sub-intervals, i.e.
\begin{equation}
P_0 = \left(1-\sum_{k'} f_{k'}\frac{\tau}{K}\right)^K\simeq e^{-\tau \sum_{k'} f_{k'}}~~~~~(K\gg 1)~~.
\end{equation}
Hence
\begin{equation}\label{gilles2}
P(k,\tau|\mathbf{x},t)\simeq f_k \,\, e^{-\tau \sum_{k'} f_{k'}}~~,
\end{equation}
which can also be re-cast as
\begin{equation}\label{gilles3}
P(k,\tau|\mathbf{x},t)\simeq \underbrace{\left(\sum_{k'}f_{k'}\right) e^{-\tau \sum_{k'} f_{k'}}}_{\text{prob. of waiting time $\tau$}} \quad\times\,\underbrace{\frac{f_k}{\sum_{k'}f_{k'}}}_{\text{prob. of process $k$}} \,\, ~~.
\end{equation}
A value of $\tau$ sampled from the above distribution of waiting times is easily obtained by noting that, if $u$ denotes a random variable uniformly distributed in $[0,1]$, then
\begin{equation}
\tau=-\frac{\ln(u)}{\sum_{k'}f_{k'}}
\end{equation}
is actually distributed according to the exponential function given in (\ref{gilles3}). This allows to formulate the GA in the following scheme:
\begin{svgraybox}
{\bf \textsf{Gillespie Algorithm} }
\begin{description}
\setlength\itemsep{0pt}
\item[Step 1:] Initialization: set initial populations for all molecular species (vector $\mathbf{x}(0)$) together with the rate $c_k$ of each process $k$ and an end-time $T$
\item[Step 2:] Evaluate reaction probabilities $f_k$ for each $k$ as well as $\sum_{k'}f_{k'}\equiv Z$
\item[Step 3:] Generate a pair $(k,\tau)$ from (\ref{gilles3})
\item[Step 4:] Update molecular populations according to the selected process $k$ and advance time by $\tau$
\item[Step 5:] Iterate from Step 2 or stop if the end-time $T$ has been reached
\end{description}
\end{svgraybox}
Fig. \ref{Fig2}B, C, E and F show how mean molecular levels obtained by the GA (markers) compare against analytic results (lines). One sees that the Fano Factor (FF) markedly peaks when molecular levels become roughly equimolar, i.e. close to the threshold where the system becomes susceptible to changes in the modulated parameter (in this case, the miRNA transcription rate or the transcription rate of ceRNA 1). The coefficient of variation (CV) also modifies its qualitative behaviour in the same range, although this feature generically appears to be less drastic (see however \cite{bosi}). This shows that when ceRNAs become susceptible and cross-talk is established, fluctuations in molecular levels become strongly correlated.
The fluctuation scenario just described is clearly connected to the establishment of miRNA-mediated crosstalk. How exactly, and how it relates to other signatures of cross-talk, is the subject of the following section.
\subsection{Quantifying miRNA-mediated crosstalk at steady state}
\label{subsec:3}
The competing endogenous RNA scenario concerns the possibility that, as a result of competition to bind miRNAs, ceRNAs could cross-regulate each other. We have so far identified two signatures that accompany the establishment of miRNA-mediated crosstalk at stationarity:
\begin{enumerate}
\item[a.] a change in the steady state level of a ceRNA following a change of the level of a competitor (i.e. a response following a perturbation);
\item[b.] an increase of connected ceRNA-ceRNA correlations.
\end{enumerate}
Both are clearly defined and testable in experiments and from data (at least in principle). Yet, despite the apparent simplicity, the reliable detection of the ceRNA mechanism in experiments or data is far from simple. The key issue lies in the fact that several mechanisms, both involving miRNAs and involving other molecular actors, potentially bear similar effects on transcripts and, as the cause differs, so do their consequences. Disentangling the competition-driven ceRNA effect from other processes is in many ways essential to be able to predict how a miRNA-ceRNA network will react to perturbations. We shall recap below how the ceRNA crosstalk scenario looks when seen through different glasses. While each allows to capture certain aspects of the ceRNA mechanism, different quantities employed to quantify crosstalk intensity focus on slightly different physical features and therefore can be useful in different situations. Understanding such differences is however crucial both for applications and for the unambiguous identification of biological drivers.
\subsubsection{Pearson correlation coefficient}
Since an increase of correlations between molecular levels accompanies the establishment of crosstalk, it is reasonable to view the Pearson correlation coefficient between two ceRNAs as a basic proxy for crosstalk intensity \cite{alau,bosi,meht}. For ceRNAs $i$ and $j$, it is defined as
\begin{equation}\label{rho}
\rho_{ij}
\frac{\avg{m_i \,m_j}-\avg{m_i}\avg{m_j}}{\sqrt{\avg{m_i^2}-\avg{m_i}^2}\sqrt{\avg{m_j^2}-\avg{m_j}^2)}}\equiv\frac{{\rm cov}(m_i,m_j)}{\sqrt{\avg{\delta m_i^2}}\,\sqrt{\avg{\delta m_j^2}}}~~,
\end{equation}
where averages are taken over random fluctuations in the steady state of a stochastic dynamics. (When the interaction network is conserved across different cellular samples and single snapshots of molecular levels are available for each sample, the $\avg{\cdots}$ average can also be taken over different samples, as long as each sample can be considered to be stationary.) Note that $-1\leq\rho_{ij}\leq 1$.
The rationale for using (\ref{rho}) as a measure of crosstalk intensity is roughly the following. In a network of $N$ ceRNA species interacting with $M$ miRNA species, both ceRNA and miRNA levels will fluctuate stochastically over time at stationarity. A large positive value of $\rho_{ij}$ points to the existence of a positive (linear) correlation between $m_i$ and $m_j$, i.e. to the fact that $m_i\simeq c m_j+d+$noise, with constants $c>0$ and $d$. In such conditions, it is reasonable to expect that an increase in the level of ceRNA $i$, whichever its origin, will divert part of the miRNA population currently targeting ceRNA $j$ to bind to $i$, thereby freeing up molecules of $j$ for translation. In practice, with a large $\rho_{ij}$, perturbations affecting ceRNA $i$ could be ``broadcast'' to ceRNA $j$ because of the miRNA-mediated statistical correlation existing between their respective levels.
The Pearson correlation coefficient between competing ceRNAs indeed attains a maximum in a specific range of values for the transcription rates, see e.g. Fig. \ref{Fig3}B.
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig3}
\end{center}
\caption{{\bf (A)} Stochastic simulation showing the free levels of two ceRNA species co-regulated by a miRNA species (not shown). Both ceRNAs are susceptible with respect to changes in the miRNA level. The transcription rate of ceRNA 1 is perturbed at the time indicated by the dashed line. ceRNA 2 responds by increasing its amount. {\bf (B)} Susceptibilities and Pearson coefficients for two ceRNAs co-regulated by a miRNA species for moderate miRNA repression strength. All three quantifiers of ceRNA crosstalk are significantly different from zero and $\chi_{12}\simeq \chi_{21}$. {\bf (C)} Same as (A) but now ceRNA 1 is fully repressed by the miRNA. Still, an increase of its transcription rate yields an upregulation of $m_2$. {\bf (D)} Same as (B) but for strong miRNA repression on ceRNA 1. Both the Pearson coefficient $\rho_{21}$ and $\chi_{12}$ (quantifying the response of ceRNA 1 to a perturbation affecting ceRNA 2) are effectively zero, whereas $\chi_{21}$ is not. Parameter values are given in Table \ref{pars}.}
\label{Fig3}
\end{figure}
Expectedly, this happens when the levels of the different molecular species become comparable (or, more precisely, when the number of miRNA binding sites becomes similar to that of miRNA molecules) \cite{bosi}. Here, ceRNA fluctuations become strongly correlated and one might expect ceRNA crosstalk to be active, so that a perturbation affecting one ceRNA will result in a shift in the level its competitor. In other words, this regime is characterized by significant crosstalk effects.
\subsubsection{Susceptibility}
A mechanistic (as opposed to statistical) quantification of the magnitude of the ceRNA effect can be obtained by computing derivatives of steady-state ceRNA levels like \cite{figl}
\begin{equation}\label{sus}
\arraycolsep=1pt\def1.8{1.9}
\begin{array}{r@{}l}
\chi_{ij} &{}= \frac{\partial \avg{m_i}}{\partial b_j}\geq 0\\
\chi_{ia} &{}= \frac{\partial \avg{m_i}}{\partial \beta_a}\leq 0
\end{array}
\end{equation}
where $b_j$ (resp. $\beta_a$) stands for the transcription rate of ceRNA $j$ (resp. miRNA $a$). We shall term quantities like (\ref{sus}) {\it susceptibilities}. In short, $\chi_{ij}$ measures the variation in the mean level of ceRNA $i$ caused by a (small) change in $b_j$. As an increase of $b_j$ leads to an increase of the level of ceRNA $j$ by titration of miRNAs away from it, $\chi_{ij}$ is bound to be non-negative. A similar straightforward interpretation applies to $\chi_{ia}$, which is non-positive since an increase of $\beta_a$ is bound to cause a decrease of $\avg{m_i}$. The central hypothesis behind Eq (\ref{sus}) is that small perturbations cause small changes in molecular levels, or, more precisely, that the latter will be proportional to the former if the perturbation is sufficiently small (linear response scenario).
Assuming no direct control of ceRNA $i$ by ceRNA $j$, a large value of $\chi_{ij}$ directly points to the existence of miRNA-mediated crosstalk in terms of a change in the level of a target upon perturbing the level of a competitor. Hence $\chi_{ij}$ focuses on the response part of the ceRNA effect rather than on the fluctuation-related aspects.
Quantities like $\chi_{ij}$ can be directly computed from the steady state conditions and in numerical simulations upon probing the system with the desired perturbation. A susceptibility-based theory of ceRNA crosstalk at steady state has indeed been presented in \cite{figl}. When quantified through $\chi_{ij}$, ceRNA crosstalk displays the following key features:
\begin{description}
\item[{\bf Selectivity}~:]When a miRNA targets multiple ceRNA species, crosstalk may occur only among a subset of them. This effect is related to the fact that different ceRNAs can have different thresholds for repression by the miRNA and is enhanced by heterogeneities in the thresholds;
\item[{\bf Directionality (asymmetry)}~:]In general, $\chi_{ij}\neq\chi_{ji}$, i.e. ceRNA $i$ may respond to a perturbation affecting ceRNA $j$ but not the reverse;
\item[{\bf Plasticity}~:]The pattern of miRNA-mediated ceRNA crosstalk, whereby ceRNA $j$ is linked to ceRNA $i$ when $\chi_{ij}$ is sufficiently large, is modulated by kinetic parameters, and particularly by miRNA levels (in other words, changes in miRNA availability modify the ceRNA crosstalk network);
\item[{\bf Dependency on stoichiometric processing}~:]If all miRNA-ceRNA complexes formed by ceRNA $j$ are degraded in a purely catalytic way, then $\chi_{ij}=0$ (i.e. stoichiometric processing is necessary for ceRNA crosstalk at stationarity).
\end{description}
Like the Pearson coefficient $\rho_{ij}$, the ceRNA-ceRNA susceptibility $\chi_{ij}$ also peaks when ceRNA crosstalk is strongest (see Fig. \ref{Fig3}B). However, the fact that susceptibilities are perturbation-specific makes their usefulness for data analysis and the interpretation of experiments less immediate compared to Pearson coefficients. Ideally, one would like to connect susceptibilities like (\ref{sus}) to simpler quantities like correlation functions. A more refined mathematical analysis of the stochastic dynamics shows that this is indeed possible.
\subsubsection{Fluctuations versus response}
It is important to understand that the physical meaning and therefore the crosstalk scenarios underlied by $\rho_{ij}$ and $\chi_{ij}$ are rather different. The fact that $\chi_{ij}$ is asymmetric under exchange of its indices (i.e. $\chi_{ij}\neq\chi_{ji}$ in general) whereas $\rho_{ij}$ is necessarily symmetric already pointed in this direction. Other subtle differences however emerge when the two quantities are compared in greater detail.
In first place, $\chi_{ij}$ can be non zero (and possibly large) even for a completely deterministic system like (\ref{uno}), as it simply measures how a target's steady state level is modulated by changes affecting the transcription rate of one of its competitors, independently of the presence of stochastic fluctuations around the steady state. In this sense, $\chi_{ij}$ focuses exclusively on the effects induced by competition. On the other hand, in absence of fluctuations $\rho_{ij}$ is identically zero.
Second, and related to this, is the fact that a large value of $\rho_{ij}$ can occur when both ceRNAs respond to fluctuations in miRNA levels (`indirect correlation'). This however does not imply that $m_i$ and $m_j$ are directly correlated. (If variables $X$ and $Y$ are both correlated with $Z$, they will be correlated too. However, in absence of a direct correlation between $X$ and $Y$, upon conditioning over the value of $Z$ one will observe that $X$ and $Y$ are uncorrelated.) The same holds in presence of extrinsic noise, in which case averages are performed over different samples rather than over time in a single sample. To see this directly, one can consider a system formed by $N$ ceRNA species (labeled $i,j,k,\ldots$) and $M$ miRNA species (labeled $a$) \cite{figl}. If transcription rates fluctuate across cells and if fluctuations are sufficiently small, ceRNA levels at steady state will be approximately given by
\begin{equation}\label{cov}
\arraycolsep=1pt\def1.8{1.9}
\begin{array}{r@{}l}
\avg{m_i} &{}\simeq \ovl{\avg{m_i}}+\sum_{j} \frac{\partial \avg{m_i}}{\partial b_j}(b_j-\ovl{b_j})+\sum_a\frac{\partial {\avg{m_i}}}{\partial \beta_a}(\beta_a-\ovl{\beta_a})\\
&{}\equiv \ovl{\avg{m_i}}+\sum_{j} \chi_{ij}\delta b_j+\sum_a\chi_{ia}\delta\beta_a ~~,
\end{array}
\end{equation}
the over-bar denoting an average over transcription rates. Assuming that transcription rates of different species are mutually independent, the Pearson correlation coefficient $\rho_{ij}$ can be seen to be given by
\begin{equation}
\label{pearson1}
\rho_{ij}=A\,\left(\sum_k \chi_{ik}\chi_{jk}\ovl{\delta b_k^2}+\sum_a \chi_{ia}\chi_{ja}\ovl{\delta \beta_a^2}\right)
\end{equation}
where $A>0$ is a constant, the index $k$ runs over ceRNAs, the index $a$ runs over miRNAs and $\ovl{\delta b_k^2}$ (resp. $\ovl{\delta \beta_a^2}$) is the variance of the transcription rate of ceRNA species $k$ (resp. miRNA species $a$). Now one sees that, if all ceRNA-ceRNA susceptibilities are zero (i.e. in absence of competition-induced crosstalk),
\begin{equation}
\rho_{ij} \propto \sum_a \chi_{ia}\chi_{ja}\ovl{\delta \beta_a^2}~~.
\end{equation}
Because ceRNAs always respond to fluctuations in miRNA levels, susceptibilities on the right-hand side are not zero. In particular, both $\chi_{ia}$ and $\chi_{ja}$ are negative, as an increase in miRNA levels causes a decrease in the level of free ceRNAs. One therefore concludes that $\rho_{ij}>0$ even though all ceRNA-ceRNA susceptibilities are nil. This explicitly shows that $\chi_{ij}$ and $\rho_{ij}$ describe {\it a priori} different crosstalk mechanisms.
A mathematical analysis of susceptibilities and fluctuations shows that crosstalk intensity ultimately depends on whether the involved ceRNAs are unrepressed, susceptible or repressed by miRNAs. In particular, it turns out that the ceRNA-ceRNA susceptibility $\chi_{ij}$ is qualitatively described by a matrix whose entries depend only on the state of repression of $i$ (the responding ceRNA) and $j$ (the perturbed one), given by \cite{figl}
\begin{equation}
\chi_{ij}\,\,=\,\,
\begin{tabular}{c c|c|c|c|}
\multicolumn{2}{c|}{~} & \multicolumn{3}{c|}{$j$ (perturbed)} \\
\multicolumn{2}{c|}{~} & \multicolumn{1}{c|}{Unrepr.} & \multicolumn{1}{c|}{Susc.} & \multicolumn{1}{c|}{Repr.} \\
\hline
\parbox[t]{3mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{$i$ (resp.)}}} & Unrepr. & $\simeq 0$ & $\simeq 0$ & $\simeq 0$\\
& Susc. & $\simeq 0$ & \cellcolor{gray!50}{$>0$} & \cellcolor{gray!50}{$>0$}\\
& Repr. & $\simeq 0$ & $\simeq 0$ & $\simeq 0$\\
\hline
\end{tabular}~~.
\end{equation}
Besides showing explicitly that $\chi_{ij}\neq \chi_{ji}$, the above matrix clarifies that a non-zero $\chi_{ij}$ (and therefore competition-driven response of $i$ to a change in the transcription rate of $j$) occurs (i) symmetrically, when both ceRNAs are susceptible to the miRNA (as in Fig. \ref{Fig3}A), and (ii) asymmetrically, when the perturbed ceRNA is repressed while the responding one is susceptible (as in Fig. \ref{Fig3}C). Along the same lines, one finds that \cite{tran}
\begin{equation}
\rho_{ij}\,\,=\,\,
\begin{tabular}{c c|c|c|c|}
\multicolumn{2}{c|}{~} & \multicolumn{3}{c|}{$j$ (perturbed)} \\
\multicolumn{2}{c|}{~} & \multicolumn{1}{c|}{Unrepr.} & \multicolumn{1}{c|}{Susc.} & \multicolumn{1}{c|}{Repr.} \\
\hline
\parbox[t]{3mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{$i$ (resp.)}}} & Unrepr. & $\simeq 0$ & $\simeq 0$ & $\simeq 0$\\
& Susc. & $\simeq 0$ & \cellcolor{gray!50}{$>0$} & $\simeq 0$\\
& Repr. & $\simeq 0$ & $\simeq 0$ & $\simeq 0$\\
\hline
\end{tabular}~~.
\end{equation}
i.e. the Pearson coefficient should expected to be significantly different from zero only when both ceRNAs are susceptible to changes in miRNA levels, as is clear by comparing Figures \ref{Fig3}B and D.
The quantitative relationship linking susceptibilities to fluctuations emerges through a more careful mathematical analysis of Eq~(\ref{due}) based on approximating the stochastic variability affecting molecular levels with a thermal-like noise. This leads to a set of results closely related to the Fluctuation-Dissipation Relations that characterize the linear-response regime of multi-particle systems in statistical physics. Specifically, one finds that, under broad conditions, susceptibilities can be expressed in terms of covariances of molecular levels or functions thereof. In particular, in Ref. \cite{tran} it is shown that
\begin{equation}\label{covar}
\arraycolsep=1pt\def1.8{1.9}
\begin{array}{r@{}l}
\chi_{ij} &{}\equiv\frac{\partial \avg{m_i}}{\partial b_j}=\gamma\,\, {\rm cov}(m_i,\log m_j)\geq 0~~,\\
\omega_{ij} &{}\equiv\frac{\partial\avg{m_i}}{\partial d_j}=-\gamma\,\,{\rm cov}(m_i,m_j)\leq 0~~,\\
\chi_{ia} &{}\equiv\frac{\partial \avg{m_i}}{\partial \beta_a}=\gamma\,\, {\rm cov}(m_i,\log \mu_a)\leq 0~~,\\
\omega_{ia} &{}\equiv\frac{\partial\avg{m_i}}{\partial \delta_a}=-\gamma\,\,{\rm cov}(m_i,\mu_a)\geq 0~~,\\
\end{array}
\end{equation}
where $\gamma>0$ is a constant. In other terms, the response $\chi_{ij}$ of $\avg{m_i}$ to a perturbation affecting the transcription rate of ceRNA $j$ is proportional to the covariance function ${\rm cov}(m_i,\log m_j)$ which incidentally, like $\chi_{ij}$, is not symmetric under the exchange of $i$ and $j$. Similarly, the bare ceRNA-ceRNA covariance ${\rm cov}(m_i,m_j)$ describes the response of $\avg{m_i}$ to (small) change of the intrinsic {\it degradation rate} $d_j$ of ceRNA $j$. Importantly, by comparing (\ref{rho}) with $\omega_{ij}$, Eq. (\ref{covar}), one sees that, perhaps unexpectedly, the Pearson coefficient $\rho_{ij}$ is related to $\omega_{ij}$ (rather than $\chi_{ij}$). (Likewise, one could calculate ceRNA-miRNA susceptibilities like $\chi_{ia}$ and $\omega_{ia}$ by evaluating bare covariances of ceRNA and miRNA levels as shown in (\ref{covar}).)
Generically, covariances are as easy to estimate from transcriptional data as Pearson coefficients, from which they only differ by the (crucial) normalization factor corresponding to the magnitude of fluctuations of individual variables. Relationships (\ref{covar}) have been used to infer different features of ceRNA crosstalk network generated by the tumor suppressor gene PTEN from transcriptional data, in particular directionality \cite{tran}. The large-scale use of such quantities might provide detailed transcriptome-wide crosstalk patterns, open for analysis and further validation.
\subsection{The role of network topology}
The topology of the miRNA-ceRNA provides an additional degree of freedom through which the effectiveness of ceRNA crosstalk can be influenced. To understand how, we assume that the miRNA-ceRNA network is sufficiently sparse and that connectivity correlations are absent. In such conditions, one can reasonably neglect ceRNA-ceRNA couplings involving more than one miRNA species and express the ceRNA-ceRNA susceptibility as
\begin{equation}\label{chiija}
\chi_{ij}\simeq\sum_a\underbrace{\frac{\partial m_i}{\partial \mu_a}\frac{\partial \mu_a}{\partial b_j}}_{\chi_{ij,a}}~~.
\end{equation}
One sees that if $\chi_{ij,a}$, i.e. the ceRNA-ceRNA susceptibility mediated by miRNA species $a$, is roughly the same for all miRNA regulators shared by $i$ and $j$, i.e. if $\chi_{ij,a}\simeq \chi_{ij}^{(0)}$ for all $a$, then $\chi_{ij}\simeq n_{ij}\chi_{ij}^{(0)}$, with $n_{ij}$ the number of miRNA species that target both ceRNAs $i$ and $j$. In other words, $\chi_{ij}$ increases with the number $n_{ij}$ of miRNA species shared by $i$ and $j$. This dependence can become especially significant in presence of strong degree correlations in the miRNA-ceRNA network, explaining why clustered networks such as those addressed in \cite{bosi} generically lead to more intense crosstalk patterns than random networks.
The role of topology is however most clearly isolated when ingredients other than strictly topological ones are as homogeneous as possible. We therefore assume that
\begin{enumerate}
\item[a.] all kinetic parameters are homogeneous (i.e. independent of the molecular species); in particular $\mu_{0,ia}\equiv \mu_0$ for all miRNA-ceRNA pairs;
\item[b.] miRNA levels are homogeneous, that is $\mu_a=\mu$ for each $a$.
\end{enumerate}
Based on these, one can show that, when the number $n_i$ (resp. $n_j$) of miRNAs targeting ceRNA $i$ (resp. ceRNA $j$) is sufficiently large, each shared miRNA contributes a quantity \cite{figl}
\begin{equation}\label{chiija2}
\chi_{ij,a}\simeq \frac{1}{d}\,\,\frac{\til{\mu}}{A+\sum_{k\in a}\frac{1}{1+n_k\til{\mu}}}\,\,\frac{1}{(1+n_i\til{\mu})^2 (1+n_j\til{\mu})}~~
\end{equation}
to the overall susceptibility Eq~(\ref{chiija}), where $\til{\mu}\equiv\mu/\mu_0$ is the miRNA level expressed in units of $\mu_0$, $A>0$ is a constant while $k\in a$ denotes the set of ceRNAs that interact with miRNA $a$. Hence $\chi_{ij,a}$ decreases (i.e. crosstalk intensity is diluted) as $n_i$ increases, as $n_j$ increases, and/or as the number of targets of miRNA $a$ increases.
This suggests that a particularly intriguing scenario arises when a large number of miRNA species target $i$ and $j$ and when $\mu\ll\mu_0$, i.e. when all ceRNA species are unrepressed by miRNAs. For simplicity, we assume the miRNA-ceRNA interaction network to be a regular bipartite graph where each ceRNA interacts with $n_i=n$ miRNAs while each miRNA interacts with $\nu_\alpha=\nu$ ceRNAs. In this case, (\ref{chiija2}) takes the form
\begin{equation}\label{chiija3}
\chi_{ij,a}\simeq \frac{1}{d}\,\,\frac{\til{\mu}}{\nu+A(1+n\til{\mu})}\,\,\frac{1}{(1+n\til{\mu})^2}~~.
\end{equation}
Now the value of $\chi_{ij,a}$ clearly depends on $\til{\mu}$. In particular, one sees that
\begin{equation}\label{casi}
\chi_{ij,a}
\begin{cases}
\ll \frac{1}{dn} & \text{for $\mu\ll\mu_0/n$}\\
\simeq \frac{1}{dn} &\text{for $\mu\simeq\mu_0/n$}\\
\ll \frac{1}{dn} & \text{for $\mu\gg\mu_0/n$}\\
\end{cases}~~.
\end{equation}
In other terms, $\chi_{ij,a}$ is maximum when miRNA levels are close to $\mu_0/n$, i.e. (for sufficiently large $n$) when each is well below the susceptibility threshold.
Formula (\ref{casi}) essentially reproduces the standard 3-regime scenario (unrepressed, susceptible, repressed) in a network context, albeit starting from the assumption that ceRNAs are unrepressed by each individual miRNA species. In this sense, it describes a ``distributed'' effect: many weakly interacting miRNA species can collectively mediate efficient ceRNA crosstalk. Recalling (\ref{chiija}), we see that when $\mu\simeq\mu_0/n$ the overall susceptibility is given by
\begin{equation}
\chi_{ij}\propto \frac{n_{ij}}{d n}~~,
\end{equation}
which becomes comparable to the self-susceptibility $\chi_{ii}$ for $n_{ij}\simeq n$. A sketch summarizing the results just described is shown in Fig. \ref{Fig5}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig5}
\end{center}
\caption{Sketch of a miRNA-ceRNA network with $N=4$ and $M=6$. Each ceRNA species is regulated by 3 miRNA species, but ceRNA pairs $(1,2)$ and $(3,4)$ share more regulators than other pairs. Crosstalk between 1 and 2 and between 3 and 4 should therefore generically be stronger than for other ceRNA pairs. On the other hand, ceRNAs 1 and 4 don't have any regulator in common. Still, they may be able to crosstalk through the chain of miRNA-mediated interactions shown in light blue.}
\label{Fig5}
\end{figure}
When connectivity correlations are not negligible and the approximation (\ref{chiija}) fails, $\chi_{ij}$ can in principle be expressed as
\begin{equation}
\chi_{ij}=\sum_{n\geq 0} \chi_{ij}^{(n)}~~,
\end{equation}
where $\chi_{ij}^{(n)}$ stands for the contribution to the $i-j$ susceptibility given by crosstalk interactions mediated by chains formed by $n$ miRNA species. Starting from the steady state conditions (\ref{eq:steadystateM}), one can compute $\chi_{ij}^{(n)}$ exactly in the limit where the stoichiometricity ratio $\alpha_{ia}$ is the same for all pairs, i.e. $\alpha_{ia}=\alpha$ for each $i$ and $a$, finding
\begin{equation}\label{totale}
\arraycolsep=1pt\def1.8{1.9}
\begin{array}{r@{}l}
\chi_{ij}^{(n)}&{}=\frac{1}{d_i}\frac{\alpha^n}{1+\sum_a\frac{\mu_a}{\mu_{0,ia}}} \, \left(\mathbf{X}^n\right)_{ij}~~,\\
X_{ij}&{}=\frac{m_i^\star}{\alpha\left(1+\sum_a\frac{\mu_a}{\mu_{0,ia}}\right)^2}\sum_{a} A_{ai}A_{aj}\frac{\mu_a^\star}{\mu_{0,ia}m_{0,ja}}\left(1+\sum_\ell \frac{m_\ell}{m_{0,\ell a}}\right)^2~~,
\end{array}
\end{equation}
where $\mathbf{X}$ is the matrix with elements $X_{ij}$ and $A_{ai}=1$ if miRNA $a$ targets ceRNA $i$ and zero otherwise. Because $\alpha<1$, one sees that the contribution coming from chains of $n$ miRNA-mediated couplings becomes smaller and smaller (exponentially fast) as $n$ increases. Eq~(\ref{totale}) shows explicitly that ceRNAs $i$ and $j$ can crosstalk even when they have no miRNA regulator in common (in which case $X_{ij}=0$), provided there is a path of miRNA-mediated interactions connecting them (as suggested e.g. in \cite{figl,itza}; see also Fig. \ref{Fig5}). Hence, clearly, the topological structure of the miRNA-ceRNA network can strongly influence the emergent crosstalk scenario. The discussion presented here does virtually nothing to address the ensuing complexity. A deeper understanding of the interplay between topological and kinetic heterogeneities might shed light on the evolutionary drivers of miRNA targeting patterns and of the ceRNA mechanism.
\subsection{Noise processing}
\subsubsection{Noise buffering in small regulatory motifs}
Together with transcription factors (TFs), miRNAs form a highly interconnected network whose structure can be decomposed in small regulatory patterns, or circuits. Few of them, hereafter call {\it motifs}, are overrepresented and thus expected to perform regulatory functions. In particular, it has been proven that all these miRNA-mediated motifs play some role in stabilizing the expression of the miRNA-target against fluctions \cite{osella11, bosia12, riba14, osella14, grigolon16}. Amongst others, a special role is performed by feedforward loops involving one miRNA, one TF and one target. Both the miRNA and the TF can play the role of the master regulator, while the target is dowregulated by the miRNA and activated or inhibited by the TF.
The incoherent version of this motif, where the TF activates the expression of miRNA and target, can couple fine-tuning of the target together with an efficient noise control \cite{osella11, grigolon16}. Intuitively, this can be understood by noting that fluctuations that propagate from TF to target and miRNA are correlated, so that an increase or decrease in the amount of miRNA will coincide with a decrease in the amount of target.
The theoretical framework for the analysis of these effects is that of the ME, which in this case takes into account five different variables, one for each of the involved molecular species (mRNA and protein for the TF, mRNA and protein for the target, and the miRNA). The transcriptional activation of miRNA and target is modelled via a non-linear increasing Hill function of the number of TF, i.e.
\begin{equation}
\label{hill_activ}
b_m(f) = \frac{b_m f^c}{h_{m}^{c}+f^{c}} ~~~~~ , ~~~~~ \beta_{\mu}(f) = \frac{\beta_{\mu} f^c}{h_{\mu}^{c}+f^{c}} ~~ ,
\end{equation}
where $b_m$ and $\beta_{\mu}$ are the transcription rates of target $m$ and miRNA $\mu$ respectively, $c$ is the Hill coefficient setting the steepness of the sigmoidal function and $h_{m}$ and $h_{\mu}$ are the dissociation constants, that specify the amount
of TF proteins $f$ at which the transcription rate is half of its maximal value ($b_m$ and $\beta_{\mu}$ respectively).
The miRNA interaction can be either modelled via a repressive Hill function of the number of miRNA molecules, i.e. as $b_f(\mu) = \frac{b_p h^{c}}{h^{c}+\mu^{c}}$, or via a titration-based mechanism. In the Hill function, $c$ is again the Hill coefficient and $h$ set the amount of miRNAs necessary to halve the maximum target translation rate $b_f$.
In the first case, one implicitly assumes that the miRNA action is catalytic (that is, the miRNA is never affected by the interaction with the target) and directs translational repression. In the second case, instead, one assumes that the miRNA action is stoichiometric, via binding and unbinding reactions (with rates $k_{m\mu}^{+}$ and $k_{m\mu}^{-}$ respectively). As long as miRNA and target mRNA are bound, the target cannot be translated. The miRNA might be affected by the interaction with the target (with recycling rate $\alpha$) and the target itself has an effective degradation rate that depends on the binding and unbinding rates and that is bigger than its intrinsic value $d_i$. In this case the miRNA actively promotes the degradation of the target. It is possible to show analytically and by numerical simulations that the maximal noise attenuation for the target is obtained for a moderate miRNA repression, independently of the way the miRNA interaction is modelled \cite{osella11}. This prediction, besides being in agreement with experimental observations of the impact of a wide
class of microRNAs on their target proteins, also suggests that an optimal noise reduction might be achieved even when the miRNA repression is diluted over multiple targets, provided these ceRNAs are not too noisy.
The analysis of data from the {\it Encyclopedia of DNA Elements} (ENCODE) \cite{encode} revealed that two other classes of miRNA-mediated circuits are enriched over the mixed network of miRNAs and TFs. One of them has a miRNAs that regulates two different genes that can eventually dimerize; the second has a miRNA that interacts with two TFs which in turn regulate the same gene. In both cases, the miRNA seems to have a role in stabilizing the relative concentration of their targets. The interesting fact is that a further enrichment appears when looking for those circuits in which there is a transcriptional connection between the two miRNA targets, i.e. one of them is a TF of the other. This TF, together with the miRNA, can in turn regulate multiple targets. This motif is again a feedforward loop where the miRNA plays the role of the master regulator and the TF and targets are ceRNAs. When modelling the motif with a titrative interaction for miRNA and target, in line with (\ref{eq:steadystateM}), and with an activatory Hill function from the TF to the target, it becomes clear that the topology of the circuit, together with the ceRNA interaction, enhances the coordination of the targets \cite{riba14}. This aspect is useful when TF and target have to maintain a fixed concentration ratio, which might be the case when they interact under a given stoichiometry.
\subsubsection{Transcriptional noise and the role of transcriptional correlations}
miRNA-mediated crosstalk can also provide a pathway to processing extrinsic noise, specifically cell-to-cell variability in transcription rates. Generalizing the lines that brought us to (\ref{cov}), one can say that if such a noise is sufficiently small, each component $\avg{x_k}$ of the steady state concentration vector $\avg{\mathbf{x}}= (\{\avg{m_i}\}_{i=1}^N,\{\avg{\mu_a}\}_{a=1}^M)$ can be written as
\begin{equation}\label{pob}
\avg{x_k} \simeq \overline{\avg{x_k}}+\sum_{s} \chi_{ks}(r_{s}-\overline{r}_{s})~~~~~,~~~~~\chi_{ks}=\frac{\partial \avg{x_k}}{\partial r_{s}}~~,
\end{equation}
where $\ovl{\avg{\mathbf{x}}}$ stands for the mean steady state vector (averaged over transcriptional noise), $r_s$ denotes the components of the vector $\mathbf{r}=(\{b_i\}_{i=1}^N,\{\beta_a\}_{a=1}^M)$ of transcription rates (including both those relative to ceRNAs and miRNAs), and the sum runs over all ceRNA and miRNA species. In turn, transcriptional noise induces fluctuations in the level of molecular species $k$ described by \cite{figl}
\begin{equation}
\label{sigma}
\sigma^2_{k}\equiv\overline{(\avg{x_k}- \overline{\avg{x_k}})^2}= \sum_{s,s'} \chi_{ks}\,\chi_{ks'}\,\Sigma_{ss'}~~,
\end{equation}
where $\boldsymbol{\Sigma}$ denotes the covariance matrix of transcription rates. If $\boldsymbol{\Sigma}$ is diagonal, i.e. if transcription rates are mutually independent, the above expression reduces to
\begin{equation}
\sigma^2_{k}=\sum_{s} \chi_{ks}^2\, \Sigma_{ss}~~.
\end{equation}
This means that, in absence of transcriptional correlations, each molecular species in the network (both ceRNAs and miRNAs) contributes a positive quantity to the overall level of noise affecting species $k$. In such conditions, the latter clearly exceeds the intrinsic noise level $\Sigma_{kk}$. In particular, large competition-driven susceptibilities (both to perturbations affecting ceRNAs and to perturbations affecting miRNAs) may cause $\sigma_k^2$ to be much larger than $\Sigma_{kk}$, eventually leading to a loss of resolution in molecular levels that will necessarily limit crosstalk effectiveness.
Interestingly, though, Eq (\ref{sigma}) suggests that the presence of transcriptional correlations (i.e. of off-diagonal terms in $\boldsymbol{\Sigma}$) can compensate for this effect \cite{figl}. For instance, negative correlations between ceRNA transcription rates tend to reduce the overall noise level affecting ceRNA $k$ with respect to the fully uncorrelated case (since both $\chi_{ks}$ and $\chi_{ks'}$ are non-negative if $k$, $s$ and $s'$ are ceRNAs). The same holds for positive correlations between the transcription rates of miRNAs and ceRNAs. In both cases, specific patterns of transcriptional correlations coupled with competition may confer a miRNA-ceRNA network the ability to buffer extrinsic noise. On the contrary, anti-correlated miRNA-ceRNA transcription rates or positively correlated ceRNA transcription rates tend to amplify extrinsic noise. These effects are displayed in Fig. \ref{Fig4}, where we show the fluctuation picture arising when all transcription rates are Gaussian distributed, with a fixed ratio between the average and the width.
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig4}
\end{center}
\caption{{\bf (A, B)} Ratio between the magnitude of fluctuations for each molecular species for an interacting ($k_{11}^+,k_{21}^+>0$) and a non-interacting ($k_{11}^+,k_{21}^+=0$) system with 2 ceRNAs and one miRNA (``normalized fluctuations'') for uncorrelated ceRNA transcription rates ($\Sigma_{12}=0$) as a function of the miRNA transcription rate. {\bf (C, D)} Ratio between the normalized fluctuations obtained for maximally anti-correlated transcription rates ($\Sigma_{12}=-1$) and for the fully uncorrelated case ($\Sigma_{12}=0$) as a function of the miRNA transcription rate. {\bf (E, F)} Mean steady state molecular levels as a function of the miRNA transcription rate. All results were obtained by averaging steady state solutions over transcriptional noise. Parameter values are reported in Table \ref{pars}.}
\label{Fig4}
\end{figure}
Uncorrelated ceRNA transcription rates lead to an enhancement of fluctuations with respect to the case in which the miRNA is absent, while anti-correlated ceRNA transcription rates can attenuate this effect.
The noise-processing capacity of crosstalk patterns, and hence ultimately their effectiveness, is therefore strongly linked to the statistics of transcription rates. We shall see below that such correlations can indeed be exploited for the stabilization of the expression levels of protein complexes via the ceRNA mechanism.
\subsubsection{Emergence of bimodal gene expression}
As shown above, one of the main properties of molecular sequestration is the possibility to obtain threshold responses and ultrasensitivity in absence of molecular cooperativity (a property found also when one or more genes are regulated by miRNAs). We also recalled that the system Eq (\ref{uno}) possesses a unique, asymptotically stable steady state \cite{flon}. However, both theoretical and experimental studies have shown that miRNAs, in peculiar conditions of stoichiometry, induce bimodal distributions in the expression level of their targets \cite{bose12, bosi, sgro}. As reviewed in \cite{tsimring14} and shown in \cite{samoilov05}, some biological systems may present bimodality just as a consequence of stochasticity and despite being monostable at the deterministic level. The titrative interaction between miRNA and targets places targets, and ceRNAs in general, into this class of systems. Indeed, when the target expression level is around the threshold established by the amounts of miRNA, if the interaction is sufficiently strong, a small fluctuation in the amount of miRNA or target molecules makes the system jump from the repressed to the unrepressed regime and viceversa. The direct outcome is a bimodal distribution of the targets around the threshold, whose modes are related to the repressed and unrepressed regimes.
The constraint of strong miRNA-target interaction can however be relaxed by introducing some extrinsic noise in the system. This scenario has been exhaustively addressed, both analytically and numerically, in \cite{delgiudice18}. Let us focus on a simple system with two ceRNAs and one miRNA. The system is described by the probability distribution $P(\mu,m_1,m_2,t|\mathbf{K})$ of observing $\mu$ molecules of miRNAs and $m_1,m_2$ molecules of mRNAs of target $1$ and $2$ at time $t$, for a given set of parameters $\mathbf{K} = \{b_1,b_2,\beta,d_1,d_2,\delta,k_{1\mu}^+,k_{2\mu}^+\}$. Such a probability distribution evolves according to the ME (\ref{eq:masterCompleta}) with $N=2$ and $M=1$. Fluctuations in $\mathbf{K}$ should be taken into account in order to obtain the full distribution at the steady state $P(\mu,m_1,m_2)$.
For sakes of simplicity, now assume that $\beta$ is the only fluctuating rate, drawn from a Gaussian distribution centered around $\avg{\beta}$, with variance $\sigma_{\beta}^{2}$ and defined for $\beta>0$. We can obtain the steady-state probability distribution $P(\mu,m_1,m_2|\beta)$ conditional on a specific $\beta$ by applying e.g. the LNA or the Gaussian approximation to the ME. Once this is done, the joint distribution $P(\mu,m_1,m_2)$ is found by performing a weighted average over all possible values of $\beta$, i.e. by applying the law of total probability: $P(\mu,m_1,m_2)=\int P(\beta) P(\mu,m_1,m_2|\beta) d\beta$.
The presence of extrinsic noise in terms of fluctuating parameters is such that the miRNA transcription rate $\beta$ is not the same for every cell as for the pure intrinsic noise case (indeed, we are extracting $\beta$ from a Gaussian distribution). This implies that picking values of $\beta$ above or below the threshold has the consequence of placing the system in the repressed or unrepressed regime respectively. Again, the outcome is a bimodal distribution, which is this time at the population level. Then, the larger the variance $\sigma_{\beta}^2$ (i.e. the extrinsic noise), the broader the ranges of expressed target explored by the left-tails of the Gaussian distribution that will superimpose in the unrepressed mode. The right-tail instead will accumulate cells in the repressed mode. This makes the threshold/noise coupling an efficient tool to filter the variability introduced by extrinsic noise.
\subsubsection{Impact on protein expression}
\label{prots}
The ability of generic regulatory systems to process noise is most crucial for the fine tuning of protein levels \cite{lope}. Interestingly, the control exterted by miRNAs on a single target has been found to be capable of buffering its expression noise \cite{goya}, especially for sufficiently low expression levels \cite{schm}. Given this scenario, one can ask whether the presence of a competitor would improve noise processing, especially at high expression, with the rationale that fluctuations affecting the target mRNA will be smaller (at fixed average) if a competitor titrates regulatory miRNAs away from it. This idea has been tested in simulations after modifying the basic model, Eq. (\ref{due}), to account for protein production \cite{cern}. This is done by simply including the extra equation
\begin{equation}
\frac{d p_i }{dt }= g_i m_i - q_i p_i~~,
\end{equation}
which, for each mRNA species $i$, describes the time evolution of the level $p_i$ of proteins of type $i$ due to synthesis (occurring at rate $g_i$ per substrate molecule) and degradation (occurring at rate $q_i$ per protein). Fluctuations affecting $p_i$ depend on the strength of the interaction between the miRNA and the target's competitor. A weak coupling is insufficient to draw miRNAs away from the target, leading (expectedly) to the same qualitative picture found in absence of the competitor. Likewise, very strong miRNA-competitor coupling leaves the target free from miRNAs, in which case its noise level is comparable to that attained in absence of miRNAs. However, for an intermediate value of the miRNA-competitor binding rate, titration by the competitor appears to be optimally tuned to reduce target fluctuations even at high expression levels (see Fig. \ref{Fig6}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig6}
\end{center}
\caption{{\bf (A)} Mean level of a protein ($p_1$) interacting with a miRNA versus the transcription rate of its mRNA ($b_1$) for different values of the miRNA-mRNA interaction strength. No competitor is present. Expression of $p_1$ gets a stronger threshold-linear behaviour as the miRNA-mRNA interaction strength increases. {\bf (B)} Relative fluctuations (CV) of $p_1$ versus $b_1$, again in absence of competition. {\bf (C)} CV of a target protein ($p_2$) as a function of the mean protein level for the case in which the protein is not interacting with a miRNA (black line, $k_{11}^+=k_{21}^-=0$), is miRNA-regulated but has no competitor (red line, $k_{11}^+=0$, $k_{21}^+=1$) and is miRNA-regulated and has a competitor (blue line, $k_{11}^+=e^{-2}$, $k_{21}^+=1$). {\bf (D)} Maximal mutual information between $p_2$ and its transcription rate $b_2$ for the three regulatory modes presented in (C), plotted as a function of the miRNA-competitor interaction strength. The ceRNA-effect provides the most efficient fine-tuning pathway for intermediate strengths. We used $g_1=0.5$/min and $q_1=0.1$/min for panels (A) and (B); $g_1 = g_2= 0.5$/min, $q_1 = q_2=0.1$/min
for panels (C) and (D). Remaining parameter values are reported in Table \ref{pars}.}
\label{Fig6}
\end{figure}
In this regime, the competitor is maximally derepressed. Remarkably, the overall behaviour of relative fluctuations is close to the Poissonian scenario obtained for an unregulated protein, implying that target derepression plays the main role in reducing fluctuations. Moreover, when crosstalk is most efficient, noise at low expression levels is still efficiently buffered with respect to the case in which miRNAs are absent. A more refined analysis shows that miRNA recycling generically provides enhanced fine tuning by increasing the effective miRNA level.
The fact that, in the human PPI network, the functional products of mRNAs targeted by the same miRNAs are more strongly connected than would be expected by chance strongly suggests that miRNA-mediated regulation, and by extension the ceRNA mechanisms, might play a role in the regulation of protein complex levels \cite{lian,yuan,sass}. In particular, protein forming the subunits of larger complexes tend to be regulated by miRNA clusters, i.e. by groups of miRNA species that are co-expressed \cite{hsu}. When competing RNAs are the substrate for the synthesis of interacting proteins, the onset of the ceRNA mechanism modifies the correlation pattern of the two sub-units, specifically changing the sign of correlations from negative (corresponding to sub-units that are not co-regulated) to positive (reflective the positive correlation that is established between ceRNAs in crosstalk conditions). Such a modification has been observed experimentally \cite{du,nada,kwon}, suggesting that it might provide a biological (albeit non-universal) signature of the ceRNA effect in action.
\subsubsection{Limits to crosstalk effectiveness}
From the previous discussion it is clear that the effectiveness of the ceRNA mechanism is dictated in large part by the relative levels of the molecular species involved and is ultimately limited by noise. An important question in this respect is whether one can characterize the optimal performance that miRNA-mediated regulation can achieve in controlling gene expression. In general, the optimal properties achievable by a regulatory circuit describe fundamental physical limits to its performance, which cannot be overcome independently of kinetic details, and point to the individual processes constituting, in some sense, the bottlenecks for regulatory effectiveness. It is clear that this requires, on one hand, a quantitative definition of `regulatory effectiveness' and, on the other, a benchmark. To fix ideas, one can focus on the system formed by a single miRNA connecting two competing RNAs. Following \cite{tkac}, a natural definition for the effectiveness of ceRNA crosstalk is represented by the degree to which one can control the level of one of the ceRNAs, say ceRNA $i$, by modulating the level of its competitor (ceRNA $j$). In a stochastic setting, the miRNA-mediated interaction linking $i$ and $j$ can be seen as a ``communication channel'' that probabilistically translates the transcription rate of $j$ into a value of $m_i$. This channel is fully described by the conditional probability density $p(m_i|b_j)$, returning a random value of $m_i$ (the contributing noise coming from all involved processes) upon presenting input $b_j$. In turn, miRNA-mediated regulation consists in processing, via $p(m_i|b_j)$ a distribution of values for $b_j$ (denoted by $p(b_j)$) into a distribution of values of $m_i$. For any given $p(m_i|b_j)$ and $p(b_j)$, the strength of the mutual dependence between these variables is quantified by the mutual information \cite{prob}
\begin{equation}
I(b_j,m_i)=\int_{b_j^\textrm{min}}^{b_j^{\textrm{max}}} db_j ~ p(b_j)\int_{{m_i^\textrm{min}}}^{{m_i^\textrm{max}}} dm_i ~ p(m_i|b_j) \log_2 \frac{p(m_i|b_j)}{p(m_i)}~~,
\end{equation}
with $p(m_i)=\int_{b_j^\textrm{min}}^{b_j^{\textrm{max}}} db_j p(m_i|b_j)p(b_j)$ the output distribution of $m_i$. Assuming that the channel is fixed, i.e. that $p(m_i|b_j)$ is given, the optimal regulatory effectiveness is obtained when the input distribution $p(b_j)$ is such that $I$ is maximized:
\begin{equation}
\max_{p(b_j)} ~ I (b_j, m_i) \equiv I_{\max} ~~.
\end{equation}
$I_{\max}$ is called the {\it capacity} in information-theoretic terms and ultimately measures how much information (in bits) can be conveyed at most from input ($b_j$) to output ($m_i$) by a given input-output relationship $p(m_i|b_j)$. In loose but intuitive terms, $I_\textrm{max}$ describes the number $\mathcal{N}$ of different values $m_i$ that can be distinguished in a reliable way given the noise, which is roughly given by $\mathcal{N}\sim 2^{I_\textrm{max}}$. If $I_\textrm{max}\simeq 0$ (note that $I\geq 0$ by definition), the noise only allows to distinguish at most one level of $m_i$; for $I_\textrm{max}\simeq 1$ two levels (high/low) can be separated; and so on.
The effectiveness of miRNA-mediated crosstalk has been characterized within the above setup starting from numerical simulations of the stochastic dynamics and using direct transcriptional regulation of $m_i$ (i.e., the capacity of the corresponding miRNA-independent regulatory channel) as the benchmark against which miRNA-mediated information flow was evaluated. In particular, the dependence of $I_\textrm{max}$ on kinetic parameters was analyzed to identify optimal parameter regions and limiting processes. The emergent scenario can be summarized as follows \cite{prob}:
\begin{enumerate}
\item As might have been expected, the capacity of miRNA-mediated regulation is optimal in a specific range of values for the target's repression strength. Intuitively, a tight control of $m_i$ based on $b_j$ requires ceRNA $i$ to be sensitive to changes in miRNA levels. Too weak (resp. too strong) repression causes ceRNA $i$ to become fully unrepressed (resp. fully repressed), so that the optimal range lies between these extremes. Quite remarkably, though, optimal ceRNA crosstalk can be more effective than direct transcriptional control.
\item In presence of significantly different catalytic degradation rates (faster for $m_i$, slower for $m_j$) ceRNA crosstalk outperforms direct transcriptional regulation. Intuitively, the above situation makes transcriptional control especially inefficient since $m_i$ is going to be strongly repressed by miRNAs. miRNA-mediated control, instead, benefits from the fact that $m_j$ can de-repress ceRNA $i$ by lifting miRNAs away from it.
\item When miRNA populations are sufficiently large and miRNA-ceRNA couplings are weak, miRNA-mediated regulation is as effective as a direct transcriptional control. This is intuitively due to the fact that, in this limit, the relative noise affecting miRNA levels becomes negligible. This removes the additional source of noise affecting the post-transcriptional channel compared to the transcriptional one, effectively making the two regulatory modes comparable.
\end{enumerate}
The outlook is that, besides generically contributing to noise buffering, ceRNA crosstalk can control of gene expression to a degree that is tightly connected to the ability of the competitor ($m_j$) to de-repress the target ($m_i$). When the controller's kinetics does not suffice to titrate miRNAs away from ceRNA $i$, miRNA-mediated regulation is ineffective. Otherwise, it provides a high (and, possibly, the highest achievable) degree of control over expression levels, especially when kinetic parameters are sufficiently heterogeneous.
\subsection{ceRNA crosstalk away from stationarity}
\label{sec:3}
\subsubsection{Equilibration times}
\begin{figure}[t]
\begin{center}
\includegraphics[width=16cm]{Fig7}
\end{center}
\caption{We consider a simple ceRNA network for an increasing number of targets ranging from 2 to 20 and a single microRNA. {\bf (A)} Equilibration time of ceRNA$_1$ when ceRNA$_2$ is induced, as a function of the miRNA transcription rate. Around threshold, we observe a critical slowing down in the response time. {\bf (B)} Same as (A), but now the response time is measured after a knock-down of one of the competitors. In this case, we observe a speed up of the response time at threshold.}
\label{Fig7}
\end{figure}
The titrative miRNA-target interaction entails both susceptibility and statistical correlation between the competing chemical species. We have seen before how all these effects become maximal at quasi equimolar ratio. One can however also study how fast the system responds to an external perturbation. To fix ideas, we will focus as usual on the case of a single miRNA targeting 2 ceRNAs. In particular, we want to quantify the time needed for a particular ceRNA (here ceRNA$_1$) to reach the new stationary state after
\begin{itemize}
\item A sudden increase of the transcriptional activity of ceRNA$_2$ at time $t=0$, i.e.
\begin{displaymath}
b_2(t=0^-)=0 \quad\mathrm{and}\quad b_2(t=0^+)=b^*
\end{displaymath}
\item A sudden decrease of the transcriptional activity of ceRNA$_2$ at time $t=0$, i.e.
\begin{displaymath}
b_2(t=0^-)=b^* \quad\mathrm{and}\quad b_2(t=0^+)=0
\end{displaymath}
\end{itemize}
We define the response time as the time needed for ceRNA$_1$ to reach half the way between the initial (before perturbation) and final (after perturbation) steady state levels. In particular one can evaluate the response times $T_\mathrm{ON}$ and $T_\mathrm{OFF}$ for both the switch-on and switch-off scenarios (i.e. for ceRNA$_2$ $\mathrm{OFF}\rightarrow\mathrm{ON}$ and $\mathrm{ON}\rightarrow\mathrm{OFF}$ respectively) by numerically integrating Eq.~(\ref{uno}) to estimate $T_\mathrm{ON/OFF}$ as the times at which the following relations hold:
\begin{eqnarray}
m_1(T_\mathrm{ON}) &=& m_1(0)+\frac12 \left(\lim_{t\rightarrow\infty}
m_1(t) - m_1(0)\right) \quad m_1(0) = m_1^\mathrm{ss}~,~m_2(0) = 0 \\
m_1(T_\mathrm{OFF}) &=& m_1(0)-\frac12 \left(m_1(0) -
\lim_{t\rightarrow\infty} m_1(t)\right) \quad m_1(0) = m_1^\mathrm{ss}~,~m_2(0) = m_2^\mathrm{ss}\,
\end{eqnarray}
In this framework we can easily study the dependence of the response times $T_\mathrm{ON/OFF}$ on the basal miRNA concentration (i.e. on $\beta_1$ in this case). Results (see Fig.~\ref{Fig7}) show a non-monotonous dependence of $T_\mathrm{ON/OFF}$ on the trascriptional activity of the miRNA. In particular, $T_\mathrm{ON}$ (resp. $T_\mathrm{OFF}$) displays a maximum (resp. a minimum) in correspondence with the threshold between the repressed and unrepressed phase.
A natural question is how the presence of more ceRNAs changes the scenario we just described for the simple one miRNA two ceRNAs network. The same {\em in silico} experiment can be generalized to an arbitrary number of ceRNAs where all but one (say ceRNA$_2$) is either knock-out or induced. Perhaps unsurprisingly (see Fig.~\ref{Fig7}), one again sees a dilution effect: upon increasing the number of ceRNAs from 2 to 20 the relevance of the effect --measured in terms of the distance between the initial and final state of the system-- becomes quantitatively less relevant.
\subsubsection{Out of equilibrium dynamics}
The out-of-equilibrium dynamics of the miRNA-ceRNA system has been studied in \cite{dyna}. The emergent crosstalk scenario is substantially richer than the stationary one. For simplicity, we shall limit ourselves to describing results obtained for a system with $N$ ceRNAs interacting with and a single miRNA species. From a physical viewpoint, the quantities
\begin{equation}
\arraycolsep=1pt\def1.8{1.8}
\begin{array}{r@{}l}
&{}\tau_0=\delta^{-1} \quad, \quad\tau_{1,i}=d_i^{-1}\quad ,\quad \tau_{2,i}=(\sigma_i+\kappa_i+k_i^-)^{-1} \\
&{}\tau_{3,i}=(\sigma_i+\kappa_i)^{-1} \quad ,\quad \tau_{4,i}=\sigma_i^{-1}~~, \quad \tau_{5,i}=\kappa_i^{-1}
\end{array}
\end{equation}
represent the relevant characteristic intrinsic time scales of this system. Based on Eqs (\ref{uno}), they represent, respectively, the mean lifetime of miRNA species $a$ ($\tau_0$)) and of ceRNA species $i$ ($\tau_{1,i}$), the mean lifetime of the complex formed by ceRNA $i$ ($\tau_{2,i}$), and the mean time required for complex degradation ($\tau_{3,i}$), stoichiometric complex degradation ($\tau_{4,i}$) and catalytic complex degradation ($\tau_{5,i}$). The features characterizing dynamical crosstalk can change depending on how these time scales are related. To get some insight, one can focus on how the system relaxes back to the steady state following a small perturbation away from it. Upon linearizing the system (\ref{uno}), one can derive equations for the deviations of each molecular species from the steady state, i.e. for the quantities
\begin{equation}
\arraycolsep=1pt\def1.8{1.5}
\begin{array}{r@{}l}
x_i(t) &{}\equiv m_i(t)-\avg{m_i} \\
y(t)&{}\equiv \mu(t)-\avg{\mu}\\
z_i(t)&{}\equiv c_i(t)-\avg{c_i}
\end{array}~~.
\end{equation}
(We have suppressed the miRNA index for sakes of simplicity.) Introducing (small) time-dependent additive perturbations of the transcription rates of the form $b^o_i(t)$ and $\beta_o(t)$, the above variables can be seen to evolve in time according to
\begin{equation}\label{dyx}
\arraycolsep=1pt\def1.8{1.8}
\begin{array}{r@{}l}
\frac{d}{dt}x_i&{}=-d_ix_i + b^o_i-k_i^+(\mu\,x_i+m_i\,y)+k_i^-z_i\\
\frac{d}{dt}y&{}=- \delta y + \beta_o-\sum_i k_i^+(\mu\,x_i+m_i\,y)+\sum_i(k_i^-+\kappa_i)z_i \\
\frac{d}{dt}z_i&{}=-(\sigma_i+k_i^-+\kappa_i)z_i + k_i^+(\mu\,x_i+m_i\,y)
\end{array}~~,
\end{equation}
This system can be analyzed in the frequency domain ($\omega$) by Fourier-transforming (\ref{dyx}). This allows to define the {\it dynamical susceptibility}
\begin{equation}\label{susk1}
\widehat{\chi_{ij}}(\omega)=\frac{\partial \widehat{x_i}}{\partial \widehat{b^o_j}}~~
\end{equation}
where $\widehat{f}$ is the Fourier transform of $f$. The general study of this quantity is possible while not straightforward \cite{detw}. However, $\widehat{\chi_{ij}}(\omega)$ can be estimated in a relatively simple way in few instructive limiting cases in which timescales are sufficiently separated. For instance, when $\tau_{3,j}\ll 1/k_j^-$ and $\tau_{1,j}<\tau_{5,j}$, complexes formed by ceRNA $j$ will typically keep miRNAs blocked for times longer than the intrinsic ceRNA degradation timescale. This may allow for ceRNA $i$ to get de-repressed and hence for the establishment of crosstalk, independently of whether stoichiometric processing takes place. Indeed one finds that, when $\kappa_j \ll \omega\ll d_j$ (i.e. for timescales intermediate between $\tau_{1,j}$ and $\tau_{5,j}$),
\begin{equation}\label{qq}
\widehat{\chi_{ij}}(\omega)\simeq
\frac{\sigma_j+\kappa_j}{\sigma_j}~\chi_{ij}~~
\end{equation}
where $\chi_{ij}=\frac{\partial\avg{m_i}}{\partial b_j}$ stands for the steady-state susceptibility \cite{dyna}. Remarkably, the quantity on the left-hand-side of Eq (\ref{qq}) can be shown to remain finite for $\sigma_j\to 0$, providing quantitative support to the observation that ceRNA crosstalk can be active dynamically even in purely catalytic systems (where no crosstalk occurs at stationarity and $\chi_{ij}$ vanishes). In other words, then, in this limiting case and in an intermediate frequency window, the dynamical susceptibility is comparable to the steady-state value and occurs even for $\sigma_j=0$. Away from this window, instead, crosstalk in this limit is weaker than it is at stationarity.
A more careful analysis shows that, in certain regimes, the dynamical response can even exceed the stationary one. This happens, for instance, when complex dissociation is much faster than other processing pathways and ceRNAs are fully repressed, implying that dynamical crosstalk can occur even between pairs of ceRNAs that could not interact at steady state \cite{dyna}. In this sense, the ceRNA mechanism out of equilibrium is substantially more complex and richer than its stationary counterpart. In addition, the possibility to modulate the time scales of different interactions allows to construct systems in which static and dynamic responses are tuned so as to ensure the correct transient activation of a specific gene and the long-term stabilization of expression levels. An example of such a coordination, based on findings related to skeletal muscle cell differentiation \cite{legn}, has bee studied in \cite{fior}.
\section{Outlook}
Mathematical models developed to elucidate the emergent features of ceRNA crosstalk have so far mainly relied on computational schemes for stochastic simulations (Gillespie algorithm) and on analytical approximations of the master equation associated to the system of interacting molecules (LNA, Gaussian, Langevin). On the other hand, a full understanding of competition-driven coupling requires, as we have seen, disentangling it from concurrent effects. Indeed, the identification of crosstalk from transcriptional data is in our view especially hard since statistical correlations between RNAs sharing a common miRNA regulator can arise just due to the fact that they both respond to fluctuating miRNA levels. Once the relationship between competition- and fluctuations-related features is clarified, ceRNA crosstalk patterns display strong intrinsic specificities like
\begin{enumerate}
\item selectivity,
\item asymmetry,
\item plasticity (i.e. sensitivity to kinetic parameters),
\item sensitivity to the degree of parameter heterogeneity, and
\item the possibility to aggregate a large number of weak interactions to significantly impact molecular levels.
\end{enumerate}
These features in turn allow for the establishment of complex noise-processing properties. Note that, unsurprisingly, some of these features characterize other competition scenarios in regulatory system (e.g. competition to bind transcription factors, $\sigma$-factors, ribosomes, etc. \cite{maur,brew,tull}).
We have reviewed these aspects together with the methods that can be employed to quantify them. Several important points might however deserve equal consideration. In first place, miRNAs can also crosstalk through ceRNAs, generating a very similar phenomenology whose impact has been, to our knowledge, far less clarified \cite{loin}. Secondly, the modeling framework we discussed ignores some kinetic steps assuming essentially that they are non rate-limiting. Still, it is known that in some cases binding to Argonaute (Ago), the catalytic component of the RNA-induced silencing complex, represents a kinetic bottleneck \cite{koll}. Likewise, crosstalk can be affected by the competition to bind Ago \cite{loin}. Third, a rich trafficking of miRNAs and their targets is known to occur between the cell nucleus and the cytoplasm, leading to remarkable localization effects whose biological significance is largely unexplored \cite{nils}. Well-mixed models like those discussed here are clearly unable to deal with such effects; spatial generalizations are mandatory \cite{levlev,levi}. Finally, the phenomenology derived from small modules can integrate in highly non-trivial ways at the scale of the transcriptome, where topology provides additional degrees of freedom to modulate crosstalk patterns. While, as shown here, some (basic) things about the role of network structure can be understood with simple calculations, a more thorough data-based analysis of these aspects would be greatly welcome.
\begin{acknowledgement}
Work supported by the European Union's Horizon 2020 research and innovation programme MSCA-RISE-2016 under grant agreement No 734439 INFERNET. We are indebted with Matteo Figliuzzi, Enzo Marinari, Matteo Marsili and Riccardo Zecchina for our fruitful and enjoyable collaboration.
\end{acknowledgement}
|
1,116,691,500,459 | arxiv | \section{Introduction}
Predicting pedestrian trajectory is an essential task for accomplishing mobility-based jobs in naturalistic environments, such as robotics navigation in crowded areas, safe autonomous driving and many other applications that require foreseeing motion in an interactive dynamical system. Existing literature has focused so far on modeling the social interactions between pedestrians by discretizing the environment as a grid of local spatial neighborhoods \cite{alahi2016social}, taking global scope of the whole scene \cite{vemula2018social} or inferring relationships between pedestrians pairwise \cite{choi2019looking}. So far, spatial neighborhoods have been the fundamental basis for considering pedestrians influence on each other, accounting only for the positional and higher-order motion features.
Other methods have used additional features such as head pose \cite{hasan2018mx,hasan2019forecasting,hasan2018seeing} to attain the visual field of attention in pedestrians in order to assess how they are related to each other.
Classical approaches \cite{hasan2018seeing,helbing1995social,yamaguchi2011you} resorted to a hardcoded quantitative threshold to define the proximal distance for neighborhoods. The use of deep learning improves the neighborhood concept by learning a custom threshold for each pedestrian \cite{kipf2018nri}.
MX-LSTM \cite{hasan2018mx} determines neighborhoods around pedestrians visually based on the fixed visual horizon of pedestrian. Within this imaginary area, pedestrians spatial states get pooled. To this end, MX-LSTM only considers the correlation between head pose and speed. In the aforementioned works, generalization can be problematic as the models rely on fixed assumptions that pertain to specific scenes. Recent works introduced additional features such as the looking angle of pedestrians \cite{hasan2018mx}. These approaches and GAN-based approaches resorted to dedicating several pooling layers for treating multiple features \cite{lisotto2019social,ridel2019scene} or multiple trajectories states \cite{amirian2019social,sadeghian2018sophie,zhang2019seabig} which will require additional transformation and pooling layers or assigning separate LSTM to each pedestrian in order to treat all the features together.
\begin{figure}
\centering
\includegraphics[width=6cm,
height=4cm, trim={0 0 0 2cm }]{images/intro_figure.png}
\caption{Illustration of adaptive deep neighborhoods selection process. The process is comprised of two grids. To the left, the static neighborhood grid $f_O$ segments the scene image into several local regions. It takes pedestrians looking angle $v_1, v_2$ (shaped as yellow cones) to stem their awareness of the static surroundings. The dynamic grid $f_S$ takes pedestrians trajectories $x_1, x_2$ along with their looking angle to stem their social interactions. The end goal is to predict future trajectories $\Tilde{x}^1_{t+l}, \Tilde{x}^2_{t+l}$ accurately given the estimation of future potential neighborhoods and the best modeling of social interactions. To the right, the output static grid has few highlighted areas, which indicates future neighborhoods where pedestrians would walk. Also note the leaning links connecting pedestrians to indicate how the existence of social influence on each other.}
\label{fig:intro_fig_visuospatial_ngh}
\end{figure}{}
In order to overcome the limitations of existing approaches, we estimate the neighborhood given the static context around pedestrian and a mixture of social cues that defines pedestrian situational awareness and social interactions. More concretely, we define an adaptive neighborhood that relies on visual locus and spatial constraints, supported by the existential correlation that associates the head pose to walking direction and intention. This is based on the observation that although people often do not articulately plan their walking trajectories, they keep their attention focused within coarse path boundaries. They also do not strictly define their neighborhoods. The neighborhood is a virtual concept to discretize the scene and depends on a mixture of social cues that indicate social interaction state and pedestrian awareness.
Our approach updates the neighborhood definition for a multi-cued context, where multiple features are encoded for pedestrians. We coin this Visuospatial neighborhoods, as such neighborhoods characterize the spatial distance around pedestrians as well as their visual attention boundaries.
Figure \ref{fig:intro_fig_visuospatial_ngh} shows that neighbourhood $\nu_4$ is highlighted with yellowish shade. This is perceived as a future neighborhood for pedestrians that are related to each other. Hence their states are to be pooled together as they share the same future neighborhood. Anticipating some region as potential neighborhood relies on where they look and their walking path.
We propose Graph-to-Kernel LSTM (G2K LSTM) that combines the spatial physical boundaries and the visual attention scope to shape each pedestrian neighborhood. G2K LSTM is an LSTM-based approach that transforms a spatio-temporal graph into the kernel to estimate the correlations between pedestrians. This correlation represents the importance of a relationship between two people, stemming from the natural correlation between their visual angle and their relative distances. In sociological and psychological proxemics \cite{hall1966hidden,bera2017sociosense}, human maintains specific distance from others to be his/her personal zone within which they feel comfortable and evade collisions. This zone is maintained by what they observe and pay attention to.
Attaining accurate estimation of such neighborhoods is a non-trivial task, due to the stochasticity pertaining to the ground-truth definition. Following a stochastic optimization, neighborhood formation is bounded to minimize the trajectory prediction errors, such that the underlying graph structure improves the social interaction modeling.
We develop neighborhood modeling mechanism based on Grid-LSTM \cite{kalchbrenner2015grid}, which is a grid mask that segments the environment into a regular-shaped grid. It encompasses the sharing mechanism between adjacent neighborhoods in the grid. This sharing determines how pedestrians social interaction is modeled in deep networks. Predicting pedestrians future trajectory can be used to further define future neighborhoods, thereby leading to a better understanding of how pedestrians are related and influenced by each other.
In summary, this paper delivers the following contributions:
\begin{itemize}
\item We introduce Grid-LSTM into pedestrian trajectory prediction to encode multiple social cues altogether. Seeding from the natural correlation found between the visual and the spatial cues mixture, the network learns a soft-attention mechanism for evaluating given social interaction importance from within the provided data. The correlation between head pose and walking direction emphasizes pedestrian walking intention and in this work we combine head pose with walking trajectories to improve social and contextual modeling of pedestrians interactions.
\item We present a deep neighborhood selection method for estimating the influence of social relationships between pedestrians. It guides the message-passing process according to a relational inference in the graph-based network that decides the routes for sharing the attention weights between pedestrian nodes.
\item Due to the aforementioned data-driven mechanisms, our approach yields state-of-the-art results on widely-tested datasets. Our models also produce consistent results across the datasets, which indicates its generalization capability to various crowd contexts.
\end{itemize}
\section{Related Work}
\paragraph{The visual field and the situational awareness.}
Existing works have shown the benefits of combining head pose with positional trajectory for prediction. The head pose is used as a substitute for the gaze direction to determine the Visual Field of Attention (VFOA) \cite{hasan2018seeing,yang2018my}.
The head pose feature correlates with the walking direction and speed, which
emits pedestrian destinations as well as their awareness of the surrounding context. The visual field of attention in pedestrians relied on assumptions that align head pose with gaze direction to fixate the attention region as pedestrian is walking. In resemblance to \cite{yang2018my}, we argue that the width of the visual field and its shape shall affect the representation of pedestrian visual awareness state and thereby their neighborhood perception.
To a large extent, when pedestrians are walking, they only consider other pedestrians who are close and can pose a direct influence. This social situation can be captured when using a narrow visual angle, i.e. 30 $\deg$. Nevertheless, pedestrian does not always look straight, they tend to tilt their heads and therefore perceive more about the environment structure and other dynamical objects around them. Using a wider sight span for each pedestrian allows a better perception of pedestrian awareness and focal attention.
\paragraph{Relational inference for neighborhood selection.}
Extensive research is conducted on relationships inference between entities of data in image segmentation \cite{scarselli2008graph,battaglia2018relational}, graph recovery for interactive physical systems \cite{webb2019factorised,kipf2018nri}. Recently, researchers engaged spatio-temporal graphs to model pedestrians relationships to each other \cite{jain2016structural,vemula2018social}. More advanced approaches such as \cite{choi2019looking,sadeghian2018sophie,xue2019location,fernando2018soft+} evaluate the relational importance between pedestrians using neural attention technique. In addition to using attention, \cite{zhang2019sr} deploys the refinery process that iterates over variants of the neighborhood until it selects the best neighbors set for each pedestrian. In our work, we similarly target neighborhood selection problems but with fewer iterations and faster recurrent cell called Grid-LSTM that requires fewer training epochs.
\paragraph{Social neighborhoods for human-human interaction.}
In the literature, this area is extensively studied as two separate disciplines: Human-human interaction and Human-space interaction. Forming neighborhoods is found to be based on a single type or a combination of both interaction types and the objective is to determine a way for combining pedestrians and discovering their influence. The approaches that socially define the basis for neighborhoods \cite{alahi2016social,cheng2018pedestrian,xue2018ss} proposed pooling techniques to effectively combine spatially proximate pedestrians, while \cite{haddad2020self} proposed influence-ruled techniques to combine pedestrians that align their motion according to each other. However, even when modeling the spatial neighborhoods with the aid of context information, existing approaches generally involve hardcoded proxemic distance for outlining neighborhood boundaries as fixed grids. While these works achieve successful results, they fail to consider dynamic environments representation which may cause failure in cases that require adaptive neighborhoods or extra cues such as pedestrian visual sight span, to ascertain the conjunction between social motion and contextual restrictions.
\section{Our Approach}
\subsection{Problem Formulation}
The problem of learning neighborhood boundaries is formulated as minimization of Euclidean errors between the predicted trajectory $\widetilde{X}$ and ground-truth trajectory $X$:
\begin{equation} \label{eqn_costfn}
\mathcal{L} = \underset{J_\theta}{argmin} ||\quad \widetilde{X} - X \quad||^2_2
\end{equation}{}
Such that $J_\theta$ refers to the network trainable parameters that minimize the loss, $\mathcal{L}$, \textit{L2-norm} of $\widetilde{X}$ and $X$.
Let $X$ be pedestrians trajectories, such that: $X$ = ${x_1, x_2, ... , x_n}$, with n pedestrians. $\widetilde{X}$ are future trajectories, $x^{i}_t$ is i-th pedestrian trajectory from time-step $t = 1$ until $t = t + obs$, given that $obs$ is observation length. We observe 8 steps of each pedestrian trajectory and predict for the next 12 steps.
Each predicted step is added to the first predicted point, $\Tilde{x}^i_{t+pred} = \Tilde{x}^i_{t+1} + \Tilde{x}^i_{t+2} + ... + \Tilde{x}^i_{t+l}$ to maintain consistency and dependency between predicted steps.
\subsection{Temporal Graphs}
Given a set of pedestrians P, we represent their trajectories over time using a temporal graph $G_t$ at each time step $t$, containing \ensuremath{\mathcal{N}} nodes, such that each pedestrian is assigned a node $n_t$ to store their position and temporal edge $e_t$ that links the node $n_{t-1}$ to $n_t$.
Our approach Graph-to-Kernel (G2K), maps temporal graphs into a kernel of fixed dimension, $K$, such that the kernel generates predicted trajectories $\widetilde{X}$ and adjacency states between the pedestrian nodes. Adjacency models the social interactions. So $J_\theta$ in Eq. \ref{eqn_costfn} refers to pedestrians associations to each others:
\begin{equation}\label{eqn:kernel}
\widetilde{X} , J_\theta = K(G_{(\mathcal{N},\nu)})
\end{equation}{}
\subsection{Social Relational Inference (SRI)}
\label{sec:sri}
In this section, we explain the SRI unit which is the proposed kernel. We detail the steps for the mechanism in which the kernel $K$ performs deep relational inference between pedestrians. SRI unit starts estimating the adjacency state for each pedestrian. It forms the adaptively-shaped social neighborhoods.
The kernel has a customized design relevant to the features set included in our model versions.
As defined earlier in Eq. \ref{eqn:kernel}, kernel $K$ generates the social interaction states and future positional predictions, so according to $MC$ model and $MCR^*$ models set, Eq. \ref{eqn:kernel} is updated as follows:
\begin{equation}
\label{mc_mcr_K}
\widetilde{X}, J_\theta = K(f_S, V, h)
\end{equation}{}
A single Grid-LSTM cell $NLSTM_{\nu}$ is used as the encoder of the social human-human interactions. It assumes that motion happens over the uniformly-divided square grid where each neighborhood is tagged by $\nu$.
SRI casts $NLSTM_{\nu}$ over the temporal graph $g_t$. It takes pedestrians trajectories, $X$, initial hidden state $h$ and generate embeddings of spatial features $f_S$. To formulate multi-cued trajectories set $T$, we firstly encode positional trajectories $X$ using two-stages transformation function $\phi$ as follows:
\begin{equation}
\mathcal{X} = \phi (X)
\end{equation}
\begin{equation}
\phi (X) = W_{ii} * (W_i * X)
\end{equation}
In our work, we coin a new term for multi-cued neighborhoods as "Visuospatial" neighborhood, that is a combination of pedestrians spatial whereabouts and their visual sight span.
To represent the Visuospatial neighborhoods, we encode the 2D head pose annotations, also called Vislets, $V$ using single-stage transformation function $\phi$ as follows:
\begin{equation}
\mathcal{V} = \phi (V)
\end{equation}
\begin{equation}
\phi (V) = W_i * V
\end{equation}
Then we concatenate the embedded cues $\mathcal{V}$ and $\mathcal{X}$ in Eq. to be fed into the GNN Grid-LSTM in Eq. \ref{eqn:ngh_glstm}:
\begin{equation}
T = (\mathcal{X}, \mathcal{V})
\end{equation}{}
\begin{equation}
\label{eqn:ngh_glstm}
f_S, h_S = NLSTM_{\nu}(T, h_S)
\end{equation}{}
We developed several models starting with the simplest model comprising of single Grid-LSTM cell with only positional trajectories. We term it as G-LSTM model.
According to G-LSTM model, kernel $K$ of Eq. \ref{eqn:kernel} is updated as follows:
\begin{equation}
\label{glstm_K}
\widetilde{X}, J_\theta = K(f_S, h)
\end{equation}
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth, height=6cm]{images/gated_neighborhood_network.png}
\caption{Full pipeline of G2K kernel. The SRI network encodes Vislets $\mathcal{V}$ and positional trajectories $\mathcal{X}$ for each pedestrian trajectory. Then maps them into social grid mask $f_S$ using $NLSTM_\nu$. The GNN network discretize static context using $NLSTM_O$ into 'Visuospatial' neighborhoods and stores pedestrian contextual awareness in $f_O$. At the consequent step, SRI takes $f_O$ and $f_S$, and maps them into the weighted adjacency matrix. This will generate the edge set $\nu$ as means of completing graph $G_t$ at time-step $t$.}
\label{fig:sri}
\end{figure}{}
After that, we develop two versions of Multi-Cued Relational inference (MCR) model: $MCR_n$ \& $MCR_{mp}$.
Hidden states sharing between pedestrians is guided by a deep relational mechanism that involves learning the social interaction effect. This effect is manifested through soft-attention that evaluates pedestrian relationships importance and chooses the more influential relationships set than other associations that can have a weaker impact on pedestrian trajectory. Consequently, the hidden states become messages passed between pedestrians based on a deep selective process that determines who are the neighbors of each pedestrian as displayed in Figure \ref{fig:sri}.
As part of the SRI network, we run GNN network to produce the final static features $\mathcal{F}$. SRI takes this output and feeds it through another transformation function $\phi$. The latter embeds the features set output by GNN unit and yields $\mathcal{F}'$:
\begin{equation}
\mathcal{F}' = \phi(\mathcal{F}) = C * ( (W_v * [f_S, \mathcal{V}] + b_v) * (W_r * \mathcal{F}) )
\end{equation}
In $MCR_{mpc}$, $\mathcal{F}$ stores the enhanced modeling, comprised of physical spatial constraints $C$ and social relative features $f_S$.
In $MCR_n$, $\mathcal{F}$ will only take the social relative features $f_S$.
SRI unit will tune the influence of each region. It deploys scaled self-attention mechanism \cite{velivckovic2017graph} (as defined in Eq. \ref{eqn:soft_attn}). Attention coefficients $\mathscr{a}$ considers the human-human interaction features and the human-space interaction features:
\begin{equation} \label{eqn:soft_attn}
a = \frac{Softmax (\exp{(\mathcal{F}')})}{\Sigma \exp{(\mathcal{F}')}}
\end{equation}{}
Finally, $MCR_n$ kernel neurally evaluate hidden states by passing through Softmax layer. This will transform pedestrians nodes states into the continuous space of the interval [0,1]:
\begin{equation} \label{eqn:softmax_fnri}
Softmax(H) := [0,1]
\end{equation}{}
Given the $i$th pedestrian trajectory features, neighborhood $\nu^i_t$ of pedestrian $i$ is defined as follows:
\begin{equation}
\nu^i_t = \{(n_i, n_j)\}^+; \quad |\nu^i_t| <= |\mathcal{N}|
\end{equation}{}
Moreover, neighborhood boundaries at pedestrian $i$ are defined by the set of edge pairs that connect pedestrian nodes to other nodes in the graph.
Note here that $\nu$ is the Greek symbol (\textit{Nu}) and it is different from the symbol $V$ which was earlier tagged to the Vislets.
Inspired by the neural factorization technique \cite{webb2019factorised} to factorize spatio-temporal graph edges, we establish a deep mechanism for neighborhoods that is aware of the static and the social constraints.
Since the best setting for neighborhoods is unknown, we estimate pedestrians relationships using their relative visual attention and spatial motion features to learn their neighborhoods.
In $MCR_{mp}$ kernel, the model directly factorizes relationships between pedestrians by passing importance weights in Eq. \ref{eqn:soft_attn} as a message with the hidden states matrix and thresholding the strong and weak relationships:
\begin{equation}
H = Softmax (f'_{\mathcal{O}_{t+1}}*H)
\end{equation}{}
For accomplishing social relational inference of pedestrians interactions, the last two steps are common among models: $MCR_n$, $MCR_{mp}$ and $MCR_{mpc}$. Using a normally-initialized linear transformation matrix $W$, the new hidden states are mapped into adjacency matrix to determine the edges in the graph which manifest pedestrians relationships with various degrees of strength:
\begin{equation}
A = W*H
\end{equation}{}
Eventually, $J_\theta$, is the updated pedestrian adjacency states $H^*_t$, which will be fed again in the following time-step to the social neighborhood encoder in Eq. \ref{eqn:ngh_glstm}:
\begin{equation}
H^*_{t} = A*H
\end{equation}{}
The resulting adjacency-based representation $H^*_{t}$ is evaluated by whether the new proposal of edge set, generates better modeling than the previous time-step edge set.
This is set as the objective cost function $J_\theta$.
\subsection{Gated Neighborhood Network (GNN)}
\label{sec:gnn}
In this section, we explain our proposed mechanism for deep neighborhoods formation.
GNN unit involves a Grid-LSTM \cite{kalchbrenner2015grid} cell $NLSTM_O$ to encode contextual interactions. It assumes the scene as fixed-shaped grid of shape $[n x n]$.
\begin{figure*}
\centering
\includegraphics[width=\linewidth, height=6cm]{images/mpc_kernel_3.png}
\caption{Gated Neighborhood Network pipeline. At the beginning, $2DCONV$ encodes a static image of the scene and forward the features into NLSTM cell which discretizes the environment into a virtual grid.}
\label{fig:gnn}
\end{figure*}{}
Figure \ref{fig:gnn} illustrates the pipeline of GNN. $2DCONV$ is a 2D convolutional layer used for encoding the contextual interaction. It runs a grid mask filter $M$ and assumes that static space neighborhoods are discretized into a square grid. The mask filter is a normally-initialized matrix to indicate that initially, all local regions are of equal influence on pedestrian motion.
\begin{equation}
C = 2DCONV(f_S, h_\mathcal{O}, M)
\end{equation}{}
Compared to literature approaches, the Social-Grid model \cite{cheng2018pedestrian} dedicates a separate Grid-LSTM \cite{kalchbrenner2015grid} for each pedestrian. In our models, we use only a single Grid-LSTM to encode a set of features for all pedestrians.
MX-LSTM \cite{hasan2018mx} applies the concept of Visual Field of Attention (VFOA) to determine pedestrians attention to each other. They hard-code pedestrians VFOA and rely only on the head pose which to some extent can validate the looking angle.
The social pooling and the visual pooling proposed in \cite{alahi2016social,hasan2018mx} respectively selects one feature modality for learning about pedestrians interactions by pooling their states into pre-determined neighborhoods.
In our work we encode head pose $V$ with static features $C$ to formulate the "Visuospatial" neighborhood representation as means of pedestrian attention to static context, using single Grid-LSTM cell $NLSTM_O$:
\begin{equation} \label{eqn:visuospatial}
f_\mathcal{O}, h_\mathcal{O} = NLSTM_O(C, V, h_\mathcal{O})
\end{equation}{}
Eq. \ref{eqn:visuospatial} introduce multi-modality concept through combining both, visual awareness state $V$ and physical spatial constraints $C$, for stemming pedestrians attention to the physical context. In other words, encoding pedestrians interaction with the static context is stored in $f_\mathcal{O}$. Later in Section \ref{sec:sri}, $f_\mathcal{O}$ will be passed with the social interaction features $f_{S}$ to calculate Adjacency state of each pedestrian.
Taking the static grid features $f_{\mathcal{O}}$ in Eq. \ref{eqn:visuospatial} are regularized by a constant factor of $\lambda$ as a means of starting with equally important static neighborhood regions:
\begin{equation}
f_{\mathcal{O}}^\ensuremath{\prime} = f_{\mathcal{O}} * f_{\lambda}
\end{equation}{}
In formal definition, given a frame $f$, we discretize it into a grid of uniformly-shaped grid $G$ of $k$ neighbourhoods. Initially, coarse neighborhoods in the grid $f_\mathcal{O}$ are assigned equal importance which will be adapted through training batches.
A final neighborhood mask $\mathcal{F}$ contains a combination of fixed and relative pedestrians trajectory features, $f_O \& f_S$. $\mathcal{F}$ can vary depending on the input features to the static mask:
\begin{equation}
\mathcal{F} = f_S * f_{\mathcal{O}}^\ensuremath{\prime}
\end{equation}{}
The static features $f_\mathcal{O}$ are weighted by soft-attention coefficient $a$ using the scaled soft-attention mechanism \cite{bahdanau2014neural}, before being neurally evaluated in Eq. \ref{eqn:soft_attn}:
\begin{equation}
f'_{\mathcal{O}_{t+1}} = a * f'_{\mathcal{O}_{t+1}} * h_\mathcal{O}
\end{equation}{}
\section{Experiments} \label{sec:exp}
\subsection{Accuracy Metrics}
We use Euclidean average errors same as in \cite{hasan2018mx,alahi2016social}, which are:
\begin{itemize}
\item Average Displacement Error (ADE):
measures prediction errors along the time-steps between the predicted trajectory and the ground-truth trajectory as follows:
\begin{equation}
\frac{\sum_{i = 1}^N \sum_{j = 1}^l ||(\widetilde{X_i^j} - X_i^j)||_2 } {N * |T|} ,
\end{equation}{}
\item Final Displacement Error (FDE): measures prediction errors at the final time-step between the predicted trajectory and the ground-truth trajectory as follows:
\begin{equation}
\frac{ \sum_{i = 1}^N ||(\widetilde{X}^{T-1}_i - X^{T-1}_i)||_2 } {N} \quad .
\end{equation}{}
\end{itemize}
\subsection{Implementation Details}
During training, we set the batch size to 16, grid size to 4 and lambda parameter to 0.0005. We set the hidden size to 128 for Grid-LSTM, the number of frequency blocks to 4, frequency skip to 4 and the number of cell units to 2. Each frame gets segmented into 8 virtual zones as indicated by the grid size and set the global neighborhood size to 32. In all experiments over our models, the features extracted from the static grid and social neighborhood vector were mapped to hidden states of length fixed at 10. With Grid-LSTM we set the training epochs to 10 as we noticed that LSTM has reached convergence within that number of iterations without any major improvements in the learning curve beyond that.
\subsection{Baseline Models and Proposed Models}
\begin{itemize}
\item \textit{S-LSTM} \cite{alahi2016social}. Dedicates LSTM for every pedestrian, and pool their states before predicting future steps. Their method only combines features of pedestrians who are found occupants of common neighborhood space. The neighborhood and occupancy grid sizes are set empirically for attaining the best results over ETH and UCY datasets.
\item \textit{MX-LSTM}. A multi-cued model that encodes Vislets and Tracklets to estimate pedestrian trajectory using LSTM. It determines pedestrians relationships using fixed visual frustum that indicates their visual attention state.
\item \textit{SR-LSTM} \cite{zhang2019sr}. A LSTM-based network inspired by message-passing graph neural networks \cite{gilmer2017neural} which improves pedestrians motion representation and social neighborhoods for predicting future trajectory over fully-connected spatio-temporal graphs.
\item \textit{G-LSTM}. Single Grid-LSTM cell that encodes pedestrian walking trajectories without additional cues. It doesn't include Sections \ref{sec:gnn} and \ref{sec:sri} in its pipeline.
\item $MC$ Multi-Cue inference.
This model combines Vislets and Tracklets to predict pedestrians trajectories. It uses Vislets to incorporate pedestrian situational awareness into their motion modeling, however, it does not differentiate pedestrians influence.
\item $MCR_n$ Multi-Cue Relational inference.
Built upon the $MC$ model, it capitalizes on its understanding of situational awareness to differentiate pedestrians importance on each other for more accurate modeling of social interactions.
\item $MCR_{mp}$ Multi-Cue Relational inference. Uses message passing mechanism through virtual static neighborhood grid that directly passes importance values which reflects pedestrians social interaction.
\item $MCR_{mpc}$. In addition to the previous design components, this model encodes the visual static feature of the scene using a 2D convolutional layer to account for the contextual interaction between pedestrian and scene structure.
\end{itemize}{}
\subsection{Quantitative Analysis}
\begin{table*}[t]
\begin{center}
\begin{tabular}{ccccccc}
\hline
Metric & Model & Zara1 & Zara2 & UCY-Univ& TownCentre & AVG\\ \hline
ADE & S-LSTM \cite{alahi2016social} & 0.68 & 0.63 & 0.62& 1.96 & 0.64 \\
& MX-LSTM \cite{hasan2018mx} & 0.59 & 0.35 & 0.49 & 1.15 & 0.48 \\
& SR-LSTM\cite{zhang2019sr} & \textbf{0.41} & \textbf{0.32} & 0.51 & 1.36 & 0.65 \\
& MC & 0.46 & 0.38 & 0.47 & 0.43& 0.44 \\
& MCR\_n & 0.47 & 0.38 & 0.47 & 0.39 & 0.43\\
& MCR\_{mp} & 0.45 & 0.38 & \textbf{0.45} & \textbf{0.34}& \textbf{0.41}\\
& MCR\_{mpc} & 0.45 & 0.38 & 0.47 & 0.39 & 0.42\\\hline
Metric& Model & Zara01 & Zara02& UCY-Univ &TownCentre & Avg \\\hline
FDE & S-LSTM \cite{alahi2016social} &1.53 & 1.43& 1.40 &3.96 &1.45 \\
&MX-LSTM \cite{hasan2018mx}&1.31 & 0.79 & 1.12 & 2.30 & 1.38 \\
& SR-LSTM\cite{zhang2019sr}&\textbf{0.90}& \textbf{0.70} &1.10& 2.47& 1.30\\
& MC&0.98& 0.84& \textbf{0.96}& 1.06& 0.96\\
& MCR\_n&1.00& 0.82& 0.97& 0.95& 0.94\\
& MCR\_{mp}&1.00& 0.93& 1.01&\textbf{0.80}& \textbf{0.94} \\
& MCR\_{mpc} & 0.95 & 0.83 & 1.03 & 1.00 & 0.95 \\\hline
\end{tabular}
\end{center}
\caption{Euclidean Errors in trajectory prediction over Crowd \& TownCentre datasets in meters. Observation length is 8 steps (3.2 seconds) and prediction length is 12 steps (4.2 seconds). }
\label{tab:quant_table}
\end{table*}{}
Table \ref{tab:quant_table} summarizes our experimental evaluation over Zara and UCY datasets.
Our models achieve state-of-the-art results in pedestrian trajectory prediction task. Taking the closest model, $MC$, it resembles the MX-LSTM design in terms of using mixture of pedestrian features, yet we are able to improve upon MX-LSTM \cite{hasan2018mx} by a margin of 8\% and 12\% in $MC$ and 10\% and 13\% in $MCR_{mp}$ over the average errors in ADE and FDE, respectively.
In $MCR_{mp}$, we were able to achieve comparable results with SR-LSTM \cite{zhang2019sr}. They rely on sharing states between pedestrians using a message-passing mechanism that carries out the social information between pedestrian nodes in a global manner, meaning that they assume a fully-connected graph to refine the social interaction modeling through graphs. In Table \ref{tab:quant_table}, by comparing SR-LSTM to our message-passing model $MCR_{mp}$, we observe that both achieve approximate prediction errors, yet SR-LSTM induce higher message passing complexity due to using fully-connected graph and performing message passes pedestrian-wise to refine the social features representation.
On the other hand, $MCR_{mp}$ encodes pedestrians awareness states into their social interaction using multiplication as previously discussed in Section \ref{sec:sri} without the need for iterative message-passing throughout the nodes. $MCR_{mp}$ model has better performance over the SR-LSTM method in Univ subset. The density of social interactions is the highest in Univ scenario with less impact from the static context on pedestrian paths. This is where the addition of social cues shows improvement in a highly interactive crowd, in comparison with the less interactive crowd as in Zara sets. The crowd in Zara is significantly influenced by the shop facade and therefore encoding the physical context features was more beneficial under these two subsets as we observe from $MCR_{mpc}$ performance.
In general, the crowd in TownCentre induce a rather linear walking pattern with less socialization and less attention paid to the contextual remarks. Best results were obtained through $MCR_{mp}$, as it focuses on relational inference given the tracklets and vislets cues. This proves that our relational inference mechanism succeeded in inferring interactions even in the less interactive situation along with the inference performance in UCY-Univ where the crowd behaves like transient groups where the walks are interrupted by conversations.
\subsection{Ablation Studies}
We performed additional ablative experiments to quantitatively illustrate the importance of main grid components and attention mechanisms, by retraining the model each time without specific layer(s). We tested the impact of two components: the Grid-LSTM handling static grid shown earlier in Figure \ref{fig:gnn} and the soft-attention mechanism explained through Eq. \ref{eqn:soft_attn}. We followed the same k-fold cross-validation used for training the original MCR model on the same datasets: Zara and UCY.
Table \ref{tab:glstm_abl} clearly illustrates the improvement of our basic model, G-LSTM, upon the NoFrustum model \cite{hasan2018mx}. Our basic model only deploys one Grid-LSTM cell and relies on pedestrians positional trajectories. This design complexity resembles the NoFrustum model which is a simpler variant of MX-LSTM work. They only use pedestrians positions with LSTM cell. The usage of Grid-LSTM attains outperforming results over Zara and UCY in a row.
We picked the NoFrustum variant as a baseline to compare with our basic model design G-LSTM, which excludes the GNN and SRI units from its pipeline and only take positional trajectories into single Grid-LSTM cell.
We tested neighborhood size on two values: 32 and 64 in the grid $G$. In our implementation, 64 local neighborhood creates a virtual neighborhood that covers a smaller area around pedestrian while 32 indicates that neighborhoods become coarser in size. Through experiments, we observed that the smaller the neighborhood, the higher the prediction error becomes. The errors increased by up to 15\% and 20\% in UCY-Univ over ADE and FDE respectively. This elevation has a special cause that is related to how our model benefits from encoding more pedestrian states together and understand that their cues are relevant to each other. However, in TownCentre, the pedestrians are far less communicative with each other, therefore, we noticed that ADE and FDE errors decreased by 10\% and 25\%, respectively.
\begin{table}[t]
\begin{center}
\begin{tabular}{ccccccc}
\hline
Metric & Model & Zara1 & Zara2 & UCY-Univ & TownCentre & AVG\\ \hline
ADE & NoFrustum \cite{hasan2018mx} & 0.63 & \textbf{0.36} & 0.51 & 1.70 & 0.50\\
& SR-LSTM$_{\{ID:2\}}$ & 0.47 & 0.38 & 0.56 & 1.36 & 0.77\\
& G-LSTM &\textbf{0.46} & 0.38 & \textbf{0.48} & \textbf{0.38} & \textbf{0.43}\\ \hline
FDE & NoFrustum \cite{hasan2018mx} & 1.40& 0.84 &1.15 & 3.40& 1.13 \\
& SR-LSTM$_{\{ID:2\}}$ & 1.07 & 0.85 & 1.27& 2.47& 1.42 \\
& G-LSTM & \textbf{0.95}& \textbf{0.83} & \textbf{1.01}& \textbf{0.89} & \textbf{0.92} \\\hline
\end{tabular}
\end{center}{}
\caption{Comparison of trajectory prediction errors between MX-LSTM NoFrustum model, SR-LSTM$_{\{ID:2\}}$ and our basic Grid-LSTM model, G-LSTM. Errors are displayed in meters. SR-LSTM$_{\{ID:2\}}$ is a model configuration which excludes social attention and selective message-passing routes between pedestrians.}
\label{tab:glstm_abl}
\end{table}{}
\paragraph{Component Impact Analysis}
Before encoding the static context features in $MCR_{mpc}$, the virtual grid $G$ was considered an additional encoding layer that only transforms the multiple features together concerning the social interaction without adding a reference to the contextual interaction. Hence, it indicated low effectiveness and appeared as unnecessary to use the static grid to achieve noticeable improvements. Accounting for the semantic features in $G$ illustrates the reduction in FDE across all datasets and this proves the positive impact of this feature on predicting pedestrian trajectory at its final point being aware of their static surrounding settings. The addition of this component only increases the running time by 0.30 seconds compared to our socially-focused models.
\subsection{Qualitative Analysis}
Figure \ref{zara_htmap} is a visualization of attention weights generated in $MCR_{mp}$. Attention weights, in this case, are distributed over the active part of the scene which includes the navigable space and social interaction between pedestrians. The dark squares include the least active areas where an obstacle is blocking motion.
Moreover, the heatmap of adjacency matrix is a depiction of each neighborhood. These are assigned weighted values to show different impact on the crowd motion. This is manifested through the zones that highlight higher impact through the lighter shades of color, such as Zara shop entrance and the walkway at the right side. The lighter shade means that throughout the observed frames, these two regions were busy and have specific influence that causes change in pedestrians direction or dynamics.
Figure \ref{zara_7815_maxadjmat} presents failure case of estimating an active neighborhood under $MCR_n$. However, $MCR_{mpc}$ relates attention to physical space features. Figure \ref{zara_8015_maxadjmat}, illustrates that the maximum attention is given to the area through which pedestrian future path is passing. This illustration proves that our model is able to effectively recognize future neighborhoods for pedestrians as these neighborhoods presents target points that were regularly visited in the scene.
\begin{figure}
\subfloat[][Attention weights from $MCR_{mp}$ over the virtual grid in Zara1 scene]{\label{zara_htmap}
\includegraphics[width=0.3\linewidth, height=0.3\linewidth]{images/zara1_6017_htmap_1.png}
}
\subfloat[][Maximum attention weight assigned to the neighborhood highlighted by the white patch, in Frame 8015 of Zara2 dataset.]{\label{zara_8015_maxadjmat}
\includegraphics[width=0.3\linewidth, height=0.3\linewidth]{images/Zara2_8015_maxwght_adj_bg_1.png}}
\subfloat[][Failure case of assigning maximum importance to neighborhoods in the scene]{\label{zara_7815_maxadjmat}
\includegraphics[width=0.3\linewidth, height=0.3\linewidth]{images/zara2_7815_max_adjmat_bg_1.png}}
\caption{Snapshots taken from our models to visualize the perception of neighborhoods relevant importance to pedestrians.}
\label{fig:qualit_mcr}
\end{figure}{}
\section{Conclusion}
In this work, we introduced a novel perspective on modeling social and contextual interactions as virtual neighborhoods with an estimation of its future relative importance values as a Gated Neighborhood Network (GNN). We developed the Social Relational Inference (SRI), a data-driven inference mechanism, to model the social interaction on graphs with no reliance on any metric assumption. Our approach outperformed state-of-the-art approaches that conduct pedestrian trajectory prediction with the help of several hand-engineered settings to model the crowd motion.
In our future work, we plan to further improve the alignment between GNN and SRI components, so as to generate more realistic perception of the neighborhoods importance. Additionally, the current approach takes 3 to 4 seconds to run predictions over moderately crowded scene consisting of around 12 pedestrians. We will investigate methods to improve the training process in deep recurrent models such that it can eliminate network component complexity.
\bibliographystyle{splncs}
|
1,116,691,500,460 | arxiv | \section{Introduction}
Let $G$ be a connected reductive algebraic group defined over a
field $k$ of characteristic not $2$, $\theta$ a $k$-involution of $G$, i.e. a
$k$-automorphism of $G$ of order two,
$H$ a $k$-open subgroup of the fixed point group
of $\theta$ and $G_k$ (resp. $H_k$) the set of $k$-rational points
of $G$ (resp. $H$). The variety $G/H$ is called a symmetric variety and $G_k/H_k$ is called a symmetric
$k$-variety.
For $k$ the real numbers or the $p$-adic numbers, the
symmetric $k$-varieties are also called reductive (real or $p$-adic) symmetric spaces or simply (real or $p$-adic)
symmetric spaces.
Symmetric spaces play an important role in many areas of mathematics and physics, but probably best known are the representations associated with these symmetric spaces which have been studied by many prominent mathematicians starting with a study of compact groups and their representations by Cartan \cite{Cartan29}, to a study of Riemannian symmetric spaces and real (and $p$-adic) groups by Harish Chandra \cite{Harish84} to a more recent study of the non Riemannian symmetric spaces (see for example \cite{Faraut79,Flensted-Jensen80,Oshima-Sekiguchi80,Ban-Schlichtkrull97,Ban-Schlichtkrull97b}) leading to a Plancherel formula in 1996 by Delorme \cite{Delorme96}. Once this Plancherel formula was obtained the attention shifted to $p$-adic symmetric spaces (see for example \cite{Delorme-Pascale14,Carmona-Delorme14,Delorme13,Helm-Helm02b,Helm-Helm05}).
In the late 1980's generalizations of these reductive symmetric spaces to other base fields started to play a role in other areas, like in the study of arithmetic subgroups (see \cite{Tong-Wang}), the study of character sheaves (see for example
\cite{Lusztig90a,Grojnowski92}), geometry (see
\cite{Procesi-Concini83,Procesi-Concini85} and
\cite{abeasis}), singularity theory (see \cite{Lusztig-Vogan83} and
\cite{Hirzebruch-Slodowy90}), and the study of Harish Chandra modules (see
\cite{Beilinson-Bernstein81} and \cite{Vogan83,Vogan82}). This prompted Helminck and Wang to commence a study of rationality properties of symmetric $k$-varieties over general base fields, see \cite{Helm-Wang93} for some first results.
To study these symmetric $k$-varieties for any field $k$, one needs a classification. The isomorphy of these symmetric $k$-varieties can be reduced to the $\Inn (G,G_k)$-isomorphy of the related $k$-involutions (see \cite{Helm-Wang93}). Here $\Inn (G,G_k)$ is the set of inner automorphisms of $G$ that leave $G_k$ invariant.
A characterization of these isomorphism
classes of the $k$-involutions was given in \cite{Helm2000}
essentially using the following 3 invariants:
\begin{enumerate}
\item classification of admissible $(\Gamma, \theta)$-indices.
\item classification of the $G_k$-isomorphism classes of
$k$-involutions of
the $k$-an\-iso\-tro\-pic kernel of $G$.
\item classification of the $G_k$-isomorphism classes of $k$-inner
elements of $G$.
\end{enumerate}
For more details, see \cite{Helm2000}. The admissible $(\Gamma,
\theta)$-indices determine most of the fine structure of the
symmetric $k$-varieties and a classification of these was included
in \cite{Helm2000} as well. For $k$ algebraically closed or $k$ the
real numbers the full classification can be found in \cite{Helm88}.
For other fields a full classification of the remaining two invariants is
still lacking. In particular the case of symmetric $k$-varieties
over the $p$-adic numbers is of interest. We note that
the above characterization was only proven for $k$ a perfect field.
In \cite{HWD04,Helm-Wu2002} a full characterization of the $\Inn (G,G_k)$-isomorphism
classes of $k$-involutions was given in the case that $G=\Sl(n, k)$ which
does not depend on any of the results in
\cite{Helm2000}. It was also shown how one may construct an outer-$k$-involution from a given non-degenerate symmetric or skew-symmetric bilinear form $\beta$ of $k^n$. Using this characterization the
possible isomorphism classes for $k$ algebraically
closed, the real numbers, the $p$-adic numbers and finite
fields were classified.
We note that much work has been done is characterizing automorphisms of $\oo(n,k,\beta)$, notably the works of Diedonne in \cite{Die51} and \cite{Die63}, Rickart in \cite{Ric50} and \cite{Ric51}, and Wonenburger in \cite{Won62}. These results consider automorphisms of $G_k$ under the action of the full automorphism group of $G_k$, so these results do not directly apply to the problem of isomorphy of $k$-involutions under the action of the subgroup $\Inn (G,G_k)$ of $\aut (G_k)$ as considered in this paper.
In this paper we study $k$-involutions of $\So(n, k, \beta)$, the special orthogonal group with respect to a symmetric bilinear form $\beta$ on $k^n$. We give a characterization
of the isomorphy classes of $k$-involutions of $\So(n, k, \beta)$, which come from inner automorphisms $\Inn_A$ of the general linear group $\Gl (n,\overline{k})$. In section 2, we state many of the important definitions and initial observations.
In section 3 we show that if a $k$-automorphism $\theta= \Inn_A$ where $A \in \Gl (n,\overline{k})$ leaves $\So(n,k,\beta)$ invariant, then
we can assume $A$ is in $\So(n,k[\sqrt{\alpha}],\beta)$ when $n$ is odd, and $A$ is in $\oo(n,k[\sqrt{\alpha}],\beta)$ when $n$ is even, where $k[\sqrt{\alpha}]$ is a quadratic extension of $k$. Further, we show that each entry of $A$ must be a $k$-multiple of $\sqrt{\alpha}$. To prove these results, we require that either $\chr(k) \ne 3$, or we require a restriction on the bilinear form $\beta$. In section 4, we have the main results of the paper. We determine which $A \in \So(n,k[\sqrt{\alpha}],\beta)$ if $n$ is odd and which $A \in \oo(n,k[\sqrt{\alpha}],\beta)$ if $n$ is even induce $k$-involutions of $\So(n,k,\beta)$ of the form $\Inn_A$. We will see that when $n$ is odd, there is only one type of these $k$-involutions, and if $n$ is even, there are four types of these $k$-involutions. For each type, we determine nice conditions that are equivalent to isomorphy of these $k$-involutions over $\So(n,k,\beta)$ if $n$ is odd and over $\oo(n,k,\beta)$ if $n$ is even. When $n \ne 3, 4, 6, 8$, the $k$-involutions of the form $\Inn_A$ will be all of the $k$-involutions of $\So(n,k,\beta)$. In section 5, we discuss the maximal number of possible isomorphy classes of $k$-involutions of the form $\Inn_A$. In section 6 we look at some explicit examples of orthogonal groups, most of which are the standard orthogonal group, when $k$ is algebraically closed, the real numbers, a finite field of order odd $q= p^m$ where $p>2$, or the $p$-adic numbers
\section{Preliminaries}
Our basic reference for
reductive groups will be the papers of Borel and Tits
\cite{Borel-Tits65}, \cite{Borel-Tits72} and also the books of
Borel \cite{Borel91}, Humphreys \cite{Humph75} and Springer
\cite{Spring81}. We shall follow their notations and terminology.
All algebraic groups and algebraic varieties are taken over an
arbitrary field $k$ (of characteristic $\neq 2$) and all algebraic
groups considered are linear algebraic groups.
Our main reference for results regarding $k$-involutions of $\Sl(n,k)$ will be \cite{HWD04}. Let
$k$ be a field of characteristic not $2$, $\bar k$ the algebraic closure of $k$,
$$\M(n,k)=\{ n\times n \text{-matrices with
entries in $k$} \}, $$
$$\Gl(n,k)= \{ A\in \M(n,k)\mid \det
(A)\neq 0\}$$ and
$$\Sl(n,k)= \{ A\in \M(n,k)\mid \det
(A)=1\}. $$ Let $k^*$ denote the multiplicative group of all the nonzero
elements, $(k^*)^2=\{a^2\mid a\in k^*\}$ and $I_n \in \M(n,k)$
denote the identity matrix. We will sometimes use $I$ instead of $I_n$ when the dimension of the identity matrix is clear.
We recall some important definitions and theorems from \cite{HWD04}.
\begin{definit}
\label{isoinv}
Let $G$ be an algebraic groups defined over a field $k$. Let $G_k$ be the $k$-rational points of $G$. Let $\aut(G, G_k)$ denote the the set of $k$-automorphisms of $G_k$. That is, $\aut(G, G_k)$ is the set of automorphisms of $G$ which fix $G_k$. We say $\theta \in \aut(G,G_k)$ is a {\it $k$-involution} if $\theta^2 = \id$ but $\theta \ne \id$. That is, a $k$-involution is a $k$-automorphism of order 2.
Choose $A \in G_k$. Then the map $\Inn_A(X) = A^{-1}XA$ is called an {\it inner $k$-automorphism of $G_k$}. We denote the set of such $k$-automorphisms by $\Inn(G_k)$. If $\Inn_A \in \Inn(G_k)$ is a $k$-involution, then we say that $\Inn_A$ is an {\it inner $k$-involution of $G_k$}.
Assume $H$ is an algebraic group defined over $k$ which contains $G$. Let $H_k$ be the $k$-rational points of $H$. Choose $A \in H$. If the map $\Inn_A(X) = A^{-1}XA$ is such that $\Inn_A \in \aut(G,G_k)$, then we say that $\Inn_A$ is an {\it inner $k$-automorphism of $G_k$ over $H$}. We denote the set of such $k$-automorphisms by $\Inn(H,G_k)$. If $\Inn_A \in \Inn(H,G_k)$ is a $k$-involution, then we say that $\Inn_A$ is an {\it inner $k$-involution of $G_k$ over $H$}.
Suppose $\theta, \tau \in \aut((G,G_K)$.We say that $\theta$ is {\it isomorphic} to $\tau$ {\it over $H_k$} if there is $\phi$ in ${\Inn(H_k)}$ such that $\tau=\phi ^{-1}\theta\phi$. Equivalently, we say that $\tau$ and $\theta$ are in the same \textit{isomorphy class over $H_k$}.
\end{definit}
In \cite{HWD04}, the isomorphy classes of the inner-$k$-involutions of $\Sl(n,k)$ were classified, and they are as follows:
\begin{theorem} \label{sltheorem1}
Suppose the $k$-involution $\theta \in \aut(\Sl(n,k))$ is of inner type. Then
up to isomorphism over $\Gl(n,k)$, $\theta$ is one of the following:
\begin{enumerate}
\item \label{sltheorem1.1} $\Inn_Y|_G$, where $Y = I_{n-i,i} \in
\Gl(n,k)$ where $i \in \left\{1, 2, \dots, \lfloor \frac
{n}{2}\rfloor \right\}$ where $$I_{n-i,i} = \left(\begin{array}{cc}I_{n-i} & 0 \\0 & -I_i\end{array}\right)$$.
\item \label{sltheorem1.2} $\Inn_Y|_G$,
where
$Y = L_{\frac n 2, x} \in \Gl(n,k)$ where $x$ is a fixed element of a coset of $k^*\slash
k^{*2}$, $x \not\equiv 1\mod k^{*2}$ and
$$L_{\frac{n}{2},x}=\begin{pmatrix} 0 & 1 & \hdots & 0 & 0 \\
x & 0 & \hdots & 0 & 0
\\ \vdots & \vdots & \ddots &
\vdots & \vdots \\ 0 & 0 & \hdots & 0 & 1\\ 0 & 0 & \hdots & x & 0
\end{pmatrix} .$$
\end{enumerate}
Note that $(ii)$ can only occur when $n$ is even.
\end{theorem}
For the purposes of this paper, we will use matrices of the form
$\left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\xI_{\frac{n}{2}} & 0\end{smallmatrix}\right)$
(and their multiples) rather than $L_{\frac n 2, x}$. Either of these serves as a member of the isomorphy class listed in the previous theorem. It will become apparent that the isomorphy classes of $k$-involutions $\So(n,k,\beta)$ over $\oo(n,k,\beta)$ are just isomorphy classes of $\Sl(n,k)$ over $\Gl(n,k)$ that have been divided into multiple isomorphy classes.
We now begin to define the notion of a special orthogonal group. Let $M$ be the matrix of a non-degenerate bilinear form $\beta$
over $k^n$
with respect to a basis $\{ e_1, \dots e_n \}$ of $k^n$.
We will say that $M$ is the matrix of $\beta$ if
the basis $\{ e_1, \dots e_n \}$
is the standard basis of $k^n$.
The typical notation for the orthogonal group is $\oo(n,k)$, which is the group $$\oo(n,k)= \{ A\in \M(n,k)\mid (Ax)^T(Ay) = x^Ty\}.$$ This group consists of the matrices which fix the standard dot product. This can be generalized to any non-degenerate symmetric bilinear $\beta$, which will yield the group $$\oo(n,k,\beta)= \{ A\in \M(n,k)\mid \beta(Ax,Ay) = \beta(x,y) \}.$$ If $M$ is the matrix of $\beta$ with respect to the standard basis, then we can equivalently say $$\oo(n,k,\beta)= \{ A\in \M(n,k)\mid A^TMA = M \}.$$ It is clear from this definition that all matrices in $\oo(n,k,\beta)$ have determinant 1 or -1. We define the {\it special orthogonal group of $\beta$} to be the group $$\So(n,k, \beta)= \oo(n,k, \beta) \cap \Sl(n,k),$$ and we define the {\it group of similitudes of $\beta$} to be the group $$\Go(n,k,\beta) = \{ A\in \M(n,k)\mid \beta(Ax,Ay) = \alpha \beta(x,y), \alpha \in k^* \}.$$ We note that $$\Go(n,k,\beta)= \{ A\in \M(n,k)\mid A^TMA = \alpha M, \alpha \in k^* \}.$$
We say two $n \times n$ matrices $A$ and $B$ are considered {\it congruent} over $k$ if there exists $Q \in \Gl(n,k)$ such that $Q^TAQ = B$. We also say that $A$ and $B$ are {\it congruent} via $Q$.
We note a couple of important facts, the first of which will be used repeatedly throughout this paper.
\begin{enumerate}
\item Symmetric matrices are congruent to diagonal matrices when $\chr(k) \ne 2$, where the entries of the diagonal matrix are are representatives of the cosets of $k^*/(k^*)^2.$
\item If $\beta_1$ and $\beta_2$ correspond to $M_1$ and $M_2$, then $\So(n,k,\beta_1)$ and $\So(n,k,\beta_2)$ are isomorphic via
$$\Phi:\So(n,k,\beta_1) \rightarrow \So(n,k,\beta_2): X \rightarrow Q^{-1}XQ$$ for some $Q \in \Gl(n,k)$ if and only if $Q^TM_1Q = M_2$ ($M_1$ and $M_2$ are congruent via $Q$).
\end{enumerate}
So, we will assume that $\beta$ is such that $M$ is diagonal. Then, to classify the $k$-involutions of an orthogonal group where $M$ is not diagonal, one can apply the characterization that will follow by simply using the isomorphism $\Phi$ given above.
We say two vectors $x,y\in k^n$ are said to be \textit{orthogonal} with respect to the bilinear form $\beta$ if $\beta(x,y)=0$. We will eventually see that orthogonal vectors play an important role in the structure of $k$-involutions of $\So(n,k,\beta)$.
Lastly, we will always assume, whether stated or not, that $n > 2$ and that $\chr(k) \ne 2$.
\section{$k$-automorphisms of $\So(n,k,\beta)$}
In this paper, we consider the $k$-involutions that lie in the group $\Inn(\Gl(n,\overline{k}),\So(n,k,\beta))$. When $n$ is odd, this group turns out to be $\Inn(\So(n,\overline{k},\beta),\So(n,k,\beta))$, and when $n$ is even, this group turns out to be $\Inn(\oo(n,\overline{k},\beta),\So(n,k,\beta))$. We will see that when $n$ is odd and $n \ne 3$, that all of the $k$-involutions of $\So(n,k,\beta)$ will be inner $k$-involutions of $\So(n,k,\beta)$ over $\So(n, \overline{k}, \beta)$, and when $n$ is even and $n \ne 4, 6, 8,$, that all of the $k$-involutions of $\So(n,k,\beta)$ will be inner $k$-involutions of $\So(n,k,\beta)$ over $\oo(n, \overline{k}, \beta)$. We will show isomorphy conditions of these $k$-involutions over $\So(n,k,\beta)$ when $n$ is odd, and over $\oo(n,k,\beta)$ when $n$ is even.
It follows from a proposition on page 191 of \cite{Borel91} that the outer $k$-automorphism group $$\out(\So(n,\overline{k},\beta)) = \aut(\So(n,\overline{k},\beta)) /\Inn(\So(n,\overline{k},\beta))$$ must be a subgroup of the diagram automorphisms of the associated Dynkin diagram. If $n = 2m+1$ and $m \ge 2$, then this Dynkin diagram is $B_m$ which has only the trivial diagram automorphism. Thus, there are no outer $k$-automorphisms of $\So(n,\overline{k},\beta)$ when $n$ is odd and $n \ge 5$. If $n = 2m$ and $m \ge 4$, then this Dynkin diagram is $D_m$. The group of automorphisms of this Dynkin diagram is $\mathbb{Z}_2$ when $m > 4$. So, when $n$ is even and $n \ge 10$, $\out(\So(n,\overline{k},\beta)) = \mathbb{Z}_2$. We will see that the outer $k$-automorphisms are of the form $\Inn_A$ where $A \in \oo(n,\overline{k},\beta)$ and $\det(A) = -1$. When $k$ is not algebraically closed, then all $k$-automorphisms of $\So(n,k,\beta)$ will still be of the form $\Inn_A$ for some $A \in \oo(n,\overline{k},\beta)$ since $\Inn_A$ must also be an $k$-automorphism of $\So(n,\overline{k},\beta).$ Thus, the characterizations that follow in this paper consider all $k$-automorphisms and $k$-involutions of $\So(n,k,\beta)$, assuming that $n \ne 3, 4, 6, 8$. In the cases where $n = 3, 4, 6, 8$, the results that follow only consider the $k$-automorphisms and $k$-involutions that can be written as $\Inn_A$ for some $A \in \Gl(n,\overline{k})$.
We now examine which $k$-automorphisms will act as the identity on $\So(n,k, \beta)$. This will prove to be useful when we classify matrix representatives for $k$-automorphisms.
\begin{lem}
Assume $n > 2$. Let $A \in \Gl(n,\overline{k})$. If $\Inn_A$ is the identity on $\So(n,k, \beta)$, then $A $ is a diagonal matrix.
\end{lem}
\begin{proof}
Suppose $A$ is such that $\Inn_A$ is the identity on $\So(n,k, \beta)$. For $1 \le r < s \le n$, let $X_{rs}$ be the diagonal matrix with all 1's, except in the $r$th and $s$th diagonal entries, where instead there are -1's. This matrix always lies in $\So(n,k, \beta)$. So, we must have $A X_{rs} = X_{rs}A.$ On the left side, the matrix is the same as $A$, but with the $r$th and $s$th columns negated. On the right side, the matrix is the same as $A$, but with the $r$th and $s$th rows negated. So, all entries of $A$ on these rows and columns which aren't in the $(r,r)$, $(r,s)$, $(s,r)$ or $(s,s)$ components must be equal to 0, since this is the only number which equals its negative. To see that the $(r,s)$ and $(s,r)$ components of $A$ must also equal 0, we can repeat this process for $X_{rt}$, where $t$ is distinct from both $r$ and $s$. (Note that this is where we use the fact that $n > 2$.) Thus, all off-diagonal elements of $A$ are 0, which means $A$ is diagonal.
\end{proof}
We want to be able to say more about the matrix $A$ when $\Inn_A$ acts as the identity. It turns out that if we make the following assumption on the orthogonal group $\So(n,k,\beta)$, then we can show that $A$ is a multiple of the identity.
\begin{definit}
Let $k$ be a field and suppose $\beta$ is a bilinear form on $k^n$ such that it has matrix representation $M$, where $M$ is diagonal with diagonal entries $m_1,...,m_n$, which are representatives in $k^*$ of cosets of $k^*/(k^*)^2$. If for each pair $m_s$ and $m_t$, $x^2+\frac{m_s}{m_t}y^2 = 1$ has a solution $(x,y)$ such that $y \ne 0$, then we call $\So(n,k,\beta)$ a {\it friendly orthogonal group}.
\end{definit}
With this new terminology in mind, we get the following result.
\begin{lem}
\label{IdentityLem}
Assume $n > 2$. Suppose $\So(n,k, \beta)$ is a friendly orthogonal group. Let $A \in \Gl(n,\overline{k})$. Then, $\Inn_A$ is the identity on $\So(n,k, \beta)$ if and only if $A = \alpha I$ for some $\alpha \in \overline{k}^*$.
\end{lem}
\begin{proof}
We know from the previous lemma that $A$ is diagonal. Let $a_i$ represent the $i$th diagonal entry of $A$. Recall that we are assuming that $M$ is diagonal. Let $m_i$ represent the $i$th diagonal entry of $M$. Then, there exists $a, b \in k$ where $b \ne 0$ such that $a^2+\frac{m_s}{m_t}b^2 = 1.$ For $1 \le i < j \le n$, let $$Y_{st} = \left(\begin{array}{cccccccccccc}1 & 0 & \cdots & & & & & & & & \cdots & 0 \\0 & 1 & & & & & & & & & & \vdots \\\vdots & & \ddots & & & & & & & & & \\ & & & 1 & & & & & & & & \\ & & & & a & 0 & \cdots & 0 & b & & & \\ & & & & 0 & 1 & & & 0 & & & \\ & & & & \vdots & & \ddots & & \vdots & & & \\ & & & & 0 & & & 1 & 0 & & & \\ & & & & -\frac{m_s}{m_t}b & 0 & \cdots & 0 & a & & & \\ & & & & & & & & & 1 & & \vdots \\\vdots & & & & & & & & & & \ddots & 0 \\0 & \cdots & & & & & & & & \cdots & 0 & 1\end{array}\right),$$ where the noteworthy entries occur in the $s$th and $t$th rows and columns. It is a simple calculation to show that $Y_{st}^TMY_{st} = M$, and that $\det(Y_{st}) =1$. So, $Y_{st} \in \So(n,k, \beta)$. Then, we know that $AY_{st} = Y_{st}A.$ By comparing both sides of this equality and inspecting the $(s,t)$ entry, we see that $ba_t = ba_s$. Since we are assuming that $b \ne 0$, then it follows that $a_t = a_s$. Since we can repeat this for all $s$ and $t$, then it is clear that $A$ is a multiple of the identity.
\end{proof}
This result is only useful if we can show that $\So(n,k,\beta)$ is commonly a friendly orthogonal group. In the following theorem, we see that most $\So(n,k,\beta)$ are friendly.
\begin{lem}
\begin{enumerate}
\item If $\chr(k) \ne 2, 3$, then $\So(n,k,\beta)$ is a friendly orthogonal group.
\item If $M = \left(\begin{array}{cccc}m_1 & & & \\ & m_2 & & \\ & & \ddots & \\ & & & m_n\end{array}\right)$ is such that $m_s \ne -m_t$ whenever $s \ne t$ and $\chr(k) \ne 2$, then $\So(n,k,\beta)$ is a friendly orthogonal group.
\end{enumerate}
\end{lem}
\begin{proof}
When $\chr(k) \ne 2$, then we see that $1 = x^2+\alpha y^2$ has solution $(x,y) = \left(\frac{\alpha-1}{\alpha+1}, \frac{2}{\alpha+1}\right)$ when $\alpha \ne -1$. When $\chr(k) \ne 3$, then we see that $1=x^2-y^2$ has solution $(x,y) = \left( \frac{5}{3}, \frac{4}{3} \right)$. Based on these two solutions, it is clear that $x^2+\frac{m_s}{m_t}y^2 = 1$ will always have a solution in $k$ if
$\chr(k) \ne 2, 3$, and also when $\frac{m_s}{m_t} \ne -1$ and $\chr(k) \ne 2$.
\end{proof}
To show that this condition on orthogonal groups is not trivial, we note a case where $\So(n,k, \beta)$ is not a friendly orthogonal group.
\begin{beisp}
Suppose $k = \mathbb{F}_3$ and that $\beta$ is such that $M = \left(\begin{array}{cc}I & 0 \\0 & -1\end{array}\right).$ Then, $\So(n,\mathbb{F}_3, \beta)$ is not a friendly orthogonal group, because there is no solution to $x^2-y^2=1$ where $y \ne 0.$
\end{beisp}
For the remainder of the paper, we will assume that all orthogonal groups are friendly.
Fix a bilinear form $\beta$ with matrix $M$. If $A \in \Gl(n,k)$ is a matrix such $M^{-1}A^TMA = \alpha I_{n}$, then $A \in \Go(n,k,\beta) = \{ A\in \M(n,k)\mid \beta(Ax,Ay) = \alpha \beta(x,y), \alpha \in k \}$, the group of similitudes of $\beta$.
We now have the following preliminary result that characterizes $k$-automorphisms of $\So(n,k, \beta)$.
\begin{lem}
Assume $n >2$. If $A \in \Gl(n,\overline{k})$, then $\Inn_A(\So(n,k,\beta)) \subseteq\So(n,k, \beta)$ if and only if $A \in \Go(n,\overline{k},\beta)$, $A = p \widetilde{A}$ where $p \in \overline{k}$, and
\begin{enumerate}
\item $\widetilde{A} \in \So(n, \overline{k}, \beta)$ if $n$ odd, or
\item $\widetilde{A} \in \oo(n, \overline{k}, \beta)$ if $n$ is even.
\end{enumerate}
\end{lem}
\begin{proof}
Suppose $A \in \Gl(n,\overline{k})$ and $\Inn_A(\So(n,k, \beta)) \subseteq \So(n,k, \beta)$. Choose $X \in \So(n,k, \beta).$ Then, $A^{-1}XA \in \So(n,k, \beta).$ So,
\begin{align*}
I &= (A^{-1}XA )^{-1}A^{-1}XA \\ & = M^{-1}(A^{-1}XA )^TMA^{-1}XA \\ & = M^{-1}A^{T}X^T(A^{-1})^T MA^{-1}XA.
\end{align*}
This implies that $$A^{-1}X^{-1} = M^{-1}A^{T}X^T(A^{-1})^T MA^{-1},$$ which means $$X^{-1} = AM^{-1}A^{T}X^T(A^{-1})^T MA^{-1}.$$ We can rewrite this as $$M^{-1}X^TM = AM^{-1}A^{T}X^T(A^{-1})^T MA^{-1}.$$ If we transpose both sides, then we see that $$MXM^{-1} = (A^{-1})^TMA^{-1}XAM^{-1}A^T.$$ Solving for the $X$ term on the left, we get that
\begin{align*}
X &= M^{-1}(A^{-1})^TMA^{-1}XAM^{-1}A^TM \\ &= (AM^{-1}A^TM)^{-1}X(AM^{-1}A^TM).\end{align*}
By Lemma \ref{IdentityLem}, it follows that $(AM^{-1}A^TM) = \alpha I$ for some $\alpha \in \overline{k}^*.$ Thus, $A \in \Go(n,\overline{k},\beta)$. Choose $p \in \overline{k}$ such that $p^2 = \alpha$. Then let $ \widetilde{A} = \frac{1}{p}A.$ It follows that $M^{-1}\widetilde{A}^TM \widetilde{A} = \frac{1}{p^2}M^{-1}A^TMA = \frac{1}{\alpha}(\alpha I) = I, $ which shows that $\widetilde{A}$ is orthogonal. That is, $\widetilde{A} \in \oo(n, \overline{k}, \beta)$. If $n$ is odd and $\det(\widetilde{A}) = -1$, then we can replace $\widetilde{A}$ with $-\widetilde{A}$, and instead have a matrix inside $\So(n, \overline{k}, \beta)$.
Since the converse is clear, we have proven the statement.
\end{proof}
In the following theorem which completes the characterization of $k$-automorphisms on $\So(n,k, \beta)$, we see that we can choose the entries of $A$ to be in $k$ or a quadratic extension of $k$. For the remainder of this paper, we will use $\sqrt{\alpha}$ to denote a fixed square root of $\alpha \in k^*$.
\begin{theorem}
\label{CharThm2So}
Assume $n >2$.
\begin{enumerate}
\item If $n$ is odd and $A$ is in $\So(n,\overline{k}, \beta)$, then $\Inn_A$ keeps $\So(n,k, \beta)$ invariant if and only if we can choose $\tilde{A} \in \So(n,k,\beta)$ such that $\Inn_{\tilde{A}} = \Inn_A$.
\item If $n$ is even and $A$ is in $\oo(n,\overline{k}, \beta)$, then $\Inn_A$ keeps $\So(n,k, \beta)$ invariant if and only if there exists $p \in \overline{k}$ and $B \in \Go(n,k,\beta)$ such that $B = pA$. Further, we can show $A \in \oo(n, k[\sqrt{\alpha}], \beta)$ where each entry of $A$ is a $k$-multiple of $\sqrt{\alpha}.$
\end{enumerate}
\end{theorem}
\begin{proof}
Let $n >2$ be arbitrary, and suppose $A$ is in $\oo(n,\overline{k},\beta)$ such that $\Inn_A$ keeps $\So(n,k, \beta)$ invariant. Let $a_{ij}$ be the $(i,j)$ entry of $A$. We break the proof into steps. At the beginning of each step, we state what we shall prove in that step.
{\bf Step One:} $a_{ri}a_{rj}+a_{si}a_{sj} \in k$ for all $i,j, r,s$.
Let $X_{rs}$ be the diagonal matrix with all entries -1, except for the $(r,r)$ and $(s,s)$ entries, which are 1. Since $M$ is diagonal, it is clear that $X_{rs} \in \oo(n,k, \beta)$. If $n$ is even, then $X_{rs} \in \So(n,k,\beta)$. If $n$ is odd, then $-X_{rs} \in \So(n,k,\beta)$. So, we know that $\Inn_A(X_{rs})$ or $-\Inn_A(X_{rs})$ must lie in $\So(n,k,\beta)$. It is also clear that $\Inn_A(I) \in \So(n,k, \beta)$. So, both $\Inn_A(X_{rs})$ and $\Inn_A(I)$ have entries in $k$. Let us examine the entries of $\Inn_A(X_{rs})$:
\begin{align*}
\Inn_A(X_{rs}) &= A^{-1}X_{rs}A = (M^{-1}A^T)(MX_{rs}A)\\
&= \left(\begin{array}{ccc}\frac{a_{11}}{m_1} & \cdots & \frac{a_{n1}}{m_1} \\\vdots & \ddots & \vdots \\\frac{a_{1n}}{m_n} & \cdots & \frac{a_{nn}}{m_n}\end{array}\right) \left(\begin{array}{ccc}-m_1a_{11} & \cdots & -m_1a_{1n} \\\vdots & & \vdots \\-m_{r-1}a_{r-1,1} & \cdots & -m_{r-1}a_{r-1,n} \\m_ra_{r1} & \cdots & m_ra_{rn} \\-m_{r+1}a_{r+1,1} & \cdots & -m_{r+1}a_{r+1,n} \\\vdots & & \vdots \\-m_{s-1}a_{s-1,1} & \cdots & -m_{s-1}a_{s-1,n} \\m_sa_{s1} & \cdots & m_sa_{sn} \\-m_{s+1}a_{s+1,1} & \cdots & -m_{s+1}a_{s+1,n} \\\vdots & & \vdots \\-m_{n}a_{n1} & \cdots & -m_{n}a_{nn}\end{array}\right)\\
&= \left( a_{ri}a_{rj}+a_{si}a_{sj}-\sum_{l \ne r,s}a_{li}a_{lj} \right)_{(i,j)}.
\end{align*}
Since $\Inn_A(X_{rs})$ and $I= \Inn_A(I)$ have entries in $k$, then so does the matrix $\Inn_A(X_{rs}+I) = I+\Inn_A(X_{rs}).$ Using a similar calculation to the above, we can see that $\Inn_A(I)+\Inn_A(X_{rs})$ has entries of the form $2a_{ri}a_{rj}+2a_{si}a_{sj}.$ It follows that $a_{ri}a_{rj}+a_{si}a_{sj} \in k$ for all $i,j, r,s$.
{\bf Step Two:} $a_{ri}a_{rj} \in k$ for all $i,j,r$.
Choose integer $t$ distinct from $r$ and $s$ such that $1 \le t \le n$. We have that $$a_{ri}a_{rj}-a_{ti}a_{tj} =(a_{ri}a_{rj}+a_{si}a_{sj})-(a_{si}a_{sj}+a_{ti}a_{tj}) \in k,$$
which means that $$a_{ri}a_{rj} = \frac{1}{2}(a_{ri}a_{rj}-a_{ti}a_{tj} )+\frac{1}{2}(a_{ri}a_{rj}+a_{ti}a_{tj} ) \in k,$$ for all $i,j,r$.
{\bf Step Three:} $a_{ir}a_{jr} \in k$ for all $i,j, r$.
Now, we consider the bilinear form $\beta_1$ which has matrix $M^{-1}$ (using the standard basis for $k^n$). We know that $X \in \So(n,\overline{k},\beta)$ if and only $X^TMX = M$. But, if that is the case for a given $X$, then it follows that $$X^{-1}M^{-1}(X^{-1})^T= M^{-1}.$$ Thus, $(X^{-1})^T \in \So(n,\overline{k},\beta_1).$ Since this is a group, then we also know that $X^T \in \So(n,\overline{k},\beta_1).$ It is then easy to see that $X \in \So(n,k,\beta)$ if and only if $X^T \in \So(n,k,\beta_1)$.
We further claim that $\Inn_{A^T}$ is a $k$-automorphism of $ \So(n,k,\beta_1)$. Suppose $Y \in \So(n,k,\beta_1)$ and consider $\Inn_{A^T}(Y) = (A^T)^{-1}YA^T.$ This matrix lies in $ \So(n,k,\beta_1)$ if and only if its inverse-transpose lies in $ \So(n,k,\beta)$. The inverse-transpose of $\Inn_{A^T}(Y)$ is $A(Y^{-1})^TA^{-1} = \Inn_{A^{-1}}((Y^{-1})^T) = (\Inn_A)^{-1}((Y^{-1})^T).$ This proves our claim.
Since $\Inn_{A^T}$ is a $k$-automorphism of $ \So(n,k,\beta_1)$, then it follows Step Two that $a_{ir}a_{jr} \in k$ for all $i,j, r$.
{\bf Step Four:} $a_{si}a_{tj}-a_{ti}a_{sj} \in k,$ for all $i,j,s,t$.
We now recall the matrices $Y_{st} \in \So(n,k,\beta)$ from the proof of Lemma \ref{IdentityLem}. So, it must be the case that $\Inn_A(Y_{st}) \in \So(n,k,\beta)$. Let us examine the entries of $\Inn_A(Y_{st})$:
\begin{align*}
\Inn_A(Y_{st}) &= A^{-1}Y_{st}A\\ &= (M^{-1}A^T)(MY_{st}A)\\
&= \left(\begin{array}{ccc}\frac{a_{11}}{m_1} & \cdots & \frac{a_{n1}}{m_1} \\\vdots & \ddots & \vdots \\\frac{a_{1n}}{m_n} & \cdots & \frac{a_{nn}}{m_n}\end{array}\right) \left(\begin{array}{ccc}m_1a_{11} & \cdots & m_1a_{1n} \\\vdots & & \vdots \\m_{s-1}a_{s-1,1} & \cdots & m_{s-1}a_{s-1,n} \\am_sa_{s1}+bm_sa_{t1} & \cdots & am_sa_{sn}+bm_sa_{tn} \\m_{s+1}a_{s+1,1} & \cdots & m_{s+1}a_{s+1,n} \\\vdots & & \vdots \\m_{t-1}a_{t-1,1} & \cdots & m_{t-1}a_{t-1,n} \\-bm_ta_{s1}+am_ta_{t1} & \cdots & -bm_ta_{sn}+am_ta_{tn} \\m_{t+1}a_{t+1,1} & \cdots & m_{t+1}a_{t+1,n} \\\vdots & & \vdots \\m_{n}a_{n1} & \cdots & m_{n}a_{nn}\end{array}\right)\\
&= \left( a_{si}(aa_{sj}+ba_{tj})+a_{ti}(-ba_{sj}+aa_{tj})+\sum_{l \ne s,t} a_{li}a_{lj} \right)_{(i,j)}.
\end{align*}
Since each of the matrix entries of $\Inn_A(Y_{ij})$ must lie in $k$ and we also know that $a_{li}a_{lj} \in k$ for $1 \le l \le n$ from Step Two, then it follows that
$$a_{si}(aa_{sj}+ba_{tj})+a_{ti}(-ba_{sj}+aa_{tj})$$
$$=a(a_{si}a_{sj})+b(a_{si}a_{tj})-b(a_{ti}a_{sj})+a(a_{ti}a_{tj}) \in k.$$
We know $b \in k$, and that $a_{si}a_{sj}$ and $a_{ti}a_{tj} \in k$ by Step Two. So, we see that $a_{si}a_{tj}-a_{ti}a_{sj} \in k,$ for all $i,j,s,t$.
{\bf Step Five:} $a_{si}a_{tj} \in k$ for all $i,j,s,t$.
{\bf Substep Five A} : $a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) = ca_{si}$ for $c \in k$.
If both $a_{si}$ and $a_{tj} \in k$, then $a_{is}a_{jt} \in k$ is clear. So, we will assume $a_{tj} \not \in k$. We will also assume that $a_{si} \ne 0$, since if $a_{si} = 0$, then it is again clear that $a_{si}a_{tj} \in k$.
From Step Four, we know that $$a_{si}a_{tj}-a_{ti}a_{sj} \in k.$$ It follows that $$a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) \in k[a_{si}].$$ Let $c = a_{si}a_{tj}-a_{ti}a_{sj}$. Then, we have $a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) = ca_{si}$ for $c \in k$.
{\bf Substep Five B} : $a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) = da_{tj}$ for $d \in k$.
We now consider $a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) = a_{si}^2a_{tj}-a_{si}a_{sj}a_{ti}$ in a different fashion. From Step Two, we know that $a_{ti}a_{tj} \in k$. Also recall that we are assuming $a_{tj} \not \in k$. Since Step Two also tells us that $a_{tj}^2 \in k$, then it follows that $a_{ti}$ and $a_{tj}$ must lie in the same coset of $k^*/ (k^*)^2$. So, there exists $p \in k$ such that $a_{ti} = pa_{tj}$. From these observations, we see that $$a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) = (a_{si}^2-a_{si}a_{sj}p)a_{tj}.$$ From Step Two, we know that $a^2_{si}, a_{si}a_{sj} \in k$. Let $d = a_{si}a_{tj}-a_{ti}a_{sj}$, and note that $d \in k$. We have just shown that $a_{si}(a_{si}a_{tj}-a_{ti}a_{sj}) = da_{tj}$ for $d \in k$.
{\bf Substep Five C:} $a_{si}a_{tj} \in k$ for all $i,j,s,t$.
Combining Substeps Five A and B, we see that $ca_{si} = da_{tj}$ for some $c, d \in k$. We prove this step by considering the two cases where $c \ne 0$ and $c = 0$.
If $c \ne 0$, then $d \ne 0$. Further, we have that $a_{si} = \frac{d}{c}a_{tj}.$ From this, we see that $a_{si}a_{tj} = \frac{d}{c}a_{tj}^2.$ Since $\frac{d}{c} \in k$ by assumption and $a_{tj}^2$ by Step Two, then we have that $a_{si}a_{tj} \in k$.
Alternatively, if $c = 0$, then since $c = a_{si}a_{tj}-a_{ti}a_{s}$, we know that $a_{si}a_{tj}=a_{ti}a_{sj}$. Recall that in Substep Five B that $a_{ti} = pa_{tj}$ for $p \in k$. Combining these facts, we have that
\begin{align*}
a_{si}a_{tj} &= a_{ti}a_{sj}\\
&= pa_{tj}a_{sj}\\
\end{align*}
By Step Three, we know that $a_{tj}a_{sj} \in k$. So, it follows that $a_{si}a_{tj} \in k$.
{\bf Step Six:} $A \in \oo(n,k[\sqrt{\alpha}],\beta)$ for some $\alpha \in k$.
From Step Five, it is clear that $k[a_{si}] = k[a_{tj}]$ for all $i,j,s,t$ (assuming that $a_{is}$ and $a_{jt}$ are both nonzero). Fix a nonzero entry $a_{si}$ of $A$. Let $\alpha = a_{is}^2$ and denote $a_{si}$ as $\sqrt{\alpha}$. We have shown that all the entries of $A$ are in $k[\sqrt{\alpha}].$ This means that $A \in \oo(n,k[\sqrt{\alpha}],\beta)$, and all of the entries of $A$ are $k$-multiples of $\sqrt{\alpha}$, as desired.
{\bf Step Seven:} If $n$ is odd, then $A \in \So(n,k,\beta)$.
If $n$ is odd, then we can replace $A$ with $-A$ to get a matrix in $\So(n,k[\sqrt{\alpha}],\beta).$ So, assume that $A \in \So(n,k[\sqrt{\alpha}],\beta).$ We now show that we do not need a quadratic extension of $k$ when $n$ is odd. Proceed by contradiction and assume $A \in \So(n,k[\sqrt{\alpha}],\beta)$ where $\sqrt{\alpha} \not \in k$. From Step Six we know that $\sqrt{\alpha}A \in \Gl(n,k)$. Then,
$$\det(\sqrt{\alpha}A ) = [\sqrt{\alpha}]^n\det(A) = \alpha^{\frac{n}{2}} \not \in k,$$
which is a contradiction. So, if $n$ is odd, then $A \in \So(n,k,\beta)$.
\end{proof}
\section{$k$-involutions of $\So(n,k, \beta)$}
We now begin to focus on $k$-involutions and their characterization, as eel as the characterization of their isomorphy classes. We will distinguish different types of $k$-involutions. First, we note that for some $k$-involutions, $\phi$, there exists $A \in \oo(n,k,\beta)$ such that $\phi = \Inn_A$, but not in all cases. Sometimes we must settle for $A \in \oo(n,k[\sqrt{\alpha}], \beta) \setminus \oo(n,k,\beta)$ where each entry of $A$ is a $k$-multiple of $\sqrt{\alpha}$.
This is not the only way in which we can distinguish between different types of $k$-involutions. If $\Inn_A$ is a $k$-involution, then $\Inn_{A^2} = (\Inn_A)^2$ is the identity map. We know from earlier that this means that $A^2 = \gamma I$ for some $\gamma \in \overline{k}.$ But, we know for certain that $A$ is orthogonal. So, $A^2$ is also orthogonal. That means that $(A^2)^TM(A^2) = M$, which implies $(\gamma I)^TM (\gamma I) = M$, which means $\gamma^2 = 1$. So, $\gamma = \pm 1.$ Thus, we can also distinguish between different types of $k$-involutions by seeing if $A^2 = I$ or $A^2 = -I$. This gives the four types of $k$-involutions, which are outlined in Table \ref{InvDefSo}.
\begin{table}[h]
\centering
\caption {The various possible types of $k$-involutions of $\So(n,k,\beta)$} \label{InvDefSo}
\begin{tabular}[t]{|c||c|c|}
\hline & $ A \in \oo(n,k,\beta)$ & $A \in \oo(n,k[\sqrt{\alpha}], \beta) \setminus \oo(n,k,\beta)$ \\
\hline \hline $A^2 = I$ & Type 1 & Type 2 \\
\hline $A^2 = -I$ & Type 3 & Type 4 \\
\hline
\end{tabular}
\end{table}
It follows from our characterization of $k$-automorphisms that when $n$ is odd, that Type 2 and Type 4 $k$-involutions do not occur. But, we also see that if $n$ is odd and $A$ is orthogonal, then $A^2$ must have determinant 1. So, we see in addition that Type 3 $k$-involutions can also only occur when $n$ is even.
In the following theorem, we show that an isomorphy class of $k$-involutions will lie neatly into exactly one of these types of $k$-involutions. First we need a lemma with a condition equivalent to isomorphism over $\oo(n,k,\beta)$.
\begin{lem}
\label{TidyLem}
Assume $\Inn_A$ and $\Inn_B$ are $k$-involutions of $\So(n,k,\beta)$ where $A$ lies in $\oo(n,k,\beta)$ or in $\oo(n,k[\sqrt{\alpha}],\beta)$ and each entry of $A$ is a $k$-multiple of $\sqrt{\alpha}$, and likewise for $B$. Then, $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$ (or $\So(n,k,\beta)$) if and only if there exists $Q \in \oo(n,k,\beta)$ (or $\So(n,k,\beta)$) such that $Q^{-1}AQ = B$ or $-B$.
\end{lem}
\begin{proof}
First assume there exists $Q \in \oo(n,k,\beta)$ (or $\So(n,k,\beta)$) such that $Q^{-1}AQ = B$ or $-B$. Then, for all $U \in \So(n,k,\beta)$, we have
\begin{align*}
\Inn_Q\Inn_A \Inn_{Q^{-1}} (U) &= Q^{-1}A^{-1}QUQ^{-1}AQ\\
&= (Q^{-1}AQ)^{-1}U(Q^{-1}AQ)\\
&= (\pm B)^{-1}U(\pm B)\\
&= B^{-1}UB\\
&= \Inn_B(U).
\end{align*}
So, $\Inn_Q\Inn_A \Inn_{Q^{-1}} =\Inn_B$. That is, $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$ (or $\So(n,k,\beta)$).
To prove the converse, we now assume that $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$ (or $\So(n,k,\beta)$). So, there exists $Q \in \oo(n,k,\beta)$ (or $\So(n,k,\beta)$) such that $\Inn_Q\Inn_A \Inn_{Q^{-1}} =\Inn_B$. Thus, for all $U \in \So(n,k,\beta)$, we have
$$Q^{-1}A^{-1}QUQ^{-1}AQ = B^{-1}UB,$$
which implies
$$BQ^{-1}A^{-1}QUQ^{-1}AQB^{-1} = U.$$
So, $Q^{-1}AQB^{-1} $ commutes with all elements of $\So(n,k,\beta)$. The center of $\So(n,k,\beta)$ is $\{I \}$ if $n$ is odd and $\{ I, -I \}$ if n is even. So, $Q^{-1}AQ = B$ or $-B$.
\end{proof}
\begin{theorem}
\label{TypesAreTidy}
Assume $\Inn_A$ and $\Inn_B$ are $k$-involutions of $\So(n,k,\beta)$ where $A$ lies in $\oo(n,k,\beta)$ or in $\oo(n,k[\sqrt{\alpha}],\beta)$ and each entry of $A$ is a $k$-multiple of $\sqrt{\alpha}$, and likewise for $B$. If $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$, then they must be $k$-involutions of the same type.
\end{theorem}
\begin{proof}
Assume $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$. By Lemma \ref{TidyLem} we can assume there exists $Q \in \oo(n,k,\beta)$ such that $Q^{-1}AQ = B$ or $-B$. This implies that
\begin{enumerate}
\item $A \in \oo(n,k,\beta)$ if and only if $B \in \oo(n,k,\beta)$; and
\item $A^2 = I$ if and only if $B^2 = I$.
\end{enumerate}
Thus, $\Inn_A$ and $\Inn_B$ must be of the same type.
\end{proof}
We introduce the notation $E(A, \lambda)$ to refer to the eigenspace of a matrix $A$ corresponding to eigenvalue $\lambda$, where we assume the vectors lie in $\overline{k}^n$. In practice, we will be concerned with basis vectors that lie either in $k$, or a particular quadratic extension of $k$.
\subsection{Type 1 $k$-involutions}
We now find a structured form for the matrices of all types of $k$-involutions. We begin with Type 1 $k$-involutions. When $n$ is odd, these are the only $k$-involutions.
\begin{lem}
\label{Type1ClassSo}
Suppose $\theta$ is a Type 1 $k$-involution of $\So(n,k,\beta)$. Then, there exists $A \in \oo(n,k,\beta)$ such that $A = X \left(\begin{smallmatrix}-I_s & 0 \\0 & I_t\end{smallmatrix}\right) X^{-1}$ where $s+t =n$, $s \le t$, and $$X = \left(\begin{matrix}x_1 & x_2 & \cdots & x_n \end{matrix}\right)\in \Gl(n,k),$$ where the $x_i$ are orthogonal eigenvectors of $A$, meaning $X^TMX$ is diagonal.
\end{lem}
\begin{proof}
Since $A^2 = I$, then all eigenvalues of $A$ are $\pm 1$. Since there are no repeated roots in the minimal polynomial of $A$, then we see that $A$ is diagonalizable. We want to construct bases for $E(A,1)$ and $E(A,-1)$ such that all the basis vectors lie in $k^n$. Let $s = \dim(E(A,-1))$ and $t = \dim(E(A,1))$, and observe that $s+t = n$ since $A$ is diagonalizable. If $s >t$, then replace $A$ with $-A$, and use this matrix instead. (It will induce the same $k$-involution.) Let $\{z_1,...,z_n\}$ be a basis for $k^n$. For each $i$, let $u_i = (A-I)z_i.$ Note that $$Au_i = A(A-I)z_i = -(A-I)z_i = -u_i.$$ So, $\{u_1,...,u_n\}$ must span $E(A,-1)$. Thus, we can appropriately choose $s$ of these vectors and form a basis for $E(A,-1)$. Label these basis vectors as $y_1,...,y_s$. We can similarly form a basis for $E(A,1)$. We shall call these vectors $y_{s+1},...,y_n$. Let $Y$ be the matrix with the vectors $y_1,...,y_n$ as its columns. Then, by construction, $$Y^{-1}AY = \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right).$$ We can rearrange to get $$A = Y \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) Y^{-1}.$$
Recall that $A^T = MAM^{-1}$, since $A \in \oo(n,k,\beta)$. So,
$$\left( Y \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) Y^{-1} \right)^T = M \left( Y \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) Y^{-1} \right) M^{-1}.$$
This implies
$$(Y^{-1})^T \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) Y^T = MY \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) (MY)^{-1}$$
which means
$$ \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) Y^TMY = Y^TMY \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) .$$
So, $Y^TMY = \left(\begin{smallmatrix}Y_1 & 0 \\0 & Y_2\end{smallmatrix}\right)$, where $Y_1$ is $s \times s$, $Y_2$ is $t \times t$, and both are symmetric. It follows that there exists $N = \left(\begin{smallmatrix}N_1 & 0 \\0 & N_2\end{smallmatrix}\right) \in \Gl(n,k)$ such that $N^TY^TMYN$ is diagonal. Let $X = YN$. Then,
\begin{align*}
X \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) X^{-1} &= YN\left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) (YN)^{-1}\\
&= Y \left(\begin{array}{cc}N_1 & 0 \\0 & N_2\end{array}\right) \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) \left(\begin{array}{cc}N_1^{-1} & 0 \\0 & N_2^{-1}\end{array}\right)Y^{-1}\\
&= Y \left(\begin{array}{cc}-I_s & 0 \\0 & I_t\end{array}\right) Y^{-1}\\
&= A,
\end{align*}
where $X^TMX$ is diagonal. It follows from this last observation that the column vectors of $X$ must be orthogonal with respect to $\beta$.
\end{proof}
Now we show conditions equivalent to isomorphy. Note that we say two matrices are $A$ and $B$ are {\it congruent} over a group $G$ if there exists $Q \in G$ such that $Q^{-1}AQ = B$.
\begin{theorem}
\label{type1lemSo}
Suppose $\theta$ and $\phi$ are two Type 1 $k$-involutions of $\So(n,k,\beta)$ where $\theta = \Inn_A$ and $\phi = \Inn_B$. Then, $A = X \left(\begin{smallmatrix}-I_{m_A} & 0 \\0 & I_{n-m_A}\end{smallmatrix}\right) X^{-1}$ and $B = Y\left(\begin{smallmatrix}-I_{m_B} & 0 \\0 & I_{n-m_B}\end{smallmatrix}\right) Y^{-1}$ where $m_A, m_B \le \frac{n}{2}$, and $$X = \left(\begin{matrix}x_1 & x_2 & \cdots & x_n \end{matrix}\right), Y = \left(\begin{matrix}y_1 & y_2 & \cdots & y_n \end{matrix}\right)\in \Gl(n,k)$$
have columns that are orthogonal eigenvectors of $A$ and $B$ respectively. We also have the diagonal matrices $$X^TMX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right)$$ and $$Y^TMY = \left(\begin{array}{cc}Y_1 & 0 \\0 & Y_2\end{array}\right).$$
The following are equivalent:
\begin{enumerate}
\item $\theta$ is isomorphic to $\phi$ over $\So(n,k,\beta)$.
\item $A$ is conjugate to $B$ or $-B$ over $\So(n,k,\beta)$.
\item $ X_1$ is congruent to $Y_1$ over $\Gl(m,k)$ and $X_2$ is congruent to $Y_2$ over $\Gl(n-m,k)$, or $ X_1$ is congruent to $Y_2$ over $\Gl(\frac{n}{2},k)$ and $X_2$ is congruent to $Y_1$ over $\Gl(\frac{n}{2},k)$.
\item $\theta$ is isomorphic to $\phi$ over $\oo(n,k,\beta)$.
\item $A$ is conjugate to $B$ or $-B$ over $\oo(n,k,\beta)$.
\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of $(i)$ and $(ii)$ follows from Lemma \ref{TidyLem}, as does the equivalence of $(iv)$ and $(v)$.
Next we show that $(ii)$ implies $(iii)$. First suppose that $Q^{-1}AQ = B$ for some $Q \in \So(n,k,\beta)$. $Q^{-1}AQ = B$ implies $$Q^{-1} X \left(\begin{array}{cc}-I_{m_A} & 0 \\0 & I_{n-m_A}\end{array}\right) X^{-1}Q = Y\left(\begin{array}{cc}-I_{m_B} & 0 \\0 & I_{n-m_B}\end{array}\right) Y^{-1}.$$ Since the matrices on both sides of the equality above must have the same eigenvalues with the same multiplicities, then we see that $m_A = m_B$. Let $m = m_A = m_B$, and recall from Theorem \ref{sltheorem1} that $$-I_{m,n-m} = \left(\begin{array}{cc}-I_{m} & 0 \\0 & I_{n-m}\end{array}\right).$$ Rearranging the previous equation, we have $$I_{m,n-m}X^{-1}QY = X^{-1}QYI_{m,n-m},$$ which tells us that $X^{-1}QY = \left(\begin{smallmatrix}R_1 & 0 \\0 & R_2\end{smallmatrix}\right),$ where $R_1 \in \Gl(m,k)$ and $R_2\in \Gl(n-m,k)$. Rearranging, we have that $QY = X\left(\begin{smallmatrix}R_1 & 0 \\0 & R_2\end{smallmatrix}\right).$ Since $Q \in \oo(n,k,\beta)$, then we know that $Q^TMQ=M$.
So,
\begin{align*}
Y^TMY &= Y^TQ^TMQY\\
&= \left(\begin{array}{cc}R_1 & 0 \\0 & R_2\end{array}\right)^T(X^TMX)\left(\begin{array}{cc}R_1 & 0 \\0 & R_2\end{array}\right).
\end{align*}
From here we see that
$Y_1= R_1^TX_1R_1$ and
$Y_2 = R_2^TX_2R_2.$
Now suppose that $Q^{-1}AQ = -B$ for some $Q \in \So(n,k,\beta)$. This implies $$Q^{-1} X \left(\begin{array}{cc}-I_{m_A} & 0 \\0 & I_{n-m_A}\end{array}\right) X^{-1}Q = Y\left(\begin{array}{cc}I_{m_B} & 0 \\0 & -I_{n-m_B}\end{array}\right) Y^{-1}.$$ Since the matrices on both sides of the equality above must have the same eigenvalues with the same multiplicities, then we see that $m_A = n-m_B$. Since $m_A, m_B \le \frac{n}{2}$, then it follows that $m_A = m_B = \frac{n}{2}$. Rearranging the previous equation, we have $$I_{\frac{n}{2},\frac{n}{2}}X^{-1}QY = X^{-1}QYI_{\frac{n}{2},\frac{n}{2}},$$ which tells us that $X^{-1}QY = \left(\begin{smallmatrix} 0 &R_1 \\ R_2 & 0 \end{smallmatrix}\right),$ where $R_1, R_2 \in \Gl(\frac{n}{2},k)$. Rearranging, we have that $QY = X\left(\begin{smallmatrix} 0 &R_1 \\ R_2 & 0 \end{smallmatrix}\right).$ Since $Q \in \oo(n,k,\beta)$, then we know that $Q^TMQ=M$.
So,
\begin{align*}
Y^TMY &= Y^TQ^TMQY\\
&=\left(\begin{array}{cc} 0 &R_1 \\ R_2 & 0 \end{array}\right)^T(X^TMX)\left(\begin{array}{cc} 0 &R_1 \\ R_2 & 0 \end{array}\right).
\end{align*}
From here we see that
$Y_2= R_1^TX_1R_1$ and
$Y_1 = R_2^TX_2R_2.$
This shows that $(ii)$ implies $(iii)$.
Now we show that $(iii)$ implies $(ii)$. Assume that $(iii)$ is the case. Specifically, assume that $R_1 \in \Gl(m,k)$ and $R_2 \in \Gl(n-m,k)$ such that $Y_1= R_1^TX_1R_1$ and
$Y_2 = R_2^TX_2R_2.$ Let $R = \left(\begin{smallmatrix}R_1 & 0 \\0 & R_2\end{smallmatrix}\right).$ So, we have $Y^TMY = R^T(X^TMX)R$. Let $Q = XRY^{-1}$. We will now show that $Q^{-1}AQ = B$ and that $Q \in \So(n,k,\beta)$.
\begin{align*}
Q^{-1}AQ &= (XRY^{-1})^{-1}A(XRY^{-1})\\
&= YR^{-1}X^{-1}AXRY^{-1}\\
&= YR^{-1}(-I_{m,n-m})RY^{-1}\\
&= Y(-I_{m,n-m})Y^{-1} = B.
\end{align*}
Next, we must show that $Q \in \So(n,k,\beta)$. We first show that $Q^TMQ = M$. Recall that $Y^TMY = R^T(X^TMX)R$. So,
\begin{align*}
Q^TMQ &= (XRY^{-1})^{T}M(XRY^{-1})\\
&= (Y^{-1})^T(R^TX^TMXR)Y^{-1}\\
&= (Y^{-1})^T(Y^TMY)Y^{-1}\\
&= M.
\end{align*}
In the event that $\det(Q) = -1$, then we can replace the first column of $X$ with its negative. This will have no effect on $R$ or $Y$, so the new $Q = XRY^{-1}$ have determinant 1, and it will still be the case that $Q^{-1}AQ = B$ and $Q^TMQ = M$. So, $Q \in \So(n,k,\beta)$.
If instead we assume that $R_1 \in \Gl(m,k)$ and $R_2 \in \Gl(n-m,k)$ such that $Y_2= R_1^TX_1R_1$ and
$Y_1 = R_2^TX_2R_2,$ then if we let $R = \left(\begin{smallmatrix} 0 &R_1 \\ R_2 & 0 \end{smallmatrix}\right)$, then we can let $Q = XRY^{-1}$ and get that $Q^{-1}AQ = -B$ and $Q \in \So(n,k,\beta)$. This shows that $(iii)$ implies $(ii)$.
We now show that $(iv)$ and $(v)$ are equivalent to the previous three conditions. First, we note that it is clear that $(i)$ implies $(iv)$. So, we need only show that $(iv)$ or $(v)$ implies one of the other three conditions. But, $(v)$ implies $(iii)$ from an argument very similar to the argument where we showed that $(ii)$ implies $(iii)$. Thus, all the conditions are equivalent.
\end{proof}
We note that this Theorem shows that isomorphy over $\So(n,k,\beta)$ and $\oo(n,k,\beta)$ are the same for Type 1 $k$-involutions. We will show in an explicit example that this does not occur in the Type 2 case. For the remaining three types of $k$-involutions, we will only find conditions for isomorphy over $\oo(n,k,\beta)$. Again, recall that these three Types of $k$-involutions only occur when $n$ is even. So, when $n$ is odd, we have isomorphy conditions over $\So(n,k,\beta)$.
\subsection{Type 2 $k$-involutions}
We have a similar characterization of the matrices and isomorphy classes in the Type 2 case. We first prove a result about that characterizes the eigenvectors in the Type 2 case.
\begin{lem}
Suppose $A \in \oo(n,k[\sqrt{\alpha}],\beta) \setminus \oo(n,k,\beta) $ induces a Type 2 $k$-involution of $\So(n,k,\beta)$ where $\sqrt{\alpha} \not \in k$. Recall that the entries of $A$ are all $k$-multiples of $\sqrt{\alpha}$. Suppose $x,y \in k^n$ such that $x+\sqrt{\alpha}y \in E(A,-1)$. Then, $x-\sqrt{\alpha}y \in E(A,1)$. Likewise, if $u,v \in k^n$ such that $u+\sqrt{\alpha}v \in E(A,1)$. Then, $u-\sqrt{\alpha}v \in E(A,-1)$. Further, $\dim(E(A,1))= \dim(E(A,-1))$.
\end{lem}
\begin{proof}
Suppose $a,b \in k$. Then, we refer to the Galois automorphism of the field $k[\sqrt{\alpha}] $ over $k$ that sends $a+\sqrt{\alpha}b $ to $a - \sqrt{\alpha}b$ as ``$\sqrt{\alpha}$-conjugation." Since it is an automorphism, it must preserve multiplication. Further, for a matrix $X = Y +\sqrt{\alpha}Z$ for $Y, Z$ with entries in $k$, we say that $Y -\sqrt{\alpha}Z$ is the ``$\sqrt{\alpha}$-conjugation" of $X$. We note that this mapping will preserve multiplication of matrices.
Since $$A(x+\sqrt{\alpha}y) = -x-\sqrt{\alpha}y,$$ then we can take the ``$\sqrt{\alpha}$-conjugation" to see that $$(-A)(x-\sqrt{\alpha}) = -x+\sqrt{\alpha}y.$$ We note that the ``$\sqrt{\alpha}$-conjugation" of $A$ is $-A$ because each entry of $A$ is a $k$-multiple of $\sqrt{\alpha}$. We can multiply both side by -1 to see $$A(x-\sqrt{\alpha}) = x-\sqrt{\alpha}y.$$ That is, $x-\sqrt{\alpha}y \in E(A,1)$. This proves the first statement. An analogous argument proves the second.
To see that $\dim(E(A,1))= \dim(E(A,-1))$ is the case, note that the first statement tells us that $\dim(E(A,1)) \le \dim(E(A,-1))$, and that the second statement tells us that $\dim(E(A,1))\ge \dim(E(A,-1))$, since ``$\sqrt{\alpha}$-conjugation" is an invertible operator on $k[\sqrt{\alpha}]^n$.
\end{proof}
We are now able to characterize the Type 2 $k$-involutions.
\begin{lem}
\label{Type2ClassSo}
Suppose $\theta$ is a Type 2 $k$-involution of $\So(n,k,\beta)$. Let $A$ be the orthogonal matrix in $\oo(n,k[\sqrt{\alpha}],\beta)$ such that $\theta = \Inn_A$. Then, $$A = \frac{-\sqrt{\alpha}}{\alpha} X \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) X^{-1}$$ where $$X = \left(\begin{array}{cccccccc}x_1 & x_2 & \cdots & x_{\frac{n}{2}} &y_1 & y_2 & \cdots & y_{\frac{n}{2}} \end{array}\right)\in \Gl(n,k),$$ where
for each $i$, we have orthogonal vectors $x_i+\sqrt{\alpha}y_i \in E(A,-1)$ and orthogonal vectors $x_i-\sqrt{\alpha}y_i \in E(A,1)$. Further, $$X^TMX = \left(\begin{array}{cc}X_1 & X_2 \\X_2 & \frac{1}{\alpha}X_1 \end{array}\right)$$ where $X_1$ and $X_2$ are diagonal matrices.
\end{lem}
\begin{proof}
We begin by constructing bases for $E(A,1)$ and $E(A,-1)$ such that all the basis vectors lie in $k[\sqrt{\alpha}]^n$. From the previous lemma, we know that $\dim(E(A,1)) = \dim(E(A,-1)) = \frac{n}{2}.$ (Note that this means that $n$ must be even for a Type 2 $k$-involution to occur.) Since $\Inn_A$ is a Type 1 $k$-involution of $\So(n,k[\sqrt{\alpha}], \beta)$, then we can apply Lemma \ref{Type1ClassSo} to find an orthogonal basis $\{ x_1+\sqrt{\alpha}y_1,...,x_{\frac{n}{2}}+\sqrt{\alpha}y_{\frac{n}{2}} \}$ of $E(A,-1)$, where $x_1,...,x_{\frac{n}{2}},y_1,...,y_{\frac{n}{2}} \in k^n$. By the previous lemma, we know that$\{x_1-\sqrt{\alpha}y_1,...,x_{\frac{n}{2}}-\sqrt{\alpha}y_{\frac{n}{2}}\}$ must be a basis for $E(A,1)$. Let $X = \left(\begin{smallmatrix}x_1 & x_2 & \cdots & x_{\frac{n}{2}} &y_1 & y_2 & \cdots & y_{\frac{n}{2}} \end{smallmatrix}\right)\in \Gl(n,k).$
We now make a couple of observations. Suppose $u = x+\sqrt{\alpha} y$ is a -1-eigenvector of $A$ such that $x,y \in k^n$. Then, we know $v = x-\sqrt{\alpha} y$ is a 1-eigenvector of $A$. Observe that
\begin{align*}
Ax &= \frac{1}{2}A(u+v)\\
&= \frac{1}{2}(-u+v)\\
&= -\sqrt{\alpha} y.
\end{align*}
It follows from this that $$Ay = -\frac{\sqrt{\alpha}}{\alpha}x.$$
Since $Ax = -\sqrt{\alpha} y$ and $Ay = -\frac{\sqrt{\alpha}}{\alpha}x$, then it follows that $$X^{-1}AX = \left(\begin{smallmatrix}0 & -\frac{\sqrt{\alpha}}{\alpha}I_{\frac{n}{2}} \\ -\sqrt{\alpha}I_{\frac{n}{2}} & 0\end{smallmatrix}\right).$$ Rearranging this, we see that $$A = -\frac{\sqrt{\alpha}}{\alpha} X \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) X^{-1}.$$
Now, we need only prove the last statement to prove the Lemma. Since $\{ x_1+\sqrt{\alpha}y_1,...,x_{\frac{n}{2}}+\sqrt{\alpha}y_{\frac{n}{2}} \}$ is an orthogonal set of vectors, then we know when $i \ne j$ that
\begin{align*}
0 &= \beta(x_i+\sqrt{\alpha}y_i, x_j+\sqrt{\alpha}y_j)\\
&= (\beta(x_i, x_j)+\alpha\beta(y_i, y_j))+\sqrt{\alpha}(\beta(x_i,y_j)+\beta(x_j,y_i)).
\end{align*}
This tells us that $$\beta(x_i, x_j)= -\alpha\beta(y_i, y_j)$$ and $$\beta(x_i,y_j)= -\beta(x_j,y_i).$$
Since vectors from $E(A,1)$ and $E(A,-1)$ are orthogonal, then we also know that
\begin{align*}
0 &= \beta(x_i+\sqrt{\alpha}y_i, x_j-\sqrt{\alpha}y_j)\\
&= (\beta(x_i, x_j)-\alpha\beta(y_i, y_j))+\sqrt{\alpha}(-\beta(x_i,y_j)+\beta(x_j,y_i)),
\end{align*}
regardless of if $i$ and $j$ are distinct or equal.
This tells us that $$\beta(x_i, x_j)= \alpha\beta(y_i, y_j)$$ and $$\beta(x_i,y_j)= \beta(x_j,y_i).$$
So, when $i \ne j$, then we know that $$ \beta(x_i, y_j) = 0,$$
$$ \beta(x_i, x_j) = 0,$$
and
$$ \beta(y_i, y_j) = 0.$$
When $i = j$, we note that $$\beta(x_i, x_i)= \alpha\beta(y_i, y_i).$$ Then, we have $$X^TMX = \left(\begin{array}{cc}X_1 & X_2 \\X_2 & \frac{1}{\alpha}X_1 \end{array}\right)$$ where $X_1$ and $X_2$ have been shown to be diagonal.
\end{proof}
We now show an example of a Type 2 $k$-involution, and apply the previous lemma to it.
\begin{beisp}
Assume that $\beta$ is the standard dot product. Then, $\Inn_A$ can be a Type 2 $k$-involution of $\So(4,\mathbb{Q})$ if $A$ is symmetric and orthogonal, since this will imply that $A^2 =I$, and if the entries of $A$ are all $k$-multiples of some $\sqrt{\alpha}$ such that $\sqrt{\alpha} \not \in k$ but $\alpha \in k$. Observe that the matrix $$A = \frac{\sqrt{3}}{3} \left(\begin{array}{cccc}0 & 1 & -1 & 1 \\1 & 0 & 1 & 1 \\-1 & 1 & 1 & 0 \\1 & 1 & 0 & -1\end{array}\right)$$ is both symmetric and orthogonal. Since each entry is the $\mathbb{Q}$-multiple of $\sqrt{3}$, then it is clear that $\Inn_A$ is a Type 2 $k$-involution of $\So(4, \mathbb{Q})$. It can be shown that $E(A,-1)$ has dimension 2. An orthogonal basis for this subspace is formed by the vectors $$v_1 = \left(\begin{array}{c}\frac{1}{2} \\\frac{1}{2} \\0 \\1\end{array}\right)+ \sqrt{3} \left(\begin{array}{c}-\frac{1}{2} \\-\frac{1}{2} \\0 \\0\end{array}\right)$$ and $$ v_2=\left(\begin{array}{c}\frac{1}{2} \\ -\frac{1}{2} \\1 \\0\end{array}\right)+ \sqrt{3} \left(\begin{array}{c}\frac{1}{2} \\ -\frac{1}{2} \\0 \\0\end{array}\right).$$
It can be shown that $$v_3 = \left(\begin{array}{c}\frac{1}{2} \\\frac{1}{2} \\0 \\1\end{array}\right)- \sqrt{3} \left(\begin{array}{c}-\frac{1}{2} \\-\frac{1}{2} \\0 \\0\end{array}\right)$$ and $$ v_4=\left(\begin{array}{c}\frac{1}{2} \\ -\frac{1}{2} \\1 \\0\end{array}\right)- \sqrt{3} \left(\begin{array}{c}\frac{1}{2} \\ -\frac{1}{2} \\0 \\0\end{array}\right) $$ are orthogonal $1$-eigenvectors of $A$, where these are the $\sqrt{3}$-conjugates of $v_1$ and $v_2$, respectively.
Following the notation of the previous lemma, we have $$X = \left(\begin{array}{cccc}\frac{1}{2} & \frac{1}{2} & -\frac{1}{2} & \frac{1}{2} \\\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} & -\frac{1}{2} \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0\end{array}\right),$$ where $X^TX = \left(\begin{smallmatrix}\frac{3}{3} & 0 & -\frac{1}{2} & 0 \\0 & \frac{3}{3} & 0 & \frac{1}{2} \\-\frac{1}{2} & 0 & \frac{1}{2} & 0 \\0 & \frac{1}{2} & 0 & \frac{1}{2}\end{smallmatrix}\right)$ and $A = -\frac{\sqrt{3}}{3}X\left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ 3I_{\frac{n}{2}} & 0\end{smallmatrix}\right) X^{-1}$ .
\end{beisp}
We now find conditions in the Type 2 case that are equivalent to isomorphy.
\begin{theorem}
\label{type2lemSo}
Suppose $\theta$ and $\phi$ are two Type 2 $k$-involutions of $\So(n,k,\beta)$ where $\theta = \Inn_A$ and $\phi = \Inn_B$. Then, $$A = -\frac{\sqrt{\alpha}}{\alpha} X \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) X^{-1} \in \oo(n,k[\sqrt{\alpha}],\beta)$$ where $$X = \left(\begin{array}{cccccccc}x_1 & x_2 & \cdots & x_{\frac{n}{2}} & y_1 & y_2 & \cdots & y_{\frac{n}{2}} \end{array}\right)\in \Gl(n,k)$$ and the $x_i +\sqrt{\alpha}y_i$ are the orthogonal basis of $E(A,-1)$, and $$X^TMX = \left(\begin{array}{cc}X_1 & X_2 \\X_2 & \frac{1}{\alpha}X_1 \end{array}\right)$$ where $X_1$ and $X_2$ are diagonal matrices,
and $$B = -\frac{\sqrt{\gamma}}{\gamma} Y \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \gamma I_{\frac{n}{2}} & 0\end{array}\right) Y^{-1}\in \oo(n,k[\sqrt{\gamma}],\beta)$$ where $$Y = \left(\begin{array}{cccccccc}\tilde{x}_1 & \tilde{x}_2 & \cdots & \tilde{x}_{\frac{n}{2}} & \tilde{y}_1 & \tilde{v}_2 & \cdots & \tilde{y}_{\frac{n}{2}} \end{array}\right)\in \Gl(n,k)$$ and the $\tilde{x}_i +\sqrt{\gamma}\tilde{y}_i$ is the orthogonal eigenvectors of $E(B,-1)$, and $$Y^TMY = \left(\begin{array}{cc}Y_1 & Y_2 \\Y_2 & \frac{1}{\gamma}Y_1 \end{array}\right)$$ where $Y_1$ and $Y_2$ are diagonal matrices,
and the following are equivalent:
\begin{enumerate}
\item $\theta$ is isomorphic to $\phi$ over $\oo(n,k,\beta)$.
\item $A$ is conjugate to $B$ or $-B$ over $\oo(n,k,\beta)$.
\item $\alpha = \gamma$ and $Y^TMY = R^TX^TMXR$ where $R = \left(\begin{smallmatrix}R_1 & R_2 \\ \alpha R_2 & R_1\end{smallmatrix}\right) \in \Gl(n,k)$, or $\alpha = \gamma$ and $Y^TMY = R^TX^TMXR$ where $R = \left(\begin{smallmatrix}R_1 & R_2 \\ -\alpha R_2 & -R_1\end{smallmatrix}\right) \in \Gl(n,k)$.
\item We can choose $X$ and $Y$ such that $\alpha = \gamma$, and for $R = \left(\begin{smallmatrix}R_1 & R_2 \\ \alpha R_2 & R_1\end{smallmatrix}\right) \in \Gl(n,k)$ we have
$$Y_1 = R_1^TX_1R_1 +\alpha R_2^TX_2R_1+\alpha R_1^TX_2R_2+\alpha R_2^TX_1R_2$$
and
$$Y_2 = R_2^TX_1R_1+R_1^TX_2R_1+\alpha R_2^TX_2R_2+R_1^TX_1R_2,$$
or for $R = \left(\begin{smallmatrix}R_1 & R_2 \\ -\alpha R_2 & -R_1\end{smallmatrix}\right) \in \Gl(n,k)$ we have
$$Y_1 = R_1^TX_1R_1 -\alpha R_2^TX_2R_1-\alpha R_1^TX_2R_2+\alpha R_2^TX_1R_2$$
and
$$Y_2 = R_2^TX_1R_1-R_1^TX_2R_1-\alpha R_2^TX_2R_2+R_1^TX_1R_2.$$
\end{enumerate}
\end{theorem}
\begin{proof}
The equivalence of $(i)$ and $(ii)$ follows from Lemma \ref{TidyLem}. So, we begin by showing that $(ii)$ implies $(iii)$. First suppose there exists $Q \in \oo(n,k,\beta)$ such that $Q^{-1}AQ = B$. So, we have $$Q^{-1} \frac{\sqrt{\alpha}}{\alpha} X \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) X^{-1} Q = \frac{\sqrt{\gamma}}{\gamma} Y \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \gamma I_{\frac{n}{2}} & 0\end{array}\right) Y^{-1}.$$ Also, we know that since $A \in \So(n,k[\sqrt{\alpha}],\beta)$ and $B\in \So(n,k[\sqrt{\gamma}],\beta)$ are congruent over $\oo(n,k,\beta)$, then $\gamma$ must be a $k$-multiple of $\alpha$. Without loss of generality, we will assume $\gamma = \alpha$. Thus, $$Q^{-1} X \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) X^{-1} Q = Y \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) Y^{-1}.$$
Rearranging, we see that $$\left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right)X^{-1}QY = X^{-1}QY \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right).$$ Let $R = X^{-1}QY $, and note that $R \in \Gl(n,k)$. Since $\left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{smallmatrix}\right) R = R \left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{smallmatrix}\right)$, then $R = \left(\begin{smallmatrix}R_1 & R_2 \\ \alpha R_2 & R_1\end{smallmatrix}\right)$. Observe that $XR = QY$. Also, observe that since $Q \in \So(n,k,\beta)$, then we know that $Q^TMQ=M$. It follows from these observations that
\begin{align*}
R^T(X^TMX)R &= (XR)^TM(XR)\\
&= (QY)^TM(QY)\\
&= Y^T(Q^TMQ)Y\\
&= Y^TMY.
\end{align*}
If instead we assume that there exists $Q \in \oo(n,k,\beta)$ such that $Q^{-1}AQ = -B$, then we can similarly show that $\alpha = \gamma$ and $Y^TMY = R^TX^TMXR$ where $R = \left(\begin{smallmatrix}R_1 & R_2 \\ -\alpha R_2 & -R_1\end{smallmatrix}\right) \in \Gl(n,k)$ for $R_1, R_2 \in \M(\frac{n}{2},k).$ This proves that $(ii)$ implies $(iii)$.
We now show that $(iii)$ implies $(ii)$. First assume $\alpha = \gamma$ and $X^TMX$ is congruent to $Y^TMY$ over $\Gl(n,k)$ where $Y^TMY = R^TX^TMXR$ for $R = \left(\begin{smallmatrix}R_1 & R_2 \\ \alpha R_2 & R_1\end{smallmatrix}\right)$, where $R_1, R_2 \in \Gl(\frac{n}{2},k)$. Let $Q = XRY^{-1}.$ Then, we observe that
\begin{align*}
Q^{-1}AQ &= (XRY^{-1})^{-1}A(XRY^{-1})\\
&= YR^{-1}(X^{-1}AX)RY^{-1}\\
&= -\frac{-\sqrt{\alpha}}{\alpha} YR^{-1} \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) RY^{-1}\\
&= -\frac{-\sqrt{\alpha}}{\alpha} Y \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) R^{-1}RY^{-1}\\
&= -\frac{-\sqrt{\alpha}}{\alpha} Y\left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) Y^{-1} = B.
\end{align*}
To show that $(ii)$ is indeed the case, we need only show that $Q \in \oo(n,k,\beta)$. By construction, we know that $Q \in \Gl(n,k)$. So, it is suffice to show $Q^TMQ = M$. But,
\begin{align*}
Q^TMQ &= (XRY^{-1})^TM(XRY^{-1})\\
&= (Y^{-1})^T(R^TX^TMXR)Y^{-1}\\
&= (Y^{-1})^T(Y^TMY)Y^{-1} = M.
\end{align*}
If we instead assume that $\alpha = \gamma$ and $X^TMX$ is congruent to $Y^TMY$ over $\Gl(n,k)$ where $Y^TMY = R^TX^TMXR$ for $R = \left(\begin{smallmatrix}R_1 & R_2 \\ -\alpha R_2 & -R_1\end{smallmatrix}\right)$, where $R_1, R_2 \in \Gl(\frac{n}{2},k)$, then if we let $Q = XRY^{-1},$ we can similarly show that $Q^{-1}AQ = -B$ and $Q \in \oo(n,k,\beta)$. This shows that $(iii)$ implies $(ii)$.
Lastly, matrix multiplication shows that $(iii)$ and $(iv)$ are equivalent.
\end{proof}
The reader will notice that in the Type 1 case, our conditions gave us isomorphy of $k$-involutions over $\So(n,k,\beta)$, but the Type 2 case gave us isomorphy of $k$-involutions over $\oo(n,k,\beta)$. In the following example, we give an example that shows that isomorphy over $\oo(n,k,\beta)$ is not the same as isomorphy over $\So(n,k,\beta)$ for Type 2 $k$-involutions.
\begin{beisp}
Consider the group $\So(4, \mathbb{F}_3)$. That is, consider the case where $k$ is the group of three elements, and the bilinear form is the standard dot product. Let $i$ denote a fixed square root of $2 =-1$. A Type 2 $k$-involution is induced by the matrix
$$A = i \left(\begin{array}{cccc}1 & 1 & 0 & 0 \\1 & 2 & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & 2\end{array}\right) \in \So(4, \mathbb{F}_3[i]).$$
By analyzing suitable eigenvectors for this matrix, we see that
$$X = \left(\begin{array}{cccc}1 & 0 & 2 & 0 \\0 & 0 & 2 & 0 \\0 & 1 & 0 & 2 \\0 & 0 & 0 & 2\end{array}\right)$$
where
$$A = i X^{-1} \left(\begin{array}{cccc}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\2 & 0 & 0 & 0 \\0 & 2 & 0 & 0\end{array}\right) X.$$
We also see that
$$X^TX = \left(\begin{array}{cc}X_1 & X_2 \\X_2 & 2X_1 \end{array}\right)$$ where $X_1 = \left(\begin{smallmatrix}1 & 0 \\0 & 1\end{smallmatrix}\right)$ and $X_2 = \left(\begin{smallmatrix}2 & 0 \\0 & 2\end{smallmatrix}\right).$
Now, we also consider the Type 2 $k$-involution of $\So(4, \mathbb{F}_3)$ that is induced by the matrix
$$B = i \left(\begin{array}{cccc}0 & 0 & 2 & 1 \\0 & 0 & 2 & 2 \\2 & 2 & 0 & 0 \\1 & 2 & 0 & 0\end{array}\right) \in \So(4, \mathbb{F}_3[i]).$$
By analyzing suitable eigenvectors for this matrix, we see that for
$$Y = \left(\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 2 & 1\end{array}\right),$$
we have
$$B = i Y^{-1} \left(\begin{array}{cccc}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\2 & 0 & 0 & 0 \\0 & 2 & 0 & 0\end{array}\right) Y.$$
We also see that
$$Y^TY = \left(\begin{array}{cc}Y_1 & Y_2 \\Y_2 & 2Y_1 \end{array}\right)$$ where $Y_1 = \left(\begin{smallmatrix}1 & 0 \\0 & 1\end{smallmatrix}\right)$ and $Y_2 = \left(\begin{smallmatrix}0 & 0 \\0 & 0\end{smallmatrix}\right).$
We have two ways of showing that these Type 2 $k$-involutions are congruent over $\oo(4,\mathbb{F}_3)$. First, we consider the matrix
$$Q = \left(\begin{array}{cccc}1 & 2 & 2 & 1 \\1 & 1 & 1 & 1 \\1 & 1 & 2 & 2 \\2 & 1 & 2 & 1\end{array}\right) \in \oo(4, \mathbb{F}_3) \setminus \So(4, \mathbb{F}_3).$$
Then, $B = Q^{-1}AQ$. This is condition $(ii)$ of the previous theorem.
Secondly, if we let $R = \left(\begin{smallmatrix}R_1 & R_2 \\ 2R_2 & R_1\end{smallmatrix}\right) \in \Gl(4,\mathbb{F}_3)$ where $R_1 = \left(\begin{smallmatrix}1 & 0 \\0 & 1\end{smallmatrix}\right)$ and $R_2 = \left(\begin{smallmatrix}1 & 1 \\2 & 1\end{smallmatrix}\right),$ then we get that $R^TY^TYR= X^TX.$ This is condition $(iii)$ of the previous theorem.
We now show that there does not exist $W \in \So(4, \mathbb{F}_3)$ such that $B = W^{-1}AW$. We proceed by contradiction and suppose that these does exist such an $W$. It then follows that $A$ and $QW^{-1} \in \oo(4, \mathbb{F}_3)$ are commuting matrices. It is a simple matter to show that matrices that commute with $A$ must be of the form
$$\left(\begin{array}{cccc}a & b & c & d \\b & a+b & d & c+d \\e & f & g & h \\f & e+f & h & g+h\end{array}\right).$$
One such matrix is
$$\left(\begin{array}{cccc}1 & 1 & 2 & 2 \\1 & 2 & 2 & 1 \\2 & 2 & 2 & 2 \\2 & 1 & 2 & 1\end{array}\right) \in \So(4, \mathbb{F}_3).$$
But, all other such orthogonal matrices differ from this matrix only in that an even number of rows and/or columns have been multiplied by $2=-1$ or an even number of rows and/or columns have been swapped. All of these actions create matrices that will also have determinant 1. Thus, all the matrices in $\oo(4, \mathbb{F}_3)$ which commute with $A$ are also members of $\So(4, \mathbb{F}_3),$ which contradicts $QW^{-1} \in \oo(4, \mathbb{F}_3)$ commuting with $A$. So, no such $W \in \So(4,\mathbb{F}_3)$ can exist, which means we cannot strengthen the above theorem by replacing $\oo(n,k,\beta)$ with $\So(n,k,\beta)$ in conditions $(i)$ and $(ii)$.
\end{beisp}
\begin{theorem}
Suppose $\Inn_A$ and $\Inn_B$ are both Type 2 $k$-involutions of $\So(n,k,\beta)$. Then, $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$ if and only if they are isomorphic over \newline $\So(n,k[\sqrt{\alpha}], \beta)$.
\end{theorem}
\begin{proof}
When viewed as Type 2 $k$-involutions of $\So(n,k,\beta)$, we can write $$A = -\frac{\sqrt{\alpha}}{\alpha} U \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) U^{-1} \text{ where } U^TMU = \left(\begin{array}{cc}U_1 & U_2 \\U_2 & \frac{1}{\alpha}U_1 \end{array}\right)$$ and $$B = -\frac{\sqrt{\alpha}}{\alpha} V \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{array}\right) V^{-1} \text{ where } V^TMV = \left(\begin{array}{cc}V_1 & V_2 \\V_2 & \frac{1}{\alpha}V_1 \end{array}\right)$$ and $U_1$, $U_2$, $V_1$ and $V_2$ are diagonal matrices.
When $\Inn_A$ and $\Inn_B$ are viewed as $k$-involutions of $\So(n,k[\sqrt{\alpha}], \beta)$, then they are Type 1 $k$-involutions. Further, we can choose $X$ and $Y\in \Gl(n,k[\sqrt{\alpha}])$ such that $A = X \left(\begin{smallmatrix}-I_{\frac{n}{2}} & 0 \\0 & I_{\frac{n}{2}}\end{smallmatrix}\right)X^{-1}$, $B = Y \left(\begin{smallmatrix}-I_{\frac{n}{2}} & 0 \\0 & I_{\frac{n}{2}}\end{smallmatrix}\right)Y^{-1}$, and
$$X_1 = \frac{1}{2}(U_1 +\sqrt{\alpha}U_2),$$
$$X_2 = \frac{1}{2}(U_1 -\sqrt{\alpha}U_2),$$
$$Y_1 = \frac{1}{2}(V_1 +\sqrt{\alpha}V_2),$$
and $$Y_2 = \frac{1}{2}(V_1 -\sqrt{\alpha}V_2).$$
This follows from the way in which $U$ and $V$ are constructed from the eigenvalues of $A$ and $B$. We need to simply have $X$ and $Y$ consist of the appropriate eigenvectors, and mandate that the last $\frac{n}{2}$ columns of $X$ and $Y$ are the $\sqrt{\alpha}$-conjugates of the first $\frac{n}{2}$ columns. (The only exception to this is that we may need to negate the first column of $X$, so that we can preserve isomorphy of $\Inn_A$ and $\Inn_B$ over $\So(n,k[\sqrt{\alpha}],\beta)$, if we are assuming that. But, this will not change the value of $X_1$.)
Now, suppose $\Inn_A$ and $\Inn_B$ are isomorphic over $\So(n,k[\sqrt{\alpha}],\beta)$ as Type 1 $k$-involutions. Then, from Theorem \ref{type1lemSo} we know that $Y_1$ is congruent to either $X_1$ or $X_2$.
In the first case, we see that
\begin{align*}
\frac{1}{2}(V_1 +\sqrt{\alpha}V_2) &= Y_1\\
&= (R_1+\sqrt{\alpha}R_2)^T X_1(R_1+\sqrt{\alpha}R_2)\\
&= (R_1+\sqrt{\alpha}R_2)^T \frac{1}{2}(U_1 +\sqrt{\alpha}U_2)(R_1+\sqrt{\alpha}R_2),
\end{align*}
where $R_1$ and $R_2$ are over $k$.
In the second case, we see that
\begin{align*}
\frac{1}{2}(V_1 +\sqrt{\alpha}V_2) &= Y_1\\
&= (R_1+\sqrt{\alpha}R_2)^T X_2(R_1+\sqrt{\alpha}R_2)\\
&= (R_1+\sqrt{\alpha}R_2)^T \frac{1}{2}(U_1 -\sqrt{\alpha}U_2)(R_1+\sqrt{\alpha}R_2),
\end{align*}
where $R_1$ and $R_2$ are over $k$.
It follows from this that
$$V_1 = R_1^TU_1R_1 +\alpha R_2^TU_2R_1+\alpha R_1^TU_2R_2+\alpha R_2^TU_1R_2$$
and
$$V_2 = R_2^TU_1R_1+R_1^TU_2R_1+\alpha R_2^TU_2R_2+R_1^TU_1R_2,$$
or
$$V_1 = R_1^TU_1R_1 -\alpha R_2^TU_2R_1-\alpha R_1^TU_2R_2+\alpha R_2^TU_1R_2,$$
and
$$V_2 = R_2^TU_1R_1-R_1^TU_2R_1-\alpha R_2^TU_2R_2+R_1^TU_1R_2.$$
The previous theorem tells us that this means that $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$. Since the converse is clear, then we have shown what was needed.
\end{proof}
\subsection{Type 3 $k$-involutions}
We now examine the Type 3 case. Recall that $\phi$ is a Type 3 $k$-involution if $\phi = \Inn_A$, where $A \in \oo(n,k,\beta)$ and $A^2 = -I$. Such matrices have eigenvalues $\pm i$, where $i$ is a fixed square root of $-1$, and are diagonalizable because the minimal polynomial has no repeated roots. We refer to the Galois automorphism that send $a+bi$ to $a-bi$ for $a, b \in k$ as complex conjugation.
We begin by proving a couple or results about the eigenvectors of such matrices.
\begin{lem}
Suppose $A \in \oo(n,k,\beta) $ induces a Type 3 $k$-involution of $\So(n,k,\beta)$. Also suppose $x,y \in k^n$ such that $x+iy \in E(A,-i)$. Then, $x-iy \in E(A,i)$. Likewise, if $u,v \in k^n$ such that $u+iv \in E(A,i)$, then $u-iv \in E(A,-i)$. Further, $\dim(E(A,i))= \dim(E(A,-i))$.
\end{lem}
\begin{proof}
Suppose $x,y \in k^n$ such that $x+iy \in E(A,-i)$. Then,
$$A(x+iy) = -i(x+iy)$$ implies
$$Ax+iAy = y-ix.$$ If we take the complex conjugate, then we see that
$$Ax-iAy = y+ix.$$ This implies
$$A(x-iy) = i(x-iy),$$ which shows that $x-iy \in E(A,i)$. A similar proof will show that if $u,v \in k^n$ such that $u+iv \in E(A,i)$, then $u-iv \in E(A,-i)$.
Since $x+iy \in E(A,-i)$ implies $x-iy \in E(A,i)$ and vice versa, then we see that $\dim(E(A,i))$ $= \dim(E(A,-i))$.
\end{proof}
\begin{lem}
\label{Type3EigenSo}
Suppose $\theta = \Inn_A$ is a Type 3 $k$-involution of $\So(n,k,\beta)$ where $A \in \oo(n,k,\beta)$. Then, we can find $x_1,...,x_{\frac{n}{2}}, y_1,...,y_{\frac{n}{2}} \in k^n$ such that the $x_j+iy_j$ are a basis for $E(A,-i)$ and the $x_j-iy_j$ are a basis for $E(A,i)$.
\end{lem}
\begin{proof}
Since $\Inn_A$ is Type 3, then we are assuming that $A \in \oo(n,k, \beta)$ and $A^2 = -I$. Note that this also means that $n$ is even. It follows that all eigenvalues of $A$ are $\pm i$. Since there are no repeated roots in the minimal polynomial of $A$, then we see that $A$ is diagonalizable. We begin by constructing bases for $E(A,i)$ and $E(A,-i)$ such that all the basis vectors lie in $k[i]^n$. Let $\{z_1,...,z_n\}$ be a basis for $k^n$. For each $j$, let $u_j = z_j+iAz_j$ Note that
\begin{align*}
Au_j &= A(z_j+iAz_j)\\
&= (A+iA^2)z_j\\
&= (A-iI)z_j\\
&= -i(z_j+iAz_j)\\
&= -iu_j.
\end{align*}
So, $\{u_1,...,u_n\}$ must span $E(A,-i)$. Thus, we can appropriately choose $\frac{n}{2}$ of these vectors and form a basis for $E(A,-i)$. Note that each of these vectors lies in $k[i]^n$. Label these basis vectors as $v_1,...,v_\frac{n}{2}$. We can write each of these vectors as $v_j = x_j+iy_j$. By the previous lemma, we know that $x_j-iy_j \in E(A,i)$. Since these vectors will be linearly independent, then they form a basis for $E(A,i)$.
\end{proof}
We are now able to prove results that characterize the matrices that induce Type 3 $k$-involutions, and then use these characterizations to find conditions on these $k$-involutions that are equivalent to isomorphy. We will have to prove our result by looking at separate cases, depending on whether or not $i$ lies in $k$. We begin by assuming that $i \in k$.
\begin{lem}
\label{Type3ClassYesSo}
Assume $i \in k$ and suppose $\theta = \Inn_A$ is a Type 3 $k$-involution of $\So(n,k,\beta)$, where $A \in \oo(n,k,\beta)$. Then, $A = X \left(\begin{smallmatrix}-iI_{\frac{n}{2}} &0 \\ 0& iI_{\frac{n}{2}}\end{smallmatrix}\right) X^{-1}$ for some $X \in \Gl(n,k),$ where $X^TMX = \left(\begin{smallmatrix}0 & X_1\\ X_1 & 0\end{smallmatrix}\right)$, where $X_1$ is a diagonal matrix.
\end{lem}
\begin{proof}
We know from Lemma \ref{Type3EigenSo} that we have bases for $E(A,-i)$ and $E(A,I)$ that lie in $k^n$. We will show that we can in fact choose bases $a_1,...,a_{\frac{n}{2}}$ for $E(A,-i) \cap k^n$ and $b_1,...,b_{\frac{n}{2}}$ for $E(A,i) \cap k^n$ such that $\beta(a_j,a_l) = 0 = \beta(b_j,b_l)$ and $\beta(a_j, b_l)$ is nonzero if and only if $j=l$. We will build these bases recursively.
First, we know that we can choose some nonzero $a_1 \in E(A,-i) \cap k^n$. Then, since $\beta$ is non degenerate, we can choose a vector $t$ such that $\beta(a_1, t) \ne 0$. We note that $E(A,-i) \oplus E(A,i) = k^n$, so we can choose $t_{-i} \in E(A,-i)\cap k^n$ and $t_i \in E(A,i)\cap k^n$ such that $t = t_{-i}+t_i$. Since $\beta(a_1, t_{-i}) = 0$, then it follows that $\beta(a_1, t_i) \in k$ is nonzero. Let $b_1 = t_i$.
Let $E_1 = \Span_k(a_1,b_1)$ and let $F_1$ be the orthogonal complement of $E_1$ in $k^n$. Since the system of linear equations $$\beta(a_1,x) = 0$$ $$\beta(b_1,x) =0$$ has $n-2$ free variables, then we see that $F_1$ has dimension $n-2$.
We now need to find $a_2 \in F_1 \cap E(A,-i)$. Similar to the construction in the previous lemma, we can choose $x \in F_1$, and let $a_2 = x+iAx$. It follows that $a_2 \in F_1 \cap E(A,-i)$. Now we want $b_2 \in F_2 \cap E(A,i)$ such that $\beta(a_2, b_2) =1$. Since $\beta|_{F_1}$ is non degenerate, then there exists some $y \in F_2$ such that $\beta(a_2,y) \ne 0$. Similar to the construction of $b_1$, we see that this implies the existence a vector $b_2$ that fits our criteria.
Now, we let $E_2 = \Span_k(a_1,a_2,b_1,b_2)$ and let $F_2$ be the orthogonal complement of $E_2$ in $k^n$. We continue this same argument $\frac{n}{2}$ times, until we have the bases that we wanted to find. Let $$X= (a_1,...,a_{\frac{n}{2}}, b_1,...,b_{\frac{n}{2}}).$$ Then, the result follows.
\end{proof}
\begin{theorem}
\label{type3lemYesSo}
Assume that $i \in k$. Then, if $\Inn_A$ and $\Inn_B$ are both Type 3 $k$-involutions of $\So(n,k,\beta)$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$.
\end{theorem}
\begin{proof}
Suppose we have two such $k$-involutions of $\So(n,k,\beta)$. Let them be represented by matrices $A,B \in \oo(n,k,\beta)$. By the previous Lemma, we can choose diagonal $X, Y \in \Gl(n,k)$ such that $$X^{-1}AX = \left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right) = Y^{-1}BY,$$ $$X^TMX = \left(\begin{array}{cc}0 & X_1\\ X_1 & 0\end{array}\right),$$ and $$Y^TMY = \left(\begin{array}{cc}0 & Y_1\\ Y_1 & 0\end{array}\right).$$
Since $X_1$ and $Y_1$ are both invertible diagonal matrices, then we can choose $R_1$ and $R_2 \in \Gl(\frac{n}{2},k)$ such that $Y_1 = R_1^TX_1R_2$. Let $R = \left(\begin{smallmatrix}R_1 & 0 \\0 & R_2\end{smallmatrix}\right)$ and $Q = XRY^{-1}$. It follows from this that $R^TX^TMXR = Y^TMY$. We will show that $Q \in \oo(n,k,\beta)$ and $Q^{-1}AQ = B$. This will then prove that $\Inn_A$ and $\Inn_B$ lie in the same isomorphy class by Lemma \ref{TidyLem}.
First we show that $Q \in \oo(n,k,\beta)$. Note that
\begin{align*}
Q^TMQ &= (XRY^{-1})^TM(XRY^{-1})\\
&= (Y^{-1})^TR^T(X^TMX)RY^{-1}\\
&= (Y^{-1})^T(Y^TMY)Y^{-1}\\
&= M,
\end{align*}
which proves this claim.
Lastly, we show that $Q^{-1}AQ = B$. We first note that $R$ and $\left(\begin{smallmatrix}-iI & 0 \\0 & iI\end{smallmatrix}\right)$ commute. Then, we see that
\begin{align*}
Q^{-1}AQ &= (XRY^{-1})^{-1}A (XRY^{-1})\\
&= YR^{-1}(X^{-1}AX)RY^{-1}\\
&= Y R^{-1}\left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right)R Y^{-1}\\
&= Y R^{-1}R\left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right) Y^{-1}\\
&= Y \left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right) Y^{-1}\\
&= B.
\end{align*}
\end{proof}
We now begin examining the case where $i \not \in k$.
\begin{lem}
\label{Type3ClassNoSo}
Assume $i \not \in k$ and suppose $\theta=\Inn_A$ is a Type 3 $k$-involution of $\So(n,k,\beta)$. Then, $A = U \left(\begin{smallmatrix}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{smallmatrix}\right) U^{-1}$ for $$U = \left(\begin{array}{cccccccccc}a_1 & a_2 & \cdots & a_\frac{n}{2} &b_1 & b_2 & \cdots & b_\frac{n}{2} \end{array}\right)\in \Gl(n,k),$$ where the $a_j+ib_j$ are a basis for $E(A,-i)$, the $a_j-ib_j$ are a basis for $E(A,i)$, and $U^TMU = \left(\begin{smallmatrix}U_1 & 0 \\ 0 & U_1\end{smallmatrix}\right)$ is a diagonal matrix.
\end{lem}
\begin{proof}
We know from Lemma \ref{Type3EigenSo} that we have bases for $E(A,-i)$ and $E(A,I)$ that lie in $k[i]^n$. We will show that we can in fact choose bases $a_1+ib_1,...,a_{\frac{n}{2}}+ib_{\frac{n}{2}}$ for $E(A,-i) \cap k[i]^n$ and $a_1-ib_1,...,a_{\frac{n}{2}}-ib_{\frac{n}{2}}$ for $E(A,i) \cap k[i]^n$ such that $\beta(a_j+ib_j, a_l-ib_l)$ is nonzero if and only if $j=l$. From this, we will be able to show that $\beta(a_j,a_l) = 0 = \beta(b_j,b_l)$ when $j \ne l$ and $\beta(a_j, b_l) = 0$ for all $j$ and $l$. We will build these bases recursively.
Recall that given any vector $x \in k^n$, we know that $x+iAx \in E(A,-i)$. We want to choose $x\in k^n$ such that $\beta(x,x) \ne 0$. (The reasons for this will become apparent.) $M$ is an invertible matrix, so there are at least $n$ instances of $e_j^TMe_l \ne 0$. If there is an instance where $j=l$, let $x=e_j$. If not, then instead we have $e_j^TMe_l = 0 = e_l^TMe_j$, and we let $x = e_j+e_l$. Then,
\begin{align*}
\beta(x,x) &= \beta(e_j+e_l, e_j+e_l)\\
&= 2 \beta(e_j,e_l)\\
&\ne 0.
\end{align*}
So, we have $x \in k^n$ such that $\beta(x,x) \ne 0$, and we have $x+iAx \in E(A,-i)$. Let $a_1 = x$ and $b_1 = Ax$. So, $a_1+ib_1 \in E(A,-i)$ and $a_1-ib_1 \in E(A,i)$. From this, it follows that
\begin{align*}
\beta(a_1+ib_1, a_1-ib_1) &= (\beta(a_1,a_1) +\beta(b_1,b_1))+i(-\beta(a_1,b_1)+\beta(a_1,b_1)\\
&= 2\beta(a_1,a_1) = 2\beta(x,x)\\
&\ne 0.
\end{align*}
Let $E_1 = \Span_{k[i]}(a_1+ib_1, a_1-ib_1) = \Span_{k[i]}(a_1,b_1)$, and let $F_1$ be the orthogonal complement of $E_1$ over $k[i]$. $F_1$ has dimension $n-2$, and $\beta|_{F_1}$ is nondegenerate. So, we can find a nonzero vector $x \in F_1 \cap k^n$ such that $\beta|_{F_1}(x,x) = 0$. So, as in the last case, let $a_2 = x$ and $b_2 = Ax$. As before, we have $\beta(a_1+ib_1, a_1-ib_1) \ne 0$.
Let $E_2 = \Span_{k[i]}(a_1,a_2,b_1,b_2)$, and let $F_2$ be the orthogonal complement of $E_2$ over $k[i]$. In this manner, we can create the bases that we noted in the opening paragraph of this proof.
Note that we always have $$0 = \beta(a_j+ib_j, a_l+ib_l) = (\beta(a_j,a_l)-\beta(b_j,b_l))+i(\beta(a_j,b_l)+\beta(b_j,a_l)),$$ and when $j \ne l$ we have $$0 = \beta(a_j+ib_j, a_l-ib_l) = (\beta(a_j,a_l)+\beta(b_j,b_l))+i(-\beta(a_j,b_l)+\beta(b_j,a_l)).$$
This tells us that when $j \ne l$ that $$\beta(a_j,b_l) = \beta(a_j,a_l) = \beta(b_j,b_l) = 0.$$ When $j = l$, we see that $\beta(b_j,b_j) = \beta(a_j,a_j)$ and that $\beta(a_j, b_j)= -\beta(b_j,a_j)$. The last of these shows that $\beta(a_j,b_l) = 0$, regardless of the values of $j$ and $l$.
Let $$U = (a_1,...,a_{\frac{n}{2}},b_1,...,b_{\frac{n}{2}}).$$ Then, it follows that $U^TMU = \left(\begin{smallmatrix}U_1 & 0 \\0 & U_1\end{smallmatrix}\right)$ where $U_1$ is a diagonal $\frac{n}{2} \times \frac{n}{2}$ matrix.
Lastly, since $b_j = Aa_j$, then it follows that $Ab_j = -a_j$. So, we have that $$A = U\left(\begin{array}{cc}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{array}\right) U^{-1}.$$
\end{proof}
We now look at an example that highlights some of these results that we have just proven in the Type 3 case.
\begin{beisp}
Assume that $\beta$ is the standard dot product. Then, $\theta$ can be a Type 3 $k$-involution of $\So(4,\mathbb{R})$ only if we can choose $A \in \oo(4,\mathbb{R})$ such that $A^2 =-I$. This means the matrix must satisfy $A^T = -A$. That is, the matrix must be skew-symmetric. Observe that the matrix $$A = \left(\begin{array}{cccc}0 & 1 & 0 & 0 \\-1 & 0 & 0 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & -1 & 0\end{array}\right) \in \oo(4, \mathbb{R})$$ is skew-symmetric, so it induces a Type 3 $k$-involution of $\So(4, \mathbb{R})$. It can be shown that $E(A,-i)$ has dimension 2. A basis for this subspace is formed by the vectors $$v_1 = \left(\begin{array}{c}0\\ 0 \\0 \\1\end{array}\right)+ i \left(\begin{array}{c}0 \\ 0 \\1 \\0\end{array}\right)$$ and $$ v_2=\left(\begin{array}{c} 0 \\ 1 \\0 \\0\end{array}\right)+ i\left(\begin{array}{c}1 \\ 0 \\ 0 \\0\end{array}\right).$$
It can be shown that $$v_3 = \left(\begin{array}{c}0\\ 0 \\0 \\1\end{array}\right)- i \left(\begin{array}{c}0 \\ 0 \\1 \\0\end{array}\right)$$ and $$ v_4=\left(\begin{array}{c} 0 \\ 1 \\0 \\0\end{array}\right)- i\left(\begin{array}{c}1 \\ 0 \\0 \\0\end{array}\right)$$ are $i$-eigenvectors of $A$, where these are the conjugates of $v_1$ and $v_2$, respectively.
Following the notation of the previous lemma, we have $$U = \left(\begin{array}{cccc}0 & 0 & 0 & 1 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\1 & 0 & 0 & 0\end{array}\right),$$ where $U^TU = I_4$ and $A = U \left(\begin{array}{cc}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{array}\right) U^{-1}$ . Using the notation of Lemma \ref{Type3ClassNoSo}, we note that $U_1 = I_2$.
\end{beisp}
We now find conditions on Type 3 $k$-involutions that are equivalent to isomorphy, in the case that $i \not \in k$.
\begin{theorem}
\label{type3lemNoSo}
Assume $i \not \in k$. Then, if $\Inn_A$ and $\Inn_B$ are both Type 3 $k$-involutions of $\So(n,k,\beta)$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$.
\end{theorem}
\begin{proof}
By the Lemma \ref{Type3ClassNoSo}, we can choose a matrix $U \in \Gl(n,k)$ such that $$A = U \left(\begin{array}{cc}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{array}\right) U^{-1}
\text{ for }
U = \left(\begin{array}{cccccccccc}a_1 & a_2 & \cdots & a_\frac{n}{2} &b_1 & b_2 & \cdots & b_\frac{n}{2} \end{array}\right)\in \Gl(n,k),$$ where the $a_j+ib_j$ are a basis for $E(A,-i)$, the $a_j-ib_j$ are a basis for $E(A,i)$, and $U^TMU = \left(\begin{smallmatrix}U_1 & 0 \\ 0 & U_1\end{smallmatrix}\right)$ is a diagonal matrix.
Let $$X = (a_1+ib_1,...,a_{\frac{n}{2}}+ib_{\frac{n}{2}}, a_1-ib_1,...,a_{\frac{n}{2}}-ib_{\frac{n}{2}}),$$ and consider $\Inn_A$ and $\Inn_B$ as $k$-involutions of $\So(n,k[i],\beta)$. By construction, we see that $X$ is a matrix that satisfies the conditions of Lemma \ref{Type3ClassYesSo} for the group $\So(n,k[i],\beta)$. We note that $X_1 = 2U_1$. We also know by Theorem \ref{type3lemYesSo} that $\Inn_A$ and $\Inn_B$ are isomorphic (when viewed as $k$-involutions of $\So(n,k[i],\beta)$) over $\oo(n,k[i],\beta)$. So, we can choose $Q_i \in \oo(n,k[i],\beta)$ such that $Q_i^{-1}AQ_i = B$. Let $Y = Q_i^{-1}X$. We now show a couple of facts about $Y$.
First, we note that since $Y$ was obtained from $X$ via row operations, then for $1 \le j \le \frac{n}{2}$, the $j$th and $\frac{n}{2}+j$th columns are $i$-conjugates of one another.
Also, note that
\begin{align*}
Y^{-1}BY &= (Q_i^{-1}X)^{-1}B (Q_i^{-1}X)\\
&= X^{-1}Q_iBQ_i^{-1}X\\
&= X^{-1}AX\\
&= \left(\begin{array}{cc} -iI_{\frac{n}{2}} & 0 \\0& iI_{\frac{n}{2}} \end{array}\right).
\end{align*}
Lastly, we see that
\begin{align*}
Y^TMY &= (Q_i^{-1}X)^TM(Q_i^{-1}X)\\
& = X^T((Q_i^{-1})^TMQ_i)X\\
&= X^TMX\\
&= \left(\begin{array}{cc} 0 & X_1 \\ X_1 & 0 \end{array}\right)\\
&= \left(\begin{array}{cc} 0 & 2U_1 \\ 2U_1 & 0 \end{array}\right).
\end{align*}
We can write $$Y = (c_1+id_1,...,c_{\frac{n}{2}}+id_{\frac{n}{2}}, c_1-id_1,...,c_{\frac{n}{2}}-id_{\frac{n}{2}})$$ where $c_j, d_j \in k^n$. So, let $$V = (c_1,...,c_{\frac{n}{2}},d_1,...,d_{\frac{n}{2}})\in \Gl(n,k).$$ It follows from what we have shown that $B = V \left(\begin{smallmatrix}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{smallmatrix}\right) V^{-1}$ where $$V^TMV = \left(\begin{array}{cc}U_1 & 0 \\ 0 & U_1\end{array}\right) = U^TMU.$$
Now, let $Q = UV^{-1}$. We will show that $Q^{-1}AQ = B$ and $Q \in \oo(n,k,\beta)$. This will prove that $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$ by Lemma \ref{TidyLem}.
We first show that $Q \in \oo(n,k,\beta)$.
\begin{align*}
Q^TMQ &= (UV^{-1})^TMUV^{-1}\\
&= (V^{-1})^T(U^TMU)V^{-1}\\
&= (V^{-1})^T(V^TMV)V^{-1}\\
&= M.
\end{align*}
Lastly, we show that $Q^{-1}AQ = B$.
\begin{align*}
Q^{-1}AQ &= (UV^{-1})^{-1}A(UV^{-1})\\
&= VU^{-1}AUV^{-1}\\
&= V\left(\begin{array}{cc}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{array}\right) V^{-1}\\
&= B.
\end{align*}
\end{proof}
Combining the results from this section, we get the following corollary.
\begin{cor}
\label{CorType3So}
If $\Inn_A$ and $\Inn_B$ are both Type 3 $k$-involutions of $\So(n,k,\beta)$ where $A, B \in \oo(n,k \beta)$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$. That is, $\So(n,k,\beta)$ has at most one isomorphy class of Type 3 $k$-involutions.
\end{cor}
If $\Inn_A$ and $\Inn_B$ are both Type 3 $k$-involutions of $\So(n,k,\beta)$ where $A, B \in \oo(n,k \beta)$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$. That is, $\So(n,k,\beta)$ has at most one isomorphy class of Type 3 $k$-involutions.
\subsection{Type 4 $k$-involutions}
We now move on to a similar characterization in the Type 4 case. First, we characterize the eigenvectors of the matrices that induce these $k$-involutions. Recall that we can choose $A \in \oo(n,k[\sqrt{\alpha}],\beta)$ such that each entry of $A$ is a $k$-multiple of $\sqrt{\alpha}$, and that we know $A^2 = -I$. We begin by proving a couple of lemmas about the eigenspaces of these matrices.
\begin{lem}
Suppose $A \in \oo(n,k[\sqrt{\alpha}],\beta) $ induces a Type 4 $k$-involution of $\So(n,k,\beta)$. Also suppose $x,y \in k^n$ such that $x+\sqrt{-\alpha}y \in E(A,-i)$. Then, $x-\sqrt{-\alpha}y \in E(A,i)$. Likewise, if $u,v \in k^n$ such that $u+\sqrt{-\alpha}v \in E(A,i)$. Then, $u-\sqrt{-\alpha}v \in E(A,-i)$. Further, $\dim(E(A,i))= \dim(E(A,-i))$.
\end{lem}
\begin{proof}
Suppose $x,y \in k^n$ such that $x+\sqrt{-\alpha}y \in E(A,-i)$. Then,
$$A(x+\sqrt{-\alpha}y) = -i(x+\sqrt{-\alpha}y)$$ which implies
$$Ax+\sqrt{-\alpha}Ay = \sqrt{\alpha}y-ix.$$
Then, complex conjugation tells us that
$$Ax-\sqrt{-\alpha}Ay = \sqrt{\alpha}y+ix,$$ which tells us that
$$A(x-\sqrt{-\alpha}y) = i(x-\sqrt{-\alpha}y).$$ A similar argument shows that if $u,v \in k^n$ such that $u+\sqrt{-\alpha}v \in E(A,i)$. Then, $u-\sqrt{-\alpha}v \in E(A,-i)$.
Since $x+\sqrt{-\alpha}y \in E(A,-i)$ implies $x-\sqrt{-\alpha}y \in E(A,i)$ and vice versa, then we see that $\dim(E(A,i))= \dim(E(A,-i))$.
\end{proof}
\begin{lem}
\label{Type4EigenSo}
Suppose $\theta = \Inn_A$ is a Type 4 $k$-involution of $\So(n,k,\beta)$ where $A \in $ \newline$\oo(n,k[\sqrt{\alpha}],\beta)$. Then, we can find $x_1,...,x_{\frac{n}{2}}, y_1,...,y_{\frac{n}{2}} \in k^n$ such that the $x+\sqrt{-\alpha}y$ are a basis for $E(A,-i)$ and the $x-\sqrt{-\alpha}y$ are a basis for $E(A,i)$.
\end{lem}
\begin{proof}
Since $\Inn_A$ is Type 4, then we are assuming that $A \in \oo(n,k[\sqrt{\alpha}], \beta)$ and $A^2 = -I$. Note that this also means that $n$ is even. It follows that all eigenvalues of $A$ are $\pm i$. Since there are no repeated roots in the minimal polynomial of $A$, then we see that $A$ is diagonalizable. We begin by constructing bases for $E(A,i)$ and $E(A,-i)$ such that all the basis vectors lie in $k[i]^n$. Let $\{z_1,...,z_n\}$ be a basis for $k^n$. For each $j$, let $u_j = (\sqrt{\alpha}A-\sqrt{-\alpha}I)z_j.$ Note that
\begin{align*}
Au_j &= A(\sqrt{\alpha}A-\sqrt{-\alpha}I)z_j\\
&= (\sqrt{\alpha}A^2-\sqrt{-\alpha}A)z_j\\
&= -i(\sqrt{\alpha}A-\sqrt{-\alpha}I)z_j\\
&= -iu_j.
\end{align*}
So, $\{u_1,...,u_n\}$ must span $E(A,-i)$. Thus, we can appropriately choose $\frac{n}{2}$ of these vectors and form a basis for $E(A,-i)$. Note that each of these vectors lies in $k[i]^n$. Label these basis vectors as $v_1,...,v_\frac{n}{2}$. We can write each of these vectors as $v_j = x_j+\sqrt{-\alpha}y_j$. By the previous lemma, we know that $x_j-\sqrt{-\alpha}y_j \in E(A,i)$, and it follows that these will be linearly independent. Since there are $\frac{n}{2}$ of them, then they form a basis for $E(A,i)$.
\end{proof}
We are now able to prove results that characterize the matrices that induce Type 4 $k$-involutions, and then use these characterizations to find conditions on these $k$-involutions that are equivalent to isomorphy. We will have separate cases, depending on whether or not $\sqrt{-\alpha}$ lies in $k$. We begin by assuming that $\sqrt{-\alpha} \in k$. Since we are also assuming that $\sqrt{\alpha} \not \in k$, then it follows from these two assumptions that $\alpha$ and $-1$ lie in the same square class of $k$. Thus, we can assume in this case that $\alpha = -1$, which means $\sqrt{-\alpha} = 1$.
\begin{lem}
\label{Type4ClassYesSo}
Assume $\sqrt{-\alpha} \in k$ and suppose $\theta=\Inn_A$ is a Type 4 $k$-involution of $\So(n,k,\beta)$. Then, $A = X \left(\begin{smallmatrix}-iI_{\frac{n}{2}} &0 \\ 0& iI_{\frac{n}{2}}\end{smallmatrix}\right) X^{-1}$ for some $X \in \Gl(n,k),$ where $X^TMX = \left(\begin{smallmatrix}0 & X_1 \\ X_1 & 0\end{smallmatrix}\right)$, and $X_1$ is diagonal.
\end{lem}
\begin{proof}
We know from Lemma \ref{Type4EigenSo} that we have bases for $E(A,-i)$ and $E(A,I)$ that lie in $k^n$. We will show that we can in fact choose bases $a_1,...,a_{\frac{n}{2}}$ for $E(A,-i) \cap k^n$ and $b_1,...,b_{\frac{n}{2}}$ for $E(A,i) \cap k^n$ such that $\beta(a_j,a_l) = 0 = \beta(b_j,b_l)$ and $\beta(a_j, b_l)$ is nonzero if and only if $j = l$. We will build these bases recursively.
First, we know that we can choose some nonzero $a_1 \in E(A,-i) \cap k^n$. Then, since $\beta$ is non degenerate, we can choose a vector $t$ such that $\beta(a_1, t) \ne 0$. We note that $E(A,-i) \oplus E(A,i) = k^n$, so we can choose $t_{-i} \in E(A,-i)\cap k^n$ and $t_i \in E(A,i)\cap k^n$ such that $t = t_{-i}+t_i$. Since $\beta(a_1, t_{-i}) = 0$, then it follows that $\beta(a_1, t_i) \in k$ is nonzero. Let $b_1 = t_i$.
Let $E_1 = \Span_k(a_1,b_1)$ and let $F_1$ be the orthogonal complement of $E_1$ in $k^n$. Since the system of linear equations $$\beta(a_1,x) = 0$$ $$\beta(b_1,x) =0$$ has $n-2$ free variables, then we see that $F_1$ has dimension $n-2$.
We now want to find $a_2 \in F_1 \cap E(A,-i)$. Similar to the construction in the previous lemma, we can choose $x \in F_1$, and let $a_2 = (\sqrt{\alpha}A-\sqrt{-\alpha}I)x$. It follows that $a_2 \in F_1 \cap E(A,-i)$. Now we want $b_2 \in F_2 \cap E(A,i)$ such that $\beta(a_2, b_2)$ is nonzero. Since $\beta|_{F_1}$ is non degenerate, then there exists some $y \in F_2$ such that $\beta(a_2,y) \ne 0$. Similar to the construction of $b_1$, we see that this implies the existence a vector $b_2$ that fits our criteria.
Now, we let $E_2 = \Span_k(a_1,a_2,b_1,b_2)$ and let $F_2$ be the orthogonal complement of $E_2$ in $k^n$. We continue this same argument $\frac{n}{2}$ times, until we have the bases that we wanted to find. Let $$X= (a_1,...,a_{\frac{n}{2}}, b_1,...,b_{\frac{n}{2}}).$$ Then, the result follows.
\end{proof}
Here is an example of a Type 4 $k$-involution when $\sqrt{-\alpha} \in k$.
\begin{beisp}
Assume that $\beta$ is the standard dot product and that $k = \mathbb{F}_3$, the field of three elements. So, the square roots of 2 are $\pm i$. Observe that the matrix $$A =i\left(\begin{array}{cccc}0 & 0 & 1 & 1 \\0 & 0 & 1 & -1 \\2 & 2 & 0 & 0 \\2 & 1 & 0 & 0\end{array}\right) \in \oo(4, \mathbb{F}_3[i])$$ satisfies the relation $A^2 = -I_4$. Since each entry of $A$ is a $\mathbb{F}_3$-multiple of $i$, then it follows that $\Inn_A$ is an $k$-involution of $\So(4,\mathbb{F}_3)$ of Type 4. A basis for $E(A,-i)$ is formed by the vectors $$v_1 = \left(\begin{array}{c}0\\ 0 \\0 \\1\end{array}\right)+ \left(\begin{array}{c} 1 \\ 2\\ 0 \\ 0\end{array}\right) = \left(\begin{array}{c} 1 \\ 2\\ 0 \\ 1\end{array}\right)$$ and $$ v_2=\left(\begin{array}{c} 0 \\ 0 \\1 \\0\end{array}\right)+ \left(\begin{array}{c} 1\\ 1\\0 \\ 0 \end{array}\right) = \left(\begin{array}{c} 1 \\ 1\\ 1 \\ 0\end{array}\right).$$
It can be shown that $$v_3 = \left(\begin{array}{c}0\\ 0 \\0 \\1\end{array}\right)- \left(\begin{array}{c} 1 \\ 2\\ 0 \\ 0\end{array}\right) = \left(\begin{array}{c} 2 \\ 1\\ 0 \\ 1\end{array}\right)$$ and $$ v_2=\left(\begin{array}{c} 0 \\ 0 \\1 \\0\end{array}\right)- \left(\begin{array}{c} 1\\ 1\\0 \\ 0 \end{array}\right) = \left(\begin{array}{c} 2 \\ 2\\ 1 \\ 0\end{array}\right)$$ are $i$-eigenvectors of $A$.
Following the notation of the previous lemma, we have $$X = \left(\begin{array}{cccc}0 & 0 & 1 & 1 \\0 & 0 & 2 &1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0\end{array}\right),$$ where $X^TX = \left(\begin{smallmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 2 & 0 \\0 & 0 & 0 & 2\end{smallmatrix}\right)$ and $A = -iX \left(\begin{smallmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\-i & 0 & 0 & 0 \\0 & -i & 0 & 0\end{smallmatrix}\right) X^{-1}$. We also note that $X_1 = I$.
\end{beisp}
Now we characterize the isomorphy classes of Type 4 $k$-involutions in the case where $\sqrt{-\alpha} \in k$.
\begin{theorem}
\label{type4lemYesSo}
Assume that $\sqrt{-\alpha} \in k$. Then, if $\Inn_A$ and $\Inn_B$ are both Type 4 $k$-involutions of $\So(n,k,\beta)$ where the entries of $A$ and $B$ are $k$-multiples of $\sqrt{\alpha}$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$.
\end{theorem}
\begin{proof}
Suppose we have two such $k$-involutions of $\So(n,k,\beta)$. Let them be represented by matrices $A,B \in \oo(n,k,\beta)$. By Lemma \ref{Type4ClassYesSo}, we can choose $X, Y \in \Gl(n,k)$ such that $$X^{-1}AX = \left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right) = Y^{-1}BY,$$ $$X^TMX = \left(\begin{array}{cc}0 & X_1\\ X_1 & 0\end{array}\right),$$ and $$Y^TMY = \left(\begin{array}{cc}0 & Y_1\\ Y_1 & 0\end{array}\right),$$ where $X_1$ and $Y_1$ are diagonal.
Since $X_1$ and $Y_1$ are both invertible diagonal matrices, then we can choose $R_1$ and $R_2 \in \Gl(\frac{n}{2},k)$ such that $Y_1 = R_1^TX_1R_2$. Let $R = \left(\begin{smallmatrix}R_1 & 0 \\0 & R_2\end{smallmatrix}\right)$ and $Q = XRY^{-1}$. It follows from this that $R^TX^TMXR = Y^TMY$. We will show that $Q \in \oo(n,k,\beta)$ and $Q^{-1}AQ = B$. This will then prove that $\Inn_A$ and $\Inn_B$ lie in the same isomorphy class by Lemma \ref{TidyLem}.
First we show that $Q \in \oo(n,k,\beta)$. By construction, the entries of $Q$ lie in $k$. Also, note that
\begin{align*}
Q^TMQ &= (XRY^{-1})^TM(XRY^{-1})\\
&= (Y^{-1})^TR^T(X^TMX)RY^{-1}\\
&= (Y^{-1})^T(Y^TMY)Y^{-1}\\
&= M,
\end{align*}
which proves $Q \in \oo(n,k,\beta)$.
Lastly, we show that $Q^{-1}AQ = B$. We first note that $R$ and $\left(\begin{smallmatrix}-iI & 0 \\0 & iI\end{smallmatrix}\right)$ commute. Then, we see that
\begin{align*}
Q^{-1}AQ &= (XRY^{-1})^{-1}A (XRY^{-1})\\
&= YR^{-1}(X^{-1}AX)RY^{-1}\\
&= Y R^{-1}\left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right)R Y^{-1}\\
&= Y R^{-1}R\left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right) Y^{-1}\\
&= Y \left(\begin{array}{cc}-iI & 0 \\0 & iI\end{array}\right) Y^{-1}\\
&= B.
\end{align*}
\end{proof}
We now examine the case where $\sqrt{-\alpha} \not \in k$.
\begin{lem}
\label{Type4ClassNoSo}
Assume $\sqrt{-\alpha} \not \in k$ and suppose $\theta=\Inn_A$ is a Type 4 $k$-involution of $\So(n,k,\beta)$. Then, $A = -\frac{\sqrt{\alpha}}{\alpha} U \left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ -\alpha I_{\frac{n}{2}} & 0\end{smallmatrix}\right) U^{-1}$ for $$U = \left(\begin{array}{cccccccccc}a_1 & a_2 & \cdots & a_\frac{n}{2} &b_1 & b_2 & \cdots & b_\frac{n}{2} \end{array}\right)\in \Gl(n,k),$$ where the $a_j+\sqrt{-\alpha}b_j$ are a basis for $E(A,-i)$, the $a_j-\sqrt{-\alpha}b_j$ are a basis for $E(A,i)$, and $U^TMU = \left(\begin{smallmatrix}U_1 & 0 \\0 & \frac{1}{\alpha}U_1\end{smallmatrix}\right)$ is diagonal.
\end{lem}
\begin{proof}
We know from Lemma \ref{Type4EigenSo} that we have bases for $E(A,-i)$ and $E(A,I)$ that lie in $k[\sqrt{-\alpha}]^n$. We will show that we can in fact choose bases $a_1+\sqrt{-\alpha}b_1,...,a_{\frac{n}{2}}+\sqrt{-\alpha}b_{\frac{n}{2}}$ for $E(A,-i) \cap k[i]^n$ and $a_1-\sqrt{-\alpha}b_1,...,a_{\frac{n}{2}}-\sqrt{-\alpha}b_{\frac{n}{2}}$ for $E(A,i) \cap k[\sqrt{-\alpha}]^n$ such that $\beta(a_j+\sqrt{-\alpha}b_j, a_l-\sqrt{-\alpha}b_l)$ is nonzero if and only if $j=l$. From this, we will be able to show that $\beta(a_j,a_l) = 0 = \beta(b_j,b_l)$ when $j \ne l$ and $\beta(a_j, b_l) = 0$ for all $j$ and $l$. We will build these bases recursively.
Given any vector $x \in k^n$, we know that $x+iAx \in E(A,-i)$. We want to choose $x\in k^n$ such that $\beta(x,x) \ne 0$. (The reasons for this will become apparent.) $M$ is an invertible matrix, so there are at least $n$ instances of $e_j^TMe_l \ne 0$. If there is an instance where $j=l$, let $x=e_j$. If instead we have $e_j^TMe_l = 0 = e_l^TMe_j$, then let $x = e_j+e_l$. We note that this works because $$\beta(x,x) = \beta(e_j+e_l, e_j+e_l) = 2 \beta(e_j,e_l) \ne 0.$$
So, we have $x \in k^n$ such that $\beta(x,x) \ne 0$, and we have $x+iAx \in E(A,-i)$. Let $a_1 = x$ and $b_1 = \frac{1}{\sqrt{\alpha}}Ax$. So, $a_1+\sqrt{-\alpha}b_1 \in E(A,-i)$ and $a_1-\sqrt{-\alpha}b_1 \in E(A,i)$. From this, it follows that
\begin{align*}
\beta(a_1+\sqrt{-\alpha}b_1, a_1-\sqrt{-\alpha}b_1) &= (\beta(a_1,a_1) +\alpha \beta(b_1,b_1))+\sqrt{-\alpha}(-\beta(a_1,b_1)+\beta(a_1,b_1)\\
&= \beta(x,x) +\alpha \beta\left( \frac{1}{\sqrt{\alpha}}Ax,\frac{1}{\sqrt{\alpha}}Ax\right)\\
&= 2\beta(x,x)\\
&\ne 0.
\end{align*}
Let $E_1 = \Span_{k[\sqrt{-\alpha}]}(a_1+\sqrt{-\alpha}b_1, a_1-\sqrt{-\alpha}b_1) = \Span_{k[\sqrt{-\alpha}]}(a_1,b_1)$, and let $F_1$ be the orthogonal complement of $E_1$ over $k[\sqrt{-\alpha}]$. $F_1$ has dimension $n-2$, and $\beta|_{F_1}$ is nondegenerate. So, we can find a nonzero vector $x \in F_1 \cap k^n$ such that $\beta|_{F_1}(x,x) = 0$. So, as in the last case, let $a_2 = x$ and $b_2 = \frac{1}{\sqrt{\alpha}}Ax$. As before, we have $\beta(a_2+\sqrt{-\alpha}b_2, a_2-\sqrt{-\alpha}b_2) \ne 0$.
Let $E_2 = \Span_{k[\sqrt{-\alpha}]}(a_1,a_2,b_1,b_2)$, and let $F_2$ be the orthogonal complement of $E_2$ over $k[\sqrt{-\alpha}]$. In this manner, we can create the bases that we noted in the opening paragraph of this proof.
Note that we always have $$0 = \beta(a_j+\sqrt{-\alpha}b_j, a_l+\sqrt{-\alpha}b_l) = (\beta(a_j,a_l)-\alpha \beta(b_j,b_l))+\sqrt{-\alpha}(\beta(a_j,b_l)+\beta(b_j,a_l)),$$ and when $j \ne l$ we have $$0 = \beta(a_j+\sqrt{-\alpha}b_j, a_l-\sqrt{-\alpha}b_l) = (\beta(a_j,a_l)+\alpha\beta(b_j,b_l))+\sqrt{-\alpha}(-\beta(a_j,b_l)+\beta(b_j,a_l)).$$
This tells us that when $j \ne l$ that $$\beta(a_j,b_l) = \beta(a_j,a_l) = \beta(b_j,b_l) = 0.$$ When $j = l$, we see that $\beta(b_j,b_j) = \frac{1}{\alpha}\beta(a_j,a_j)$ and that $\beta(a_j, b_j)= -\beta(b_j,a_j)$. The last of these shows that $\beta(a_j,b_l) = 0$, regardless of the values of $j$ and $l$.
Let $$U = (a_1,...,a_{\frac{n}{2}},b_1,...,b_{\frac{n}{2}}).$$ Then, it follows that $U^TMU = \left(\begin{smallmatrix}U_1 & 0 \\0 & \frac{1}{\alpha}U_1\end{smallmatrix}\right)$ where $U_1$ is a diagonal $\frac{n}{2} \times \frac{n}{2}$ matrix.
Lastly, since $b_j = \frac{1}{\sqrt{\alpha}}Aa_j$, then it follows that $Ab_j = -\frac{1}{\sqrt{\alpha}}a_j$. So, we have that $A = -\frac{\sqrt{\alpha}}{\alpha}U \left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ -\alpha I_{\frac{n}{2}} & 0\end{smallmatrix}\right) U^{-1}$.
\end{proof}
Here is an example of a Type 4 $k$-involution in the case that $\sqrt{-\alpha} \not \in k$.
\begin{beisp}
Assume that $\beta$ is the standard dot product. Observe that the matrix $$A = \frac{\sqrt{2}}{2}\left(\begin{array}{cccc}0 & 0 & 1 & 1 \\0 & 0 & 1 & -1 \\-1 & -1 & 0 & 0 \\-1 & 1 & 0 & 0\end{array}\right) \in \oo(4, \mathbb{Q}[\sqrt{2}])$$ is such that $A^2 = -I_4$. Since each entry of $A$ is a $\mathbb{Q}$-multiple of $\sqrt{2}$, then it follows that $\Inn_A$ is an $k$-involution of $\So(4,\mathbb{Q})$ of Type 4. It can be shown that $E(A,-i)$ has dimension 2. A basis for this subspace is formed by the vectors $$v_1 = \left(\begin{array}{c}0\\ 0 \\0 \\1\end{array}\right)+ \sqrt{-2} \left(\begin{array}{c} -\frac{1}{2} \\ \frac{1}{2}\\ 0 \\ 0\end{array}\right)$$ and $$ v_2=\left(\begin{array}{c} 0 \\ 0 \\1 \\0\end{array}\right)+ \sqrt{-2} \left(\begin{array}{c} -\frac{1}{2} \\ -\frac{1}{2}\\0 \\ 0 \end{array}\right).$$
It can be shown that$$v_3 = \left(\begin{array}{c}0\\ 0 \\0 \\1\end{array}\right)- \sqrt{-2} \left(\begin{array}{c} -\frac{1}{2} \\ \frac{1}{2}\\ 0 \\ 0\end{array}\right)$$ and $$ v_4=\left(\begin{array}{c} 0 \\ 0 \\1 \\0\end{array}\right)- \sqrt{-2} \left(\begin{array}{c} -\frac{1}{2} \\ -\frac{1}{2}\\0 \\ 0 \end{array}\right)$$ are $i$-eigenvectors of $A$, where these are the conjugates of $v_1$ and $v_2$, respectively.
Following the notation of the previous lemma, we have $$U = \left(\begin{array}{cccc}0 & 0 & -\frac{1}{2} & -\frac{1}{2} \\0 & 0 & \frac{1}{2} &- \frac{1}{2} \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0\end{array}\right),$$ where $U^TU = \left(\begin{smallmatrix}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & \frac{1}{2} & 0 \\0 & 0 & 0 & \frac{1}{2}\end{smallmatrix}\right)$ and $A = -\frac{\sqrt{2}}{2}U \left(\begin{smallmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\-\sqrt{2} & 0 & 0 & 0 \\0 & -\sqrt{2} & 0 & 0\end{smallmatrix}\right) U^{-1}$ . We also note that $U_1 = I$.
\end{beisp}
We now find conditions on Type 4 $k$-involutions that are equivalent to isomorphy in the case where $\sqrt{-\alpha} \not \in k$.
\begin{theorem}
\label{type4lemNoSo}
Assume $\sqrt{-\alpha} \not \in k$. Then, if $\Inn_A$ and $\Inn_B$ are both Type 4 $k$-involutions of $\So(n,k,\beta)$ where $A, B \in \oo(n,k[\sqrt{\alpha}], \beta)$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$.
\end{theorem}
\begin{proof}
By Lemma \ref{Type4ClassNoSo}, we can choose a matrix $U \in \Gl(n,k)$ such that $$A = -\frac{\sqrt{\alpha}}{\alpha} U \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ -\alpha I_{\frac{n}{2}} & 0\end{array}\right) U^{-1}$$ for $$U = \left(\begin{array}{cccccccccc}a_1 & a_2 & \cdots & a_\frac{n}{2} &b_1 & b_2 & \cdots & b_\frac{n}{2} \end{array}\right),$$ where the $a_j+\sqrt{-\alpha}b_j$ are a basis for $E(A,-i)$, the $a_j-\sqrt{-\alpha}b_j$ are a basis for $E(A,i)$, and $U^TMU = \left(\begin{smallmatrix}U_1 & 0 \\0 & \frac{1}{\alpha}U_1\end{smallmatrix}\right)$ is diagonal.
Consider $\Inn_A$ and $\Inn_B$ as $k$-involutions of $\So(n,k[\sqrt{-\alpha}],\beta)$. If $k[\sqrt{-\alpha}] = k[\sqrt{\alpha}]$, then these are Type 3 $k$-involutions of $\So(n,k[\sqrt{-\alpha}],\beta)$, since $A$ and $B$ would have entries in the field, and $i \in k[\sqrt{-\alpha}]$. Otherwise, if $k[\sqrt{-\alpha}] \ne k[\sqrt{\alpha}]$, then these are Type 4 $k$-involutions where $ \sqrt{-\alpha} \in k[\sqrt{-\alpha}]$.
Let
$$X = (a_1+\sqrt{-\alpha}b_1,...,a_{\frac{n}{2}}+\sqrt{-\alpha}b_{\frac{n}{2}}, a_1-\sqrt{-\alpha}b_1,...,a_{\frac{n}{2}}-\sqrt{-\alpha}b_{\frac{n}{2}}).$$
By construction, we see that $X$ is a matrix that satisfies the conditions of Lemma \ref{Type3ClassNoSo} or Lemma \ref{Type4ClassYesSo} for the group $\So(n,k[\sqrt{\alpha}],\beta)$. We note that $X_1 = 2U_1$. We also know by Corollary \ref{CorType3So} or Theorem \ref{type4lemYesSo} that $\Inn_A$ and $\Inn_B$ are isomorphic (when viewed as $k$-involutions of $\So(n,k[\sqrt{-\alpha}],\beta)$) over $\oo(n,k[\sqrt{-\alpha}],\beta)$. So, we can choose $Q_{\alpha} \in \oo(n,k[\sqrt{-\alpha}],\beta)$ such that $Q_{\alpha}^{-1}AQ_{\alpha} = B$. Let $Y = Q_{\alpha}^{-1}X$. Since $Y$ is constructed by doing row operations on $X$, then we can write
$$Y = (c_1+\sqrt{-\alpha}d_1,...,c_{\frac{n}{2}}+\sqrt{-\alpha}d_{\frac{n}{2}}, c_1-\sqrt{-\alpha}d_1,...,c_{\frac{n}{2}}-\sqrt{-\alpha}c_{\frac{n}{2}}),$$
where $c_j, d_j \in k^n$. We now show a couple of facts about $Y$.
First, we note that since $Y$ was obtained from $X$ via row operations, then for $1 \le j \le \frac{n}{2}$, the $j$th and $\frac{n}{2}+j$th columns are $i$-conjugates of one another.
Next, we observe that
\begin{align*}
Y^{-1}BY &= (Q_{\alpha}^{-1}X)^{-1}B (Q_{\alpha}^{-1}X)\\
&= X^{-1}Q_{\alpha}BQ_{\alpha}^{-1}X \\
&= X^{-1}AX\\
&= \left(\begin{array}{cc} -iI_{\frac{n}{2}} & 0 \\0& iI_{\frac{n}{2}} \end{array}\right).
\end{align*}
Lastly, we see that
\begin{align*}
Y^TMY &= (Q_{\alpha}^{-1}X)^TM(Q_{\alpha}^{-1}X)\\
&= X^T((Q_{\alpha}^{-1})^TMQ_{\alpha})X\\
&= X^TMX\\
&= \left(\begin{array}{cc} 0 & X_1 \\ X_1 & 0 \end{array}\right)\\
&= \left(\begin{array}{cc} 0 & 2U_1 \\ 2U_1 & 0 \end{array}\right).
\end{align*}
Let $$V = (c_1,...,c_{\frac{n}{2}},d_1,...,d_{\frac{n}{2}}) \in \Gl(n,k).$$ It follows from what we have shown that $B = -\frac{\sqrt{\alpha}}{\alpha}V \left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ -\alpha I_{\frac{n}{2}} & 0\end{smallmatrix}\right) V^{-1}$ where $V^TMV = \left(\begin{smallmatrix}U_1 & 0 \\ 0 & \frac{1}{\alpha}U_1\end{smallmatrix}\right) = U^TMU$.
Now, let $Q = UV^{-1}$. We will show that $Q^{-1}AQ = B$ and $Q \in \oo(n,k,\beta)$. This will prove that $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$ by Lemma \ref{TidyLem}.
We first show that $Q \in \oo(n,k,\beta)$.
\begin{align*}
Q^TMQ &= (UV^{-1})^TMUV^{-1}\\
&= (V^{-1})^T(U^TMU)V^{-1}\\&
= (V^{-1})^T(V^TMV)V^{-1}\\
&= M.
\end{align*}
Lastly, we show that $Q^{-1}AQ = B$.
\begin{align*}
Q^{-1}AQ &= (UV^{-1})^{-1}A(UV^{-1})\\
&= VU^{-1}AUV^{-1}\\
&=-\frac{\sqrt{\alpha}}{\alpha}V \left(\begin{array}{cc}0 & I_{\frac{n}{2}} \\ -\alpha I_{\frac{n}{2}} & 0\end{array}\right) V^{-1}\\
&= B.
\end{align*}
\end{proof}
Combining the results from this section, we get the following corollary.
\begin{cor}
\label{CorType4So}
If $\Inn_A$ and $\Inn_B$ are both Type 4 $k$-involutions of $\So(n,k,\beta)$ where $A, B \in \oo(n,k[\sqrt{\alpha}], \beta)$, then $\Inn_A$ and $\Inn_B$ are isomorphic over $\oo(n,k,\beta)$. That is, $\So(n,k,\beta)$ has at most $|k^*/(k^*)^2|-1$ isomorphy classes of Type 4 $k$-involutions.
\end{cor}
\section{Maximal Number of Isomorphy classes}
From the work we have done, it follows that the maximum number of isomorphy classes of $k$-involutions of $\So(n,k,\beta)$ over $\oo(n,k,\beta)$ is a function of the number of square classes of $k$, and the number of congruency classes of invertible diagonal matrices over $k$. We first define the following formulas.
\begin{definit}
Let $\tau_1(k) = |k^*/(k^*)^2|-1$ and $\tau_2(m,k)$ be the number of congruency classes of invertible symmetric matrices of $\Gl(m,k)$ over $\Gl(m,k)$.
Let $C_1(n,k,\beta)$, $C_2(n,k,\beta)$, $C_3(n,k,\beta)$ and $C_4(n,k,\beta)$ be the number of isomorphy classes of $\So(n,k,\beta)$ $k$-involutions over $\oo(n,k,\beta)$ of types 1, 2, 3, and 4, respectively.
\end{definit}
From our previous work, we have the following:
\begin{cor}
\begin{enumerate}
\item If $n$ is odd, then $$C_1(n,k,\beta) \le \left( \sum_{m=1}^{\frac{n-1}{2}} \tau_2(n-m,k)\tau_2(m,k) \right).$$ If $n$ is even, then
\begin{align*}
C_1(n,k,\beta) &\le \left( \sum_{m=1}^{\frac{n}{2}-1} \tau_2(n-m,k)\tau_2(m,k) \right)+\left(\begin{array}{c}\tau_2(\frac{n}{2},k) \\2\end{array}\right)+\tau_2 \left(\frac{n}{2},k\right)\\
&= \left( \sum_{m=1}^{\frac{n}{2}-1} \tau_2(n-m,k)\tau_2(m,k) \right)+\frac{\tau(\frac{n}{2},k)(\tau(\frac{n}{2},k)+1)}{2}.\\
\end{align*}
\item If $n$ is even, then
\begin{align*}
C_2(n,k,\beta) &\le \tau_1(k)\left(\left(\begin{array}{c}\tau_2(\frac{n}{2},k) \\2\end{array}\right)+\tau_2 \left(\frac{n}{2},k\right)\right)\\
&=\tau_1(k)\left( \frac{\tau(\frac{n}{2},k)(\tau(\frac{n}{2},k)+1)}{2}\right) .\\
\end{align*}
\item If $n$ is even, then $$C_3(n,k,\beta) \le 1 .$$
\item If $n$ is even, then $$C_4(n,k,\beta) \le \tau_1(k) .$$
\item If $n$ is odd, then $C_2(n,k,\beta) = C_3(n,k,\beta) = C_4(n,k,\beta) = 0$.
\end{enumerate}
\end{cor}
We now list values of $\tau_1$ and $\tau_2$ for a few classes of fields.
\begin{table}[h]
\centering
\caption { Some values of $\tau_1(k)$ } \label{tau1}
\begin{tabular}[t]{|c||c|c|c|c|c|}
\hline k & $\overline{k}$ &$\mathbb{R}$ &$ \mathbb{F}_q$, $2 \not | q$ &$ \mathbb{Q}_p$, $p \ne 2$ & $ \mathbb{Q}_2$ \\
\hline $\tau_1(k)$ & 0 & 1 & 1 & 3 & 7 \\
\hline
\end{tabular}
\end{table}
\begin{table}[h]
\centering
\caption { Some values of $\tau_2(m,k)$ } \label{tau2}
\begin{tabular}[t]{|c||c|c|c|c|c|}
\hline k & $\overline{k}$ &$\mathbb{R}$ &$ \mathbb{F}_q$, $2 \not | q$ \\
\hline $\tau_2(m,k)$ & 1 & m+1 & $2$ \\
\hline
\end{tabular}
\end{table}
For the $\mathbb{Q}_p$, $\tau_2$ is a bit more difficult. Here we have
$$\tau_2(m,\mathbb{Q}_p) = \left\{\begin{array}{c}1+\cdots \left(\begin{array}{c}3 \\m\end{array}\right),\hspace{.4 cm} m \le 3 \\2^3, \hspace{2.1 cm} m \ge 3\end{array}\right.$$
when $p \ne 2$ and
$$\tau_2(m,\mathbb{Q}_2) = \left\{\begin{array}{c}1+\cdots \left(\begin{array}{c}7 \\m\end{array}\right),\hspace{.4 cm} m \le 7 \\2^7, \hspace{2.1 cm} m \ge 7\end{array}\right. .$$
Based on these values of $\tau_1$ and $\tau_2$, it is a straightforward matter to compute the maximal value of $C_j(n,k,\beta)$ for the fields mentioned above. We do so explicitly for the fields $\overline{k}$, $\mathbb{R}$, and $\mathbb{F}_q$ where $2 \not | q$.
\begin{cor}
\label{MaxIsomClassesSo}
Suppose $k = \overline{k}$
\begin{enumerate}
\item If $n$ is odd, then $C_1(n,\overline{k},\beta) \le \frac{n-1}{2}.$ If $n$ is even, then $C_1(n,\overline{k},\beta) \le \frac{n}{2}.$
\item $C_2(n,\overline{k},\beta) =0.$
\item If $n$ is odd, then $C_3(n,\overline{k},\beta) = 0$. If $n$ is even, then $C_3(n,\overline{k},\beta) \le 1. $
\item $C_4(n,\overline{k},\beta) =0.$
\end{enumerate}
Now suppose $k = \mathbb{R}$
\begin{enumerate}
\item If $n$ is odd, then
\begin{align*}
C_1(n, \mathbb{R},\beta) &\le \sum_{m=1}^{\frac{n-1}{2}} (m+1)(n-m+1)\\
&= \frac{1}{12}(n^3+6n^2-n-6).\\
\end{align*}
If $n$ is even, then
\begin{align*}
C_1(n, \mathbb{R},\beta) &\le \left(\sum_{m=1}^{\frac{n}{2}-1} (m+1)(n-m+1)\right) + \left(\begin{array}{c}\frac{n}{2}+1 \\2\end{array}\right) + \frac{n}{2}+1\\
&= \frac{1}{12}(n^3+6n^2+2n).\\
\end{align*}
\item If $n$ is odd, then $C_2(n, \mathbb{R},\beta) =0.$ If $n$ is even, then
\begin{align*}
C_2(n, \mathbb{R},\beta) &\le \left(\begin{array}{c}\frac{n}{2}+1 \\2\end{array}\right) + \frac{n}{2}+1\\
&= \frac{1}{8}(n^2+6n+8).\\
\end{align*}
\item If $n$ is odd, then $C_3(n, \mathbb{R},\beta) = 0$. If $n$ is even, then $C_3(n, \mathbb{R},\beta) \le 1. $
\item If $n$ is odd, then $C_4(n, \mathbb{R},\beta) = 0$. If $n$ is even, then $C_4(n, \mathbb{R},\beta) \le 1. $
\end{enumerate}
Lastly, suppose $k = \mathbb{F}_q$ such that $2 \not | q$.
\begin{enumerate}
\item If $n$ is odd, then $C_1(n,\mathbb{F}_q,\beta) \le 2n-6.$ If $n$ is even, then $C_1(n,\mathbb{F}_q,\beta) \le2n-1.$
\item If $n$ is odd, then $C_2(n,\mathbb{F}_q,\beta) =0.$ If $n$ is even, then $C_2(n, \mathbb{F}_q,\beta) \le 3.$
\item If $n$ is odd, then $C_3(n,\mathbb{F}_q,\beta) = 0$. If $n$ is even, then $C_3(n,\mathbb{F}_q,\beta) \le 1. $
\item If $n$ is odd, then $C_4(n, \mathbb{F}_q,\beta) = 0$. If $n$ is even, then $C_4(n, \mathbb{F}_q,\beta) \le 1. $
\end{enumerate}
\end{cor}
\section{Explicit Examples}
\subsection{Algebraically Closed Fields}
We now find the exact number of isomorphy classes for some $\So(n,k, \beta)$. We begin by looking at the case where $k = \overline{k}$. Note that all symmetric non degenerate bilinear forms are congruent to the dot product over an algebraically closed field.
\begin{cor}
Assume $k = \overline{k}$. If $\theta$ is an $k$-involution of $\So(n,k)$, then $\theta$ is isomorphic to $\Inn_A$ where $A = \left(\begin{smallmatrix}-I_{m} & 0 \\0 & I_{n-m}\end{smallmatrix}\right)$ and $0 \le m < \frac{n}{2},$ or $A = \left(\begin{smallmatrix}0 & -I_{\frac{n}{2}} \\I_{\frac{n}{2}} & 0\end{smallmatrix}\right)$.
\end{cor}
\begin{proof}
Since $k$ is algebraically closed, we know that all $k$-involutions of $\So(n,k)$ are of Type 1 or 3. We first consider the Type 1 case. We will now find a representative matrix $A$ for each isomorphy class of Type 1 $k$-involutions. Suppose $\theta$ is a Type 1 $k$-involution. We will find a representative matrix $A$ for the isomorphic class containing $\theta$. We know we can assume $A \in \oo(n,k)$. Further, by Lemma \ref{Type1ClassSo} we can write $A = X \left(\begin{smallmatrix}-I_{m} & 0 \\0 & I_{n-m}\end{smallmatrix}\right) X^{-1}$, where we know $X^TX$ is diagonal. Since $k = \overline{k}$, then we also assume that $X^TX$ must be congruent to $I_n$. Since we are looking for a representative $A$ of our isomorphy class, we may assume $X^TX = I_n$, and we can choose $X = I_n$. This means $A = \left(\begin{smallmatrix}-I_{m} & 0 \\0 & I_{n-m}\end{smallmatrix}\right)$ is a representative of our isomorphy class.
We see that Type 3 $k$-involutions will exist since $J = \left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\-I_{\frac{n}{2}} & 0\end{smallmatrix}\right)$ will induce a Type 3 $k$-involution. Thus, there is one isomorphy class of Type 3 $k$-involutions.
\end{proof}
We note that in this case, that the maximal number of isomorphy classes do in fact exist. That is, in Corollary \ref{MaxIsomClassesSo}, for the case where $k = \overline{k}$, we have equality in every statement.
\subsection{The Standard Real Orthogonal Group}
We now examine the case where $\beta$ is the standard dot product, and $k = \mathbb{R}$.
\begin{cor}
If $\theta$ is an $k$-involution of $\So(n,\mathbb{R})$, then $\theta$ is isomorphic to $\Inn_A$ where $A = \left(\begin{smallmatrix}-I_{m} & 0 \\0 & I_{n-m}\end{smallmatrix}\right)$ and $0 \le m \le \frac{n}{2},$ or $A = \left(\begin{smallmatrix}0 & -I_{\frac{n}{2}} \\I_{\frac{n}{2}} & 0\end{smallmatrix}\right)$. There are no Type 2 or Type 4 $k$-involutions for this group.
\end{cor}
\begin{proof}
We first consider the Type 1 case. We will find a representative matrix $A$ for each isomorphy class of Type 1 $k$-involutions. Suppose $\theta$ is a Type 1 $k$-involution. We will find a representative matrix $A$ for the isomorphy class containing $\theta$. We know we can assume $A \in \oo(n,k)$. Further, by Lemma \ref{Type1ClassSo} we can write $A = X \left(\begin{smallmatrix}-I_{m} & 0 \\0 & I_{n-m}\end{smallmatrix}\right) X^{-1}$, where we know $X^TX$ is congruent to a diagonal where the diagonal entries are all 1's and -1's. Since we are looking for a representative of our isomorphy class, let us assume we have $X^TX$ is equal to this diagonal matrix. We see that there can be no $-1$'s in the diagonal matrix since $k = \mathbb{R}$. So, we assume $X^TX = I_n$, which means we can choose $X = I_n$. So, $A = \left(\begin{smallmatrix}-I_{m} & 0 \\0 & I_{n-m}\end{smallmatrix}\right)$ is a representative of our isomorphy class.
We proceed by contradiction to show that there are now Type 2 $k$-involutions of $\So(n,\mathbb{R})$. Suppose $\theta$ is a Type 2 $k$-involution. We want to find $A$ such that $\theta = \Inn_A$, By Lemma \ref{type2lemSo} we can write $A = -\frac{\sqrt{\alpha}}{\alpha} X \left(\begin{smallmatrix}0 & I_{\frac{n}{2}} \\ \alpha I_{\frac{n}{2}} & 0\end{smallmatrix}\right) X^{-1}$ where $X^TX = \left(\begin{smallmatrix}X_1 & 0 \\ 0 & \frac{1}{\alpha}X_1\end{smallmatrix}\right)$ is diagonal. We recall that $\alpha \in \mathbb{R}^*$ but $\sqrt{\alpha} \not \in \mathbb{R}^*.$ So, $\alpha$ must be a negative number, and we can choose $\alpha = -1$. That is, $X^TX = \left(\begin{smallmatrix}X_1 & 0 \\ 0 & -X_1\end{smallmatrix}\right)$. But, this is a contradiction, because when $k= \mathbb{R}$, there does not exist any nonzero vectors $x$ such that $x^Tx \le 0$, so the whole diagonal of $X^TX$ must be positive, which is not possible. This shows that there are no Type 2 $k$-involutions in this case. In a similar way, we can show that there are also no Type 4 $k$-involutions in this case.
We know that there is at most one isomorphy class of Type 3 $k$-involutions by Corollary \ref{MaxIsomClassesSo}. Since $A = \left(\begin{smallmatrix}0 & -I_{\frac{n}{2}} \\ I_{\frac{n}{2}} & 0\end{smallmatrix}\right) $induces a Type 3 $k$-involution, then $A$ is a representative of the only Type 3 isomorphy class.
\end{proof}
Unlike the algebraically closed case, we note that in this case, that the maximal number of isomorphy classes do not exist. That is, in Corollary \ref{MaxIsomClassesSo}, for the case where $k = \mathbb{R}$, we have an explicit example where we do not have equality. In fact, given that we have seen that the Type 1 and 3 cases must exist for this group, we actually have the minimal number of isomorphy classes possible.
\subsection{Orthogonal Groups of $\mathbb{F}_q$}
We begin by examining the Type 1 $k$-involutions where $k = \mathbb{F}_q$ and $q = p^h$ for all cases where $p \ge 3$. This is a complete classification of the $k$-involutions when $n$ is odd. We note that for these fields we have $|(k^*)^2| = 2$. So, we will use 1 and $\delta_q$ as representatives of of the distinct field square classes. Based on properties of symmetric matrices over $k = \mathbb{F}_q$, we know that up to congruence, there are two possibilities for $M$: either $M = I_n$ or $M = \left(\begin{smallmatrix}I_{n-1} & 0 \\0 & \delta_q\end{smallmatrix}\right)$.
\begin{theorem}
Assume that $M = I_n$. Suppose $\theta$ is a Type 1 $k$-involution of $\So(n,\mathbb{F}_q)$. Then $\theta$ is isomorphic to $\Inn_A$ where we can write $A = I_{n-m,m}$ for $0 \le m \le \frac{n}{2}$ or $$A = \left(\begin{array}{cccc}-I_{m-1} & 0 & 0 & 0 \\0 & 1-2\frac{a^2}{\delta_q} & 0 & \frac{2ab}{\delta_q} \\0 & 0 & I_{n-m-1} & 0 \\0 & \frac{2ab}{\delta_q} & 0 & 1-2\frac{b^2}{\delta_q} \end{array}\right)$$ for $0 \le m \le \frac{n}{2}$, where $\delta_q$ is a nontrivial non-square in $\mathbb{F}_q$ where $a^2+b^2 = \delta_q$ and $a,b \in \mathbb{F}_q$.
Now, assume that $M = \left(\begin{smallmatrix}I_{n-1} & 0 \\0 & \delta_q\end{smallmatrix}\right)$. Suppose $\theta$ is an $k$-involution of $\So(n,\mathbb{F}_q, \beta)$. Then $\theta$ is isomorphic to $\Inn_A$ where we can write $$A = I_{n-m,m} \text{ \hspace{.2cm} or \hspace{.2cm} } A = \left(\begin{array}{cccc}-I_{n-m-1} & & \\ & I_{m} & \\ & & -1 \end{array}\right)$$ for $0 \le m \le \frac{n}{2}$.
\end{theorem}
\begin{proof}
We will use the equivalent conditions of Lemma \ref{type1lemSo} to prove that the matrices listed above will distinctly be representatives of the isomorphy classes of the $k$-involutions of $\So(n,\mathbb{F}_q)$. For future reference, fix $a, b \in \mathbb{F}_q$ such that $a^2+b^2 = \delta_q$
If $\theta$ is a Type 1 $k$-involution, then by Lemma \ref{Type1ClassSo} we can choose a matrix $A$ such that $\theta = \Inn_A$ and we can write $A = X \left(\begin{smallmatrix}-I_s & 0 \\0 & I_t\end{smallmatrix}\right) X^{-1},$ where $s+t = n$ and $$X^TMX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right)$$ must be diagonal, and $X_1$ is an $s \times s$ matrix, and $X_2$ is a $t \times t$ matrix. It is a well known fact that any diagonal matrix over $\mathbb{F}_q$ must be congruent to either $I_n$ or $\left(\begin{smallmatrix}I_{n-1} & 0 \\0 & \delta_q\end{smallmatrix}\right)$ where $\delta_q$ is some fixed non-square in $\mathbb{F}_q$. So, from the equivalent conditions in Theorem \ref{type1lemSo} it is known that $X_1$ and $X_2$ must each be congruent to $I$ or $\left(\begin{smallmatrix}I & 0 \\0 & \delta_q\end{smallmatrix}\right)$ (sizing the matrices appropriately). Further, since $\det(X^TX) = (\det(X))^2$ is a square, we observe that $X_1$ and $X_2$ must be simultaneously congruent to either $I$ or $\left(\begin{smallmatrix}I & 0 \\0 & \delta_q\end{smallmatrix}\right)$ (again, sizing appropriately).
Since we are searching for a representative of the congruence class, it can be assumed that $X^TMX$ is either $I$ or $\left(\begin{smallmatrix}I & 0 & 0 & 0 \\0 & \delta_q & 0 & 0 \\0 & 0 & I & 0 \\0 & 0 & 0 & \delta_q\end{smallmatrix}\right)$. These are the only possibilities, and also they must correspond to distinct isomorphy classes of Type 1 $k$-involutions under the conditions of Theorem \ref{type1lemSo}.
{\bf Case 1}: $\beta$ is the standard dot product, and $M = I$.
{\bf Subcase 1.1}: $X^TX = I$.
We can let $X = I$, which means $A = \left(\begin{smallmatrix}-I_s & 0 \\0 & I_t\end{smallmatrix}\right)$ is the representative of the isomorphy class.
{\bf Subcase 1.2}: $X^TX = \left(\begin{smallmatrix}I & 0 & 0 & 0 \\0 & \delta_q & 0 & 0 \\0 & 0 & I & 0 \\0 & 0 & 0 & \delta_q\end{smallmatrix}\right)$.
We can let $$X = \left(\begin{array}{cccc}I & 0 & 0 & 0 \\0 & a & 0 & b \\0 & 0 & I & 0 \\0 & -b & 0 & a\end{array}\right).$$ It follows from this that $$A = \left(\begin{array}{cccc}-I_{m-1} & 0 & 0 & 0 \\0 & 1-2\frac{a^2}{\delta_q} & 0 & \frac{2ab}{\delta_q} \\0 & 0 & I_{n-m-1} & 0 \\0 & \frac{2ab}{\delta_q} & 0 & 1-2\frac{b^2}{\delta_q} \end{array}\right)$$ is a representative of the isomorphy class.
{\bf Case 2}: $\beta$ is such that $M = \left(\begin{smallmatrix}I_{n-1} & 0 \\0 & \delta_q\end{smallmatrix}\right).$
{\bf Subcase 2.1}: $X^TMX = M$.
In the first case, since we are looking for a representative of our congruence class, we can assume $X^T\left(\begin{smallmatrix}I_{n-1} & 0 \\0 & \delta_q\end{smallmatrix}\right)X = \left(\begin{smallmatrix}I_{n-1} & 0 \\0 & \delta_q\end{smallmatrix}\right)$. This means we can assume $X=I$ choose $A = \left(\begin{smallmatrix}-I_s & 0 \\0 & I_t\end{smallmatrix}\right)$ as the representative of the isomorphy class.
{\bf Subcase 2.2}: $X^TMX = \left(\begin{smallmatrix}I_{\frac{n}{2}-1} & 0 & 0 \\0 & \delta_q & 0 \\0 & 0 & I_{\frac{n}{2}}\end{smallmatrix}\right).$
We can choose $X = \left(\begin{smallmatrix}I_{s-1} & & & \\ & 0 & & 1 \\ & & I_{t-1} & \\ & 1 & & 0\end{smallmatrix}\right)$. This gives representative $A = \left(\begin{smallmatrix}-I_{s-1} & & \\ & I_t & \\ & & -1 \end{smallmatrix}\right).$
\end{proof}
By counting the number of isomorphy classes from this Theorem, its clear that if $n$ is odd, then $C_1(n,\mathbb{F}_q,\beta) = n+1,$ and if $n$ is even, then $C_1(n,\mathbb{F}_q,\beta) = n+2.$
For the remaining three types of $k$-involutions, we restrict our attention to the case where $\beta$ is the standard dot product. Recall that $n$ must be even. In this case, it is clear that $A = \left(\begin{array}{cc}0 & I_n \\-I_n & 0\end{array}\right)$ will induce a Type 3 $k$-involution, and that $C_3(n, k)= 1$.
We know that $C_2(n,\mathbb{F}_q) \le 3$ and $C_4(n,\mathbb{F}_q) \le 1$. We will specifically look at the cases where $q = $3, 5, and 7. For these cases, we see that we have existence of both Type 2 and Type 4 $k$-involutions via the matrices in Table \ref{So_Fp}.
\begin{table}[h]
\centering
\caption { Type 2 and Type 4 examples for $\So(4,\mathbb{F}_p$) } \label{So_Fp}
\begin{tabular}[t]{|c||c|c|}
\hline $k$ & Type 2 & Type 4 \\
\hline $\mathbb{F}_3$ &$ i \left(\begin{smallmatrix}1 & 1 & 0 & 0 \\1 & 2 & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & 2\end{smallmatrix}\right)$ & $i\left(\begin{smallmatrix}1 & 2 & 0 & 0 \\1 & 1 & 0 & 0 \\0 & 0 & 1 & 2 \\0 & 0 & 1 & 1\end{smallmatrix}\right) $ \\
\hline $\mathbb{F}_5$ & $\sqrt{2}\left(\begin{smallmatrix}1 & 1 & 0 & 0 \\1 & 4 & 0 & 0 \\0 & 0 & 1 & 1 \\0 & 0 & 1 & 4\end{smallmatrix}\right) $ & $\sqrt{2}\left(\begin{smallmatrix}1 & 4 & 0 & 0 \\1 & 1 & 0 & 0 \\0 & 0 & 1 & 4 \\0 & 0 & 1 & 1\end{smallmatrix}\right) $ \\
\hline $\mathbb{F}_7$ & $\sqrt{3}\left(\begin{smallmatrix}1 & 3 & 0 & 0 \\3 & 6 & 0 & 0 \\0 & 0 & 1 & 3 \\0 & 0 & 3 & 6\end{smallmatrix}\right) $ & $\sqrt{3}\left(\begin{smallmatrix}1 & 4 & 0 & 0 \\3 & 1 & 0 & 0 \\0 & 0 & 1 & 4 \\0 & 0 & 3 & 1\end{smallmatrix}\right) $ \\
\hline
\end{tabular}
\end{table}
We note that these examples will all generalize to higher dimensions, so it is clear that for these fields that whenever $n$ is even, $C_2(n,\mathbb{F}_q), C_4(n,\mathbb{F}_q) \ge 1$. So, for these three specific fields, we know that $C_4(n,\mathbb{F}_q) =1$, and that the number of isomorphy classes of Type 4 $k$-involutions are maximized. But, for $\So(4, \mathbb{F}_p)$ where $p =$ 3, 5, and 7, we have done computations in Maple which use the conditions of Theorem \ref{type2lemSo} that show that$C_2(4, \mathbb{F}_p) = 1$. So, the number of Type 2 isomorphy classes is not maximized in these cases. While we have been unable to prove this up to this point, we believe that this is a pattern that would continue. That is, we have the following conjecture:
\begin{conj}
Suppose that $\So(n,k)$ is a finite orthogonal group and that $n$ is even. Then, $C_2(n,k)=C_4(n,k) = 1$
\end{conj}
\subsection{$p$-adic numbers}
We now turn our attention to the case where $k= \mathbb{Q}_p$. We will assume $M = I_n$. We show a classification of the possible isomorphy classes of the Type 1 $k$-involutions of $\So(n, \mathbb{Q}_p)$ where $p >2$, using Lemma \ref{type1lemSo}. Note that if $n$ is odd and $n \ne 3$, then this all of the possible isomorphy classes of the $k$-involutions of $\So(n, \mathbb{Q}_p)$. We note that we say ``possible" because we don't show existence, but rather we use our characterization of Type 1 $k$-involutions to show which classes may exist. It still remains to be shown which of these possible classes does exist.
We first state a result from \cite{Jones} about symmetric matrices with entries from the p-adic numbers.
\begin{lem}
\label{padicSo}
Symmetric matrices $M_1$ and $M_2$ with entries in $\mathbb{Q}\sb{p}$ are congruent if and only if
$$\det(M_1)=\gamma^{2}\det(M_2)
\hspace{1 em} and \hspace{1 em} c\sb{p}(M_1)=c\sb{p}(M_2) $$
where $c_p(M)$ denotes the Hasse symbol of matrix $M$.
\end{lem}
We use this to prove a result that is an extension of Theorem \ref{type1lemSo} in the case that $k= \mathbb{Q}_p$.
\begin{cor}
\label{padicCor}
Assume the hypotheses of Theorem \ref{type1lemSo}. Statements $(i)$ through $(v)$ of Theorem \ref{type1lemSo} are equivalent the following condition:
There exists some $\gamma \in \mathbb{Q}_p$ such that
$$\det(X_1) =\gamma^2\det(Y_1), \hspace{.2cm} \det(X_2) =\gamma^2\det(Y_2), \hspace{.2cm} c_p(X_1)=c_p(Y_1), \hspace{.2cm} \& \hspace{.2cm} c_p(X_2)=c_p(Y_2)$$ or
$$\det(X_1) =\gamma^2\det(Y_2), \hspace{.2cm} \det(X_2) =\gamma^2\det(Y_1), \hspace{.2cm} c_p(X_1)=c_p(Y_2), \hspace{.2cm} \& \hspace{.2cm} c_p(X_2)=c_p(Y_1).$$
\end{cor}
\begin{proof}
We note that this condition is equivalent to $(iii)$ of Theorem \ref{type1lemSo} by Lemma \ref{padicSo}.
\end{proof}
Corollary \ref{padicCor} gave us conditions on the square class of the determinant and the Hasse symbol to classify the isomorphy classes for $\So(n, \mathbb{Q}_p)$. Using these conditions, we have classified all of the possible isomorphy classes of Type 1 $k$-involutions based on what the values of $X_1$ and $X_2$ would be for a representative of the congruency class in Tables \ref{so-Qp-table1} and \ref{so-Qp-table2}. We note that each isomorphy class is determined by the triple $(\det(X_1) = \det(X_2), c_p(X_1),c_p(X_2)).$ To show that each of these possible congruency classes exists, one would need to find a matrix $X$ such that $X^TX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right).$ This would then determine $A$. In the case where $-1 \not \in (\mathbb{Q}_p^*)^2$, this will always be the case. To see that this is true, note that $X^TX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right)$ will always be a symmetric matrix with a determinant that is in the same square class as 1. When $-1 \not \in(\mathbb{Q}_p^*)^2$, all such matrices are such that $c_p\left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right) = 1$ is the case. So, $\left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right)$ will be congruent to $I_n$, which gives us the existence of $X$ such that $X^TX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right)$. In the case where $-1 \in (\mathbb{Q}_p^*)^2$, then it is possible that $c_p\left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right) = -1$. For these cases, it is not clear (to the authors) that there exists $X$ such that $X^TX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right)$.
\begin{table}[h]
\centering
\caption {$X_1$ and $X_2$ values when $k= \mathbb{Q}_p$, $p>2$, and $-1\in (\mathbb{Q}_p^*)^2$} \label{so-Qp-table1}
\begin{tabular}[t]{|c|c|c|c|c|c|c|c|}
\hline $X_1$ &$X_2$ & $\det(X_1)$ and $\det(X_2)$ &$c_p(X_1)$ &$c_p(X_2)$ \\
\hline $I_n$ & $I_n$ & 1 & 1& 1 \\
\hline $I_n$ & \tiny{$ \left(\begin{array}{cccc}I_{n-3} & 0 & 0 & 0 \\0 & p & 0 & 0 \\0 & 0 & N_p & 0 \\0 & 0 & 0 & pN_p\end{array}\right)$} & 1 & 1& -1 \\
\hline \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & p\end{array}\right)$} & \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & p\end{array}\right)$} & $p$ & 1& 1 \\
\hline \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & p\end{array}\right)$} & \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & N_p & 0 \\0 & 0 & pN_p\end{array}\right)$} & $p$ & 1& -1 \\
\hline \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & N_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & N_p\end{array}\right)$} & $N_p$ & 1& 1 \\
\hline \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & N_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & pN_p\end{array}\right)$} & $N_p$& 1& -1 \\
\hline \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & pN_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & pN_p\end{array}\right)$} & $pN_p$ & 1& 1 \\
\hline \tiny{$ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & pN_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & N_p\end{array}\right)$} & $pN_p$& 1& -1 \\
\hline \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & N_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & N_p\end{array}\right)$} & $pN_p$& -1& -1 \\
\hline \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & pN_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & pN_p\end{array}\right)$} & $N_p$& -1& -1 \\
\hline \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & N_p & 0 \\0 & 0 & pN_p\end{array}\right)$} & \tiny{$ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & N_p & 0 \\0 & 0 & pN_p\end{array}\right)$} & $p$& -1& -1 \\
\hline \tiny{$ \left(\begin{array}{cccc}I_{n-3} & 0 & 0 & 0 \\0 & p & 0 & 0 \\0 & 0 & N_p & 0 \\0 & 0 & 0 & pN_p\end{array}\right)$} & \tiny{$\left(\begin{array}{cccc}I_{n-3} & 0 & 0 & 0 \\0 & p & 0 & 0 \\0 & 0 & N_p & 0 \\0 & 0 & 0 & pN_p\end{array}\right)$ }& $1$ & -1& -1 \\
\hline
\end{tabular}
\end{table}
\begin{table}[h]
\centering
\caption {$X_1$ and $X_2$ values when $k= \mathbb{Q}_p$, $p>2$ and $-1\not \in(\mathbb{Q}_p^*)^2$} \label{so-Qp-table2}
\begin{tabular}[t]{|c|c|c|c|c|c|c|c|}
\hline $X_1$ &$X_2$ & $\det(X_1)$ and $\det(X_2)$ &$c_p(X_1)$ &$c_p(X_2)$ \\
\hline $I_n$ & $I_n$ & 1 & 1& 1 \\
\hline $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & p\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & p\end{array}\right)$ & $p$ & -1& -1 \\
\hline $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & p\end{array}\right)$ & $ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & N_p & 0 \\0 & 0 & pN_p\end{array}\right)$ & $p$ &- 1& 1 \\
\hline $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & N_p\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & N_p\end{array}\right)$ & $N_p$ & 1& 1 \\
\hline $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & pN_p\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & pN_p\end{array}\right)$ & $pN_p$ & -1& -1 \\
\hline $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & pN_p\end{array}\right)$ & $ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & N_p\end{array}\right)$ & $pN_p$& -1& 1 \\
\hline $ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & N_p\end{array}\right)$ & $ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & p & 0 \\0 & 0 & N_p\end{array}\right)$ & $pN_p$& 1& 1 \\
\hline $ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & N_p & 0 \\0 & 0 & pN_p\end{array}\right)$ & $ \left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & N_p & 0 \\0 & 0 & pN_p\end{array}\right)$ & $p$& 1& 1 \\
\hline
\end{tabular}
\end{table}
We now assume that $p = 2$, and we construct a classification of the Type 1 $k$-involutions. We again note that if $n$ is odd and $n \ne 3$, then this is a complete classification. We see that $\pm 1$, $\pm 2$, $\pm 3$ and $\pm 6$ are representatives for all of the the distinct square classes of $(\mathbb{Q}_2^*)^2$. For this case, we have not constructed tables with complete classifications of the two sets of isomorphy classes. Instead, we have constructed a table, Table \ref{so-Qp-table5}, where there is a diagonal matrix over $\mathbb{Q}_2$ for each possible pair of determinant square class and value of Hasse symbol. A potential isomorphy class is determined by choosing for $X_1$ and $X_2$ any pair of matrices on this table where the two given matrices have determinants in the same square class. So, given the different possible Hasse symbol values, there are at most 24 isomorphy classes of Type 1 $k$-involutions. As in some of the previous cases, it is not immediately clear that there does or does not exist a matrix $X$ in each of these cases such that $X^TX = \left(\begin{array}{cc}X_1 & 0 \\0 & X_2\end{array}\right).$
\begin{table}[h]
\centering
\caption { $X_1$ and $X_2$ values when $k= \mathbb{Q}_2$ } \label{so-Qp-table5}
\begin{tabular}[t]{|c|c|c|}
\hline $\det(Y)$ square class & $c_f(Y) = 1$ &$c_p(Y) = -1$ \\
\hline $1$ & $ I_n$ & $ \left(\begin{array}{cccc}I_{n-3} & 0 & 0 & 0 \\0 & -2 & 0 & 0 \\0 & 0 & 3 & 0 \\0 & 0 & 0 & -6\end{array}\right)$ \\
\hline $-1$ & $\left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & 2 & 0 \\0 & 0 & -2\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & -1\end{array}\right)$ \\
\hline $ 2$ & $\left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & -1 & 0 \\0 & 0 & -2\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & 2\end{array}\right)$ \\
\hline $ -2$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & -2\end{array}\right)$ & $ \left(\begin{array}{cccc}I_{n-3} & 0 & 0 & 0 \\0 & -1 & 0 & 0 \\0 & 0 & -3 & 0 \\0 & 0 & 0 & -6\end{array}\right)$ \\
\hline $ 3$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & 3\end{array}\right)$ & $\left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 6\end{array}\right)$ \\
\hline $ -3$ & $\left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & -1 & 0 \\0 & 0 & -3\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & -3\end{array}\right)$ \\
\hline $ 6$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & 6\end{array}\right)$ & $\left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & 2 & 0 \\0 & 0 & 3\end{array}\right)$ \\
\hline $ -6$ & $\left(\begin{array}{ccc}I_{n-2} & 0 & 0 \\0 & -1 & 0 \\0 & 0 & 6\end{array}\right)$ & $ \left(\begin{array}{cc}I_{n-1} & 0 \\0 & -6\end{array}\right)$ \\
\hline
\end{tabular}
\end{table}
\clearpage
|
1,116,691,500,461 | arxiv | \section{Introduction}
All topological spaces under consideration are assumed to be $T_1$ and Hausdorff.
A \emph{Dowker space} is a normal topological space whose product with the unit interval is not normal.
Dowker \cite{Dowker_C.H.} raised the question of their very existence, and
gave a useful characterization of these spaces.
The first consistent example of such a space was given by Rudin \cite{Rudin_souslin_line_dowker_space},
who constructed a Dowker space of size $\aleph_1$, assuming the existence of a Souslin tree.
Later on, in \cite{Rudin_first_Dowker_ZFC_example}, Rudin constructed another Dowker space, this time in ZFC, and of cardinality $ (\aleph_\omega)^{\aleph_0}$.
Two decades later, Balogh \cite{Balogh_space} gave a $\zfc$ construction of a Dowker space of size $2^{\aleph_0}$, and Kojman and Shelah \cite{MR1605988} gave a $\zfc$ construction of a Dowker space of size $\aleph_{\omega+1}$.
A question remaining of focal interest ever since is whether $\zfc$ proves the existence of a \emph{small} Dowker space.
One of the sleekest consistent constructions of a Dowker space of size $\aleph_1$
may be found in de Caux's paper \cite{de_Caux_space}, assuming the combinatorial principle $\clubsuit$.
Whether $\clubsuit$ implies the existence of a Souslin tree
was asked by Juh\'asz around 1987 and remains open to this date.
For a comprehensive survey on Dowker spaces, we refer the reader to \cite{Rudin_chapter_handbook_of_set_topology}, \cite{Szepytycki_and_Weiss_review} and \cite{Dowker_Open_Quesions}.
An \emph{$S$-space} is a regular topological space which is hereditary separable but not hereditary Lindel\"of.
Whether such a space exists was asked at the late 1960's by Hajnal and Juh\'asz and independently by Countryman.
Along the years many consistent constructions of $S$-spaces were found,
many of which are due to Kunen and his co-authors \cite{two_more_S_spaces,MR667660,MR1234990,MR2023413,MR2492307,MR3690760,MR3962617}.
Rudin showed that the existence of a Souslin tree yields an $S$-space \cite{rudin_S_space_Souslin},
and even an $S$-space which is Dowker \cite{rudin_separable_dowker_space}.
Juhász, Kunen, and Rudin \cite{two_more_S_spaces} gave an example from $\ch$ of a first countable, locally compact
$S$-space, known as the \emph{Kunen Line},
as well as an example from $\ch$ of a first countable, $S$-space which is Dowker.
In the other direction, Kunen \cite{MR0440487} proved that, assuming $\ma_{\aleph_1}$, there are no strong $S$-spaces (that is, spaces all of whose finite powers are $S$-spaces),
Szentmikl\'{o}ssy \cite{MR588860} proved that, assuming $\ma_{\aleph_1}$, there are no compact $S$-spaces,
and Todor\v{c}evi\'{c} \cite{Stevo_No_S_Space} proved that, assuming $\pfa$, there are no $S$-spaces whatsoever.
For a comprehensive survey on $S$-spaces, we refer the reader to \cite{MR588816}, \cite{MR776626} and \cite{MR2713439}.
An \emph{$O$-space} is an uncountable regular topological space all of whose uncountable open sets are co-countable.
Note that any $O$-space is an $S$-space of size $\aleph_1$ all of whose closed sets are $G_\delta$ (i.e., the space is \emph{perfect}),
and hence not Dowker (cf. \cite[pp.~248]{MR467673}, \cite[\S2]{CH_with_no_Ostaszweski_spaces} and \cite[\S5]{MR1934262}).
This class of spaces is named after Adam Ostaszewski who constructed in \cite{Ostaszewski_space}, assuming $\clubsuit$, a normal,
locally compact, non-Lindel\"of $O$-space. He also showed that, assuming $\ch$, the space can be made countably compact.
A few years later, Mohammed Dahroug, who was a Ph.D. student of William Weiss at the University of Toronto,
constructed a first-countable, locally compact,
non-Lindel\"of $O$-space from a Souslin tree.
He also showed that, assuming $\ch$, the space can be made countably compact and normal.
Dahroug's work was never typed down.
The purpose of this paper is to formulate a combinatorial principle that follows both from $\clubsuit$ and from the existence of a Souslin tree,
and is still strong enough to yield an $S$-space which is Dowker, as well as a normal $O$-space. We call it $\clubsuit_{\ad}$. The exact definition may be found in Definition~\ref{clubsuit AD definition} below.
The main results of this paper read as follows.
\begin{thma} \begin{enumerate}
\item For every infinite cardinal $\lambda$, if there exists a $\cf(\lambda)$-complete $\lambda^+$-Souslin tree,
then for every partition $\mathcal S$ of $E^{\lambda^+}_{\cf(\lambda)}$ into stationary sets,
$\clubsuit_{\ad}(\mathcal S,{<}{\cf(\lambda)})$ holds.
\item For every regular uncountable cardinal $\kappa$, if there exists a regressive $\kappa$-Souslin tree,
then for every partition $\mathcal S$ of $E^\kappa_\omega$ into stationary sets,
$\clubsuit_{\ad}(\mathcal S,{<}\omega)$ holds.
\end{enumerate}
\end{thma}
\begin{thmb} Suppose that $\mathcal S$ is an infinite partition of some non-reflecting stationary subset of a regular uncountable cardinal $\kappa$.
If $\clubsuit_{\ad}(\mathcal S,2)$ holds,
then there exists a Dowker space of cardinality $\kappa$.
\end{thmb}
Note that for every infinite regular cardinal $\lambda$, $E^{\lambda^+}_\lambda$ is a non-reflecting stationary subset of $\lambda^+$.
\begin{thmc} If $\clubsuit_{\ad}(\{\omega_1\},1)$ holds, then there exists a collectionwise normal non-Lindel\"of $O$-space.
\end{thmc}
\begin{thmd} If $\clubsuit_{\ad}(\{E^{\lambda^+}_{\lambda}\},1)$ holds for an infinite regular cardinal $\lambda$,
then there exists a collectionwise normal Dowker space of cardinality $\lambda^+$, having hereditary density $\lambda$
and Lindel\"of degree $\lambda^+$.
\end{thmd}
\subsection{Organization of this paper} In Section~\ref{section2}, we formulate the guessing principle $\clubsuit_{\ad}(\mathcal S,{<}\theta)$ and its refinement $\clubsuit_{\ad}(\mathcal S,\mu,{<}\theta)$,
prove that $\clubsuit(S)$ entails a strong instance of $\clubsuit_{\ad}(\mathcal S,{<}\omega)$,
and that it is consistent that $\clubsuit_{\ad}(\{\omega_1\},{<}\omega)$ holds, but $\clubsuit(\omega_1)$ fails.
It is also shown that the weakest instance $\clubsuit_{\ad}(\{\kappa\},1,1)$ fails for $\kappa$ weakly compact,
and may consistently fail for $\kappa:=\omega_1$.
The proof of Theorem~A will be found there.
In Section~\ref{sectionladdersystemspace}, we present a $\clubsuit_{\ad}(\mathcal S,1,2)$-based construction of a Dowker space which is moreover a ladder-system space.
This covers scenarios previously considered by Good, Rudin and Weiss, in which the Dowker spaces constructed were not ladder-system spaces.
The proof of Theorem~B will be found there.
In Section~\ref{Collectionwisesection}, we present two $\clubsuit_{\ad}(\{E^{\lambda^+}_\lambda\},\lambda,1)$-based constructions of collectionwise normal spaces of small hereditary density and large Lindel\"of degree.
These spaces are not ladder-system spaces, rather, they are de Caux type spaces.
The proof of Theorems C and D will be found there.
In Section~\ref{normalsquare}, we comment on a construction of Szeptycki of an $\aleph_2$-sized Dowker space with a normal square,
assuming that $ \diamondsuit^*(S) $ holds for some stationary $S\s E^{\omega_2}_{\omega_1}$.
Here, it is demonstrated that the construction may be carried out from a weaker assumption which is known to be consistent with the failure of $\clubsuit(E^{\omega_2}_{\omega_1})$.
\subsection{Notation}\label{notationsubsection}
The \emph{hereditary density number} of a topological space $\mathbb X$, denoted $\hd(\mathbb X)$, is the least infinite cardinal $\lambda$ such that each subspace of $\mathbb X$ contains a dense subset of cardinality at most $\lambda$.
The \emph{Lindel\"of degree} of $\mathbb X$, denoted $\Lin(\mathbb X)$ is the least infinite cardinal $\lambda$ such that every open cover of $\mathbb X$ has a subcover of cardinality at most $\lambda$.
For an accessible cardinal $\kappa$ and a cardinal $\lambda<\kappa$, we write $\log_\lambda(\kappa):=\min\{\chi\mid \lambda^\chi\ge\kappa\}$.
$\reg(\kappa)$ denotes the set of all infinite regular cardinal below $\kappa$.
For a set of ordinals $C$, we write $ \acc^+(C):=\{ \alpha<\sup(C) \mid \sup(C\cap \alpha)=\alpha>0 \} $,
$ \acc(C):=\{ \alpha\in C \mid \sup(C\cap \alpha)=\alpha>0 \} $ and $\nacc(C):=C\setminus\acc(C)$.
For ordinals $ \alpha<\gamma$, denote $ E^{\gamma}_\alpha := \{\beta<\gamma \mid \cf(\beta)=\alpha \} $ and define $E^{\gamma}_{\neq\alpha}, E^{\gamma}_{<\alpha}, E^{\gamma}_{\le\alpha}, E^{\gamma}_{>\alpha}, E^{\gamma}_{\ge\alpha}$ similarly.
For a set $A$ and a cardinal $\theta$, write $[A]^\theta := \{ B\subseteq A \mid |B|=\theta \}$ and define $[A]^{<\theta}$ similarly.
For two sets $A$ and $B$, we write $A\s^* B$ to express that either $A=\emptyset$ or $A\setminus\alpha\s B$ for some $\alpha\in A$.
For a poset $(P,\lhd)$ and an element $x\in P$, we write $x_\downarrow:=\{ y\in P\mid y\lhd x\}$ and $x^\uparrow:=\{ y\in P\mid x\lhd y\}$.
For a family $\mathcal A$ of subsets of some ordinal, we let $\mup(\mathcal A):=\sup\{ \min(a) \mid a\in \mathcal A,~a\neq \emptyset \}$.
\subsection{Conventions} Throughout the paper, $\kappa$ stands for a regular uncountable cardinal,
and $\lambda$ stands for an infinite cardinal.
\section{A new guessing principle}\label{section2}
We commence by recalling some classic guessing principles.
\begin{definition}\label{principles} For a stationary subset $ S\subseteq \kappa $:
\begin{enumerate}
\item $ \diamondsuit^*(S) $ asserts the existence of a sequence $ \langle \mathcal A_\alpha \mid \alpha\in S \rangle $ such that:
\begin{itemize}
\item for all $ \alpha\in S $, $ \mathcal A_\alpha \subseteq \mathcal P( \alpha) $ and $ |\mathcal A_\alpha|\le|\alpha|$;
\item for every $B\s \kappa$, there exists a club $ C\subseteq \kappa $ such that $C\cap S \subseteq \{\alpha\in S\mid B\cap\alpha\in \mathcal A_\alpha \}$.
\end{itemize}
\item $ \diamondsuit(S) $ asserts the existence of a sequence $ \langle A_\alpha \mid \alpha\in S \rangle $ such that:
\begin{itemize}
\item for all $ \alpha\in S $, $ A_\alpha \subseteq \alpha $;
\item for every $B\s \kappa$, the set $\{\alpha\in S\mid B\cap\alpha=A_\alpha \}$ is stationary.
\end{itemize}
\item $ \clubsuit(S) $ asserts the existence of a sequence $ \langle A_\alpha\mid \alpha\in S \rangle $ such that:
\begin{itemize}
\item\label{Definiton clubsuit - Clause A_alpha} for all $ \alpha\in S\cap \acc(\kappa) $, $ A_\alpha $ is a cofinal subset of $\alpha$ of order type $\cf(\alpha)$;
\item\label{Definiton clubsuit - Clause guess} for every cofinal subset $ B\subseteq \kappa$, the set $\{\alpha\in S \mid A_\alpha \subseteq B \}$ is stationary.
\end{itemize}
\item \label{clubsuit_J(S)} $\clubsuit_{J}(S)$ asserts the existence of a matrix $\langle A_{\alpha,i}\mid\alpha\in S,~i<\cf(\alpha)\rangle$ such that:
\begin{itemize}
\item\label{clubsuits_J_ordertype_omega} For all $ \alpha\in S\cap\acc(\kappa) $,
$ \langle A_{\alpha,i} \mid i<\cf(\alpha) \rangle $ is a sequence of pairwise disjoint cofinal subsets of $ \alpha$, each of order-type $\cf(\alpha)$;
\item\label{clubsuits_J_unboundedsubset} For every cofinal subset $B\s\kappa$, the following set is stationary:
$$\{\alpha\in S\mid\forall i<\cf(\alpha)[\sup(B\cap A_{\alpha,i})= \alpha]\}.$$
\end{itemize}
\end{enumerate}
\end{definition}
\begin{remark}\label{diamondsuit iff clubsuit and ch}\label{one cohen implies clubsuit_J}
The principle $\diamondsuit^*$ was introduced by Kunen and Jensen in \cite{jensen1969some},
the principle $\diamondsuit$ was introduced by Jensen in \cite{Jensen_V=L_Diamond},
the principle $\clubsuit$ was introduced by Ostaszewski in \cite{Ostaszewski_space},
and the principle $\clubsuit_J$ was introduced by Juh\'{a}sz in \cite{Juhasz_clubsuit_ostaszewski} (under the name $(t)$).
It is not hard to see that for stationary $S'\s S\s\kappa$, $\diamondsuit^*(S')\implies\diamondsuit(S)\implies\clubsuit(S)\implies\clubsuit_J(S)$.
Devlin (see \cite[p.~507]{Ostaszewski_space}) proved that $\diamondsuit(S)\iff \clubsuit(S) + \kappa^{<\kappa}=\kappa$.
In \cite{Juhasz_clubsuit_ostaszewski}, Juh\'{a}sz
proved that $\clubsuit_J(\omega_1)$ is adjoined by the forcing to add a Cohen real, and proved that the former suffices for the construction of an Ostaszewski space.
\end{remark}
To present our new guessing principle, we shall first need the following definition.
\begin{definition}\label{almost-disjoint ladder-system} For a set of ordinals $S$:
\begin{enumerate}
\item A sequence $\langle A_\alpha\mid \alpha\in S\rangle$
is said to be an \emph{$\ad$-ladder system} iff the two hold:
\begin{itemize}
\item For all $\alpha\in S\cap\acc(\kappa)$, $A_\alpha$ is a cofinal subset of $\alpha$;
\item For all two distinct $\alpha,\alpha'\in S$, $\sup(A_\alpha\cap A_{\alpha'})<\alpha$.
\end{itemize}
\item A sequence $\langle \mathcal A_\alpha\mid \alpha\in S\rangle$
is said to be an \emph{$\ad$-multi-ladder system} iff the two hold:
\begin{itemize}
\item\label{clubsuit AD, Clause family softer} For all $\alpha\in S\cap\acc(\kappa)$, $\mathcal A_\alpha$ is a nonempty family consisting of pairwise disjoint cofinal subsets of $\alpha$;
\item\label{clubsuit AD, Clause almost disjoint} For all two distinct $A,A'\in\bigcup_{\alpha\in S}\mathcal A_\alpha$, $\sup(A\cap A')<\sup(A)$.
\end{itemize}
\end{enumerate}
\end{definition}
Now, we are ready to present the new guessing principle.
\begin{definition}\label{clubsuit AD definition} For a family $\mathcal S$ of stationary subsets of $\kappa$, $\clubsuit_{\ad}(\mathcal S,{<}\theta)$
asserts the existence of an $\ad$-multi-ladder system $\vec{\mathcal A}=\langle \mathcal A_{\alpha}\mid\alpha\in \bigcup\mathcal S\rangle$ such that:
\begin{enumerate}
\item\label{clubsuit AD, Clause family} For every $\alpha\in\bigcup\mathcal S$, $|\mathcal A_\alpha|=\cf(\alpha)$;
\item\label{clubsuit AD, Clause cofinal set} For every $\mathcal B\s[\kappa]^{\kappa}$ with $|\mathcal B|<\theta$,
and every $S\in\mathcal S$,
the following set is stationary:
$$G(S,\mathcal B):=\{\alpha\in S\mid \forall (A,B)\in\mathcal A_\alpha\times\mathcal B~[\sup(A\cap B)= \alpha]\}.$$
\end{enumerate}
\end{definition}
\begin{conv}\label{clubsuit AD simple definition} We write $\clubsuit_{\ad}(\mathcal S,\theta)$ for $\clubsuit_{\ad}(\mathcal S,{<}(\theta+1))$,
and $\clubsuit_{\ad}(S)$ for $\clubsuit_{\ad}(\{S\},1)$.
\end{conv}
\begin{remark} For any $\chi\in\reg(\kappa)$
and any stationary $S\s E^\kappa_\chi$, $\clubsuit_J(S) \implies \clubsuit_{\ad}(S)$.
\end{remark}
\begin{figure}[H]
\centering
\begin{tikzcd}[row sep=scriptsize, column sep=scriptsize]
\diamondsuit(S) \arrow[d]\arrow[rr]{} & {} & \diamondsuit(\omega_1)\arrow[dd] \\
\clubsuit(S) \arrow[d] & {} \text{Cohen real}\arrow[dl]\arrow[dr] & {} \\
\clubsuit_J(S) \arrow[d] & {} & \exists\aleph_1\text{-Souslin Tree}\arrow[d] \\
\clubsuit_{\ad}(S) & {} & \forall\mathcal S~\clubsuit_{\ad}(\mathcal S,{<}\omega)\arrow[ll] \\
\end{tikzcd}
\caption{Diagram of implications between the combinatorial principles under discussion, at the level of $\omega_1$.}
\end{figure}
To motivate Definition~\ref{almost-disjoint ladder-system}, let us point out
two easy facts concerning disjointifying $\ad$ systems.
\begin{prop}\label{Lemma - non-reflecting stat, diagonlization of initial seg}
Suppose that $S$ is a non-reflecting stationary subset of $\kappa$, and $\vec A=\langle A_\alpha \mid \alpha\in S \rangle$ is an $\ad$-ladder system.
Then there exists a sequence of functions $\langle f_\xi\mid \xi<\kappa\rangle$ such that, for every $\xi<\kappa$:
\begin{enumerate}
\item $f_\xi$ is a regressive function from $S\cap\xi$ to $\xi$;
\item the sets in $\langle A_\alpha\setminus f_\xi(\alpha) \mid \alpha\in S\cap \xi\rangle$ are pairwise disjoint.
\end{enumerate}
\end{prop}
\begin{proof} We recursively construct a sequence $\langle f_{\xi}\mid \xi<\kappa \rangle$ such that for every $\xi<\kappa$, Clauses (1) and (2) above hold true.
$\br$ The cases where either $\xi=0$ or $\xi=\beta+1$ for $\beta\notin S$ are straightforward.
$\br$ Suppose $\xi=\beta+1$ with $\beta\in S$ for which $f_\beta$ has already been defined. Define $g:S\cap\beta\rightarrow\beta$ via $g(\alpha):=\sup(A_\beta\cap A_\alpha)$.
As $\vec A$ is an $\ad$-ladder system, $g$ is regressive. In effect, we may define a regressive function $f_\xi:S\cap\xi\rightarrow\xi$ via $f_\xi(\alpha):=\max\{f_\beta(\alpha),g(\alpha)\}$ for $\alpha\in S\cap \beta$,
and $f_\xi(\beta):=0$. Notice that by the recursive assumption the function $f_\xi$ is as sought.
$\br$ Suppose $\xi\in\acc(\kappa)$ for which $\langle f_\beta\mid\beta<\xi\rangle$ has already been defined. As $S$ is non-reflecting we may fix a club $C\subseteq \xi$ disjoint from $S$.
For every $\alpha<\xi$, set $\alpha^-:=\sup(C\cap\alpha)$ and $\alpha^+:=\min(C\setminus \alpha)$. Note that, for every nonzero $\alpha\in S\cap\xi$, $\alpha^-<\alpha<\alpha^+$.
Define a regressive function $f_\xi:S\cap \xi\rightarrow \xi$ via $f_\xi(\alpha):=\max\{f_{\alpha^+}(\alpha),\alpha^-\}$. Notice that by the recursive hypothesis, the function $f_\xi$ is as sought.
\end{proof}
\begin{prop}\label{Proposition - disjointify multi-ladder system} Suppose that $S$ is a subset of $E^\kappa_{\ge\lambda}$ and $\vec{\mathcal A}=\langle \mathcal A_\alpha\mid \alpha\in S\rangle$
is an $\ad$-multi-ladder system. For any $\mathcal B\s\bigcup_{\alpha\in S}\mathcal A_\alpha$ with $|\mathcal B|\le\lambda$,
there exists a function $f:\mathcal B\rightarrow\kappa$ such that:
\begin{enumerate}
\item for every $B\in\mathcal B$, $f(B)\in B$;
\item the sets in $\langle B\setminus f(B) \mid B\in\mathcal B\rangle$ are pairwise disjoint.
\end{enumerate}
\end{prop}
\begin{proof} Given $\mathcal B\s\bigcup_{\alpha\in S}\mathcal A_\alpha$ with $|\mathcal B|\le\lambda$,
fix an injective enumeration $\langle B_\xi \mid \xi<|\mathcal B|\rangle$ of $\mathcal B$.
By the hypothesis on $\vec{\mathcal A}$, for every pair $\zeta<\xi<|\mathcal B|$,
$$\sup(B_\xi\cap B_\zeta)<\sup(B_\xi)\in E^\kappa_{\ge\lambda}\s E^\kappa_{>\xi},$$
so that $\sup_{\zeta<\xi}\sup(B_\xi\cap B_\zeta)<\sup(B_\xi)$.
It follows that we may define a function $f:\mathcal B\rightarrow\kappa$ via:
$$f(B_\xi):=\begin{cases}\min(B_\xi),&\text{if }\xi=0;\\
\min\{\beta\in B_\xi\mid \sup\{\sup(B_\xi\cap B_\zeta)\mid \zeta<\xi\}<\beta\},&\text{otherwise}.
\end{cases}$$
Evidently, $f$ is as sought.
\end{proof}
Our next lemma shows, in particular, that for any $\chi\in\reg(\kappa)$
and any stationary $S\s E^\kappa_\chi$,
$\clubsuit(S) \implies \clubsuit_{\ad}(\{S\},{<}\omega)$.
The reverse implication does not hold in general, as established by Corollary~\ref{clubadvsclub} below.
The proof of the lemma will make use of the following fact.
\begin{fact}[Brodsky-Rinot, {\cite[\S3]{paper23}}]\label{clubfacts}
For any stationary $S\s\kappa$, all of the following are equivalent:
\begin{enumerate}
\item $ \clubsuit(S) $;
\item \label{many clubsuit from one} there exists a partition $ \langle S_i\mid i<\kappa \rangle $ of $ S $ into pairwise disjoint stationary sets such that $ \clubsuit(S_i ) $ holds for each $ i<\kappa $;
\item \label{matrix clubsuit from one}
for any (possibly finite) cardinal $\theta$ such that $\kappa^\theta=\kappa$, there exists a matrix $ \langle A_{\alpha,\tau} \mid \alpha\in S,~\tau\le \theta \rangle $ such that,
for every sequence $\langle A_\tau \mid \tau\leq\theta \rangle$ of cofinal subsets of $\kappa$, the following set is stationary in $\kappa$:
$$\{ \alpha\in S \mid \forall \tau\leq \theta~[A_{\alpha,\tau}\subseteq A_\tau\cap \alpha \ \& \ \sup(A_{\alpha,\tau})=\alpha] \} .$$
\item \label{matrix clubsuit <omega hitting}
there exists a sequence $\langle X_\alpha \mid \alpha\in S\rangle$ such that:
\begin{itemize}
\item for every $\alpha\in S\cap \acc(\kappa)$, $X_\alpha\subseteq [\alpha]^{<\omega} $ with $\mup(X_\alpha)=\alpha$;\footnote{Recall that $\mup$ was defined in Subsection~\ref{notationsubsection}.}
\item\label{mupclub} for every $X\subseteq [\kappa]^{<\omega}$ with $\mup(X)=\kappa$, the following set is stationary:
$$\{ \alpha\in S \mid X_\alpha \subseteq X\}.$$
\end{itemize}
\end{enumerate}
\end{fact}
\begin{lemma}\label{lemma216} Suppose that $\clubsuit(S)$ holds for some stationary $S\s E^\kappa_\chi$ with $\chi\in\reg(\kappa)$.
Then there exists a partition $\mathcal S$ of $S$ into $\kappa$ many stationary sets
for which $\clubsuit_{\ad}(\mathcal S,{<}\omega)$ holds as witnessed by a sequence $\vec{\mathcal A}=\langle \mathcal A_\alpha\mid\alpha\in S\rangle$
with $\otp(A)=\chi$ for all $A\in\bigcup_{\alpha\in S}\mathcal A_\alpha$.
\end{lemma}
\begin{proof} By Fact~\ref{clubfacts}\eqref{many clubsuit from one}, fix a partition $ \langle S_i\mid i<\kappa \rangle $ of $S$ into pairwise disjoint stationary sets such that $ \clubsuit(S_i ) $ holds for each $ i<\kappa $.
Set $\mathcal S:=\{ S_i\mid i<\kappa\}$.
Next, for each $i<\kappa$, let $\langle X_\alpha \mid \alpha\in S_i\rangle$ be a sequence as in Fact~\ref{clubfacts}\eqref{mupclub}.
Fix a surjection $h:\chi\rightarrow\chi$ such that $|h^{-1}\{j\}|=\chi$ for all $j<\chi$.
To simplify the upcoming argument let us agree to write, for any two nonempty sets of ordinals $a,b$,
``$a<b$'' iff $\alpha<\beta$ for all $(\alpha,\beta)\in a\times b$.
Let $\alpha\in S\cap\acc(\kappa)$.
Recall that $X_\alpha\subseteq [\alpha]^{<\omega} $ and $\mup(X_\alpha)=\alpha$.
Fix a strictly increasing sequence of ordinals $\langle \alpha_\zeta\mid \zeta<\chi\rangle$ that converges to $\alpha$.
Now, by recursion on $\zeta<\chi$, we construct a sequence $\langle x_\zeta \mid \zeta<\chi\rangle$ such that, for every $\zeta<\chi$:
\begin{enumerate}
\item $ x_\zeta \in X_\alpha$, and
\item\label{Clause inc. min} for every $\xi<\zeta$, $(x_\xi\cup\{\alpha_\zeta\})<x_\zeta$.
\end{enumerate}
Suppose $\zeta<\chi$ and that $\langle x_\xi \mid \xi<\zeta \rangle$ has already been defined.
Evidently, $\eta:=\sup(\{ \max(x_\xi) \mid \xi<\zeta \}\cup\{\alpha_\zeta\})$ is $<\alpha$.
So, as $\mup(X_\alpha)=\alpha$, we may let $x_\zeta:= x$ for some $x\in X_\alpha$ with $\min(x)>\eta$.
This completes the construction. By Clause~(2), $\langle x_\zeta\mid \zeta<\chi\rangle$ is $<$-increasing,
and $\mup\{ x_\zeta \mid \zeta<\chi \} = \alpha$.
Finally, for every $j<\chi$, let $X^j_{\alpha}:=\{x_\zeta\mid \zeta<\chi, h(\zeta)=j\}$,
and $A^j_\alpha:=\biguplus X^j_\alpha$.
\begin{claim} $\langle A^j_\alpha\mid j<\chi\rangle$ is a sequence of pairwise disjoint cofinal subsets of $\alpha$, each of order-type $\chi$.
\end{claim}
\begin{proof} Let $j<\chi$.
As $\mup\{ x_\zeta \mid \zeta<\chi, h(\zeta)=j \} = \alpha$,
and as $\langle x_\zeta\mid \zeta<\chi, h(\zeta)=j \rangle$ is a $<$-increasing $\cf(\alpha)$-sequence of finite sets,
we infer that $\sup(A^j_\alpha)=\alpha$,
and that, for every $\beta<\alpha$,
$\otp(A^j_\alpha\cap \beta)<\chi$. Altogether, $\otp(A^j_\alpha)=\chi$.
Also, since $\langle x_\zeta\mid \zeta<\chi\rangle$ consists of pairwise disjoint sets,
the elements of $\langle A^j_\alpha\mid j<\chi\rangle$ are pairwise disjoint.
\end{proof}
For every $\alpha\in S$, set $\mathcal A_\alpha:=\{ A_\alpha^j\mid j<\chi \}$.
It immediately follows from the preceding claim that $\vec{\mathcal A}:=\langle \mathcal A_\alpha \mid \alpha \in S \rangle $ is an $\ad$-multi-ladder system.
\begin{claim}For every finite $\mathcal B\s[\kappa]^{\kappa}$ and every $i<\kappa$,
the following set is stationary:
$$G(S_i,\mathcal B):=\{\alpha\in S_i\mid \forall (A,B)\in\mathcal A_\alpha\times\mathcal B~[\sup(A\cap B)= \alpha]\}.$$
\end{claim}
\begin{proof}
Suppose that $\langle B_n\mid n<m\rangle$ is a finite sequence of cofinal subsets of $\kappa$.
For each $n<m$, fix an injective enumeration $\langle b_{n,\iota} \mid \iota<\kappa \rangle$ of $B_n$.
Set $X:=\{ \{ b_{n,\iota} \mid n<m\} \mid \iota< \kappa \}$ and notice that $X\subseteq [\kappa]^{<\omega}$ with $\mup(X)=\kappa$.
In effect, for every $i<\kappa$, the set $T_i:=\{ \alpha \in S_i \mid X_\alpha \subseteq X \}$ is stationary.
Let $i<\kappa$. We claim that $T_i\s G(S_i,\mathcal B)$. To see this, let $\alpha\in T_i$ and $(A,B)\in\mathcal A_\alpha\times\mathcal B$ be arbitrary.
Fix $j<\chi$ and $n<m$ such that $A=A^j_\alpha$ and $B=B_n$.
As $X^j_{\alpha}\s X_\alpha\s X$, by the definition of $X$, for every $x\in X^j_\alpha$, $x\cap B_n\neq\emptyset$. As $\mup(X^j_\alpha)=\alpha$,
it follows that $\sup(A_\alpha^j\cap B_n)=\alpha$.
\end{proof}
This completes the proof.
\end{proof}
We conclude this subsection by formulating a three-cardinal variant of $\clubsuit_{\ad}$:
\begin{definition}\label{threecardinalsvariant}
$\clubsuit_{\ad}(\mathcal S,\mu,{<}\theta)$ asserts the existence of a system
$\vec{\mathcal A}=\langle \mathcal A_{\alpha}\mid\alpha\in \bigcup\mathcal S\rangle$ as in Definition~\ref{clubsuit AD definition},
but in which Clause~(1) is replaced by the requirement that, for every $\alpha\in\bigcup S$, $|\mathcal A_\alpha|=\mu$.
We write $\clubsuit_{\ad}(\mathcal S,\mu,\theta)$ for $\clubsuit_{\ad}(\mathcal S,\mu,{<}(\theta+1))$.
\end{definition}
It is clear that $\clubsuit_{\ad}(\mathcal S,\omega,{<}\theta)$ follows from $\clubsuit_{\ad}(\mathcal S,{<}\theta)$.
Also, the following lemma is obvious.
\begin{lemma}\label{adone} For a family $\mathcal S$ of stationary subsets of $\kappa$, $\clubsuit_{\ad}(\mathcal S,1,2)$ holds
iff there exists an $\ad$-ladder system $\langle A_{\alpha}\mid\alpha\in \bigcup\mathcal S\rangle$ such that,
for all $B_0,B_1\in[\kappa]^{\kappa}$ and $S\in\mathcal S$,
the set $\{\alpha\in S\mid \sup(A_\alpha\cap B_0)=\sup(A_\alpha\cap B_1)=\alpha\}$ is stationary.\qed
\end{lemma}
\subsection{Interlude on Souslin trees}\label{Guess Construction subsection}
Recall that a poset $\mathbf T=(T,{<_T})$ is a \emph{$\kappa$-Souslin tree} iff all of the following hold:
\begin{itemize}
\item $|T|=\kappa$;
\item $(T,{<_T})$ has no chains or antichains of size $\kappa$;
\item for every $x\in T$, $(x_\downarrow,<_T)$ is well-ordered.
\end{itemize}
For every $x\in T$, denote $\h(x):=\otp(x_\downarrow,<_T)$.
For every $A\s\kappa$, let $T\restriction A:=\{ x\in T\mid \h(x)\in A\}$.
Note that, for every $\alpha<\kappa$, $T_\alpha:=\{ x\in T\mid \h(x)=\alpha\}$ and $T\restriction\alpha$ have size ${<}\kappa$.
The next well-known lemma shows that Souslin trees are similar to Luzin spaces in the sense that every large subset of a Souslin tree is \emph{somewhere dense}.
\begin{lemma}[folklore]\label{Souslin_denseness}
Suppose $\mathbf T=(T,{<_T})$ is a $\kappa$-Souslin tree and $B\subseteq T $ is a subset with $|B|=\kappa$.
Then there exists $w\in T$ such that $w^\uparrow\cap B$ is cofinal in $w^\uparrow$.
\end{lemma}
\begin{proof} Let $X$ denote the collection of all $x\in T$ such that $x^\uparrow\cap B$ is empty.
Let $A\s X$ be a maximal antichain in $X$.
As $\mathbf T$ is a $\kappa$-Souslin tree, $|A|<\kappa$, so we may find a large enough $\delta<\kappa$ such that $A\s T\restriction\delta$.
As $|B|=\kappa>|T\restriction(\delta+1)|$, let us fix $b\in B$ with $\h(b)>\delta$.
Finally, let $w$ denote the unique element of $T_\delta$ with $w<_T b$.
\begin{claim} $w^\uparrow\cap B$ is cofinal in $w^\uparrow$.
\end{claim}
\begin{proof} Suppose not. Then there must exist some $x\in X$ with $w\le_T x$.
As $A$ is a maximal antichain in $X$, we may find $\bar x\in A$ which is comparable with $x$.
As $\h(\bar x)<\delta\le\h(x)$, it follows that $\bar x<_T x$. As $\h(w)=\delta$ and $w\le_T x$, it follows that $\bar x<_Tw\le_T x$.
In particular, $\bar x<_Tw<_Tb$, so that $b\in\bar x^\uparrow\cap B$, contradicting the fact that $\bar x\in X$.
\end{proof}
This completes the proof.
\end{proof}
\begin{definition} A $\kappa$-Souslin tree $\mathbf T=(T,{<_T})$ is said to be:
\begin{itemize}
\item \emph{normal} iff for any $x\in T$ and $\alpha<\kappa$ with $\h(x)<\alpha$, there exists $y\in T_\alpha$ with $x<_T y$;
\item \emph{$\mu$-splitting} iff every node in $T$ admits at least $\mu$-many immediate successors,
that is, for every $x\in T$, $|\{ y\in T\mid x<_T y, \h(y)=\h(x)+1\}|\ge\mu$;
\item \emph{prolific} iff every $x\in T$ admits at least $\h(x)$-many immediate successors;
\item \emph{$\chi$-complete} iff any $<_T$-increasing sequence of elements from $T$, and of length ${<}\chi$, has an upper bound in $T$;
\item \emph{regressive} iff there exists a map $\rho:T\restriction\acc(\kappa)\rightarrow T$ satisfying the following:
\begin{itemize}
\item for every $x\in T\restriction\acc(\kappa)$, $\rho(x)<_T x$;
\item for all $\alpha\in\acc(\kappa)$ and $x,y\in T_\alpha$,
if $\rho(x)<_T y$ and $\rho(y)<_T x$, then $x=y$;
\end{itemize}
\item \emph{ordinal-based} iff $T=\kappa$ and, for all $x,y\in T$, if $\h(x)<\h(y)$, then $x\in y$.
\end{itemize}
A subset $B\s T$ is said to be an \emph{$\alpha$-branch} iff $(B,<_T)$ is linearly ordered and $\{\h(x)\mid x\in B\}=\alpha$;
it is said to be \emph{vanishing} iff it has no upper bound in $T$.
\end{definition}
\begin{definition} A $\lambda^+$-Souslin tree is said to be \emph{maximally-complete} iff it is $\chi$-complete
for $\chi:=\log_\lambda(\lambda^+)$.
\end{definition}
Note that the existence of a $\cf(\lambda)$-complete $\lambda^+$-Souslin tree
is equivalent to the conjunction of ``$\lambda^{<\cf(\lambda)}=\lambda$'' and ``there is a maximally-complete $\lambda^+$-Souslin tree''.
\begin{prop}[folklore]\label{ordinalsbased} For cardinals $\chi,\mu<\cf(\kappa)=\kappa$, if there exists a $\kappa$-Souslin tree which is $\chi$-complete (resp.~regressive),
then there exists an ordinal-based $\mu$-splitting, normal, prolific
$\kappa$-Souslin tree which is $\chi$-complete (resp.~regressive).
\end{prop}
\begin{proof} Suppose $\mathbf T=(T,{<_T})$ is a $\kappa$-Souslin tree.
By a standard fact (see \cite[Lemma~2.4]{rinot20}), we may fix a club $E\subseteq \kappa$ such that $(T\restriction E, <_T)$ is normal and splitting.
Consider the set $D:=\{\alpha<\kappa\mid \otp(E\cap\alpha)=\mu^\alpha\}$ which is a subclub of $E$.
It is clear that $\mathbf T':=(T\restriction D, <_T)$ is a normal $\kappa$-Souslin tree.
\begin{claim} \begin{enumerate}
\item $\mathbf T'$ is prolific and
$\mu$-splitting;
\item if $\mathbf T$ is $\chi$-complete, then so is $\mathbf T'$;
\item if $\mathbf T$ is regressive, then so is $\mathbf T'$.
\end{enumerate}
\end{claim}
\begin{proof} (1) Fix an arbitrary node $x$ of $\mathbf T'$, so that $x\in T\restriction D$.
Let $\delta:=\min(D\setminus(\h_{\mathbf T}(x)+1))$.
As $\delta\in D\s E$ and $(T\restriction E, <_T)$ is normal, let us fix $z\in T_\delta$ with $x<_T z$.
Let $e:=\{\varepsilon\in E\mid \h(x)<\varepsilon<\delta\}$.
Note that from $\otp(E\cap\delta)=\mu^\delta$, it follows that $\otp(e)=\delta$ and $|e|\ge\mu$.
For every $\varepsilon\in e$, let $y_\varepsilon$ denote the unique element of $T_\varepsilon$ satisfying $y_\varepsilon <_T z$,
and denote $\varepsilon^+:=\min(e\setminus(\varepsilon+1))$.
Then, using the fact that $(T\restriction E, <_T)$ is normal and splitting,
for every $\varepsilon\in e$, pick $\hat y_\varepsilon\in T_{\varepsilon^+}$ such that $y_\varepsilon<_T \hat y_\varepsilon$ and $\hat y_\varepsilon\neq y_{\varepsilon^+}$,
and then pick $z_\varepsilon\in T_\delta$ with $\hat y_\varepsilon<_T z_\varepsilon$.
Then $\{ z_\varepsilon\mid \varepsilon \in e\}$ consists of $|e|$-many immediate successors of $x$ in $\mathbf T'$.
(2) Since $D$ is closed.
(3) Suppose $\rho:T\restriction\acc(\kappa)\rightarrow T$ witnesses that $\mathbf T$ is regressive. Define $\rho':T\restriction\acc(D)\rightarrow T\restriction D$ as follows.
Given $\alpha\in\acc(D)$ and $x\in T_\alpha$, let $\delta:=\min(D\setminus(\h_{\mathbf T}(\rho(x))+1))$,
and then let $\rho'(x)$ be the unique $y<_T x$ with $\h_{\mathbf T}(y)=\delta$.
It is clear that $\rho'$ witnesses that $\mathbf T'$ is regressive.
\end{proof}
Recursively define a sequence of injections $\langle \pi_\alpha:T_\alpha\rightarrow\kappa\mid \alpha\in D\rangle$ such that for, every $\alpha\in D$:
\begin{itemize}
\item For every $\alpha'\in D\cap\alpha$, $\im(\pi_{\alpha'})\cap\im(\pi_\alpha)=\emptyset$;
\item $\biguplus\{\im(\pi_{\alpha'})\mid \alpha'\in D\cap(\alpha+1)\}$ is an ordinal.
\end{itemize}
Evidently, $\pi:=\bigcup_{\alpha\in D}\pi_\alpha$ is an injection from $T\restriction D$ onto $\kappa$.
Let ${\lhd}:=\{ (\pi(x),\pi(y))\mid (x,y)\in{<_T}\}$. Then $(\kappa,{\lhd})$ is an ordinal-based $\kappa$-Souslin tree order-isomorphic to $\mathbf T'$.
\end{proof}
A richer introduction to abstract transfinite trees, Aronszajn trees and Souslin trees may be found in Section~2 of \cite{paper23},
and a comprehensive treatment of the consistency of existence of Souslin trees may be found in Section~6 of the same paper.
For our purpose, it suffices to mention the following fact:
\begin{fact}\label{fact216} For an infinite cardinal $\lambda$ satisfying $\diamondsuit(\lambda^+)$:\footnote{Note that, by \cite{Sh:922}, for any \emph{uncountable} cardinal $\lambda$, $\diamondsuit(\lambda^+)$ holds iff $2^\lambda=\lambda^+$.}
\begin{enumerate}
\item \cite{Jensen_V=L_Diamond} Assuming $\lambda^{<\lambda}=\lambda$, if $\diamondsuit(E^{\lambda^+}_\lambda)$ holds, then there exists a $\lambda$-complete $\lambda^+$-Souslin tree;
\item \cite{paper37} Assuming $\lambda^{<\lambda}=\lambda$, if $\square(\lambda^+,{<}\lambda)$ holds, then there exists a $\lambda$-complete $\lambda^{+}$-Souslin tree;
\item \cite{paper24,paper51} Assuming $\lambda^{\aleph_0}=\lambda$ or $\lambda\ge\beth_\omega$ or $\mathfrak b\le\lambda<\aleph_\omega$,
if $\square(\lambda^+)$ holds, then there exists a maximally-complete $\lambda^+$-Souslin tree and there exists a regressive $\lambda^+$-Souslin tree;
\item\cite{paper31} If $\square_{\lambda^+}$ holds, then there exists a $\lambda^+$-complete $\lambda^{++}$-Souslin tree.
\end{enumerate}
\end{fact}
We now introduce a new characteristic of Souslin trees.
\begin{definition}[The levels of vanishing branches] For a $\kappa$-Souslin tree $\mathbf T=(T,{<_T})$, let $V(\mathbf T)$ denote the set of all $\alpha\in\acc(\kappa)$
such that, for every $x\in T\restriction\alpha$, there exists a vanishing $\alpha$-branch containing $x$.
\end{definition}
It follows from a theorem of Shelah \cite{Sh:624} that it is consistent that for some Mahlo cardinal $\kappa$, there exists a $\kappa$-Souslin tree $\mathbf T$ for which $V(\mathbf T)=\emptyset$.
\begin{lemma}\label{regressivecharacteristic} Suppose that $\mathbf T=(T,{<_T})$ is a normal $2$-splitting regressive $\kappa$-Souslin tree.
Then $V(\mathbf T)\supseteq E^\kappa_\omega$.
\end{lemma}
\begin{proof} Towards a contradiction, suppose that $\alpha\in E^\kappa_\omega\setminus V(\mathbf T)$.
Fix $x\in T\restriction\alpha$ such that every $\alpha$-branch $B$ with $x\in B$ has an upper bound in $T$.
Fix a strictly increasing sequence of ordinals $\langle \alpha_n\mid n<\omega\rangle$ that converges to $\alpha$, and $\alpha_0:=\h(x)$.
We shall recursively construct an array $\langle x_t\mid t\in{}^{<\omega}2\rangle$ in such a way that $x_t\in T_{\alpha_{|t|}}$.
Set $x_\emptyset:=x$. Now, for every $t\in{}^{<\omega}2$ such that $x_t$ has already been defined, since $\mathbf T$ is $2$-splitting and normal,
we may find $y\neq z$ in $T_{\alpha_{|t|+1}}$ with $x<_T y,z$; then, let $x_{t{}^\smallfrown\langle 0\rangle}:=y$ and $x_{t{}^\smallfrown\langle 1\rangle}:=z$.
Next, given $t\in{}^{\omega}2$, let $B_t:=\{ y\in T\restriction\alpha\mid \exists n<\omega(y<_T x_{t\restriction n})\}$.
As $B_t$ is an $\alpha$-branch containing $x$, it must have a bound $b_t\in T$.
Clearly, $\h(b_t)\ge\alpha$, and we may moreover assume that $\h(b_t)=\alpha$.
Note that the construction secures that, for all $t\neq t'$ in ${}^\omega2$, $b_t\neq b_{t'}$.
Let $\rho:T\restriction\acc(\kappa)\rightarrow T$ be a witness to the fact that $\mathbf T$ is regressive.
Next, for every $t\in{}^\omega2$, fix a large enough $n_t<\omega$ such that $\rho(b_t)<_T x_{t\restriction n_t}$.
By the pigeonhole principle, we may now fix $s\in{}^{<\omega}2$ such that $\{t\in{}^\omega2\mid t\restriction n_t=s\}$ is uncountable.
Pick $t\neq t'$ in ${}^\omega2$ such that $t\restriction n_{t}=s=t'\restriction n_{t'}$.
Then, $\rho(b_t)<_T x_s<_Tb_{t'}$ and $\rho(b_{t'})<_T x_s<_T b_t$,
contradicting the fact that $b_t\neq b_{t'}$.
\end{proof}
\begin{remark} It follows that if $\mathbf T$ is a normal $2$-splitting coherent $\kappa$-Souslin tree, then $V(\mathbf T)=E^\kappa_\omega$.
A consistent construction of such a tree may be found in \cite[Proposition~2.5]{paper22}.
\end{remark}
\begin{lemma}\label{vanishcomplete}
Suppose that $\mathbf T=(T,{<_T})$ is a normal maximally-complete $\lambda$-splitting $\lambda^+$-Souslin tree.
Then $V(\mathbf T)\supseteq E^{\lambda^+}_{\chi}$ for the (regular) cardinal $\chi:=\log_\lambda(\lambda^+)$.
\end{lemma}
\begin{proof} Towards a contradiction, suppose that $\alpha\in E^{\lambda^+}_{\chi}\setminus V(\mathbf T)$.
Fix $x\in T\restriction\alpha$ such that every $\alpha$-branch $B$ with $x\in B$ has an upper bound in $T$.
Fix a strictly increasing and continuous sequence of ordinals $\langle \alpha_\epsilon\mid \epsilon<\chi\rangle$ that converges to $\alpha$, and $\alpha_0:=\h(x)$.
Very much like the proof of Lemma~\ref{regressivecharacteristic}, we may recursively construct an array $\langle x_t\mid t\in{}^{<\chi}\lambda\rangle$ in such a way that:
\begin{itemize}
\item $x_\emptyset=x$;
\item for all $t\in{}^{<\chi}\lambda$, $x_t\in T_{\alpha_{\dom(t)}}$;
\item for all $t,s\in {}^{<\chi}\lambda$, if $t\s s$, then $x_t<_T x_s$;
\item for all $t\in{}^{<\chi}\lambda$ and $i<j<\lambda$, $x_{t{}^\smallfrown\langle i\rangle}\neq x_{t{}^\smallfrown\langle j\rangle}$.
\end{itemize}
For each $t\in{}^{\chi}\lambda$, find $b_t\in T_\alpha$ such that, for every $\epsilon<\chi$, $x_{t\restriction\epsilon}<_T b_t$.
Then, $\{ b_t\mid t\in{}^{\chi}\lambda\}$ is an antichain of size $\ge\lambda^+$ in $\mathbf T$. This is a contradiction.
\end{proof}
\begin{remark} It follows that if $\mathbf T$ is a normal $\lambda$-complete $\lambda$-splitting $\lambda^+$-Souslin tree, then
$V(\mathbf T)=E^{\lambda^+}_{\lambda}$.
\end{remark}
\subsection{Deriving our guessing principle from a Souslin tree}
\begin{theorem}\label{thm111}
Suppose that $\mathbf T=(T,{<_T})$ is an ordinal-based $\chi$-splitting $\kappa$-Souslin tree,
and $\theta\le\chi$ is a cardinal satisfying $\kappa^{<\theta}=\kappa$.
Then, for every collection $\mathcal S$ of pairwise disjoint stationary subsets of $V(\mathbf T)\cap E^\kappa_{\chi}$,
there exists an $\ad$-multi-ladder system $\langle \mathcal A_{\alpha}\mid\alpha\in E^\kappa_{\chi}\rangle$
satisfying the following.
For every $\mathcal B\s[\kappa]^{\kappa}$ with $|\mathcal B|<\theta$,
every $S\in\mathcal S$,
and every cardinal $\mu<\kappa$,
the following set is stationary:
$$G_{\ge\mu}(S,\mathcal B):=\{\alpha\in S\mid |\mathcal A_\alpha|\ge\mu\ \&\ \forall (A,B)\in\mathcal A_\alpha\times\mathcal B~[\sup(A\cap B)= \alpha]\}.$$
\end{theorem}
\begin{proof}
As $\mathbf T$ is $\chi$-splitting and prolific, for each $w\in T$, we may fix an injective sequence $\langle w_i\mid i<\max\{\chi,\h(w)\}\rangle$ consisting of immediate successors of $w$.
As $\kappa^{<\theta}=\kappa$, we may fix an injective enumeration $\langle W_\eta\mid \eta<\kappa\rangle$ of all subsets $W$ of $T$ such that:
\begin{itemize}
\item $0<|W|<\theta$, and
\item $\h\restriction W$ is a constant function whose sole value is in $[\chi,\kappa)$.
\end{itemize}
\begin{claim}\label{claim1112} For every $\alpha\in E^\kappa_{\chi}$,
there exists a cofinal subset $A_\alpha$ of $\alpha$ which is an antichain in $\mathbf T$.
\end{claim}
\begin{proof} Fix an arbitrary $\alpha\in E^\kappa_{\chi}$,
and let $X$ be an arbitrary cofinal subset of $\alpha$ of order-type $\chi$.
If there exists some $\epsilon<\kappa$ such that $|X\cap T_\epsilon|=\chi$,
then we are done by letting $A_\alpha:=X\cap T_\epsilon$. Thus, hereafter assume this is not the case,
and pick a sequence $\langle x^j\mid j<\chi\rangle$
of elements of $X$ for which $\langle \h(x^j)\mid j<\chi\rangle$ is strictly increasing.
For notational simplicity, let us assume that $\{ x^j\mid j<\chi\}=X$.
Now, there are two cases to consider:
$\br$ Suppose that there exists a node $w\in T$ such that, for every $i<\chi$, there exists $j_i<\chi$ such that $x^{j_i}$ extends $w_i$.
Recalling that $\langle w_i\mid i<\chi\rangle$ is an injective sequence of immediate successors of $w$,
we infer that $A_\alpha:=\{ x^{j_i}\mid i<\chi\}$ is an antichain as sought.
$\br$ Suppose not. In particular, for each $j<\chi$ (using $w:=x^j$), we may fix $i_j<\chi$
such that $(x^j)_{i_j}$ is not extended by any element of $X$.
We claim that $A_\alpha:=\{ (x^j)_{i_j}\mid j<\chi\}$ is as sought.
For every $j<\chi$, we have $\h(x^j)<\h((x^j)_{i_j})<\h(x^{j+2})$,
so that $x^j\in (x^j)_{i_j}\in x^{j+2}$, and hence $A_\alpha$ is yet another cofinal subset of $\alpha$.
To see that $A_\alpha$ is an antichain, suppose that there exists a pair $j<j'$
such that $(x^j)_{i_j}$ is comparable with $(x^{j'})_{i_{j'}}$.
As $\h((x^{j})_{i_j})=\h(x^j)+1<\h(x^{j'})+1=\h((x^{j'})_{i_{j'}})$,
it follows that $(x^j)_{i_j}$ is extended by $(x^{j'})_{i_{j'}}$,
and in particular, $(x^j)_{i_j}$ is extended by $x^{j'}$ which is an element of $X$, contradicting the choice of $i_j$.
\end{proof}
Next, suppose that we are given a collection $\mathcal S$ of pairwise disjoint stationary subsets of $V(\mathbf T)\cap E^\kappa_{\chi}$.
As $T=\kappa$ and $|T\restriction\alpha|<\kappa$ for all $\alpha<\kappa$, $C:=\{ \alpha<\kappa\mid \alpha=T\restriction\alpha\}$ is a club in $\kappa$.
Let $\langle S_\eta\mid \eta<\kappa\rangle$ be a sequence of pairwise disjoint subsets of $E^{\kappa}_{\chi}\cap\acc(C)$ satisfying:
\begin{itemize}
\item For every $S\in\mathcal S$, $S_\eta\cap S$ is stationary;
\item For every $\eta<\kappa$, $\min(S_\eta)$ is greater than the unique element of $\{\h(w)\mid w\in W_\eta\}$,
which we hereafter denote by $\epsilon_\eta$.
\end{itemize}
Let $R:=E^\kappa_{\chi}\setminus\biguplus_{\eta<\kappa}S_\eta$.
For every $\alpha\in R$, we appeal to Claim~\ref{claim1112} and
pick a cofinal subset $A_\alpha\s\alpha$ which is an antichain in $\mathbf T$.
By possibly thinning out, we may also assume that $\otp(A_\alpha)=\cf(\alpha)$.
Then, we set $\mathcal A_\alpha:=\{A_\alpha\}$.
Next, let $\alpha\in \biguplus_{\eta<\kappa}S_\eta$ be arbitrary.
Let $\eta<\kappa$ be the unique ordinal such that $\alpha\in S_\eta$.
As $\alpha\in V(\mathbf T)$,
for every $w\in W_\eta$ and every $i<\epsilon_\eta$, we may find a subset $A^w_{\alpha,i}$ of $T$ such that:
\begin{enumerate}
\item $A^w_{\alpha,i}$ is a chain with minimal element $w_i$;
\item $\{ \h(x)\mid x\in A^w_{\alpha,i}\}=\alpha\setminus\epsilon_\eta+1$;
\item there exists no $z\in T_\alpha$ such that, for all $x\in A^w_{\alpha,i}$, $x<_T z$.
\end{enumerate}
Finally, let $\mathcal A_\alpha:=\{ A_{\alpha,i}\mid i<\epsilon_\eta\}$, where $A_{\alpha,i}:=\bigcup_{w\in W_\eta}A^w_{\alpha,i}$.
\begin{claim}\label{Claim - Guess disjoint cofinal}
$ \langle A_{\alpha,i} \mid i<\epsilon_\eta\rangle $ is a sequence of pairwise disjoint cofinal subsets of $\alpha$.
\end{claim}
\begin{proof} Let $w,u\in W_\eta$ and $i,j<\epsilon_\eta$ and suppose that $x\in A^w_{\alpha,i}\cap A^{u}_{\alpha,j}$.
By Clause~(1) above, $x$ extends both $w_i$ and $u_{j}$.
But $w_i$ and $u_{j}$ are predecessors of $x$ at the same level $T_{\epsilon_\eta+1}$, so that $w_i=u_j$ and it easily follows that $(w,i)=(u,j)$.
For all $w\in W_\eta$ and $i<\epsilon_\eta$, it follows from $\alpha\in C$ and Clause~(2) above that $A^w_{\alpha,i}\s\alpha$,
so that $A_{\alpha,i}\s\alpha$.
In addition, as $\alpha\in\acc(C)$, if we pick any $w\in W_\eta$, then it follows from Clause~(2) above that $\sup\{ \beta\in C\mid A^w_{\alpha,i}\cap T_\beta\neq\emptyset\}=\alpha$,
and hence $A_{\alpha,i}$ is cofinal in $\alpha$.
\end{proof}
\begin{claim}\label{Claim - Guess almost disjoint} Let $A,A'\in\bigcup_{\alpha\in E^{\kappa}_{\chi}}\mathcal A_\alpha$ with $A\neq A'$. Then $\sup(A\cap A')<\sup(A)$.
\end{claim}
\begin{proof} Let $\alpha,\alpha'$ be such that $A\in\mathcal A_\alpha$ and $A'\in\mathcal A_{\alpha'}$.
As the elements of $\mathcal A_\alpha$ are pairwise disjoint, we may assume that $\alpha\neq\alpha'$.
If $\alpha'<\alpha$, then $\sup(A\cap A')\le\alpha'<\alpha$, so assume that $\alpha'>\alpha$.
$\br$ If $\alpha'\in R$, then $\otp(A\cap A')\le\otp(\alpha\cap A')<\otp(A')=\cf(\alpha)$, so that $\sup(A\cap A')<\alpha$.
$\br$ If $\alpha'\notin R$ and $\alpha\in R$,
then $A$ is antichain, while $A'$ is the union of ${<}\theta$ many chains,
so that $|A\cap A'|<\theta\le\chi=\cf(\alpha)$,
and again $\sup(A\cap A')<\alpha$.
$\br$ If $\alpha,\alpha'\notin R$, then let $\eta,\zeta,i,j$ be such that $A=\bigcup_{w\in W_\eta}A^w_{\alpha,i}$
and $A'=\bigcup_{u\in W_\zeta}A^u_{\alpha,j}$.
Since $\max\{|W_\eta|,|W_\zeta|\}<\theta\le\chi=\cf(\alpha)$, it suffices to show that for each $w\in W_\eta$ and $u\in W_\zeta$,
$\sup(A^w_{\alpha,i}\cap A^u_{\beta,j})<\alpha$.
But the latter follows from Clause~(3) above together with the fact that $\alpha\in C$.
\end{proof}
Thus, we are left with verifying the following.
\begin{claim}\label{Claim - Guess hitting}
Suppose $\mathcal B=\{B_\tau\mid \tau<\theta'\}$ is a family of cofinal subsets of $\kappa$, with $\theta'<\theta$.
Suppose $S\in\mathcal S$ and $\mu<\kappa$ is some cardinal.
Then the set $G_{\ge\mu}(S,\mathcal B)$ is stationary.
\end{claim}
\begin{proof} Recalling Lemma~\ref{Souslin_denseness}, for each $\tau<\theta'$, we may fix $w^\tau\in T$ such that $(w^\tau)^\uparrow\cap B_\tau$ is cofinal in $(w^\tau)^\uparrow$.
Since $\mathbf T$ is normal, we may extend the said elements to ensure that $\h\restriction\{ w^\tau\mid \tau<\theta'\}$ is a constant function
whose sole value is some $\epsilon<\kappa$ with $\epsilon\ge\max\{\chi,\mu\}$.
It follows that there exists (a unique) $\eta<\kappa$ such that $W_\eta=\{ w^\tau\mid \tau<\theta'\}$.
Next, since $\mathbf T$ has no antichains of size $\kappa$, we may fix a sparse enough club $C'\s C$ with $\min(C')>\epsilon_\eta$ such that,
for every pair of ordinals $\gamma<\beta$ from $C'$ and every $w^\tau\in W_\eta$,
the set $B_\tau\cap((w^\tau)^\uparrow)\setminus(T\restriction\gamma)$ contains a maximal antichain in itself which is a subset of $T\restriction\beta$.
Consider the stationary set $\Gamma:=S_\eta\cap\acc(C')$.
Now, let $\alpha\in\Gamma$ be arbitrary. By Claim~\ref{Claim - Guess disjoint cofinal}, $|\mathcal A_\alpha|=|\epsilon_\eta|\ge\mu$.
Next, let $\tau<\theta'$ and $A\in\mathcal A_\alpha$ be arbitrary.
Find $i<\epsilon_\eta$ such that $A=A_{\alpha,i}=\bigcup_{w\in W_\eta}A^w_{\alpha,i}$. In particular, $A\supseteq A^{w_\tau}_{\alpha,i}$.
As $\sup(C'\cap\alpha)=\alpha$, it thus suffices to show that for every $\gamma\in C'\cap\alpha$, $\sup(A^{w^\tau}_{\alpha,i}\cap B_\tau)\ge\gamma$.
For this, let $\gamma\in C'\cap\alpha$ be arbitrary. Let $\beta:=\min(C'\setminus(\gamma+1))$.
Let $x$ denote the unique element of $A^{w^\tau}_{\alpha,i}\cap T_\beta$. As $w^\tau<_T (w^\tau)_i\le_T x$, we may find $a\in B_\tau$ with $x\le_T a$.
As $\gamma<\beta$ is a pair of elements of $C'$, it follows that there exists $a'\in B_\tau$ with $a'\le_T a$ such that $\gamma\le \h(a')<\beta$.
As $x,a'\le_T a$ and $\h(a')>\epsilon_\eta=\h(w^\tau)$, it follows that $a'\in A^{w^\tau}_{\alpha,i}\setminus (T\restriction\gamma)$, as sought.
\end{proof}
This completes the proof of Theorem~\ref{thm111}.
\end{proof}
\begin{cor}\label{Cor - regressive Souslin implies almost disjoint clubsuit omega} Suppose that $\mathbf T$ is a $\kappa$-Souslin tree. For every $\chi\in\reg(\kappa)$ and
every collection $\mathcal S$ of pairwise disjoint stationary subsets of $E^{\kappa}_{\chi}$,
any of the following implies that $\clubsuit_{\ad}(\mathcal S, {<}\chi)$ holds:
\begin{enumerate}
\item $\kappa=\lambda^+$, $\lambda^{\aleph_0}>\lambda$ and $\chi=\aleph_0$;
\item $\kappa=\lambda^+$, $\chi=\log_\lambda(\lambda^+)$ and $\mathbf T$ is maximally-complete;
\item $\chi=\aleph_0$ and $\mathbf T$ is regressive.
\end{enumerate}
\end{cor}
\begin{proof} (1) This follows from Clause~(2).
(2) Appeal to Proposition~\ref{ordinalsbased},
Lemma~\ref{vanishcomplete} and Theorem~\ref{thm111} using $(\kappa,\mu,\theta):=(\lambda^+,\chi,\chi)$
to get a sequence $\langle \mathcal A_{\alpha}\mid\alpha\in E^{\lambda^+}_{\chi}\rangle$.
The only thing that possibly does not fit is that there may be $\alpha\in\bigcup\mathcal S$ for which $|\mathcal A_\alpha|\neq\chi$.
But this is easy to fix:
\begin{itemize}
\item[$\br$] For any $\alpha\in \bigcup\mathcal S$ such that $|\mathcal A_\alpha|>\chi$,
replace $\mathcal A_\alpha$ by some subset of it of size $\chi$.
\item[$\br$] For any $\alpha\in \bigcup\mathcal S$ such that $|\mathcal A_\alpha|<\chi$,
pick $A\in\mathcal A_\alpha$ and replace $\mathcal A_\alpha$ by some partition of $A$ into $\chi$ many sets.
\end{itemize}
(3) Appeal to Proposition~\ref{ordinalsbased},
Lemma~\ref{regressivecharacteristic} and Theorem~\ref{thm111} using $(\mu,\theta):=(\omega,\omega)$
to get a sequence $\langle \mathcal A_{\alpha}\mid\alpha\in E^{\kappa}_{\omega}\rangle$.
The only thing that possibly does not fit is that there may be $\alpha\in\bigcup\mathcal S$ for which $|\mathcal A_\alpha|\neq\omega$.
But we may fix it as in the previous case.
\end{proof}
Theorem~A now follows immediately.
\begin{cor}\label{Souslin implies almost disjoint clubsuit} \begin{enumerate}
\item If there exists a $\cf(\lambda)$-complete $\lambda^+$-Souslin tree,
then for every partition $\mathcal S$ of $E^{\lambda^+}_{\cf(\lambda)}$ into stationary sets,
$\clubsuit_{\ad}(\mathcal S,{<}{\cf(\lambda)})$ holds.
\item If there exists a regressive $\kappa$-Souslin tree,
then for every partition $\mathcal S$ of $E^\kappa_\omega$ into stationary sets,
$\clubsuit_{\ad}(\mathcal S,{<}\omega)$ holds.\qed
\end{enumerate}
\end{cor}
\begin{cor}\label{w1souslin} If there exists an $\omega_1$-Souslin tree, then $\clubsuit_{\ad}(\{\omega_1\},{<}\omega)$ holds.\qed
\end{cor}
\begin{cor}\label{clubadvsclub} It is consistent with $\ch$ that $\clubsuit_{\ad}(\{\omega_1\},{<}\omega)$ holds, but $\clubsuit(\omega_1)$ fails.
\end{cor}
\begin{proof} Start with a model of $\gch+\neg\diamondsuit(\omega_1)$ (e.g., Jensen's model \cite{Jensen_The_Souslin_Problem} of $\gch$ with no $\aleph_1$-Souslin trees).
Now, force to add a single Cohen real and work in the corresponding extension.
As this is a countable forcing, $\ch$ still holds and $\diamondsuit(\omega_1)$ still fails,
so that by Devlin's theorem (see Remark~\ref{diamondsuit iff clubsuit and ch}), $\clubsuit(\omega_1)$ fails as well.
Finally, by a theorem of Shelah \cite{MR768264}, the forcing to add a Cohen real introduces an $\omega_1$-Souslin tree,
so that, by the preceding corollary, $\clubsuit_{\ad}(\{\omega_1\},{<}\omega)$ holds.
\end{proof}
\begin{cor} The assertion that $\clubsuit_{\ad}(\mathcal S,{<}\omega)$ holds
for every partition $\mathcal S$ of $\omega_1$ into stationary sets is consistent with any of the two:
\begin{enumerate}
\item $\clubsuit_J(\omega_1)$ fails;
\item $\clubsuit_J(S)$ fails for some stationary $S\s\omega_1$, and $\ch$ holds.
\end{enumerate}
\end{cor}
\begin{proof} For a stationary subset $S$ of $\omega_1$, let $\unif(S,2)$ assert
that for every sequence of functions $\vec f=\langle f_\alpha:A_\alpha\rightarrow2\mid \alpha\in \acc(\omega_1)\cap S\rangle$ where
each $A_\alpha$ is a cofinal subset of $\alpha$ of order-type $\omega$,
there exists a function $f:\omega_1\rightarrow2$ that uniformizes $\vec f$, i.e., for every $\alpha\in S\cap\acc(\omega_1)$, $\Delta(f,f_\alpha):=\{ \beta\in A_\alpha\mid f(\beta)\neq f_\alpha(\beta)\}$ is finite.
\begin{claim} $\clubsuit_J(S)$ refutes $\unif(S,2)$.
\end{claim}
\begin{proof} Suppose that $\langle A_{\alpha,i}\mid\alpha\in S,~i<\omega\rangle$ is as in Definition~\ref{principles}(4).
In particular, for every $\alpha\in S\cap\acc(\omega_1)$, $A_\alpha:= A_{\alpha,0}\uplus A_{\alpha,1}$ is a cofinal subset of $\alpha$ of order-type $\omega$,
and we may define a function $f_\alpha:A_\alpha\rightarrow2$ via $f_\alpha(\beta):=0$ iff $\beta\in A_{\alpha,0}$.
Towards a contradiction, suppose that there exists a function $f:\omega_1\rightarrow 2$ that uniformizes $\vec f:=\langle f_\alpha\mid \alpha\in S\cap\acc(\omega_1)\rangle$.
By the pigeonhole principle, pick $j<2$ for which $B:=\{\beta<\omega_1\mid f(\beta)=j\}$ is uncountable.
Now, fix $\alpha\in S\cap\acc(\omega_1)$ such that $\sup(B\cap A_{\alpha,i})=\alpha$ for all $i<\omega$.
Let $i:=1-j$. Pick $\beta\in B\cap A_{\alpha,i}\setminus \Delta(f,f_\alpha)$.
Then $j=f(\beta)=f_\alpha(\beta)=i$. This is a contradiction.
\end{proof}
(1) As made clear by the proof of \cite[Theorem~5.2]{DvSh:65},
it is possible to force $\unif(\omega_1,2)$ via a finite support iteration of Knaster posets. In particular,
if we start with a ground model with an $\aleph_1$-Souslin tree, then we can force
$\unif(\omega_1,2)$ without killing the tree.
Now appeal to Corollary~\ref{Cor - regressive Souslin implies almost disjoint clubsuit omega}(1).
(2) In \cite[Theorems 2.1 and 2.4]{Sh:64}, $\diamondsuit(\omega_1)$ is shown to be consistent with the existence of a
stationary subset $S\s\omega_1$ on which $\unif(S,2)$ holds.
Now, appeal to Fact~\ref{fact216}(1) and Corollary~\ref{Cor - regressive Souslin implies almost disjoint clubsuit omega}(1).
\end{proof}
We conclude this subsection by stating an additional result, this time concerning the three-cardinal variant of $\clubsuit_{\ad}$ (recall Definition~\ref{threecardinalsvariant}):
\begin{theorem} Suppose that there exists a $\kappa$-Souslin tree.
Then $\clubsuit_{\ad}(\mathcal S,1,1)$ holds for some collection $\mathcal S$ of $\kappa$-many pairwise disjoint stationary subsets of $\kappa$.
If $\kappa$ is a successor cardinal, then moreover $\bigcup\mathcal S=E^\kappa_\chi$ for some cardinal $\chi\in\reg(\kappa)$.\qed
\end{theorem}
We omit the proof due to the lack of applications of $\clubsuit_{\ad}(\mathcal S,1,1)$, at present.
\subsection{The consistency of the negation of our guessing principle}
By Lemma~\ref{lemma216}, for a stationary set $S$ consisting of points of some fixed cofinality,
$\clubsuit(S)$ entails $\clubsuit_{\ad}(\{S\},{<}\omega)$. The next theorem shows that the restriction to a fixed cofinality is crucial.
\begin{theorem} If $\kappa$ is weakly compact, then for any $S$ with $\reg(\kappa)\s S\s\kappa$, $\clubsuit_{\ad}(\{S\},1,1)$ fails.
\end{theorem}
\begin{proof} Suppose that $\vec A=\langle A_\alpha\mid\alpha\in S\rangle$ is a $\clubsuit_{\ad}(\{S\},1,1)$-sequence,
with $\reg(\kappa)\s S\s\kappa$.
We define a $C$-sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$, as follows:
$\br$ Let $C_0:=\emptyset$.
$\br$ For every $\alpha<\kappa$, let $C_{\alpha+1}:=\{\alpha\}$.
$\br$ For every $\alpha\in\acc(\kappa)\setminus\reg(\kappa)$,
fix a closed and cofinal $C_\alpha\s\alpha$ with $\otp(C_\alpha)=\cf(\alpha)<\min(C_\alpha)$.
$\br$ For every $\alpha\in\reg(\kappa)$, let $B_\alpha:=\nacc(A_\alpha)$ and finally let $C_\alpha:={B_\alpha}\uplus{\acc^+(B_\alpha)}$.
Note that $B_\alpha$ is a cofinal subset of $A_\alpha$, and that $C_\alpha$ is a closed and cofinal subset of $\alpha$.
Towards a contradiction, suppose that $\kappa$ is weakly compact.
So, by \cite[Theorem~1.8]{TodActa}, we may fix a club $C\s\kappa$ such that, for every $\delta<\kappa$,
for some $\alpha(\delta)<\kappa$, $C\cap\delta=C_{\alpha(\delta)}\cap\delta$.
Consider the club $D:=\{\delta\in\acc(\kappa)\mid \otp(C\cap\delta)=\delta\}$.
\begin{claim} For every $\delta\in D$, $\alpha(\delta)\in \reg(\kappa)$.
\end{claim}
\begin{proof} Let $\delta\in D$. Since $C_{\alpha(\delta)}\cap\delta=C\cap\delta$ is infinite, $\alpha(\delta)\in\acc(\kappa)$.
Now, if $\alpha(\delta)\in\acc(\kappa)\setminus\reg(\kappa)$, then $\delta=\otp(C_{\alpha(\delta)}\cap\delta)\le\otp(C_{\alpha(\delta)})<\min(C_{\alpha(\delta)})$,
which is an absurd.
\end{proof}
Evidently, $B:=\nacc(C)$ is a cofinal subset of $\kappa$.
So, by the choice of $\vec A$,
$G:=\{\delta\in S\cap D\mid \sup(B\cap A_\delta)=\delta\}$ is stationary.
Pick a pair of ordinals $\delta_0<\delta_1$ from $G$.
For each $i<2$, since $\alpha(\delta_i)\in\reg(\kappa)$,
$$B\cap\delta_i=\nacc(C)\cap\delta_i=\nacc(C_{\alpha(\delta_i)})\cap\delta_i= B_{\alpha(\delta_i)}\cap\delta_i\s A_{\alpha(\delta_i)}.$$
As $\sup(B\cap A_{\delta_i})=\delta_i$ and $B\cap\delta_i\s A_{\alpha(\delta_i)}$, $\sup(A_{\alpha(\delta_i)}\cap A_{\delta_i})=\delta_i$,
so, since $\vec A$ is an AD-ladder system, it must be the case that $\alpha(\delta_i)=\delta_i$.
Altogether, $B\cap\delta_0\s A_{\delta_0}\cap A_{\delta_1}$, so that $\delta_0>\sup( A_{\delta_0}\cap A_{\delta_1})\ge\sup(B\cap\delta_0)=\delta_0$. This is a contradiction.
\end{proof}
An ideal $\mathcal I$ consisting of countable sets is said to be a \emph{P-ideal}
iff every countable family of sets in the ideal admits a pseudo-union in the ideal. That is,
for every sequence $\langle X_n \mid n<\omega\rangle$ of elements of $\mathcal I$,
there exists $X\in\mathcal I$ such that $X_n\setminus X$ is finite for all $n<\omega$.
\begin{definition}[Todor\v{c}evi\'{c}, \cite{MR1809418}] The P-ideal dichotomy ($\pid$) asserts that for every P-ideal $\mathcal I$ consisting of countable subsets of some set $Z$, either:
\begin{enumerate}
\item there is an uncountable $B\s Z$ such that $[B]^{\aleph_0}\s\mathcal I$, or
\item there is a sequence $\langle B_n\mid n<\omega\rangle$ such that $\bigcup_{n<\omega}B_n=Z$ and, for each $n<\omega$, $[B_n]^{\aleph_0}\cap\mathcal I=\emptyset$.
\end{enumerate}
\end{definition}
We denote by $\pid_{\aleph_1}$ the restriction of the above principle to $Z:=\aleph_1$.
This special case was first introduced and studied by Abraham and Todor\v{c}evi\'{c} in \cite{MR1441232}, and was denoted there by $({}^\ast)$.
\begin{lemma}\label{pfa implies clubsuit_AD(omega_1) fails} Suppose that $\pid_{\aleph_1}$ holds and $\mathfrak b>\omega_1$.
Then, for any stationary $S\s \omega_1$, $\clubsuit_{\ad}(\{S\},1,1)$ fails.
\end{lemma}
\begin{proof} Towards a contradiction, suppose that $S\s\omega_1$ is stationary, and that $\vec A=\langle A_\alpha \mid\alpha\in S\rangle$ is a $\clubsuit_{\ad}(\{S\},1,1)$-sequence.
Let $$\mathcal I:=\{ X\in[\omega_1]^{\le\aleph_0}\mid \forall\alpha\in\acc(\omega_1)\cap S[A_\alpha\cap X\text{ is finite}]\}.$$
It is clear that $\mathcal I$ is an ideal.
\begin{claim} $\mathcal I$ is a P-ideal.
\end{claim}
\begin{proof} Let $\vec X=\langle X_n \mid n<\omega\rangle$ be a sequence of elements of $\mathcal I$.
We need to find a pseudo-union of $\vec X$ that lies in $\mathcal I$.
As $\mathcal I$ is downward closed, we may assume that $\langle X_n\mid n<\omega\rangle$ consists of pairwise disjoint sets.
Fix a bijection $e:\omega\leftrightarrow\biguplus_{n<\omega}X_n$. Then, for all $\alpha\in S$, define a function $f_\alpha:\omega\rightarrow\omega$ via
$$f_\alpha(n):=\min\{ m<\omega\mid X_n\cap A_\alpha\s e``m\}.$$
As $\mathfrak b>\omega_1$, let us fix a function $f:\omega\rightarrow\omega$ such that $f_\alpha<^* f$ for all $\alpha\in S$.
Set $X:=\biguplus\{ X_n\setminus e[f(n)]\mid n<\omega\}$. Clearly, for every $n<\omega$, $X_n\setminus X$ is a subset of $e[f(n)]$, and, in particular, it is finite.
Towards a contradiction, suppose that $X\notin\mathcal I$. Fix $\alpha\in\acc(\omega_1)\cap S$ such that $A_\alpha\cap X$ is infinite.
Since $X\s\biguplus_{n<\omega}X_n$, but $A_\alpha\cap X_n$ is finite for all $n<\omega$, we may find a large enough $n<\omega$
such that $A_\alpha\cap X\cap X_n\neq\emptyset$ and $f_\alpha(n)<f(n)$.
Pick $\beta\in A_\alpha\cap X\cap X_n$.
By the definition of $f_\alpha$, $\beta\in e[f_\alpha(n)]$. But $f_\alpha(n)<f(n)$,
so that $\beta\in e[f(n)]$, contradicting the fact that $\beta\in X$.
\end{proof}
\begin{claim} Let $B\s\omega_1$ be uncountable.
\begin{enumerate}
\item There exists $X\in[B]^{\aleph_0}$ with $X\notin\mathcal I$;
\item There exists $X\in[B]^{\aleph_0}$ with $X\in\mathcal I$.
\end{enumerate}
\end{claim}
\begin{proof} As $B$ is uncountable, $G:=\{\alpha\in \acc(\omega_1)\cap S\mid \sup(A_\alpha\cap B)=\alpha\}$ is stationary.
(1) Fix arbitrary $\alpha\in G$.
Then $X:=A_\alpha\cap B$ is an element of $[B]^{\aleph_0}\setminus\mathcal I$.
(2) Let $\langle \alpha_n\mid n<\omega\rangle$ be some increasing sequence of elements of $G$.
For every $n<\omega$, let $\langle \alpha^m_n\mid m<\omega\rangle$ be the increasing enumeration of some cofinal subset of $A_{\alpha_n}\cap B$.
Furthermore, we require that, for all $n<\omega$, $\alpha_n<\alpha_{n+1}^0$.
Set $\beta:=\sup_{n<\omega}\alpha_n$.
As $\vec A$ is an $\ad$-ladder system, for every $\alpha\in S\cap\acc(\omega_1)\setminus\beta$,
we may define a function $f_\alpha:\omega\rightarrow\omega$ via:
$$f_\alpha(n):=\min\{ m<\omega\mid A_{\alpha_n}\cap A_\alpha\s \alpha^m_n\}.$$
As $\mathfrak b>\omega_1$, let us fix a function $f:\omega\rightarrow\omega$ such that $f_\alpha<^* f$ for all $\alpha\in S$.
Set $X:=\{ \alpha_n^{f(n)}\mid 0<n<\omega\}$. For every $n<\omega$, the interval $(\alpha_n,\alpha_{n+1})$ contains a single element of $X$,
so that $X$ is a cofinal subset of $\beta$ with $\otp(X)=\omega$.
In particular, $X\in[B]^{\aleph_0}$.
Towards a contradiction, suppose that $X\notin\mathcal I$. Fix $\alpha\in\acc(\omega_1)\cap S$ such that $A_\alpha\cap X$ is infinite.
Clearly, $\alpha\ge\beta$. So, we may find $k<\omega$ such that, for every integer $n\ge k$, $f_\alpha(n)<f(n)$.
As $A_\alpha\cap X$ is infinite, let us now pick a positive integer $n\ge k$ such that $\alpha_n^{f(n)}\in A_\alpha$.
Recalling that $\{\alpha_n^m\mid m<\omega\}\s A_{\alpha_n}$,
we altogether infer that $\alpha_n^{f(n)}\in A_{\alpha_n}\cap A_\alpha\s \alpha_n^{f_\alpha(n)}$.
In particular, $\alpha_n^{f(n)}<\alpha_n^{f_\alpha(n)}$,
contradicting the fact that $f_\alpha(n)<f(n)$.
\end{proof}
Altogether, $\mathcal I$ is a P-ideal for which the two alternatives of $\pid_{\aleph_1}$ fail. This is a contradiction.
\end{proof}
\begin{cor} \begin{enumerate}
\item $\pfa$ implies that $\clubsuit_{\ad}(\{\omega_1\},1,1)$ fails;
\item $\clubsuit_{\ad}(\{\omega_1\},1,1)$ does not follow from the existence of an almost Souslin tree.
\end{enumerate}
\end{cor}
\begin{proof} (1) It is well-known that $\pfa$ implies $\pid+\ma_{\aleph_1}$. In particular, it implies $\pid_{\aleph_1}$ together with $\mathfrak b>\omega_1$.
(2) Almost Souslin trees were defined in \cite[\S3]{MR548979}.
In \cite{MR4128470} and \cite{MR4105604} one can find models of $\pid$ with $\mathfrak p>\omega_1$ (in particular, $\mathfrak b>\omega_1$),
in which there exists an Aronszajn tree which is almost Souslin.
\end{proof}
\begin{question}\label{mavsad}
Does $\ma_{\aleph_1}$ imply that $\clubsuit_{\ad}(\omega_1)$ fails?
\end{question}
\begin{question} Is $\ch$ consistent with the failure of $\clubsuit_{\ad}(\omega_1)$?
\end{question}
Note that a combination of the main results of \cite{Juhasz_clubsuit_ostaszewski,CH_with_no_Ostaszweski_spaces}
implies that $\ch$ is consistent with the failure of $\clubsuit_J(\omega_1)$.
\section{A Ladder-system Dowker space}\label{sectionladdersystemspace}
In \cite{Good_Dowker_large_cardinals}, Good constructed a Dowker space of size $\kappa^+$ using $\clubsuit(S,2),$ where $S$ is a non-reflecting stationary subset of $E^{\kappa^+}_{\omega}$.
We won't define the principle $\clubsuit(S,2)$,
but only mention that, by Fact~\ref{clubfacts}\eqref{matrix clubsuit from one}, it is no stronger than $\clubsuit(S)$.
In this section, a ladder-system Dowker space of size $\kappa $ is constructed under the assumption of $\clubsuit_{\ad}(\mathcal S,1,2)$, where $\mathcal S$ is an infinite partition of some non-reflecting stationary subset of $\kappa$.
By Lemma~\ref{lemma216}, $\clubsuit(S)$ implies $\clubsuit_{\ad}(\mathcal S,{<}\omega)$, which surely implies $\clubsuit_{\ad}(\mathcal S,1,2)$,
so, our construction in particular gives a ladder-system Dowker space in Good's scenario.
It also gives ladder-system Dowker spaces in scenarios considered by Rudin \cite{kappa_Dowker_from_souslin_Rudin} and Weiss \cite{MR628595},
as explained at the end of this section.
\medskip
The constructions in this and in the next section are motivated by the following lemma.
\begin{lemma}\label{Lemma - Dowker general argument}
Suppose that $\mathbb X=( X,\tau)$ is a normal Hausdorff topological space of size $\kappa$,
having no two disjoint closed sets of size $\kappa$.
If there exists a $\s$-decreasing sequence $\langle D_n \mid n<\omega \rangle$ of closed sets of cardinality $\kappa$ such that $\bigcap_{n<\omega} D_n = \emptyset$,
then $\mathbb X$ is Dowker.
\end{lemma}
\begin{proof}
Recall that, by \cite{Dowker_C.H.}, a space is Dowker iff it is Hausdorff, normal and
there is a $\s$-decreasing sequence $\langle D_n \mid n<\omega \rangle$ of closed sets with $ \bigcap_{n<\omega}D_n=\emptyset $,
such that, for every sequence $\langle U_n \mid n<\omega \rangle$ of open sets, if $ D_n\subseteq U_n $ for every $n<\omega$,
then $ \bigcap_{n<\omega}U_n\neq\emptyset $.
Now suppose that there exists a $\s$-decreasing sequence $\langle D_n \mid n<\omega \rangle$ of closed sets of cardinality $\kappa$ such that $\bigcap_{n<\omega} D_n = \emptyset$.
Suppose that $\langle U_n \mid n<\omega\rangle $ is a sequence of open sets such that $ D_n\subseteq U_n $ for every $n<\omega$.
For every $n<\omega$, $F_n:=X\setminus U_n$ is a closed set disjoint from $D_n$, and hence of cardinality $<\kappa$.
So, as $\omega<\cf(\kappa)=\kappa$, $\bigcup_{n<\omega}F_n$ has size less then $\kappa$.
In particular, $X\setminus \bigcup_{n<\omega}F_n\neq\emptyset$.
Altogether, $ \bigcap_{n<\omega}U_n=\bigcap_{n<\omega} X\setminus F_n=X\setminus \bigcup_{n<\omega} F_n\neq\emptyset$.
Recalling that $\mathbb X$ is normal, we altogether infer that $\mathbb X$ is Dowker.
\end{proof}
\medskip\noindent\textbf{The space.} Suppose that $S$ is a stationary subset of $\acc(\kappa)$,
$\mathcal S$ is a partition of $S$ with $|\mathcal S|=\aleph_0$,
and $\clubsuit_{\ad}(\mathcal S,1,2)$ holds.
Fix an $\ad$-ladder system $\langle A_{\alpha}\mid\alpha\in S\rangle$ as in Lemma~\ref{adone}.
Fix an injective enumeration $\langle S_{n+1} \mid n<\omega \rangle $ of $\mathcal S$, and let $S_0:=\kappa\setminus S$.
As $\langle S_n\mid n<\omega\rangle$ is a partition of $\kappa$,
for each $\alpha<\kappa$, we may let $n(\alpha)$ denote the unique $n<\omega$ such that $\alpha\in S_n$.
For each $n<\omega$, let $ W_n:=\bigcup_{i\leq n}S_i $.
Finally, define a sequence $\vec L=\langle L_\alpha\mid\alpha<\kappa\rangle$ via:
$$L_\alpha:=\begin{cases}
W_{n(\alpha)-1}\cap A_\alpha,&\text{if }n(\alpha)>0\ \&\ \sup(W_{n(\alpha)-1}\cap A_\alpha)=\alpha;\\
\emptyset,&\text{otherwise.}
\end{cases}$$
\begin{lemma}\label{lemma61}
\begin{enumerate}
\item\label{lemma - ladder-system club sequence - Clause W_n is open} For all $n<\omega$ and $\alpha\in S_{n+1}$, $L_\alpha\s W_n$;
\item $\bar S:=\{ \alpha\in\acc(\kappa)\mid \sup(L_\alpha)=\alpha\}$ is a stationary subset of $S$;
\item\label{adofladders} For all $\alpha\neq\alpha'$ from $\bar S$, $\sup(L_\alpha\cap L_{\alpha'})<\alpha$;
\item\label{lemma - ladder-system club sequence - Clause hitting} For all $B_0,B_1\in[\kappa]^\kappa$, there exists $m<\omega$ such that, for every $n\in\omega\setminus m$, the following set is stationary:
\[ \{ \alpha\in S_n \mid \sup(L_\alpha\cap B_0)=\sup(L_\alpha\cap B_1)=\alpha] \}.\]
\end{enumerate}
\end{lemma}
\begin{proof} (2) This follows from Clause~(4).
(3) For all $\alpha\neq\alpha'$ from $\bar S$, $\sup(L_\alpha\cap L_{\alpha'})\le\sup(A_\alpha\cap A_{\alpha'})<\alpha$;
(4) Given two cofinal subsets $B_0,B_1$ of $\kappa$,
find $m_0,m_1<\omega$ be such that $|B_0\cap S_{m_0}|=|B_1\cap S_{m_1}|=\kappa$. Evidently, $m:=\max\{n_0,n_1\}+1$ is as sought.
\end{proof}
Now, consider the ladder-system space $\mathbb X=(\kappa,\tau)$ which is determined by $\vec L$ (equivalently, determined by $\vec L\restriction\bar S$).
This means that a set $U\s\kappa$ is $\tau$-open iff, for every $\alpha\in U$, $L_\alpha\s^* U$.
Put differently, if we denote $N^\epsilon_\alpha:=(L_\alpha\setminus\epsilon)\cup\{\alpha\}$,
then, for every $\alpha<\kappa$, every neighborhood of $\alpha$ covers some element from
$\mathcal N_\alpha:=\{ N^{\epsilon}_\alpha\mid \epsilon<\alpha \}$.
\begin{definition}For any set of ordinals $N$, denote $N^-:=N\cap\sup(N)$.
\end{definition}
Note that, for all $\alpha<\kappa$ and $N\in\mathcal N_\alpha$, $N^-$ is either empty or a cofinal subset of $\alpha$.
\begin{lemma}\label{t1lemma}
The space $\mathbb X$ is $ T_1 $.
\end{lemma}
\begin{proof}
Let $x$ be an element of the space $\mathbb X$.
To see that $\kappa\setminus \{x\}$ is $\tau$-open,
notice that for every $y\in X$, $\mathcal N_y$ is a chain such that $\bigcap\mathcal N_y=\{y\}$.
In particular, for every $y\in X\setminus\{x\}$,
there exists $N_y\in\mathcal N_y$ with $N_y\s\kappa\setminus\{x\}$.
\end{proof}
\begin{lemma}\label{lemma35}
\begin{enumerate}
\item $\kappa\setminus\bar S$ is a discrete subspace of size $\kappa$;
\item For every $\xi<\kappa$, $\xi$ is $\tau$-open;
\item For every $\xi\in\kappa\setminus\bar S$, $\xi$ is $\tau$-closed;
\item For every $n<\omega$, $W_n$ is $\tau$-open.
\end{enumerate}
\end{lemma}
\begin{proof} (4) By Lemma~\ref{lemma61}\eqref{lemma - ladder-system club sequence - Clause W_n is open}.
\end{proof}
\begin{lemma}\label{another lambda^+ dowker - no two disjoint big closed sets clubsuit dowker}
There are no two disjoint $\tau$-closed subsets of cardinality $\kappa$.
\end{lemma}
\begin{proof} Towards a contradiction,
suppose that $K_0$ and $K_1$ are two disjoint $\tau$-closed subsets of cardinality $\kappa$.
Using Lemma~\ref{lemma61}\eqref{lemma - ladder-system club sequence - Clause hitting},
let us fix $n<\omega$ such that $\sup(L_\alpha\cap K_0)=\sup(L_\alpha\cap K_1)=\alpha$.
As both $K_0$ and $K_1$ are $\tau$-closed, this implies that $\alpha\in K_0$ and $\alpha\in K_1$, contradicting the fact $K_0$ and $K_1$ are disjoint.
\end{proof}
Following the terminology coined in \cite{MR44519} and \cite{MR2099600}, we introduce the following.
\begin{definition} The sequence $\vec L$ is said to be \emph{almost $\mathcal P_0$} iff, for every $\xi<\kappa$ and every function $c:\bar S\cap\xi\rightarrow\omega$,
there exists a function $c^*:\xi\rightarrow\omega$ such that, for every $\alpha\in\bar S\cap\xi$, $c^*\restriction L_\alpha$ is eventually constant with value $c(\alpha)$.
\end{definition}
\begin{lemma}\label{normal clubsuit dowker} If $\vec L$ is almost $\mathcal P_0$, then $\mathbb X$ is normal and Hausdorff.
\end{lemma}
\begin{proof} Suppose that $\vec L$ is almost $\mathcal P_0$.
Let $K_0$ and $K_1$ be disjoint $\tau$-closed subsets of $\kappa$.
By Lemma~\ref{another lambda^+ dowker - no two disjoint big closed sets clubsuit dowker}, at least one of them is bounded, say $K_0$.
Using Lemma~\ref{lemma35}(1), fix a large enough $\xi\in\kappa\setminus\bar S$ such that $K_0\subseteq \xi$.
Note that by Clauses (2) and (3) of Lemma~\ref{lemma35}, $\xi$ is clopen.
Now, set $K^0_0:=K_0$ and $K^0_1:=K_1$.
\begin{claim} Suppose $n<\omega$ and that $K^n_0$ and $K^n_1$ are disjoint $\tau$-closed sets with $K^n_0\s\xi$.
Then there exist disjoint $\tau$-closed sets $K^{n+1}_0$ and $K^{n+1}_1$ with $K^{n+1}_0\s\xi$ such that for all $i<2$:
\begin{enumerate}
\item $K^n_i\s K^{n+1}_i$;
\item for every $\alpha\in K^n_i\cap\xi$, $L_\alpha\s^* K^{n+1}_i$.
\end{enumerate}
\end{claim}
\begin{proof}
For every $i<2$, define $c_i:\bar S\cap\xi\rightarrow2$ via $c_i(\alpha):=1$ iff $\alpha\in K^n_i$.
Now, as $\vec L$ is almost $\mathcal P_0$, for each $i<2$, we may fix a function $c^*_i:\xi\rightarrow2$
such that, for every $\alpha\in\bar S\cap \xi$, $c_i^*\restriction L_\alpha$ is eventually constant with value $c_i(\alpha)$.
For each $i<2$, set $$K^{n+1}_i:=K^n_i\cup\{\gamma\in\xi\setminus K^n_{1-i}\mid c^*_i(\gamma)=1\ \&\ c^*_{1-i}(\gamma)=0\}.$$
Evidently $K_i^n\s K_i^{n+1}$. It is also easy to see that $K^{n+1}_0\cap K^{n+1}_1=\emptyset$.
Let $i<2$. To see that $K_i^{n+1}$ is $\tau$-closed,
fix an arbitrary nonzero $\alpha<\kappa$ such that $\sup(K_i^{n+1}\cap L_\alpha)=\alpha$,
and we shall prove that $\alpha\in K_i^{n+1}$. As $K_i^n\s K_i^{n+1}$, we may avoid trivialities, and assume that $\alpha\notin K_i^{n}$,
so that $\alpha$ belongs to the set
$$\{\gamma\in\xi\setminus K^n_{1-i}\mid c^*_i(\gamma)=1\ \&\ c^*_{1-i}(\gamma)=0\}.$$
Altogether, $\alpha\in(\bar S\cap \xi)\setminus K_i^n$, which must mean that $c_i(\alpha)=0$.
Fix a large enough $\epsilon<\alpha$ such that $c_i^*\restriction(L_\alpha\setminus\epsilon)$ is eventually constant with value $0$.
Then $\sup(K_i^{n+1}\cap L_\alpha)\le\epsilon<\alpha$, contradicting the choice of $\alpha$.
Finally, to verify Clause~(2), fix arbitrary $i<2$ and $\alpha\in K_i^n\cap\xi\cap\bar S$.
By the definition of the two functions, $c_i(\alpha)=1$ and $c_{1-i}(\alpha)=0$.
So, there exists a large enough $\epsilon<\alpha$ such that $c_i^*\restriction(L_\alpha\setminus\epsilon)$
is a constant function with value $1$,
and $c_{1-i}^*\restriction(L_\alpha\setminus\epsilon)$
is a constant function with value $0$. In effect, $L_\alpha\setminus\epsilon\s K_i^{n+1}$.
\end{proof}
By an iterative application of the preceding claim, we obtain a sequence of pairs $\langle (K_0^n,K_1^n)\mid n<\omega\rangle$.
Set $U_0:=\bigcup_{n<\omega}K_0^n$ and $U_1:=(\kappa\setminus \xi)\cup\bigcup_{n<\omega}K_1^n$.
By Clause~(2) of the preceding claim and the fact that $\xi$ is clopen, $U_i$ is open for each $i<2$.
Thus, we are left with verifying the following.
\begin{claim} $U_0\cap U_1=\emptyset$.
\end{claim}
\begin{proof} Suppose not, and pick $\alpha\in U_0\cap U_1$.
Notice that $U_0\subseteq \xi$, hence we can find $n_0,n_1<\omega$ such that $\alpha\in K_0^{n_0}\cap K^{n_1}_1$.
Then, for $n:=\max\{n_0,n_1\}$, we get that $\alpha\in K_0^n\cap K_1^n$, contradicting the fact that $K_0^n$ and $K_1^n$ are disjoint.
\end{proof}
This completes the proof of normality. Since $\mathbb X$ is $T_1$, it also follows that it is Hausdorff.
\end{proof}
\begin{cor}\label{clubsuit_AD_space_not_countably_paracompact}
If $\vec L$ is almost $\mathcal P_0$, then $\mathbb X$ is Dowker.
\end{cor}
\begin{proof}
For every $n<\omega$, set $D_n:=\kappa\setminus W_n$.
Then $\langle D_n \mid n<\omega \rangle$ is a $\s$-decreasing sequence of closed sets of cardinality $\kappa$ such that $\bigcap_{n<\omega} D_n = \emptyset$.
So, by Lemmas \ref{normal clubsuit dowker}, \ref{another lambda^+ dowker - no two disjoint big closed sets clubsuit dowker}
and \ref{Lemma - Dowker general argument}, $\mathbb X$ is Dowker.
\end{proof}
\begin{lemma}\label{lemma62} Each of the following two imply that $\vec L$ is almost $\mathcal P_0$:
\begin{enumerate}
\item $\ma({<}\kappa)$ holds and $\otp(L_\alpha)=\omega$ for all $\alpha\in\bar S$;
\item $\bar S$ is a non-reflecting stationary set.
\end{enumerate}
\end{lemma}
\begin{proof}
(1) This follows immediately from \cite[{\S}II, Theorem~4.3]{Proper_and_improper_forcing_Shelah_Book}.
(2) By Lemma~\ref{lemma61}\eqref{adofladders} and Proposition~\ref{Lemma - non-reflecting stat, diagonlization of initial seg},
we may fix a sequence $\langle f_\xi\mid \xi<\kappa\rangle$ such that, for every $\xi<\kappa$:
\begin{itemize}
\item $f_\xi$ is a regressive function from $\bar S\cap\xi$ to $\xi$;
\item the sets in $\langle L_\alpha\setminus f_\xi(\alpha) \mid \alpha\in\bar S\cap \xi\rangle$ are pairwise disjoint.
\end{itemize}
Now, given a nonzero $\xi<\kappa$, let $f^+_\xi:\xi\rightarrow\xi$ denote the function
such that $f^+_\xi(\alpha)=f_\xi(\alpha)$ for all $\alpha\in\bar S\cap\xi$,
and $f^+_\xi(\alpha)=0$ for all $\alpha\in\xi\setminus\bar S$.
Evidently, for every $\beta<\xi$, $\{ \alpha<\xi\mid \beta\in L_\alpha\setminus f^+_\xi(\alpha)\}$ is a subset of $\bar S$ containing at most one element.
So, for any function $c:\bar S\cap\xi\rightarrow\omega$, we may define a corresponding function $c^*:\xi\rightarrow\omega$ via:
$$c^*(\beta):=\begin{cases}
c(\alpha),&\text{if }\beta\in L_\alpha\setminus f^+_\xi(\alpha);\\
0,&\text{otherwise}.
\end{cases}$$
A moment's reflection makes it clear that $c^*$ is as sought.
\end{proof}
\begin{cor} Suppose that $\mathcal S$ is an infinite partition of some non-reflecting stationary subset of a regular uncountable cardinal $\kappa$.
If $\clubsuit_{\ad}(\mathcal S,1,2)$ holds,
then there exists a ladder-system Dowker space of cardinality $\kappa$.\qed
\end{cor}
\begin{remark} The preceding is the Introduction's Theorem~B.
\end{remark}
\begin{cor} If ${\ma}+ \clubsuit(E^{\mathfrak c}_\omega)$ holds, then there exists a ladder-system Dowker space over $\mathfrak c$.
In particular, if $\ma$ holds and $\mathfrak c$ is the successor of a cardinal of uncountable cofinality,
then there exists a ladder-system Dowker space over $\mathfrak c$.
\end{cor}
\begin{proof} The first part follows from Lemmas \ref{lemma216} and \ref{lemma62}(1).
For the second part, note that if $\ma$ holds and $\mathfrak c=\lambda^+$,
then $2^\lambda=\lambda^+$, so if, moreover, $\lambda$ is a cardinal of uncountable cofinality,
then by the main result of \cite{Sh:922}, $\diamondsuit(E^{\lambda^+}_\omega)$ holds. In particular, in this case, $\clubsuit(E^{\mathfrak c}_\omega)$ holds.
\end{proof}
\begin{remark} In \cite{MR628595}, Weiss proved that if $\ma$ and $\diamondsuit(E^{\mathfrak c}_\omega)$ both hold, then there exists a locally compact, first countable, separable, real compact, Dowker space of size $\mathfrak c$.
\end{remark}
\begin{cor}
If there exists a $\lambda$-complete $\lambda^+$-Souslin tree,
then there exists a ladder-system Dowker space over $\lambda^+$.
\end{cor}
\begin{proof}
By Corollary~\ref{Souslin implies almost disjoint clubsuit}, Lemma~\ref{lemma62}(2), and
the fact that $E^{\lambda^+}_\lambda$ is a non-reflecting stationary subset of $\lambda^+$.
\end{proof}
\begin{remark} In \cite{kappa_Dowker_from_souslin_Rudin}, Rudin constructed a Dowker space of size $\lambda^+$ from a $\lambda^+$-Souslin tree, for $\lambda$ regular.
\end{remark}
\begin{cor}
If there exist a regressive $\kappa$-Souslin tree and a non-reflecting stationary subset of $E^\kappa_\omega$,
then there exists a ladder-system Dowker space over $\kappa$.
\end{cor}
\begin{proof}
By Corollary~\ref{Cor - regressive Souslin implies almost disjoint clubsuit omega} and Lemma~\ref{lemma62}(2).
\end{proof}
\begin{remark} It is well-known (see \cite{Jensen_V=L_Diamond} or \cite{paper22}) that if $\kappa>\aleph_0$, $\square_\kappa$ holds and $2^\kappa=\kappa^+$,
then there exists a regressive $\kappa$-Souslin tree and there exists a non-reflecting stationary subset of $E^{\kappa^+}_\omega$.
In \cite{Good_Dowker_large_cardinals},
Good proved that if $\kappa>\aleph_0$, $\square_\kappa$ holds and $2^\kappa=\kappa^+$,
then there exists a Dowker space over $\kappa^+$ which is first countable, locally countable, locally compact, zero-dimensional, and collectionwise normal that is of scattered length $\omega$.
\end{remark}
\section{de Caux type spaces}\label{Collectionwisesection}
\subsection{Collectionwise normality}
In this short subsection, we present a sufficient condition for a certain type of topological space to be collectionwise normal.
This will be used in verifying that the spaces constructed later in this section are indeed collectionwise normal.
\begin{definition} Let $\mathbb X=(X,\tau)$ be a topological space.
\begin{itemize}
\item A sequence $\langle K_i\mid i<\theta\rangle$ of subsets of $ X $ is said to be \emph{discrete} iff for every $ x\in X $, there is an open neighborhood $U$ of $x$ such that $\{ i<\theta\mid U\cap K_i\neq\emptyset\}$ contains at most one element.
\item The space $ \mathbb X $ is said to be \emph{collectionwise normal} iff for every discrete sequence $ \langle K_i \mid i<\theta\rangle $ of closed sets,
there exists a sequence $ \langle U_i\mid i<\theta\rangle$ of pairwise disjoint open sets such that $ K_i\s U_i $ for all $i<\theta$.
\end{itemize}
\end{definition}
\begin{remark} Note that any collectionwise normal space is normal.
\end{remark}
Let $\mathbb X=(X,\tau)$ be some topological space determined by a sequence of weak neighborhoods, $\langle \mathcal N_x \mid x\in X \rangle$.
This means that a subset $ U \s X$ is $\tau$-open iff for any $x\in U $, there is $N\in\mathcal N_x$ with $N\s U$.
\begin{lemma}\label{general normal lemma}
Suppose that $\theta$ is some nonzero cardinal and that $\langle K_{i} \mid {i}<\theta\rangle$ is a discrete sequence of $\tau$-closed sets,
$O$ is a $\tau$-open set covering $\bigcup_{0<i<\theta}K_i$,
and there exists a transversal $\langle N_x\mid x\in X\rangle\in\prod_{x\in X}\mathcal N_x$ such that:
\begin{enumerate}[label=(\alph*)]
\item for all $x\in O$, $N_{x}\subseteq O$;
\item\label{disjoint tails general} for all $x\in O$ and $x'\in X\setminus\{x\}$, $N_{x}\cap N_{x'}\s\{x,x'\}$;
\item\label{tails outside H,K general} for all $x\in X$ and ${i}<\theta$, if $N_x\cap K_{i}\neq \emptyset$, then $x\in K_{i}$.
\end{enumerate}
Then there exists a sequence $\langle U_{i} \mid {i}<\theta \rangle$ of pairwise disjoint $\tau$-open sets such that $K_0\subseteq U_0$,
and $K_i\subseteq U_i\subseteq O$ for all nonzero $i<\theta$.
\end{lemma}
\begin{proof}
By recursion on $n<\omega$, we construct a matrix $\langle U_i^n \mid {i}<\theta,~n<\omega \rangle $, as follows:
\begin{itemize}
\item[$\br$] For each ${i}<\theta$, set $U^0_{i}:=K_{i}$.
\item[$\br$] For every $n<\omega$ such that $\langle U_i^n \mid {i}<\theta\rangle$ has already been defined, set
$U^{n+1}_{{i}}:= \bigcup\{N_x \mid x \in U_i^n\}$ for each $i<\theta$.
\end{itemize}
Evidently, $U_{i}:=\bigcup_{n<\omega}U_i^n$ is an open set covering $K_{i}$.
\begin{claim}\label{U is bounded by xi general} Let $i$ with $0<i<\theta$. Then $U_{i}\subseteq O$.
\end{claim}
\begin{proof} We have $U_{i,0}=K_{i}\s O$.
In addition, for every $n<\omega$ such that $U_i^n\s O$,
Clause~(a) implies that $N_x\subseteq O$ for every $x \in U_i^n$,
and hence $U_{i}^{n+1}\s O$.
\end{proof}
\begin{claim}
The sets in the sequence $\langle U_{i} \mid {i}<\theta \rangle$ are pairwise disjoint.
\end{claim}
\begin{proof} Suppose not. Fix ${i}\neq{i}'$ in $\theta$ such that $U_{{i}}\cap U_{i'}\neq \emptyset $.
Let $n:=\min \{ k<\omega \mid U_{{i}}^k\cap U_{{i}'} \neq \emptyset \}$,
and then let $n':=\min\{ k<\omega \mid U_i^n\cap U_{{i}'}^k\neq \emptyset \}$.
\begin{subclaim} $\min\{n,n'\}>0$.
\end{subclaim}
\begin{proof}
First, as $U_{{i}}^0=K_{i}$ is disjoint from $U_{{i}'}^0=K_{{i}'}$, we infer that $(n,n')\neq(0,0)$.
$\br$ If $n=0$ and $n'>0$, then let $y\in K_{i}\cap U_{{i}'}^{n'}$.
It follows that there exists some $x \in U_{{i}'}^{n'-1}$ such that $y\in K_{i}\cap N_x $.
By Clause~\ref{tails outside H,K general}, then, $x \in K_{i}$.
So $x \in U_{{i}}^0\cap U_{{i}'}^{n'-1}$, contradicting the minimality of $n'$.
$\br$ If $n>0$ and $n'=0$, then let $y\in U_i^n\cap K_{{i}'}$.
It follows that there exists some $x \in U_{{i}}^{n-1}$ such that $N_x\cap K_{{i}'}\neq\emptyset$.
By Clause~\ref{tails outside H,K general}, then, $x \in K_{{i}'}$.
So $x \in U_{{i}}^{n-1}\cap U_{{i}'}^0$, contradicting the minimality of $n$.
\end{proof}
It follows that, for all $y\in U_i^n$, either $y\in U_{{i}}^{n-1}$ or $y\in N_x $ for some $x \in U_{{i}}^{n-1}\setminus\{y\}$.
Likewise, for all $y\in U_{i'}^{n'}$, either $y\in U_{i'}^{n'-1}$ or $y\in N_x $ for some $x \in U_{i'}^{n'-1}\setminus\{y\}$.
Now, by the choice of the pair $(n,n')$, let us fix $y\in U_i^n\cap U_{{i}'}^{n'}$. There are four cases to consider:
\begin{enumerate}
\item $y\in U_{{i}}^{n-1}\cap U_{{i}'}^{n'-1}$.
In this case, $U_{{i}}^{n-1}\cap U_{{i}'} \neq \emptyset$, contradicting the minimality of $n$.
\item $y\in N_{x}\cap U_{{i}'}^{n'-1}$ for some $x \in U_{{i}}^{n-1}\setminus\{y\}$.
In this case, $N_{x}\subseteq U_i^n$, so $U_i^n\cap U_{{i}'}^{n'-1} \neq \emptyset$, contradicting the minimality of $n'$.
\item $y\in U_{{i}}^{n-1}\cap N_{x}$ for some $x\in U_{{i}'}^{n'-1}\setminus\{y\}$.
In this case, $N_{x}\subseteq U_{{i}'}^{n'}$, so $U_{i}^{n-1}\cap U_{{i}'} \neq \emptyset$, contradicting the minimality of $n$.
\item There exist $x \in U_{{i}}^{n-1}\setminus\{y\}$ and $x'\in U_{{i}'}^{n'-1}\setminus\{y\}$ such that $y \in N_{x}\cap N_{x'}$.
Equivalently, there exist $x \in U_{{i}}^{n-1}$ and $x'\in U_{{i}'}^{n'-1}$ such that $y \in(N_{x}\cap N_{x'})\setminus\{x,x'\}$.
In this case, there are two subcases:
\begin{enumerate}
\item[$\br$] If $x =x '$, then $x \in U_{i}^{n-1}\cap U_{{i}'}^{n'-1}$, contradicting the minimality of $n$.
\item[$\br$] If $x \neq x '$, then, by Claim~\ref{U is bounded by xi general}, either $x \in O$ or $x'\in O$.
This is in contradiction with Clause~\ref{disjoint tails general}.
\end{enumerate}
\end{enumerate}
Altogether, $\{U_{i}\mid {i}<\theta\}$ is a family of pairwise disjoint sets as sought.
\end{proof}
This completes the proof.
\end{proof}
\subsection{$O$-space}\label{higherSspaceSection}
This subsection is dedicated to proving Theorem~C:
\begin{theorem}\label{ospace}
Suppose that $\clubsuit_{\ad}(\{\omega_1\},\omega,1)$ holds. Then there exists a collectionwise normal,
non-Lindel\"of $O$-space.
\end{theorem}
\begin{remark}
Note that unlike Dahroug's construction that defines a topology over the $\aleph_1$-Souslin tree,
here the topology will be defined over $\omega_1$. Thus, when taken together with Corollary~\ref{w1souslin}, this appears to yield the first ``linear'' construction of an $O$-space from an $\aleph_1$-Souslin tree.
\end{remark}
\begin{remark} The arguments of this subsection immediately generalize to yield a collectionwise normal higher $O$-space from $\clubsuit_{\ad}(\{E^{\lambda^+}_\lambda\},\lambda,1)$.
The focus on the case $\lambda=\omega$ is just for simplicity.
\end{remark}
Let $\langle \mathcal A_{\alpha}\mid\alpha<\omega_1\rangle$ be a $\clubsuit_{\ad}(\{\omega_1\},\omega,1)$-sequence.
For each $\alpha\in\acc(\omega_1)$, fix an injective enumeration $\{ A_{\alpha+i}\mid i<\omega\}$ of the elements of $\mathcal A_\alpha$.
For every infinite $\xi<\omega_1$, as $\mathcal B_\xi:=\{ A_\beta\mid \omega\le\beta<\xi+\omega\}$ is a countable subset of $\bigcup_{\alpha\in E^{\omega_1}_\omega}\mathcal A_\alpha$,
we may appeal to Proposition~\ref{Proposition - disjointify multi-ladder system} to fix a function $f_\xi:\mathcal B_\xi\rightarrow\omega_1$ such that:
\begin{enumerate}
\item for every $B\in\mathcal B_\xi$, $f_\xi(B)\in B$;
\item the sets in $\langle B\setminus f_\xi(B) \mid B\in\mathcal B_\xi\rangle$ are pairwise disjoint.
\end{enumerate}
\begin{definition} For every $\beta<\omega_1$, let $\alpha_\beta:=\min\{\alpha\le\beta\mid \exists i<\omega(\beta=\alpha+i)\}$.
\end{definition}
We are now ready to define our topological space $\mathbb X=(\omega_1,\tau) $.
For all $\beta<\omega$, let $\mathcal N_\beta:=\{\{\beta\}\}$.
For all infinite $\beta<\omega_1$ and $\epsilon<\alpha_\beta$, denote $N_\beta^\epsilon:=(A_{\beta}\setminus\epsilon)\cup\{\beta\}$,
and then set $\mathcal N_\beta:=\{ N_\beta^\epsilon\mid \epsilon<\alpha_\beta\}$.
Now, a subset $ U \s \omega_1$ is $\tau$-open iff for any $\beta\in U $,
there is $N\in\mathcal N_\beta$ with $N\s U$.
It is easy to check that $\mathbb X $ is a $ T_1 $ topological space
and that, for every $\xi<\omega_1$, $\xi$ is $\tau$-open.
\begin{definition}
For any set of ordinals $N$, denote $N^-:=N\cap\sup(N)$.
\end{definition}
Note that for all $\beta<\omega_1$ and $N\in\mathcal N_\beta$, $N^-$ is a cofinal subset of $\alpha_\beta$.
\begin{remark} The topology of the space from the previous section
is such that a set $U$ is open iff, for every $\beta\in U$, $L_\beta\s^* U$,
and the topology of the space here is such that a set $U$ is open iff, for every $\beta\in U$, $A_\beta\s^* U$.
The approach seems identical, but there is a subtle difference:
in the previous section, for every ordinal $\beta$, we had $\sup(L_\beta)\in\{0,\beta\}$,
whereas here, for every ordinal $\beta$, we have $\sup(A_\beta)=\alpha_\beta\in\beta+1$.
\end{remark}
\begin{lemma}
For every $\alpha\in\acc(\omega_1)$, $\cl([\alpha,\alpha+\omega))=\omega_1\setminus\alpha$.
\end{lemma}
\begin{proof}
Let $A:=\cl([\alpha,\alpha+\omega))$.
We prove by induction on $ \delta\in\acc(\omega_1)\setminus\alpha $ that $ [\delta,\delta+\omega)\subseteq A$.
$\br$ For $\delta=\alpha$, trivially $ [\alpha,\alpha+\omega) \subseteq A$.
$\br$ Suppose $\delta\in\acc(\omega_1)\setminus\alpha$ and $[\delta,\delta+\omega)\s A$.
Put $\delta':=\delta+\omega$ and we shall prove that $[\delta',\delta'+\omega)\s A$.
Let $\beta \in [\delta',\delta'+\omega)$ be arbitrary.
For each $N\in \mathcal N_\beta$, we have $\sup(N\cap [\delta,\delta+\omega))=\delta+\omega$, and hence $\beta\in\cl([\delta,\delta+\omega))\s A$.
$\br$ Suppose that $\delta\in\acc(\acc(\omega_1)\setminus \alpha)$ and, for every $\gamma\in\acc(\delta)\setminus\alpha$, $[\gamma,\gamma+\omega)\s A$.
Therefore, $[\alpha,\delta) \subseteq A$.
For every $\beta \in [\delta,\delta+\omega)$ and $N\in \mathcal N_\beta$, $\sup(N\cap [\alpha,\delta))=\delta$,
and hence $\beta\in\cl([\alpha,\delta))\s A$.
\end{proof}
\begin{cor}\label{closed sets are countable or co countable}
\begin{enumerate}
\item $(\omega_1,\tau)$ is hereditary separable;
\item\label{Clause - S-space closed sets countable or co count.} Every $ \tau $-closed set is either countable or co-countable.
\end{enumerate}
\end{cor}
\begin{proof}
Let $B$ be an arbitrary uncountable subset of $\omega_1$.
By Clause~(\ref{clubsuit AD, Clause cofinal set}) of Definition~\ref{clubsuit AD definition},
we can find an $ \alpha\in\acc(\omega_1)$ such that $ \sup(A_{\alpha+i}\cap B)=\alpha $
for all $ i<\omega$.
Let $D:=B\cap\alpha$.
If follows that, for every $\beta\in[\alpha,\alpha+\omega)$ and every $N\in\mathcal N_\beta$,
$\sup(N\cap D)=\alpha$. So, $ [\alpha,\alpha+\omega) \subseteq \cl(D)$
and hence $\omega_1\setminus\alpha=\cl([\alpha,\alpha+\omega))\subseteq \cl(D)$.
\begin{enumerate}
\item As $B\setminus(\omega_1\setminus\alpha)=D$, it follows that $\cl(B)=\cl(D)$,
so that $D$ is a countable dense subset of the subspace $B$.
\item If $B$ is moreover closed, then $(\omega_1\setminus\alpha)\s B$, so that $B$ is co-countable. \qedhere
\end{enumerate}
\end{proof}
\begin{lemma} $\mathbb X$ is Hausdorff and collectionwise normal.
\end{lemma}
\begin{proof} As $\mathbb X$ is $T_1$, it suffices to verify that it is collectionwise normal.
Let $\vec K=\langle K_i \mid i<\theta \rangle$ be an arbitrary discrete sequence of closed sets,
for some cardinal $\theta$. To avoid trivialities, assume that $\theta\ge 2$.
\begin{claim} $\theta\le\omega$.
\end{claim}
\begin{proof} Otherwise, set $B:=\{\beta_i\mid i<\omega_1\}$ for some transversal $\langle \beta_i\ \mid i<\omega_1\rangle\in\prod_{i<\omega_1}K_i$.
As $B$ is necessarily uncountable, using Clause~(\ref{clubsuit AD, Clause cofinal set}) of Definition~\ref{clubsuit AD definition},
we may fix $\alpha\in\acc(\omega_1)$ such that $\sup(A_\alpha\cap B)=\alpha$.
Then any open neighborhood of $\alpha$ meets infinitely many elements of $\vec K$,
contradicting its discreteness.
\end{proof}
By Corollary~\ref{closed sets are countable or co countable}\eqref{Clause - S-space closed sets countable or co count.},
the sequence $\vec K$ contains at most one uncountable set.
By possibly re-indexing, we may assume that $\{ i<\theta\mid |K_i|=\aleph_1\}\s\{0\}$.
Now, as $\theta$ is countable,
we may find a large enough $\xi\in\acc(\omega_1)$ such that $K_i\subseteq \xi$ for all nonzero $i<\theta$.
\begin{claim}\label{ostaszewski tails claim}
There exists a sequence $\langle N_\beta\mid \beta<\omega_1\rangle\in\prod_{\beta<\omega_1}\mathcal N_\beta$
such that:
\begin{enumerate}[label=(\alph*)]
\item for all $\beta<\xi$, $N_\beta\subseteq\xi$;
\item for all $\beta<\xi$ and $\beta'\in\omega_1\setminus\{\beta\}$, $N_{\beta}^-\cap N_{\beta'}^-=\emptyset$;
\item for all $\beta<\omega_1$ and $i<\theta$, if $N_\beta\cap K_i\neq \emptyset$ then $\beta \in K_i$.
\end{enumerate}
\end{claim}
\begin{proof} There are three cases to consider:
$\br$ For every $\beta<\omega$, just set $N_\beta :=\{\beta\}$.
$\br$ For every $\beta\in \omega_1\setminus(\xi+\omega)$, as $\alpha_\beta>\xi$, we may let
$$\epsilon:=\max(\{\xi,\sup(A_\beta \cap K_0)\}\cap\alpha_\beta),$$
and then set $N_{\beta} := N_{\beta}^{\epsilon+1}$.
$\br$ Suppose $\beta$ is not of the above form. In particular, $A_\beta\in\mathcal B_\xi$
and $f_\xi(A_\beta)<\alpha_\beta$.
As $\vec K$ is discrete, let us pick an open neighborhood $U$ of $\beta$ which
for which $I:=\{ i<\theta\mid U\cap K_i\neq\emptyset\}$ contains at most one element.
Find $\varepsilon<\alpha_\beta$ such that $A_{\beta}\setminus \varepsilon\subseteq U$,
and then let
$$\epsilon:=\begin{cases}
\max(\{f_\xi(A_\beta),\varepsilon\}),&\text{if }I=\emptyset;\\
\max(\{f_\xi(A_\beta),\varepsilon,\sup(A_{\beta}\cap K_i)\}\cap\alpha_\beta),&\text{if }I=\{i\}.
\end{cases}$$
Finally, set $N_{\beta} := N_{\beta}^{\epsilon+1}$.
We omit the proof that $\langle N_\beta\mid\beta<\omega_1\rangle$ is as sought,
because a similar verification is given in details in the proof of Claim~\ref{Dowker tails claim} below.
\end{proof}
It now follows from Lemma~\ref{general normal lemma} that there exists a sequence $\langle U_i \mid i<\theta \rangle$
of pairwise open sets such that $K_i\subseteq U_i$ for all $i<\theta$.
\end{proof}
\subsection{A Dowker space with small hereditary density}\label{Subsection - Dowker Space of size lambda^+ }
In \cite{Rudin_souslin_line_dowker_space}, Rudin constructed a Dowker space of size $\aleph_1$ from a Souslin tree.
In \cite{rudin_S_space_Souslin}, she constructed an $S$-space of size $\aleph_1$ from a Souslin tree.
In \cite{rudin_separable_dowker_space}, she constructed an $S$-space of size $\aleph_1$ which is Dowker from a Souslin tree,
and in \cite{kappa_Dowker_from_souslin_Rudin}, she constructed a Dowker space of size $\lambda^+$ from a $\lambda^+$-Souslin tree, for $\lambda$ regular.
In \cite{de_Caux_space}, de Caux constructed a Dowker space of size $\aleph_1$ assuming $\clubsuit(\omega_1)$. Here we roughly follow de Caux's general approach,
but using $\clubsuit_{\ad}$, instead. So this gives a simultaneous generalization of de Caux's result and Rudin's result.
\begin{theorem} Suppose that $\clubsuit_{\ad}(\{E^{\lambda^+}_{\lambda}\},\lambda,1)$ holds for an infinite regular cardinal $\lambda$.
Then there exists a collectionwise normal Dowker space $\mathbb X$ of cardinality $\lambda^+$
such that $\hd(\mathbb X)=\lambda$ and $\Lin(\mathbb X)=\lambda^+$.
\end{theorem}
\begin{remark} The preceding is the Introduction's Theorem~D.
\end{remark}
Let $\langle \mathcal A_{\alpha}\mid\alpha\in E^{\lambda^+}_{\lambda}\rangle$ be a $\clubsuit_{\ad}(\{E^{\lambda^+}_{\lambda}\},\lambda,1)$-sequence.
For each $\alpha\in E^{\lambda^+}_{\lambda}$, fix an injective enumeration $\{ A^{j,n}_{\alpha+i}\mid i<\lambda, j\leq n<\omega\}$ of the elements of $\mathcal A_\alpha$.
For every $\xi\in\lambda^+\setminus\lambda$, as $\mathcal B_\xi:=\{ A^{j,n}_\beta\mid \lambda\le\beta<\xi+\lambda, j\le n<\omega\}$ is a subset of $\bigcup_{\alpha\in E^{\lambda^+}_\lambda}\mathcal A_\alpha$ of size $\lambda$,
we may appeal to Proposition~\ref{Proposition - disjointify multi-ladder system} to fix a function $f_\xi:\mathcal B_\xi\rightarrow\xi$ such that:
\begin{enumerate}
\item For every $B\in\mathcal B_\xi$, $f_\xi(B)\in B$;
\item The sets in $\langle B\setminus f_\xi(B) \mid B\in\mathcal B_\xi\rangle$ are pairwise disjoint.
\end{enumerate}
\begin{definition} For every $\beta<\lambda^+$, let $\alpha_\beta:=\min\{\alpha\le\beta\mid \exists i<\lambda(\beta=\alpha+i)\}$.
\end{definition}
We are now ready to define a topology $\tau$ on the set $X:=\lambda^+\times\omega$.
For all $x\in \lambda\times \omega$, just let $\mathcal N_{x}:=\{\{x\}\}$.
For all $x=(\beta,n)$ in $X$ with $\beta\ge\lambda$, denote
$N_{x}^\epsilon:=\{x\}\cup\bigcup_{j\leq n}((A^{j,n}_{\beta}\setminus\epsilon)\times\{j\})$,
and then set $\mathcal N_x:=\{ N_x^\epsilon\mid \epsilon<\alpha_\beta\}$.
Finally, a subset $ U \s \lambda^+\times\omega$ is $\tau$-open iff for any $x\in U $,
there is $N\in\mathcal N_x$ with $N\s U$.
It is easy to check that $\mathbb X:=(X, \tau) $ is a $ T_1 $ topological space.
\begin{definition}
For any subset $N\s \lambda^+\times\omega$, denote:
\begin{itemize}
\item $N^-:=\{(\gamma,j)\in N\mid \exists (\beta,n)\in N\,(\gamma<\beta)\}$;
\item $N^j:=\{ \gamma\mid (\gamma,j)\in N^-\}$ for any $j<\omega$.
\end{itemize}
\end{definition}
The following is obvious.
\begin{lemma}\label{Fact - neighborhoods behave as expected}
For all $x=(\beta,n)$ in $X$, $N\in\mathcal N_x$, and $j\leq n$, $N^j$ is a cofinal subset of $\alpha_\beta$ and $N\subseteq (\beta+1)\times(n+1)$.
In particular, for all $\delta<\lambda^+$ and $n<\omega$:
\begin{itemize}
\item $\delta\times n$ is $\tau$-open;
\item $\lambda^+\times(\omega\setminus n)$ is $\tau$-closed.\qed
\end{itemize}
\end{lemma}
So $\{ \delta\times\omega\mid \delta<\lambda^+\}$ witnesses that $\Lin(\mathbb X)=\lambda^+$.
\begin{lemma}\label{Dowker closure ost}
For every $\alpha\in E^{\lambda^+}_{\lambda}$ and $k<\omega$, $$(\lambda^+\setminus(\alpha+\lambda))\times (\omega\setminus k)\s \cl([\alpha,\alpha+\lambda)\times\{k\}).$$
\end{lemma}
\begin{proof} Denote $I_{\alpha,j}:=[\alpha,\alpha+\lambda)\times\{j \}$ and $F_{\alpha,j}:=\cl(I_{\alpha,j})$.
\begin{claim}\label{Dowker closure ost - claim inner induction}
Let $\alpha\in E^{\lambda^+}_{\lambda}$ and $j<\omega$. Then
$(\lambda^+\setminus\alpha)\times \{j\}\s F_{\alpha,j}$.
\end{claim}
\begin{proof}
We prove by induction on $ \delta\in E^{\lambda^+}_{\lambda}\setminus\alpha $ that $ I_{\delta,j}\subseteq F_{\alpha,j}$.
$\br$ For $\delta=\alpha$, trivially $I_{\delta,j} \subseteq F_{\alpha,j}$.
$\br$ Suppose that $\delta\in E^{\lambda^+}_{\lambda}\setminus(\alpha+\lambda)$ is an ordinal such that, for every $\gamma\in E^\delta_\lambda\setminus\alpha$, $I_{\gamma,j}\s F_{\alpha,j}$.
Therefore, $[\alpha,\delta)\times\{j\} \subseteq F_{\alpha,j}$.
Let $x\in I_{\delta,j}$. Since, for all $N\in \mathcal N_x$,
$N^j$ is a cofinal subset of $\delta$, $N\cap( [\alpha,\delta)\times\{j\})\neq\emptyset$.
Therefore, $x\in\cl([\alpha,\delta)\times\{j\})\s F_{\alpha,j}$.
\end{proof}
Let $\alpha\in E^{\lambda^+}_{\lambda}$ and $k<\omega$. Let $j\ge\kappa$ be some integer;
we need to prove that $(\lambda^+\setminus(\alpha+\lambda))\times\{j\}\s F_{\alpha,k}$.
By the preceding claim, it suffices to prove that $F_{\alpha+\lambda,j}\s F_{\alpha,k}$.
But the latter is a closed set, so it suffices to prove that $I_{\alpha+\lambda,j}\s F_{\alpha,k}$.
Let $x\in I_{\alpha+\lambda,j}$ be arbitrary.
For each $N\in \mathcal N_x$, as $k\le j$, $N^k$ is a cofinal subset of $\alpha+\lambda$,
and then $N\cap I_{\alpha,k}\neq\emptyset$.
Therefore, $x\in\cl(I_{\alpha,k})=F_{\alpha,k}$, as sought.
\end{proof}
\begin{cor}\label{corollary - Dowker space clubsuit_AD set of big size} For any $B\s X$ of size $\lambda^+$:
\begin{enumerate}
\item There exists $D\s B$ with $|D|=\lambda$ such that $B\s \cl(D)$;
\item\label{Clause - B closed contain a tail of the space} If $B$ is $\tau$-closed, then there is $(\beta,k)\in\lambda^+\times\omega$ such that $(\lambda^+\setminus\beta)\times(\omega\setminus k)\s B$.
\end{enumerate}
\end{cor}
\begin{proof} Given $B$ as above,
fix some $k<\omega$ such that $|B\cap (\lambda^+\times \{k\})|=\lambda^+$.
By Clause~(\ref{clubsuit AD, Clause cofinal set}) of Definition~\ref{clubsuit AD definition},
we can find an $ \alpha\in E^{\lambda^+}_{\lambda}$ such that $\dom((A^{j,n}_{\alpha+i}\times \{k \})\cap B)$ is cofinal in $\alpha$
for all $ i<\lambda$ and $j\leq n<\omega$.
Let $D:=B\cap(\alpha\times \{k\})$.
If follows that, for every $x\in[\alpha,\alpha+\lambda)\times\{k\}$ and every $N\in\mathcal N_x$,
$\dom(N\cap D)$ is cofinal in $\alpha$.
So, $ [\alpha,\alpha+\lambda)\times\{k\} \subseteq \cl(D)$
and hence $(\lambda^+\setminus(\alpha+\lambda))\times(\omega\setminus k) \s \cl([\alpha,\alpha+\lambda)\times\{k\})\subseteq \cl(D)$.
\begin{enumerate}
\item As $|B\setminus\cl(D)|\le|X\setminus\cl(D)|\le\lambda$, we see that $D\cup(B\setminus\cl(D))$ is a dense subset of $B$ of cardinality $\lambda$.
\item If $B$ is $\tau$-closed, then for $\beta:=\alpha+\lambda$,
$(\lambda^+\setminus\beta)\times(\omega\setminus k)\s\cl(D)\s B$.\qedhere
\end{enumerate}
\end{proof}
\begin{cor}\label{lambda^+ Dowker space - no two disjoint closed uncountable sets}
$\hd(\mathbb X)\le\lambda$ and there are no two disjoint closed subspaces of $\mathbb X$ of cardinality $\lambda^+$.
\end{cor}
\begin{proof}
By Corollary~\ref{corollary - Dowker space clubsuit_AD set of big size}\eqref{Clause - B closed contain a tail of the space}.
\end{proof}
\begin{lemma}\label{normality429} The space $\mathbb X$ is Hausdorff and collectionwise normal.
\end{lemma}
\begin{proof} As $\mathbb X$ is $T_1$, it suffices to verify that it is collectionwise normal.
Fix a nonzero cardinal $\theta$ and a discrete sequence $\vec K=\langle K_i \mid i<\theta \rangle$ of closed sets.
It follows from Clause~(\ref{clubsuit AD, Clause cofinal set}) of Definition~\ref{clubsuit AD definition}
that $\theta\le\lambda$, so, using Corollary~\ref{lambda^+ Dowker space - no two disjoint closed uncountable sets} and by possibly re-indexing, we
may find a large enough $\xi\in E^{\lambda^+}_\lambda$ such that $K_i\subseteq \xi\times\omega$ for all nonzero $i<\theta$.
Recall that $O:=\xi \times \omega$ is an open set.
\begin{claim}\label{Dowker tails claim}
There exists a sequence $\langle N_x\mid x\in X\rangle\in\prod_{x\in X}\mathcal N_x$
such that:
\begin{enumerate}[label=(\alph*)]
\item for all $x\in O$, $N_{x}\subseteq O$;
\item\label{dowker disjoint tails} for all $x\in O$ and $x'\in X\setminus\{x\}$, $N_{x}^-\cap N_{x'}^-=\emptyset$;
\item\label{dowker tails outside H,K} for all $x\in X$ and $i<\theta$, if $N_x\cap K_i\neq \emptyset$ then $x \in K_i$.
\end{enumerate}
\end{claim}
\begin{proof} Recalling Lemma~\ref{Fact - neighborhoods behave as expected},
we should only worry about requirements (b) and (c).
Let $x=(\beta,n)$ in $X$. There are three cases to consider:
$\br$ If $\beta<\lambda$, then set $N_x :=\{x\}$. Evidently, requirement~(c) is satisfied.
$\br$ If $\beta\ge\xi+\lambda$, then $\alpha_\beta>\xi$, so we let
$$\epsilon:=\max(\{\xi,\sup(\dom[(\bigcup\nolimits_{j\leq n}( A^{j,n}_{\beta}\times \{j\}))\cap K_0])\}\cap\alpha_\beta),$$
and then set $N_x := N_x^{\epsilon+1}$. Evidently, $N_x\cap O=\emptyset$.
In particular, for all $i<\theta$, if $N_x\cap K_i\neq\emptyset$, then $i=0$ and $\sup(\dom[(\bigcup\nolimits_{j\leq n}( A^{j,n}_{\beta}\times \{j\}))\cap K_0])=\alpha_\beta$,
which means that $x\in K_0$, since the latter is closed.
So, requirement~(c) is satisfied.
$\br$ If $\lambda\le\beta<\xi+\lambda$, then $A_\beta^{j,n}\in\mathcal B_\xi$ for all $j\le n$,
so that $\phi_x:=\max\{ f_\xi(A^{j,n}_\beta)\mid j\le n\}$ is $<\alpha_\beta\le\xi$.
As $\vec K$ is discrete, let us pick an open neighborhood $U$ of $x$
for which $I:=\{ i<\theta\mid U\cap K_i\neq\emptyset\}$ contains at most one element,
and then find a large enough $\varepsilon\in\alpha_\beta\setminus\phi_x$ such that $N^{\varepsilon}_x\subseteq U$.
Let
$$\epsilon:=\begin{cases}
\varepsilon,&\text{if }I=\emptyset;\\
\max(\{\varepsilon,\sup(\dom[(\bigcup_{j\leq n}( A^{j,n}_{\beta}\times \{j\}))\cap K_i])\}\cap\alpha_\beta),&\text{if }I=\{i\},
\end{cases}$$
and then set $N_x:=N_x^{\epsilon+1}$.
Evidently, $N_x\s U$. So, for all $i<\theta$, if $N_x\cap K_i\neq\emptyset$, then $I=\{i\}$ and $\sup(\dom[(\bigcup\nolimits_{j\leq n}( A^{j,n}_{\beta}\times \{j\}))\cap K_i])=\alpha_\beta$,
which means that $x\in K_i$. So, requirement~(c) is satisfied.
We are left with verifying that $\langle N_x\mid x\in X\rangle$ satisfies requirement~(b).
Fix arbitrary $x\in O$ and $x'\in X\setminus \{x\}$.
Say, $x=(\beta,n)$ and $x'=(\beta',n')$.
Note that if $\beta<\lambda$, then $N_x^-$ is empty,
and if $\beta'\ge\xi+\lambda$, then $\min(\dom(N_{x'}))>\xi>\beta=\sup(\dom(N_x))$.
Therefore, if $N_x^-\cap N_{x'}^-\neq\emptyset$, then $\lambda \le \beta,\beta'<\xi+\lambda$.
Thus, assume that $\lambda \le \beta,\beta'<\xi+\lambda$, and fix $\epsilon\ge\phi_x$ and $\epsilon'\ge\phi_{x'}$ such that
\begin{itemize}
\item $N_{x}^-=\bigcup_{j\leq n}((A^{j,n}_{\beta}\setminus(\epsilon+1))\times\{j\})$, and
\item $N_{x'}^-=\bigcup_{j\leq n'}((A^{j,n'}_{\beta'}\setminus(\epsilon'+1))\times\{j\})$.
\end{itemize}
So, if $N_x^-\cap N_{x'}^-\neq\emptyset$, then there exists $j\le\min\{n,n'\}$ such that $(A^{j,n}_{\beta}\setminus\phi_x)\cap (A^{j,n'}_{\beta'}\setminus\phi_{x'})\neq\emptyset$.
In particular, $(A^{j,n}_{\beta}\setminus f_\xi(A^{j,n}_{\beta}))\cap (A^{j,n'}_{\beta'}\setminus f_\xi(A^{j,n'}_{\beta'}))\neq\emptyset$,
contradicting the fact that $(\beta,n)\neq (\beta',n')$.
\end{proof}
It now follows from Lemma~\ref{general normal lemma} that there exists a sequence $\langle U_i \mid i<\theta \rangle$
of pairwise open sets such that $K_i\subseteq U_i$ for all $i<\theta$.
\end{proof}
\begin{lemma}
The space $\mathbb X$ is Dowker.
\end{lemma}
\begin{proof}
Denote $D_n:=\lambda^+\times (\omega\setminus n)$. Notice that $\langle D_n \mid n<\omega \rangle $ is a $\subseteq$-decreasing sequence of $\lambda^+$-sized $\tau$-closed sets such that $\bigcap_{n<\omega} D_n = \emptyset$.
By Corollary~\ref{lambda^+ Dowker space - no two disjoint closed uncountable sets}, there are no two disjoint closed sets of cardinality $\lambda^+$.
So, by Corollary~\ref{normality429} and Lemma~\ref{Lemma - Dowker general argument}, the space $(X,\tau)$ is Dowker.
\end{proof}
\section{A Dowker space with a normal square}\label{normalsquare}
In \cite{diamond_omega_2_LS_doweker}, Szeptycki proved that, assuming $ \diamondsuit^*(S) $ for a stationary $S\s E^{\omega_2}_{\omega_1}$,
there exists a ladder-system over a subset of $S$ whose corresponding ladder-system space is a Dowker space having a normal square.
As seen in Section~\ref{sectionladdersystemspace}, the hypothesis may be reduced to $ \clubsuit_{\ad}(\{E^{\omega_2}_{\omega_1}\},1,2) $
and still give a ladder-system whose corresponding space is Dowker,\footnote{Recall that
by Remark~\ref{diamondsuit iff clubsuit and ch} and Lemma~\ref{lemma216},
for $S\s E^{\omega_2}_{\omega_1}$, $\diamondsuit^*(S)\implies\diamondsuit(E^{\omega_2}_{\omega_1})\implies \clubsuit(E^{\omega_2}_{\omega_1})\implies \clubsuit_{\ad}(E^{\omega_2}_{\omega_1},\omega_1,{<}\omega)\implies \clubsuit_{\ad}(E^{\omega_2}_{\omega_1},1,2) $.}
since $E^{\omega_2}_{\omega_1}$ is a non-reflecting stationary subset of $\omega_2$.
But what about the normal square?
In this short section, we point out that the $\diamondsuit^*$ hypothesis may be reduced to an assumption in the language of the Brodsky-Rinot proxy principle (see Definition~\ref{proxyprinciple} below).
For the rest of this section, let $\lambda$ denote an infinite regular cardinal.
\begin{prop}\label{prop61} $\p_{\lambda}^-(\lambda^+,2,\sqleft{\lambda^+},\lambda,\{E^{\lambda^+}_{\lambda}\})$
entails the existence of a ladder-system over a subset of $E^{\lambda^+}_{\lambda}$
whose corresponding ladder-system space $(\lambda^+,\tau)$ is a Dowker space having a normal square.
\end{prop}
The point is that the hypothesis of the preceding already follows
from $\diamondsuit(E^{\lambda^+}_{\lambda})$, but it is also consistent with its failure (see Clauses (1) and (11) of \cite[Theorem~6.1]{paper23}).
\begin{definition}[Brodsky-Rinot, \cite{paper23}]\label{proxyprinciple} For a family $\mathcal S\s\mathcal P(\kappa)$,
and a cardinal $\theta<\kappa$,
$\p_\xi^-(\kappa,2,\sqleft{\kappa},\theta,\mathcal S)$ asserts the existence of a sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$
such that:
\begin{itemize}
\item for every $\alpha\in\acc(\kappa)$, $C_\alpha$ is a club in $\alpha$ of order-type $\le\xi$;
\item for every $S\in\mathcal S$ and every sequence $\langle B_i \mid i < \theta \rangle$ of cofinal subsets of $\kappa$,
there exist stationarily many $\alpha \in S$ such that, for all $i < \theta$,
$$\sup\{\delta\in B_i\cap\alpha\mid \min(C_\alpha\setminus(\delta+1))\in B_i\}=\alpha.$$
\end{itemize}
\end{definition}
Note that for every $\mathcal S\s\mathcal P(E^{\lambda^+}_{\lambda})$, any $\p_{\lambda}^-(\lambda^+,2,\sqleft{\lambda^+},\theta,\mathcal S)$-sequence
witnesses the validity of $\clubsuit_{\ad}(\mathcal S,1,\theta)$.
\begin{fact}[Brodsky-Rinot, {\cite[Theorem~4.15]{paper23}}]\label{Fact Good proxy} For a family $\mathcal S\s\mathcal P(\kappa)$,
and a cardinal $\theta<\kappa$,
$\p_\xi^-(\kappa,2,\sqleft{\kappa},\theta,\mathcal S)$ entails the existence of a sequence $\langle C_\alpha\mid\alpha<\kappa\rangle$
such that:
\begin{enumerate}
\item for every $\alpha\in\acc(\kappa)$, $C_\alpha$ is a club in $\alpha$ of order-type $\le\xi$;
\item\label{Fact Good proxy - Clause hitting} for every $S\in\mathcal S$ and every sequence $\langle \mathcal B_i \mid i < \theta \rangle$ with $\mathcal B_i\s[\kappa]^{<\omega}$ and $\mup(\mathcal B_i)=\kappa$ for all $i<\theta$,
there exist stationarily many $\alpha \in S$ such that, for all $i<\theta$,
$$\mup\{ x\in\mathcal B_i\mid x\s C_\alpha\}=\alpha.$$
\end{enumerate}
\end{fact}
For an ordinal $\alpha$, let us say that a subset $x$ of the product $\alpha\times\alpha$ is \emph{dominating}
iff for every $(\beta,\gamma)\in(\alpha,\alpha)$, there exists $(\beta',\gamma')\in x$ with $\beta\le\beta'$ and $\gamma\le\gamma'$.
Now, we are ready to prove the main lemma.
\begin{lemma}\label{lemma - ladder-system club sequence with square}
Suppose that $\p_{\lambda}^-(\lambda^+,2,\sqleft{\lambda^+},\lambda,\{E^{\lambda^+}_{\lambda}\})$ holds.
Then there exist a partition $\langle S_n\mid n<\omega\rangle$ of $\lambda^+$ into stationary sets and a sequence $\langle s_\alpha\mid \alpha\in E^{\lambda^+}_\lambda \rangle $ such that:
\begin{enumerate}
\item\label{lemma - ladder-system club sequence with square - Clause type} For each $\alpha\in E^{\lambda^+}_\lambda$, $s_\alpha$ is either empty or a cofinal subset of $\alpha$ of order-type $\lambda$;
\item\label{lemma - ladder-system club sequence with square - Clause bigcup S_i is open} For all $n<\omega$ and $\alpha\in S_{n+1}$, $s_\alpha\subseteq \bigcup_{i\leq n} S_i$;
\item\label{lemma - ladder-system club sequence with square - Clause hitting} For every $k<\omega$, every $\lambda$-sized subfamily $\mathcal F \subseteq [\bigcup_{i\leq k} S_i]^{\lambda^+}$,
and every $n\in\omega\setminus(k+1)$, the following set is stationary:
$$ \{ \alpha\in S_n \mid \forall F\in\mathcal F~[ \sup(s_\alpha\cap F)=\alpha] \};$$
\item\label{lemma - ladder-system club sequence with square - Clause square hitting}
For every two dominating subsets $B_0,B_1$ of $\lambda^+\times\lambda^+$, there exists $m<\omega$ such that for every $n\in\omega\setminus m$ the following set is stationary:
$$ \{ \alpha\in S_n \mid \forall i<2~[(s_\alpha\times s_\alpha)\cap B_i\text{ dominates }(\alpha,\alpha)] \}.$$
\end{enumerate}
\end{lemma}
\begin{proof}
Let $\vec C=\langle C_\alpha\mid\alpha<\lambda^+\rangle$ be a $\p_{\lambda}^-(\lambda^+,\lambda^+,\sqleft{\lambda^+},\lambda,\{E^{\lambda^+}_\lambda\})$-sequence.
Let $\mathcal I$ denote the collection of all $T\s E^{\lambda^+}_\lambda$
such that $\vec C$ is not a $\p_{\lambda}^-(\lambda^+,\lambda^+,\sqleft{\lambda^+},\lambda,\{T\})$-sequence.
Evidently, $\mathcal I$ is a $\lambda^+$-complete ideal over $E^{\lambda^+}_\lambda$. So, by Ulam's theorem,
$\mathcal I$ is not weakly $\lambda^+$-saturated.
This means that exists a sequence $\langle S_\iota\mid \iota<\lambda^+\rangle$ of pairwise disjoint subsets of $E^{\lambda^+}_\lambda$,
such that, for each $\iota<\lambda^+$,
$\vec C$ is a $\p_{\lambda}^-(\lambda^+,\lambda^+,\sqleft{\lambda^+},\lambda,\{S_\iota\})$-sequence.
In particular, we may fix a family $\mathcal S$ consisting of $\aleph_0$-many pairwise disjoint stationary subsets of $E^{\lambda^+}_{\lambda}$
such that $\p_{\lambda}^-(\lambda^+,\lambda^+,\sqleft{\lambda^+},\lambda,\mathcal S)$ holds and yet $S_0:=\lambda^+\setminus\bigcup\mathcal S$ is stationary.
Fix an injective enumeration $\langle S_{n+1}\mid n<\omega\rangle$ of $\mathcal S$.
For every $\alpha\in E^{\lambda^+}_{\lambda}$, let $n(\alpha)$ be such that $\alpha\in S_{n(\alpha)}$.
For each $n<\omega$, let $ W_n:=\bigcup_{i\leq n}S_i $.
Now, let $\vec D= \langle D_\alpha\mid\alpha<\lambda^+\rangle$ be a $\p_{\lambda}^-(\lambda^+,\lambda^+,\sqleft{\lambda^+},\lambda,\{S_{n+1} \mid n<\omega \})$-sequence as in Fact \ref{Fact Good proxy}.
For every $\alpha\in E^{\lambda^+}_{\lambda}$, let
$$s_\alpha:=\begin{cases}
W_{n(\alpha)-1}\cap D_\alpha,&\text{if }n(\alpha)>0\ \&\ \sup(W_{n(\alpha)-1}\cap D_\alpha)=\alpha;\\
\emptyset,&\text{otherwise.}
\end{cases}$$
We claim that the sequence $\langle s_\alpha \mid \alpha \in E^{\lambda^+}_{\lambda}\rangle$ is as sought.
Notice that Clauses (\ref{lemma - ladder-system club sequence with square - Clause type}) and (\ref{lemma - ladder-system club sequence with square - Clause bigcup S_i is open})
hold by our very construction, as $D_\alpha$ has order-type $\lambda$ for every $\alpha\in E^{\lambda^+}_{\lambda}$.
\begin{claim} Let $k<\omega$ and $\mathcal F \subseteq [W_k]^{\lambda^+}$ be a family of size $\lambda$.
For every integer $n>k$, $\{ \alpha\in S_n \mid \forall F\in\mathcal F~[ \sup(s_\alpha\cap F)=\alpha] \}$ is stationary.
\end{claim}
\begin{proof} Let $\{\mathcal B_i\mid i<\lambda\}$ be some enumeration of $\{ [F]^1 \mid F\in\mathcal F\}$.
Evidently, for every $i<\lambda$, $\mathcal B_i\subseteq[\lambda^+]^{<\omega}$ and $\mup(\mathcal B_i)=\lambda^+$.
Now, by Fact~\ref{Fact Good proxy}, Clause~(\ref{Fact Good proxy - Clause hitting}),
for every $n<\omega$, the set $G_n$ of all $\alpha \in S_n$ such that, for all $i<\lambda$,
$$\mup\{ x\in\mathcal B_i\mid x\subseteq D_\alpha\}=\alpha,$$
is stationary.
In particular, for all $n>k$, $\alpha\in G_n$ and $F\in\mathcal F$:
\[\sup(s_\alpha\cap F)=\sup(W_{n(\alpha)-1}\cap D_\alpha\cap F)=\alpha.\qedhere\]
\end{proof}
\begin{claim} Let $B_0,B_1$ be two dominating subsets of $\lambda^+\times\lambda^+$.
Then there exists some $m<\omega$ such that for every $n\in\omega\setminus m$ the following set is stationary:
$$ \{ \alpha\in S_n \mid \forall i<2~[(s_\alpha\times s_\alpha)\cap B_i\text{ dominates }(\alpha,\alpha)] \}.$$
\end{claim}
\begin{proof} For every $\epsilon<\lambda^+$ and $i<2$, fix $(\xi^i_\epsilon,\zeta^i_\epsilon)\in B_i$ with $\min\{\xi^i_\epsilon,\zeta^i_\epsilon\}>\epsilon$.
For every $n<\omega$, let
$$\mathcal B_n:=\{\{\xi^0_\epsilon,\zeta^0_\epsilon,\xi^1_\epsilon,\zeta^1_\epsilon\}\mid \epsilon<\lambda^+, \max\{n(\xi^0_\epsilon),n(\zeta^0_\epsilon),n(\xi^1_\epsilon),n(\zeta^1_\epsilon)\}=n\}.$$
By the pigeonhole principle, we may find $m<\omega$ such that $|\mathcal B_{m}|=\lambda^+$. In particular,
$\mathcal B_{m}\s[\lambda^+]^{<\omega}$ and $\mup(\mathcal B_{m})=\lambda^+$.
Now, by Fact~\ref{Fact Good proxy}, Clause (\ref{Fact Good proxy - Clause hitting}),
for every $n<\omega$, the set $G_n$ of all $\alpha \in S_n$ such
$$\mup\{ x\in\mathcal B_m\mid x\subseteq D_\alpha\}=\alpha,$$
is stationary.
In particular, for all $n>m$, $\alpha\in G_n$ and $i<2$,
$(s_\alpha)^2\cap B_i=(D_\alpha\cap W_{n(\alpha)-1})^2\cap B_i$ and it dominates $(\alpha,\alpha)$.
\end{proof}
This completes the proof.
\end{proof}
At this point, the proof of Proposition~\ref{prop61} continues exactly as in \cite[Theorem~4]{diamond_omega_2_LS_doweker},
using the sequences $\langle S_n\mid n<\omega\rangle$ and $\langle s_\alpha\mid\alpha<\lambda^+\rangle$ constructed in the preceding lemma.
\begin{remark} Lemma~\ref{lemma - ladder-system club sequence with square} also implies that
$\p_{\omega}^-(\omega_1,2,\sqleft{\omega_1},\omega,\{\omega_1\})$ is a weakening of $\clubsuit^*$ sufficient for the constructions of \cite{MR1429177}.
\end{remark}
\section{Acknowledgements}
Some of the results of this paper come from the second author's M.Sc.~thesis written under the supervision of the first author at Bar-Ilan University.
We are grateful to Bill Weiss for kindly sharing with us a scan of Dahroug's handwritten notes with the construction of an Ostaszewski space from a Souslin tree and $\ch$.
Our thanks go to Tanmay Inamdar for many illuminating discussions,
and to Istv\'an Juh\'asz for reading a preliminary version of this paper and providing a valuable feedback.
We also thank the referee for a useful feedback.
Both authors were partially supported by the Israel Science Foundation (grant agreement 2066/18).
The first author was also partially supported by the European Research Council (grant agreement ERC-2018-StG 802756).
\newcommand{\etalchar}[1]{$^{#1}$}
|
1,116,691,500,462 | arxiv | \section{Introduction}
The \emph{maximum-entropy sampling problem}, a fundamental problem in optimal statistical design,
was formally introduced in \cite{SW} and then applied in many areas such as
the re-design of environmental-monitoring networks (see \cite{ZidekSunLe2000}, for example).
In the Gaussian case, the problem can be cast as
\begin{align*}
z(C,s)\!:=\! \max \{\ldet C[S,S] ~\!\!\!:~\!\!\!~ S\subseteq N, |S|=s\},\tag{MESP}\label{MESP}
\end{align*}
where ldet denotes the natural logarithm of the determinant, $C$ is an $n\times n$ covariance matrix (of Gaussian random variables), positive integer $s<n$ with $s\leq \rank(C)$,
$N:=\{1,...,n\}$, and for subsets $S\subseteq N$ and $T\subseteq N$ ($1\le |S|\le n, 1\le |T|\le n$), $C[S,T]$ denotes the submatrix of $C$ having rows indexed by $S$ and columns indexed by $T$.
Of course we should assume that $s\leq \rank(C)$, otherwise $z(C,s)=-\infty$.
We further assume that $C[j,j]>0$ for all $j\in N$,
because if we had any $C[j,j]=0$, then such a $j$
could not be in any feasible solution of \ref{MESP}
having objective value greater than $-\infty$.
In the constrained version CMESP, we also have $m\geq 0$ side constraints: $\sum_{j\in S}a_{ij}\leq b_i$, for $1\leq i \leq m$.
In the environmental-monitoring application of \ref{MESP}, we collect time-series observations from $n$ environmental monitoring stations, and we prepare a sample covariance matrix $C$ (see \cite{Al-ThaniLee1,Rpackage}, for details on how this can be done effectively).
In many situations, keeping all $n$ of the monitoring stations running is too costly, as so we wish to
select a subset of size $s$, and continue monitoring only at them. Maximizing the
``differential entropy'' of the $s$ sites, is a means of choosing the $s$-subset with maximum information.
In the U.S.A., for example, there are approximately $n=250$ stations that comprise the
NADP (National Acidic Deposition Program) maintains the NTN (National Trends Network); see \cite{NADPNTN}).
Unfortunately, \ref{MESP} is NP-hard (see \cite{KLQ}). Successful exact approaches are based on a branch-and-bound framework (see \cite{KLQ}). The original approach of \cite{KLQ} uses the so-called ``eigenvalue bound'' (the sum of the logs of the $s$ greatest eigenvalues of $C$)
as an upper bound, and there has subsequently been considerable work on extensions and alternative bounding methods.
\cite{LeeConstrained} extended the eigenvalue bound to CMESP, \cite{HLW} gave a ``combinatorial mask'' technique for improving the eigenvalue bound for \ref{MESP}, \cite{LeeWilliamsILP} further leveraged these
ideas using integer-linear optimization, dynamic programming, and optimal matchings, and then
\cite{AnstreicherLee_Masked,BurerLee} developed the general ``masking'' technique to further improve
the eigenvalue bound. In another vein, bounding techniques based on convex-relaxation
for \ref{MESP} (which easily apply as
well to CMESP), were developed; see \cite{AFLW_IPCO,AFLW_Using,Anstreicher_BQP_entropy,Kurt_linx}.
Although no bounding technique wins on all instances, probably the current best is the so-called ``linx bound'' of \cite{Kurt_linx}, which has several nice properties.
In recent work, \cite{chen_mixing} gave a general technique for
``mixing'' more than one bound that is based on convex relaxation.
\textbf{Notation.}
$\Diag(x)\in \mathbb{R}^{n\times n}$ denotes the square diagonal matrix with diagonal elements as the elements of $x\in\mathbb{R}^n$. $\diag(X)\in\mathbb{R}^n$ denotes the vector obtained from the diagonal
elements of matrix $X\in \mathbb{R}^{n\times n}$.
$\circ $ is the Hadamard (i.e., element-wise) product.
$J_n$ is the square all-ones matrix of order $n$.
$I_n$ is the identity matrix of order $n$.
$\mathbf{e_n}$ is the $n$-vector of all-ones.
$\mathbf{e}^n_i$ denotes the $i$-th standard unit vector in $\mathbb{R}^n$. Often we will omit the size $n$ if it is clear from the context.
If $S$ is a set, $|S|$ denotes the cardinality of the set.
{\bf Scaling}. An important general technique for potentially improving some of the
entropy upper bounds is ``scaling'',
based on the simple observation that
for a positive constant $\gamma$, and $S$
with $|S|=s$, we have that
\[
\det (\gamma C)[S,S] = \gamma^s \det C[S,S].
\]
With this identity, we can easily see that
\[
z(C,s) = z(\gamma C,s) -s \log \gamma~.
\]
So upper bounds for $z(\gamma C,s)$
yield upper bounds for $z(C,s)$,
shifting by $-s \log \gamma$. This idea was first exploited in \cite{AFLW_IPCO,AFLW_Using}.
It is worth noting that most bounding methods are \emph{not} invariant under
scaling; that is the bound does \emph{not} generally shift by $-s\log \gamma$ (a notable exception being the eigenvalue bound).
{\bf Masking}. Another important technique for potentially improving some of the
entropy upper bounds is ``masking''.
Given a positive integer $n$, a \emph{mask} (also known as a ``correlation matrix'')
is an $n\times n$ symmetric positive-semidefinite matrix
with $\diag(M)=\mathbf{e}$. We denote the set of order-$n$ masks as $\mathcal{M}_n$.
Masking for \ref{MESP}, introduced in full generality in \cite{AnstreicherLee_Masked},
is based on the observation that for any $S\subset N$ and mask $M$,
\[
\det C[S,S] \leq \det (C\circ M)[S,S].
\]
That is, masking cannot decrease entropy. Therefore,
for any mask $M\in\mathcal{M}_n$, we have
\[
z(C,s) \leq z(C\circ M,s).
\]
\noindent
This implies that upper bounds for $ z(C\circ M,s)$
are also upper bounds for $z(C,s)$.
\medskip
{\bf The linx bound}.
\cite{Kurt_linx} proposed the very-successful \emph{linx bound}
$\linx (C,s):= $
\begin{equation} \label{basiclinx}
\max\{ f(C,s;x) ~:~ x\in P(n,s)\},
\end{equation}
where
$f(C,s;x):= \frac{1}{2} \ldet(C\Diag(x)C+\Diag(\mathbf{e}-x))$, and
$ P(n,s):=\{x\in \mathbb{R}^n ~:~ \mathbf{e}^\top x=s,~ 0\le x\le \mathbf{e}\}$.
Applying a scaling parameter $\gamma>0$, we arrive at the
\emph{scaled linx bound} $\linx (C,s;\gamma):=$
\begin{equation} \label{gammalinx}
\max\{ f(C,s;\gamma;x) ~\!:\!~ x\in P(n,s)\},
\end{equation}
where $f(C,s;\gamma;x):=$
\[
\textstyle \frac{1}{2} \left(\ldet(\gamma C \Diag(x)C
+\Diag(\mathbf{e}-x))-s\log \gamma \right).
\]
We are interested in finding optimal scaling parameters in the context of the scaled linx bound.
Because \eqref{gammalinx} is an ``exact relaxation'' (i.e., the objective
of the relaxation on an $x\in\{0,1\}^n$ is exactly $\ldet C[S,S]$ for
$S$ equal to the support of $x$),
for every scaling parameter $\gamma>0$,
the following useful fact (true for any exact relaxation)
is easy to see.
\begin{prop}\label{powersetopt}
If $\hat x\in \{0,1\}^n$ is an optimal solution of \eqref{gammalinx} for $\gamma=\hat \gamma$, then
$\hat \gamma$ is optimal. That is,
$
\linx (C,s; \hat \gamma)=\min_{\gamma>0}\linx (C,s; \gamma).
$
\end{prop}
An important result from \cite{chen_mixing} is that by replacing the scaling parameter $\gamma$ with $e^{\psi}$, the linx bound (as well as the the BQP bound of \cite{Anstreicher_BQP_entropy}) is convex in $\psi$, and so a locally-optimal scaling parameter is globally optimal (see \cite{chen_mixing} for details and an algorithmic approach to optimizing the scaling parameter
based on this).
In what follows, we further study the effect of the scaling parameter on the linx bound,
and we demonstrate the benefit masking can have on the linx bound.
Applying a mask $M\in\mathcal{M}_n$, we arrive at the \emph{scaled and masked linx bound}
$\linx (C,s;M,\gamma):=$
\begin{equation} \label{maskgammalinx}
\max\{ f(C,s;M,\gamma;x) ~\!:\!~ x\in P(n,s)\},
\end{equation}
where
\begin{align*}
&f(C,s;M,\gamma;x)\! :=
\textstyle \!\frac{1}{2} \!
\left(\ldet(\!\gamma(C \!\circ\! M)\!\Diag(\!x\!)\!(C \!\circ\! M)\right.\\
&\left.\quad+\!\Diag(\mathbf{e}\!-\!x))\!-\!s\log \gamma \right).
\end{align*}
We refer to the mask $M$ as optimal for $\linx (C,s;M,\gamma)$, if it minimizes its value among all $M\in\mathcal{M}_n$.
For brevity, we write $\linx (C,s; M) := \linx (C,s; M, 1)$ .
We note that masking does not generally produce an exact relaxation, so Proposition \ref{powersetopt}
does not extend to a sufficient condition for optimal masks.
{\bf Main results and organization.}
A main goal of ours is to demonstrate the strong potential for masking to improve the linx bound.
We do this by exhibiting sequences $\{C_k,s_k;M_k,\gamma_k, \hat{\gamma}_k\}_{k=1}^\infty$, with $C_k\succeq 0$ of order $n_k$, $M_k\in\mathcal{M}_{n_k}$,
$n_k\to\infty$, such that $\linx(C_k,s_k;\gamma_k)-\linx(C_k,s_k;M_{k},\hat \gamma_k)\geq \alpha_k$, where $\alpha_k$ grows linearly with $n_k$. First we do this for $\gamma_k=\hat \gamma_k=1$ (i.e., no scaling). Then, at the expense of a worse lower bound $\alpha_k$,
we do this when $\gamma_k$ and $\hat \gamma_k$ are the optimal scale factors.
To get such lower bounds on the gap $\linx(C_k,s_k;\gamma_k)-\linx(C_k,s_k;M_{n_k},\hat \gamma_k)$,
we need a good lower bound on $\linx(C_k,s_k;\gamma_k)$ and a good upper bound on $\linx(C_k,s_k;M_{n_k},\hat \gamma_k)$.
In fact, to establish these gaps, we will take $M_{n_k}=I_{n_k}$, and so to get our needed upper bounds,
we use an exact characterization of the linx bound and the optimal scaling for the linx bound,
when $C$ is diagonal (useful because $C_{n_k}\circ I_{n_k}$ is diagonal).
bn
In \S\ref{sec:gap}, we establish that using a mask but no scaling parameter (i.e., $\gamma=1$), the best-case improvement in the
linx bound is at least linear in $n$; specifically, $\approx .0312n$.
In \S\ref{sec:gamma}, we study the behavior of the linx bound
as we vary the scaling parameter $\gamma>0$. It was already established that
the linx bound is convex in $\log(\gamma)$ (see \cite{chen_mixing}).
We establish the limiting behavior,
as $\gamma$ goes to 0 and to infinity. When $s=\rank(C)$, the limit
as $\gamma$ goes to infinity can be better than any finite choice of $\gamma$;
in this case, we establish that the limit can be calculated by solving
a single convex optimization problem. In \S\ref{sec:gammagap}, we establish that using a mask and \emph{optimal} scaling parameters, the best-case improvement in the
linx bound remains at least linear in $n$; specifically, $\approx .024n$. \S\ref{sec:final} contains some final remarks.
\section{Linear gap for the linx bound}\label{sec:gap}
Our main goal in this section is to establish a linear lower bound on the
best-case gap between the linx bound and the masked linx bound, giving a good justification
for considering mask optimization.
Specifically, we will give a sequence $\{C_n,s_n;M_n\}$, for all even positive integers $n$, with $C_n\succeq 0$ of order $n$, and $M_n\in\mathcal{M}_{n}$, such that $\linx(C_n,s_n)-\linx(C_n,s_n;M_{n})\geq \frac{1}{4}\log\left(\frac{4}{3}\right) n$. In fact, we will take $s_n:={\textstyle\frac{n}{2}}$, and $M_n:=I_n$. Because we use $M_n:=I_n$, we will
have $\linx(C_n,s_n;M_{n})=\linx(\Diag(d_{(n)}),s_n)$, where $d_{(n)}=\diag(C_n)$.
Hence it is useful to characterize, in general, the optimal solution of
\eqref{basiclinx} when $C$ is diagonal. Additionally, beyond our own use in the present work, we believe that such a
characterization can be useful in future work on gaps for the linx bound.
Without loss of generality, we assume that $C:=\Diag(d)$ where $d\in\mathbb{R}^n$ and $d_1\!\geq\! d_2\!\geq\! \cdots\! \geq\! d_n\!>\!0$. Then
\[
f(C,s; x) = {\textstyle\frac{1}{2}} \log
\prod_{i\in N}
\left(
(d_i^2-1)x_i+1
\right)
.
\]
\begin{lem} \label{lem:propxhat}
Let $C:=\Diag(d)$, where $d\in\mathbb{R}^n$ satisfies $d_1\geq d_2\geq \cdots \geq d_n > 0$. There exists an optimal solution $\hat x$ of \eqref{basiclinx} such that $\hat x_1\ge \hat x_2 \ge \cdots \ge \hat x_n$ and $\hat x_i=\hat x_j$, for all $i,j\in N$, such that $d_i=d_j$.
\end{lem}
\proof{Proof.}
Clearly, \eqref{basiclinx} has an optimal solution $\hat x$. And
\begin{align*}
&\left((d_i^2-1)\hat x_i+1\right)\left((d_j^2-1)\hat x_j+1\right)\\
&-\!\left((d_i^2-1)\hat x_j+1\right)\!\left((d_j^2-1)\hat x_i+1\right)\\
&\!=\! (d_i^2-d_j^2)(\hat{x}_i-\hat{x}_j).
\end{align*}
If $d_i>d_j$, from the identity above we see that $\hat x_i\geq \hat x_j$, otherwise, by exchanging components $i$ and $j$ of $\hat x$, we would increase the objective value of \eqref{basiclinx}.
If $d_i=d_j$, let $\delta:= \hat x_i + \hat x_j $. Then,
\begin{align*}
&\left((d_i^2-1)\hat x_i+1\right)\left((d_j^2-1)\hat x_j+1\right)\\
& \quad = \left((d_i^2-1)\hat x_i+1\right)
\left((d_j^2-1)(\delta -{ \hat x_i})+1\right)
\end{align*}
In this case, if $d_i=1$, the above function is constant. Otherwise, it is a univariate concave quadratic in $\hat{x_i}$, and its maximum is uniquely attained at $ \hat x_i =\delta/2$. Therefore, by setting $ \hat x_i = \hat x_j =\delta/2$, the maximum of the function is always attained. \Halmos
\endproof
\begin{defn}
We refer to an optimal solution $\hat x$ of \eqref{basiclinx} which satisfies the properties in Lemma \ref{lem:propxhat} as a uniform optimal solution.
\end{defn}
\begin{lem}\label{lem:conc}
Let $C:=\Diag(d)$, where $d\in\mathbb{R}^n$ satisfies $d_1\geq d_2\geq \cdots \geq d_n > 0$ and $0\leq x\leq \mathbf{e}$. Then $f(C,s;x)$ strictly increases with $x_i$, if $d_i>1$, does not change with $x_i$, if $d_i=1$, and strictly decreases with $x_i$, if $d_i<1$. Furthermore, $f(C,s;\cdot)$ is concave in $[0,1]^n$ and strictly concave if $d_i\neq 1$, for all $i\in N$.
\end{lem}
\proof{Proof.}
For all $i\in N$, $d_i>0$ and $0\leq x_i\leq 1$ implies that
$(d_i^2-1)x_i+1 > 0$.
Then, for $i\in N$,
\begin{align*}
&\frac{\partial f(C,s;x)}{\partial x_i}=\frac{d_i^2-1}{2 (
\!(d_i^2-1)x_i\!+\!1
)}\left\{\begin{array}{c}\!< 0, \mbox{ if }\! d_i<1,\\\!=0,\mbox{ if }\! d_i=1,\\\!>0,\mbox{ if }\! d_i>0,\end{array}\right.\\
&\frac{\partial^2 f(C,s;x)}{\partial x_i^2}=\frac{-(d_i^2-1)^2}{2 (\!
(d_i^2-1)x_i\!+\!1)^2}\left\{\begin{array}{c}\!< 0,\! \mbox{ if }\! d_i\neq 1,\\\!=0,\!\mbox{ if }\! d_i=1,\end{array}\right.\\
& \mbox{and}\\
&\frac{\partial^2 f(C,s;x)}{\partial x_i\partial x_j}=0, \mbox{ for } 1\leq i\neq j \leq n. \Halmos
\end{align*}
\endproof
\begin{rem}
From Lemmas \ref{lem:propxhat} and \ref{lem:conc}, we see that if $C=\Diag(d)$, with $0<d_i\neq 1, \forall i\in N$, then
\eqref{basiclinx} has a unique optimal solution, which is a uniform optimal solution.
\end{rem}
Next, we establish necessary conditions for $\hat x$ to be a uniform optimal solution for
\eqref{basiclinx} when $C$ is diagonal, based on
checking a finite set of feasible directions; we could also
get these conditions from the KKT conditions for \eqref{basiclinx},
also establishing their sufficiency, but our approach is simpler and suits our purpose.
\begin{lem}\label{diagoptcondition}
Let $C:=\Diag(d)$, where $d\in\mathbb{R}^n$ satisfies $d_1\geq d_2\geq \cdots \geq d_n > 0$.
Let $\hat x$ be a uniform optimal solution of \eqref{basiclinx}. For $1\leq i<j\leq n$, we have
\begin{equation}
\frac{d_j^2-1}{(d_j^2-1)\hat x_j+1}\le \frac{d_i^2-1}{(d_i^2-1)\hat x_i+1}. \label{cond1}
\end{equation}
Additionally, if $1 > \hat x_i \geq \hat x_j > 0$, then
\begin{equation}
\frac{d_j^2-1}{(d_j^2-1)\hat x_j+1}= \frac{d_i^2-1}{(d_i^2-1)\hat x_i+1}. \label{cond2}
\end{equation}
\end{lem}
\proof{Proof.}
\eqref{cond1} is clear when $\hat x_i=\hat x_j$, from the fact that $d_i\geq d_j> 0$.
So we may assume that $\hat x_i>\hat x_j$. In this case $\mathbf{e}_j-\mathbf{e}_i$ is a feasible direction
for $\hat{x}$ relative to \eqref{basiclinx}. Because $\hat{x}$ is optimal for \eqref{basiclinx},
we must have that $ \nabla f(C,s;\hat x)^\top (\mathbf{e}_j-\mathbf{e}_i)\leq 0$, which is equivalent to \eqref{cond1}.
\eqref{cond2} follows from the fact that, in this case, $\mathbf{e}_i-\mathbf{e}_j$ is also a feasible
direction for $\hat{x}$ relative to \eqref{basiclinx}. \Halmos
\endproof
We have a corollary of Lemma \ref{diagoptcondition} for two special cases: $ d_n>1$ and $d_1<1$. We will see later that the characterization of the optimal solution in general can be reduced to the characterization of the optimal solution in these two special cases.
\begin{cor}\label{diagoptconditionspecial}
Let $C:=\Diag(d)$, where $d\in\mathbb{R}^n$ satisfies either $d_1\geq d_2\geq \cdots \geq d_n>1$ or $1>d_1\geq d_2\geq \cdots \geq d_n> 0$.
Let $\hat x$ be a uniform optimal solution of \eqref{basiclinx}. Then,
\begin{equation}\label{coreq1}
\hat x_i-\hat x_j\le \frac{1}{d_j^2-1}-\frac{1}{d_i^2-1}.
\end{equation}
Additionally, if $1 > \hat x_i \geq \hat x_j > 0$, then
\begin{equation}\label{coreq2}
\hat x_i-\hat x_j=\frac{1}{d_j^2-1}-\frac{1}{d_i^2-1}.
\end{equation}
\end{cor}
\proof{Proof.}
If either $d_n>1$ or $d_1<1$, we have $d_i^2-1\neq 0, \forall i\in N$. Also, both $d_i^2-1$ and $d_j^2-1$ have the the same sign $\forall i, j\in N$. Together with $(d_i^2-1)\hat x_i+1>0, \forall i\in N$, we have that \eqref{cond1} and \eqref{cond2} equals \eqref{coreq1} and \eqref{coreq2}, respectively. \Halmos
\endproof
To characterize an optimal solution of \eqref{basiclinx} when $C$ is diagonal, we first establish a lemma that characterizes an optimal solution of \eqref{basiclinx} in the two special cases discussed in Corollary~\ref{diagoptconditionspecial}.
\begin{lem}\label{diagopt1}
Let $C:=\Diag(d)$, where $d\in\mathbb{R}^n$ satisfies either $d_1\geq d_2\geq \cdots \geq d_n >1$ or $1>d_1\geq d_2\geq \cdots \geq d_n > 0$. Let $\hat x$ be a uniform optimal solution of \eqref{basiclinx} for a given $0<s<n$. We have,
\begin{itemize}
\item[(i)] if $ \frac{1}{d_{s+1}^2-1}-\frac{1}{d_s^2-1}\ge 1$,
then
\[
\hat x_i:=
\left\{
\begin{array}{ll}
1,&\hbox{for $1\leq i \leq s$,}\\
0,&\hbox{for $s+1\leq i \leq n$,}
\end{array}
\right.
\]
\item[(ii)] if $\frac{1}{d_{s+1}^2-1}-\frac{1}{d_s^2-1} < 1$,
then $0<\hat x_s <1$, and
\begin{equation}\label{eqlemma24}
\hat x_i:=
\left\{
\begin{array}{ll}
\min\left\{1,~\hat x_s+\frac{1}{d_{s}^2-1}-\frac{1}{d_{i}^2-1}\right\},\\[6pt]
\qquad \qquad\qquad\hbox{for $1\leq i \leq s-1$,}\\
\max\left\{0,~\hat x_s+\frac{1}{d_{s}^2-1}-\frac{1}{d_{i}^2-1}\right\},\\[6pt]
\qquad \qquad\qquad \hbox{for $s+1\leq i \leq n.$}
\end{array}
\right.
\end{equation}
\end{itemize}
\end{lem}
\proof{Proof.}
We have already shown that under the hypotheses, \eqref{basiclinx} has a unique optimal solution $\hat x$, where $\hat x_1\ge \hat x_2\ge \cdots\ge \hat x_n$. Thus, for $(i)$, we only need to show that $\hat x_s=1$.
Suppose that $\hat x_s<1$; then
$1>\hat x_s\ge \hat x_{s+1}>0$, i.e., $\hat x_{s}-\hat x_{s+1}<1\le \frac{1}{d_{s+1}^2-1}-\frac{1}{d_{s}^2-1}$, which violates the necessary condition \eqref{coreq2} in Corollary~\ref{diagoptconditionspecial}.
For $(ii)$, we see that if $\hat x_s=1$, then $\hat x_{s+1}=0$ and $\textstyle \hat x_{s}-\hat x_{s+1}=1>\textstyle\frac{1}{d_{s+1}^2-1}-\textstyle \frac{1}{d_s^2-1}$, which violates the
necessary
condition \eqref{coreq1} in Corollary~\ref{diagoptconditionspecial}. If $\hat x_s=0$, then $\sum_{i=1}^n \hat x_i\le \sum_{i=1}^{s-1} \hat x_i\le s-1$, which contradicts the feasibility of $\hat{x}$. Therefore, $0<\hat x_s <1$. Finally, by the necessary
conditions in Corollary~\ref{diagoptconditionspecial}, the other parts of $(ii)$ must hold. \Halmos
\endproof
In case $(ii)$ in Lemma~\ref{diagopt1},
we can solve the equation $\mathbf{e}^\top x =s$ for $\hat x_s$:
\begin{align*}
&\sum_{i=1}^{s-1} \min\left\{1,~ \hat x_s+\frac{1}{d_{s}^2-1}-\frac{1}{d_{i}^2-1}\right\} +\hat x_s\\
&\quad
+ \sum_{i=s+1}^n \max\left\{0,~ \hat x_s+\frac{1}{d_{s}^2-1}-\frac{1}{d_{i}^2-1}\right\}=s~,
\end{align*}
where $0<\hat x_s <1$. Note that the left-hand side of this equation is increasing, piecewise linear, and continuous in $\hat x_s$, so the equation is easy to solve. Once $\hat x_s$ is determined, all $\hat x_i$, with $ i\neq s$ are also uniquely determined by \eqref{eqlemma24}.
Finally, we have the following characterization of optimal solutions when $C$ is diagonal.
\begin{thm}\label{diagoptall}
Let $C:=\Diag(d)$, where $d\in\mathbb{R}^n$ satisfies $d_1\!\geq\! d_2\!\geq\! \cdots\! \geq\! d_n \!> \!0$. Let $L:= \{i\in N : d_i <1\}$, $E:= \{i\in N : d_i =1\}$, and $G:= \{i\in N : d_i >1\}$. Then
$\hat x$ defined below is an optimal solution for \eqref{basiclinx}.
\begin{itemize}
\item[(i)] If $s\le |G|$, let $\tilde x\in \mathbb{R}^{|G|}$ be the optimal solution of
$\widetilde{(1.1)}$,
which is \eqref{basiclinx}
with $(C,s)$ replaced by $(\tilde{C},\tilde{s})$ as follows:
$\tilde C:=\Diag(\tilde d)$, where $\tilde d\in \mathbb{R}^{|G|}, \tilde d_i:=d_i, 1\le i\le |G|$, and $\tilde s:=s$.
Then,
\[
\hat x_i:=
\left\{
\begin{array}{ll}
\tilde x_i ,&\hbox{for $i\in G$,}\\
0,&\hbox{for $i\in E\cup L$.}
\end{array}
\right.
\]
\item[(ii)] If $|G|< s\le |G\cup E|$, let $\tilde x_i, i\in E$ be any value such that $0\le \tilde x_i\le 1$ and $\sum_{i\in E} \tilde x_i = s-|G|$. Then,
\[
\hat x_i:=
\left\{
\begin{array}{ll}
1 ,&\hbox{for $i\in G$,}\\
\tilde x_i,&\hbox{for $i\in E,$}\\
0 ,&\hbox{for $i\in L.$}
\end{array}
\right.
\]
\item[(iii)] If $|G\cup E|<s$, let $\tilde x\in \mathbb{R}^{|L|}$ be the optimal solution of
$\widetilde{(1.1)}$,
which is \eqref{basiclinx}
with $(C,s)$ replaced by $(\tilde{C},\tilde{s})$ as follows:
$\tilde C:=\Diag(\tilde d)$ such that $\tilde d\in \mathbb{R}^{|L|}, \tilde d_i=d_{i+|G\cup E|}, 1\le i\le |L|$, and $\tilde s:=s-|G\cup E|$. Then,
\[
\hat x_i:=
\left\{
\begin{array}{ll}
1 ,&\hbox{for $i\in G\cup E$,}\\
\tilde x_i ,&\hbox{for $i\in L$.}
\end{array}
\right.
\]
\end{itemize}
\end{thm}
\proof{Proof.}
We will prove $(i)$ in detail; $(ii)$ and $(iii)$ can be proved in a similar manner. The feasibility of $\hat x$ is obvious. We will prove that $\hat x$ is optimal as well. Let us assume otherwise, i.e, we assume that $x^*$ is an optimal solution to \eqref{basiclinx} and \begin{equation}\label{hyp}f(C,s; x^*)> f(C,s;\hat x).\end{equation}
We first claim that $x_i^*=0$, $\forall i\in E\cup L$.
Otherwise let $x_i^*\! >\!0$, for some $i \!\in\! E\cup L$. Then, as $s\le |G|$, by feasibility of $x^*$, we have $x_j^*\!< \!1$ for some $ j \in G$. Therefore $e_j -e_i $ is a feasible direction from $x^*$ in \eqref{basiclinx}. However, by Lemma \ref{lem:conc}, we have that $\nabla f(C,s; x^*)^\top (e_j- e_i )>0$, contradicting the optimality of $x^*$.
Now, define
$\tilde{x}^*\in \mathbb{R}^{|G|}$ such that $\tilde{x}^*_i=x^*_i$, $\forall i\in G$. As $x_i^*\!=\!0$, $\forall i\in E\cup L$, it is straightforward to see that $\tilde{x}^*$ is feasible to $\widetilde{(1.1)}$. Thus $f(\tilde{C},\tilde{s};\tilde{x})\ge f(\tilde{C},\tilde{s};\tilde{x}^*)$. Note also that $\hat x_i\!=\!0$, $\forall i\!\in\! E\cup L$, so $f(C,s;\hat{x})=f(\tilde{C},\tilde{s};\tilde{x})\ge f(\tilde{C},\tilde{s};\tilde{x}^*)=f(C,s;x^*)$, contradicting \eqref{hyp}.
\Halmos
\endproof
Having characterized an optimal solution for \eqref{basiclinx} when $C $ is diagonal, we will now consider the more general case where $C$ is any positive-semidefinite matrix and establish a simple lower bound on $\linx (C,s)$, by considering the eigenvalues of $C$.
\begin{lem} \label{lb}
For any positive-semidefinite order-$n$ matrix $C$ and integer $0<s<n$,
\begin{align*}
\linx (C,s)\ge {\textstyle\frac{1}{2}} \log
\prod_{i=1}^n \textstyle \left(\frac{s}{n}\lambda_i^2+1-\frac{s}{n}\right)
,
\end{align*}
where $\lambda_1,...,\lambda_n$ are the eigenvalues of $C$.
\end{lem}
\proof{Proof.}
We diagonalize $C$: That is, we choose an orthogonal matrix $Q$
so that $Q^\top C Q= \Lambda := \Diag (\lambda_1,\lambda_2,...,\lambda_n)$.
Let $\bar x=\frac{s}{n}\mathbf{e}$. Then
\begin{align*}
\linx (C,s)\ge &f(C,s;\bar x)\\
=&\textstyle\frac{1}{2}\ldet \left(\frac{s}{n}C^2+\left(1-\frac{s}{n}\right)I\right)\\
=&\textstyle\frac{1}{2}\ldet \left(\frac{s}{n}Q\Lambda^2 Q^\top +\left(1-\frac{s}{n}\right)I\right)\\
=&\textstyle\frac{1}{2}\ldet \left(\frac{s}{n}\Lambda^2+\left(1-\frac{s}{n}\right)I\right)\\
=&{\textstyle\frac{1}{2}}\log
\prod_{i=1}^n \textstyle \left(\frac{s}{n}\lambda_i^2+1-\frac{s}{n}\right)
. \Halmos
\end{align*}
\endproof
We now study the efficacy of using a mask $M$ for the linx bound (versus choosing $M=J$). We will show that there is a sequence of $\{C_n\}_{n\in \mathcal{I}}$ such that $\linx (C_n,{\textstyle\frac{n}{2}})-\linx (C_n,{\textstyle\frac{n}{2}}; I)\ge \frac{1}{4}\log(\frac{4}{3}) n\approx .0312 n$. The results shows that by choosing an appropriate mask $M$ different from $J$, we can decrease the linx bound by at least an amount that is linear in $n$.
Recall that we have characterized an optimal solution of \eqref{basiclinx} when $C$ is diagonal in Theorem~\ref{diagoptall} and a lower bound of $\linx(C,s)$ when $C$ is any positive-semidefinite matrix in Lemma~\ref{lb}. Note that $C\circ I$ is diagonal. Then we have the following gap.
\begin{align}
&\linx (C,s)-\linx (C,s; I)\ge \label{maskgapineq}\\
&{\textstyle\frac{1}{2}}\!\log
\!\prod_{i=1}^n \!\!\left(\textstyle \frac{s}{n}\lambda_i^2\!+\!1\!-\!\frac{s}{n}\right)
\!-\! {\textstyle\frac{1}{2}}\!\log
\!\prod_{i=1}^n\!\!\left( d_i^2\hat x_i\!+\!1\!-\!\hat x_i\right)
\nonumber
\end{align}
where $\lambda_i, i\in N$ are the eigenvalues of $C$, $d_i, i\in N$ are diagonal elements of $C$, and $\hat x$ is an optimal solution of \eqref{basiclinx} with $C$ replaced by $C\circ I$.
We will employ this lower bound on the gap in what follows.
Before presenting our main result, we will characterize the optimal mask when $n=2$ and $s=1$.
We will use this to construct a
gap between $\linx (C,s)$ and $\linx (C,s; I)$ that is linear in the order of $C$.
\begin{thm}\label{2orderoptmask}
Let $C_2:=\left(\begin{array}{cc}
a & c \\
c& b
\end{array}\right)$ be positive-semidefinite
where we assume, without loss of generality, $a\geq b$. Let
$M_2^*=\left(\begin{array}{cc}
1 &m^* \\
m^*& 1
\end{array}\right)$
be an optimal mask for $\linx (C_2,1; M_2)$. We have,
\[
\begin{array}{cl}
(i)&\!\! if\, c=0, \, then \, m^* \, can \, be\, any \,value\, in [-1,1];\\
(ii)& \!\!if \;\frac{ab-1}{c^2}\ge 1, \;then \;m^*\; can \; be\;\pm 1;\\
(iii)&\!\! if \;\frac{ab-1}{c^2}\le 0, \;then \;m^*\; can \; be\; 0;\\
(iv)&\!\! if \;0<\frac{ab-1}{c^2}<1, \;then\; m^*\; can \; be\; \pm \sqrt{\frac{ab-1}{c^2}}.
\end{array}
\]
\end{thm}
\proof{Proof.}
Let $M_2:=\left(\begin{array}{cc}
1 &m \\
m& 1
\end{array}\right) \in \mathcal{M}_2$. Let $m^*=\argmin_{-1\le m\le 1}\left\{ \left(c^2m^2+1-ab\right)^2 \right\}$.
Considering that $x_1+x_2=1$, we obtain
\begin{align*}
&\ldet(\!(C_2\circ M_2)\!\Diag(\!x\!)\!(C_2\circ M_2)\!+\!I_2\!-\!\Diag(\!x\!)\!)\\
=&\log (\!(c^2m^2+1-ab)^2 x_1 x_2 +(ax_1+bx_2)^2)\\
\ge & \log (\!(c^2(m^*)^2+1-ab)^2 x_1 x_2 +(ax_1+bx_2)^2)\\
=& \ldet(\!(C_2\circ M_2^*)\!\Diag(\!x\!)\!(C_2\circ M_2^*)\!+\!I_2\!-\!\Diag(\!x\!)\!),
\end{align*}
which implies that $M_2^*$ is an optimal mask.
The values of $m^*$ in cases $(i-iv)$ can be easily obtained from $m^*=\argmin_{-1\le m\le 1}\left\{ \left(c^2m^2+1-ab\right)^2 \right\}$.
\Halmos
\endproof
For simplicity of the following discussions, we introduce the next lemma.
\begin{lem}\label{2optgap}
With the same hypotheses and notations as Theorem \ref{2orderoptmask}, $g(a,b,c):=\exp(2\linx(C_2,1;M_2^*))$. Define \[\Delta\linx(C,s;M):=\linx(C,s)-\linx (C,s; M).\] Then
\[
\textstyle\Delta \linx (C_2,1; M_2^*)
\ge\textstyle\frac{1}{2}\log\frac{(c^2+1-ab)^2+(a+b)^2}{4 g(a,b,c)}.
\]
\end{lem}
\proof{Proof.}
Let $\lambda_1\ge \lambda_2\ge 0$ be the two eigenvalues of $C_2$. Considering that
$\lambda_1+\lambda_2=a+b$ and $\lambda_1\lambda_2=ab-c^2$, the result follows directly from Lemma \ref{lb}.
\Halmos
\endproof
Note that for cases $(i-ii)$ in Theorem \ref{2orderoptmask}, there is no mask better than $J_2$. So we focus on cases $(iii-iv)$.
For case $(iv)$, we can calculate from the proof of Theorem \ref{2orderoptmask} that $\linx(C_2,1;M_2^*)=\frac{1}{2}\log\left(a^2\right)$. By Lemma \ref{2optgap}, we have
\[
\textstyle\Delta\linx(C_2,1; M_2^*)
\ge\textstyle\frac{1}{2}\log\frac{(c^2+1-ab)^2+(a+b)^2}{4a^2}.
\]
Note that $\frac{ab-1}{c^2}>0$ and $a\geq b$ imply $a>1$.
Further, $a>1$, $\frac{ab-1}{c^2}<1$ and $ab\geq c^2$
imply $0< c^2+1-ab\le 1 <a$. So,
\begin{align*}
\textstyle\frac{1}{2}\log\frac{(c^2+1-ab)^2+(a+b)^2}{4a^2}< {\textstyle\frac{1}{2}}\log\left( {\textstyle\frac{5}{4}}\right).
\end{align*}
Moreover, by choosing $a=b=c>1$, we are in case $(iv)$, and the
gap becomes $\frac{1}{2}\log(1+1/4a^2)$, which we can make as
close to $\frac{1}{2}\log(5/4)$
as we like.
For case $(iii)$, the optimal mask is $I_2$, and we can find a greater gap than we could for case $(iv)$. We prove this and our main result in the following theorem.
\begin{thm}\label{maskgap}
There is an infinite sequence of positive-semidefinite matrices $\{C_n\}_{n\in \mathcal{I}}$ such that
\begin{align*}
\linx \left(C_n,{\textstyle\frac{n}{2}}\right)\!-\!\linx \left(C_n,{\textstyle\frac{n}{2}}; I\right)\ge {\textstyle\frac{1}{4}}\log\left( {\textstyle\frac{4}{3}}\right) n, \, n\in \mathcal{I},
\end{align*}
where $\mathcal{I}$ is the set of even integers. Moreover, for $n=2$, this is the maximum
possible lower bound on the gap that can be achieved using the lower bound from Lemma \ref{lb}.
\end{thm}
\begin{rem}
As we have indicated above in our analysis of case (iv),
and proceeding similarly to
how we proceed below, we can also get linear gaps with masks that are
different from the identity mask, albeit with a worse constant (strictly less than $\frac{1}{4}\log(\frac{5}{4})$).
\end{rem}
\proof{Proof.}(Theorem \ref{maskgap})
First, consider $n=2,s=1$. We use the same notations as in Theorem \ref{2orderoptmask}
and consider its case $(iii)$, where $ab\le 1$, so that the optimal mask is $I_2$. In the following, we will use $\hat x$ to denote the optimal solution of \eqref{maskgammalinx} for $C:=C_2$, $s=1$, $M:=I_2$, and $\gamma=1$;
so $\linx(C_2,1;I_2)=\linx(C_2,1;I_2,1)=f(C_2,1;I_2,1;\hat x)$.
We have two sub-cases to analyze:
\begin{itemize}
\item[$(i)$] $a\ge 1\ge b$: by Theorem~\ref{diagoptall}, $\hat x=(1,0)^\top$ is an optimal solution and $\linx(C_2,1;I_2)=$ $\frac{1}{2}\log\left(a^2\right)$. By Lemma \ref{2optgap}, we have
\begin{equation}\label{eqloga2}
\textstyle\Delta\linx(C_2,1;I_2) \ge \textstyle\frac{1}{2}\log\frac{(c^2+1-ab)^2+(a+b)^2}{4a^2}.
\end{equation}
Note that $(c^2+1-ab)^2+(a+b)^2\le 5a^2$, so $\frac{(c^2+1-ab)^2+(a+b)^2}{4a^2}\le \frac{5}{4}$. The equality can be obtained when $a=b=c=1$.
\item[$(ii)$] $1>a\ge b$: there are still two sub-cases: \[\textstyle \frac{1}{b^2-1}-\frac{1}{a^2-1}\ge 1 \mbox{ and } \frac{1}{b^2-1}-\frac{1}{a^2-1}< 1.\]
\setlength{\leftmarginii}{0.0cm}
\begin{itemize}
\item[$\bullet$] If $\frac{1}{b^2-1}-\frac{1}{a^2-1}\ge 1$, then by Theorem~\ref{diagoptall}, we also have $\hat x=(1,0)^\top$ and $\linx(C_2,1;I_2)=\frac{1}{2}\log\left(a^2\right)$. Thus, \eqref{eqloga2} also holds.
From $\frac{1}{b^2-1}-\frac{1}{a^2-1}\ge 1$ and $b^2\ge 0$, we can see that ${\textstyle\frac{1}{2}}\le a^2<1$ and $b\le \sqrt{2-\frac{1}{a^2}}$. Together with $c^2\le ab<1$, letting $t:=\frac{1}{a^2}\in (1,2]$, we get
\begin{align*}
&\textstyle \max \frac{(c^2+1-ab)^2+(a+b)^2}{4a^2} =
\textstyle \max \frac{1+\left(a+\sqrt{2-\frac{1}{a^2}}\right)^2}{4a^2}\\
&\textstyle=\max \frac{1}{4} +\frac{3}{4} t-\frac{1}{4}t^2+{\textstyle\frac{1}{2}}\sqrt{2t-t^2}\\
& \textstyle\le \max \frac{1}{4} +\frac{3}{4} t-\frac{1}{4}t^2+\max {\textstyle\frac{1}{2}}\sqrt{2t-t^2}\\
& \textstyle=\frac{13}{16}+\frac{1}{2}< \frac{4}{3}.
\end{align*}
In the next sub-case, we will build a $\frac{1}{2}\log\left(\frac{4}{3}\right)$ gap, so the gap in the present sub-case is sub-optimal.
\item[$\bullet$] If $\frac{1}{b^2-1}-\frac{1}{a^2-1}<1$, then by Theorem~\ref{diagoptall},
$\hat x=\frac{1}{2}\left(1+\frac{1}{b^2-1}-\frac{1}{a^2-1},1-\frac{1}{b^2-1}+\frac{1}{a^2-1}\right)^\top$
is an optimal solution, and we have
\begin{align*}
\linx (C_2,1;& I_2)
\!=\! \textstyle \frac{1}{2}\!\log\!\left(\!\frac{1}{4}\!\left(a^2\!+\frac{a^2-1}{b^2-1}\right)\!\left(b^2\!+\frac{b^2-1}{a^2-1}\right)\!\right).
\end{align*}
By Lemma \ref{2optgap}, we have
\[
\textstyle \Delta\linx (C_2,1; I_2) \ge \textstyle {\textstyle\frac{1}{2}}\log\frac{(c^2+1-ab)^2+(a+b)^2}{\left(a^2+\frac{a^2-1}{b^2-1}\right)\left(b^2+\frac{b^2-1}{a^2-1}\right)}.
\]
We claim that
\[
\textstyle
\frac{(c^2+1-ab)^2+(a+b)^2}{\left(a^2+\frac{a^2-1}{b^2-1}\right)\left(b^2+\frac{b^2-1}{a^2-1}\right)}\leq \frac{1+(a+b)^2}{\left(a^2+\frac{a^2-1}{b^2-1}\right)\left(b^2+\frac{b^2-1}{a^2-1}\right)}\leq \frac{4}{3}.
\]
The first inequality holds because $c^2\leq ab<1$ and the second holds for being equivalent to
\[
\textstyle (1\!-\!2ab)^2+(a\!-\!b)^2+4(a^2\!-\!b^2)\left(\frac{1}{b^2-1}-\frac{1}{a^2-1}\right) \geq 0,
\]
We get equality in both with $a=b=c=\frac{\sqrt{2}}{2}$.
\end{itemize}
\end{itemize}
\vspace{0.1in}
\noindent
In the analysis above, we see that we can create the largest gap in the last case. Therefore, we define \[\textstyle C_2:=\left(\begin{array}{cc}
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\
\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2}
\end{array}\right).\] Then $\lambda_1=\sqrt 2,\lambda_2=0$, the optimal solution for $C_2\circ I_2$ is $\left({\textstyle\frac{1}{2}},{\textstyle\frac{1}{2}}\right)^\top$, and
\begin{align*}
&\linx (C_2,1)-\linx (C_2,1; I_2)\\
&\ge {\textstyle\frac{1}{2}}\log\left(\left({\textstyle\frac{1}{2}}\cdot (\sqrt 2)^2+1-{\textstyle\frac{1}{2}}\right)\left({\textstyle\frac{1}{2}}\cdot 0^2+1-{\textstyle\frac{1}{2}}\right)\right)\\
&\textstyle
-{\textstyle\frac{1}{2}}\log\left(\!\!\left({\textstyle\frac{1}{2}}\cdot \left(\!\frac{\sqrt{2}}{2}\right)^2\!+\!1-{\textstyle\frac{1}{2}}\!\right)\!\!\!\left(\!{\textstyle\frac{1}{2}}\cdot \left(\frac{\sqrt{2}}{2}\right)^2\!+\!1-{\textstyle\frac{1}{2}}\!\right)\!\!\right)\\
& =\textstyle {\textstyle\frac{1}{2}}\log\left(\frac{4}{3}\right).
\end{align*}
For $n=2k$, we construct a block-diagonal matrix $C_n$ with $k={\textstyle\frac{n}{2}}$ blocks, and each block is such a $C_2$ matrix. Then we take $s={\textstyle\frac{n}{2}}$. In this way, $C_n$ has $k$ eigenvalues of $\sqrt{2}$ and $k$ eigenvalues of $0$. Also, all diagonal elements of $C_n$ are $\frac{\sqrt{2}}{2}$. By \eqref{maskgapineq}, we have
\begin{align*}
&\linx \left(C_n,{\textstyle\frac{n}{2}}\right)-\linx \left(C_n,{\textstyle\frac{n}{2}}; I\right)\\
& \textstyle \ge \frac{n}{4}\log\left(\left({\textstyle\frac{1}{2}} (\sqrt 2)^2+1-{\textstyle\frac{1}{2}}\right)\left({\textstyle\frac{1}{2}} (0)^2+1-{\textstyle\frac{1}{2}}\right)\right)\\
& ~ \textstyle -\frac{n}{4}\!\log\!\left(\!\!\left(\!{\textstyle\frac{1}{2}} \left(\frac{\sqrt{2}}{2}\right)^2+1-{\textstyle\frac{1}{2}}\right)\!\!\!\left(\!{\textstyle\frac{1}{2}} \left(\frac{\sqrt{2}}{2}\right)^2+1-{\textstyle\frac{1}{2}}\right)\!\!\right)\\
&
=\textstyle \frac{1}{4}\log\left(\frac{4}{3}\right)n. \Halmos
\end{align*}
\endproof
\section{Optimal scaling parameter: some special cases and general behavior}\label{sec:gamma}
In this section, we first show how an appropriate scaling parameter $\gamma$ can help improve the linx bound by forcing one optimal solution of \eqref{gammalinx} to lie in $\{0,1\}^n$ when $C$ is diagonal or $C$ is non-singular of order $2$. Next, we show the following results: $(i)$ if $s<\rank(C)$, then an optimal scaling parameter $\gamma$ for \eqref{gammalinx} can always be obtained, $(ii)$ if $s=\rank(C)$ and $\hat \gamma$ is an optimal scaling parameter for linx, then so is any $\gamma\geq \hat\gamma$, $(iii)$ if $s>\rank(C)$, there is no optimal $\gamma$. In fact, in this case we show that the linx-bound has the nice property of recognizing the behavior of \ref{MESP}, it tends to minus infinity as $\gamma$ tends to infinity.
\begin{prop}\label{diagoptgamma}
For diagonal positive-definite matrix $C:=\Diag \{d_1,...,d_n\}$, where $d_1\ge ...\ge d_n> 0$ and $0<s<n$, the scaling parameter $\hat \gamma=\frac{1}{d_s^2}$ forces an optimal solution of \eqref{gammalinx} to lie in $\{0,1\}^n$.
Therefore $\hat \gamma$ is an optimal scaling parameter.
\end{prop}
\proof{Proof.}
Note that
\[
f(C,s; \gamma; x)={\textstyle\frac{1}{2}} \log
\prod_{i=1}^n (\gamma d_i^2x_i+1-x_i)
-{\textstyle\frac{1}{2}} s\log\gamma.
\]
Partition $N$ as $N=L'\cup E'\cup G'$, where $\gamma d_i^2<1,i\in L'$; $\gamma d_i^2=1, i\in E'$; $\gamma d_i^2>1, i\in G'$. As we have seen in Lemma \ref{lem:conc}, $f(C,s; \gamma; x)$ strictly decreases with $x_i, i\in L'$, does not change with $x_i, i\in E'$ and strictly increases with $x_i, i\in G'$. So, if there is a $\gamma>0$ such that $|G'|\le s$ while $|E'\cup G'|\ge s$. By Theorem \ref{diagoptall},
$(e_s^\top ,0)^\top $ is an optimal solution for \eqref{gammalinx} which lies in $\{0,1\}^n$. In fact, $\hat \gamma:=\frac{1}{d_s^2}$ is such a
scaling
parameter. Therefore, by Proposition ~\ref{powersetopt}, $\hat \gamma$ is optimal.
\Halmos
\endproof
\begin{prop}\label{2optgamma}
Let $C_2:=\left(\begin{array}{cc}
a & c \\
c& b
\end{array}\right)$ be positive-definite
where we assume, without loss of generality, $a\geq b$. Let $s=1$.
Then the scaling parameter $\hat \gamma=\frac{a^2-c^2}{(ab-c^2)^2}$ forces an optimal solution of \eqref{gammalinx} to lie in $\{0,1\}^2$. Therefore $\hat \gamma$ is an optimal scaling parameter.
\end{prop}
\proof{Proof.}
We have
\begin{align*}
&f(C_2,1; x)\\
&={\textstyle\frac{1}{2}} \ldet (C_2 \Diag (x) C_2+I_2-\Diag (x)) \\
&={\textstyle\frac{1}{2}} \!\log (\!(c^2+1-ab)^2 x_1 x_2 +(ax_1+bx_2)^2).
\end{align*}
Because $f(C_2,1;x)$ is concave, and the null space of $\mathbf{e}_2$ is $\{(t,-t)^\top ~:~ t\in \mathbb{R}\}$, to prove that one optimal solution lies in $\{0,1\}^2$ (in particular, we assume this optimal solution is $\hat x=(1,0)^\top$), we only need to prove
\[
\left.\frac{f(C_2,1; \hat x - t(e_1-e_2))}{\partial t}\right|_{t=0} \le 0
\]
which is equivalent to
\begin{equation}\label{eq:diff}
\frac{\partial f(C_2,1; \hat x)}{\partial x_1}- \frac{\partial f(C_2,1; \hat x)}{\partial x_2}\ge 0,
\end{equation}
and finally
$
2(a^2-c^2)-(c^2-ab)^2-1\ge 0.
$
Because $f(C_2,s; \gamma; x)=f(\sqrt{\gamma} C_2,s; x)+{\textstyle\frac{1}{2}}\log\gamma$, we have that
\[
\frac{\partial f(C_2,1;\gamma; \hat x)}{\partial x_1}- \frac{\partial f(C_2,1; \gamma;\hat x)}{\partial x_2}\ge 0,
\]
is equivalent to
\[
\frac{\partial f(\sqrt{\gamma}C_2,1; \hat x)}{\partial x_1}- \frac{\partial f(\sqrt{\gamma}C_2,1; \hat x)}{\partial x_2}\ge 0,
\]
and finally
\begin{equation} \label{2ordersuffcondi}
2(a^2-c^2)\gamma-(c^2-ab)^2\gamma^2-1\ge 0.
\end{equation}
The left-hand side of \eqref{2ordersuffcondi} is maximized by $\hat\gamma=\frac{a^2-c^2}{(ab-c^2)^2}$ and the corresponding value is $\frac{(a^2-c^2)^2}{(ab-c^2)^2}-1$ which is nonnegative because $a^2\ge ab>c^2$. Note that if $a>b$ then $\frac{(a^2-c^2)^2}{(ab-c^2)^2}-1>0$, which means there is an $\epsilon>0$ such that for any $\gamma\in [\hat\gamma-\epsilon,\hat\gamma+\epsilon]$, $\frac{(a^2-c^2)^2}{(ab-c^2)^2}-1\ge 0$ and $\hat x$ is an optimal solution, i.e., the optimal scaling parameter is not unique. Finally, by Proposition \ref{powersetopt}, $\hat \gamma$ is optimal. \Halmos
\endproof
Not unexpectedly, there also exists a large and simple class of $C$ where no optimal solution of \eqref{gammalinx} lies in $\{0,1\}^n$ for any $0<s<n$ and any scaling parameter $\gamma$, as we will see in Theorem \ref{neverinpowerset}.
First, note that when $C=\tau_1 I$, for any $\tau_1>0$, in all cases of Theorem~\ref{diagoptall}, $\hat x:= \frac{s}{n} \mathbf{e}$ is an optimal solution of \eqref{basiclinx}. The same observation can be extracted from Lemma \ref{lem:propxhat}. In fact, this observation is also a special case of the following result, which follows immediately from the concavity of $f(C,s;x)$ and its symmetry in this case.
\begin{prop}
\label{tauijopt}
Suppose that $\tau_1>0$, $\tau_2 \ge 0$,and $0<s<n$ integer.
Let $C=\tau_1 I+\tau _2 J$, then $\hat x=\frac{s}{n}\mathbf{e}$ is an optimal solution for \eqref{basiclinx}.
\end{prop}
\begin{thm}\label{neverinpowerset}
For any order $n\ge 3$, any $0<s<n$ and $C=\tau_1 I+\tau_2 J$, $\tau_1>0$, $\tau_2> 0$, and for any scaling parameter $\gamma>0$, the optimal solution of \eqref{gammalinx} cannot lie in $\{0,1\}^n$.
\end{thm}
\proof{Proof.}
By Proposition~\ref{tauijopt}, one optimal solution for \eqref{gammalinx} is $\frac{s}{n} \mathbf{e}$ under the setting of this theorem. From the proof of [Theorem 21, \cite{chen_mixing}], we have that if
\begin{align}
& \gamma C\Diag (y)C-\Diag (y)\neq 0,\label{strictconcavecondi}\\
& \hbox{ when $\mathbf{e}^\top y=0$, $-\mathbf{e}\le y\le \mathbf{e}$, $y\neq 0$}, \nonumber
\end{align}
then $f(C,s;\gamma;x)$ is strictly concave with a unique optimal solution on the feasible region of \eqref{gammalinx}. Because $\frac{s}{n}\mathbf{e}$ is already optimal in this case, we see that the optimal solution cannot lie in $\{0,1\}^n$. Now we prove that \eqref{strictconcavecondi} holds.
Substituting $\tau_1 I+\tau_2 J$ for $C$ in \eqref{strictconcavecondi} and dividing by $\gamma$, we get
\begin{align}
& \textstyle
\left(\tau_1^2-\frac{1}{\gamma}\right)\Diag (y)+\tau_1\tau_2(\mathbf{e}y^\top +y\mathbf{e}^\top )\neq 0 \label{strictconcavecondi2}
\end{align}
It is easy to see that if \eqref{strictconcavecondi2} is \emph{not} satisfied, then $y_i+y_j=0$ for all $i\not=j$.
But this cannot be true when $n\geq 3$ for $y\not=0$. \Halmos
\endproof
Interestingly, there is also a very simple example for which no $\gamma>0$ can be an optimal scale factor :
\begin{prop}\label{2nooptgamma}
For $C:=J_2$, $s:=1$, there is no optimal scaling factor $\gamma$ for \eqref{gammalinx}. In fact, for all $\gamma>0$,
\[
\textstyle
\linx (J_2,1; \gamma)={\textstyle\frac{1}{2}}\log \left(1+\frac{1}{4\gamma}\right).
\]
which monotonically decreases as $\gamma$ increases.
\end{prop}
\proof{Proof.}
\begin{align*}
& \linx (J_2,1;\gamma)\\
& = {\textstyle\frac{1}{2}}\max _{\begin{array}{c}
\scriptstyle x_1+x_2=1 \\
\scriptstyle 0\le x_1, x_2\le 1
\end{array}}\left\{\log\left(\gamma (x_2+x_1)(2-x_1-x_2)\right.\right.\\[-25pt]
&\qquad\qquad\qquad\qquad \left. \left.
+(1-x_1)(1-x_2)\right)-\log \gamma
\right\}\\[5pt]
& \le {\textstyle\frac{1}{2}}\max_{\begin{array}{c}
\scriptstyle x_1+x_2=1 \\
\scriptstyle 0\le x_1, x_2\le 1
\end{array}}\left\{
\log\left(\vphantom{\left(\textstyle\frac{2-x_1-x_2}{2}\right)^2}\right.\gamma (x_2+x_1)(2-x_1-x_2)\right.\\[-20pt]
&\qquad\qquad\qquad\qquad\qquad\left.\left. +\left(\textstyle\frac{2-x_1-x_2}{2}\right)^2\right)-\log \gamma
\right\}\\
&\textstyle = \textstyle\frac{1}{2}\left(\log \left(\gamma+\frac{1}{4}\right)-\log\gamma\right) = {\textstyle\frac{1}{2}}\log \left(1+\frac{1}{4\gamma}\right).
\end{align*}
Note that both maximums are achieved at $x_1=x_2=1/2$, and so the inequality is an equation.
\Halmos
\endproof
Based on Proposition~\ref{2nooptgamma}, our interest is in what cases, we are guaranteed to have a finite optimal scaling parameter $\gamma$. In fact, a broad sufficient condition is $s<\rank (C)$ by the following theorem.
\begin{thm}\label{ssmallrank}
For all positive-semidefinite $C$ and $0<s<n$, we have
$$
\lim_{\gamma\rightarrow 0} \linx (C,s; \gamma)=+\infty.
$$
If we further assume that $s<\rank(C)$, then
$$
\lim_{\gamma\rightarrow +\infty} \linx (C,s; \gamma)=+\infty.
$$
\end{thm}
\proof{Proof.}
For all $\gamma>0$, by setting $\bar x:=\frac{s}{n} \mathbf{e}$, we have
\begin{align*}
&\linx (C,s; \gamma)\ge f(C,s; \gamma; \bar x)\\
& \textstyle \quad ={\textstyle\frac{1}{2}} \left(\ldet \left(\gamma\frac{s}{n}C^2+I-\frac{s}{n}I\right)-s\log \gamma\right).
\end{align*}
When $\gamma\rightarrow 0$, $\gamma\frac{s}{n}C^2+I-\frac{s}{n}I\rightarrow \left(1-\frac{s}{n}\right)I$. So
\begin{align*}
&\textstyle \lim_{\gamma\rightarrow 0} \ldet \left(\gamma\frac{s}{n}C^2+I-\frac{s}{n}I\right)= n\log \left(1-\frac{s}{n}\right), \\
&\hbox{and } \lim_{\gamma\rightarrow 0}-s\log\gamma=+\infty.
\end{align*}
Therefore, $\lim_{\gamma\rightarrow 0}\linx (C,s; \gamma)= +\infty$.
But we can also write $ f(C,s; \gamma; \bar x)=$
\[
\textstyle \frac{1}{2}\left(\ldet \left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)\!+\!(n\!-\!s)\log\gamma\right).
\]
Note that
$\lim_{\gamma\rightarrow +\infty}(n-s)\log\gamma=+\infty$. Further,
if $C$ is non-singular, then
\begin{align*}
\lim_{\gamma\rightarrow +\infty} \textstyle\ldet \left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)= \ldet \left(\frac{s}{n}C^2\right).
\end{align*}
So we can conclude that $\lim_{\gamma\rightarrow +\infty} \linx (C,s; \gamma)=+\infty$, when $C$ is nonsingular.
If $C$ is singular, then we have
\begin{align*}
\textstyle
\lim_{\gamma\rightarrow +\infty}\ldet \left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)= -\infty,
\end{align*}
and we cannot immediately conclude anything useful.
So we proceed differently.
When $s<\rank (C)$, without loss of generality,
we can write $C=Q\Lambda Q^\top$, where $Q$ is orthogonal and $\Lambda:=\Diag(\lambda_1,...,\lambda_n)$ with $\lambda_1\ge \lambda_2\ge ...\ge \lambda_n\ge 0$. We have $\lambda_i\neq 0$ for $i\le \rank (C)$ and $\lambda_i=0$ for $i> \rank (C)$.
By
L'H\^optital's
rule,
\begin{align*}
&\ \lim_{\gamma\rightarrow +\infty} \textstyle\frac{\ldet \left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)}{(n-s)\log\gamma}\\
=&\lim_{\gamma\rightarrow +\infty} \textstyle\frac{\partial \left(\ldet \left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)\right)/\partial \gamma}{\partial \left((n-s)\log\gamma\right) /\partial \gamma}\\
=& \lim_{\gamma\rightarrow +\infty}\textstyle\frac{\text{tr}\left(\left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)^{-1}\left(1-\frac{s}{n}\right)I\right)\frac{-1}{\gamma^2}}{(n-s)\frac{1}{\gamma}} \\
=& \lim_{\gamma\rightarrow +\infty}\textstyle\frac{-1}{n\gamma} \text{tr}\left(\left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)^{-1}\right)\\
=& \lim_{\gamma\rightarrow +\infty}\textstyle\frac{-1}{n} \text{tr}\left(\left(\gamma\frac{s}{n}C^2+\left(1-\frac{s}{n}\right)I\right)^{-1}\right)\\
=& \lim_{\gamma\rightarrow +\infty}\textstyle\frac{-1}{n} \text{tr}\left(\left(\gamma\frac{s}{n}Q\Lambda^2Q^\top+\left(1-\frac{s}{n}\right)QQ^\top\right)^{-1}\right)\\
=&\lim_{\gamma\rightarrow +\infty}\textstyle\frac{-1}{n} \text{tr}\left(Q\left(\gamma\frac{s}{n}\Lambda^2+\left(1-\frac{s}{n}\right)I\right)^{-1}Q^\top\right)\\
=&\textstyle\frac{-1}{n-s}\text{tr}\left(\text{Diag}\left\{\mathbf{0}_{1\times \rank (C)},\mathbf{e}_{n-\rank (C)}^\top\right\}\right)\\
=&\textstyle-\frac{n-\rank (C)}{n-s}.
\end{align*}
This means for every $\epsilon>0$, there exists $\gamma_{\epsilon}>0$ such that when $\gamma>\max\{\gamma_{\epsilon},1\}$,
\begin{align*}
&\textstyle
{\textstyle\frac{1}{2}}\left(\ldet \left(\frac{s}{n}C^2+\frac{1}{\gamma}\left(1-\frac{s}{n}\right)I\right)+(n-s)\log\gamma\right)\\
& \textstyle\quad \ge \frac{1}{2}\left(-\frac{n-\rank (C)}{n-s}-\epsilon+1\right)(n-s)\log\gamma.
\end{align*}
So
\vspace{-20pt}
\begin{align*}
&\lim_{\gamma\rightarrow +\infty} \linx (C,s; \gamma)\\
\ge & \lim_{\epsilon\rightarrow 0}\lim_{\gamma\rightarrow +\infty}\textstyle\left(-\frac{n-\rank (C)}{n-s}-\epsilon+1\right)(n-s)\log\gamma\\
= & \lim_{\epsilon\rightarrow 0}\lim_{\gamma\rightarrow +\infty}\textstyle\left(\frac{\rank (C)-s}{n-s}-\epsilon\right)(n-s)\log\gamma\\
=& +\infty. \Halmos
\end{align*}
\endproof
\begin{cor}
For all positive-semidefinite $C$ and $0<s<n$ where $s<\rank(C)$, we can find a finite optimal scaling parameter $\hat \gamma$ such that
\begin{align*}
\linx (C,s;\hat\gamma)=\min_{\gamma>0} \linx (C,s; \gamma).
\end{align*}
\end{cor}
\proof{Proof.}
By \cite{chen_mixing}, if we replace $\gamma$ with $e^{\psi}$, then $\linx (C,s; e^{\psi})$ is convex in $\psi$ and by Theorem~\ref{ssmallrank},
\begin{align*}
\lim_{\psi\rightarrow -\infty} \linx (C,s; e^{\psi})=\lim_{\gamma\rightarrow 0} \linx (C,s; \gamma)&=+\infty\\
\lim_{\psi\rightarrow +\infty} \linx (C,s; e^{\psi})=\lim_{\gamma\rightarrow +\infty} \linx (C,s; \gamma)&=+\infty.
\end{align*}
We can conclude that a minimizing $\hat{\psi}$ exists, and then we have the
minimizer $\hat \gamma:=e^{\hat{\psi}}$.
\Halmos
\endproof
When $s=\rank(C)$, the following result establishes that $\lim_{\gamma\rightarrow \infty}\linx (C,s; \gamma)$ exists and is finite, and $\linx (C,s; \gamma)$ is monotonically non-increasing in $\gamma$. This implies that if $\hat \gamma>0$ is optimal, then all $\gamma\ge \hat \gamma$ are optimal.
\begin{thm}\label{seqrank}
When $s=\rank(C)$, without loss of generality, we can write $C$ as $C=Q\Lambda Q^\top $, where $Q$ is orthogonal and $\Lambda: =\Diag (\lambda_1,...,\lambda_n)$ with $\lambda_1\ge...\ge\lambda_s> \lambda_{s+1}=... =\lambda_n=0$. Denote $\Lambda_s:=\Diag (\lambda_1,...,\lambda_s)$. Denote $P=Q^\top \Diag (x)Q$, $P_s$ as the principle sub-matrix of $P$ indexed by $(1,...,s)$, $P_{n-s}$ as the principle sub-matrix of $P$ indexed by $(s+1,...,n)$ and $P_{s,n-s}$ as the sub-matrix of $P$ with rows indexed by $(1,...,s)$ and columns indexed by $(s+1,...,n)$ ($P_{n-s,s}$ similarly).
Then the value $\lim_{\gamma\rightarrow +\infty}\linx (C,s; \gamma)$ exists and is the optimal value of the following convex program:
\begin{equation}\label{limcvx}
\begin{array}{cl}
\max &{\textstyle\frac{1}{2}} \!\left(\ldet \left(\Lambda_{s} P_{s} \Lambda_{s}\right)\!+\!\ldet \left(I_{n-s}-P_{n-s}\right)\! \right)\\ \text { s.t. } & \mathbf{e}^\top x=s \\ & 0 \leq x \leq 1.
\end{array}
\end{equation}
Furthermore, $\linx (C,s; \gamma)$ is monotonically non-increasing in $\gamma$.
\end{thm}
\proof{Proof.}
By the conditions,
\begin{align*}
& \ldet (\gamma C \Diag (x)C+I-\Diag (x))-s\log \gamma\\
& \quad =\ldet (\gamma \Lambda P \Lambda +I-P)-s\log\gamma.
\end{align*}
Because $C,s$ are fixed, let $F_s(\gamma;x):=\gamma \Lambda_s P_s \Lambda_s +I_s-P_s$ be the principle sub-matrix of $\gamma \Lambda P \Lambda +I-P$ indexed by $(1,...,s)$. We first prove that for any $\gamma>0$ and any $x$ feasible, $F_s(\gamma;x)$ is positive-definite so that we can use Schur complement formula to represent the determinant of $\gamma \Lambda P \Lambda +I-P$.
The construction of $P$ implies its eigenvalues are $\{x_1,x_2,...,x_n\}$ so all eigenvalues of $P$ lie in $[0,1]$. Because $P_s$ is a principle sub-matrix of $P$, by [Theorem 4.3.17, \cite{horn2012matrix}], all eigenvalues of $P_s$ lie in $[0,1]$. Decompose $P_s$ as $P_s=\hat Q \hat \Lambda \hat Q^\top $ where $\hat Q$ is orthogonal and $\hat \Lambda$ is the diagonal matrix of eigenvalues of $P_s$. In particular, all elements of $\diag(\hat \Lambda)$ are in $[0,1]$. Let $\hat C=\hat{Q}^\top \Lambda_s\hat Q$, then
\begin{align*}
F_s(\gamma;x)=\hat{Q}\left(\gamma \hat C\hat \Lambda \hat C+I_s-\hat \Lambda\right)\hat{Q}^\top .
\end{align*}
Because $\Lambda_s$ is positive-definite, so is $\hat C$. By [Lemma 20, \cite{chen_mixing}], $F_s(\gamma;x)$ is positive-definite for any $\hat\Lambda$ where $0\le \diag (\hat \Lambda)\le \mathbf{e}$.
We only need to consider $x$ in the feasible region such that $\gamma \Lambda P \Lambda +I-P$, (equivalently, $\gamma C \Diag (x)C+I-\Diag (x)$) is positive-definite.
So we assume that $\gamma \Lambda P \Lambda +I-P$ is positive-definite in the following.
Then the Schur complement of $\gamma \Lambda P \Lambda +I-P$ in $F_s(\gamma;x)$, which is $I_{n-s}-P_{n-s} -P_{n-s,s}F_s(\gamma;x)^{-1}P_{s,n-s}$, is also positive-definite. Furthermore, $P_{n-s,s}F_s(\gamma;x)^{-1}P_{s,n-s}$ is positive-semidefinite by the positive-definiteness of $F_s(\gamma;x)$ and we get that $I_{n-s}-P_{n-s}$ is positive-definite. On the other hand, because the feasible region of $x$ is compact, and the objective value of \eqref{limcvx} is upper bounded, the optimal value of \eqref{limcvx} is attainable by some $x$ such that the corresponding $\Lambda_s P_s \Lambda_s$ and $I_{n-s}-P_{n-s}$ are positive-definite. So we justify the definition of \eqref{limcvx}, and
\begin{align*}
& \ldet (\gamma \Lambda P \Lambda +I-P)-s\log\gamma\\
&= \ldet (F_s(\gamma;x)) -s\log\gamma\\
& +\!\ldet ( I_{n-s}\!-\!P_{n-s} \!-\!P_{n-s,s}F_s(\gamma;x)^{-1}\!P_{s,n-s} )\\
&\textstyle = \ldet \left(\Lambda_s P_s \Lambda_s +\frac{1}{\gamma}(I_s-P_s)\right)\\
&+\!\ldet (I_{n-s}\!-\!P_{n-s}\!-\!P_{n-s,s}F_s(\gamma;x)^{-1}\!P_{s,n-s})\!.
\end{align*}
Denote the optimal solution of \eqref{limcvx} as $x^*$ and $P^*=Q^\top \Diag(x^*) Q$. We claim that $\lim_{\gamma\rightarrow +\infty}\linx (C,s;\gamma)=$
\begin{align*}
{\textstyle\frac{1}{2}}\left( \ldet (\Lambda_s P^*_s \Lambda_s )+\ldet (I_{n-s}-P^*_{n-s})\right).
\end{align*}
We now prove this claim. In fact, for any $x$ feasible to \eqref{gammalinx} such that $\gamma \Lambda P \Lambda +I-P$ is positive-definite and that $F_s(\gamma;x)$ is positive-definite, we have
\begin{align}\label{infub}
&\textstyle \ldet (\Lambda_s P_s \Lambda_s+\frac{1}{\gamma}(I_s-P_s))\\
&\textstyle +\ldet (I_{n-s}\!-\!P_{n-s}\!-\!P_{n-s,s}F_s(\gamma;x)^{-1}\!P_{s,n-s})\nonumber\\
& \textstyle \le \ldet (\Lambda_s P_s \Lambda_s+\frac{1}{\gamma}I_s)+\ldet (I_{n-s}-P_{n-s}).\nonumber
\end{align}
We further assume that $\Lambda_s P_s \Lambda_s$ is positive-definite otherwise the right-hand-side of \eqref{infub} goes to minus infinity as $\gamma$ goes to infinity because $\ldet (I_{n-s}-P_{n-s})$ is clearly upper bounded by a uniform finite number for any $x$.
Decompose $\Lambda_s P_s \Lambda_s$ as $Q'\Lambda'Q'^\top $ where $Q'$ is orthogonal and $\Lambda': =\Diag (\lambda_1',...,\lambda_s')$ where $\lambda_1'\ge \lambda_2'\ge\ldots\ge\lambda_s'>0$ is the diagonal matrix of eigenvalues of $\Lambda_s P_s \Lambda_s$. Then
\begin{align*}
\textstyle \ldet \left(\Lambda_s P_s \Lambda_s +\frac{1}{\gamma}I_s\right)=\log\left(\displaystyle\prod_{i=1}^s \textstyle \left(\lambda'_i+\frac{1}{\gamma}\right)\right).
\end{align*}
Because every element of $\Lambda_s P_s \Lambda_s$ is bounded by a uniform number for any $x$, by Gershgorin circle theorem, $\lambda'_i,i\in\{1,...,s\}$ are bounded by a uniform number for all $x$. We pick a positive number $L_1>0$, when $\gamma\ge L_1$, there is a compact set $\mathcal{H}\subset \mathbb{R}^s$ (independent of $\gamma$) such that for all $x$ feasible to \eqref{gammalinx}, $(\lambda'_1+\frac{1}{\gamma},...,\lambda_s'+\frac{1}{\gamma})^\top$ as well as $(\lambda'_1,...,\lambda_s')^\top$ belongs to $\mathcal{H}$. Because the function $\prod_{i=1}^s y_i$ is continuous differentiable in $y$ on $\mathbb{R}^s$, it is Lipschitz continuous on $\mathcal{H}$, then, $\exists$ $L_2>0$ such that
\begin{align*}
\left|\prod_{i=1}^s \textstyle \left(\lambda'_i+\frac{1}{\gamma}\right)-\displaystyle \prod_{i=1}^s \lambda'_i\right|\leq \textstyle \frac{L_2\sqrt{s}}{\gamma}.
\end{align*}
Because $I_{n-s}-P_{n-s}$ is positive-definite and every element is bounded by a uniform number for any $x$, there exists $L_3>0$,
\begin{align*}
0<\det (I_{n-s}-P_{n-s})\le L_3.
\end{align*}
With the above arguments, when $\gamma\ge L_1$, we have
\begin{align*}
&\textstyle \det\left(\Lambda_s P_s \Lambda_s +\frac{1}{\gamma}I_s\right)\det (I_{n-s}\!-\!P_{n-s})\\
&= \left(\prod_{i=1}^s \textstyle\left(\lambda'_i+\frac{1}{\gamma}\right)\right)\det (I_{n-s}\!-\!P_{n-s})\\
&\le \left(\prod_{i=1}^s \lambda'_i+ \textstyle\frac{L_2\sqrt{s}}{\gamma}\right) \det (I_{n-s}\!-\!P_{n-s})\\
& = \det (\Lambda_s P_s \Lambda_s)\!\det (I_{n-s}\!-\!P_{n-s})\\&\qquad +\textstyle\frac{L_2\sqrt{s}}{\gamma}\!\det (I_{n-s}\!-\!P_{n-s})\\
& \le \det (\Lambda_s P^*_s \Lambda_s)\!\det (I_{n-s}\!-\!P^*_{n-s})\\ &\qquad +\textstyle\frac{L_2\sqrt{s}}{\gamma}\!\det (\!I_{n-s}\!-\!P_{n-s})\\
&\!\le\! \det (\Lambda_s P^*_s \Lambda_s)\!\det (I_{n-s}\!-\!P^*_{n-s})\!+\!\textstyle\frac{L_2 L_3\sqrt{s}}{\gamma}.
\end{align*}
For any $x$ such that $\Lambda_s P_s \Lambda_s$ is singular, because the eigenvalues of $\Lambda_s P_s \Lambda_s$ are upper bounded uniformly for all $x$ feasible, clearly there is some $L_4>0$ such that when $\gamma\ge L_4$, any such $x$ cannot be an optimal solution for \eqref{gammalinx}.
Because $\log(\cdot)$ is monotonically increasing, the above implies that when $\gamma\ge \max\{L_1,L_4\}$, we have
\begin{align*}
&\linx (C,s; \gamma)\\
&=\max_{\begin{array}{c}
\scriptstyle \mathbf{e}^\top x=s, \\
\scriptstyle 0\le x\le \mathbf{e}
\end{array}} \textstyle {\textstyle\frac{1}{2}}\left(\ldet \left(\Lambda_s P_s \Lambda_s+\frac{1}{\gamma}(I_s-P_s)\right)\right.\\
&\textstyle \left. +
\vphantom{\ldet \left(\Lambda_s P_s \Lambda+\frac{1}{\gamma}(I_s-P_s)\right)}
\ldet (I_{n-s}-P_{n-s}-P_{n-s,s}F_s(\gamma;x)^{-1}P_{s,n-s})\right)\\
&\leq {\textstyle\frac{1}{2}}\log \left(\det (\Lambda_s P^*_s \Lambda_s)\det (I_{n-s}-P^*_{n-s})\right.\\
&\qquad \left.+\textstyle \frac{L_2 L_3\sqrt{s}}{\gamma}\right).
\end{align*}
Taking limits on both sides, we have
\begin{align*}
&\lim_{\gamma\rightarrow +\infty}\linx (C,s; \gamma)\leq \\
&\lim_{\gamma\rightarrow +\infty} \!{\textstyle\frac{1}{2}}\!\log \!\left(\!\det (\Lambda_s P^*_s \Lambda_s)\!\det (I_{n-s}\!-\!P^*_{n-s})\right.\\
&\qquad \left.+\textstyle \frac{L_2L_3\sqrt{s}}{\gamma}\right)\\
&= {\textstyle\frac{1}{2}}\left(\ldet (\Lambda_s P^*_s \Lambda_s )+\ldet (I_{n-s}-P^*_{n-s})\right).
\end{align*}
On the other hand, the optimal solution $x^*$ of \eqref{limcvx} is feasible to \eqref{gammalinx} and we have proved before that $\Lambda_s P^*_s \Lambda_s$ and $I_{n-s}-P^*_{n-s}$ are positive-definite, we have $\lim_{\gamma\rightarrow \infty} F_s(\gamma;x^*)^{-1}=O_{n}$ where $O_{n}$ is an all-zeros order-$n$ matrix and $\linx (C,s; \gamma)\ge f (C,s; \gamma; x^*)$. Furthermore,
\begin{align*}
& \lim_{\gamma\rightarrow +\infty}\linx (C,s; \gamma)
\ge \lim_{\gamma\rightarrow +\infty}f (C,s; \gamma; x^*)\\
&= \lim_{\gamma\rightarrow +\infty}{\textstyle\frac{1}{2}}\left(\ldet \left(\Lambda_s P^*_s \Lambda_s \textstyle+\frac{1}{\gamma}(I_s-P^*_s)\right)\right.\\
& \left. +\ldet (I_{n-s}-P^*_{n-s}-P^*_{n-s,s}F_s(\gamma;x^*)^{-1}P^*_{s,n-s})\right)\\
&= \textstyle\frac{1}{2}\left( \ldet (\Lambda_s P^*_s \Lambda_s )+\ldet (I_{n-s}-P^*_{n-s})\right).
\end{align*}
In all, we have
$\displaystyle \lim_{\gamma\rightarrow +\infty}\!\!\linx (C,s; \gamma)$
\[
=\!{\textstyle\frac{1}{2}}\!\!\left( \ldet (\Lambda_s P^*_s \Lambda_s )\!+\!\ldet (I_{n-s}\!-\!P^*_{n-s})\right)\!.
\]
Finally, because $\linx (C,s; e^{\psi})$ is convex in $\psi$ (see [Theorem 18, \cite{chen_mixing}])
and has a finite limit as $\psi\rightarrow +\infty$, we can conclude that $\linx (C,s; e^{\psi})$ is non-increasing in
$\psi$, and hence $\linx (C,s; \gamma)$ is non-increasing in $\gamma$.
\Halmos
\endproof
At the outset, we assumed $s\leq \rank(C)$.
Of course, the case where $s>\rank (C)$ is a bit strange because the optimal value of \ref{MESP} is always $-\infty$. But by the following theorem, the linx-bound problem can recognize these cases.
\begin{thm}\label{slarrank}
If $s>\rank (C)$, then
$
\lim_{\gamma\rightarrow +\infty}
$
$
\linx (C,s; \gamma)
$
$
=-\infty,
$
and there is no optimal $\gamma$.
\end{thm}
\proof{Proof.}
Let $r=\rank (C)$. We use similar notations as in Theorem~\ref{seqrank}, but with the a little difference. Here we have $\Lambda_r: =\Diag\{\lambda_1,\lambda_2,\ldots,\lambda_r\}$ and $P_{r}$, $P_{n-r}$, $P_{n-r,r}$, $P_{r,n-r}$ similarly because $r<s$. Then
\begin{align*}
&\ldet (\gamma C \Diag (x)C+I-\Diag (x))-s\log \gamma\\
&=\textstyle \ldet \left(\Lambda_r P_r \Lambda_r +\frac{1}{\gamma}(I_r-P_r)\right)-(s-r)\log\gamma\\
&\quad +\ldet (I_{n-r}-P_{n-r}-P_{n-r,r}F_r(\gamma;x)^{-1}P_{r,n-r}).
\end{align*}
We consider the convex program
\begin{align} \label{rankcvxprogram}
&\textstyle \max~{\textstyle\frac{1}{2}}\left(\ldet \left(\Lambda_{r} P_{r} \Lambda_{r} \right)\!+\!\ldet (I_{n-r}\!-\!P_{n-r})\!\right)\\
&\text{s.t.~} \mathbf{e}^\top x=s,~ 0\le x\le 1.\nonumber
\end{align}
Similar to that in Theorem~\ref{seqrank}, \eqref{rankcvxprogram} is well-defined. Denote the optimal solution of \eqref{rankcvxprogram} as $x^*$ and we have corresponding $P_r^*$ and $P_{n-r}^*$, by similar arguments as in Theorem~\ref{seqrank}, there exists $L_1, L_2, L_3, L_4 >0$ such that when $\gamma\ge \max\{L_1, L_4\}$,
\begin{align*}
&\lim_{\gamma\rightarrow +\infty} \linx (C,s; \gamma)\\
& = \lim_{\gamma\rightarrow +\infty}\max_{\begin{array}{c}
\scriptstyle \mathbf{e}^\top x=s, \\[-3pt]
\scriptstyle 0\le x\le \mathbf{e}
\end{array}} {\textstyle\frac{1}{2}}\left(\ldet \left(\Lambda_r P_r \Lambda_r +\textstyle \frac{1}{\gamma}(I_r-P_r)\right)\right.\\[-3pt]
&\left. \qquad +\ldet (I_{n-r}-P_{n-r}-P_{n-r,r}F_r(\gamma;x)^{-1}P_{r,n-r})\right.\\[-3pt]
&\left.\vphantom{\ldet \left(\Lambda_r P_r \Lambda_r +\textstyle \frac{1}{\gamma}(I_r-P_r)\right)}\qquad -(s-r)\log\gamma\right)\\[-3pt]
&\le \lim_{\gamma\rightarrow +\infty}{\textstyle\frac{1}{2}} \left(\vphantom{\textstyle\frac{L_2 L_3 \sqrt{r}}{\gamma}}
\ldet (\Lambda_r P^*_r \Lambda_r )+\ldet (I_{n-r}-P^*_{n-r})\right.\\
& \left. \qquad +\textstyle\frac{L_2 L_3 \sqrt{r}}{\gamma}-(s-r)\log\gamma\right)
= -\infty. \Halmos
\end{align*}
\endproof
\section{Linear gap under optimal scaling}\label{sec:gammagap}
In Theorem~\ref{maskgap}, we constructed an infinite sequence $\{C_n\}_{n\in \mathcal{I}}$ where by choosing mask $I$, we decreased the linx bound by an amount that is at least linear in $n$ (specifically, $\approx .0312n$).
This is even the case when we choose optimal scaling parameters $\gamma$ (separately), with some sacrifice
in the constant.
\begin{thm}\label{optgammalingap}
There is an infinite sequence of positive-semidefinite matrices $\{C_n\}_{n\in \mathcal{I}}$, such that
\begin{align*}
\min_{\gamma>0}\linx \left(C_n,{\textstyle\frac{n}{2}};\gamma\right)-\min_{\bar \gamma>0}\linx \left(C_n,{\textstyle\frac{n}{2}}; I, \bar \gamma\right)\ge b n
\end{align*}
for some positive scalar $b\ge 0.024036$.
\end{thm}
\proof{Proof.}
We consider a crafted sequence of $C_n$. Assuming $n=4k$, and $C_n$ is block diagonal with $k$ blocks as $\left(\begin{array}{cc}
1 & c_1 \\
c_1 & 1
\end{array}\right)$ and $k$ blocks as $\left(\begin{array}{cc}
1 & c_2 \\
c_2 & 1
\end{array}\right)$ where $c_1\neq c_2$, $c_1^2\leq 1$, $c_2^2\leq 1$.
By Lemma~\ref{lb},
\begin{align*}
&\linx\left(C_n,{\textstyle\frac{n}{2}}; \gamma\right)\\
&\ge \sum_{i=1}^{k}\left(\textstyle\frac{1}{2}\log\left(\frac{(1-c_1^2)^2}{4}\gamma+\frac{1+c_1^2}{2}+\frac{1}{4\gamma}\right)\right.\\
&\left.\qquad\qquad+\textstyle\frac{1}{2}\log \left(\frac{(1-c_2^2)^2}{4}\gamma+\frac{1+c_2^2}{2}+\frac{1}{4\gamma}\right)\right)\\
&=\textstyle k \left({\textstyle\frac{1}{2}}\log\left(\frac{(1-c_1^2)^2}{4}\gamma+\frac{1+c_1^2}{2}+\frac{1}{4\gamma}\right)\right.\\
&\qquad\qquad\left. + \textstyle\frac{1}{2}\log \left(\frac{(1-c_2^2)^2}{4}\gamma+\frac{1+c_2^2}{2}+\frac{1}{4\gamma}\right)\right).
\end{align*}
If $c_1^2, c_2^2<1$, the minimum of $\frac{(1-c_1^2)^2}{4}\gamma+\frac{1+c_1^2}{2}+\frac{1}{4\gamma}$ is $1$, achieved by the unique minimizer $\hat \gamma_1=1-c_1^2$,
and the minimum of $\frac{(1-c_2^2)^2}{4}\gamma+\frac{1+c_2^2}{2}+\frac{1}{4\gamma}$ is $1$, achieved by the unique minimizer $\hat \gamma_2=1-c_2^2$. If $c_1^2=1$, then no matter what value $\gamma$ is, $\frac{(1-c_2^2)^2}{4}\gamma+\frac{1+c_2^2}{2}+\frac{1}{4\gamma}$ is always greater than $1$. The case for $c_2^2=1$ is similar.
Thus we can choose $c_1^2\not= c_2^2$, then for all possible values of $c_1, c_2$,
\begin{align}\label{sec4lingap}
b_{c_1,c_2} \!:= \min_{\gamma>0} {\textstyle} \sum_{i=1}^2 \textstyle \log\!\left(\frac{(1-c_i^2)^2}{4}\gamma+\frac{1+c_i^2}{2}+\frac{1}{4\gamma}\right) \!> \!0.
\end{align}
Then we have
\begin{align*}
\min_{\gamma>0}\linx\left(C_n,{\textstyle\frac{n}{2}}; \gamma\right)\ge \textstyle\frac{1}{2} k b_{c_1,c_2}=\frac{b_{c_1,c_2}}{8} n.
\end{align*}
On the other hand, by Proposition~\ref{diagoptgamma}, we have $\min_{\bar \gamma}\linx\left(C_n,{\textstyle\frac{n}{2}};I,\bar \gamma\right)=\linx\left(C_n,{\textstyle\frac{n}{2}};I, 1\right)=0$.
So,
\begin{align*}
&\min_{\gamma}\linx\left(C_n,{\textstyle\frac{n}{2}}; \gamma\right)-\min_{\bar \gamma}\linx\left(C_n,{\textstyle\frac{n}{2}}; I,\bar \gamma\right)\\
&\qquad \ge
\textstyle \frac{b_{c_1,c_2}}{8} n.
\end{align*}
Let $b=\frac{b_{c_1,c_2}}{8} $, we get what we want. In particular, if we set $c_1=0, c_2=1$, the optimal $\gamma$ for \eqref{sec4lingap} is $\hat \gamma=\frac{1+\sqrt{3}}{2}$ and $\frac{b_{0,1}}{8}=$
\begin{align*}
& \textstyle
\frac{1}{8}\!\left(\log\left(1\!+\!\frac{1}{2(1+\sqrt{3})}\right)\!+\!\log\left({\textstyle\frac{1}{2}}\!+\!\frac{1}{2(1+\sqrt{3})}\!+\!\frac{1+\sqrt{3}}{8}\right)\!\right)\\[-3pt]
&\qquad \ge 0.024036. \Halmos
\end{align*}
\endproof
\section{Final Remarks}\label{sec:final}
Our technical results establish the strong potential for
masking to improve on the (scaled) linx bound.
So the next logical step is to work on optimizing the mask in this context. Similar work was carried out successfully for the spectral bound (see \cite{AnstreicherLee_Masked} and \cite{BurerLee}), where nonconvexity and nondifferentiability were the main difficulties to overcome. In the context of the linx bound, even at smooth points, it is not easy to get a handle on the necessary derivative information. There is also the potential to incorporate the ``mixing'' technique of \cite{chen_mixing} on top of mask optimization.
We are currently working in this direction, and we plan to report on algorithmic results (with experimentation on benchmark data) for mask optimization in a future paper.
\bibliographystyle{alpha}
|
1,116,691,500,463 | arxiv | \section{Introduction}
Advanced vehicle control algorithms are crucial for the development of safe and reliable \textcolor{black}{autonomous driving (AD)} applications to reduce road accidents and causalities~\cite{DO2021104856}. Conventional control designs such as PID and linear state feedback often serve in low-level feedback loops of industrial automotive applications. Albeit having low computational complexity, the performance of such controllers is limited in safety-critical traffic scenarios such as emergency collision avoidance. Recently, nonlinear model predictive control for autonomous driving applications has been studied and shown promising results, mainly from academic research~\cite{soncdc2019, 8754713}. On the other side, due to the computational complexity and limited resources required by numerical optimization, NMPC has not been commonly considered in industrial autonomous driving control platforms, \textcolor{black}{with most contributions often omitting one of the fundamental XiL stages~\cite{Ferreau2017}}.
The main contribution of this paper is an efficient development framework that can be used in the automotive industry to design and safely deploy real-time NMPC on road vehicles. \textcolor{black}{Two test scenarios are created for the intended trajectory tracking application:}
\begin{itemize}
\item Autonomous valet parking: to deal with safety around suddenly appearing pedestrians and vehicles. The driving takes place at a low speed, around 10kph. The controller is tested in a private parking area.
\item Lane keeping: to avoid collision with obstacles in city or highway scenarios. NMPC is tested on proving ground in Aldenhoven, Germany at high speeds around 60kph.
\end{itemize}
We present a framework for embedded NMPC in autonomous driving applications, providing a number of benefits: First, it is built upon a detailed nonlinear predictive model, capturing the system delays, actuator saturation and look-ahead capabilities to generate feasible trajectories that avoid dangerous driving situations. Second, constraints and objectives can be set for specific performance and driver comfort. NMPC is deployed in the loop with the vehicle, without any resampling, as a low-level controller calculating a control policy every iteration respecting hard real-time constraints. Real-time operation implies a dependency on the logical explanation of the solution and most importantly on the numerical optimization scheme execution time~\cite{RTControl}.
The development framework is motivated by the R\&D ADAS team of Siemens, with focus on testing real-time NMPC strategies in different standard scenarios throughout XiL stages~\cite{sontra2017}. XiL verification is well-established as an effective means to develop safe and secure industrial automotive systems, as illustrated in Figure~\ref{fig:DevelopmentFramework}. In this systems engineering process, the control system is first tested in simulation. MiL validation is conducted with high-fidelity multi-physics simulators using Simcenter Amesim with disturbances and parameter mismatches to test the algorithm's robustness. Once validated, C/C++ code is auto-generated and tested (SiL). Real-time performance is validated in a realistic virtual traffic environment with physics-based sensors in Simcenter Prescan. Then the generated code is integrated into ECU hardware (HiL) and eventually deployed in physical road-vehicle (VeHiL). The plug-and-play framework requires limited workforce for the user from design to deployment on the vehicle. Real and virtual environment combined in XiL is attractive, as it helps reducing testing costs as performance is assessed without test sites and costly sensors/obstacles. Tuning campaigns are facilitated, development and implementation cycle time are decreased. \textcolor{black}{This paper presents one demonstration of the NMPC approach applicability in XiL with a decreasing risk factor as we progress through the different stages.} Although tested with a specific toolbox and solver, the framework allows for an ease of reproducibility and expansion to other C/C++ capable toolboxes, solvers, \textcolor{black}{optimal control problem (OCP)}, car models and scenarios.
Hardware setup comprises a Ford Focus vehicle, perception and localization sensors (GPS, Radars, Camera, LiDAR), driving robot (Anthony Best Dynamics) for actuating steering (SR), throttle (AR) and brake systems (BR), and an embedded platform dSPACE MicroAutobox III as the NMPC computation module. They are represented in Figure~\ref{Communication_Confidential}.
The paper is organized as follows. Section II discusses some background on NMPC for autonomous driving. Section III presents the hardware deployment of optimization solvers for collision avoidance application. Validation results with both virtual and real obstacles are given in Section IV.%
%
\begin{figure}
\centering
\includegraphics[width=0.9\columnwidth]{development_framework.png}
\caption{XiL development framework}
\label{fig:DevelopmentFramework}
\end{figure}%
\section{Background}
This section presents the vehicle model and the developed NMPC formulation for trajectory following. We then provide background on SQP solvers and explain the collision avoidance planning algorithm.
\subsection{Bicycle Model}
Car dynamics are represented with a real-time feasible model such as the 6 DoF bicycle model. Controller validation with a 15 DoF model is later performed in a MiL framework. From experimental validation, the model defined below is satisfactory for lane keeping and emergency scenarios, assuming no effect of roll and pitch on lateral dynamics.%
%
{\small \begin{align}
\label{Eq:BicycleModelStates}
\Dot{v}_x &= \frac{1}{M}(F_{xf} cos\delta + F_{xr} -F_{yf} sin\delta - F_{res} + M\Dot{\psi} v_y), \nonumber\\
\Dot{v}_y &= \frac{1}{M}(F_{xf} sin\delta + F_{yr} + F_{yf} cos\delta - M\Dot{\psi} v_x), \\
\Dot{\omega} &= \frac{1}{I_z}(L_f (F_{yf} cos\delta + F_{xf} sin\delta) - L_r F_{yr}), \nonumber\\
\Dot{X} &= v_x cos\psi - v_y sin\psi, \nonumber\\
\Dot{Y} &= v_x sin\psi + v_y cos\psi, \nonumber\\
\Dot{\psi} &= \omega. \nonumber
\end{align}}%
Linear tire model approximates the lateral forces assuming small slip angles using cornering stiffness $K_f$ and $K_r$. The longitudinal and lateral forces are computed as:%
{\small \begin{equation}
\label{Eq:BicycleModelForces}
F_{xf} = F_{xr} = 0.5 \frac{t_r T_{max}}{R}, F_{yf} = K_f \alpha_f, F_{yr} = K_r \alpha_r.
\end{equation}}%
%
The front and rear slip angles can be defined as follows:%
%
{\small \begin{equation}
\label{Eq:BicycleModelSlipAngle}
\alpha_f = -\tan^{-1}\Big(\frac{\Dot{\psi}L_f + v_y}{v_x}\Big) + \delta,
\alpha_r = \tan^{-1}\Big(\frac{\Dot{\psi}L_r - v_y}{v_x} \Big) .
\end{equation}}%
%
\textcolor{black}{Resistance in the longitudinal direction is modeled as the sum of rolling resistance and air drag:}%
%
{ \begin{equation}
\label{Eq:BicycleModelDrag}
F_{res} = C_{r0} + C_{r2}v_x^2.
\end{equation}}%
%
\noindent\textcolor{black}{The body reference frame has its origin at the CoG with the X-axis pointing to the front of the car. The position of this frame is defined by X,Y in the global frame. We use the body frame to express forces and moments in the car dynamics because the inertia remains constant in it. Therefore, the kinematics ODE in $X, Y, \psi$ are expressed in the Cartesian global frame as in~\eqref{Eq:BicycleModelStates}. All the variables and model parameters are summarized in Table~\ref{table:BicycleModel_Variables}.}
{ \begin{table}
\centering
\scriptsize
\textcolor{black}{ \begin{tabular}{c c c }
\hline
State \& Parameter & Description & Unit \\ [0.5ex]
\hline
$v_x$, $v_y$ & Body frame longitudinal and lateral velocities & \si{\meter}$\cdot$\si{\per\second}\\
\hline
$\omega$ & Body frame yaw rate & \si{\radian}$\cdot$\si{\per\second}\\
\hline
X,Y & Global Cartesian coordinates & \si{\meter}\\
\hline
$\psi$ & Global vehicle heading (yaw) & \si{\radian}\\
\hline
M, $I_z$ & Total mass and Inertia& \si{\kilogram}, \si{\kilogram}$\cdot$\si{\square\meter}\\
\hline
$L_f$,$L_r$ & CoG's distance from front and rear axles & \si{\meter}\\
\hline
$F_{xf}$,$F_{xr}$ & Local front and rear axles longitudinal forces & \si{\newton}\\
\hline
$\delta$, $t_r$& Steering angle and normalized throttle & \si{\radian}, 1 \\
\hline
$T_{max}$& Maximum engine torque & \si{\newton}$\cdot$\si{\meter}\\
\hline
R& Wheel radius & \si{\meter}\\
\hline
$C_{r0}$, $C_{r2}$ & Zero and second order friction parameters & \si{\newton}, \si{\kilogram}$\cdot$\si{\per\meter}\\
\hline
\end{tabular}}
\caption{Bicycle model states and parameters}
\label{table:BicycleModel_Variables}
\end{table}
}
\subsection{Nonlinear Model Predictive Control}
NMPC directly controls the car by computing the normalized throttle $t_r$ with respect to the maximum engine force, and the body frame steering angle $\delta$. A receding horizon scheme of $N$ steps is used. \textcolor{black}{A set of nonlinear difference equations $f_d$ is obtained by applying a Runge-Kutta $4^{th}$ order method to the dynamics $\Dot{x} = f(x,u)$ in~\eqref{Eq:BicycleModelStates}.} The NLP then optimizes over the discrete-time OCP for trajectory following:%
%
\textcolor{black}{{\small \begin{align}
\label{Eq:NMPC_Formulation}
\begin{split}
\min_{x(0),\dots, x(N),u(0),\dots, u(N-1)} \sum_{k=0}^{N-1} l_k(x_{k},u_{k}) &+ V_{N}(x_{N})\\
\textrm{subject to: } x_{0} = x(0) \\
x(k+1) = f_d(x(k),u(k)) \qquad &k=0,\dots,N \\
v_{x,min} \leq v_x(k) \leq v_{x,max} \qquad &k=0,\dots,N-1\\
v_{y,min} \leq v_y(k) \leq v_{y,max} \qquad &k=0,\dots,N-1\\
\omega_{min} \leq \omega(k) \leq \omega_{max} \qquad &k=0,\dots,N-1\\
X,Y \in \mathcal{D} \qquad &k=0,\dots,N-1\\
\delta_{min} \leq \delta \leq \delta_{max} \qquad &k=0,\dots,N-1\\
-1 \leq t_r \leq 1 \qquad &k=0,\dots,N-1\\
e_N \in \chi_N. \qquad & \\
\end{split}
\end{align}}}%
%
The stage cost $l_k(x_{k},u_{k})$ with \textcolor{black}{$u = [\delta, t_r]$} is defined as:%
%
{\small \begin{align}
l_k(x_k, u_k) = (x(k) - x_{ref}(k)) ^ T Q (x(k) - x_{ref}(k))\\
\;\;\; +\; u(k) ^ T R u(k) + \Delta u(k) ^ T S\Delta u(k), \nonumber
\end{align}}%
%
with $Q \in\mathbb{R}^{6\times 6} \succeq 0, R \in\mathbb{R}^{2\times 2} \succ 0, S\in\mathbb{R}^{2\times 2} \succeq 0,$ and $x_{ref}(k) = [v_{ref}(k), 0,0,X_{ref}(k), Y_{ref}(k),\psi_{ref}(k) ]$ is the reference trajectory from the planner. \textcolor{black}{The lateral velocity and yaw rate are regulated to a zero reference. In this application, constraints on the input rates are relaxed and added to the cost function by penalizing large control temporal differences $\Delta u = u(k+1) - u(k) $. $V_N(x_N)$ is a quadratic terminal cost on the tracking error with a bigger cost matrix than Q. $\chi_f$ is the terminal set on the tracking error $e_N = x(N) - x_{ref}(N)$.}
\textcolor{black}{Moreover, $\mathcal{D}$ is the set of left and right boundaries that define a safe driving corridor for obstacle avoidance as seen in Figure~\ref{fig:Planner}.
}
\subsection{Collision avoidance planner}
\textcolor{black}{The local planner is provided with the original reference path and velocity profile $v_{x,ref}$. It chooses a portion of the global path to track $x_{ref}(k) =[v_{x,ref}(k), 0, 0, X_c(k), Y_c(k), \psi_c(k)]$ for $k \in [k^*,\dots,k^* + N]$ by localizing the car with respect to the closest point on the trajectory such that: $k^* = \underset{k}{\arg\min} (X-X_c(k))^2 + (Y - Y_c(k))^2$. Moreover, the planner receives obstacle information and adjusts the boundaries of the driveable area $\mathcal{D}$ creating a driving corridor taking all obstacles into account as seen in Figure~\ref{fig:Planner}, all while adjusting the reference to be within $\mathcal{D}$. The lateral and longitudinal safe distances create a no-go box-zone $\mathcal{D}^{'}$ around the different obstacles and are tunable online to mimic different reaction times or distances. The size of $\mathcal{D}$ depends on the speed and reaction time. For scenarios presented in this paper, 1.2 to 1.5 seconds of safe duration proved to be sufficient to react to sudden obstacles. }
\subsection{SQP and QP solvers}
Sequential quadratic programming is a popular optimization
method thanks to its ability to handle highly nonlinear problems and to efficiently warm-start~\cite{NoceWrig06}. In this application, an SQP with QRQP quadratic solver from CasADi is used~\cite{NLPCodeGen}. QRQP is based on the sparsity exploiting active-set method and the derivatives are produced by the CasADi toolbox~\cite{Andersson2019}, an open source symbolic software framework for nonlinear programming.
This paper does not provide a benchmark/comparison of different solvers for automotive applications but rather implements an out-of-the-box solver to demonstrate the framework. The choice of toolbox and solver is just a placeholder for any other library written in C/C++ or with C code generation capabilities~\cite{verschueren2020acados, PolyMPC}.%
%
\begin{figure}
\centering
\includegraphics[width=0.65\columnwidth]{planner_v2.png}
\caption{\textcolor{black}{Planner: closest point localizer and safe corridor}}
\label{fig:Planner}
\end{figure}%
%
\section{Embedded NMPC Implementation}
This section deals with the deployment steps on dSPACE MicroAutobox III (MABX-III) hardware for an embedded control application and validates MiL/HiL in the virtual environment of Prescan. First, the controller is prepared for a real-time environment on a platform with a C/C++ compiler: MABX-III has an ARM Cortex A-15 processor, operates on a 2GB DDR4 RAM with 64MB flash memory for real-time application, and runs RTOS. Deployment to MABX-III is done via Simulink interface. This chapter explains how the tailored NMPC is C-code generated, tested in Simulink, and deployed as a standalone library in runtime. Turning the optimization function into C-code enhances performance with no callbacks into Matlab environment, and with static memory allocation of the block's internal states on compile time. Second, the generated code is validated in MiL simulation with Amesim as in Figure~\ref{fig:MiLBlockDiagram}.
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{Chapter_dSPACE_FIGURES/Control_BlockDiagram_2.png}
\caption{MiL: closed-loop control structure}
\label{fig:MiLBlockDiagram}
\end{figure}
\subsection{Code generation for standalone NMPC}
\begin{figure}[b]
\centering
\includegraphics[width=8cm]{Figures/ocp_to_deploy_3.png}
\caption{HiL: NMPC deployment steps}
\label{fig:StepsForDeplyoment}
\end{figure}
NMPC for trajectory following is written in Matlab using CasADi. The challenge resides in code generating the Simulink model to be deployed on MABX-III and the proposed procedure is depicted in Figure~\ref{fig:StepsForDeplyoment}. Particularly for this application, the C source codes are compiled using MEX (with MSVC: Microsoft Visual Studio or MinGW64 compilers). Other compilers such as GCC could be used depending on the target platform. For an OCP written in C/C++, the code can be called directly using \textit{S-function builder}. The MEX executable is compiled from the source codes, linked with the toolbox's implementation code, and is called using an \textit{S-Function block}. Therefore, the MEX binary is a self-contained library. The following comments are to be stated for this project implementation: \textcolor{black}{1) CasADi code generation toolbox is used to generate the NMPC source code containing the evaluation of the different steps in an SQP algorithm and construction of the QP subproblems 2) The wrapper for the NMPC source code solves~\eqref{Eq:NMPC_Formulation} given the current state estimates and reference, and outputs an open-loop primal solution and the first control action.}
\subsection{MiL in a virtual environment}
The first test in the XiL process, evaluates the closed-loop performance, with high fidelity vehicle dynamics (15DoF) from Amesim, with noise and parameter mismatch. Simulations are carried out to test the NMPC in trajectory following and emergency collision avoidance applications in an ISO 3888-1 standard double lane change scenario at 80kph. \textcolor{black}{Results presented in Figure~\ref{fig:amesim_ISODLC_XYYaw}, show a high-performance tracking response.} NMPC is solved to convergence and satisfies the real-time constraints. The system profiler (Simulink or MSVC profiler), indicates that the code generated NMPC execution time averages at 2.4ms. The effect of code generation on NMPC's execution time is presented in Table~\ref{table:Profiling}. Since MABX-III has limited computational power compared to the host PC (with 32GB RAM), \textcolor{black}{it is necessary to understand how execution time scales on the platform for real-time performance on MABX-III}. After running several HiL tests, it is found that execution time is increased between 7 and 10 times on MABX-III. Therefore, the first validation for real-time performance before deployment is to operate the NMPC on the host PC with an execution time below 4.5ms for a sample time of 40ms. Otherwise, NMPC is real-time incapable and needs to be redesigned. %
\begin{figure
\centering
\includegraphics[width=0.7\columnwidth]{Chapter_dSPACE_FIGURES/XYYawVx_amesim_ISODLC.tikz}
\caption{MiL: vehicle states (solid) and reference (dashed)}
\label{fig:amesim_ISODLC_XYYaw}
\end{figure}%
{ \begin{table}[b]
\centering
\scriptsize
\begin{tabular}{|m{2.3cm} | m{2.1cm}| m{2.3cm} |}
\hline
Type: & No code generation & Code generation\newline
(per evaluation) \\ [0.5ex]
\hline\hline
Total & 22.0ms & 2.4ms (2.4ms) \\
\hline\hline
QP & 2.00ms & 0.218ms (0.109ms)\\
\hline
Line search & 2.00ms & 0.218ms (0.109ms)\\
\hline
Cost and constraints & 2.00ms & 0.218ms (0.109ms)\\
\hline
Gradient & 1.00ms & 0.109ms (0.109ms)\\
\hline
Hessian & 8.00ms & 0.87ms (0.436ms)\\
\hline
Jacobian & 5.00ms & 0.545ms (0.27ms)\\
\hline
\end{tabular}
\caption{Profiling optimization sub-functions' evaluation time with and without code generation on Host PC}
\label{table:Profiling}
\end{table}}%
\subsection{HiL in a virtual environment}
HiL validation is performed for the same scenario as in MiL, however, controls are applied to a vehicle physically lifted off the ground. Controller real-time capabilities and control signal smoothness are the main targets. Two important aspects are required before deployment: a. all Simulink blocks are code generatable, b. NMPC formulation is code optimized for the quickest execution time. Code generating the optimization function speeds up the evaluation time from 4 to 10 times as compared to the Matlab evaluation~\cite{Andersson2019}. The following properties are used in the NMPC compilation:%
\begin{itemize}
\item Solver: SQP method with QRQP (Active-Set method)
\item Maximum number of SQP: 50 and QP: 100 iterations
\item Integration type: Runge-Kutta 4 (RK4) with 4 steps
\item Hessian approximation: Exact
\item OCP method: Direct (discretize then optimize)
\item Shooting method: Multiple shooting
\item Sample time: 40ms (25Hz controller)
\item Prediction horizon: N = 30
\item Primal and dual infeasibility threshold: $1e^{-06}$ and $1e^{-04}$
\end{itemize}%
Possible improvements to reduce computation time are:
\begin{enumerate
\item Warm start as SQP is heavily affected by initialization
\item Reformulate the OCP to relax non critical active constraints, add slack variables or add them to the cost
\item Scale the problem in order to improve conditioning
\item Reformulate the OCP in a matrix form and minimize nested for-loops to facilitate derivative calculations
\end{enumerate}
The second test towards full vehicle deployment is a validation of the NMPC on MABX-III for highway and cut-in scenarios. The car is visualized in Prescan through communication via ROS~\cite{288} as in Figure~\ref{Communication_Confidential}. \textcolor{black}{NMPC operates as the low-level controller running at 25Hz as the optimal policy is applied on the driving robot. HiL test is validated with the NMPC block operating in real-time on MABX-III, with an execution time around 22ms.}
\section{Vehicle Hardware in the Loop (VeHiL)}
Real-time optimal control implementation for HiL and VeHiL requires communication among hardware and software, presented in this chapter and summarized in Figure~\ref{Communication_Confidential}. \textcolor{black}{The results of the physical testing campaigns in a private parking area and on proving ground are also presented.}
For HiL testing, the vehicle in Figure~\ref{Communication_Confidential} is replaced by a simulator. Therefore, the communication architecture allows the user to go from offline simulation to online MiL/HiL and finally to VeHiL with the exact same NMPC code.
\begin{figure}%
\centering
\includegraphics[width=0.7\columnwidth]{Chapter_Communication/Communication_Confidential5_jpeg}
\caption{VeHiL: communication and vehicle framework}
\label{Communication_Confidential}
\end{figure}%
\subsection{VeHiL: Parking validation}
\begin{figure}%
\centering
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/TrajectoryFollow/AVP_Testing/ExecutionTime_2.tikz}
\caption{Parking VeHiL: NMPC execution time (solid) and sample time (dashed)}
\label{fig:Testing_ExecutionTime}
\end{figure}%
\begin{figure
\centering
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/TrajectoryFollow/AVP_Testing/XYYaw_Pos_2.tikz}
\caption{Parking VeHiL: X-Y and X-Yaw states (vehicle states: solid, reference trajectory: dashed, Obstacle: box)}
\label{fig:Testing_XY}
\end{figure}%
\begin{figure
\centering
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/TrajectoryFollow/AVP_Testing/ControlAction_ObsDistance_2.tikz}
\caption{\textcolor{black}{Parking VeHiL: obstacle and NMPC commands}}
\label{fig:Testing_ObstacleDetection}
\end{figure}%
In this testing campaign, the vehicle is integrated in the loop with the embedded controller for parking scenarios, in presence of obstacles. The original reference trajectory is generated from human driving around the parking area at 10kph, without obstacles. MiL with Amesim and HiL with the virtual environment are first validated for this scenario. \textcolor{black}{Although safe, autonomous driving was uncomfortable with jerky throttle and aggressive steering. This could be caused by the stalling engine torque at low speeds and the layout of the parking area that included an upward slope, which were not accounted for in the NMPC dynamics.} Figure~\ref{Communication_Confidential} shows a real-time visualization of the physical Ego car generated by colleagues at the ADAS group in Siemens: \textcolor{black}{MABX-III receives the vehicle coordinates from IMU and GPS and communicates with Prescan for a virtual representation of the testing site. Moreover, Ego vehicle controlled by NMPC can be seen in grey avoiding a virtual construction person.}
Figure~\ref{fig:Testing_XY} shows the autonomously driven path after the controller tuning campaign for this AD scenario. NMPC commands the robot as the $X=0m$ line is crossed (blue circle). Those plots demonstrate the NMPC capabilities in both accurate tracking and collision avoidance. The planner shifts the reference laterally from the original one for a lane change at 1.5m. NMPC reacts quickly to an obstacle only detected within 10m of distance. Finally, Figure~\ref{fig:Testing_ObstacleDetection} shows the obstacle detected range and the NMPC commands in steering robot angle (SR), throttle pedal position (AR), and brake (BR). The obstacle information is not fed to the NMPC before a distance-to-collision of 10 meters, to simulate emergency obstacle avoidance. \textcolor{black}{The unsmooth throttle behavior between 30 and 40s is caused by the vehicle deceleration in the upward slope. This can be tackled by tuning, a more accurate model, or by online parameter adaptation. Nevertheless, the task was still accomplished with real-time performance as the embedded NMPC solves with an execution time of 14ms for a horizon of 1.2seconds as shown in Figure~\ref{fig:Testing_ExecutionTime}.}
\subsection{VeHiL: Proving ground validation}
\begin{figure}
\centering
\includegraphics[height = 4cm,width=0.9\columnwidth]{dlc_aldenhoven_jpeg}
\caption{Proving ground VeHiL: obstacle avoidance}
\label{fig:dlc_andenhoven}
\end{figure}
The second part of the campaign took place in Germany on a secure testing site as shown in the shots of Figure~\ref{fig:dlc_andenhoven} and consisted of an \textcolor{black}{Adaptive Cruise Control (ACC)} for lane-keeping at 60kph over 500 meters with collision avoidance using a dummy vehicle. \textcolor{black}{In order to test the NMPC at its boundaries of sudden obstacle emergence, the safe duration parameter, presented in Figure~\ref{fig:Planner} was reduced online from 1.8 to 1.2 seconds of reaction time (20 meters of safe distance before and after the obstacle at 60kph).} This step, shows the benefits of deploying such a controller in automotive applications as the duration is insufficient for the human driver to take control and avoid an accident. Results of this testing phase, shown in Figures~\ref{fig:AldenDay2_ExecutionTime} through \ref{fig:NMPCControl_Aldenhoven}, prove the NMPC quickly reacted while satisfying all dynamic and actuator constraints. The controller smoothly corrects the initial positioning error off the centerline and performs a cruise control until obstacle detection. The car immediately recovers the original lane after avoiding the obstacle. NMPC's execution time averaged at 22ms, for a sampling time of 40ms, hence running in real-time on the embedded platform as in Figure~\ref{fig:AldenDay2_ExecutionTime}. The velocity tracking error is similar to the first phase, at almost 0.5m/s as in Figure~\ref{fig:AldenDay2_XYYaw} and is mainly due to longitudinal model mismatches.
As from Figures~\ref{fig:Testing_ObstacleDetection} and \ref{fig:NMPCControl_Aldenhoven}, the jerky throttle control could be a result of the robot delay and the step change in the spatial reference causing some constraints to become active.
Numerical convergence was achieved within at most 2 SQP and 1 to 3 QP iterations, for a total computation time of 22ms and 14ms on average for the 60kph and 10kph scenarios respectively. \textcolor{black}{Warm starting the primal variables, taking into consideration the driving corridor, significantly cut down execution time.} The SQP solver used in this project scenarios was efficient and satisfactory in real-time optimal control, handling non-linearities and converging to the optimal solution. The scenarios were carried out in a MiL framework and resulted in control policies similar to VeHiL, creating a potential real to simulation flow. This shows the developed framework's importance as most of the tuning, OCP reformulation, and real-time capabilities were validated with the high fidelity model, with little costs, zero incidents, and safe vehicle integration. \textcolor{black}{XiL cycle testing improves scalability and sensitivities to parameters as it allows testing with various driving scenarios and traffic situations all while injecting disturbances and noises in simulation and hardware.}
\begin{figure
\centering
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/AldenhovenTesting/Alden_Day2/Execution_Time_2.tikz}
\caption{Proving ground VeHiL: NMPC execution time (solid) and sample time (dashed)}
\label{fig:AldenDay2_ExecutionTime}
\end{figure}%
\begin{figure}%
\centering
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/AldenhovenTesting/Alden_Day2/XYYaw_Pos_2.tikz}
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/AldenhovenTesting/Alden_Day2/Vx_2.tikz}
\caption{Proving ground VeHiL: vehicle X-Y, X-Yaw, longitudinal speed (vehicle states: solid, reference trajectory: dashed, NMPC activation: blue, Obstacle: box)}
\label{fig:AldenDay2_XYYaw}
\end{figure}%
\begin{figure
\centering
\includegraphics[width=0.7\columnwidth]{MATLAB_PLOTS/AldenhovenTesting/Alden_Day2/SR_Pos_2.tikz}
\caption{Proving ground VeHiL: NMPC control actions}
\label{fig:NMPCControl_Aldenhoven}
\end{figure}%
\subsection{Problems and possible improvements}%
\begin{itemize}
\item The framework allows for online manual parameter tuning, however, it would be beneficial to include an auto-tuner to facilitate performance matching
\item Code generation is beneficial for the implementation of rapid prototyping such as in this project, nevertheless, it often results in very large source codes that are hard to debug rendering the detailed function profiling more complex and one could just avoid code generation
\end{itemize}
\section{Conclusion}
This paper presents a development framework for designing, validating, and implementing a real-time optimal controller for autonomous driving applications. The validation process satisfies the automotive industry requirements as it progresses from ISO standards scenarios, to the different XiL applications. The framework allows testing in real and/or virtual environments using high-fidelity dynamics and sensors in Simcenter software.
The NMPC approach is demonstrated with the case of trajectory control as it is deployed as a low-level controller in the real-time applications at 25Hz, showing the potential capabilities of this controller type for collision and accident avoidance and ACC. \textcolor{black}{Trajectory control is one relevant use case of the applicability of real-time NMPC, however, other use cases could be tested using the same XiL development framework.} Embedded numerical optimization is deployed on the platform and applications can be extended to more complex OCP or other optimization based formulations such as optimal planning. The framework was tested and validated on a Ford Focus in the parking area and on proving grounds, in presence of obstacles and with a strict requirement on real-time calculations.
\bibliographystyle{ieeetr}
|
1,116,691,500,464 | arxiv | \section*{SUMMARY}
Artificial Intelligence (AI) applications can profoundly impact society. Recently, there has been extensive interest in studying how scientists design AI systems for general tasks. However, it remains an open question about whether the AI systems developed in this way can work as expected in different regional contexts while simultaneously empowering local people. How can scientists co-create AI systems with local communities to address regional concerns? This article contributes new perspectives in this under-explored direction at the intersection of data science, AI, citizen science, and human-computer interaction. Through case studies, we discuss challenges in co-designing AI systems with local people, collecting and explaining community data using AI, and adapting AI systems to long-term social change. We also consolidate insights into bridging AI research and citizen needs, including evaluating the social impact of AI, curating community datasets for AI development, and building AI pipelines to explain data patterns to laypeople.
\section*{INTRODUCTION}
Artificial Intelligence (AI) techniques are typically engineered with the goals of high performance and accuracy. Recently, AI algorithms have been integrated into diverse and real-world applications. It has become an important topic to explore the impact of AI on society from a people-centered perspective~\cite{shneiderman2020bridging}. Previous works in citizen science have identified methods of using AI to engage the public in research, such as sustaining participation, verifying data quality, classifying and labeling objects, predicting user interests, and explaining data patterns~\cite{franzen2021machine,ceccaroni2019opportunities,lotfian2021partnership,mcclure2020artificial}. These works investigated the challenges regarding how scientists design AI systems for citizens to participate in research projects at a large geographic scale in a generalizable way, such as building applications for citizens globally to participate in completing tasks. In contrast, we are interested in another area that receives significantly less attention:
\begin{itemize}[noitemsep,topsep=0pt]
\item How can scientists co-create AI systems ``with'' local communities to address context-specific concerns and influence a particular geographical region?
\end{itemize}
Our perspective is based on applying AI in Community Citizen Science~\cite{hsu2020human,chari2017promise} (CCS), a framework to create social impact through community empowerment at an intensely place-based local scale. We define ``community'' as a group of people who are indirectly or directly affected by issues in civil society and are dedicated to making sure that these issues are recognized and resolved. We define ``social impact'' as how a project influences the society and local communities that are affected by social or environmental issues. We define ``community empowerment'' as a process of yielding agency to communities so that they can use technology, data, and informed rhetoric to create and disseminate evidence to advocate for social and policy changes. The CCS framework, a branch of citizen science~\cite{shirk2012public,irwin2001constructing}, is beneficial in co-creating solutions and driving social impact with communities that pursue the Sustainable Development Goals~\cite{fritz2019citizen}. Based on the literature and our experiences in co-creating AI systems with citizens, this article provides critical reflections regarding this under-explored topic for data science, AI, citizen science, and human-computer interaction fields. We discuss challenges and insights in connecting AI research closely to social issues and citizen needs, using prior works as examples.
\subsection*{How CCS Links to Other Frameworks}
Community Citizen Science emphasizes close collaborations among stakeholders when tackling local concerns. It is inspired by community-based participatory research~\cite{wallerstein2006using} and popular epidemiology~\cite{brown1993public}, where citizens directly engage in gathering data and extracting knowledge from these data for advocacy and activism. Examples involve co-designing technology for local watershed management~\cite{preece2019interaction}, understanding water quality with local communities~\cite{carroll2019empowering,jollymore2017citizen}, and using geo-information tools to monitor noise and earthquakes~\cite{carton2017citizen}. CCS intends to extend previous frameworks' scope to Sustainable Development Goals, especially the goal of sustainable cities and communities. This article discusses using CCS to integrate AI in-the-wild and local regions, which is different from those that conducted studies in living lab environments (such as the work by~\citet{alavi2018hide}) or in online communities (such as the work by~\citet{brambilla2014community}).
Additionally, Community Citizen Science is related to Action Research~\cite{susman1978assessment}, RtD (research through design)~\cite{zimmerman2007research}, Service Design~\cite{zomerdijk2010service}, and the PACT framework (participatory approach to enable capabilities in communities)~\cite{bondi2021envisioning}. Extending Action Research, CCS encourages scientists to immerse themselves in the field by taking on a social role and conducting research from a first-person view. Complementing RtD that creates prototypes as proof-of-concept, CCS develops functional systems that can be deployed and used by local people. Unlike Service Design, citizens’ roles extend beyond service consumers to co-designers who co-create knowledge and systems with scientists and other stakeholders. The PACT framework and CCS share the same goal of co-designing AI systems to address critical societal issues, while CCS has an additional goal that needs to be achieved simultaneously: empowering local communities to catalyze social impact.
\begin{figure*}[t]
\centering
\includegraphics[width=2.05\columnwidth]{fig/prior_work}
\vspace{-0.1cm}
\caption{Case studies of Community Citizen Science projects that involve co-designing AI tools with local communities: (a) the air pollution monitoring project~\cite{hsu2017community} that empowered the Pittsburgh community to collect air pollution evidence in the local region for taking action, (b) the Smell Pittsburgh project~\cite{hsu2020smell} that invites citizens to report pollution odors and use the data as evidence to conduct air pollution studies, (c) the RISE project~\cite{hsu2021project} that enables citizens and scientists to annotate industrial smoke emissions and build an AI model to recognize pollution events. These cases were approved by the ethical committee of the university that hosted the projects.}
\label{fig:prior_work}
\end{figure*}
\section*{CHALLENGES}
Due to its region-based characteristics, Community Citizen Science often involves many local stakeholders—including communities, citizens, scientists, designers, and policy-makers—with complex relationships. CCS creates the space for the stakeholders to reveal underlying difficulties and locally-oriented action plans in tackling social concerns that are hard to uncover in traditional technology-oriented and researcher-centered approaches. But, stakeholders often have divergent and even contradicting values, which results in conflicts that pose challenges when designing, engineering, deploying, and evaluating AI systems.
Based on case studies (Figure~\ref{fig:prior_work}) of co-creating AI systems with local people, we outline three major challenges:
\begin{itemize}[noitemsep,topsep=0pt]
\item Co-designing AI systems with local communities
\item Collecting and explaining community data using AI
\item Adapting AI systems to long-term social changes
\end{itemize}
These challenges come mainly from the conflicts of interest between local communities (e.g., citizens and community groups) and university researchers (e.g., designers and scientists). AI researchers pursue knowledge to advance science, while local communities often desire social change. Such conflicts of interest among these two groups can lead to tensions, socio-technical gaps, and mismatched expectations when co-designing and engineering AI systems. For instance, local communities need functional and reliable systems to collect evidence, but AI researchers may be interested in producing system prototypes only to prove concepts or answer their research questions. Local community concerns can be urgent and timely, and citizens need to take practical actions that can have an immediate and effective social impact, such as public policy change. But, scientists need to produce knowledge using rigorous methods and publish papers in the academic community, often requiring a long reviewing and publication cycle.
\subsection*{Co-Designing AI Systems with Local Communities}
Challenges exist in community co-design, especially when translating multi-faceted community needs to implementable AI system features without using research-centered methods. The current practice to design AI systems is mainly centered on researchers instead of local people. Popular research methods—such as participatory design workshops, interviews, surveys—are normally used to help designers and scientists understand research questions. Although these methods enable researchers to better control the research process, essentially, university researchers are privileged and in charge of the conversations, leading to inappropriate power dynamics that can hinder trust and exacerbate inequality~\cite{klein2011dismantling,harrington2019deconstructing}. For example, during our informal conversations with local people that suffer from environmental concerns in our air quality monitoring project~\cite{hsu2017community}, many expressed feelings that scientists often treated them as experimental subjects (but not collaborators) in research studies. This imbalanced power relationship leads to difficulties in initiating conversations with citizens during our community outreach efforts.
Also, community data and knowledge are hyperlocal, which indicates that their underlying meanings ground closely to the local region and can be difficult to grasp for researchers who are not a part of the local community~\cite{carroll2015reviving}. For example, citizen-organized meetings to discuss community actions are often dynamic and context-dependent, which is not designed nor structured for research purposes. To collect research data that represents community knowledge, the current intensive procedure, such as video or audio recording, can make citizens feel uncomfortable. One alternative is to be a part of the community, to design solutions with them, join their actions, and perform ethnographic observations. For instance, researchers can better understand local community needs by actively participating in regional citizen group meetings and daily conversations with citizens. However, such in-depth community outreach approaches take tremendous personal effort, which can be unmotivating or even infeasible due to the limited academic research cycle and research-oriented academic tenure awarding system~\cite{klein2011dismantling,wallerstein2006using}.
\subsection*{Collecting and Explaining Community Data Using AI}
Challenges arise in data collection, analysis, and interpretation also due to conflicts of interest among scientists and citizens. Scientists are looking for rigorous procedures, but citizens seek evidence for action. Local communities are often frustrated by the formal scientific research procedure to prove the adverse impact of risk~\cite{brown1993public}, such as finding evidence of how pollution negatively affects health. Traditional environmental risk assessment models require a causal link between the risk and the outcome with statistical significance before taking action, which can be very difficult to achieve due to complex relationships between local people and their environments~\cite{bidwell2009community}. As a result, citizens collect their own community data (as defined by~\citet{carroll2018strengthening}, such as smoke emission photos from a nearby factory) as an alternative in order to prove their hypotheses. But, from scientists’ point of view, such strong assumption-driven evidentiary collection can lead to biases since the collection, annotation, and analysis of community data are conducted in a manner that strongly favors the assumption. One example is confirmation bias, where citizens are incentivized to search for information and provide data that confirms their prior beliefs~\cite{nickerson1998confirmation}, such as a high tendency to report odors related to pollution events~\cite{hsu2020smell}. Based on our experiences, it is extremely difficult to address or eliminate such biases when analyzing and interpreting how local social or environmental concerns affect communities.
Furthermore, researchers need to evaluate the social impact of AI systems to understand if the community co-design approach is practical. However, it is hard to determine if the intervention of AI systems actually influences the local people and leads to social changes by statistically analyzing community data. One difficulty is that local communities may have the implicit cognitive bias to overestimate and overstate the effect of the intervention since they are deeply involved in the co-creation of the AI system~\cite{norton2012ikea}. Moreover, it can be infeasible to conduct randomized experiments to prove the effectiveness of the intervention of AI on local communities~\cite{hsu2020human}. Controlling volunteer demographics and participation levels can be unethical when analyzing impact among different groups of people. Community Citizen Science treats local people as collaborators rather than participants. Researchers in CCS take the supporting role to assist communities using technology, instead of supervising and overseeing the entire project~\cite{hsu2020human}. Therefore, citizens join the CCS project at will and are not recruited like typical research studies. AI systems, in this case, are deployed in the wild with real consequences rather than a controlled test-bed environment that is designed for hypothesis testing. It remains an open research question as to how to integrate social science when studying the impact of AI systems~\cite{sloane2019ai}.
\subsection*{Adapting AI Systems to Long-Term Social Changes}
Conflicts of interest in the diverged values of citizens and scientists can lead to challenges in adapting AI systems to long-term social changes. The relationship between local people and AI systems is a feedback loop, which is similar to the concept that human interactions with architectural infrastructure are a continuous adaptation process that spans over long periods of time~\cite{alavi2019temporality}. When embedded in the social context, AI systems interact with citizens daily as community infrastructure. Communities are dynamic and frequently evolve their agenda to adapt to the social context changes. This means that the AI systems also need to adjust to such changes in local communities continuously. For instance, as we understand more about the real-life effects of the deployed AI systems on local people, we may need to fine-tune the underlying machine learning model using local community data. We may also need to improve the data analysis pipeline and strategies for interpreting results to fit local community needs in taking action. We may even need to stop the AI system from intervening in the local community under certain conditions. Such adaptation at scale requires ongoing commitment from researchers, designers, and developers to continuously maintain the infrastructure, involve local people in assessing the impact of AI, adjust the behavior of AI systems, and support communities in taking action to advocate for social changes~\cite{sloan2020participation}.
However, it is very challenging to estimate and obtain the required resources to sustain such long-term university-community engagement with local people~\cite{koekkoek2021unraveling}, especially in financially supporting local community members for their efforts. Typical research procedures can be laborious in data collection and analysis, and engineering AI systems with local people requires tremendous community outreach effort to establish mutual trust. In our experiences, applying and evaluating AI in Community Citizen Science relies heavily on an environment that has a sustainable fundraising mechanism in community organizations and universities. For example, funding is needed to hire software engineers that can maintain AI systems as community infrastructure in the long term, which can be hard to achieve in the current academic grant instruments and funding cycles.
The success of Community Citizen Science also depends on sustainable participation, which requires high levels of altruism, high awareness of local issues, and sufficient self-efficacy among local people. But, the complexity of the underlying machine learning techniques can affect the willingness to participate. On one side is whether the automation technique is trustworthy. In our experiences, local communities often perceive AI as a mysterious box that can be questionable and is not always guaranteed to work. Hence, citizens’ willingness to provide data can be low, but AI systems that employ machine learning and computer vision need data to be functional. On the other side, ``what citizens think the AI system can do'' does not match ``what the AI system can actually do'', resulting in socio-technical gaps and pitfalls for actual usage~\cite{roberts2021common}. In our experiences, local communities often have high expectations about what AI techniques can do for them, for example, automatically determining if an industrial site is violating environmental regulations. However, in practice, the AI system may only identify whether a factory emits smoke and degrades the air quality through sensors and cameras, which requires additional human efforts to verify if the pollution event is indeed a violation.
\section*{BRIDGE AI RESEARCH AND CITIZEN NEEDS}
University researchers typically lead the development of AI systems using a researcher-centered approach, where they often have more power over local communities (especially underserved ones) in terms of scientific authority and available resources. This unequal power relationship can result in a lack of trust and cause harm to underserved communities~\cite{koekkoek2021unraveling,harrington2019deconstructing}. An underlying assumption of this researcher-centered approach is that designers and scientists can put themselves in the situation of citizens and empathize with local people’s perspectives. However, university researchers are in a privileged situation in terms of socio-economic status and may come from other geographical regions or cultures, which means it can be very challenging for researchers to understand local people’s experiences fully and authentically~\cite{brown1993public}. Only by admitting this weakness and recognizing the power inequality can researchers truly respect community knowledge and be sincerely open-minded in involving local communities—especially those impacted by the problems the most—in the center of the design process when creating AI systems. Beyond being ``like'' the local people and designing solutions ``for'' them, researchers need to be ``with'' people who are affected by local concerns to co-create historicity and ensure that the AI systems are created to be valuable and beneficial to them~\cite{bennett2019promise,sloan2020participation}.
The critical role of creating social impact lies in local people and their long-term perseverance in advocating for changes. We believe that scientists need to collaborate with local people to address pressing social concerns genuinely, and even further, to immerse themselves into the local context and become citizens, hence ``scientific citizens'' (as defined by~\citet{irwin2001constructing}). However, pursuing academic research and addressing citizen concerns require different (even contradicting) efforts and can be difficult to achieve at the same time. Academic research requires contributing papers with scientific knowledge primarily to the research community, while citizen concerns typically involve many other stakeholders in a large and regional socio-technical system. It remains an open question how scientists and citizens can collaborate effectively under such dynamic, hyperlocal, and place-based conditions~\cite{bozzon2015needs}.
To move forward, we propose three viable Community Citizen Science approaches about how AI designers and scientists can conduct research and co-create social impact with local communities:
\begin{itemize}[noitemsep,topsep=0pt]
\item Evaluate AI's social impact as empirical contributions
\item Curate community data as dataset contributions
\item Build AI pipelines as methodological contributions
\end{itemize}
These approaches produce empirical, dataset, and methodological contributions respectively to the research community, as defined by~\citet{wobbrock2016research}. To the local people, these approaches establish a long-term fair university-community partnership in addressing community concerns, increase literacy in collecting community data, and equip communities with AI tools to interpret data. CCS projects will succeed when designers and scientists see themselves as citizens, and in turn, when local communities and citizens see themselves as innovators. It is essential for all parties to collaborate around the lived experiences of one another and listen to each other’s voices with humility and respect.
\subsection*{AI's Social Impact as Empirical Contributions}
Lessons learned from previously deployed AI systems in other contexts cannot be simply applied in the current one, as local communities have various cultures, behaviors, beliefs, values, and characteristics~\cite{sloane2019ai}. Hence, it is essential to understand and document how scientists can co-design AI systems with local communities and co-create long-term social impact in diverse contexts. It is also important to study the effectiveness and impact of various AI interventions with different design criteria in sustaining participation, affecting community attitude, and empowering people. The Community Citizen Science framework provides a promising path toward these goals. Implications of collaborating with local people in co-designing AI interventions, creating long-term impacts, and tackling the conflicts of interest among stakeholders can be strong empirical contributions to the academic community~\cite{sloan2020participation}. The data-driven evidence and the interventions that are produced by AI systems can impact the local region in various ways, including increasing residents’ confidence in addressing concerns, providing convincing evidence, or rebalancing power relationships among stakeholders.
For instance, our air pollution monitoring project documented the co-design process regarding how designers translated citizen needs and local knowledge into implementable AI system features, as recognized by the Human-Computer Interaction community and published on ACM CHI~\cite{hsu2017community}. This work shows researchers how we co-created an AI system to support citizens in collecting air pollution evidence and how local communities used the evidence to take action. For example, in the computer vision model for finding industrial smoke emissions in videos, the feature vectors are handcrafted according to the behaviors and characteristics of smoke, which are provided by community knowledge. Also, the communities decide the areas in the video that require image processing. The decision of having high precision in the prediction (instead of high recall) is also a design choice by local people for quickly determining severe environmental violations. Another example is our study of push notifications which are generated by an AI model to predict the presence of bad odor in the city~\cite{hsu2020smell}. The finding from the study explains how sending certain types of push notifications to local citizens is related to the increase of their engagement level, such as contributing more smell reports or browsing more data.
Although one may not simply duplicate the collaboration ecosystem in these contexts due to unique characteristics in the local communities, our projects can be seen as case studies in specific settings. Our air quality monitoring case provides insights to researchers working on similar problems in other contexts about integrating technology reliably into their settings, as cited by~\citet{ottinger2017crowdsourcing}. Moreover, the case also helped researchers understand and categorize different modes of community empowerment, as cited by~\citet{schneider2018empowerment}.
\subsection*{Community Data as Dataset Contributions}
Data work is critical in building and maintaining AI systems~\cite{sambasivan2021everyone}, as modern AI models are powered by large and constantly changing datasets. When addressing local concerns with the support of AI systems, researchers often need to finetune existing models or build new pipelines to fit local needs. This requires collecting data in a specific regional context and may introduce new tasks to the AI research field. Community Citizen Science provides a sustainable way to co-create high-quality regional datasets while simultaneously increasing citizens’ self-efficacy in addressing local problems. Based on our experiences, co-creating publicly available community data can also facilitate citizens’ sense of ownership of the collaborative work. Such value of community empowerment links AI research closely to social impact and public good.
Besides the value of increasing citizens’ data literacy, the collected real-world data, the data collection approach, and the data processing pipeline can be combined into a significant dataset contribution to the academic community in creating robust AI models. Such community datasets are gathered in the wild with local populations over a long-term period to reflect the regional context, which complements the datasets obtained using crowdsourcing approaches (such as Amazon Mechanical Turk) in a broader context. In this way, community datasets provide values for AI researchers to validate if AI models trained on general datasets can work as expected in different regional contexts. Also, the accompanying software for data labeling can contribute reusable computational tools to the research community that investigates data annotation strategies.
For example, our RISE project presented a novel video dataset for the smoke recognition task, which can help other researchers develop better computer vision models for similar tasks, as recognized by the Artificial Intelligence community and published on AAAI~\cite{hsu2021project}. Our project demonstrated the approach of collaborating with citizens affected by air pollution to annotate videos with industrial smoke emissions at large scale under various weather and lighting conditions. The dataset was used to train a computer vision model to recognize industrial smoke emissions, which allowed community activists to curate a list of severe pollution events as evidence to conduct studies and advocate for enforcement action. Another example is the Mosquito Alert project that curates and labels a large mosquito image dataset with local people using a mobile application~\cite{pataki2021deep}. The dataset is built with local community knowledge and is used to train a mosquito recognition model to support the local public health agency in disease management. Besides its social impact, the Mosquito Alert project advances science by providing a real-world dataset for researching different mosquito recognition models, as cited by~\citet{adhane2021deep}.
\subsection*{AI Pipelines as Methodological Contributions}
In Community Citizen Science, there is a need to unite expertise from the local communities and scientists to build AI pipelines using machine learning to assist data labeling, predict trends, or interpret patterns in the data. An example is to forecast pollution and find evidence of how pollution affects the living quality in a local region. Although the concept of machine learning is common among computer scientists, it can look mysterious to citizens.
Thus, during public communication and community outreach, researchers often need to visualize analysis results and explain the statistical evidence for local residents, which is highly related to the Explainable AI (XAI) and interpretable machine learning research~\cite{doshi2017towards}. However, current XAI research mainly focuses on making AI understandable for experts rather than laypeople and local communities~\cite{cheng2019explaining}. This creates a unique research opportunity to study co-design methods and software engineering workflows of translating AI models’ predictions and their internal decision-making process into human-intelligible insights in the hyperlocal context~\cite{shneiderman2020bridging,miller2019explanation,burrell2016machine}. We believe the pipeline of such translation into explainable evidence can be a methodological contribution to the academic community, which provides a way to deal with the challenge of predicting future trends and interpreting similar types of real-world data. Also, the implemented machine learning pipeline and the design insights of developing the pipeline can contribute reusable computational tools and novel software engineering workflows to the research communities that study Explainable AI and its user interfaces.
For instance, our Smell Pittsburgh project used machine learning to explain relationships between citizen-contributed odor reports and air quality sensor measurements, contributing to a methodological pipeline of translating AI predictions, as recognized by the Intelligent User Interface community and published on ACM TiiS~\cite{hsu2020smell}. In this way, the pollution patterns became visible for public scrutiny and debate. Another example is the xAire project that co-designed solutions with local schools and communities to collect Nitrogen Dioxide data~\cite{perello2021large}. The air measurements were analyzed with asthma cases in children using a statistical machine learning model. The community outreach and public communication enabled laypeople to make sense of how Nitrogen Dioxide posed a risk to local community health. The pipelines in these two examples produced meaningful patterns for citizens to understand and communicate about how pollution impacts the local region. They also informed researchers about how to process, wrangle, analyze, and interpret urban data in order to explain insights to laypeople.
\section*{NEXT STEPS}
We have explained major challenges in co-designing AI systems with local people and empowering them to create broader social impact. We also proposed Community Citizen Science approaches to simultaneously addressing local societal issues and advancing science. Computing research communities have made steps to recognize the impact of technology and AI on society, such as establishing a separate track for paper evaluation (e.g., the AAAI Special Track on AI for Social Impact). We urge the computing research communities to go further and acknowledge social impact as a type of formal contribution in scientific inquiry and paper publication. Promoting this kind of contribution can be a turning point to encourage scientists to link research to society and ultimately make university research socially responsible for the public good. Co-creating AI systems and developing reusable tools with local communities in the long term allows scientists and designers to explore real-world challenges and solution spaces for various AI techniques, including machine learning, computer vision, and natural language processing.
We also urge universities to integrate social impact into the evaluation criteria of the tenure roadmap of the academic professorship as the ``service'' pillar of the university that contributes to the public good. We envision that applying Community Citizen Science when co-designing AI systems can advance science, build public trust in AI research through genuine reciprocal university-community partnership, and directly support community action to impact society. In this way, we may fundamentally change how universities, organizations, and companies partner with their neighbors to pursue shared prosperity in the future of Community-Empowered Artificial Intelligence.
\section*{ACKNOWLEDGMENT}
We greatly appreciate the support of this work from the European Commission under the EU Horizon 2020 framework (grant number 101016233), within project PERISCOPE (Pan-European Response to the Impacts of COVID-19 and Future Pandemics and Epidemics), and from the Dutch Research Council (NWO) within the project Designing Rhythms for Social Resilience (grant number 314-99-300).
\section*{AUTHOR CONTRIBUTIONS}
This section uses the Contributor Roles Taxonomy (CRediT); Conceptualization, Y.C.H., T.H.K.H., H.V., A.M., I.N., and A.B.; Methodology, Y.C.H. and T.H.K.H.; Investigation, Y.C.H. and T.H.K.H.; Writing - Original Draft, Y.C.H.; Writing - Review \& Editing, Y.C.H., T.H.K.H., H.V., A.M., I.N., and A.B.; Supervision, A.B.; Project Administration, Y.C.H.; Funding Acquisition, A.B.
\section*{DECLARATION OF INTERESTS}
The authors declare no competing interests.
\section*{DATA AND CODE AVAILABILITY}
This research paper did not use data or code.
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1,116,691,500,465 | arxiv | \section{Introduction}
After Hawking's famous work \cite{Hawking:1974rv} - the black holes radiate - known as {\it{Hawking effect}}, it is now well understood that it is related to the event horizon of a black hole. A closely related effect is the {\it{Unruh effect}} \cite{Unruh:1976db}, where a similar type of horizon is experienced by a uniformly accelerated observer on the Minkowski space-time. A unified description of them was first put forwarded by Deser and Levin \cite{Deser:1997ri,Deser:1998xb} which was a sequel to an earlier attempt \cite{Narnhofer:1996zk}. This is called the global embedding Minkowskian space (GEMS) approach. In this approach, the relevant detector in curved space-time (namely Hawking detector) and its event horizon map to the Rindler detector in the corresponding flat higher dimensional embedding space \cite{Goenner,Rosen} and its event horizon.
Then identifying the acceleration of the Unruh detector, the Unruh temperature was calculated. Finally, use of the Tolman relation \cite{Tolman} yields the Hawking temperature. Subsequently, this unified approach to determine the Hawking temperature using the Unruh effect was applied for several black hole space-times \cite{Kim:2000ct,Tian:2005yj,Brynjolfsson:2008uc,Hong:2003xz}. However the results were confined to four dimensions and the calculations were done case by case, taking specific black hole metrics. It was not clear whether the technique was applicable to complicated examples like the Kerr-Newman metric which lacks spherical symmetry.
The motivation of this paper is to give a modified presentation of the GEMS approach that naturally admits generalization. Higher dimensional black holes with different metrics, including Kerr-Newman, are considered.
Using this new embedding, the local Hawking temperature (Unruh temperature) will be derived. Then the Tolman formula leads to the Hawking temperature.
We shall first introduce a new global embedding which embeds only the ($t-r$)-sector of the curved metric into a flat space. It will be shown that this embedding is enough to derive the Hawking result using the Deser-Levin approach \cite{Deser:1997ri,Deser:1998xb}, instead of the full embedding of the curved space-time. Hence we might as well call this the reduced global embedding. This is actually motivated from the fact that an $N$-dimensional black hole metric effectively reduces to a $2$ -dimensional metric (only the ($t-r$)-sector) near the event horizon by the dimensional reduction technique \cite{Robinson:2005pd,Carlip:1998wz,Iso:2006ut,Umetsu:2009ra} (for examples see Appendix 1). Furthermore, this $2$-dimensional metric is enough to find the Hawking quantities if the back scattering effect is ignored. Several spherically symmetric static metrics will be exemplified. Also, to show the utility of this reduced global embedding, we shall discuss the most general solution of the Einstein gravity - Kerr-Newman space-time, whose full global embedding is difficult to find. Since the reduced embedding involves just the two dimensional ($t-r$)-sector, black holes in arbitrary dimensions can be treated.
In this sense our approach is valid for any higher dimensional black hole.
The organization of the paper is as follows. In section 2 we shall find the reduced global embedding of several black hole space-times which are spherically symmetric. In the next section the power of this approach will be exploited to find the Unruh/Hawking temperature for the Kerr-Newman black hole. Finally, we shall give our concluding remarks.
\section{Reduced global embedding}
A unified picture of Hawking effect \cite{Hawking:1974rv} and Unruh effect \cite{Unruh:1976db} was established by the global embedding of a curved space-time into a higher dimensional flat space \cite{Deser:1998xb}.
Subsequently, this unified approach to determine the Hawking temperature using the Unruh effect was applied for several black hole space-times \cite{Kim:2000ct,Tian:2005yj}, but usually these are spherically symmetric. For instance, no discussion on the Kerr-Newman black hole has been given, because it is difficult to find the full global embedding.
Since the Hawking effect is governed solely by properties of the event horizon, it is enough to consider the near horizon theory. As already stated, this is a two dimensional theory obtained by dimensional reduction of the full theory. Its metric is just the ($t-r$)-sector of the original metric.
In the following sub-sections we shall find the global embedding of the near horizon effective $2$-dimensional theory. Then the usual local Hawking temperature will be calculated. Technicalities are considerably simplified and our method is general enough to include different black hole metrics.
\subsection{Schwarzschild metric}
Near the event horizon the physics is given by just the two dimensional ($t-r$) -sector of the full Schwarzschild metric \cite{Robinson:2005pd}:
\begin{eqnarray}
ds^2 = g_{tt}dt^2 + g_{rr}dr^2 = \Big(1-\frac{2m}{r}\Big)dt^2 -\frac{dr^2}{1-\frac{2m}{r}}.
\label{1.04}
\end{eqnarray}
It is interesting to see that this can be globally embedded in a flat $D=3$ space as,
\begin{eqnarray}
ds^2 = (dz^0)^2 - (dz^1)^2 - (dz^2)^2
\label{1.32}
\end{eqnarray}
by the following relations among the flat and curved coordinates:
\begin{eqnarray}
&&z^0_{out} = \kappa^{-1} \Big(1-\frac{2m}{r}\Big)^{1/2} \textrm{sinh}(\kappa t),\,\,\,\
z^1_{out} = \kappa^{-1} \Big(1-\frac{2m}{r}\Big)^{1/2} \textrm{cosh}(\kappa t),
\nonumber
\\
&&z^0_{in} = \kappa^{-1} \Big(\frac{2m}{r} - 1\Big)^{1/2} \textrm{cosh}(\kappa t),\,\,\,\
z^1_{in} = \kappa^{-1} \Big(\frac{2m}{r} - 1\Big)^{1/2} \textrm{sinh}(\kappa t),
\nonumber
\\
&&z^2 = \int dr \Big(1+\frac{r_Hr^2 + r_H^2r + r_H^3}{r^3}\Big)^{1/2},
\label{1.33}
\end{eqnarray}
where the surface gravity $\kappa=\frac{1}{4m}$ and the event horizon is located at $r_H=2m$. The suffix ``$in$'' (``$out$'') refer to the inside (outside) of the event horizon while variables without any suffix imply that these are valid on both sides of the horizon. We shall follow these notations throughout the paper. Now if a detector moves according to constant $r$ (Hawking detector) outside the horizon in the curved space, then the corresponding Unruh detector moves on the constant $z^2$ plane and it will follow the hyperbolic trajectory
\begin{eqnarray}
\Big(z^1_{out}\Big)^2 - \Big(z^0_{out}\Big)^2 = 16 m^2 \Big(1 - \frac{2m}{r}\Big) = \frac{1}{{\tilde{a}}^2}.
\label{hyper}
\end{eqnarray}
This shows that the Unruh detector is moving in the ($z^0_{out}, z^1_{out}$) flat plane with a uniform acceleration ${\tilde{a}}= \frac{1}{4m}\Big(1 - \frac{2m}{r}\Big)^{-1/2}$. Then, according to Unruh \cite{Unruh:1976db}, the accelerated detector will see a thermal spectrum in the Minkowski vacuum with the local Hawking temperature given by,
\begin{eqnarray}
T = \frac{\hbar {\tilde{a}}}{2\pi} = \frac{\hbar}{8\pi m} \Big(1 - \frac{2m}{r}\Big)^{-1/2}.
\label{localtemp}
\end{eqnarray}
So we see that with the help of the reduced global embedding the local Hawking temperature near the horizon can easily be obtained.
Now the temperature measured by any observer away from the horizon can be obtained by using the Tolman formula \cite{Tolman} which ensures constancy between the product of temperatures and corresponding Tolman factors measured at two different points in space-time. This formula is given by \cite{Tolman}:
\begin{eqnarray}
\sqrt{g_{tt}}~ T = \sqrt{g_{0_{tt}}}~ T_0
\label{new1}
\end{eqnarray}
where, in this case, the quantities on the left hand side are measured near the horizon whereas those on the right hand side are measured away from the horizon (say at $r_0$). Since away from the horizon the space-time is given by the full metric, $g_{0_{tt}}$ must correspond to the $dt^2$ coefficient of the full (four dimensional) metric.
For the case of Schwarzschild metric $g_{tt} = 1-2m/r$, $g_{0_{tt}} = 1-2m/r_0$. Now the Hawking effect is observed at infinity ($r_0 = \infty$), where $g_{0_{tt}} = 1$. Hence, use of the Tolman formula (\ref{new1}) immediately yields the Hawking temperature:
\begin{eqnarray}
T_0 = {\sqrt{g_{tt}}}~ T = \frac{\hbar}{8\pi m }.
\label{hawkingtemp}
\end{eqnarray}
Thus, use of the reduced embedding instead of the embedding of the full metric is sufficient to get the answer.
\subsection {Reissner-Nordstr$\ddot{\textrm{o}}$m metric}
In this case, the effective metric near the event horizon is given by \cite{Robinson:2005pd},
\begin{eqnarray}
ds^2 = \Big(1 - \frac{2m}{r} + \frac{e^2}{r^2}\Big)dt^2 - \frac{dr^2}{1 - \frac{2m}{r} + \frac{e^2}{r^2}}.
\label{1.34}
\end{eqnarray}
This metric can be globally embedded into the $D=4$ dimensional flat metric as,
\begin{eqnarray}
ds^2 = (dz^0)^2 - (dz^1)^2 - (dz^2)^2 + (dz^3)^2
\label{1.35}
\end{eqnarray}
where the coordinate transformations are:
\begin{eqnarray}
&&z^0_{out} = \kappa^{-1} \Big(1-\frac{2m}{r} + \frac{e^2}{r^2}\Big)^{1/2} \textrm{sinh}(\kappa t),\,\,\,\
z^1_{out} = \kappa^{-1} \Big(1-\frac{2m}{r} + \frac{e^2}{r^2}\Big)^{1/2} \textrm{cosh}(\kappa t),
\nonumber
\\
&&z^0_{in} = \kappa^{-1} \Big(\frac{2m}{r} - \frac{e^2}{r^2} - 1\Big)^{1/2} \textrm{cosh}(\kappa t),\,\,\,\
z^1_{in} = \kappa^{-1} \Big(\frac{2m}{r} - \frac{e^2}{r^2} - 1\Big)^{1/2} \textrm{sinh}(\kappa t),
\nonumber
\\
&&z^2 = \int dr \Big[1+\frac{r^2(r_+ + r_-) + r_+^2(r + r_+)}{r^2(r-r_-)}\Big]^{1/2},
\nonumber
\\
&&z^3 = \int dr \Big[\frac{4r_+^5r_-}{r^4(r_+ - r_-)^2}\Big]^{1/2}.
\label{1.36}
\end{eqnarray}
Here in this case the surface gravity $\kappa = \frac{r_+ - r_-}{2r_+^2}$ and $r_{\pm}=m\pm\sqrt{m^2-e^2}$. The black hole event horizon is given by $r_H=r_+$. Note that for $e=0$, the above transformations reduce to the Schwarzschild case (\ref{1.33}).
The Hawking detector moving in the curved space outside the horizon, following a constant $r$ trajectory, maps to the Unruh detector on the constant ($z^2,z^3$) surface. The trajectory of the Unruh detector is given by
\begin{eqnarray}
\Big(z^1_{out}\Big)^2 - \Big(z^0_{out}\Big)^2 = \Big(\frac{r_+ - r_-}{2r_+^2}\Big)^{-2} \Big(1-\frac{2m}{r} + \frac{e^2}{r^2}\Big)=\frac{1}{{\tilde{a}}^2}.
\label{RN1}
\end{eqnarray}
This, according to Unruh \cite{Unruh:1976db}, immediately leads to the local Hawking temperature $T=\frac{\hbar {\tilde{a}}}{2\pi}=\frac{\hbar(r_+ - r_-)}{4\pi r_+^2\sqrt{1-2m/r+e^2/r^2}}$ which was also obtained from the full global embedding \cite{Deser:1998xb}. Again, since in this case $g_{0_{tt}} = 1-2m/r_0 + e^2/r_0^2$ which reduces to unity at $r_0=\infty$ and $g_{tt} = 1-2m/r+e^2/r^2$, use of Tolman formula (\ref{new1}) leads to the standard Hawking temperature $T_0=\sqrt{g_{tt}}~ T=\frac{\hbar(r_+ - r_-)}{4\pi r_+^2}$.
\subsection{Schwarzschild-AdS metric}
Near the event horizon the relevant effective metric is \cite{Robinson:2005pd},
\begin{eqnarray}
ds^2 = \Big(1-\frac{2m}{r}+\frac{r^2}{R^2}\Big)dt^2 - \frac{dr^2}{\Big(1-\frac{2m}{r}+\frac{r^2}{R^2}\Big)},
\label{1.37}
\end{eqnarray}
where $R$ is related to the cosmological constant $\Lambda= -1/R^2$.
This metric can be globally embedded in the flat space (\ref{1.35}) with the following coordinate transformations:
\begin{eqnarray}
&&z^0_{out} = \kappa^{-1} \Big(1-\frac{2m}{r} + \frac{r^2}{R^2}\Big)^{1/2} \textrm{sinh}(\kappa t),\,\,\
z^1_{out} = \kappa^{-1} \Big(1-\frac{2m}{r} + \frac{r^2}{R^2}\Big)^{1/2} \textrm{cosh}(\kappa t),
\nonumber
\\
&&z^0_{in} = \kappa^{-1} \Big(\frac{2m}{r} - \frac{r^2}{R^2} - 1\Big)^{1/2} \textrm{cosh}(\kappa t),\,\,\,\
z^1_{in} = \kappa^{-1} \Big(\frac{2m}{r} - \frac{r^2}{R^2} - 1\Big)^{1/2} \textrm{sinh}(\kappa t),
\nonumber
\\
&&z^2 = \int dr \Big[1+\Big(\frac{R^3 + R r_H^2}{R^2 + 3r_H^2}\Big)^2 \frac{r^2r_H+rr_H^2 + r_H^3}{r^3(r^2+rr_H+r_H^2+R^2)}\Big]^{1/2},
\nonumber
\\
&&z^3 = \int dr \Big[\frac{(R^4 + 10R^2r_H^2+9r_H^4)(r^2+rr_H+r_H^2)}{(r^2+rr_H+r_H^2+R^2)(R^2+3r_H^2)^2}\Big]^{1/2}
\label{1.38}
\end{eqnarray}
where the surface gravity $\kappa=\frac{R^2+3r_H^2}{2r_HR^2}$ and the event horizon $r_H$ is given by the root of the equation $1-\frac{2m}{r_H} + \frac{r^2_H}{R^2}=0$.
Note that in the $R\rightarrow\infty$ limit these transformations reduce to those for the Schwarzschild case (\ref{1.33}). We observe that the Unruh detector on the ($z^2,z^3$) surface (i.e. the Hawking detector moving outside the event horizon on a constant $r$ surface) follows the hyperbolic trajectory:
\begin{eqnarray}
\Big(z^1_{out}\Big)^2 - \Big(z^0_{out}\Big)^2 = \Big(\frac{R^2+3r_H^2}{2r_HR^2}\Big)^{-2}\Big(1-\frac{2m}{r} + \frac{r^2}{R^2}\Big)=\frac{1}{{\tilde{a}}^2}
\label{ADS1}
\end{eqnarray}
leading to the local Hawking temperature $T=\frac{\hbar {\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\Big(1-\frac{2m}{r}+\frac{r^2}{R^2}\Big)^{1/2}}$. This result was obtained earlier \cite{Deser:1998xb}, but with more technical complexities, from the embedding of the full metric.
It may be pointed out that for the present case, the observer must be at a finite distance away from the event horizon, since the space-time is asymptotically AdS. Therefore, if the observer is far away from the horizon ($r_0>>r$) where $g_{0_{tt}}=1-2m/r_0+r_0^2/R^2$, then use of (\ref{new1}) immediately leads to the temperature measured at $r_0$:
\begin{eqnarray}
T_0 = \frac{\hbar\kappa}{2\pi\sqrt{1-2m/r_0 + r_0^2/R^2}}.
\label{new2}
\end{eqnarray}
Now, this shows that $T_0\rightarrow 0$ as $r_0\rightarrow \infty$; i.e. no Hawking particles are present far from horizon.
\section{Kerr-Newman metric}
So far we have discussed a unified picture of Unruh and Hawking effects using our reduced global embedding approach for spherically symmetric metrics, reproducing standard results. However, our approach was technically simpler since it involved the embedding of just the two dimensional near horizon metric. Now we shall explore the real power of this new embedding.
The utility of the reduced embedding approach comes to the fore for the Kerr-Newman black hole which is not spherically symmetric. The embedding for the full metric, as far as we are aware, is not done in the literature.
The effective $2$-dimensional metric near the event horizon is given by \cite{Iso:2006ut,Umetsu:2009ra},
\begin{eqnarray}
ds^2 = \frac{\Delta}{r^2+a^2}dt^2 - \frac{r^2+a^2}{\Delta}dr^2,
\label{Kerr1}
\end{eqnarray}
where
\begin{eqnarray}
&&\Delta = r^2-2mr+a^2+e^2 = (r-r_+)(r-r_-);\,\,\,\ a=\frac{J}{m};
\nonumber
\\
&&r_\pm = m\pm\sqrt{m^2-a^2-e^2}.
\label{Kerr2}
\end{eqnarray}
The event horizon is located at $r=r_+$. This metric can be embedded in the following $D=5$-dimensional flat space:
\begin{eqnarray}
ds^2=\Big(dz^0\Big)^2 -\Big(dz^1\Big)^2-\Big(dz^2\Big)^2 + \Big(dz^3\Big)^2 + \Big(dz^4\Big)^2,
\label{Kerr3}
\end{eqnarray}
where the coordinate transformations are
\begin{eqnarray}
&&z^0_{out} = \kappa^{-1} \Big(1-\frac{2mr}{r^2+a^2} + \frac{e^2}{r^2+a^2}\Big)^{1/2} \textrm{sinh}(\kappa t),\,\,\,\
z^1_{out} = \kappa^{-1} \Big(1-\frac{2mr}{r^2+a^2} + \frac{e^2}{r^2+a^2}\Big)^{1/2} \textrm{cosh}(\kappa t),
\nonumber
\\
&&z^0_{in} = \kappa^{-1} \Big(\frac{2mr}{r^2+a^2} - \frac{e^2}{r^2+a^2} - 1\Big)^{1/2} \textrm{cosh}(\kappa t),\,\,\,\
z^1_{in} = \kappa^{-1} \Big(\frac{2mr}{r^2+a^2} - \frac{e^2}{r^2+a^2} - 1\Big)^{1/2} \textrm{sinh}(\kappa t),
\nonumber
\\
&&z^2 = \int dr \Big[1+\frac{(r^2+a^2)(r_+ + r_-) + r_+^2(r + r_+)}{(r^2+a^2)(r-r_-)}\Big]^{1/2},
\nonumber
\\
&&z^3 = \int dr \Big[\frac{4r_+^5r_-}{(r^2+a^2)^2(r_+ - r_-)^2}\Big]^{1/2},
\nonumber
\\
&&z^4 = \int dr a\Big[\frac{r_+ + r_-}{(a^2+r_-^2)(r_- - r)} + \frac{4(a^2 + r_+^2)(a^2-r_+r_- + (r_+ +r_-)r)}{(r_+ - r_-)^2 (a^2 + r^2)^3}
\nonumber
\\
&&+ \frac{4r_+r_-(a^2+2r_+^2)}{(r_+ - r_-)^2(a^2+r^2)^2} + \frac{rr_- - a^2 + r_+(r+r_-)}{(a^2+r_-^2)(a^2+r^2)}\Big]^{1/2}.
\label{Kerr4}
\end{eqnarray}
Here the surface gravity $\kappa = \frac{r_+-r_-}{2(r_+^2 + a^2)}$.
For $e=0, a=0$, as expected, the above transformations reduce to the Schwarzschild case (\ref{1.33}) while only for $a=0$ these reduce to the Reissner-Nordstr$\ddot{\textrm{o}}$m case (\ref{1.36}).
As before, the trajectory adopted by the Unruh detector on the constant ($z^2,z^3,z^4$) surface corresponding to the Hawking detector on the constant $r$ surface is given by the hyperbolic form,
\begin{eqnarray}
\Big(z^1_{out}\Big)^2 - \Big(z^0_{out}\Big)^2 = \kappa^{-2}\Big(1-\frac{2mr}{r^2+a^2} + \frac{e^2}{r^2+a^2}\Big)=\frac{1}{{\tilde{a}}^2}.
\label{Kerr5}
\end{eqnarray}
Hence the local Hawking temperature is
\begin{eqnarray}
T=\frac{\hbar {\tilde{a}}}{2\pi}=\frac{\hbar\kappa}{2\pi\sqrt{\Big(1-\frac{2mr}{r^2+a^2} + \frac{e^2}{r^2+a^2}\Big)}}.
\label{Kerr6}
\end{eqnarray}
Finally, since $g_{tt} = 1-\frac{2mr}{r^2+a^2} + \frac{e^2}{r^2+a^2}$ (corresponding to the near horizon reduced two dimensional metric) and $g_{0_{tt}}=\frac{r_0^2-2mr_0+a^2+e^2-a^2 {\textrm{sin}}^2\theta}{r_0^2+a^2{\textrm{cos}}^2\theta}$ (corresponding to the full four dimensional metric), use of the Tolman relation (\ref{new1}) leads to the Hawking temperature
\begin{eqnarray}
T_0=\frac{\sqrt{g_{tt}}}{\sqrt{(g_0{_{tt}})_{r_0\rightarrow\infty}}}~T = \frac{\hbar\kappa}{2\pi} = \frac{\hbar(r_+-r_-)}{4\pi(r_+^2 + a^2)},
\label{Kerr7}
\end{eqnarray}
which is the well known result \cite{Iso:2006ut}.
\section{Conclusion}
We provide a new approach to the study of Hawking/Unruh effects including their unification, initiated in \cite{Deser:1997ri,Deser:1998xb,Narnhofer:1996zk}, popularly known as global embedding Minkowskian space-time (GEMS). Contrary to the usual formulation \cite{Deser:1997ri,Deser:1998xb,Narnhofer:1996zk,Kim:2000ct,Tian:2005yj,Brynjolfsson:2008uc}, the full embedding was avoided. Rather, we required the embedding of just the two dimensional ($t-r$)-sector of the theory. This was a consequence of the fact that the effective near horizon theory is basically two dimensional. Only near horizon theory is significant since Hawking/Unruh effects are governed solely by properties of the event horizon.
This two dimensional embedding ensued remarkable technical simplifications whereby the treatment of more general black holes (e.g. those lacking spherical symmetry like the Kerr-Newman) was feasible. Also, black holes in any dimensions were automatically considered since the embedding just required the ($t-r$)-sector.
|
1,116,691,500,466 | arxiv | \section{Introduction}
A complex behavior of the normal state transport properties in the copper high temperature superconductors (HTSC) have been studied for a long time. Nevertheless, up to now there is no consistent description for some basic transport properties of copper HTSC as, for example, for an unusual temperature dependence of the Hall and longitudinal resistivity. The study of the transport properties of copper HTSC is often complicated by a phase separation and other microstructure peculiarity of studied crystals. This has led to a significant evolution in the views of microscopic electronic properties through the continuous improvements in investigation methods and sample quality. (see for example recent review \cite{1_nature14165} and references therein). In particular, it is strongly influenced by the development of the angle resolved photoemission spectroscopy (ARPES) techniques \cite{2_RevModPhys.75.473} which brought invaluable information about the existence of some specific features of the Fermi surface (FS) structure. Now it is clear that some of the anomalous normal state properties can be due to local kinetics of excitations at some regions of FS \cite{3_PhysRevB.88.041104}.
The members of a new family of HTSC, the iron superconductors \cite{4_JACS063355c}, show sometimes very similar anomalous behavior of the normal state transport properties. First of all, the Hall constants, similar to the Hall constants of the copper HTSC, demonstrate a large temperature variation, often with a sharp change in slope below 100 -- 150 K, as was observed for LiFeAs \cite{5_PhysRevB.84.064512}, Ba(FeMe)${}_{2}$As${}_{2}$ (Me=Co,Cu) \cite{6_PhysRevB.80.054517} and other superconducting series. Besides, the Hall resistance and longitudinal magnetoresistance may significantly deviate, correspondingly, from linear and quadratic in magnetic field dependencies as was observed for BaFe${}_{2}$As${}_{2}$ \cite{7_PhysRevB.84.184514}.
In general, the iron superconductors are multiband semimetals. The FS is formed by iron $d$-electrons \cite{8_Ann.Phys.583.8}. The resistivity of compounds is relatively high ( 0.1 -- 10 m${}\Omega$ cm ) and probably can be explained by peculiarities of $d$ electrons in these compounds \cite{9_PhysRevB.87.024504}. The recent ARPES studies of the iron superconductors revealed importance of electronic correlations in achieving a high $T_{c}$'s \cite{PhysRevX.4.031041, PhysRevLett.115.256403}. The observed pronounced reduction of the overall bandwidth on going to superconductivity, band-selective impurity scattering \cite{PhysRevX.4.031041} and other common phenomenon of iron superconductors need further understanding.
Similar to copper HTSCs and many other superconducting families the sample microstructure or inhomogeneity often hinders the study of the transport properties of iron superconductors. For example, superconducting (Na,K)Fe${}_{2}$Se${}_{2}$ crystals in normal state show phase separation a metal and semiconducting phases which leads to a formation of domains with a large variation of a local conductivity \cite{10_J.A.P116.043904}. Fortunately, these complexities are minimized for high quality superconducting crystals of FeSe and substituted compositions \cite{11_PhysRevB.88.174512, 12_PNAS105.14262} which show superconductivity in a stoichiometric form.
We have studied temperature and field dependence of the resistivity and Hall effect for the series FeSe${}_{1-x}$S${}_{x}$ x=0.04, 0.09 and 0.19. The sample FeSe${}_{0.81}$S${}_{0.19}$ shows no transition to an orthorhombic phase at low temperatures. Comparison of the transport properties of this sample with the transport properties of other compositions allowed us to reveal the highly mobile electron component that occurred only in an orthorhombic phase which presumably originates from the local FS regions and probably exists in many other orthorhombic phases of iron superconductors.
\section{Experiment}
The crystals of FeSe${}_{1-x}$S${}_{x}$ with x=0.04 and 0.09 were prepared using the KCl/AlCl${}_{3}$ flux technique \cite{13_CrystEngComm12.1989}. For the sample with x=0.19 the modified method was used described elsewhere \cite{Chareev2016,Chareev2016.2}. The chemical composition of the crystals was studied with the energy dispersive micro analysis system. The composition measurements were done at three points for four average size crystals from the each growth batch. The statistical error in sulfur content was about 5\% for all three batches.
Electrical measurements were done on cleaved rectangular samples with lengths in the range of 0.5 -- 2 mm, widths about 0.5 mm and thicknesses in the range of 0.01 -- 0.05 mm. The crystal dimensions were measured using a Zeiss binocular microscope. Used method yields up to 30 -- 50\% of systematic errors in absolute values of resistivity. This error is mainly due to a thickness determination and it does not increase during the transformation from resistivity to conductivity components.
Electric contacts were made by sputtering of Au/Ti layers with a precisely machined mechanical mask. The current electrodes were 0.1 mm wide lines along small sides of the bar. Potential and Hall 0.1$\times$0.1 mm$^{2}$ electrodes were connected to a sample holder with a 0.025 mm gold wire using H20E silver epoxy. The distance between Hall contacts was 0.5 mm. The distances between current electrodes were 1.5 mm for the sample with x= 0.9 and 1 mm for the samples with x=0.04 and 0.19. Therefore, the corresponding ratios of length to width were 3.0 and 2.0 which give 0.98 and 0.93 for the correction factor due to the Hall potential shortening by extended current electrodes \cite{15_jan1957galvamomagnetic}. We did not apply these corrections to our data considering it to be insignificant.
DC magnetoresistance and Hall effect measurements were done using QD PPMS and EDX options of MPMS 7T with Keithley 2400 and Keithley 2192. Resistance measurements in pulsed magnetic fields were done up to 25 -- 30 T using 100 kHz AC modulation. The pulse rise time was about 8 msec.
\section{Results}
\begin{figure}[h]
\includegraphics[scale=0.5,angle=0]{Fig1c.eps}
\caption{\label{fig1} (Color online). Temperature dependence of $\rho_{xx}$ for FeSe${}_{1-x}$S${}_{x}$ (x=0.04, 0.09, 0.19) single crystals.Two lines show the linear and quadratic functions of temperature. }
\end{figure}
The studied samples show the superconducting transitions at temperatures 10.25, 10.45 and 8.45 K for x=0.04, 0.09 and 0.19 correspondingly. A detailed description of the superconducting properties for these compositions was published elsewhere \cite{16_JLTP}, scanning tunneling microscopy and spectroscopy measurements for x=0.04, 0.09 were reported in Ref. \cite{PhysRevB.92.235113}. Here we discuss normal state transport properties of these samples. Fig.~\ref{fig1} shows a log-log plot of resistivity versus temperature for the studied FeSe${}_{1-x}$S${}_{x}$ (x=0.04, 0.09, 0.19) samples. Two lines in the plot show slopes of the linear and quadratic power dependencies for the sake of comparison. Discussed above a large systematic error in the sample thickness measurements could offset the curves from the actual positions in the plot.
The shapes of curves reflect qualitative changes in the substance properties under substitution. First of all, the curve corresponding to x=0.19 does not have a kink below 100 K which occurs at the transition to an orthorhombic structure. The absence of the transition in this sample is in agreement with the decrease of the corresponding transition temperature observed for other two samples from 82 K for x=0.04 to 69 K for x=0.09. The similar suppression of structural transition in Fe(SeS) series was also reported by other authors \cite{17_PhysRevB.92.121108}.
Another peculiarity of $R(T)$ curves is a crossover from a low temperature metal behavior to a saturation at room temperatures. Since the pure FeSe shows a similar saturation that turns into $R(T)$ decrease above 350 K, it can reflect the importance of activated carriers in transport properties of FeSe${}_{1-x}$S${}_{x}$ at high temperatures. Consequently, Gorkov's-Tetelbaum model that describe carriers activation in copper HTSC\cite{18_PhysRevB.77.180511} can be applied to this series of iron superconductors. As it follows from the curves crossing in Fig.~\ref{fig1}, an isovalent substitution of Se by S moves the inflection point of $R(T)$ curve to higher temperatures which may be due to either the increase of the corresponding activation energy or the decrease in the band population.
Both the high temperature metal to semiconductor crossover and the low temperature kink due to a structural transition decrease the value of the residual resistance ratio (RRR) determined for $R(T)$ dependencies. Therefore, RRR underestimates the quality of the crystals. The resistance of FeSe${}_{0.81}$S${}_{0.19}$ decreases from 300 to 10 K near 15 times reflecting a good quality of the crystal. Furthermore, following the idea of this ratio it is important to note that all $R(T)$ curves plotted in Fig.~\ref{fig1} are well parallel to each other almost the whole range. It ensures that presented below analysis of the electronic properties of FeSe${}_{1-x}$S${}_{x}$ series was done for the crystals of the same quality.
Our analysis was based on fitting the experimental data with the two or three band model. To extract parameters it is convenient first to calculate conductivity components from the measured resistivity components. For tetragonal crystals :
\begin{eqnarray}
\sigma_{xx}=\sigma_{yy}=\frac{\rho_{xx}}{(\rho_{xx}^{2}+\rho_{xy}^{2})} \nonumber\\
\sigma_{xy}=\sigma_{yx}=\frac{\rho_{xy}}{(\rho_{xx}^{2}+\rho_{xy}^{2})}
\nonumber
\end{eqnarray}
where $\sigma_{ij}$ are conductivity tensor components and $\rho_{ij}$ are resistivity tensor components. As we described before, the Hall and potential electrodes of our samples were done with precise mask and the main systematic error in measured resistivity components are due to the sample cross-section measurements which equally rescale $\rho_{xy}$ and $\rho_{xx}$ component. Besides, measurements were done in a persistent mode at every $B$, with a current commutation to cancel effect of thermo, contact and other EMF. Mixing of components due to misalignment of potential electrodes and current direction was compensated by extracting of odd and even in magnetic field components for $\rho_{xy}$ and $\rho_{xx}$ correspondingly. Therefore, we do not expect any additional distortion or error multiplication in the analyzed data due to this transformation.
The conductivities of bands are additive and within relaxation-time approximation for an arbitrary number of bands we can write
\begin{eqnarray}
\sigma_{xx}=F_{R}(B)\equiv\sum_{i=1}^{l}\frac{\lvert\sigma_{i}\rvert}{(1+\mu_{i}^{2}B^2{})}\nonumber\\
\sigma_{xy}=F_{H}(B)\equiv\sum_{i=1}^{l}\frac{\sigma_{i}\mu_{i}B}{(1+\mu_{i}^{2}B^2{})}\nonumber\\
\sigma_{i}=en_{i}\mu_{i}\nonumber
\end{eqnarray}
where $i$ is a band index, $e$ is the charge of the electron, $\sigma_{i}$ is a conductivity at $B=0$, $\mu_{i}$ is a mobility, $n_{i}$ is a currier concentration and $l$ is a number of bands. Fitting procedure determines $\mu_{i}$ and $n_{i}$ by minimizing the sum:
\begin{equation}
\sum_{k=1}^{N}{\bigg[\bigg({\frac{\sigma_{xx}[k]- F_{R}(B[k])}{\sigma_{xx}[k]}\bigg)}^{2}+\bigg({\frac{\sigma_{xy}[k]- F_{H}(B[k])}{\sigma_{xy}[k]}\bigg)}^{2} \bigg]} \nonumber
\end{equation}
where $\sigma_{xx}[k]$, $\sigma_{xy}[k]$ and $B[k]$ are the values of $\sigma_{xx}$, $\sigma_{xy}$ and $B$ at experimental point $k$ and $N$ is the number of the measured points. For the purposes of consistency, we used described fitting method for all samples and for both the two and the three band fitting. We limited our fitting to temperatures lower or equal to 100 K because at high temperatures the field dependence of $\rho_{xx}$ is too weak to be resolved in the used magnetic field range. For the same reason we used compensated (equal carrier concentration for the hole and electron bands) two band model at 100 K for all samples.
\subsection{FeSe${}_{0.81}$S${}_{0.19}$}
\begin{figure}[ht]
\includegraphics[scale=0.5,angle=0]{Fig2c.eps}
\caption{\label{fig2}(Color online). (a) Hall coefficient as a function of temperature for FeSe${}_{0.81}$S${}_{0.19}$. Inset: Magnetic-field dependence of $\rho_{xy}$ at selected temperatures. (b) The transverse magnetoresistance MR=$(\rho_{xx}(B)-\rho_{xx}(0))/\rho_{xx}(0)$ ( $B \parallel \boldsymbol{c}$) plotted versus $B^{2}$ at temperatures between 12 - 100 K. Inset: MR versus $(B/\rho_{xx}(0))^{2}$ ( Kohler plot) for the same data. }
\end{figure}
The longitudinal and Hall resistances in a low temperature tetragonal phase behave distinctly different from corresponding resistances of orthorhombic samples. It is in good agreement with the recent report on a quantum critical point near x=0.17 in FeSe${}_{1-x}$S${}_{x}$ series \cite{Hosoi19072016}. First off all, the orthorhombic FeSe${}_{1-x}$S${}_{x}$ (x=0.04, 0.09) samples show a strongly nonlinear dependence of $\rho_{xy}(B)$ at low temperatures. The $\rho_{xx}(B)$ dependencies also have features which lead, in particular, to a violation of the Kohler's rule, as it will be discussed below. In contrast, the tetragonal at low temperatures composition FeSe${}_{0.81}$S${}_{0.19}$ is closer in properties to a simple compensated semimetal. In support of this assertion, the inset of Fig.~\ref{fig2} (a) shows $\rho_{xy}(B)$ in fields up to 5 T for selected temperatures. All curves are linear which allows to determine the Hall constants at every temperature. The Hall constant shows a strong temperature dependence which is plotted in Fig.~\ref{fig2} (a). The plotted in Fig.~\ref{fig2} (b) $\rho_{xx}(B)$ dependencies follow $B^{2}$ law and the Kohler's rule is passably satisfied as seen from the inset of Fig.~\ref{fig2} (b).
Our analysis has shown that the measured transport properties of FeSe${}_{0.81}$S${}_{0.19}$ can be comprehensively described in the framework of the simple two band model. This method allows extracting two band concentrations and mobilities without any additional restrictions on their values. We limited our fitting to temperatures lower or equal to 100 K because of at higher temperatures the field dependence of $\rho_{xx}$ is too weak to be resolved in the used magnetic field range.
The Fig.~\ref{fig3} shows a temperature dependence of carrier concentrations (top) and inverse mobilities (bottom) extracted for FeSe${}_{0.81}$S${}_{0.19}$. The numerical values are listed in Table~\ref{Table1}. The concentration for electron and hole bands are near temperature independent and close to each other. The inverse mobilities show a good linearity without signs of a low temperature saturation that confirms the high quality of the crystal.
\begin{figure}[ht]
\includegraphics[scale=0.5,angle=0]{Fig3c.eps}
\caption{\label{fig3}(Color online). The temperature depandence of the carrier density (top panel) and the inverse mobility (bottom panel) for holes and electrones in FeSe${}_{0.81}$S${}_{0.19}$ extracted from the simple two band model. }
\end{figure}
\subsection{FeSe${}_{0.96}$S${}_{0.04}$}
\begin{figure}[h]
\includegraphics[scale=0.5,angle=0]{Fig4c.eps}
\caption{\label{fig4}(Color online). (a) Magnetic-field dependence of $\rho_{xy}$ for FeSe${}_{0.96}$S${}_{0.04}$ at temperatures between 12 - 50 K. (b) MR versus $(B/\rho_{xx}(0))^{2}$ ( Kohler plot) for FeSe${}_{0.96}$S${}_{0.04}$ at temperatures between 12 - 50 K and magnetic-field up to 7 T. The straight line highlights the plot curvature. Inset: MR versus $B^{2}$ at 12 K measured in pulsed fields up to 25 T. }
\end{figure}
The Fig.~\ref{fig4} shows the low temperature $\rho_{xy}(B)$ dependencies (a) and the Kohler's plot (b) under $B$ up to 7 T for the sample with x=0.04. The inset of Fig.~\ref{fig4}(b) shows the high field $\rho_{xx}(B^{2})$ curve measured up to 25 T at 12 K. The Kohler's rule is not satisfied as it follows from the mismatch of the curve measured at 50 K and curves measured at 12 and 25 K. This conclusion in agreement with the recent study \cite{PhysRevB.93.180503} of Kohler's rule violations for undoped FeSe under pressure. The nonlinear behavior of $\rho_{xy}(B)$ is a clear evidence for the presence of the carriers with a mobility satisfying $\mu_{i}B \approx 1$ in the measured field range. The $\rho_{xx}(B)$ curve measured at 12 K deviates significantly from $B^{2}$ law below 3 -- 4 T while it follows $B^{2}$ law in higher fields up to 25 T where MR reaches 600 \% as it shown in the inset of Fig. ~\ref{fig4} (b). It gives reasonable ground to assume existence of a highly mobile band along with the typical for a simple semimetal compensated main hole and electron bands with a considerably lower mobility. The violation of the Kohler's rule in this case may indicate that the properties of the observed highly mobile carriers is a temperature dependent. The conductivity of the highly mobile band is evidently suppressed in relatively low magnetic fields. From Fig. ~\ref{fig4} (b) we can roughly estimate the suppressed part of conductivity near 5 -- 10 \% by the relative amplitude of the low field hump in $\rho_{xx}(B)$ at 12 K. Therefore if we suppose a factor 10 for the mobilities ratio than the density of states for the highly mobile band is only near 1 \% of the overall carrier density.
To extract band parameters we fitted the experimental data for FeSe${}_{0.96}$S${}_{0.04}$ with the three band model. All data obtained are listed in Table~\ref{Table1}. The experimental and simulated curves for conductivities at 12, 25 and 50 K are plotted in Fig.~\ref{fig5}. The fit quality is good for all temperatures. The results confirms the assumption concerning low number of the highly mobile carriers. The obtained value of the carrier density for the highly mobile electrons is near 1$\times$10$^{18}$ cm$^{-3}$ which is two order lower than for the main bands.
\begin{figure}[h]
\includegraphics[scale=0.5,angle=0]{Fig5cr.eps}
\caption{\label{fig5}(Color online). The simultaneous fit of $\sigma_{xy}$ (a) and $\sigma_{xx}$ (b) for FeSe${}_{0.96}$S${}_{0.04}$ using the three band model (solid lines). Open circles are experimental data. The dotted lines give the best two band model fit for 12 K data. ($\mu_{n}$=2100 cm$^{2}$/Vs, $\mu_{p}$=1900 cm$^{2}$/Vs, $n_{n}$=7.7 10$^{19}$ cm$^{-3}$, $n_{p}$=9.1 10$^{19}$ cm$^{-3}$) }
\end{figure}
\subsection{FeSe${}_{0.91}$S${}_{0.09}$}
\begin{figure}[h]
\includegraphics[scale=0.5,angle=0]{Fig6c.eps}
\caption{\label{fig6}(Color online). (a) Magnetic-field dependence of $\rho_{xy}$ for FeSe${}_{0.91}$S${}_{0.09}$ at temperatures between 12 - 50 K. (b) MR versus $B^{2}$ for FeSe${}_{0.91}$S${}_{0.09}$. Inset: MR versus B at 12 K for $B \parallel \boldsymbol{c}$ and $B \parallel \boldsymbol{ab}$}
\end{figure}
The Fig.~\ref{fig6} (a) and (b) show correspondingly $\rho_{xy}(B)$ and $\rho_{xx}(B^{2}$) for the sample with x=0.09. For this sample the value of MR at 12 K in 7 T is near 20 \% which is an intermediate value between 45 \% observed for the sample with x=0.04 and 10 \% for the sample with x=0.19 that confirms systematic changes in electron properties of Fe(SeS) series with sulfur substitution. The low temperature curves are noticeably nonlinear that, similar to the sample with x=0.04, shows the presence of highly mobile carriers. A distinctive feature of the sample with x=0.09 is a negative MR observed at low temperatures. It is likely an occasional property of the particular crystal but it is possibly not the exception.
The inset of Fig.~\ref{fig6} (b) shows dependence of the MR on a magnetic field for the two field orientations. It is clear seen an isotropic behavior of the MR that means that the observed negative MR originates to a scattering suppression. It is usually a magnetic impurity scattering but for FeSe${}_{0.91}$S${}_{0.09}$ the negative MR occurs only in a very narrow temperature range near the onset of superconducting transition (not shown). Based on the last property we suppose that the observed negative MR can originate to the suppression of scattering by superconducting order parameter fluctuations.
The data extracting procedure for this sample was the same as for the sample with x=0.04. Despite the data for FeSe${}_{0.91}$S${}_{0.09}$ is more noisy than for other sample due to the crystal dimensional factor, the fit quality was good for all temperatures. The extracted parameters are listed in Table~\ref{Table1}. Qualitatively, the extracted parameters for the sample FeSe${}_{0.91}$S${}_{0.09}$ are in a good agreement with the parameters for FeSe${}_{0.96}$S${}_{0.04}$.
\begin{table*}[h]
\caption{\label{Table1} The results of data fitting using the simple two band model for
the sample with x=0.19, three band model for samples with x=0.09 and 0.04 at 12, 25 and 50 K and compensated two band model ($n_{h1}=n_{e1}$ ) for samples with x=0.09 and 0.04 at 100 K }
\begin{ruledtabular}
\begin{tabular}{cccccccc}
& & \multicolumn{2}{c}{e1}&\multicolumn{2}{c}{h1}&\multicolumn{2}{c}{e2}\\
$x$ & $T $ & $\mu_{n}$ & $n_{n}$ & $\mu_{p}$ & $n_{p}$ & $\mu_{n}$ & $n_{n}$ \\
& (K) & (cm$^{2}$/Vs) & (10$^{19}$ cm${}^{-3}$) & (cm$^{2}$/Vs) & (10$^{19}$ cm$^{-3}$) & (cm$^{2}$/Vs) & (10$^{19}$ cm$^{-3}$) \\
\hline
0.19&12&571&53.4&352&52.9 \\
&25&360&48.7&233&48.6 \\
&50&172&52.6&123&52.6 \\
&100&67&48.0&75&48.0 \\
0.09&12&670&7.0&630&8.0&2100&0.10 \\
&25&410&7.0&430&7.0&1600&0.10 \\
&50&170&10.0&190&12.0&800&0.09 \\
&100&44&27.0&46&27.0 \\
0.04&12&950&12.0&905&15.0&4775&0.12 \\
&25&675&10.0&575&13.0&3398&0.10 \\
&50&265&15.0&298&14.0&1563&0.05 \\
&100&56&38.0&60&38.0 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Discussion and conclusion}
The presence of electrons with considerably higher mobility than those of holes is well known phenomenon for iron-based superconductors. It is usually manifest itself as a low temperature anomaly in field dependence of a resistivity. For example, BaFe${}_{2}$As${}_{2}$ family demonstrate anomalous low temperature field dependence for both a longitudinal and Hall resistance \cite{7_PhysRevB.84.184514}. Extracted values of mobility at 5 K were 4500 and 1500 cm$^{2}$/Vs for electron bands and 1800 cm$^{2}$/Vs for a hole band. Another example is a recent paper on FeSe \cite{20_PhysRevLett.115.027006}, where transport property was measured up to 88 T. Authors reports 1843 and 457 cm$^{2}$/Vs for electrons and 623 cm$^{2}$/Vs for holes at 10 K.
Some authors consider a large difference in mobility as a property relating to whole FS pockets. For example, the study of the Hall effect and resistivity in BaFe${}_{2}$As${}_{2}$ family \cite{21_PhysRevLett.103.057001, 22_PhysRevB.80.140508} showed that electron carriers dominate the transport properties. On this basis the disparity of the electron and hole scattering rates were concluded as universal properties of iron superconductors. Our results for Fe(SeS) series are completely refute this assumption. First of all, the low temperature $\rho_{xy}(B)$ and $\rho_{xx}(B)$ dependencies for the superconducting tetragonal sample FeSe${}_{0.81}$S${}_{0.19}$ do not have anomaly in low magnetic fields. Moreover, this sample behave as a simple two band semimetal. It means that the highly mobile electron band occurs exclusively in an orthorhombic phase. Further, the value of the carrier concentration for the highly mobile electron band is very low in comparison with other bands. It rather corresponds to local ''hot spots'' than to whole FS pockets.
Our results for the main electron and hole bands of orthorhombic samples are close to the reported for FeSe in Ref. \cite{20_PhysRevLett.115.027006} values while for the highly mobile band our value of mobility is significantly higher and, consequently, carrier concentration is very different. We suspect that the value 1843 cm$^{2}$/Vs was an underestimation due to pure quality of measured crystals. Reported $\rho_{xy}(B)$ curves are almost coincide at 20 and 30 K \cite{20_PhysRevLett.115.027006} which for our mind can be explained only by a high residual resistance of the sample. For our orthorhombic samples the extremums of $\rho_{xy}(B)$ curves are distinctly moves to low fields with temperature decrease from 25 to 12 K (see Fig.~\ref{fig4}(a) and Fig.~\ref{fig6}(a) ) in agreement with a mobility increase.
Our results for orthorhombic samples are in a good quantitative and qualitative agreement with the results obtained for a high quality FeSe single crystal in Ref. \cite{26_PhysRevB.90.144516} by a mobility spectrum analysis . In the cited paper authors interpret a minority
band with ultrafast carrier mobility as originating either from the Dirac cone or the large anisotropy of FS's. We argue that the large anisotropy of carriers near some local points of FS can be explained by the approach of the Van Hove singularity point to the Fermi level as a result of a tetragonal to orthorhombic transition.
The FS reconstruction at a tetragonal to orthorhombic transition have been intensively studied by both theoretical and experimental methods since the discovery of the iron superconductors. The calculations for FeSe show that in a generic electronic structure consisting of hole pockets/cylinders in the middle of the Brulluen zone (BZ) and electron pockets/cylinders at the border of BZ, the main changes occur at the border of BZ and only minor in the center of BZ \cite{23_PhysRevB.91.214503}. The lifting of degeneracy of $d_{xz}/d_{yz}$ orbitals changes the electron-like cylinders at the border of BZ from round to elliptical form and thus causes the FS shrinking in some directions. We expect that due to a corrugation the cylinder necks can became very thin or even collapse. In both cases the Van Hove singularity which is near the center of the electron cylinders will approach the Fermi level energy. The radius of the electron cylinders necks which can be used as a rough estimate for the distance between the Fermi level and the Van Hove singularity point where determined experimentally. The recent Shubnikov-de Haas oscillation measurements \cite{PhysRevB.90.144517} for FeSe gave the values of the electron neck diameter in the range of a few meV that means that the Van Hove singularity is in a conduction band even at low temperatures. The existence of the Van Hove singularity at the Fermi level in FeSe was also discussed in Ref.\cite{24_PhysRevLett.113.237001, 25_PhysRevLett.115.106402}.
Another interesting peculiarity of obtained results is an evolution of the carrier density of the main bands at a tetragonal to an orthorhombic transition. Our analysis (Table~\ref{Table1}), in agreement with other analyses in the framework of a semiclassical transport theory \cite{20_PhysRevLett.115.027006, 26_PhysRevB.90.144516}, shows the decrease in the electron and hole concentrations at the transition point by several times. It may reflect the overall instability of electronic structure and can be directly related to the bandwidth reductions observed in iron-based superconductors series \cite{PhysRevX.4.031041, PhysRevLett.115.256403}.
In conclusion, our results demonstrate that a tetragonal to an orthorhombic transition cause an emergence of the highly mobile electrons originating to the local regions of the Fermi surface.
\begin{acknowledgments}
This work was supported in part from the Ministry of Education and Science of the Russian Federation in the framework of Increase Competitiveness Program of NUST 'MISiS' (№ К2-2015-075 and № K4-2015-020) and by Act 211 of the Government of Russian Federation, agreement № 02.A03.21.0006. We acknowledge support from Russian Foundation for Basic Research Grants № 14-02-00111, 14-02-00245, 14-02-92693, 15-03-99628A, 15-52-45037, ofi-m-16-29-03266, 16-02-00021 and 16-03-00463.
\end{acknowledgments}
\bibliographystyle{apsrev4-1}
|
1,116,691,500,467 | arxiv | \section{Introduction}
\label{sec:intro}
\begin{figure
\centering
\includegraphics[width=0.5\textwidth]{feature1.png}
\caption{\label{fig:feature} An illustration of the feature scattering mechanism. The classical inflationary trajectory is featureless. However, because of the quantum fluctuation of the isocurvature direction, hidden features may be encountered. At such an encounter, the inflationary trajectory gets scattered, the inflaton losses kinetic energy and thus isocurvature fluctuation converts to curvature fluctuation. }
\end{figure}
In the recent years, observations of the cosmic microwave background (CMB) radiation fluctuations by the {\it Wilkinson Microwave Anisotropy Probe} ({\it WMAP}) and {\it Planck} satellite have led to a precise measurement of temperature fluctuations on the sky from the largest scales down to arcmin scales \cite{Hinshaw13,Planck16}. The temperature anisotropy is found to be highly Gaussian and ``statistically isotropic'' in the sense that nearly all statistical proprieties of the temperature anisotropy can be described by the angular power spectrum $C^{TT}_{\ell}$~\cite{Planck23}. However, it was found since {\it WMAP} 1-year data that there is a deep cold spot ($\Delta T \simeq -120\,$K) in the southern Galactic hemisphere along the direction ($l=209^{\rm o},\,b=-57^{\rm o}$) with angular radius $\theta \simeq 10^{\rm o}$~\cite{Vielva04,Cruz07}, which is further confirmed by {\it Planck} nominal mission data~\cite{Planck23}. The cold spot is highly non-Gaussian in the sense that the probability of the cold spot existing in the statistical isotropic Gaussian universe is less than $0.1$ percent~\cite{Gurzadyan14}.
Since then, the cold spot feature in the CMB map has invoked many observational and theoretical investigations. Initially, it was suggested that the unsubtracted foreground contamination might be responsible for the apparent non-Gaussian features~\cite{Chiang06,Tojeiro06}, but later studies~\cite{Cruz06,Planck23} showed that the significance of cold spot is not affected by Galactic residues in the region of the spot. It was also proposed that a spherically symmetric void with radius $\sim 300\,$Mpc at redshift $z=1$ can produce a large and deep CMB cold spot through the late-time integrated Sachs-Wolfe effect~(ISW)~\cite{Inoue06} (also known as the Rees-Sciama effect~\cite{Rees68}). Later studies~\cite{Szapudi14,Finelli14} with the galaxy survey data~\cite{Kovacs14} did find such a supervoid of size $r\simeq 195\,$Mpc with density contrast $\delta_{0}\simeq -0.1$ at redshift $z=0.16$ align with the cold spot direction. However, more detailed following-up studies~\cite{Nadathur14,Zibin14} showed that the Rees-Sciama effect produced by such a void is several orders of magnitude lower than the linear ISW effect therefore is not able to account for the observed feature.
The interesting non-Gaussian feature of cold-spot also invokes theorists to investigate the plausible explanation from the early universe. By considering various cosmological defects in the early universe, Refs.~\cite{Cruz07-sci,Cruz08} proposed that a cosmic ``texture'' (i.e. a concentration of stress-energy and a time-varying gravitational potential due to the symmetry-breaking phase transition) can generate hot and cold spots on the last-scattering surface, with the fundamental symmetry-breaking scale found to be $\phi_{0}\sim 10^{15}\,$GeV. However, by applying the Bayesian method to {\it WMAP} full-sky data, Ref.~\cite{Feeney12} did not find strong evidence of the texture model, neither completely rule out the possibility (at 95\% confidence level). It was also proposed that cosmic bubble collision, predicted by eternal inflation theories, can induce the density perturbation between our bubble and others, which can give arise to the localized features in the CMB~\cite{Aguirre07,Gurzadyan13}. But more detail data analysis~\cite{Feeney13} showed that the expected number of bubble collision is too few to account for the features in the CMB. Alternatively, a cold spot may follow from a different trajectory during multi-stream inflation~\cite{Li:2009sp, Afshordi:2010wn} or modulated reheating after multi-field inflation~\cite{Sanchez14}.
The above physical or astronomical interpretations of cold spot either fail at some level, or require fine tuning or exotic scenarios of the early universe. Economically, some cosmologists would prefer to interpret the cold spot merely as a ``$3\sigma$'' statistical fluke. In this paper, we will provide a natural and physically plausible explanation of the cold spot, through multiple-field inflation. If the inflationary trajectory is scattered by a feature hidden in the isocurvature direction, the inflaton loses some energy and thus inflation tends to be longer. Therefore, it is possible that only a small portion of the sky hits the feature due to stochastic fluctuations, then that local patch of the universe experiences longer period of inflation and thus produces a cold spot. The mechanism is illustrated in Fig.~\ref{fig:feature}.
This paper is organized as follows. In Section~\ref{sec:cold-spot-from}, we provide an explicit example of feature scattering, from massless isocurvature directions. Our predictions are compared with the measurement of the cold-spot in {\it Planck}'s {\tt SMICA} map. In Section~\ref{sec:mass-isoc-direct}, we consider isocurvature directions with mass $m\sim H$. We conclude in Section~\ref{sec:concl-disc} and discuss possible future directions. Throughout the paper, the unit ${\rm M}_{\rm pl} = 1/\sqrt{8 \pi G} = 1$ is used unless otherwise stated.
\section{Cold spot from feature scattering}
\label{sec:cold-spot-from}
\subsection{Potential choice and method of calculation}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.53\textwidth]{neg_kloss}
\hspace{0.03\textwidth}
\includegraphics[width=0.4\textwidth]{neg_traj}
\caption{\label{fig:neg_traj} Left panel: $\partial_N \phi$ as a function of $N$. The red dashed line is for without feature scattering, and the blue solid line is with feature scattering. Note that the $\phi$ field loses its kinetic energy very sharply at the 4-th e-fold (where the feature scattering is happening), and the kinetic energy slowly recovers after of order 1 e-fold. Right panel: The kinetic energy $K(\phi)/H_*^2 = (\partial_N \phi)^2$ and $K(\chi)/H_*^2 = (\partial_N \chi)^2$. The color on the plot denotes e-folding number. One finds that about 0.1\% of the $\phi$ field kinetic energy transfers to $\chi$. Here we have taken $\chi_* = H_* / (2\pi)$ right before hitting the feature.}
\end{figure*}
\begin{figure*}[htbp]
\centering
\centerline{
\includegraphics[width=0.35\textwidth]{neg_chi_DT}
\includegraphics[width=0.35\textwidth]{osc_chi_DT}
\includegraphics[width=0.35\textwidth]{pos_chi_DT}}
\caption{\label{fig:neg_chi_DT} The temperature fluctuation as a function of $\chi_* - \chi_0$. The left, middle and right panels are for parameters of Benchmark 1,~2,~3 respectively.}
\end{figure*}
\begin{figure*}[htbp]
\centering
\centerline{
\includegraphics[width=0.45\textwidth]{neg_theta_DT}
\includegraphics[width=0.45\textwidth]{neg_theta_DT_eg}}
\centerline{\includegraphics[width=0.45\textwidth]{osc_theta_DT}
\includegraphics[width=0.45\textwidth]{osc_theta_DT_eg}}
\centerline{\includegraphics[width=0.45\textwidth]{pos_theta_DT}
\includegraphics[width=0.45\textwidth]{pos_theta_DT_eg}}
\caption{\label{fig:neg_theta_DT} Left column: The cold spot profile (we have rotated the map such that $\theta=0$ corresponds to the center of the cold spot). The gray bars are 1000 simulations of fluctuations of the feature model. The green bars are simulation of isotropic Gaussian fluctuations (5000 simulated maps). Right column: five best-fitting cold spot profiles from the Benchmark models. The red dot is the actual cold spot from the {\it Planck} {\tt SMICA} map for both panels. The three rows from top to bottom are the Benchmark models 1,~2,~3 respectively. Here the Sachs-Wolfe approximation is used when numerically fitting the bubble profile. The Sachs-Wolfe approximation works well only on large scales. Thus the fitting of the first two points with simulation may not be accurate enough. But one can see that the error bars of those two points from simulation are also large so that the inaccuracy should not affect the physical result significantly.}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=1\textwidth]{neg_eg}
\includegraphics[width=1\textwidth]{osc_eg}
\includegraphics[width=\textwidth]{pos_eg}
\caption{\label{fig:neg_eg} (From top to bottom.) A selection of the best-fitting maps for Benchmark 1,~2~3 respectively. The unit on the plot legend is $\mu$K.}
\end{figure*}
In this section, we provide an explicit example of feature scattering to illustrate the physical mechanism. The generalization to other inflationary potentials should be straightforward. Consider inflation with two fields, both have standard kinetic term, and the potential is given by
\begin{eqnarray}
\label{eq:pot}
V &=& V_\mathrm{sr}(\phi) + \delta V~, \nonumber \\
\delta V &=& A \exp \left[ -\frac{(\phi-\phi_0)^2}{2\sigma_\phi^2} - \frac{(\chi-\chi_0)^2}{2\sigma_\chi^2}\right] ~.
\end{eqnarray}
The classical initial trajectory is chosen to be on $\phi$ direction, i.e. $\chi=0$.
Here $V_\mathrm{sr}(\phi)$ is the slow-roll part of the potential. Our analysis is not sensitive in the form of $V_\mathrm{sr}(\phi)$, as long as it successfully drives slow-roll inflation. For illustration purpose, we choose a small field potential for $V_\mathrm{sr}(\phi)$:
\begin{align}
V_\mathrm{sr}(\phi) = V_0 \left( 1 - \frac{\mu^4}{\phi^4} \right)~.
\end{align}
Such a potential can be derived from brane dynamics of string theory \cite{Dvali:1998pa}. We will nevertheless not import anything other than this potential from string theory, but consider it as a phenomenological example.
To fit the observations, we will take $\mu=0.01$ and $V_0 \simeq 4.89 \times 10^{-14}$ \cite{Ma:2013xma}. At the horizon crossing scale of the cold spot, the inflaton field is $\phi_* \simeq 0.153$ for $N_{\ast} \simeq 4$ (counting from the start of the observable stage of inflation), and the Hubble parameter is $H_* \simeq 1.28 \times 10^{-7}$. The feature $\delta V$ is put at $\phi_0 = \phi_*$.
In the isocurvature direction, the initial value of the field $\chi_*$ has no classical preferred value. We set $\langle \chi_* \rangle = 0$ (one can always shift the $\chi_0$ parameter to satisfy this requirement if needed). The quantum fluctuation of $\chi_*(\mathbf{x})$ will be crucial in the analysis. Following the cosmic perturbation theory, $\chi_*(\mathbf{x})$ has a scale invariant power spectrum in Fourier space and we normalize the power spectrum to be $P_\chi = [H/(2\pi)]^2$. In position space, $\chi_*(\mathbf{x})$ behaves as a Gaussian random field. As we will show, the nonlinear mapping from $\chi_*(\mathbf{x})$ to $\Delta T(\mathbf{x})$ generates the cold spot on the CMB.
To explore the parameter space of $\delta V$, we will choose three sets of parameters as three illustrative examples of feature scattering.
\begin{itemize}
\item Benchmark 1: negative short feature. We set $A = -1.1 \times 10^{-5} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 10 H_* / (2\pi)$, $\sigma_\chi = 0.7 H_* / (2\pi)$ and $\chi_0 = 4 H_* / (2\pi)$. The procedure of the calculation is also introduced in this subsection.
\item Benchmark 2: negative long feature. We set $A = -1.4 \times 10^{-5} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 50 H_* / (2\pi)$, $\sigma_\chi = 0.7 H_* / (2\pi)$ and $\chi_0 = 5.2 H_* / (2\pi)$. The differences from Benchmark 1 are discussed.
\item Benchmark 3: positive feature. We set $A = 1.0 \times 10^{-4} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 50 H_* / (2\pi)$, $\sigma_\chi = 0.5 H_* / (2\pi)$ and $\chi_0 = 3.5 H_* / (2\pi)$. The differences from Benchmarks 1 and 2 are discussed.
\end{itemize}
Note that in the above benchmarks, the randomness in the potential needs not to be large. In fact, $\delta V / V \sim 10^{-5}$ is already significant enough to bring observable difference. This is because, the isocurvature fluctuation control whether the inflation trajectory hit the feature. And once the trajectory hits the feature, the change of curvature perturbation will be order $\zeta \sim \delta V / V \sim 10^{-5}$. As a result, our mechanism is extremely sensitive to small features in the potential.
The $\delta N$ formalism \cite{Starobinsky:1986fxa, Sasaki:1995aw, Lyth:2004gb} will be used to investigate the cosmological perturbations from the feature scattering. The $\delta N$ formalism uses the observation that different Hubble patches during inflation can be approximated as different local FRW universes. Thus the cosmological observables can be calculated by exploring those local FRW universes. Especially, the curvature perturbation in the uniform energy density slice can be calculated by \footnote{See the appendix of \cite{Chen:2007gd} for clarification of sign conventions.}
\begin{align}
\zeta = \delta N~,
\end{align}
where $\delta N$ is the difference of e-folding number between an initially-flat slice and a finally-uniform energy density slice. The curvature perturbation $\zeta$ is then converted to the CMB temperature anisotropy following the standard theory of CMB. In this paper we will use the Sachs-Wolfe approximation $\delta T/T \simeq - \zeta/5 = - \delta N/5$. Intuitively, the relation can be understood as:
\begin{align}
& \mbox{Inflaton scattered by feature} \nonumber \\
& \rightarrow \mbox{smaller kinetic energy in $\phi$ direction} \nonumber \\
& \rightarrow \mbox{ larger $\delta N$} \nonumber \\
& \rightarrow \mbox{later reheating} \nonumber \\
& \rightarrow \mbox{less dilution of energy} \nonumber
\\
& \rightarrow \mbox{higher energy density} \nonumber \\
& \rightarrow \mbox{deeper gravitational potential} \nonumber \\
& \rightarrow \mbox{lower CMB temperature}~.
\end{align}
More precise relation between $\delta T/T$ and $\zeta$ can be studied by solving the Boltzmann equations at recombination.
Note that $\delta N$ has two sources: the quantum fluctuation of $\phi$ and the quantum fluctuation of $\chi$. Those independent sources can be studied independently, and they needs to be added together to make the total curvature fluctuation. In the numerical calculation, we will focus on investigating $\delta N$ as a function of $\chi$ but we will add back a contribution of Gaussian random field to account for the contribution from $\phi$ at the last map-making stage.
The angular size of the cold spot is around 10 degrees. Converting to horizon crossing time of comoving wave number, this corresponds to $\ell \sim 20$. Note that in the observed CMB temperature power spectrum, there is indeed a dip at $\ell \sim 20$. It will be interesting to do a combined analysis on both signals, from feature scattering.
In the following subsections, we will explore different parameter space of Eq.~(\eqref{eq:pot}) with three benchmarks, and calculate the e-folding number as functions of the position of isocurvature field $\delta N = \delta N(\chi)$.
\subsection{Benchmark 1: Negative short feature}
In this subsection, we consider a negative short feature in the potential. We set $A = -1.1 \times 10^{-5} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 10 H_* / (2\pi)$, $\sigma_\chi = 0.7 H_* / (2\pi)$ and $\chi_0 = 4 H_* / (2\pi)$. The size of the feature in the $\phi$ direction is relatively small (compared to the case of Benchmark 2). As a result, the $\sigma$ field does not have enough time to oscillate inside the negative feature.
Before getting to the implications to the late universe, it is intuitive to check what was happening during inflation.
As an example, we consider the $\chi$ field value when hitting the feature to be $\chi_* - \chi_0 = H_*/(2\pi)$. The time evolution of $\partial_N \phi$ is plotted on the left panel of Fig.~\ref{fig:neg_traj}. One can find from the plot that, at $N_* \simeq 4$ (where feature scattering happens)
\begin{align}
\left| \frac{{\rm d}\phi}{{\rm d} N} \right| \sim 4 \times 10^{-4}~.
\end{align}
When the inflationary trajectory hits the feature, there is a change in the velocity of the inflaton $\phi$. At first, $|{\rm d}\phi/{\rm d}N|$ increase because $\phi$ falls into a potential well. However, two other effects immediately follow: First, the obtained energy is returned because $\phi$ has to climb out of the potential well; second, the potential well can be considered as a scattering center, such that a similar amount of energy is transferred to the $\chi$ direction (as evidenced in the right panel of Fig.~\ref{fig:neg_traj}). As a result, there is a overall loss of velocity of the inflaton, with the value read from the plot
\begin{align}
\delta \left| \frac{{\rm d}\phi}{{\rm d}N} \right| \sim - 4 \times 10^{-7} \sim
- 10^{-3} \left| \frac{{\rm d}\phi}{{\rm d}N} \right|~.
\end{align}
In words, the inflaton field has lost $0.1\%$ of its kinetic energy because of hitting the feature in the potential \footnote{Here we do not need to consider the change of potential energy before/after the feature scattering, because we only need to compare the change of energy before and after a very sharp scattering process, and the change of potential energy is negligible.}. Note that the inflaton always loses its kinetic energy because of scattering off a feature. Thus a cold spot instead of a hot spot is predicted \footnote{When the potential is positive, the inflaton actually gains kinetic energy at the very beginning, when it falls into the potential well. However, the gained energy is returned after a time interval that is much smaller than Hubble time. Thus the effect of gaining the kinetic energy is negligible when we integrate to obtain the e-folding number. On the other hand, the loss of kinetic energy through feature scattering needs order of 1 e-fold to recover, and thus it is the dominate effect.}.
Note that the loss of kinetic energy takes of order one e-fold to recover, because this is the time scale to reach the inflationary attractor solution. To be more precise, as shown in the left panel of Fig.~\ref{fig:neg_traj}, it takes about roughly $0.2$ e-fold for the inflaton to recover its kinetic energy. As a result \footnote{The $\delta N$ here is from the quantum fluctuation of $\chi$. The part from $\phi$ will be added at the map-making stage.},
\begin{align}
\delta N = \frac{\delta\phi}{\left| \frac{{\rm d}\phi}{{\rm d}N} \right| + \delta \left| \frac{{\rm d}\phi}{{\rm d}N} \right|}
= \frac{0.2 \times \left| \frac{{\rm d}\phi}{{\rm d}N} \right|}
{\left| \frac{{\rm d}\phi}{{\rm d}N} \right| + \delta \left| \frac{{\rm d}\phi}{{\rm d}N} \right|}
\sim 2 \times 10^{-4}~.
\end{align}
From the Sachs-Wolfe approximation, the temperature fluctuation $\delta T/T \simeq -\zeta/5 = -\delta N/5$. Thus we get $\delta T\simeq -100 \mu$K.
Carrying on the above analysis for general values of $\chi_* - \chi_0$, numerically the temperature fluctuation as a function of $\chi_* - \chi_0$ is plotted in the left panel of Fig.~\ref{fig:neg_chi_DT}. We then simulate 1000 maps and rotate those maps such that they center at the cold spot. In the left panel of the top row of Fig.~\ref{fig:neg_theta_DT}, we bin the pixels of the map as a function of $\theta$, and then plot the 0.5, 0.16 and 0.84 quantiles (corresponding to the central value and 1$\sigma$ uncertainty if the probability distribution were Gaussian) of the binned pixel temperature. In the right panel of the top row of Fig.~\ref{fig:neg_theta_DT}, we plot the top five best-fitting cold spot profiles. Three best-fitting examples are given in the top row of Fig.~\ref{fig:neg_eg}.
\subsection{Benchmark 2: Negative long feature}
In this subsection we consider a negative long feature. Inside a ``long'' feature, there is enough time for the $\chi$ field to oscillate. We set $A = -1.4 \times 10^{-5} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 50 H_* / (2\pi)$, $\sigma_\chi = 0.7 H_* / (2\pi)$ and $\chi_0 = 5.2 H_* / (2\pi)$. Note that the feature size in the $\phi$ direction is 5 times longer than that of Benchmark 1.
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.55\textwidth]{osc_kloss}
\hspace{0.03\textwidth}
\includegraphics[width=0.4\textwidth]{osc_traj}
\caption{\label{fig:osc_traj} The same as Fig.~\ref{fig:neg_traj} but for Benchmark model 2. Note that the feature in the inflationary potential is long enough to hold a few oscillations in this case.}
\end{figure*}
Finer structures develop in the case of long feature. If we have chosen an initial value $\chi_* - \chi_0 = H_*/(2\pi)$, we find similar behavior to Fig.~\ref{fig:neg_traj}. However, for some other initial values, for example, $\chi_* - \chi_0 = 1.4 H_*/(2\pi)$, the $\chi$ field oscillates inside the feature. The situation is illustrated in Fig.~\ref{fig:osc_traj}. As a result, some amount of $\chi$ kinetic energy is returned to $\phi$. For some special values, almost all kinetic energy is returned. Thus oscillatory patterns develop, as shown in the middle panel of Fig.~\ref{fig:neg_chi_DT}.
The oscillations in $\delta T(\chi)$ leads to ring objects in $\delta T(\mathbf{x})$. The cold spot in this parameter space is not completely cold, but instead has nested rings of cold and hot patterns. There is no evidence of such patterns in the actual CMB cold spot. There has been a debate if there are other ring patterns in the sky \cite{Gurzadyan:2011ac}. The Benchmark 2 parameters provide an explanation for such rings (though the shape is not perfectly spherical). However, it is shown that the appearance of such rings is due to an inappropriately chosen power spectrum, and thus not actually in the sky \cite{Eriksen:2011fi}. Thus we will not tune the parameters to fit such features here.
The general and best-fitting cold-spot profiles, and sample spots are plotted in the middle row of Figs.~\ref{fig:neg_theta_DT} and \ref{fig:neg_eg}, respectively.
\subsection{Benchmark 3: Positive feature}
In this subsection we consider positive feature scattering. We set $A = 1.0 \times 10^{-4} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 50 H_* / (2\pi)$, $\sigma_\chi = 0.5 H_* / (2\pi)$ and $\chi_0 = 3.5 H_* / (2\pi)$. With positive potential, the $\chi$ direction has a run-away solution and can never be bounded. So there is no oscillation in this case.
The $\delta T(\chi)$ dependence is not as sharp as the previous two benchmarks. Thus the cold spot does not have as clear a boundary as those previous cases. Because of the not-so-sharp $\delta T(\chi)$ dependence, to get a cold enough center of the spot, we have to increase $A$. As a result, the cold spots generated by Benchmark 3 typically (though not always) have very cold tinny cores.
The general and best-fitting cold-spot profiles, and sample spots are plotted in the bottom row of Figs.~\ref{fig:neg_theta_DT} and \ref{fig:neg_eg}, respectively.
Considering a very sharp and cold core is formed for positive features, it is unlikely that such features can fit the realistic cold spot. But considering that the sharpness of the cold core depends on model parameters. The cold core may appear at $\ell > 2000$ scales and free streaming may erase the sharpness.
\section{Massive isocurvature directions}
\label{sec:mass-isoc-direct}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.45\textwidth]{mas_theta_DT}
\includegraphics[width=0.45\textwidth]{mas_theta_DT_eg} \caption{\label{fig:mas_theta_DT} The same as Fig.~\ref{fig:neg_theta_DT} but for massive isocurvature model $m_{\chi}=0.9H_{\ast}$.}
\end{figure*}
\begin{figure*}[htbp]
\centering
\includegraphics[width=\textwidth]{mas_eg}
\caption{\label{fig:mas_eg} The same as Fig.~\ref{fig:neg_eg} but for model $m_\chi = 0.9 H_*$.}
\end{figure*}
The feature scattering not only applies to massless isocurvature directions, but also to marginally massive ones. As long as the mass of the isocurvature direction is not much greater than $H$, the energy scale for inflationary fluctuations, the isocurvature direction can still be excited to explore its field space. During inflation, fields with $m_\chi \sim H$ arise naturally \cite{Chen:2009we, Chen:2009zp, Baumann:2011nk, Chen:2012ge}. Thus it is worthwhile to investigate feature scattering of such marginally massive fields, with an additional mass term in the potential
\begin{align}
V_\mathrm{mass}(\chi) = \frac{1}{2} m_\chi^2 \chi^2~.
\end{align}
Note that the process of feature scattering happens on a time scale much shorter than Hubble. Thus as long as $m_\chi$ is not much great than $H_*$, $m_\chi$ does not enter the dynamics of feature scattering. However,
the spectral index of a massive field is significantly different from a scale invariant spectrum, with spectral index
\begin{align}
n_\chi-1 = \mathrm{Re} \left( \frac{3}{2} - \sqrt{\frac{9}{4}-\frac{m^2}{H^2}} \right)~.
\end{align}
Thus the mass of the field controls how many e-folds can the isocurvature direction randomly travel. For example, when $m = 0.9 H_*$, the $\chi$ field has a spectrum $n_\chi - 1 = 0.3$.
In the numerical example, we set $m = 0.9 H_*$, $A = -1.1 \times 10^{-5} V_0 \mu^4 / \phi_*^4$, $\sigma_\phi = 10 H_* / (2\pi)$, $\sigma_\chi = 0.7 H_* / (2\pi)$ and $\chi_0 = 4.4 H_* / (2\pi)$. In other words, we have only tuned $\chi_0$ to be slightly greater, and consider a massive $\chi$ field. Otherwise the parameters coincide with the Benchmark 1 of massless case.
The general and best-fitting cold-spot profiles, and sample spots are plotted in Figs.~\ref{fig:mas_theta_DT} and \ref{fig:mas_eg}, respectively.
As one can see from Fig.~\ref{fig:mas_eg}, there are more small-scale structures compared to the case of massless Benchmark 1.
\section{Conclusion and discussion}
\label{sec:concl-disc}
To conclude, we propose a scattering mechanism of inflaton due to the features in the inflation potential, during which the isocurvature fluctuations are converted into curvature fluctuations. The curvature fluctuations direction loses kinetic energy due to the scatters in isocurvature direction, therefore the number of e-folds becomes larger in some region of the universe. We find that the cold spot can be well explained by such a mechanism, and the spot profile reasonably fits the CMB cold spot without fine tuning of the inflationary parameters. Before ending up the paper, we would like to mention a few future directions:
\begin{itemize}
\item Beyond the Sachs-Wolfe approximation. Currently we did not use the full Boltzmann code to calculate the CMB transfer function. The Sachs-Wolfe approximation works well for exploring the coarse-grained cold-spot profile but is not enough for probing the fine structures inside the spot. One can use the full radiation transfer function to carry on a more detailed analysis in the future.
\item Verifying the approximations of the $\delta N$ formalism. The $\delta N$ formalism assumes that the horizon-crossing amplitude of the $\phi$ and $\chi$ quantum fluctuations are Gaussian and uncorrelated. In the case of feature scattering, this is not rigorously true because of the turning of trajectory. However, note that the transfer of the inflaton kinetic energy is tinny (0.1\% in the studied example), the correction from sub-horizon physics should be suppressed by this small fraction.
Having that said, it remains interesting to see if the calculation from $\delta N$ formalism can be verified by first principle calculation. However, such a calculation is challenging. The first principle calculation of cosmological perturbations is known as the in-in formalism. But in the in-in formalism, the primary calculatables are the correlation functions. In other words, one starts from the Gaussian two point correlation function and studies small departure from that. But the cold spot is highly non-Gaussian and localized object, and thus is not easily captured by the correlation functions from the in-in formalism.
\item Extra species/symmetry point (ESP) \cite{Kofman:2004yc, Battefeld:2011yj} as scattering centers. In our current examples, the inflaton kinetic energy is lost into the collective motion of the isocurvature direction. It may also be possible that the kinetic energy of the inflaton is lost into hidden ESPs in the isocurvature direction. It is interesting to explore such possibilities.
\item Connection between the feature scattering regime and the multi-stream regime: By carefully arranging the position of the feature, it is possible that different part of the universe follows different classical trajectories, separated by temporary domain walls. This is known as the multi-stream inflation \cite{Li:2009sp, Afshordi:2010wn}, which is also a possible explanation of the cold spot. Feature scattering and multi-stream corresponds to different regimes of the multi-field parameter space. If the features on the inflationary potential are random, we expect the feature scattering to be more typical (whereas highly random features in multi-stream inflation cause disasters and provide a constraint on multi-field inflation \cite{Duplessis:2012nb}).
\item Connection between the cold spot and the dip of CMB temperature power spectrum at $\ell \sim 20$. We argue they may come from the same origin -- feature scattering during inflation. With our current toy potential given at Eq.~(\eqref{eq:pot}), we are unable to explain this power deficit at the same time. However, a more general shaped potential may be able to explain both the deficit and the cold spot.
\item String theory model building. It is believed that string theory has a landscape of complicated vacuum structures \cite{Bousso:2000xa, Susskind:2003kw}. It would be interesting to build string landscape models for feature scattering. Especially, our mechanism assumes that $\sigma_\phi$ and $\sigma_\chi$ have value of order $H$. This assumption is made at the phenomenological level in this paper. Nevertheless, it is interesting to search for theoretical motivations for such a potential. For inflation to work, the inflationary perturbations, in the sense of effective field theory, should have enough range of validity. Especially, the UV cutoff of the theory should be at least of order $H$. In the marginal case where the UV cutoff of the effective field theory is just of order $H$, new features are expected when the field rolls a distance of order $H$. In addition, inspired by quasi-single field inflation \cite{Chen:2009we, Chen:2009zp}, the mass parameters during inflation should be naturally of order $H$. Thus once there are many mass parameters in the landscape, such as some mass matrix, variations will be introduced in the potential with field range of order $H$. We leave detailed investigation of those possibilities to future works.
\item Careful model comparison between our feature scattering mechanism and the $\Lambda$CDM base model. We have 5 parameters in our illustrative model. The parameters are chosen to be the convenient ones for inflation model building, but not optimized to fit the data as a phenomenological model. It remains interesting to propose a more economical model with fewer number of parameters, and perform model comparison between our mechanism and standard inflation model.
\end{itemize}
\section*{Acknowledgments}
We would like to thank the helpful discussion with Richard Battye and Clive Dickinson. YW was supported by Grant HKUST4/CRF/13G issued by the Research Grants Council (RGC) of Hong Kong, a Starting Grant of the European Research Council (ERC STG grant 279617), and the Stephen Hawking Advanced Fellowship. YZM acknowledges support from an ERC Starting Grant (no.~307209).
|
1,116,691,500,468 | arxiv | \section{Introduction}
Egocentric action anticipation \cite{damen2020epic} is receiving increasing attention recently, which aims to anticipate what the subject to do next based on the recordings from egocentric cameras. Different from the third-person action anticipation, it actually records what the subject observes and performs high-level perception of in the brain. Associating past sensory input with future actions is a fundamental step for understanding human cognition mechanisms.
It is a challenging problem since future events are highly uncertain, and there exist several possible diverse predictions based on the observation of the past \cite{furnari2018leveraging}. It is difficult to establish an explicit model between the past and the future, as the sensory input (e.g. visual observation) may have asynchronous casual effect on the next action and the future is of multi-modality in nature. Directly arranging the sensory input as a sequential order and feeding it to some conventional temporal modelling architectures (e.g. RNN) may tend to ignore the effects contributed by some relatively old experiences. In our submission, we adopted the Transformer to dynamically fuse information across time, modalities, and \textit{verb} \& \textit{noun} branches.
\begin{figure}[tp]
\centering
\includegraphics[width=\linewidth]{diagram.pdf}
\caption{Overview of our hierarchical Transformer-based fusion framework. Our framework is a cascade of several singular blocks. In each block, the temporal self-attention (TSA) module aims to model long-range temporal information, capturing asynchronous effect for the action anticipation. The cross-modality attention (CMA) module aims to fuse information across modalities via Transformer-based attention mechanism. The symbiotic attention (SA) module serves for the mutual interaction between \textit{verb} and \textit{noun} branches with the goal of benefiting each other.}
\label{fig:framework}
\end{figure}
On the other hand, each label of egocentric actions in Epic-Kitchen is formulated as a \{\textit{verb, noun}\} pair. The combination of different \textit{verbs} and \textit{nouns} would lead to thousands of candidates~\cite{furnari2018leveraging}. Similar to the ``long-tailed'' distribution in many real-world applications, the majority of actions only occur very few times. Such imbalanced distribution would decrease the generalization capability of trained model on rare classes. In this report, we adopted a state-of-the-art method, Equalization Loss~\cite{tan2020equalization}, to handle the long-tailed distribution problem.
\section{Methods}
We directly adopted the multi-modality feature provided by RULSTM~\cite{damen2020rescaling, furnari2020rolling}, which consists of features from three modalities, rgb $F_{rgb}$, flow $F_{flow}$, and object $F_{obj}$. $F_{rgb}$ and $F_{flow}$ were extracted from pretrained TSN models~\cite{wang2016temporal} on the action recognition task. $F_{obj}$ was formed by the object probability score predicted by pretrained FasterRCNN model~\cite{ren2015faster}. Each input $F\in\mathbb{R}^{N\times D_f}$ denotes the feature vector with a dimensionality of $D_f$ extracted from $N$ frames, (3.5-1)s before the beginning of the actions
Our key idea is to exploit Transformer based attention mechanisms to fuse information from temporal dimension, different modalities, as well as verb/noun branches. The overall framework is illustrated in Fig.~\ref{fig:framework} and the details of each basic component are given below.
\subsection{Temporal Self-Attention (TSA)}
Instead of applying conventional network architectures for temporally modelling like LSTM/GRU, we applied Transformer~\cite{vaswani2017attention} to better model the long-range temporal relationship by attention mechanisms. The input feature vector is added by sinusoidal positional embedding to incorporate the positional information. It transforms the input feature to a set of queries ($\mathbf{Q}$), keys ($\mathbf{K}$) and values ($\mathbf{V}$) via linear projection. Subsequently, the attention weights computed from the normalized dot product of $\mathbf{Q}$ and $\mathbf{K}$ are applied to aggregate values, as formulated in Eq.~\ref{eq:2}. It subsequently applies add \& norm operations to enable residual connections, as formulated in Eq.~\ref{eq:3}. Subsequently, non-linear feedforward MLPs followed by add \& norm residual connections are applied, as in Eq.~\ref{eq:4}.
\begin{equation}
\small
\mathbf{Q}=\mathbf{F}\mathbf{W}^q, \mathbf{K}=\mathbf{F}\mathbf{W}^k, \mathbf{V}=\mathbf{F}\mathbf{W}^v
\label{eq:1}
\end{equation}
where $\mathbf{W}^q\in\mathbb{R}^{D_f\times D_q}$, $\mathbf{W}^k\in\mathbb{R}^{D_f\times D_k}$, $\mathbf{W}^v\in\mathbb{R}^{D_f\times D_v}$ denote corresponding linear projection matrices.
\begin{equation}
\small
\mathbf{A} = softmax\left (\frac{\mathbf{Q}\mathbf{K}^T}{\sqrt{D_k}}\right )\mathbf{V}
\label{eq:2}
\end{equation}
\begin{equation}
\small
\mathbf{F}^{'} = layer\_norm(\mathbf{A} + \mathbf{F}^{in})
\label{eq:3}
\end{equation}
\begin{equation}
\small
\mathbf{F}^{out} = layer\_norm(\textbf{MLP}(\mathbf{F}') + \mathbf{F}^{'})
\label{eq:4}
\end{equation}
\subsection{Cross-Modality Attention (CMA)}
To make use of the complementary information encoded in different modalities, we introduced a cross-modality attention (CMA) mechanism, which is expected to capture asynchronous yet relevant information across modalities. Inspired by the fusion method proposed in \cite{prakash2021multi}, we concatenate $F_{rgb}$ $F_{flow}$ $F_{obj}$ into a feature with a shape of $N\times \sum D_f$, and then apply the CMA module to aggregate features across time.
\subsection{Symbiotic Attention (SA)}
Similar to previous action recognition/anticipation works, we utilized two branches to predict \textit{verb} and \textit{noun} separately. However, it is not appropriate to consider \textit{verb} and \textit{noun} as two independent variables to be predicted by two independent branches, since they share mutual contextual information \cite{wang2020symbiotic}. The awareness of the next active object provides the prior probability for predicting the next verb, whereas predicting the next verb would help recognize the next object to be manipulated. Therefore, we incorporated another Transformer module for the interaction between \textit{verb} and \textit{noun} branches. This module, referred to as Symbiotic Attention (SA) module, applied Transformer network to process concatenated feature input with a shape of $2N\times \sum D_f$.
\subsection{Cascaded Architecture}
Based on the TSA, CMA, and SA modules, the illustration of our network architecture is given in Fig.~\ref{fig:framework}. It firstly processes the input of each modality by their corresponding TSA modules. Subsequently, the CMA modules in both branches fuse features across multiple modalities, followed by a SA module performing interactions between both branches. Finally, the features extracted from two branches are concatenated together and fed into another TSA module to predict the action.
We developed a cascaded architecture with the repetition of the same block, whereas the output of each block is extracted for prediction. In practice, the block number n is set as 2.
\subsection{Equalization Loss}
To deal with the long tailed distribution, we adopted the Equalization Loss proposed in \cite{tan2020equalization}. It proposed a simple yet effective loss aimed at protecting the learning of rare classes by randomly neglecting the updating of rare classes when the target is a majority class. The loss function is modified from cross-entropy loss, and its formulation is shown as below,
\begin{equation} \small
L_{SEQL}=-\sum_{j=1}^{c}y_{j}\log(\tilde{p}_{j})
\end{equation}
\begin{equation} \small
\tilde{p}_{j}=\frac{e^{z_{j}}}{\sum_{k=1}^{c}\tilde{w}_{k}e^{z_{k}}}
\end{equation}
\begin{equation} \small
\tilde{w}_{k}=1-\beta T_{\lambda}(y_{k})(1-y_{k})
\end{equation}
where $\beta$ is random binary variable with a probability of $\gamma$ to be 1 and otherwise 0. $T_{\lambda}(y_{k})$ is a threshold function determining whether $y_{k}$ is a majority class by predefined occurrence frequency threshold.
\begin{table*}[h]
\small
\centering
\caption{Results of Ablation Studies on Validation Set.}
\begin{tabular}{@{}lccccccccc@{}}
\toprule
\multicolumn{1}{c}{\multirow{2}[0]{*}{Method}} & \multicolumn{3}{c}{Overall (\%)} & \multicolumn{3}{c}{Unseen (\%)} & \multicolumn{3}{c}{Tail (\%)} \\ \cmidrule(l){2-10}
& Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action \\
\midrule
RULTSM\cite{damen2020rescaling} & 27.76 & 30.76 & 14.04 & 28.78 & \underline{27.22} & \textbf{14.15} & 19.77 & 22.02 & 11.14 \\
\midrule
TSA-RGB & 33.23 & \underline{32.65} & 13.71 & 28.65 & 20.61 & 10.23 & 29.12 & \underline{31.41} & 13.34 \\
TSA-Flow & 24.19 & 17.02 & 6.74 & 30.61 & 15.74 & 6.01 & 19.33 & 15.46 & 5.72 \\
TSA-Obj & 25.37 & 29.51 & 9.93 & 28.39 & 22.19 & 7.06 & 21.26 & 28.09 & 9.51 \\
w/o CMA & 31.46 & 31.92 & 14.90 & \underline{34.10} & 23.47 & 10.22 & 26.37 & 30.14 & \underline{14.56} \\
w/o SA & \textbf{35.78} & 32.18 & 12.93 & 29.79 & 17.56 & 10.51 & \textbf{32.08} & 31.01 & 12.43 \\
w/o Equal & 27.65 & 31.34 & 14.16 & 27.49 & 25.25 & 12.61 & 20.92 & 25.60 & 11.98 \\
\midrule
Proposed-Single & 33.60 & 32.54 & \underline{15.05} & 33.05 & 25.43 & 11.96 & 29.04 & 31.03 & 14.39 \\
Proposed-Ensemble & \underline{35.04} & \textbf{35.49} & \textbf{16.60} & \textbf{34.64} & \textbf{27.26} & \underline{13.83} & \underline{30.08} & \textbf{33.64} & \textbf{15.53} \\
\bottomrule
\end{tabular}
\label{tab:ablation}
\end{table*}
\begin{table*}[h]
\small
\centering
\caption{Results of Testing Set on LeaderBoard.}
\begin{tabular}{@{}lccccccccc@{}}
\toprule
\multicolumn{1}{c}{\multirow{2}[0]{*}{Method}} & \multicolumn{3}{c}{Overall (\%)} & \multicolumn{3}{c}{Unseen (\%)} & \multicolumn{3}{c}{Tail (\%)} \\ \cmidrule(l){2-10}
& Verb & Noun & Action & Verb & Noun & Action & Verb & Noun & Action \\
\midrule
RULSTM-RGB & 24.69 & 26.38 & 10.45 & 17.88 & 23.16 & 9.13 & 17.32 & 16.79 & 7.39 \\
RULSTM-Flow & 21.24 & 18.12 & 7.36 & 17.27 & 18.95 & 6.86 & 13.54 & 9.44 & 4.97 \\
RULSTM-OBJ & 13.93 & 15.17 & 3.96 & 14.05 & 20.41 & 5.79 & 6.18 & 5.37 & 1.85 \\
RULSTM-Fusion & 25.25 & 26.69 & 11.19 & 19.36 & 26.87 & 9.65 & 17.56 & 15.97 & 7.92 \\
Proposed-Single & 37.13 & 30.19 & 12.44 & 29.72 & 20.87 & 10.57 & 34.53 & 28.42 & 9.74 \\
Proposed-Ensemble & 36.15 & 32.20 & 13.39 & 27.60 & 24.24 & 10.05 & 32.06 & 29.87 & 11.88 \\
\bottomrule
\end{tabular}
\label{tab:result}
\end{table*}
\section{Implementation Details}
The whole model was implemented with Pytorch and trained on a single RTX 2080 Ti GPU. The batch size was set as 128 and we applied SGD optimizer with a learning rate of 0.01 and a momentum of 0.9.
The implementation details can be found in \url{https://github.com/guxiao0822/trans_action}.
To participate in the challenge, we developed an ensemble of three trained models based on our proposed method together with the baseline {RULSTM-Fusion} to achieve performance gains from their complementary information.
\section{Results and Discussion}
Following the evaluation guideline of this challenge\footnote{\url{https://competitions.codalab.org/competitions/25925}}, the Mean Top-5 Recall Metric is used. First of all, to demonstrate the effectiveness of different modules proposed, we conducted ablation study on the validation subset with the results shown in Table~\ref{tab:ablation}. The TSA-RGB/Flow/Obj refers to the variant only applying TSA with their corresponding single-modality feature as input. w/o CMA, SA denote the variants with CMA, SA module removed respectively. w/o Equal replaces the Equalization Loss by the conventional cross-entropy loss. It can be observed that overall the complete method performs well.
For the test set, The final results of our single model and the ensemble version are given in Table~\ref{tab:result}, together with the results of the baseline method RULSTM~\cite{furnari2020rolling}. As shown in Table~\ref{tab:result}, for our single model, our method competes against the baseline methods regarding most metrics. Especially for the tail classes, a significant improvement can be observed. The ensemble of our models and RULSTM\_Fusion leads to slight improvement in terms of some metrics, especially for the result of Tail \textit{action}. It is also noteworthy that our proposed method ranked 1st for \textit{verb} in all three (sub)sets.
We noticed marginally preferable results reported by some other teams in terms of \textit{action} as shown in the Leaderboard. Future work should be targeted at further exploring the symbiotic relationship between \textit{verb} and \textit{noun} for the improvement of \textit{action} classification. Modelling the temporal transition of different actions as well as the label distribution to handle label uncertainty should also be taken into consideration.
{\small
\bibliographystyle{ieee}
|
1,116,691,500,469 | arxiv | \section{Introduction} \label{introduction}
After the remarkable discovery of Cachazo, He, and Yuan (CHY) \cite{Cachazo:2013hca, Cachazo:2013iea, Cachazo:2014xea} of general compact formulae for tree-level amplitudes as integrals over the space of punctured Riemann spheres localized over the so-called \textit{scattering equations}, it became an immediate issue how to consider fermions or supersymmetry into the formalism. Shortly after these original findings, ambitwistor strings \cite{Mason:2013sva, Berkovits:2013xba} were found to naturally give rise to the CHY formulae and provided a natural framework to consider supersymmetric generalizations of the latter. In this regard, compact amplitudes formulae with supersymmetry have been constructed in four, six, and ten-eleven dimensions in \cite{Geyer:2014fka, Geyer:2018xgb, Geyer:2019ayz} making use of twistor variables instead of the standard superspace variables.
On another line of developments, a novel formulation of the ten-dimensional massless superparticle in terms of twistor-like variables was introduced by the authors in a complementary work \cite{Sepulveda:2020kjc}. This formulation was found by looking for a first-principles description of a twistor-like construction in ten dimensions introduced by Berkovits in \cite{Berkovits:2009by}, in which a set of ``pure spinor twistor'' variables
\begin{equation}\label{ztwistors}
\mathcal{Z}^{I} = (\lambda^{\alpha}, \mu_{\alpha}, \Gamma^{m}), \ \ \bar{\mathcal{Z}}_{I} = (\bar{\mu}_{\alpha}, - \bar{\lambda}
^{\alpha}, \bar{\Gamma}^{m}),
\end{equation}
where $m=1,\ldots,10$, $\alpha=1,\ldots,16$, fulfilling
\begin{equation} \label{definingconstraints}
(\lambda \gamma^{m} \lambda) = 0, \ \ \ \lambda \mu = 0, \ \ \ (\lambda \gamma^{mn} \mu) + 4\Gamma^{m} \Gamma^{n} = 0, \ \ \ (\lambda \gamma_{m})_{\alpha} \Gamma^{m}=0
\end{equation}
were used in an attempt to generalize standard four-dimensional twistor constructions \cite{Witten:2003nn, Berkovits:2004hg, Roiban:2004vt, Britto:2004ap, Britto:2005fq} to ten dimensions, with pure spinors taking the role of higher dimensional twistors, which is a natural proposal as argued in \cite{Hughston1, Hughston2, Hughston3, Berkovits:2004bw, Boels:2009bv}. \pagebreak
In this work we will present an ambitwistor worldsheet theory based on the previously mentioned description of the superparticle \cite{Sepulveda:2020kjc}. The worldsheet theory completes the physical realization of \cite{Berkovits:2009by} when arbitrary interactions are considered, and there are resemblances with the previously mentioned models \cite{Geyer:2014fka, Geyer:2018xgb, Geyer:2019ayz}. For instance, both constructions use a twistorial representation that makes supersymmetry manifest, and as we shall see, vertex operators take similar forms. The ambitwistor string considered here is constructed by replacing time derivatives by antiholomorphic derivatives $\bar{\partial}$ and the worldline by a Riemann sphere, in accordance with the ideas of \cite{Mason:2013sva} to construct ambitwistor worldsheet actions. As in the superparticle, the worldsheet variables by definition will be required to satisfy \eqref{definingconstraints}, and the system will be subjected to a set of constraints
\begin{align}
B & \coloneqq (\lambda\gamma^{m}\bar{\lambda})\bar{\Gamma}_{m} - (\bar{\lambda}\gamma^{m}\bar{\lambda})\Gamma_{m}, \label{Bconstraint}\\[1.0ex]
J & \coloneqq \bar{\mu}_{\alpha} \lambda^{\alpha} - \bar{\lambda}^{\alpha} \mu_{\alpha} + \bar{\Gamma}^{m} \Gamma_{m} + j_{c} \label{Jconstraintintro}, \\[1.0ex]
\tilde{\mathcal{M}}^{abc} & \coloneqq (\lambda \gamma^{[a}) N^{bc]} + \frac{1}{12}(\tilde{q} \gamma^{abc} \tilde{q}) \label{splconstraint}
\end{align}
apart from the corresponding Virasoro constraint $T=0$. Here, $j_{c}$ is a quantum correction to be determined, and $a,b,c=1, \ldots, 5$ are $SU(5)$ indices. Notice that the constraints $B$ and $J$ have both been considered previously in \cite{Berkovits:2009by} and emerged naturally in the context of the superparticle, but the importance of the constraint $\tilde{\mathcal{M}}^{abc}$ -the independent
components of the super-Pauli-Lubanski three-form- to properly describe the degrees of freedom of the $D=10$ Brink-Schwarz superparticle in the twistor framework was pointed out in \cite{Sepulveda:2020kjc}. The worldsheet action so constructed would then lead to an interesting resolution to a conjecture proposed by Berkovits in \cite{Berkovits:2009by}; the worldsheet action, rather than related at first sight to the standard superstring, would be related to an ambitwistor string theory.
We construct a heterotic version of the model just introduced by coupling the system to a current algebra. Remarkably, the whole set of constraints $(B,J,T,\tilde{\mathcal{M}}^{abc})$ gives rise to an anomaly-free worldsheet model when the current algebra central charge is 16, as in the $E_{8}\times E_{8}$ or $SO(32)$ heterotic superstrings. Analogously, the Type IIB version constructed out of a simple extension of \eqref{ztwistors} and \eqref{definingconstraints} will also present a vanishing central charge.
In \cite{Sepulveda:2020kjc}, the pure spinor twistor formulation of the $D=10$ superparticle was found through a field redefinition of a superparticle model developed by Berkovits in \cite{Berkovits:1990yc}. The corresponding ambitwistor string constructed from \cite{Berkovits:1990yc} has been studied by Berkovits, Mason, and one of the authors in \cite{Berkovits:2019bbx}, where it was shown that in light-cone gauge the model is equivalent to the light-cone RNS ambitwistor string. Due to the close relation between the latter model and the ambitwistor string constructed in this work from \cite{Sepulveda:2020kjc}, we begin warming-up by finding the BRST operator for \cite{Berkovits:2019bbx} before constructing the worldsheet model with pure spinor twistor variables. This provides a first step onto covariant quantization of the ambitwistor string in \cite{Berkovits:2019bbx}.
In virtue of \eqref{definingconstraints}, the operator product expansions (OPEs) satisfied by the pure spinor twistor worldsheet variables do not correspond to those of a free theory. We thus need to resort to the tools of interacting two-dimensional conformal field theories (2D CFTs) as outlined in \cite{DiFrancesco:1997nk, Bais:1987dc} to compute OPEs between different operators of interest. Notice that the same set of tools have been used in \cite{Oda:2005sd, Oda:2007ak} in the context of the standard pure spinor formalism in order to reproduce the corresponding OPEs as originally found in \cite{Berkovits:2000fe}. We set up an analogous construction for the model developed here, and we find the corresponding expressions for the stress-energy tensor $T$, the Lorentz generator $N^{mn}$, and the projective weight operator $J$, all of which develop corrections similarly as in \cite{Oda:2005sd, Oda:2007ak}. In particular, we find that $J$ develops anomalies at quantum level.
As we will see, the model we present in this work has many similarities with the ordinary pure spinor ambitwistor string. Indeed, to make these similarities transparent we shall fix only the $B$-constraint \eqref{Bconstraint} and leave other symmetries unfixed, from which a corresponding BRST operator can be constructed. Moreover, the latter procedure will also be instrumental to see how the momentum conservation delta function arises in the pure spinor twistor model.
As in standard superstring theory, we write scattering amplitudes as correlation functions of vertex operators in pictures $0$ and $-1$. The picture $-1$ vertex operators are obtained from the corresponding superparticle wavefunction \cite{Sepulveda:2020kjc}, first considered in the original work \cite{Berkovits:2009by}, and we construct picture $0$ vertex operators through a picture raising operation. Integrated vertex operators share similar properties as those of other ambitwistor models \cite{Mason:2013sva, Berkovits:2013xba, Geyer:2014fka, Geyer:2018xgb, Geyer:2019ayz, Berkovits:2019bbx}, being localized in a set of delta functions that lead to momentum conservation and that localize over the scattering equations $k_{i} \cdot P_{(z_{i})} = 0$ \cite{Cachazo:2013hca, Cachazo:2013iea, Cachazo:2014xea}. Vertex operators are written in terms of integrals over ``auxiliary'' $SU(5)$ vectors $\epsilon^{\dot{a}}, \bar{\epsilon}_{\dot{a}}$ that are introduced by noticing that the momentum $P^{m} = (\lambda \gamma^{m} \bar{\lambda})$ satisfies a ``twistor-like constraint'' $(\lambda \gamma_{m})_{\alpha}P^{m} =0$ \cite{Berkovits:2015yra} and it is then left invariant under an $SU(5)$ subgroup of the complexified ten-dimensional Wick-rotated spacetime. We then associate an additional $SU(5)$ index to our variables which allows us to write down explicit expressions for the pictures $0$ and $-1$ vertex operators.
The amplitude prescription is a standard proposal consisting of integrating over all independent components of our fields and modding out by the killing vectors redundancies. As usual, we have to mod out by $\textrm{SL}(2,\mathbb{C})$ arising from reparametrizations, but we will also need to mod out by the $\textrm{GL}(1)$ associated to \eqref{Jconstraintintro} and the killing vectors associated to \eqref{splconstraint}. As we shall see, this prescription will turn out to be related to the corresponding prescription found in the standard pure spinor ambitwistor model \cite{Berkovits:2013xba}: the measure is tantamount to that of the pure spinor formalism $\langle (\lambda \gamma^{m}\theta) (\lambda \gamma
^{n}\theta) (\lambda \gamma^{p}\theta) (\theta \gamma_{mnp} \theta)\rangle = 1$, and both unintegrated as well as integrated vertex operators can be related, on the support of the incidence relations and the delta functions appearing in the vertex operators, to their counterparts in the standard pure spinor formalism. We conclude from this observation that our amplitude prescription indeed gives the correct correlators.
This work is organized as follows: In section \ref{section2} we review the ambitwistor model arising from Berkovits' superparticle model \cite{Berkovits:1990yc} and write the covariant BRST operator left as an open problem from \cite{Berkovits:2019bbx}. Sections \ref{section3} and \ref{section4} contain the main results of this paper. In section \ref{section3} we construct the pure spinor twistor ambitwistor model from the superparticle formulation of \cite{Sepulveda:2020kjc}, we construct the corresponding BRST operator, and write down vertex operators in pictures $-1$ and $0$. In section \ref{section4} we discuss the scattering amplitudes prescription along with its relation to the standard pure spinor formalism. We conclude in section \ref{section5} with discussions and some directions for further research. In Appendix \ref{AppendixA} we provide a quick review to the tools used to compute OPEs in interacting 2D CFTs.
\section{The Non-Pure Spinor Description of Ambitwistor Strings} \label{section2}
In this section we consider an ambitwistor model first described in \cite{Berkovits:2019bbx}, where it was shown that the model correctly reproduced the spectrum of the RNS ambitwistor string. However, the analysis was performed in light-cone gauge and a covariant analysis was left as an open problem owing to the reducibilities present in the constraints. Due to the relation between this model and the one we will describe in section \ref{section3} (via a field redefinition explained in \cite{Sepulveda:2020kjc}), we warm-up in this section constructing the BRST operator for the model described in \cite{Berkovits:2019bbx}, which in principle provides the means for a covariant analysis.
We start defining $\Lambda^{\alpha}$, $\Omega_{\beta}$ to be ten-dimensional Majorana-Weyl spinors of opposite chirality, and $\psi^{m}$ to be a ten-dimensional fermionic vector. We use letters from the beginning/middle of the Greek/Latin alphabet to denote ten-dimensional spinor/vector indices. The relation between these variables and the standard superspace variables is simply given by:
\begin{equation}
\Omega_{\alpha} = (\gamma_{m}\Lambda)_{\alpha}X^{m} + \psi^{m}(\gamma_{m}\theta)_{\alpha}, \ \ \ \ \psi^{m} = (\Lambda\gamma^{m}\theta),
\end{equation}
where $(\gamma^{m})_{\alpha\beta}$, $(\gamma^{m})^{\alpha\beta}$ are the ten-dimensional Pauli matrices satisfying the standard Dirac algebra $(\gamma^{(m})_{\alpha \gamma}(\gamma^{n)})^{\gamma \beta} = \eta^{mn}\delta_{\alpha}^{\beta}$ as well as the special identity $(\gamma^{m})_{(\alpha\beta}(\gamma_{m})_{\delta)\epsilon} = 0$ valid in ten dimensions.
The former variables were originally introduced in \cite{Berkovits:1990yc} in order to construct a model of the ten-dimensional massless superparticle with only first-class constraints. The corresponding heterotic ambitwistor string was later introduced in \cite{Berkovits:2019bbx}, with an action given by
\begin{equation}\label{heteroriginalaction}
S = \int \Big[ \frac{1}{2} \Lambda \bar{\partial} \Omega - \frac{1}{2} \Omega \bar{\partial} \Lambda - \frac{1}{2} \psi_{m} \bar{\partial} \psi^{m} + \xi_{\alpha}G^{\alpha} + \kappa T_{F} \Big] + S_{J},
\end{equation}
where $S_{J}$ stands for a current algebra and $\xi_{\alpha}$ and $\kappa$ are Lagrange multipliers enforcing a set of constraints with two levels of reducibility:
\begin{align}
& T_{F} \coloneqq (\Lambda \gamma^{m} \Lambda) \psi_{m}, \label{TFconstraint} \\[1.0ex]
& G^{\alpha} \coloneqq (\Lambda \gamma^{m} \Lambda) (\gamma_{m} \Omega)^{\alpha} - 2 \Lambda^{\alpha} (\Lambda \Omega) + \psi^{m} \psi^{n} (\gamma_{m} \gamma_{n} \Lambda)^{\alpha}= 0, \label{Gconstraint} \\[1.0ex]
& H^{m} \coloneqq (\Lambda\gamma^{m}G) - 2\psi^{m}T_{F} = 0, \label{one-reducibility} \\[1.0ex]
\label{two-reducibility}
&(\Lambda \gamma^{m} \Lambda) H_{m} = 0.
\end{align}
The stress-energy tensor for this worldsheet model is:
\begin{equation}
T_{B} = \frac{1}{2} \Lambda^{\alpha} \partial \Omega_{\alpha} - \frac{1}{2} \Omega_{\alpha} \partial \Lambda^{\alpha} - \frac{1}{2} \psi^{m} \partial \psi_{m} + T_{J},
\end{equation}
where $T_{J}$ is the stress-energy tensor associated to the current algebra.
The OPEs satisfied by the canonical variables are
\begin{equation}\label{standardOPEs}
\big\llangle \Lambda^{\alpha}_{(z)} \Omega_{\beta (w)} \big\rrangle = \frac{\delta^{\alpha}_{\beta}}{ (z-w)}, \ \ \ \ \big\llangle \psi^{m}_{(z)} \psi^{n}_{(w)} \big\rrangle = \frac{\eta^{mn}}{(z-w)},
\end{equation}
where we have introduced the notation $\llangle \rrangle$ to mean the singular terms in an OPE. Whenever there is no room for confusion we will just refer to OPE to mean that we are interested in the corresponding singular terms.
The OPEs/algebra satisfied by the constraints $(T_{B}, T_{F}, G^{\alpha})$ are readily found to be
\begin{align} \label{constraintalgebra}
\big\llangle G^{\alpha}_{(z)}T_{F (\omega)} \big\rrangle & = - \frac{2}{(z-w)} \Lambda^{\alpha} T_{F (w)}, \\[1.0ex]
\big\llangle T_{B (z)}G^{\alpha}_{(w)} \big\rrangle & = \frac{3}{2(z-w)^{2}} G^{\alpha}_{(w)} + \frac{1}{(z-w)} \partial G^{\alpha}_{(w)}, \\[1.0ex]
\big\llangle T_{B(z)}T_{F(w)} \big\rrangle & = \frac{3}{2(z-w)^{2}} T_{F (w)} + \frac{1}{(z-w)} \partial T_{F(w)}, \\[1.0ex]
\big\llangle T_{F(z)}T_{F(w)} \big\rrangle &= 0, \\[1.0ex]
\big\llangle T_{B(z)}T_{B(w)} \big\rrangle &= \frac{-\frac{11}{2} + \frac{c_{J}}{2}}{(z-w)^{4}} + \frac{2}{(z-w)^{2}}T_{B(w)} + \frac{1}{(z-w)}\partial T_{B(w)}, \\[1.0ex]
\big\llangle G^{\alpha}_{(z)}G^{\beta}_{(w)} \big\rrangle &= -\frac{4}{(z-w)} \Lambda^{[\alpha}G^{\beta]}_{(w)} - \frac{56}{(z-w)^{2}} \Lambda^{\alpha}\Lambda^{\beta}_{(w)} - \frac{36}{(z-w)} \partial \Lambda^{\beta} \Lambda^{\alpha}_{(w)} \nonumber \\[1.0ex] & \hspace{-2.2cm} - \frac{20}{(z-w)}\partial \Lambda^{\alpha} \Lambda^{\beta}_{(w)} + \frac{16}{(z-w)^{2}} (\gamma^{m})^{\alpha \beta} (\Lambda \gamma_{m} \Lambda)_{(w)} + \frac{16}{(z-w)} (\gamma_{m})^{\alpha \beta} (\partial \Lambda \gamma^{m} \Lambda)_{(w)}. \label{GGconstraintalgebra}
\end{align}
As already noticed in \cite{Berkovits:2019bbx}, the above construction is not limited to the heterotic case and one can readily generalize to the Type IIB case. \\
\textbf{The BRST Operator.} Due to the reducibilities \eqref{one-reducibility} and \eqref{two-reducibility} the BRST operator for the model \eqref{heteroriginalaction} is non-trivial to find. In general, for worldline systems, when one is in presence of a set of constraints $G_{a_{0}}$, \ $a_{0} = 1, \ldots, m_{0},$ that have a set of reducibilities: $$Z_{a_{1}}^{\ a_{0}}G_{a_{0}} = 0, \ \ \ \ \ \ a_{1}=1,\ldots,m_{1},$$ which may themselves be reducible: $$Z_{a_{k}}
^{\ a_{k-1}}Z_{a_{k-1}}^{\ a_{k-2}} = C_{a_{k}}^{\ a_{k-2}a_{0}}G_{a_{0}}, \ \ \ k=1, \ldots, L, \ \ \ a_{k} = 1, \ldots, m_{k},$$
one introduces a ghost-for-ghost pair $(\eta_{a_{k}}, \rho_{a_{k}})$ with total ghost number $(k+1,-(k+1))$ for each level of reducibility, in addition to the standard ghosts $(\eta_{a_{0}}, \rho_{a_{0}})$ associated to the constraints $G_{a_{0}}$. A general prescription \cite{Henneaux:1992ig} to write down the BRST operator is given by:
\begin{equation}
Q = \eta^{a_{0}}G_{a_{0}} + \eta^{a_{k}}Z_{a_{k}}^{\ a_{k-1}}\rho_{a_{k-1}} + \bar{Q},
\end{equation}
where $\bar{Q}$ stands for further terms in $Q$ containing at least two $\eta$'s and one $\rho$ or two $\rho$'s and one $\eta$ and that are constructed such that $Q$ is nilpotent.
It is straightforward to adapt the previous construction to the case of a 2D CFT. We introduce, apart from the standard reparametrization ghosts $(c,b)$ and the ghosts $(g_{\alpha},f^{\alpha}), \, (\gamma, \beta)$ associated to the constraints $G^{\alpha}$ and $T_{F}$ respectively, a pair of (bosonic) ghosts-for-ghosts $(s_{m},t^{m})$ and (fermionic) ghosts-for-ghosts $(\eta, \rho)$. One then finds the BRST current to be:
\begin{equation} \label{originalmodel-brstcurrent}
q(z) = \Big( cT + b c \partial c + q_{sp} -16 \Lambda^{\alpha} \partial g_{\alpha} \Big)(z),
\end{equation}
where
\begin{align} \label{originalmodel-superparticle-brstcurrent}
q_{sp}(z) = &\bigg(g_{\alpha} G^{\alpha} + \gamma T_{F} + s_{m}(\Lambda \gamma^{m} f) + s_{m} (2 \psi^{m} \beta) + \eta (\Lambda \gamma_{m} \Lambda) t^{m} \nonumber \\[1.0ex]
& \ \ + 2 (\Lambda^{\alpha} g_{\alpha})(g_{\beta} f^{\beta})
- 2 (\Lambda^{\alpha} g_{\alpha})(\gamma \beta)
-2 \big[ \Lambda^{\alpha} (\gamma_{n})_{\alpha \beta} (\gamma^{m})^{\beta \gamma} g_{\gamma} \big] s_{m}t^{n} \nonumber \\[1.0ex]
& \hspace{4.2cm} + 4 (\Lambda^{\alpha} g_{\alpha})\eta \rho + 2 \eta^{nm} s_n s_m \rho - \eta \beta^{2} \bigg)(z).
\end{align}
We have separated the contributions associated to the Virasoro constraint to those associated to the superparticle terms in \cite{Sepulveda:2020kjc}. Single contraction contributions in the $q_{sp}(z) q_{sp}(w)$ OPE vanish in a tantamount computation to that of the superparticle \cite{Sepulveda:2020kjc}. Notice, however, that new contributions could in principle arise in the simple poles due to the expansion of double contraction terms which could render the BRST operator non-nilpotent. There are two of these type of contributions: those which contain two $g$-ghosts, and those that are proportional to the $s_{m}$ ghost-for-ghost. For instance, a double contraction of the $(g,f)$ ghosts in the $2 (\Lambda^{\alpha} g_{\alpha})(g_{\beta} f^{\beta})$ term with itself gives a simple pole contribution of:
\begin{equation*}
4 [(\Lambda g)(g f)] (z) [(\Lambda g)(g f)](w) \xrightarrow[\begin{subarray}{l} \text{($g$,$f$)} \\ \text{($g$,$f$)} \end{subarray}]{\text{}} \frac{\Big[ 52 (\partial \Lambda g)(\Lambda g) + 60 (\Lambda \partial g)(\Lambda g) \Big] (w)}{(z-w)},
\end{equation*}
where the variables below the arrow stand for the fields that we have contracted, and the right side is the corresponding contribution to the simple pole. This is an example of a contribution arising from the expansion of a double contraction and that contains two $g$-ghosts. The other non-zero contributions of this type are:
\begin{align}
& 4 (\Lambda g)(\gamma \beta)_{(z)}(\Lambda g)(\gamma \beta)_{(w)} \xrightarrow[\begin{subarray}{l} \text{($\gamma$,$\beta$)}\\ \text{($\gamma$,$\beta$)} \end{subarray}]{\text{}} -4\frac{\Big[ (\partial \Lambda g)(\Lambda g) + (\Lambda \partial g)(\Lambda g) \Big] (w)}{(z-w)}, \nonumber \\
& 4 (\Lambda g)(\eta \rho)_{(z)}(\Lambda g)(\eta \rho)_{(w)} \xrightarrow[\begin{subarray}{l} \text{($\eta$,$\rho$)}\\ \text{($\eta$,$\rho$)} \end{subarray}]{\text{}} 16\frac{\Big[ (\partial \Lambda g)(\Lambda g) + (\Lambda \partial g)(\Lambda g)\Big] (w)}{(z-w)}, \nonumber \\
& 4 (\Lambda \gamma_{n} \gamma^{m} g)s_{m}t^{n}_{(z)} (\Lambda \gamma_{p} \gamma^{q} g)s_{q}t^{p}_{(w)} \xrightarrow[\begin{subarray}{l} \text{($s$,$t$)} \\ \text{($s$,$t$)} \end{subarray}]{\text{}} \frac{\big[ 16(\Lambda \gamma^{m} \Lambda)(\partial g \gamma_{m} g) - 16(\partial \Lambda)(\Lambda g) - 80 (\Lambda \partial g) (\Lambda g)\big](w)}{(z-w)}. \nonumber
\end{align}
There is also a contribution from the double poles in the $g_{\alpha}G^{\alpha}(z)g_{\beta}G^{\beta}(w)$ OPE that can be read from the $G^{\alpha}G^{\beta}$ OPE and expanding $z$ around $w$:
\begin{equation*}
\frac{16\big[ (\partial \Lambda g)(\Lambda g)\big](w)}{(z-w)} - \frac{56\big[ (\Lambda \partial g)(\Lambda g) \big](w)}{(z-w)} + \frac{16\big[ (\Lambda \gamma^{m} \Lambda)(\partial g \gamma_{m} g)\big](w)}{(z-w)}.
\end{equation*}
Adding all these type of contributions, one gets:
\begin{equation} \label{ccdoublepole}
\frac{ \big[ 32(\Lambda \gamma^{m} \Lambda) (\partial g \gamma_{m} g) + 64(\partial \Lambda g)(\Lambda g) - 64(\Lambda \partial g)(\Lambda g)\big](w)}{(z-w)}.
\end{equation}
In order for the BRST operator to be nilpotent we must cancel these contributions. Quite remarkably, this issue is fully-handled just by the last term in \eqref{originalmodel-brstcurrent} which corresponds to a normal ordering term that has been added to consider these contributions. Specifically, the $-16\Lambda^{\alpha} \partial g_{\alpha}$ term in \eqref{originalmodel-brstcurrent} introduces further OPEs with $q_{sp}$, namely:
\begin{align}
& -16 \Lambda^{\alpha}\partial g_{\alpha (z)} g_{b}G^{b}_{(w)} + \textrm{cross-term} \xrightarrow[\text{($\Lambda$,$\Omega$)}]{\text{}} \frac{ \big[ -32(\Lambda \gamma^{m} \Lambda)(\partial g \gamma_{m} g) + 64 (\Lambda \partial g)(\Lambda g) \big](w)}{(z-w)}, \\
& -32 \Lambda^{\alpha}\partial g_{\alpha (z)} (\Lambda g)(gf)_{(w)} + \textrm{cross-term} \xrightarrow[\text{($\Lambda$,$\Omega$)}]{\text{}} -\frac{64\Big[ (\partial \Lambda g)(\Lambda g)\Big](w)}{(z-w)},
\end{align}
which precisely cancel \eqref{ccdoublepole} and no terms with two $g$-ghosts remain. Furthermore, the single term $-16\Lambda^{\alpha} \partial g_{\alpha}$ also handles the second type of contributions proportional to $s_{m}$. All these contributions are given by:
\begin{align}
& 2 \gamma (\Lambda \gamma^{m} \Lambda) \psi_{m (z)} s_{n}\psi^{n}\beta_{(w)} + \textrm{cross-term} \xrightarrow[\begin{subarray}{l} \text{($\psi$,$\psi$)} \\ \text{($\beta$,$\gamma$)} \end{subarray}]{\text{}} \frac{[4(\partial \Lambda \gamma^{m} \Lambda)s_{m} - 2(\Lambda \gamma^{m} \Lambda) \partial s_{m}](w)}{(z-w)}, \nonumber \\[1.5ex]
- & 16\Lambda \partial g_{(z)} s_{m}(\Lambda \gamma^{m} f)_{(w)} + \textrm{cross-term} \xrightarrow[\text{($g$,$f$)}]{\text{}} -\frac{16(\Lambda \gamma^{m} \Lambda) \partial s_{m} (w)}{(z-w)}, \nonumber \\[1.5ex]
- & 2(\Lambda \gamma_{n}\gamma^{m} g) s_{m}t^{n}_{(z)} s_{p}(\Lambda \gamma^{p} f)_{(w)} + \textrm{cross-term} \xrightarrow[\begin{subarray}{l} \text{($s$,$t$)} \\ \text{($g$,$f$)} \end{subarray}]{\text{}} -\frac{16(\Lambda \gamma^{m} \Lambda) \partial s_{m} (w)}{(z-w)}, \nonumber \\
& g_{\alpha}G^{\alpha}_{(z)}s_{m}(\Lambda \gamma^{m} f)_{(w)} + \textrm{cross-term} \xrightarrow[\begin{subarray}{l} \text{($\Lambda$,$\Omega$)} \\ \text{($g$,$f$)} \end{subarray}]{\text{}} \frac{[14(\Lambda \gamma^{m} \Lambda) \partial s_{m} - 28(\partial \Lambda \gamma^{m} \Lambda) s_{m}](w) }{(z-w)}, \nonumber \\[1.0ex]
& 2 \eta^{mn}s_{m}s_{n}\rho_{(z)}\eta(\Lambda \gamma^{p} \Lambda)t^{p}_{(w)} + \textrm{cross-term} \xrightarrow[\begin{subarray}{l} \text{($s$,$t$)} \\ \text{($\eta$,$\rho$)} \end{subarray}]{\text{}} \frac{[4(\Lambda \gamma^{m} \Lambda) \partial s_{m} - 8(\partial \Lambda \gamma^{m} \Lambda) s_{m}](w) }{(z-w)}. \nonumber
\end{align}
As it is easy to see, the term $-16\Lambda^{\alpha} \partial g_{\alpha}$ again takes care of all these contributions and the final result is a total derivative:
\begin{equation}
-16 \frac{\partial[(\Lambda \gamma^{m} \Lambda)s_{m}](w)}{(z-w)},
\end{equation}
which is sufficient for the BRST operator to be nilpotent.
Clearly, we also need to consider the presence of the Virasoro constraint in the full BRST current:
\begin{equation}
T = T_{B} + T_{\textrm{gh}},
\end{equation}
where $T_{\textrm{gh}}$ corresponds to the stress energy tensor of all ghost fields others than the $(b,c)$ system. This is straightforward to accommodate with the superparticle contributions recalling that the BRST current has conformal weight one.
\section{The Pure Spinor Twistor Ambitwistor String} \label{section3}
In this section we construct an ambitwistor worldsheet model based on the description of the $D=10$ massless superparticle developed in \cite{Sepulveda:2020kjc}. We start defining the model and considering the OPEs for many quantities of interest in analogy with the standard pure spinor formalism \cite{Berkovits:2000fe, Oda:2005sd, Oda:2007ak}. We take special care of the fact that the OPEs satisfied by our variables do not correspond to those of a free theory. After a convenient gauge fixing procedure, a simple BRST operator is constructed and physical states are defined.
\subsection{The Worldsheet Model}
The variables from which the worldsheet model is going to be defined are given by
\begin{equation}
\mathcal{Z}^{I} = (\lambda^{\alpha},\, \mu_{\alpha},\, \Gamma^{m}), \ \ \ \ \ \bar{\mathcal{Z}}_{I} = (\bar{\mu}_{\alpha},\, -\bar{\lambda}^{\alpha}, \bar{\Gamma}^{m}),
\end{equation}
where $\lambda^{\alpha}$ is a pure spinor, $\bar{\lambda}^{\alpha}$ is a 16-component spinor, and $\Gamma^{m}$ is a fermionic vector. $\bar{\mathcal{Z}}_{I}$ will correspond to the canonical conjugates to the variables defining $\mathcal{Z}^{I}$. By definition the variables are required to solve:
\begin{align} \label{psconstraints}
S^{m} & \coloneqq \lambda\gamma^{m}\lambda = 0, \\[1.0ex]
D & \coloneqq \lambda\mu = 0, \\[1.0ex] \Phi^{mn} & \coloneqq (\lambda\gamma^{mn}\mu) + 4\Gamma^{m}\Gamma^{n} = 0, \\[1.0ex] E_{\alpha} & \coloneqq (\lambda\gamma^{m})_{\alpha}\Gamma_{m} = 0, \label{psconstraints2}
\end{align}
and are related to the standard superspace variables through the ``incidence relations'':
\begin{equation} \label{incidencerelationsps}
\mu_{\alpha} = (\gamma_{m} \lambda)_{\alpha} X^{m} + \Gamma^{m}(\gamma_{m} \theta), \ \ \ \ \ \Gamma^{m} = (\lambda \gamma^{m} \theta).
\end{equation}
In virtue of \eqref{psconstraints}-\eqref{psconstraints2}, physical quantities must be invariant under the gauge transformations:
\begin{align}
\delta \bar{\mu}_{\alpha} & = \mu_{\alpha} d + (\gamma^{mn} \mu)_{\alpha} \phi_{mn} + (\gamma^{m} \epsilon)_{\alpha} \Gamma_{m} + (\gamma^{m} \lambda)_{\alpha} s_{m}, \\ \delta \bar{\lambda}^{\alpha} & = -\lambda^{\alpha} d - (\lambda \gamma^{mn})^{\alpha} \phi_{mn}, \\ \delta \bar{\Gamma}_{m} & = - 8 \phi_{mn} \Gamma^{n} + (\lambda \gamma_{m} \epsilon),
\end{align}
where $s_{m}$, $d$, $\phi_{mn}$, $\epsilon_{\alpha}$ are gauge parameters associated to \eqref{psconstraints}-\eqref{psconstraints2} respectively.
The OPEs satisfied by these variables are non-trivial considering the relations \eqref{psconstraints}-\eqref{psconstraints2} which effectively render the theory interacting. The OPEs that we have to consider are given by:
\begin{align}
\llangle \bar{\mu}_{\alpha (z)} \lambda^{\beta}_{(w)} \rrangle &= \frac{-1}{(z-w)}\Big(\delta^{\beta}_{\alpha} - K_{\alpha}^{\ \beta} \Big)_{(w)} \, , \label{OPElambdaw} \\[0.4ex]
\llangle \bar{\lambda}^{\beta}_{(z)} \mu_{\alpha (w)} \rrangle & = \frac{1}{(z-w)}K_{\alpha (w)}^{\ \beta}\,, \\[0.4ex]
\llangle \bar{\mu}_{\alpha (z)} \mu_{\beta (w)} \rrangle & = \frac{1}{(z-w)}\Big( Y_{\alpha}\mu_{\beta} + Y_{\beta}\mu_{\alpha} - \frac{1}{2}\gamma^{m}_{\alpha\beta}(Y \gamma_{m}\mu) \Big)_{(w)}\,, \\[0.4ex]
\llangle \bar{\mu}_{\alpha (z)} \Gamma^{m}_{(w)} \rrangle & = \frac{1}{2(z-w)}\Big((\gamma^{p} \gamma^{m} Y)_{\alpha}\Gamma_{p}\Big)_{(w)}\,, \\[0.4ex]
\llangle \bar{\Gamma}^{n}_{(z)} \Gamma^{m}_{(w)} \rrangle & = \frac{1}{2 (z-w)}(\lambda \gamma^{m}\gamma^{n} Y)_{(w)}\,, \\[0.4ex]
\llangle \bar{\Gamma}^{m}_{(z)} \mu_{\alpha (w)} \rrangle & = \frac{-1}{(z-w)}\Big( (\gamma^{p}\gamma^{m} Y)_{\alpha}\Gamma_{p}\Big)_{(w)}, \label{OPEmugammabar}
\end{align}
where we have defined the projector
\begin{equation}
K_{\alpha}^{\ \beta} = \frac{1}{2}(\lambda \gamma_{s})_{\alpha}(\gamma^{s} Y)^{\beta},
\end{equation}
with
\begin{equation}
Y_{\alpha} = \frac{\nu_{\alpha}}{(\lambda \nu)},
\end{equation}
and $\nu_{\alpha}$ a fixed pure spinor so that $Y \gamma^{m} Y = 0$.
These OPEs can be found by requiring that the OPE between any single conjugate variable $\bar{\lambda}^{\alpha}$, $\bar{\mu}_{\alpha}$, or $\bar{\Gamma}^{m}$ has a vanishing OPE with the corresponding constraints \eqref{psconstraints}-\eqref{psconstraints2}. Notice that this construction is similar in fashion to that of the standard pure spinor formalism as formulated in \cite{Oda:2005sd, Oda:2007ak}. In order to deal with the OPEs \eqref{OPElambdaw}-\eqref{OPEmugammabar} we have to be careful with the fact that the coefficients are non-constant and thus we have to resort to the tools outlined in Appendix \ref{AppendixA} to proceed. In particular, we have to be careful with the ordering of the different operators when constructing the theory. As explained in Appendix \ref{AppendixA}, we define the normal-ordered product
\begin{equation} \label{NO}
(AB)_{(w)} = \oint_{w} \frac{\mathrm{d}x}{(x-w)} A_{(x)} B_{(w)},
\end{equation}
which let us consistently separate the finite terms from the divergent ones as $z \rightarrow w$ in $A_{(z)}B_{(w)} = \llangle A_{(z)} B_{(w)}\rrangle + (A_{(z)} B_{(w)})$.
The ambitwistor worldsheet model that we are going to consider here is based on a model of the $D=10$ massless superparticle developed by the authors in a complementary paper \cite{Sepulveda:2020kjc}. The pure spinor twistor heterotic ambitwistor string action is defined as:
\begin{eqnarray}\label{psambiaction}
S &=& \int d^{2}z\,\bigg(\bar{\mathcal{Z}}_{I}\bar{\partial}\mathcal{Z}^{I} + \zeta J + \chi B + \Upsilon_{abc}\tilde{\mathcal{M}}^{abc}\bigg) + S_{J},
\end{eqnarray}
where $S_{J}$ stands for the current algebra action, and $\zeta$, $\xi$, and $\Upsilon_{abc}$ are Lagrange multipliers enforcing the (classical) constraints:
\begin{align}
J_{0} & = \bar{\mu}_{\alpha}\lambda^{\alpha} - \bar{\lambda}^{\alpha}\mu_{\alpha} + \bar{\Gamma}_{m}\Gamma^{m}, \label{defJ}\\
B & = (\lambda\gamma^{m}\bar{\lambda})\bar{\Gamma}_{m} - (\bar{\lambda}\gamma^{m}\bar{\lambda})\Gamma_{m}, \\
\tilde{\mathcal{M}}^{abc} & = (\lambda\gamma^{[a}\bar{\lambda})N^{bc]} + \frac{1}{12}(\tilde{q}\gamma^{abc}\tilde{q}), \label{defMabc}
\end{align}
where $a,b,c$ are $SU(5)$ fundamental indices. $N^{mn}$ and $\tilde{q}_{\alpha}$ are the super-Lorentz generators given by
\begin{eqnarray}
N_{0}^{mn} &=& -\frac{1}{2}(\bar{\mu} \gamma^{mn} \lambda) + \frac{1}{2}(\bar{\lambda}\gamma^{mn}\mu) - 2(\bar{\Gamma}^{[m}\Gamma^{n]}), \label{defNmn}\\
\tilde{q}_{\alpha} &=& (\gamma^{m}\bar{\lambda})_{\alpha}\Gamma_{m} - \frac{1}{2}(\gamma^{m}\lambda)_{\alpha}\bar{\Gamma}_{m}. \label{defqtilde}
\end{eqnarray}
We have written a subindex $0$ whenever we have taken a quantity without considering correction terms due to quantum-mechanical effects.
In order to consider the corresponding corrections to the operators, we need to be careful with the corresponding ordering of the operators according to \eqref{NO} and ask for the correct OPEs to be fulfilled. For instance, when one considers the corrected Lorentz generator $N^{mn}$:
\begin{equation}
N^{mn} = -\frac{1}{2}(\bar{\mu} \gamma^{mn} \lambda) + \frac{1}{2}(\bar{\lambda} \gamma^{mn} \mu) -2(\bar{\Gamma}^{[m}\Gamma^{n]}) - \partial \big(Y \gamma^{mn} \lambda \big),
\end{equation}
where the parenthesis stand as well for the normal ordering \eqref{NO}, one finds that the singular terms in the $N^{mn}_{(z)}N^{pq}_{(w)}$ OPE are given by:
\begin{equation}
\big\llangle N^{mn}_{(z)} N^{pq}_{(w)} \big\rrangle = -3 \frac{(\eta^{mq} \eta^{np} - \eta^{mp} \eta^{nq})}{(z-w)^{2}}+ \frac{N^{mp} \eta^{nq} - N^{np} \eta^{mq} + N^{nq} \eta^{mp} - N^{mq} \eta^{np}}{(z-w)},
\end{equation}
which means that $N^{mn}$ generates a level $k=-3$ current algebra. The total derivative is responsible of cancelling out all spurious terms that would make the current $N^{mn}$ not satisfy a Kac-Moody algebra (see appendix \ref{AppendixA} for further details).
We can perform a similar analysis for the projective weight operator $J$ and the stress energy tensor $T$. Considering correction terms and using the OPEs \eqref{OPElambdaw}-\eqref{OPEmugammabar} we find that:
\begin{align}
J & = \bar{\mu}_{\alpha}\lambda^{\alpha} - \mu_{\alpha}\bar{\lambda}^{\alpha} + \bar{\Gamma}_{m}\Gamma^{m} - 4(Y_{\alpha}\partial\lambda^{\alpha}), \\[1.0ex]
T & = -\bar{\mu}_{\alpha}\partial\lambda^{\alpha} + \bar{\lambda}^{\alpha}\partial \mu_{\alpha} - \bar{\Gamma}_{m}\partial\Gamma^{m} + 2\partial(Y_{\alpha}\partial\lambda^{\alpha}),
\end{align}
with the full set of non-trivial OPEs given by:
\begin{align}
& \llangle T_{(z)}T_{(w)} \rrangle = \frac{22+ c_{J}}{{2 (z-w)^{4}}} + \frac{2T_{(w)}}{(z-w)^{2}} + \frac{\partial T_{(w)}}{(z-w)}, \label{TTope}\\[1.0ex]
& \llangle J_{(z)} J_{(w)} \rrangle = - \frac{3}{(z-w)^{2}}, \label{JJope}\\[1.0ex] & \llangle T_{(z)}J_{(w)} \rrangle = -\frac{7}{(z-w)^{3}} + \frac{J_{(w)}}{(z-w)^{2}} + \frac{\partial J_{(w)}}{(z-w)}, \label{TJope}\\[1.0ex]
& \llangle T_{(z)}N^{mn}_{(w)} \rrangle = \frac{N^{mn}_{(w)}}{(z-w)^{2}} + \frac{\partial N^{mn}_{(w)}}{(z-w)}. \label{TNope}
\end{align}
In particular, the Lorentz generator $N^{mn}$ behaves properly as a primary field, $J$ has a regular OPE with $N^{mn}$, and $J$ exhibits a conformal anomaly of $-7$. Interestingly, the quantum corrections can be arranged in a unique way such that the Lorentz covariance of the OPEs \eqref{TTope}-\eqref{TNope} is preserved. That is, $Y$ spinors appear in the OPEs only in the quantum corrections to $J$, $N^{mn}$ and $T$. Compare this, for instance, to the OPEs of the variables defining the model \eqref{OPElambdaw}-\eqref{OPEmugammabar}.
The components of the constraint $\tilde{\mathcal{M}}^{abc}$ in \eqref{defMabc} are not all independent. Indeed, careful inspection of $\tilde{\mathcal{M}}^{abc}$ leads us to the conclusion that it satisfies the following reducibility relations:
\begin{align}
H^{abcd} & \coloneqq P^{[a}\tilde{\mathcal{M}}^{bcd]} = -\frac{1}{24}\epsilon^{abcde}\tilde{q}_{e}\lambda^{+}B\label{red1},\\[1.0ex]
H^{abcde} & \coloneqq P^{[a}H^{bcde]} = 0\label{red2}.
\end{align}
As done for $J$, $N^{mn}$, and $T$, one can also compute the quantum corrections for $\tilde{\mathcal{M}}^{abc}$ by requiring that it behaves as a rank three tensor under Lorentz transformations. However, since this will not be relevant for our study and it is always possible to choose a frame where the quantum correction vanishes, we will ignore this ambiguity.
The central charge can be calculated for the whole system in the usual way. Using that ($J$, $B$, $\tilde{\mathcal{M}}^{abc}$) are conformal weight $(1, 2, 2)$ operators, together with eqns. \eqref{red1}, \eqref{red2}, one obtains:
\begin{equation}
c_{het.} = 22 + 10 - 10 - 26 - 2 + 26 - 10\times 26 + 5\times 74 - 1\times 146 + c_{J}.
\end{equation}
As in the $SO(32)$ or $E_{8}\times E_{8}$ heterotic strings, one finds that the total central charge vanishes when $c_{J} = 16$.
It is straightforward to generalize the above construction for the Type IIB case. The action is given by
\begin{equation} \label{psambiactiontypeIIB}
S = \int d^{2}z\,\bigg(\bar{\hat{\mathcal{Z}}}_{I}\bar{\partial}\hat{\mathcal{Z}}^{I} + \zeta J + \chi B + \hat{\chi}\hat{B} + \Upsilon_{abc}\mathcal{\tilde{M}}^{abc}\bigg),
\end{equation}
where the supertwistor variables $\hat{\mathcal{Z}}^{I} = (\lambda^{\alpha}, \mu_{\alpha}, \Gamma^{m}, \hat{\Gamma}^{m})$, $\bar{\hat{\mathcal{Z}}}_{I} = (\bar{\mu}_{\alpha}, -\bar{\lambda}^{\alpha}, \bar{\Gamma}^{m}, \bar{\hat{\Gamma}}^{m})$ are defined to satisfy the analogs of \eqref{psconstraints}-\eqref{psconstraints2}:
\begin{align}
& \lambda\gamma^{m}\lambda = 0, \ \ \ \lambda\mu = 0, \ \ \ (\lambda\gamma^{mn}\mu) + 4\Gamma^{m}\Gamma^{n} = 0, \\[1.0ex] & (\lambda\gamma^{m})_{\alpha}\Gamma_{m} = 0, \hspace{1.3cm} (\lambda\gamma^{m})_{\alpha}\Gamma_{m} = 0.
\end{align}
The constraints $\hat{B}$, $\tilde{M}^{abc}$ are defined by
\begin{eqnarray}
\tilde{B} &=& (\lambda\gamma^{m}\bar{\lambda})\bar{\tilde{\Gamma}}_{m} - (\bar{\lambda}\gamma^{m}\bar{\lambda})\tilde{\Gamma}_{m}, \\[1.0ex]
\tilde{M}^{abc} &=& (\lambda\gamma^{[a}\bar{\lambda})N^{bc]} - \frac{1}{12}(\tilde{q}\gamma^{abc}\tilde{q}) - \frac{1}{12}(\hat{\tilde{q}}\gamma^{abc}\hat{\tilde{q}}), \label{mtildetypeIIB}
\end{eqnarray}
where $\hat{\tilde{q}}_{\alpha}$, $\tilde{q}_{\alpha}$, $N^{mn}$ are the type IIB super-Lorentz generators. As before, one can show that the constraint $\tilde{M}^{abc}$ satisfies reducibility relations similar to those in eqns. \eqref{red1}, \eqref{red2}.
The total central charge is then easily shown to vanish:
\begin{equation}
c_{IIB} = 22 + 10 - 10 - 10 - 26 - 2 + 26 + 26 - 10\times 26 + 5\times 74 - 1\times 146 = 0 .
\end{equation}
\subsection{Construction of Physical States}
Physical states are defined via a BRST operator that will be constructed by applying the Faddeev-Popov method for the constraint $B$ and keeping the volume of the remaining symmetry groups as factors dividing out the path integral measure. In this manner, the BRST current reads
\begin{equation}\label{brstb}
q{(z)} = \gamma B{(z)} = (\eta e^{\phi} B){(z)},
\end{equation}
where $(\beta,\gamma)$ are the ghosts for the symmetry $B$, which have been fermionized through the standard procedure:
\begin{equation}
\gamma = \eta e^{\phi}, \ \ \ \beta = \partial \xi e^{-\phi},
\end{equation}
and $(\eta,\xi)$, $\phi$ are $bc$-type, linear dilaton CFTs, respectively. These definitions give rise to the picture charge defined by
\begin{equation}
N_{p} = \int dz\, (\xi\eta - \partial\phi).
\end{equation}
In order to write down vertex operators we will introduce additional $SU(5)$ indices on the pure spinor twistor variables which accounts for the symmetry group under which the twistor-like constraint $(\lambda \gamma^{m})_{\alpha}P^{m}=0$ is left invariant. This construction is essentially similar to the ones developed in \cite{Cheung:2009dc,Boels:2012ie,Geyer:2018xgb,Geyer:2019ayz} where spinor helicity variables describing massless states are assigned an additional little group index. Hence, the twistor variables now take the form $(\lambda_{r\,\dot{a}}^{\alpha}, \bar{\pi}_{r}^{\beta\,\dot{b}})$, where $\dot{a}$, $\dot{b}$ are $SU(5)$ vector indices, and satisfy
\begin{equation} \label{lambda-pibar-andk}
\lambda_{r\,\dot{a}}^{\alpha}(\gamma^{m})_{\alpha\beta}\bar{\pi}_{r}^{\beta\,\dot{b}} = k_{r}^{m}\delta^{\dot{b}}_{\dot{a}}.
\end{equation}
Using these variables one can define the following ten-dimensional spinors:
\begin{equation}
\lambda^{\alpha}_{r} \coloneqq \lambda^{\alpha}_{r\,\dot{a}}\epsilon^{\dot{a}}_{r}, \ \ \ \hspace{2mm}\bar{\pi}^{\alpha}_{r} \coloneqq \bar{\pi}_{r}^{\alpha\,\dot{a}}\bar{\epsilon}_{r\,\dot{a}},
\end{equation}
where $\epsilon^{\dot{a}}_{r}$, $\bar{\epsilon}_{r\,\dot{a}}$ are $SU(5)$ vectors. Notice that these spinors satisfy the relation
\begin{equation}
\lambda^{\alpha}_{r}(\gamma^{m})_{\alpha\beta}\bar{\pi}_{r}^{\beta} = k_{r}^{m}(\epsilon^{\dot{a}}_{r}\bar{\epsilon}_{r\,\dot{a}}),
\end{equation}
so in order to recover the more standard twistor relation $\lambda^{\alpha}_{r}(\gamma^{m})_{\alpha\beta}\bar{\pi}^{\beta}_{r} = k^{m}_{r}$ we integrate the $\epsilon_{r}$- and $\bar{\epsilon}_{r}$-variables in the vertex operator expression along with a delta function ensuring an appropriate localization. The super-Yang-Mills vertex operator at picture -1 then reads
\begin{align}\label{vertexoperator}
U^{(-1)}_{r} & = \int d^{2}z_{r}\,d^{5}\epsilon_{r}\,d^{5}\bar{\epsilon}_{r}\, \bar{\delta}(\epsilon_{r}^{\dot{a}}\bar{\epsilon}_{r\,\dot{a}} - 1)\bar{\delta}^{10}\bigg(\frac{\lambda_{ab}}{\lambda^{+}}(z_{r}) - \frac{\lambda_{ab\,\dot{a},r}\epsilon_{r}^{\dot{a}}}{\lambda^{+}_{r\,\dot{a}}\epsilon^{\dot{a}}_{r}}\bigg) \,\phi^{\epsilon}_{r,K}(\mathcal{Z})e^{-\phi}J^{K}\nonumber\\
&= \int d^{2}z_{r}\,\Sigma^{\epsilon}_{r}\,\phi^{\epsilon}_{r,K}(\mathcal{Z})e^{-\phi}J^{K},
\end{align}
where $\Sigma^{\epsilon}_{r}$ is defined to be
\begin{equation}\label{deltafunction}
\Sigma^{\epsilon}_{r} = \int d^{5}\epsilon_{r}\,d^{5}\bar{\epsilon}_{r}\,\bar{\delta}(\epsilon_{r}^{\dot{a}}\bar{\epsilon}_{r\,\dot{a}} - 1)\bar{\delta}^{10}\bigg(\frac{\lambda_{ab}}{\lambda^{+}}(z_{r}) - \frac{\lambda_{ab\,\dot{a},r}\epsilon_{r}^{\dot{a}}}{\lambda_{r\,\dot{a}}^{+}\epsilon_{r}^{\dot{a}}}\bigg),
\end{equation}
$\lambda_{ab}$, $\lambda^{+}$ are respectively the fully antisymmetric ten-dimensional, scalar $SU(5)$ components of $\lambda^{\alpha}$, the superscript $(-1)$ stands for the picture charge, the $J^{K}$ furnish a current algebra, and $\phi^{\epsilon}_{r,K}(\mathcal{Z})$ is given by
\begin{align}
& \phi^{\epsilon}_{r,K}(\mathcal{Z}) = \bigg( \bar{s}_{r,K} + 2\Gamma_{m}a_{-\,r,K}^{m} - 4\Gamma_{m}\Gamma_{n}s_{r,K}^{mn} \nonumber \\
& +\frac{1}{12} (\bar{\pi}_{r}\gamma_{mnpqr}\bar{\pi}_{r})\Gamma^{m}\Gamma^{n}\Gamma^{p}h^{q}a^{r}_{+\,r,K} - \frac{1}{24} (\bar{\pi}_{r}\gamma_{mnpqr}\bar{\pi}_{r})\Gamma^{m}\Gamma^{n}\Gamma^{p}\Gamma^{q}h^{r}s_{r,K}\bigg)e^{\mu_{a}\bar{\pi}_{r}^{a}\frac{\lambda^{+}_{r}}{\lambda^{+}}},\label{twistorsuperfield}
\end{align}
which coincides with the pure spinor twistor superparticle wavefunction of \cite{Sepulveda:2020kjc, Berkovits:2009by} on the support of the delta functions of \eqref{deltafunction}. In \eqref{twistorsuperfield} the gluon polarization is given by $a_{K}^{m} = a_{-\,K}^{m} + a_{+\,K}^{m}$, with $(\bar{\pi}\gamma_{m})_{\alpha}a_{-\,K}^{m} = 0$, $(\lambda\gamma_{m})a^{m}_{+\,K} = 0$, and the gluino polarization has been split into the form $\chi^{\alpha}_{K} = \bar{\pi}^{\alpha}\bar{s}_{K} + (\gamma_{mn}\lambda)^{\alpha}s^{mn}_{K} + \lambda^{\alpha}s_{K}$, with $(\bar{\pi}\gamma_{m})_{\alpha}s^{mn}_{K} = 0$. The vector $h
^{m}$ is constant and satisfies $h^{m}(\lambda\gamma_{m}\bar{\pi}) = 1$. Moreover, we have set eleven components of $\bar{\pi}^{\alpha}$ to zero using the fact that $\lambda^{\alpha}$ is a pure spinor, so that one is left with only five components, namely $\bar{\pi}^{a}$, which transform in the fundamental of $SU(5)$. One can then consider $\bar{\pi}^{\alpha}$ to be a pure spinor, and so \eqref{twistorsuperfield} is independent of the choice of $h^{m}$.
As a check, notice that one can recover the standard exponential bosonic contribution $e^{k_{r} \cdot X}$ to the vertex operator after using the incidence relations \eqref{incidencerelationsps} and the ten-dimensional delta function in \eqref{vertexoperator}. Indeed, one finds: $\mu_{a}\bar{\pi}^{a \, \dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}}\lambda^{+}_{r\,\dot{a}}\epsilon^{\dot{a}}_{r}/\lambda^{+} = X^{b} \lambda_{ab\,\dot{a},r}\epsilon_{r}^{\dot{a}}\bar{\pi}^{a \, \dot{b}}_{r} \bar{\epsilon}_{r \, \dot{b}} + \lambda^{+}_{r\,\dot{a}}\epsilon^{\dot{a}}_{r}\bar{\pi}_{r}^{b\,\cdot{b}}\bar{\epsilon}_{\dot{b}}X_{b}= k_{r\, a}X^{a} \epsilon_{r}^{\dot{a}} \bar{\epsilon}_{r \, \dot{a}} + k^{r\,b}X_{b}\epsilon^{\dot{a}}_{r}\bar{\epsilon}_{\dot{a}} = k_{r} \cdot X$, where we used \eqref{lambda-pibar-andk} in the second equality, and that the exponential appears on the support of the single $\bar{\delta}(\epsilon_{r}^{\dot{a}}\bar{\epsilon}_{r\,\dot{a}} - 1)$ delta function in \eqref{vertexoperator} to write down the last equality. As we will see in the next section, the delta function \eqref{deltafunction} will also give rise to the correct ten-dimensional momentum conservation delta function and the standard CHY scattering equations, thus providing further evidence on the validity of our proposal \eqref{vertexoperator}.
In order to construct vertex operators in different pictures, we define a picture-raising operator in the usual way:
\begin{equation}
Z \coloneqq \{Q,\xi\} = e^{\phi}B.
\end{equation}
We will be particularly interested in the picture number zero vertex to discuss scattering amplitudes. This is easily calculated to be
\begin{align}
U^{(0)}_{r} & = \int d^{2}z_{r}\, \Sigma^{\epsilon}_{r} \, \bigg[\lim_{w\rightarrow z_{r}}Z(w) \phi_{r,K}^{\epsilon}(\mathcal{Z}(z_{r}))e^{-\phi}J^{K}\bigg] =
\int d^{2}z_{r}\,\Sigma^{\epsilon}_{r} \,B_{-1}(\phi_{r,K}^{\epsilon}(\mathcal{Z}))J^{K}. \label{picturenumberzerotwistorvertex}
\end{align}
The bosonic and fermionic sectors of the picture number zero vertex operator then read
\begin{eqnarray}\label{gluonpicturezero}
U^{(0)}_{bos.\,r} &=& \int\,d^{2}z_{r}\, \Sigma^{\epsilon}_{r}\,\bigg(2(\lambda\gamma_{m}\bar{\lambda})a^{m}_{-\,r,K}\nonumber\\
&& + 2M_{mn}(\lambda_{r}\gamma^{m}\bar{\pi}_{r})a^{n}_{-\,r,K} + \frac{1}{4}(\bar{\pi}_{r}\gamma_{mnpqs}\bar{\pi}_{r})(\lambda\gamma^{m}\bar{\lambda})\Gamma^{n}\Gamma^{p}h^{q}a^{s}_{+\,r,K} \nonumber\\
&& + \frac{1}{12}(\bar{\pi}_{r}\gamma_{npqst}\bar{\pi}_{r})\bar{\Gamma}^{m}\Gamma^{n}\Gamma^{p}\Gamma^{q}h^{s}(\lambda_{r}\gamma_{m}\bar{\pi}_{r})a^{t}_{+\,r,K}\bigg)J^{K}e^{\mu_{a}\bar{\pi}^{a \, \dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}} \lambda^{+}_{r\,\dot{a}}\epsilon^{\dot{a}}_{r}/\lambda^{+}}, \nonumber \\[1.0ex]
U^{(0)}_{fer.\,r} &=& \int\,d^{2}z_{r}\, \Sigma^{\epsilon}_{r} \, \bigg(\bar{\Gamma}^{m}(\lambda_{r}\gamma_{m}\bar{\pi}_{r})\bar{s}_{r,K}\nonumber\\&& - 8(\lambda\gamma^{m}\bar{\lambda})\Gamma^{n}s_{mn,K}- 4\bar{\Gamma}^{m}\Gamma_{n}\Gamma_{p}(\lambda_{r}\gamma_{m}\bar{\pi}_{r})s^{np}_{r,K} - \frac{1}{6}(\bar{\pi}_{r}\gamma_{mnpqs}\bar{\pi}_{r})(\lambda\gamma^{m}\bar{\lambda})\Gamma^{n}\Gamma^{p}\Gamma^{q}h^{s}s_{r,K}\nonumber\\
&& - \frac{1}{24}(\bar{\pi}_{r}\gamma_{npqst}\bar{\pi}_{r})\bar{\Gamma}^{m}\Gamma^{n}\Gamma^{p}\Gamma^{q}\Gamma^{s}h^{t}(\lambda_{r}\gamma_{m}\bar{\pi}_{r})s_{t,K}\bigg)e^{\mu_{a}\bar{\pi}^{a \, \dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}}\lambda^{+}_{r\,\dot{a}}\epsilon^{\dot{a}}_{r}/\lambda^{+}}. \label{picturenumberzerooperators}
\end{eqnarray}
Here, $M^{mn} \coloneqq 2\bar{\Gamma}^{[m}\Gamma^{n]}$, satisfies the same Kac-Moody algebra inside correlation functions as the RNS fermionic Lorentz currents; namely,
\begin{equation}
\big \llangle M^{mn}_{(z)}M^{pq}_{(w)} \big \rrangle= \frac{\eta^{mp}\eta^{nq} - \eta^{mq}\eta^{np}}{(z-w)^{2}} + \frac{\eta^{mp}M^{nq} - \eta^{mq}M^{np} + \eta^{nq}M^{mp} - \eta^{np}M^{mq}}{(z-w)}. \label{KacMoodyforMmn}
\end{equation}
This follows from the presence of the delta functions \eqref{deltafunction} and careful manipulations using the techniques outlined in Appendix \ref{AppendixA}.
\section{Scattering Amplitudes} \label{section4}
The tree level $N$-point correlation function will be defined to be
\begin{eqnarray}\label{amplitudeprescription}
\mathcal{A}_{N} &=& \int \frac{d^{3}\gamma_{0}\,d^{11}\lambda\,d^{5}\mu\,d^{5}\Gamma}{SL(2,\mathbb{C})\times GL(1)\times \mathbb{M}}\,\prod_{i=1}^{3} U_{i}^{(-1)} \prod_{j=4}^{N} U_{j}^{(0)},
\end{eqnarray}
where the groups $SL(2,\mathbb{C})$, $GL(1)$, $\mathbb{M}$ correspond to the symmetry groups whose generators are given by $T$, $J$, $\tilde{\mathcal{M}}^{abc}$, respectively. As is well-known, the number of killing vectors of $SL(2,\mathbb{C})$ and $GL(1)$ are three and one, respectively. On the other hand, the number of generators for $\mathbb{M}$ will be calculated here from the effective counting of zero modes for the system $\tilde{\mathcal{M}}^{abc}$ considering its reducibilities\footnote{Although we do not provide a rigorous proof for this statement, it seems to follow from a natural definition of the path integral measure in the presence of reducible symmetries. More explicitly, if a symmetry group $G$ is reducible under a subgroup $H$, we interpret its effective action into the path integral as: $<\ldots> = \int \frac{1}{\frac{Vol [G]}{Vol [H]}}\ldots$.}. Thus, one finds that $\mathbb{M}$ has $10\cdot 3 - 5\cdot 5 + 1\cdot 7 = 12$ killing vectors, which as we will see is exactly the number of zero modes needed to get the correct ten-dimensional momentum conservation delta function. Furthermore, the presence of three vertices in picture number -1 and the rest in picture number zero adequately saturate the three zero modes of the bosonic ghost $\gamma$. The measure associated to the twistor variable $\lambda^{\alpha}$ is the same as the one appearing in ordinary pure spinor strings \cite{Berkovits:2004px}, while for $\Gamma^{m}$ one has
\begin{eqnarray}\label{measureforGamma}
\int d^{5}\Gamma &=& \frac{1}{5!}(\lambda\gamma^{mnpqr}\lambda)\frac{\partial}{\partial\Gamma^{m}}\frac{\partial}{\partial\Gamma^{n}}\frac{\partial}{\partial\Gamma^{p}}\frac{\partial}{\partial\Gamma^{q}}\frac{\partial}{\partial\Gamma^{r}},
\end{eqnarray}
which respects the constraints \eqref{psconstraints2}.
One can also view $\lambda^{\alpha}$ together with the $GL(1)$ symmetry as being a projective pure spinor. The integration measure for a projective pure spinor variable has been studied in \cite{Berkovits:2004bw}, and for the $D=10$ case reads
\begin{equation}\label{10dprojectivemeasure}
[d^{10}\lambda] = \frac{\epsilon_{\alpha_{1}\ldots\alpha_{16}}}{(\lambda^{\alpha}C_{\alpha})^{3}}d\lambda^{\alpha_{1}}\wedge\ldots\wedge d\lambda^{\alpha_{10}}\lambda^{a_{11}}(\gamma^{m}C)^{\alpha_{12}}(\gamma^{n}C)^{\alpha_{13}}(\gamma^{p}C)^{\alpha_{14}}(\gamma_{mnp})^{\alpha_{15}\alpha_{16}},
\end{equation}
where $C_{\alpha}$ is a constant spinor. Using the fact that $\lambda^{\alpha}$ is a pure spinor, one readily shows that the projective measure \eqref{10dprojectivemeasure} is independent of the choice of $C_{\alpha}$, and therefore is Lorentz-invariant.
Let us now see how the momentum conservation delta function emerges from \eqref{amplitudeprescription}. Using \eqref{lambda-pibar-andk} and the gauge described below \eqref{twistorsuperfield}, we can write
\begin{equation} \label{su5momenta}
\lambda^{+}_{r \, \dot{a}} \epsilon^{\dot{a}}_{r} \bar{\pi}^{a \, \dot{b}}_{r} \bar{\epsilon}_{r \, \dot{b}} = k_{r}^{a} \epsilon^{\dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}}, \ \ \ \lambda_{r \, ab \, \dot{a}} \epsilon^{\dot{a}}_{r} \bar{\pi}^{b \, \dot{b}}_{r} \bar{\epsilon}_{r \, \dot{b}} = k_{r \, a} \epsilon^{\dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}},
\end{equation}
which means that the left hand sides can be replaced by $k^{a}_{r}$ or $k_{r \, a}$ correspondingly, on the support of the single $\bar{\delta}(\epsilon_{r}^{\dot{a}}\bar{\epsilon}_{r\,\dot{a}} - 1)$ delta function in \eqref{deltafunction}. This is useful after integrating out the zero modes associated to the $\mu_{a}$ field, where one is left with
\begin{equation}
\delta^{(5)}\bigg(\sum_{r} \bar{\pi}^{a \, \dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}}\lambda^{+}_{r\,\dot{b}}\epsilon^{\dot{b}}_{r}/\lambda^{+} \bigg) = \lambda^{+} \delta^{(5)} \bigg( \sum_{r} \bar{\pi}^{a \, \dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}}\lambda^{+}_{r\,\dot{b}}\epsilon^{\dot{b}}_{r} \bigg) = \lambda^{+} \delta^{(5)}\bigg(\sum_{r} k^{a}_{r}\bigg),
\end{equation}
where we used the single delta function in \eqref{deltafunction} to write the last equality. Thus, we see that we have recovered five of the ten momentum conservation delta functions. The remaining five conditions follow from the observation that one has $N$ + 10 integration variables, $12 + 3 = 15$ killing vectors associated with $\mathbb{M}$ and $SL(2,\mathbb{C})$, and $N$ delta functions in \eqref{amplitudeprescription}, which means one is left with $N -(N + 10 - 15) = 5$ delta functions. These are exactly the five delta functions imposing the remaining momentum conservation conditions.
The integration over the non-zero modes of $\mu_{a}$ gives rise to the standard scattering equations. This follows from the standard procedure of taking the exponential in the vertex operators into the action as sources for $\bar{\lambda}^{a}$. After setting $\lambda^{+} = 1$, one finds that:
\begin{equation}
\bar{\lambda}^{a}(z) = \sum_{r=1}^{N} \frac{\bar{\pi}^{a \, \dot{a}}_{r} \bar{\epsilon}_{r \, \dot{a}}\lambda^{+}_{r\,\dot{b}}\epsilon^{\dot{b}}_{r}}{(z-z_{r})}.
\end{equation}
Using this expression, one can compute the momentum $P^{m}(z) = (P^{a}(z), P_{a}(z))$. These $SU(5)$ components read
\begin{equation}
P^{a}(z) = \sum_{r=1}^{N}\frac{k_{r}^{a}}{z-z_{r}}, \ \ \ P_{a}(z) = \sum_{r=1}^{N}\frac{\lambda_{ab}(z_{r})\bar{\pi}_{r}^{b \dot{a}}\bar{\epsilon}_{r\,\dot{a}}\lambda^{+}_{r\,\dot{b}}\epsilon^{\dot{b}}_{r}}{z-z_{r}},
\end{equation}
where for $P^{a}(z)$ the first of \eqref{su5momenta} was used and that one is on the support of \eqref{deltafunction}. Using the ten-dimensional delta in \eqref{deltafunction}, one can work out $P_{a}(z)$:
\begin{eqnarray}
P^{a}(z) = \sum_{r=1}^{N}\frac{k_{r}^{a}}{z-z_{r}}, \ \ \ P_{a}(z) = \sum_{r=1}^{N}\frac{\lambda_{r\,ab\,\dot{c}}\epsilon_{r}^{\dot{c}}\bar{\pi}^{b\dot{b}}\bar{\epsilon}_{r\,\dot{b}} }{z-z_{r}} = \sum_{r=1}^{N}\frac{k_{r\,a}}{z-z_{r}},
\end{eqnarray}
where the second of \eqref{su5momenta} was used along with the single delta in \eqref{deltafunction}. Thus, one concludes that the delta function \eqref{deltafunction} is proportional to the standard CHY delta function $\bar{\delta}(k_{r}\cdot P(z_{r}))$.
We now move on to discuss the dependence of \eqref{amplitudeprescription} on the external polarization and momentum data. The simplest case, the 3-point function, has already been shown to correctly reproduce the standard super-Yang-Mills 3-point function in \cite{Berkovits:2009by}. We then focus here on the general case. To see that the amplitudes prescription \eqref{amplitudeprescription} indeed describes $D=10$ super-Yang-Mills interactions, we notice a close relationship between the ambitwistor string constructed in this work and the infinite tension limit of ordinary pure spinor superstrings \cite{Berkovits:2013xba}. This relation can be seen as follows: using the incidence relations \eqref{incidencerelationsps} and the constraints \eqref{psconstraints}, we see that the measure \eqref{measureforGamma} is nothing but the ordinary pure spinor measure
\begin{equation} \label{standardpurespinormeasure}
\langle(\lambda\gamma^{m}\theta)(\lambda\gamma^{n}\theta)(\lambda\gamma^{p}\theta)(\theta\gamma_{mnp}\theta)\rangle = 1,
\end{equation}
up to some proportionality factor. The current algebra systems will certainly provide the same contributions in both models, so we concentrate on the sector containing the twistor superfield. It is straightforward to show that $\phi_{r,K}(Z) = V(\lambda,\theta)\,e^{k \cdot X}$ on the support of the incidence relations and the delta function \eqref{deltafunction}, where $V(\lambda,\theta) = \lambda^{\alpha}A_{\alpha}(\theta)$ is the usual pure spinor unintegrated vertex operator. Under analogous statements, the picture number zero twistor vertex is the same as the pure spinor integrated vertex operator $U(x,\theta) = [P^{m}A_{m}(\theta) + d_{\alpha}W^{\alpha}(\theta) + \frac{1}{2}N^{mn}F_{mn}(\theta)]e^{k \cdot X}$. This is essentially a direct consequence of the fact that $B$ can be shown to be proportional to the pure spinor $b$-ghost, and so the picture raising operation \eqref{picturenumberzerotwistorvertex} is nothing but the standard relation between the integrated and unintegrated vertex operators $U = \{b, V\}$. Thus, we conclude that both correlators must give the same dependence on external momentum and polarization data.
As a check, let us consider the fully gluonic correlator. The potential difference between the present model and the ordinary pure spinor formalism lies in the picture number zero vertex \eqref{picturenumberzerotwistorvertex} and the standard integrated vertex operator. The latter has the $\theta$-expansion
\begin{equation}\label{psintegratedvertex}
U(x,\theta) = \bigg[P^{m}a_{m} + \frac{1}{2}\bigg(p\gamma^{mn}\theta + \lambda\gamma^{mn}w\bigg)k_{m}a_{n} + \ldots\bigg]e^{k \cdot X},
\end{equation}
where $\ldots$ means higher-order terms in $\theta^{\alpha}$ which can be ignored as far as scattering amplitudes is concerned by $(p_{\alpha},\theta^{\beta})$ charge conservation. The only difference between the picture number zero twistor vertex \eqref{picturenumberzerooperators} and \eqref{psintegratedvertex} is the Lorentz current inside the parenthesis in \eqref{psintegratedvertex}, which forms a level 1 Kac-Moody algebra. However, the current $M^{mn}$ in \eqref{picturenumberzerooperators} is also a level 1 Kac-Moody current algebra system as discussed below eqn. \eqref{KacMoodyforMmn}. Higher-order terms in $\Gamma^{m}$ in \eqref{picturenumberzerotwistorvertex} can also be ignored because of $(\Gamma^{m}, \bar{\Gamma}_{n})$ charge conservation, or alternatively, by using supersymmetry arguments, and so the OPEs of both models are identical. One then concludes that both correlators are equivalent to each other.
\section{Discussions and Future Directions} \label{section5}
In this work, a new description of ambitwistor strings which makes use of a set of variables and a set of reducible constraints first introduced in \cite{Berkovits:2009by, Sepulveda:2020kjc} has been presented. A detailed quantum-mechanical analysis was performed, and after introducing the ghost system associated to the fermionic symmetry $B$, a simple BRST operator was constructed. At this stage, one might wonder why one just fixes one of the gauge symmetries and leaves all the others unfixed. It turns out that if one does so, the BRST operator takes a similar form as the one found in \cite{Sepulveda:2020kjc} in the context of the superparticle. Explicitly:
\begin{eqnarray}\label{psQ}
Q &=& c T_{\textrm{full}} + \sigma J + \gamma B + f_{abc}\tilde{\mathcal{M}}^{abc} + \sigma \gamma\beta + s_{abcd} \Big[ \tilde{f}^{abc}P^{d} + \frac{1}{4!}\epsilon^{abcde}\tilde{q}_{e}\lambda^{+}\beta \Big] \nonumber \\[0.5ex]
&& + t_{abcde}P^{a}\tilde{s}^{bcde} + \frac{1}{5!}(\lambda^{+})^{2}\epsilon^{abcde}t_{abcde}\beta^{2},
\end{eqnarray}
where ghost-for-ghost fields have been introduced for each reducibility that the set of constraints satisfies. Thus the zeroth generation of constraints $J$, $B$, $T$, $\tilde{\mathcal{M}}^{abc}$ requires the introduction of the ghosts $(\sigma, \tilde{\sigma})$, $(\beta, \gamma)$, $(b,c)$, $(f_{abc}, \tilde{f}^{abc})$, respectively. The reducibility relations $\eqref{red1}$, \eqref{red2} in turn imply the presence of the ghosts-for-ghosts $(s_{abcd}, \tilde{s}^{abcd})$, $(t_{abcde},\tilde{t}^{abcde})$. Moreover, $T_{\textrm{full}}$ in \eqref{psQ} means the full stress-energy tensor
\begin{eqnarray}
T_{\textrm{full}} &=& T + \frac{1}{2}T_{bc} + T_{\beta\gamma} + T_{\sigma\tilde{\sigma}} + T_{f_{3}\tilde{f}_{3}} + T_{s_{4}\tilde{s}_{4}} + T_{t_{5}\tilde{t}_{5}},
\end{eqnarray}
with
\begin{align}
&T_{bc} = -\partial b c - 2 b\partial c, \hspace{3.3cm} T_{\beta\gamma} = -\partial\beta \gamma - 2\beta\partial\gamma, \nonumber \\
&T_{\sigma\tilde{\sigma}} = - \tilde{\sigma}\partial\sigma, \hspace{4.3cm}
T_{f_{3}\tilde{f}_{3}} = -\partial f^{abc} f_{abc} - 2 \tilde{f}^{abc}\partial f_{abc}, \nonumber\\
&T_{s_{4}\tilde{s}_{4}} = -2\partial \tilde{s}^{abcd} s_{abcd} - 3\tilde{s}^{abcd}\partial s_{abcd}, \ \ \ T_{t_{5}\tilde{t}_{5}} = -3\partial\tilde{t}^{abcde} t_{abcde} - 4\tilde{t}^{abcde}\partial t_{abcde}.
\end{align}
The BRST operator thus constructed is nilpotent up to terms proportional to $\sigma \partial\sigma$ and $\sigma \partial^{2}c$. These terms arise from contributions involving double contractions with $\sigma\gamma\beta$ in \eqref{psQ} as well as similar contributions related to the anomalous behaviour of $J$, displayed in eqns. \eqref{JJope}, \eqref{TJope}. It is intriguing to see that although the total central charge vanishes, the anomalous behavior of $J$ spoils the nilpotency of the BRST operator. It would be very interesting to study this issue further in the future.
In this work we also found vertex operators in different pictures and they turned out to have the same structure as the ones proposed in \cite{Berkovits:2009by} inspired by the ordinary pure spinor formalism. Here, integrated vertex operators emerged naturally from a simple picture raising operation. Unlike the construction discussed in \cite{Berkovits:2009by}, we dressed up all vertices with a delta function that fixed the dependence of the pure spinor variable in terms of external momentum data. Such a construction relied on the redundant symmetry arising from the twistor-like constraint $(\lambda\gamma^{m})_{\alpha}P_{m} = 0$. The vertices so constructed then resemble those proposed in \cite{Geyer:2019ayz}, and it would thus be interesting to study if the ideas presented in this work could lead to a better understanding of the BRST and critical structure of the ten-dimensional model presented in \cite{Geyer:2019ayz}.
A correlation function measure was then defined in the usual way by integrating over the zero modes of the worldsheet twistor fields and quotienting by the respective symmetry groups of the model. The integration over the modes of $\mu_{a}$ then yielded the 10-dimensional momentum conservation delta function, and the delta functions \eqref{deltafunction} realized the usual scattering equations. Likewise, the polarization and momentum dependence of the correlator \eqref{amplitudeprescription} straightforwardly followed from the close relation between the standard pure spinor formalism and the model constructed here. As a result, the heterotic pure spinor twistor correlator \eqref{amplitudeprescription} gives the same results as the ones obtained from ordinary RNS or pure spinor ambitwistor strings.
The generalization to Type IIB is immediate. The only change to do is to replace the current algebra system by a twistor superfield depending on $\hat{\Gamma}^{m}$ \eqref{psambiactiontypeIIB} and different polarization vectors and spinors. NS-NS, NS-R, R-NS, R-R states are then obtained from the tensor product of the two twistor superfields, and the equivalence of the amplitude \eqref{amplitudeprescription} with standard results easily follows from the fermionic measures for $\Gamma^{m}$ and $\hat{\Gamma}^{m}$ as in \eqref{measureforGamma}. One might also try to describe further CHY models using different matter systems in the pure spinor twistor action \eqref{psambiaction} in a similar manner as done in \cite{Casali:2015vta}. We pretend to explore this further in future work.
\acknowledgments
We are thankful to Nathan Berkovits for discussions and comments on the draft. M.G. would like to thank Renann Jusinskas, Oliver Schlotterer and Lionel Mason for valuable and enlightening discussions. D.G.S would like to thank the Abdus Salam International Centre for Theoretical Physics, ICTP-SAIFR/IFT- UNESP, FAPESP grant 2016/01343-7, CAPES-PROEX, and Perimeter Institute for partial financial support. M.G. was supported by the European Research Council under ERC-STG-804286 UNISCAMP. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science, and Economic Development, and by the Province of Ontario through the Ministry of Research and Innovation.
|
1,116,691,500,470 | arxiv | \section{Introduction}
Given the advent of Graphics Processing Units (GPUs), deep convolutional neural networks (CNNs) with billions of floating number multiplications could receive speed-ups and make important strides in a large variety of computer vision tasks, \eg image classification~\cite{VGG,krizhevsky2012imagenet}, object detection~\cite{ren2015faster}, segmentation~\cite{long2015fully}, and human face verification~\cite{wen2016discriminative}. However, the high-power consumption of these high-end GPU cards (\eg 250W+ for GeForce RTX 2080 Ti) has blocked modern deep learning systems from being deployed on mobile devices, \eg smart phone, camera, and watch. Existing GPU cards are far from svelte and cannot be easily mounted on mobile devices. Though the GPU itself only takes up a small part of the card, we need many other hardware for supports, \eg memory chips, power circuitry, voltage regulators and other controller chips. It is therefore necessary to study efficient deep neural networks that can run with affordable computation resources on mobile devices.
Addition, subtraction, multiplication and division are the four most basic operations in mathematics. It is widely known that multiplication is slower than addition, but most of the computations in deep neural networks are multiplications between float-valued weights and float-valued activations during the forward inference. There are thus many papers on how to trade multiplications for additions, to speed up deep learning. The seminal work~\cite{courbariaux2015binaryconnect} proposed BinaryConnect to force the network weights to be binary (\eg -1 or 1), so that many multiply-accumulate operations can be replaced by simple accumulations. After that, Hubara~\etal~\cite{hubara2016binarized} proposed BNNs, which binarized not only weights but also activations in convolutional neural networks at run-time. Moreover, Rastegari~\etal~\cite{rastegari2016xnor} introduced scale factors to approximate convolutions using binary operations and outperform~\cite{hubara2016binarized,rastegari2016xnor} by large margins. Zhou~\etal~\cite{zhou2016dorefa} utilized low bit-width gradient to accelerate the training of binarized networks. Cai~\etal~\cite{cai2017deep} proposed an half-wave Gaussian quantizer for forward approximation, which achieved much closer performance to full precision networks.
Though binarizing filters of deep neural networks significantly reduces the computation cost, the original recognition accuracy often cannot be preserved. In addition, the training procedure of binary networks is not stable and usually requests a slower convergence speed with a small learning rate. Convolutions in classical CNNs are actually cross-correlation to measure the similarity of two inputs. Researchers and developers are used to taking convolution as a default operation to extract features from visual data, and introduce various methods to accelerate the convolution, even if there is a risk of sacrificing network capability. But there is hardly no attempt to replace convolution with another more efficient similarity measure that is better to only involve additions. In fact, additions are of much lower computational complexities than multiplications. Thus, we are motivated to investigate the feasibility of replacing multiplications by additions in convolutional neural networks.
\begin{figure*}[t]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.48\linewidth]{figs/ann.png} &
\quad \includegraphics[width=0.48\linewidth]{figs/cnn.png} \\
(a) Visualization of features in AdderNets &(b) Visualization of features in CNNs \\
\end{tabular}
\caption{Visualization of features in AdderNets and CNNs. Features of CNNs in different classes are divided by their angles. In contrast, features of AdderNets tend to be clustered towards different class centers, since AdderNets use the $\ell_1$-norm to distinguish different classes. The visualization results suggest that $\ell_1$-distance can served as a similarity measure the distance between the filter and the input feature in deep neural networks}
\label{Fig:visualfea}
\vspace{-1em}
\end{figure*}
In this paper, we propose adder networks that maximize the use of addition while abandoning convolution operations. Given a series of small template as ``filters’’ in the neural network, $\ell_1$-distance could be an efficient measure to summarize absolute differences between the input single and the template as shown in Figure~\ref{Fig:visualfea}. Since subtraction can be easily implemented through addition by using its complement code, $\ell_1$-distance could be a hardware-friendly measure that only has additions, and naturally becomes an efficient alternative of the convolution to construct neural networks. An improved back-propagation scheme with regularized gradients is designed to ensure sufficient updates of the templates and a better network convergence. The proposed AdderNets are deployed on several benchmarks, and experimental results demonstrate AdderNets’ advantages in accelerating inference of deep neural networks while preserving comparable recognition accuracy to conventional CNNs.
This paper is organized as follows. Section~\ref{sec:related} investigates related works on network compression. Section~\ref{sec:method} proposes Adder Networks which replace the multiplication in the conventional convolution filters with addition. Section~\ref{sec:experi} evaluates the proposed AdderNets on various benchmark datasets and models and Section~\ref{sec:conclu} concludes this paper.
\section{Related works}\label{sec:related}
To reduce the computational complexity of convolutional neural networks, a number of works have been proposed for eliminating useless calculations.
Pruning based methods aims to remove redundant weights to compress and accelerate the original network. Denton~\etal~\cite{SVD} decomposed weight matrices of fully-connected layers into simple calculations by exploiting singular value decomposition (SVD). Han~\etal~\cite{han2015deep} proposed discarding subtle weights in pre-trained deep networks to omit their original calculations without affecting the performance. Wang~\etal~\cite{wang2016cnnpack} further converted convolution filters into the DCT frequency domain and eliminated more floating number multiplications. In addition, Hu~\etal~\cite{Trimming} discarded redundant filters with less impacts to directly reduce the computations brought by these filters. Luo~\etal~\cite{luo2017thinet} discarded redundant filters according to the reconstruction error. He~\etal~\cite{he2017channel} utilized a LASSO regression to select important channels by solving least square reconstruction. Zhuang~\etal~\cite{zhuang2018discrimination} introduce additional losses to consider the discriminative power of channels and selected the most discriminative channels for the portable network.
Instead of directly reducing the computational complexity of a pre-trained heavy neural network, lot of works focused on designing novel blocks or operations to replace the conventional convolution filters. Iandola~\etal~\cite{iandola2016squeezenet} introduced a bottleneck architecture to largely decrease the computation cost of CNNs. Howard~\etal~\cite{howard2017mobilenets} designed MobileNet, which decompose the conventional convolution filters into the point-wise and depth-wise convolution filters with much fewer FLOPs. Zhang~\etal~\cite{zhang2018shufflenet} combined group convolutions~\cite{ResNeXt} and a channel shuffle operation to build efficient neural networks with fewer computations. Hu~\etal~\cite{hu2018squeeze} proposed the squeeze and excitation block, which focuses on the relationship of channels by modeling interdependencies between channels, to improve the performance at slight additional computational cost. Wu~\etal~\cite{wu2018shift} presented a parameter-free ``shift" operation with zero flop and zero parameter to replace conventional filters and largely reduce the computational and storage cost of CNNs. Zhong~\etal~\cite{zhong2018shift} further pushed the shift-based primitives into channel shift, address shift and shortcut shift to reduce the inference time on GPU while keep the performance. Wang~\etal~\cite{Versatile} developed versatile convolution filters to generate more useful features utilizing fewer calculations and parameters.
Besides eliminating redundant weights or filters in deep convolutional neural networks, Hinton~\etal~\cite{hinton2015distilling} proposed the knowledge distillation (KD)
scheme, which transfer useful information from a heavy teacher network to a portable student network by minimizing the Kullback-Leibler divergence between their outputs. Besides mimic the final outputs of the teacher networks, Romero~\etal~\cite{romero2014fitnets} exploit the hint layer to distill the information in features of the teacher network to the student network. You~\etal~\cite{you2017learning} utilized multiple teachers to guide the training of the student network and achieve better performance. Yim~\etal~\cite{yim2017gift} regarded the relationship between features from two layers in the teacher network as a novel knowledge and introduced the FSP (Flow of Solution Procedure) matrix to transfer this kind of information to the student network.
Nevertheless, the compressed networks using these algorithms still contain massive multiplications, which costs enormous computation resources. As a result, subtractions or additions are of much lower computational complexities when compared with multiplications. However, they have not been widely investigated in deep neural networks, especially in the widely used convolutional networks. Therefore, we propose to minimize the numbers of multiplications in deep neural networks by replacing them with subtractions or additions.
\section{Networks without Multiplication}\label{sec:method}
Consider a filter $F\in \mathbb{R}^{d\times d\times c_{in}\times c_{out}}$ in an intermediate layer of the deep neural network, where kernel size is $d$, input channel is $c_{in}$ and output channel is $c_{out}$. The input feature is defined as $X\in \mathbb{R}^{H\times W \times c_{in}}$, where $H$ and $W$ are the height and width of the feature, respectively. The output feature $Y$ indicates the similarity between the filter and the input feature,
\begin{equation}
\small
Y(m,n,t) = \sum_{i=0}^{d}\sum_{j=0}^{d}\sum_{k=0}^{c_{in}} S\big(X(m+i,n+j,k), F(i,j,k,t)\big), \label{fcn:conv}
\end{equation}
where $S(\cdot, \cdot)$ is a pre-defined similarity measure. If cross-correlation is taken as the metric of distance, \ie $S(x, y) = x \times y$, Eq. (\ref{fcn:conv}) becomes the convolution operation. Eq. (\ref{fcn:conv}) can also implies the calculation of a fully-connected layer when $d=1$. In fact, there are many other metrics to measure the distance between the filter and the input feature. However, most of these metrics involve multiplications, which bring in more computational cost than additions.
\subsection{Adder Networks}
We are therefore interested in deploying distance metrics that maximize the use of additions. $\ell_1$ distance calculates the sum of the absolute differences of two points’ vector representations, which contains no multiplication. Hence, by calculating $\ell_1$ distance between the filter and the input feature, Eq. (\ref{fcn:conv}) can be reformulated as
\begin{equation}
\small
Y(m,n,t) = -\sum_{i=0}^{d}\sum_{j=0}^{d}\sum_{k=0}^{c_{in}} \vert X(m+i,n+j,k) - F(i,j,k,t)\vert. \label{fcn:l1}
\end{equation}
Addition is the major operation in $\ell_1$ distance measure, since subtraction can be easily reduced to addition by using complement code. With the help of $\ell_1$ distance, similarity between the filters and features can be efficiently computed.
Although both $\ell_1$ distance (Eq. (\ref{fcn:l1}) and cross-correlation in Eq. (\ref{fcn:conv}) can measure the similarity between filters and inputs, there are some differences in their outputs. The output of a convolution filter, as a weighted summation of values in the input feature map, can be positive or negative, but the output of an adder filter is always negative. Hence, we resort to batch normalization for help, and the output of adder layers will be normalized to an appropriate range and all the activation functions used in conventional CNNs can then be used in the proposed AdderNets. Although the batch normalization layer involves multiplications, its computational cost is significantly lower than that of the convolutional layers and can be omitted. Considering a convolutional layer with a filter $F\in \mathbb{R}^{d\times d\times c_{in}\times c_{out}}$, an input $X\in \mathbb{R}^{H\times W \times c_{in}}$ and an output $Y\in \mathbb{R}^{H'\times W' \times c_{out}}$, the computation complexity of convolution and batch normalization is $\mathcal{O}(d^2c_{in}c_{out}HW)$ and $\mathcal{O}(c_{out}H'W')$, respectively. In practice, given an input channel number $c_{in}=512$ and a kernel size $d=3$ in ResNet~\cite{he2016deep}, we have $\frac{d^2c_{in}c_{out}HW}{c_{out}H'W'}\approx 4068$. Since batch normalization layer has been widely used in the state-of-the-art convolutional neural networks, we can simply upgrade these networks into AddNets by replacing their convolutional layers into adder layers to speed up the inference and reduces the energy cost.
Intuitively, Eq. (\ref{fcn:conv}) has a connection with template matching~\cite{brunelli2009template} in computer vision, which aims to find the parts of an image that match the template. $F$ in Eq. (\ref{fcn:conv}) actually works as a template, and we calculate its matching scores with different regions of the input feature $X$. Since various metrics can be utilized in template matching, it is natural that $\ell_1$ distance can be utilized to replace the cross-correlation in Eq. (\ref{fcn:conv})
\subsection{Optimization}
Neural networks utilize back-propagation to compute the gradients of filters and stochastic gradient descent to update the parameters. In CNNs, the partial derivative of output features $Y$ with respect to the filters $F$ is calculated as:
\begin{equation}
\frac{\partial Y(m,n,t)}{\partial F(i,j,k,t)} = X(m+i,n+j,k),
\end{equation}
where $i\in [m,m+d]$ and $j \in [n,n+d]$. To achieve a better update of the parameters, it is necessary to derive informative gradients for SGD. In AdderNets, the partial derivative of $Y$ with respect to the filters $F$ is:
\begin{equation}
\frac{\partial Y(m,n,t)}{\partial F(i,j,k,t)} = \mbox{sgn} (X(m+i,n+j,k) - F(i,j,k,t)), \label{fcn:l1bp}
\end{equation}
where $\mbox{sgn}(\cdot)$ denotes the sign function and the value of the gradient can only take +1, 0, or -1.
Considering the derivative of $\ell_2$-norm
\begin{equation}
\frac{\partial Y(m,n,t)}{\partial F(i,j,k,t)} = X(m+i,n+j,k) - F(i,j,k,t), \label{fcn:l2bp}
\end{equation}
Eq. (\ref{fcn:l1bp}) can therefore lead to a signSGD~\cite{bernstein2018signsgd} update of $\ell_2$-norm. However, signSGD almost never takes the direction of steepest descent and the direction only gets worse as dimensionality grows~\cite{bernstein2018convergence}. It is unsuitable to optimize the neural networks of a huge number of parameters using signSGD. Therefore, we propose using Eq. (\ref{fcn:l2bp}) to update the gradients in our AdderNets. The convergence of taking these two kinds of gradient will be further investigated in the supplementary material. Therefore, by utilizing the full-precision gradient, the filters can be updated precisely.
\iffalse
To further investigate the convergence of taking these two kinds of gradients, we make two propositions as follows.
\begin{proposition}
Denote an input patch as $x\in \mathbb{R}^n$ and a filter as $f\in \mathbb{R}^n$, the optimization problem is:
\begin{equation}
\mbox{arg}\min_f \vert x-f\vert. \label{optim}
\end{equation}
Given a fixed learning rate $\alpha$, this problem basically cannot converge to the optimal value using sign grad (Eq. (~\ref{fcn:l1bp})) via gradient descent.
\label{prop1}
\end{proposition}
\begin{proof}
The optimization problem~\ref{optim} can be rewritten as:
\begin{equation}
\mbox{arg}\min_{f_1,...,f_n} \sum_{i=1}^{n}\vert x_i-f_i\vert,
\end{equation}
where $x=\left\{x_1,...,x_n\right\}, f=\left\{f_1,...,f_n\right\}$. The update of $f_i$ using gradient descent is:
\begin{equation}
f_i^{j+1} = f_i^{j} - \alpha \mbox{sgn}(f_i^j-x_i),
\end{equation}
where $f_i^j$ denotes the $f_i$ in $j$th iteration. Without loss of generality, we assume that $f_i^0<x_i$. So we have:
\begin{equation}
f_i^{j+1} = f_i^{j} + \alpha = f_i^{j-1} + 2\alpha = ... = f_i^{0}+(j+1)\alpha,
\end{equation}
when $f_i^{j}<x_i$. Denote $t = \mbox{arg}\max_j f_i^j<x_i$, we have $f_i^{t+1}>=x_i$. If $f_i^{t+1}=f_i^0+(t+1)\alpha=x_i$ (\ie $\frac{(x_i-f_i^0)}{\alpha}=t+1$), $|f_i-x_i|$ can converge to the optimal value 0. However, if $f_i^{t+1}>x_i$, we have
\begin{equation}
\small
f_i^{t+2} = f_i^{t+1} - \alpha \mbox{sgn}(f_i^{t+1}-x_i) = f_i^0 + (t+1)\alpha - \alpha = f_i^t
\end{equation}
Similarly, we have $f_i^{t+3}=f_i^{t+1}$. Therefore, the inequality holds:
\begin{equation}
f_i^{t+2k} = f_i^{t} < x_i < f_i^{t+2k+1} ,k\in\mathbb{N}^+
\end{equation}
which demonstrate that the $f_i$ cannot converge and have an error of $x_i-f_i^t$ or $x_i-f_i^t$. The $f_i^{j}$ can converge to $x_i$ if and only if $\frac{(x_i-f_i^0)}{\alpha}\in\mathbb{Z}$, which is a strict constraint since $x_i,f_i,\alpha\in\mathbb{R}$. Moerover, the $f$ can converge to $x$ if and only if $\frac{(x_i-f_i^0)}{\alpha}\in\mathbb{Z}$ for each $f_i\in f$. The difficulty of converge increases when the number $n$ grows. In neural networks, the dimension of filters is can be very large. Therefore, problem~\ref{optim} basically cannot converge to its optimal value.
\end{proof}
The aim of filters is to find the most relevant part of input features, which meets the goal of Eq. (\ref{optim}). The $\alpha$ (\ie the learning rate of neural networks) can be seen as fixed when using multi-step learning rate, which is widely used in the training. According to the Proposition 1, if we use the sign gradient, the AdderNets will achieve a poor performance.
\begin{proposition}
For the optimization problem~\ref{optim}, $f$ can converge to the optimal value using full-precision gradient (Eq. (\ref{fcn:l2bp})) with a fixed learning rate $\alpha$ via gradient descent when $\alpha<1$.
\label{prop2}
\end{proposition}
\begin{proof}
The optimization problem~\ref{optim} can be rewritten as:
\begin{equation}
\mbox{arg}\min_{f_1,...,f_n} \sum_{i=1}^{n}\vert x_i-f_i\vert,
\end{equation}
where $x=\left\{x_1,...,x_n\right\}, f=\left\{f_1,...,f_n\right\}$. The update of $f_i$ using gradient descent is:
\begin{equation}
f_i^{j+1} = f_i^{j} - \alpha (f_i^j-x_i),
\end{equation}
where $f_i^j$ denotes the $f_i$ in $j$th iteration. If $f_i^j<x_i$, then we have the inequality:
\begin{equation}
f_i^{j+1} = f_i^{j} - \alpha (f_i^j-x_i) = (1-\alpha)f_i^j + \alpha x_i < x_i,
\end{equation}
and $f_i^{j+1}<f_i^j$. Without loss of generality, we assume that $f_i^0<x_i$. Then $f_i^j$ is monotone and bounded with respect to $j$, so the limit of $f_i^j$ exists and $\lim_{j \to +\infty } f_i^j\leq x_i$. Assume that $\lim_{j \to +\infty} f_i^j = l < x_i$. For $\epsilon = \alpha (x_i-l) $, there exists $k$ subject to $l-f_i^k<\epsilon$. Then we have:
\begin{equation}
\begin{aligned}
f_i^{k+1}&= f_i^k+\alpha(x_i-f_i^k)\geq
f_i^k+\alpha(x_i-l)\\&> l-\epsilon + alpha(x_i-l) = l,
\end{aligned}
\end{equation}
which is a contradiction. Therefore, $\lim_{j \to +\infty} f_i^j \geq x_i$. Finally, we have $\lim_{j \to +\infty} f_i^j = x_i$, \ie $f$ can converge to the optimal value.
\end{proof}
Therefore, by utilizing the full-precision gradient, the filters can be updated precisely.
\fi
Besides the gradient of the filters, the gradient of the input features $X$ is also important for the update of parameters. Therefore, we also use the full-precision gradient (Eq. (\ref{fcn:l2bp})) to calculate the gradient of $X$. However, the magnitude of the full-precision gradient may be larger than +1 or -1. Denote the filters and inputs in layer $i$ as $F_i$ and $X_i$. Different from $\frac{\partial Y}{\partial F_i}$ which only affects the gradient of $F_i$ itself, the change of $\frac{\partial Y}{\partial X_i}$ would influence the gradient in not only layer $i$ but also layers before layer $i$ according to the gradient chain rule. If we use the full-precision gradient instead of the sign gradient of $\frac{\partial Y}{\partial X}$ for each layer, the magnitude of the gradient in the layers before this layer would be increased, and the discrepancy brought by using full-precision gradient would be magnified. To this end, we clip the gradient of $X$ to $[-1,1]$ to prevent gradients from exploding. Then the partial derivative of output features $Y$ with respect to the input features $X$ is calculated as:
\begin{equation}
\small
\frac{\partial Y(m,n,t)}{\partial X(m+i,n+j,k)} = \mbox{HT}(F(i,j,k,t)-X(m+i,n+j,k)) .
\label{fcn:l2bpx}
\end{equation}
where $\mbox{HT}(\cdot)$ denotes the HardTanh function:
\begin{equation}
\mbox{HT}(x) =
\left\{
\begin{aligned}
& x \quad&\mbox{if} \quad -1<x<1,\\
& 1 &x>1,\\
& -1 &x<-1.
\end{aligned}
\right.
\end{equation}
\subsection{Adaptive Learning Rate Scaling}\label{sec:2.3}
In conventional CNNs, assuming that the weights and the input features are independent and identically distributed following normal distribution, the variance of the output can be roughly estimated as:
\begin{equation}
\begin{aligned}
Var[Y_{CNN}]& = \sum_{i=0}^{d}\sum_{j=0}^{d}\sum_{k=0}^{c_{in}} Var[X\times F] \\
&= d^2c_{in} Var[X]Var[F].
\end{aligned}
\label{varcnn}
\end{equation}
If variance of the weight is $Var[F]= \frac{1}{d^2c_{in}}$, the variance of output would be consistent with that of the input, which will be beneficial for the information flow in the neural network. In contrast, for AdderNets, the variance of the output can be approximated as:
\begin{equation}
\begin{aligned}
Var[Y_{AdderNet}] &=
\sum_{i=0}^{d}\sum_{j=0}^{d}\sum_{k=0}^{c_{in}} Var[| X -F|] \\
&= \sqrt{\frac{\pi}{2}}d^2c_{in} (Var[X]+Var[F]),
\end{aligned}
\label{varadd}
\end{equation}
when $F$ and $X$ follow normal distributions. In practice, the variance of weights $Var[F]$ is usually very small~\cite{glorot2010understanding}, \eg $10^{-3}$ or $10^{-4}$ in an ordinary CNN. Hence, compared with multiplying $ Var[X]$ with a small value in Eq. (\ref{varcnn}), the addition operation in Eq. (\ref{varadd}) tends to bring in a much larger variance of outputs in AdderNets.
We next proceed to show the influence of this larger variance of outputs on the update of AdderNets. To promote the effectiveness of activation functions, we introduce batch normalization after each adder layer. Given input $x$ over a mini-batch $\mathcal{B} = \left\{x_{1}, \cdots, x_{m}\right\}$, the batch normalization layer can be denoted as:
\begin{equation}
y = \gamma \frac{x-\mu_{\mathcal{B}}}{\sigma_{\mathcal{B}}} + \beta,
\end{equation}
where $\gamma$ and $\beta$ are parameters to be learned, and $\mu_{\mathcal{B}}= \frac{1}{m}\sum_i x_i$ and $\sigma^2_{\mathcal{B}}= \frac{1}{m}\sum_i (x_i-\mu_{\mathcal{B}})^2$ are the mean and variance over the mini-batch, respectively. The gradient of loss $\ell$ with respect to $x$ is then calculated as:
\begin{equation}
\small
\frac{\partial \ell}{\partial x_i} = \sum_{j=1}^{m} \frac{\gamma}{m^2 \sigma_{\mathcal{B}}} \left\{ \frac{\partial \ell}{\partial y_i} - \frac{\partial \ell}{\partial y_j}[ 1 + \frac{(x_i-x_j)(x_j- \mu_{\mathcal{B}})}{\sigma_{\mathcal{B}} }] \right\}.
\label{fcn:bn}
\end{equation}
Given a much larger variance $Var[Y] = \sigma_{\mathcal{B}}$ in Eq. (\ref{varadd}), the magnitude of the gradient w.r.t $X$ in AdderNets would be much smaller than that in CNNs according to Eq. (\ref{fcn:bn}), and then the magnitude of the gradient w.r.t the filters in AdderNets would be decreased as a result of gradient chain rule.
\begin{algorithm}[t]
\caption{The feed forward and back propagation of adder neural networks.}
\label{alg1}
\begin{algorithmic}[1]
\REQUIRE
An initialized adder network $\mathcal{N}$ and its training set $\mathcal{X}$ and the corresponding labels $\mathcal{Y}$, the global learning rate $\gamma$ and the hyper-parameter $\eta$.
\REPEAT
\STATE Randomly select a batch $\{(\mbox{x},\mbox{y})\}$ from $\mathcal{X}$ and $\mathcal{Y}$;
\STATE Employ the AdderNet $\mathcal{N}$ on the mini-batch: $\mbox{x} \rightarrow \mathcal{N}(\mbox{x})$;
\STATE Calculate the full-precision derivative $\frac{\partial Y}{\partial F}$ and $\frac{\partial Y}{\partial X}$ for adder filters using Eq. (\ref{fcn:l2bp}) and Eq. (\ref{fcn:l2bpx});
\STATE Exploit the chain rule to generate the gradient of parameters in $\mathcal{N}$;
\STATE Calculate the adapative learning rate $\alpha_l$ for each adder layer according to Eq. (\ref{fcn:lr2}).
\STATE Update the parameters in $\mathcal{N}$ using stochastic gradient descent.
\UNTIL convergence
\ENSURE A well-trained adder network $\mathcal{N}$ with almost no multiplications.
\end{algorithmic}
\end{algorithm}
\begin{table}[h]
\centering
\caption{The $\ell_2$-norm of gradient of weight in each layer using different networks at 1st iteration.}
\begin{tabular}{|c|c|c|c|}
\hline
\textbf{Model} & \textbf{Layer 1} & \textbf{Layer 2} & \textbf{Layer 3} \\
\hline
\hline
AdderNet & 0.0009 & 0.0012 & 0.0146 \\
\hline
CNN & 0.2261 & 0.2990 & 0.4646 \\
\hline
\end{tabular}
\label{tab:weight}
\end{table}
Table~\ref{tab:weight} reports the $\ell_2$-norm of gradients of filters $\Vert F \Vert _2$ in LeNet-5-BN using CNNs and AdderNets on the MNIST dataset during the 1st iteration. LeNet-5-BN denotes the LeNet-5~\cite{lenet} adding an batch normalization layer after each convolutional layer. As shown in this table, the norms of gradients of filters in AdderNets are much smaller than that in CNNs, which could slow down the update of filters in AdderNets.
\begin{table*}[t]
\centering
\caption{Classification results on the CIFAR-10 and CIFAR-100 datasets.}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\textbf{Model}& \textbf{Method} & \textbf{\#Mul.} & \textbf{\#Add.} & \textbf{XNOR} & \textbf{CIFAR-10} & \textbf{CIFAR-100} \\
\hline
\hline
&BNN & 0 & 0.65G & 0.65G & 89.80\% & 65.41\% \\
\cline{2-7}
VGG-small&AddNN & 0 & 1.30G &0 & 93.72\% & 72.64\% \\
\cline{2-7}
&CNN & 0.65G& 0.65G& 0 &93.80\% & 72.73\% \\
\hline
\hline
&BNN & 0 &41.17M & 41.17M & 84.87\% & 54.14\% \\
\cline{2-7}
ResNet-20&AddNN & 0 & 82.34M& 0 & 91.84\% & 67.60\% \\
\cline{2-7}
&CNN & 41.17M & 41.17M & 0 &92.25\% & 68.14\% \\
\hline
\hline
&BNN & 0 &69.12M & 69.12M & 86.74\% & 56.21\% \\
\cline{2-7}
ResNet-32&AddNN & 0 & 138.24M& 0 & 93.01\% & 69.02\% \\
\cline{2-7}
&CNN & 69.12M & 69.12M & 0 &93.29\% & 69.74\% \\
\hline
\end{tabular}
\label{tab:cls}
\vspace{-1.0em}
\end{table*}
A straightforward idea is to directly adopt a larger learning rate for filters in AdderNets. However, it is worth noticing that the norm of gradient differs much in different layers of AdderNets as shown in Table~\ref{tab:weight}, which requests special consideration of filters in different layers. To this end, we propose an adaptive learning rate for different layers in AdderNets. Specifically, the update for each adder layer $l$ is calculated by
\begin{equation}
\Delta F_l = \gamma \times \alpha_l \times \Delta L(F_l),
\label{fcn:lr1}
\end{equation}
where $\gamma$ is a global learning rate of the whole neural network (\eg for adder and BN layers), $\Delta L(F_l)$ is the gradient of the filter in layer $l$ and $\alpha_l$ is its corresponding local learning rate. As filters in AdderNets act subtraction with the inputs, the magnitude of filters and inputs are better to be similar to extract meaningful information from inputs. Because of the batch normalization layer, the magnitudes of inputs in different layers have been normalized, which then suggests a normalization for the magnitudes of filters in different layers. The local learning rate can therefore be defined as:
\begin{equation}
\alpha_l = \frac{\eta\sqrt{k}}{\Vert \Delta L(F_l)\Vert_2},
\label{fcn:lr2}
\end{equation}
where $k$ denotes the number of elements in $F_l$, and $\eta$ is a hyper-parameter to control the learning rate of adder filters. By using the proposed adaptive learning rate scaling, the adder filters in different layers can be updated with nearly the same step. The training procedure of the proposed AdderNet is summarized in Algorithm~\ref{alg1}.
\section{Experiment}\label{sec:experi}
In this section, we implement experiments to validate the effectiveness of the proposed AdderNets on several benchmark datasets, including MNIST, CIFAR and ImageNet. Ablation study and visualization of features are provided to further investigate the proposed method. The experiments are conducted on NVIDIA Tesla V100 GPU in PyTorch.
\subsection{Experiments on MNIST}\label{sec:clas}
To illustrate the effectiveness of the proposed AdderNets, we first train a LeNet-5-BN~\cite{lenet} on the MNIST dataset. The images are resized to $32\times32$ and are pro-precessed following~\cite{lenet}. The networks are optimized using Nesterov Accelerated Gradient (NAG), and the weight decay and the momentum were set as $5\times10^{-4}$ and 0.9, respectively. We train the networks for 50 epochs using the cosine learning rate decay~\cite{loshchilov2016sgdr} with an initial learning rate 0.1. The batch size is set as 256. For the proposed AdderNets, we replace the convolutional filters in LeNet-5-BN with our adder filters. Note that the fully connected layer can be regarded as a convolutional layer, we also replace the multiplications in the fully connect layers with subtractions. We set the hyper-parameter in Eq. (\ref{fcn:lr2}) to be $\eta=0.1$, which achieves best performance compared with other values from the pool $\left\{1,\frac{1}{2},\frac{1}{5},\frac{1}{10},\frac{1}{20}\right\}$.
The convolutional neural network achieves a $99.4\%$ accuracy with $\sim$435K multiplications and $\sim$435K additions. By replacing the multiplications in convolution with additions, the proposed AdderNet achieves a 99.4\% accuracy, which is the same as that of CNNs, with $\sim$870K additions and almost no multiplication
In fact, the theoretical latency of multiplications in CPUs is also larger than that of additions and subtractions. There is an instruction table~\footnote{\url{www.agner.org/optimize/instruction_tables.pdf}} which lists the instruction latencies, throughputs and micro-operation breakdowns for Intel, AMD and VIA CPUs. For example, in VIA Nano 2000 series, the latency of float multiplication and addition is 4 and 2, respectively. The AdderNet using LeNet-5 model will have $\sim$1.7M latency while CNN will have $\sim$2.6M latency in this CPU. In conclusion, the AdderNet can achieve similar accuracy with CNN but have fewer computational cost and latency. Noted that CUDA and cuDNN optimized adder convolutions are not yet available, we do not compare the actual inference time.
\subsection{Experiments on CIFAR}
We then evaluate our method on the CIFAR dataset, which consist of $32\times32$ pixel RGB color images. Since the binary networks~\cite{zhou2016dorefa} can use the XNOR operations to replace multiplications, we also compare the results of binary neural networks (BNNs). We use the same data augmentation and pro-precessing in He~\etal~\cite{he2016deep} for training and testing. Following Zhou~\etal~\cite{zhou2016dorefa}, the learning rate is set to 0.1 in the beginning and then follows a polynomial learning rate schedule. The models are trained for 400 epochs with a 256 batch size. We follow the general setting in binary networks to set the first and last layers as full-precision convolutional layers. In AdderNets, we use the same setting for a fair comparison. The hyper-parameter $\eta$ is set to 0.1 following the experiments on the MNIST dataset.
The classification results are reported in Table~\ref{tab:cls}. Since computational cost in batch normalization layer, the first layer and the last layer are significantly less than other layers, we omit these layers when counting FLOPs. We first evaluate the VGG-small model~\cite{cai2017deep} in the CIFAR-10 and CIFAR-100 dataset. As a result, the AdderNets achieve nearly the same results (93.72\% in CIFAR-10 and 72.64\% in CIFAR-100) with CNNs (93.80\% in CIFAR-10 and 72.73\% in CIFAR-100) with no multiplication. Although the model size of BNN is much smaller than those of AdderNet and CNN, its accuracies are much lower (89.80\% in CIFAR-10 and 65.41\% in CIFAR-100). We then turn to the widely used ResNet models (ResNet-20 and ResNet-32) to further investigate the performance of different networks. As for the ResNet-20, Tte convolutional neural networks achieve the highest accuracy (\ie 92.25\% in CIFAR-10 and 68.14\% in CIFAR-100) but with a large number of multiplications (41.17M). The proposed AdderNets achieve a 91.84\% accuracy in CIFAR-10 and a 67.60\% accuracy in CIFAR-100 without multiplications, which is comparable with CNNs. In contrast, the BNNs only achieve 84.87\% and 54.14\% accuracies in CIFAR-10 and CIFAR-100. The results in ResNet-32 also suggest that the proposed AdderNets can achieve similar results with conventional CNNs.
\begin{table*}[t]
\centering
\caption{Classification results on the ImageNet datasets.}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\textbf{Model} & \textbf{Method} & \textbf{\#Mul.} & \textbf{\#Add.} & \textbf{XNOR} & \textbf{Top-1 Acc.} & \textbf{Top-5 Acc.} \\
\hline
\hline
& BNN & 0 & 1.8G & 1.8G & 51.2\% & 73.2\% \\
\cline{2-7}
ResNet-18&AddNN & 0 & 3.6G &0 & 67.0\% & 87.6\% \\
\cline{2-7}
&CNN & 1.8G& 1.8G& 0 & 69.8\% & 89.1\% \\
\hline
\hline
& BNN & 0 & 3.9G & 3.9G & 55.8\% & 78.4\% \\
\cline{2-7}
ResNet-50&AddNN & 0 & 7.7G &0 & 74.9\% & 91.7\% \\
\cline{2-7}
&CNN & 3.9G& 3.9G& 0 & 76.2\% & 92.9\% \\
\hline
\end{tabular}
\vspace{-1.0em}
\label{tab:ImageNet}
\end{table*}
\begin{figure*}[t]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\linewidth]{figs/tmf.png} &
\quad \includegraphics[width=0.45\linewidth]{figs/convf.png} \\
(a) Visualization of filters of AdderNets &(b) Visualization of filters of CNNs \\
\end{tabular}
\caption{Visualization of filters in the first layer of LeNet-5-BN on the MNIST dataset. Both of them can extract useful features for image classification.}
\vspace{-1.0em}
\label{Fig:visualfilters}
\end{figure*}
\subsection{Experiments on ImageNet}
We next conduct experiments on the ImageNet dataset~\cite{krizhevsky2012imagenet}, which consist of $224\times224$ pixel RGB color images. We use ResNet-18 model to evaluate the proposed AdderNets follow the same data augmentation and pro-precessing in He~\etal~\cite{he2016deep}. We train the AdderNets for 150 epochs utilizing the cosine learning rate decay~\cite{loshchilov2016sgdr}. These networks are optimized using Nesterov Accelerated Gradient (NAG), and the weight decay and the momentum are set as $10^{-4}$ and 0.9, respectively. The batch size is set as 256 and the hyper-parameter in AdderNets is the same as that in CIFAR experiments.
Table~\ref{tab:ImageNet} shows the classification results on the ImageNet dataset by exploiting different nerual networks. The convolutional neural network achieves a 69.8\% top-1 accuracy and an 89.1\% top-5 accuracy in ResNet-18. However, there are 1.8G multiplications in this model, which bring enormous computational complexity. Since the addition operation has smaller computational cost than multiplication, we propose AdderNets to replace the multiplications in CNNs with subtractions. As a result, our AdderNet achieve a 66.8\% top-1 accuracy and an 87.4\% top-5 accuracy in ResNet-18, which demonstrate the adder filters can extract useful information from images. Rastegari~\etal~\cite{rastegari2016xnor} proposed the XNOR-net to replace the multiplications in neural networks with XNOR operations. Although the BNN can achieve high speed-up and compression ratio, it achieves only a 51.2\% top-1 accuracy and a 73.2\% top-5 accuracy in ResNet-18, which is much lower than the proposed AdderNet. We then conduct experiments on a deeper architecture (ResNet-50). The BNN could only achieve a 55.8\% top-1 accuracy and a 78.4\% top-5 accuracy using ResNet-50. In contrast, the proposed AdderNets can achieve a 74.9\% top-1 accuracy and a 91.7\% top-5 accuracy, which is closed to that of CNN (76.2\% top-1 accuracy and 92.9\% top-5 accuracy).
\subsection{Visualization Results}
\textbf{Visualization on features.} The AdderNets utilize the $\ell_1$-distance to measure the relationship between filters and input features instead of cross correlation in CNNs. Therefore, it is important to further investigate the difference of the feature space in AdderNets and CNNs. We train a LeNet++ on the MNIST dataset following~\cite{centerloss}, which has six convolutional layers and a fully-connected layer for extracting powerful 3D features. Numbers of neurons in each convolutional layer are 32, 32, 64, 64, 128, 128, and 2, respectively. For the proposed AdderNets, the last fully connected layers are replaced with the proposed add filters.
The visualization results are shown in Figure~\ref{Fig:visualfea}. The convolutional neural network calculates the cross correlation between filters and inputs. If filters and inputs are approximately normalized, convolution operation is then equivalent to calculate cosine distance between two vectors. That is probably the reason that features in different classes are divided by their angles in Figure~\ref{Fig:visualfea}. In contrast, AdderNets utilize the $\ell_1$-norm to distinguish different classes. Thus, features tend to be clustered towards different class centers. The visualization results demonstrate that the proposed AdderNets could have the similar discrimination ability to classify images as CNNs.
\textbf{Visualization on filters.} We visualize the filters of the LeNet-5-BN network in Figure~\ref{Fig:visualfilters}. Although the AdderNets and CNNs utilize different distance metrics, filters of the proposed adder networks (see Figure~\ref{Fig:visualfilters} (a)) still share some similar patterns with convolution filters (see Figure~\ref{Fig:visualfilters} (b)). The visualization experiments further demonstrate that the filters of AdderNets can effectively extract useful information from the input images and features.
\textbf{Visualization on distribution of weights.} We then visualize the distribution of weights for the 3th convolution layer on LeNet-5-BN. As shown in Figure~\ref{Fig:his}, the distribution of weights with AdderNets is close to a Laplace distribution while that with CNNs looks more like a Gaussian distribution. In fact, the prior distribution of $\ell_1$-norm is Laplace distribution~\cite{stigler1986history} and that of $\ell_2$-norm is Gaussian distribution~\cite{rennie2003l2} and the $\ell_2$-norm is exactly same as the cross correlation, which will be analyzed in the supplemental material.
\begin{figure*}[t]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.46\linewidth]{figs/lambda1.png} &
\quad \includegraphics[width=0.46\linewidth]{figs/lambda2.png} \\
(a) Accuracy &(b) Loss
\end{tabular}
\caption{Learning curve of AdderNets using different optimization schemes. FP and Sgn gradient denotes the full-precision and sign gradient. The proposed adaptive learning rate scaling with full-precision gradient achieves the highest accuracy (99.40\%) with the smallest loss.}
\label{Fig:abl}
\vspace{-1.0em}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth]{figs/weight1.PNG}
\caption{Histograms over the weights with AdderNet (left) and CNN (right). The weights of AdderNets follow Laplace distribution while those of CNNs follow Gaussian distribution. }
\label{Fig:his}
\vspace{-1.0em}
\end{figure}
\subsection{Ablation Study}
We propose to use a full-precision gradient to update the filters in our adder filters and design an adaptive learning rate scaling for deal with different layers in AdderNets. It is essential to evaluate the effectiveness of these components. We first train the LeNet-5-BN without changing its learning rate, which results in 54.91\% and 29.26\% accuracies using full-precision gradient and sign gradient, respectively. The networks can be hardly trained since its gradients are very small. Therefore, it is necessary to increase the learning rate of adder filters.
We directly increase the learning rate for filters in AdderNets by 100, which achieves best performance with full-precision gradient compared with other values from the pool $\left\{10,50,100,200,500\right\}$. As shown in Figure~\ref{Fig:abl}, the AdderNets using adaptive learning rate (ALR) and increased learning rate (ILR) achieve 97.99\% and 97.72\% accuracy with sign gradient, which is much lower than the accuracy of CNN (99.40\%). Therefore, we propose the full-precision gradient to precisely update the weights in AdderNets. As a result, the AdderNet with ILR achieves a 98.99\% accuracy using the full-precision gradient. By using the adaptive learning rate (ALR), the AdderNet can achieve a 99.40\% accuracy, which demonstrate the effectiveness of the proposed ALR method.
\begin{table}[h]
\centering
\caption{The impact of parameter $\eta$ using LeNet-5-BN on the MNIST dataset.}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\eta$ & 1 & 0.5 & 0.2 & 0.1 & 0.05 \\
\hline
\hline
Acc. (\%) & 99.26 &99.30&99.35& 99.40 & 99.32 \\
\hline
\end{tabular}
\label{tab:impact}
\end{table}
\textbf{Impact of parameters.} As discussed above, the proposed adaptive learning rate scaling has a hyper-parameter: $\eta$. We then test its impact on the accuracy of the student network by conducting the experiments on the MNIST dataset. We use LeNet-5-BN as the backbone of AdderNet. Other experimental settings are same as mentioned in Sec.~\ref{sec:clas}. It can be seen from Table~\ref{tab:impact} that the AdderNets trained utilizing the adaptive learning rate scaling achieves the highest accuracy (99.40\%) when $\eta$ = 0.1. Based on the above analysis, we keep the setting of hyper-parameters for the proposed method.
\section{Conclusions}\label{sec:conclu}
The role of classical convolutions used in deep CNNs is to measure the similarity between features and filters, and we are motivated to replace convolutions with more efficient similarity measure. We investigate the feasibility of replacing multiplications by additions in this work. An AdderNet is explored to effectively use addition to build deep neural networks with low computational costs. This kind of networks calculate the $\ell_1$-norm distance between features and filters. Corresponding optimization method is developed by using regularized full-precision gradients. Experiments conducted on benchmark datasets show that AdderNets can well approximate the performance of CNNs with the same architectures, which could have a huge impact on future hardware design. Visualization results also demonstrate that the adder filters are promising to replace original convolution filters for computer vision tasks. In our future work, we will investigate quantization results for AdderNets to achieve higher speed-up and lower energy comsumption, as well as the generality of AdderNets not only for image classification but also for detection and segmentation tasks.
\section*{Acknowledgement}
We thank anonymous reviewers for their helpful comments. This work is supported by National Natural Science Foundation of China under Grant No. 61876007, 61872012, National Key R\&D Program of China (2019YFF0302902), Beijing Academy of Artificial Intelligence (BAAI), and Australian Research Council under Project DE-180101438.
{\small
\bibliographystyle{ieee_fullname}
|
1,116,691,500,471 | arxiv | \section{Introduction}
\subsection*{Background}
This paper concerns dif\/ferential geometry in dimensions one to four, and (primarily) four kinds of geometric structure,
one in each dimension, governed by four nonlinear integrable dif\/ferential equations.
Associated to each manifold carrying one of these geometric structures, and to each Lie group, is an integrable gauge
theory, generalizing a~well-known gauge theory on f\/lat space: the selfdual Yang--Mills equation on ${\mathbb{R}}^4$ or
${\mathbb{R}}^{2,2}$ and the gauge f\/ield equations arising as reductions by a~(nondegenerate) group of translations to
lower-dimensional f\/lat spaces.
Specif\/ically, reduction by a~single non-null translation gives the Bogomolny equation for monopoles on ${\mathbb{R}}^3$,
or its Lorentzian analogue on ${\mathbb{R}}^{2,1}$~\cite{AtHi:gmm,Ward:iss}.
Reduction by two such translations yields the Hitchin equation for Higgs pairs on a~Riemann surface, harmonic maps from
a~Riemann surface to a~Lie group, or the principal chiral model on a~two-dimensional space-time~\cite{Hit:sde,Ward:iss}.
Reduction by three non-null translations leads to the Nahm equation~\cite{Nahm:eqns}.
In addition to the physical motivation, the selfdual Yang--Mills equation has attracted interest because of its good
integrability properties, which are inherited by the Bogomolny, Hitchin and Nahm equations, and their analogues.
Further integrable systems may be obtained by reducing the selfdual Yang--Mills equation by other groups of conformal
transformations of ${\mathbb{R}}^4$ or ${\mathbb{R}}^{2,2}$, and many such reductions have been
investigated~\cite{MaWo:ist}.
For example, hyperbolic monopoles arise from the reduction to three dimensions by a~rotation, the Ernst equation is
a~reduction to two dimensions by a~translation and rotation, while reductions to one dimension may be interpreted as
isomonodromic deformation problems with four poles, governed (in the generic case) by the Schlesinger
equation~\cite{Hit:tem,MaWo:ist,MMW:sbm}.
Twistor theory gives one explanation for this integrability: there is a~Ward correspondence (see~\cite{AtWa:iag})
between solutions of the selfdual Yang--Mills equation on f\/lat space and holomorphic vector bundles on (suitable open
subsets of) $\CP3$.
This suggests that the selfdual Yang--Mills equation will continue to be integrable on other spaces $M$ so long as there
is still a~Ward correspondence between solutions and holomorphic vector bundles on some complex $3$-mani\-fold~$Z$, the
\emph{twistor space} of~$M$.
Such curved twistor spaces~$Z$ were introduced by Penrose~\cite{Pen:nlg} to study selfdual vacuum metrics.
(Note that in Euclidean signature, selfdual Ricci-f\/lat metrics are locally hyperk\"ahler.)
Deep relationships between gauge f\/ield equations and selfdual vacuum metrics have been observed in a~number of places.
In~\cite{Ward:suc}, Ward considered gauge theories on ${\mathbb{R}}^{4-\ell}$ with the gauge group being a~transitive
group of dif\/feomorphisms of an $\ell$-manifold $\Sigma^\ell$.
He focused on the group of dif\/feomorphisms preserving a~f\/ixed volume form on $\Sigma^\ell$ and observed that gauge
f\/ields then give rise to selfdual vacuum metrics on ${\mathbb{R}}^{4-\ell}\mathbin{{\times}}\Sigma^\ell$.
When $\ell=1$, the group of volume preserving dif\/feomorphisms of a~circle or a~line is $\Un(1)$ or ${\mathbb{R}}$ acting
by translation, so the monopoles are Abelian, and Ward's construction reduces to the Gibbons--Hawking
Ansatz~\cite{GiHa:gmi}.
For $\ell=2$, f\/inite-dimensional subgroups of $\SDiff(\Sigma^2)$ yield other interesting constructions of selfdual
vacuum metrics~\cite{DMW:2dg}, while $\ell=3$ gives the Ashtekar--Jacobson--Smolin description of selfdual vacuum
metrics in terms of the Nahm equation~\cite{AJS:hfe}.
One can also view the (closely related) Mason--Newman formulation~\cite{MaNe:eym} as the case $\ell=4$.
In Ward's construction, gauge f\/ields on lower-dimensional f\/lat spaces give rise to curved $4$-manifolds.
On the other hand, the example of hyperbolic monopoles shows that the lower-dimensional spaces can also be curved.
One of the goals of this paper is to place these miscellaneous results and observations in a~geo\-metric framework which
simultaneously unif\/ies and generalizes them.
Selfdual vacuum metrics do not provide the right setting for this.
For example, the natural generalization of the Gibbons--Hawking Ansatz to Abelian monopoles on hyperbolic space is
LeBrun's hyperbolic Ansatz for scalar-f\/lat K\"ahler metrics~\cite{LeBr:cp2}, and even on ${\mathbb{R}}^{4-\ell}$, one
f\/inds that without the volume-preserving condition, Ward's construction leads to hypercomplex, rather than hyperk\"ahler,
structures~-- see Hitchin~\cite{Hit:hcm} and Joyce~\cite{Joy:esd} (or Dunajski~\cite{Dun:tph}) for the analogue of the
Ashtekar--Jacobson--Smolin and Mason--Newman description respectively.
In order to incorporate these constructions, it is essential to take into account a~second basic feature of the selfdual
Yang--Mills equation, in addition to integrability: \emph{conformal invariance}.
The selfdual Yang--Mills equation makes sense on any oriented conformal $4$-manifold $M$, and the purely conformal part
of Penrose's nonlinear graviton shows that there is a~curved twistor space $Z$ as long as the conformal structure on $M$
is selfdual~\cite{AHS:sd4, Pen:nlg}.
\looseness=-1
The thesis of this paper is that reductions of selfdual conformal geometry to lower dimensions provide ``integrable
background geometries'' on which curved versions of the Bogomolny, Hitchin and Nahm equations (with good integrability
properties) can be def\/ined.
This thesis can be illustrated by the three-dimensional case: the Jones--Tod correspondence~\cite{JoTo:mew} shows that
the reduction of a~selfdual conformal structure to three dimensions is an Einstein--Weyl structure, and selfdual
Yang--Mills f\/ields reduce to generalized monopoles on such Einstein--Weyl spaces, pla\-cing Euclidean and hyperbolic
monopoles in a~common framework.
Selfdual spaces with symmetry over a~given Einstein--Weyl space are built out of Abelian monopoles, and this provides
many constructions (generalizing the Gibbons--Hawking and hyperbolic Ans{\"a}tze) of hyperk\"ahler, scalar-f\/lat
K\"ahler and selfdual Einstein metrics, and also of hypercomplex structures~\cite{CaPe:sdc,GaTo:hms,Hit:cme,LeBr:cp2}.
The role of gauge f\/ields and Abelian monopoles in Ward's construction and the Jones--Tod construction respectively
suggests an underlying principle relating gauge f\/ield equations and constructions of selfdual spaces.
This is amplif\/ied by the following two generalizations of the Jones--Tod construction.
First, in~\cite{Cal:sde}, the Jones--Tod construction and Ward's construction on ${\mathbb{R}}^3$ were simultaneously
generalized by considering the Einstein--Weyl Bogomolny equation with the gauge group acting transitively by
dif\/feomorphisms on a~circle or a~line.
The volume preserving case gives the usual Jones--Tod correspondence between selfdual spaces with symmetry and Abelian
monopoles on Einstein--Weyl spaces, but more general gauge groups lead to new constructions of selfdual conformal
structures and metrics, including, in special cases, hyperk\"ahler and selfdual Einstein metrics.
Second, note that the Jones--Tod construction gives rise to a~procedure for constructing a~new selfdual space from an
invariant selfdual Maxwell f\/ield on a~given selfdual space with one-dimensional symmetry group: the selfdual Maxwell
f\/ield descends to an Abelian monopole on the quotient Einstein--Weyl space, out of which a~new selfdual space may be
built.
This was generalized, by Maszczyk, Mason and Woodhouse~\cite{MMW:sbm}, to any freely acting symmetry group, using
a~construction they call the ``switch map''~\cite{MaWo:ist}: given a~selfdual space with freely acting group of
conformal transformations $H$, and an invariant selfdual Yang--Mills connection on a~bundle $P$ with a~gauge group $G$
of the same dimension as $H$, the quotient of $P$ by $H$ is another selfdual space, with symmetry group $G$.
Hence, for example, $T^3$-invariant $\SU(2)$ Yang--Mills f\/ields on ${\mathbb{R}}^4$ give rise to selfdual conformal
structures with $\SU(2)$ symmetry, such as the scalar-f\/lat K\"ahler, hypercomplex and selfdual Einstein metrics
of~\cite{Dan:sfk,Hit:tem,Hit:hcm,PePo:kzs,Tod:p3}.
\subsection*{Overview} Ward's construction, the (generalized) Jones--Tod correspondence, and the switch map all point to
the following framework for dimensional reduction of selfdual conformal geometry and the selfdual Yang--Mills equation.
\begin{enumerate}\itemsep=0pt
\item[(i)]
There are background geometries in each dimension less than four obtained by dimensional reduction of the selfduality
condition for conformal structures.
\item[(ii)]
The nonlinear dif\/ferential equations def\/ining these background geometries have the surprising feature that they do not
depend on the symmetry group that one reduces by.
Therefore, including selfdual conformal geometry, there are only four kinds of (non\-de\-ge\-ne\-rate) background geometry.
\item[(iii)]
Instead, the symmetry group enters as a~gauge group for a~gauge theory def\/ined on the background geometry, and the gauge
f\/ield equation is the dimensional reduction of the selfdual Yang--Mills equation.
Hence the gauge f\/ield equations play a~remarkable dual role: solutions give rise both to selfdual conformal
$4$-manifolds and also to selfdual Yang--Mills f\/ields on such manifolds.
\end{enumerate}
In this paper, the above framework is established and studied in full generality.
Furthermore, the dif\/ferent geometries are related not just by symmetry reduction, but by a~more general form of
dimensional reduction, of which Ward's construction and the generalized Jones--Tod construction are examples
(cf.~also~\cite{GKPS:frsd}).
Such constructions also relate the lower-dimensional geometries to each other.
More precisely:
\begin{itemize}\itemsep=0pt
\item
For $1\leqslant k<k+\ell\leqslant4$, $(k+\ell)$-dimensional geometries are obtained from a~$k$-dimensional geometry by
solving the gauge f\/ield equation on that background where the gauge group acts transitively by dif\/feomorphisms on an
$\ell$-manifold.
\end{itemize}
In four dimensions, the gauge f\/ields and background geometries are, of course, selfdual Yang--Mills f\/ields on selfdual
spaces, while in three dimensions, one obtains monopoles on Einstein--Weyl spaces.
The one and two-dimensional stories are new, although these structures have been implicitly studied in many places, at
least in special cases~\cite{HYMO:aag,Taf:2dr} and the twistor theory of the two-dimensional geometry has been developed
independently by Donaldson and Fine~\cite{DoFi:tasd,Fin:tasde}.
I~f\/irst describe brief\/ly the two-dimensional geometries, in the form obtained by reduction from Euclidean signature.
The two reductions from Kleinian signature $(2,2)$ are similar.
Given a~complex line bundle ${\mathcal{W}}$ over a~Riemann surface $N$ with a~Hermitian metric on
${\mathcal{W}}^{*}\otimes TN$, the geometric structure is a~triple consisting of a~$\Un(1)$-connection $a$ on
${\mathcal{W}}^{*}\otimes TN$, a~section $\psi$ of ${\mathcal{W}}^{*}$ and a~section ${\mathcal{C}}$ of
${\mathcal{W}}^{*}\otimes{\mathcal{W}}^{*}\otimes TN$ satisfying the following equations:
\begin{gather*}
\overline\partial{}^a {\mathcal{C}}=0,
\qquad
\overline\partial{}^a \psi=-3{\mathcal{C}}\overline\psi,
\qquad
{*F^a}=|\psi|^2-2|{\mathcal{C}}|^2.
\end{gather*}
The f\/lat geometry is obtained by setting ${\mathcal{W}}=TN$, with the trivial connection on ${\mathcal{W}}^{*}\otimes
TN$ and ${\mathcal{C}}=\psi=0$.
This is simply a~Riemann surface with no additional structure.
The general two-dimensional background geometry was found in joint work with Lionel Mason~\cite{CaMa:svm}.
The gauge f\/ields on this background, with gauge group $G$, are pairs $(A,\Phi)$ consisting of a~$G$-connection $A$ and
a~section $\Phi$ of ${\mathcal{W}}^{*}\otimes\lie{g}_N$, where $\lie{g}_N$ is the associated Lie algebra bundle.
These pairs satisfy the following equations:
\begin{gather*}
F^A-[\Phi,\overline\Phi]=\psi\wedge\overline\Phi+\overline\psi\wedge\Phi,
\qquad
\overline\partial{}^{a,A}\Phi={\mathcal{C}}\overline\Phi.
\end{gather*}
On the f\/lat geometry, where ${\mathcal{C}}=0=\psi$, these are the Hitchin equations for stable pairs, but in the general
case, the equations are coupled to the background geometry.
In one dimension, the background geometry is governed by a~symmetric traceless $(3\times 3)$-matrix ${\mathcal{B}}$
satisfying the Riccati equation ${\mathcal{B}}_r=2({\mathcal{B}}^2)_0$, where $r$ is an af\/f\/ine coordinate and the
subscript zero denotes the traceless part.
More invariantly, ${\mathcal{B}}$ is a~section of $T^{*} C\otimes\Sym_0{\mathcal{E}}$ where ${\mathcal{E}}$ is a~rank
$3$ conformal vector bundle over a~curve $C$, with $\wedge^3{\mathcal{E}}=(TC)^3$, and one f\/ixes compatible
connections on ${\mathcal{E}}$ and $TC$ to def\/ine the $r$-derivative ${\mathcal{B}}_r$.
The gauge f\/ields on this background are sections $\Phi$ of ${\mathcal{E}}^{*}\otimes\lie{g}_C$ (where $\lie{g}_C$ is
a~Lie algebra bundle over $C$) satisfying the equation $\Phi_r-{*[\Phi,\Phi]}={\mathcal{B}}\mathinner{\cdot}\Phi$, where
$*$ denotes the star operator on ${\mathcal{E}}$ and one f\/ixes a~$G$-connection to def\/ine $\Phi_r$.
The f\/lat geometry is the trivial solution ${\mathcal{B}}=0$, in which case this gauge f\/ield equation reduces to the Nahm
equation.
There is one further important property of the Jones--Tod correspondence that continues to hold in the general
framework: it is constructive, i.e., a~selfdual conformal structure is \emph{explicitly} determined by an Einstein--Weyl
structure together with a~monopole, and conversely.
The correspondence between additional monopoles and selfdual Yang--Mills f\/ields is equally explicit.
The same remarks hold for the switch map and Ward's construction.
I therefore present explicit formulae for the constructions of the paper.
For convenience of exposition, I will concentrate on the reductions from Euclidean signature and the notation will be
adapted to this case.
However, the \emph{non-null} reductions from Kleinian signature $(2,2)$ are completely analogous, as are such reductions
of complex geometries: I indicate throughout the nondegeneracy assumptions that need to be made, and any changes in
notation that are needed.
On the other hand, it remains an interesting open project to study the integrable background geometries arising from
\emph{null} reductions: only two such reductions are considered here.
\subsection*{User guide}
This is a~long paper to read from start to f\/inish, so I give a~detailed guide to the sections, both to draw attention to
the highlights and to enable the reader to dip into the paper more easily.
The general theory of the paper is developed in Sections~\ref{s:sdbg}, \ref{s:gfe}, \ref{s:sgf} and~\ref{s:bgf}.
Section~\ref{s:sdbg} concerns selfdual spaces with a~freely acting symmetry group and presents the background geometry
equations in dimensions $1$--$3$.
In each case the main result, Theorems~\ref{th:ric},~\ref{th:sv} or~\ref{th:JnT}, identif\/ies the selfduality condition on
a~conformal structure with the background geometry equations: in particular Theorem~\ref{th:JnT} is the Jones--Tod
correspondence~\cite{JoTo:mew}.
Section~\ref{s:gfe} deals with invariant selfdual Yang--Mills f\/ields on selfdual spaces with a~freely acting symmetry group.
Here the gauge f\/ield equations on the background geometries are computed.
The main general theorems are in Sections~\ref{s:sgf} and~\ref{s:bgf}, where the inverse constructions to
Section~\ref{s:sdbg} are established and generalized.
Theorem~\ref{th:sdgf} proves that selfdual spaces may be constructed from solutions to gauge f\/ield equations on
$k$-dimensional background geometries where the gauge group acts transitively on a~$(4-k)$-manifold: in the case of
a~group acting on itself by the regular representation, this theorem reconstructs the selfdual spaces with freely acting
symmetry group of Section~\ref{s:sdbg}~-- although, as mentioned already, in the general case, the group need not even be
f\/inite-dimensional.
Theorem~\ref{th:bgf} generalizes all this to gauge groups acting on $\ell$-manifolds, giving explicit constructions of
$(k+\ell)$-dimensional background geometries from gauge f\/ields on $k$-dimensional geometries for $k+\ell\leqslant4$.
Unfortunately, the calculations here are too complicated to present in full.
However, I do provide explicit formulae: in examples arising in practice, it is usually not too hard to verify that the
$(k+\ell)$-dimensional geometry satisf\/ies the background equations, once one has a~formula.
The relation between selfdual spaces, and the three-dimensional background geometries, Einstein--Weyl spaces, is
a~longstanding one, and as a~consequence the latter have been extensively studied~\cite{Cal:sde,CaPe:sdc,CaTo:emh,
Gau:swe,GaTo:hms, Hit:cme,JoTo:mew,LeBr:cp2,Tod:p3,Tod:sew,Ward:sut}.
By contrast, the background geometries in one and two dimensions, Riccati spaces, and spinor-vortex spaces, although
implicitly underlying previous work, have only been introduced and investigated relatively recently (see,
e.g.,~\cite{CaMa:svm,DoFi:tasd}).
Assuming that one is not interested in zero-dimensional dif\/ferential geometry (see Remark~\ref{r:zdg}), then these
geometries form the foundation for the more well-known higher-dimensional structures (using Theorems~\ref{th:sdgf}
and~\ref{th:bgf}).
In Section~\ref{s:rs}, the geometry of Riccati spaces is described.
Although there are only six Riccati spaces up to local isomorphism, they have a~rather rich structure, which is most
easily revealed in a~complexif\/ied setting, since only three of the Riccati spaces arise as reductions from Euclidean
signature.
The six solutions correspond to the six types of quartic polynomial, i.e., the f\/ive types of conf\/iguration of four
points on $\CP1$ together with the zero polynomial.
These types are denoted (I,~II, III, D, N, 0), following the well-known application of this classif\/ication to Weyl tensors.
It is easy to see that the gauge f\/ield equation on the trivial (0) Riccati space is the Nahm equation.
One can also observe that the gauge f\/ield equations on the nontrivial Riccati spaces are equivalent to (strong)
isomonodromy equations for a~connection with four poles on~$\CP1$, the conf\/iguration of the poles corresponding to the
type of Riccati space.
This leads to a~uniform Lax pair for these problems, which reduces to the usual Lax pair for the Nahm equation in the
trivial case.
The theory of this paper would be very dry without examples and applications, so I intersperse the main
development with \emph{interludes}, which motivate or illustrate the theory, yet are, to varying degrees,
self-contained.
The f\/irst interlude, Section~\ref{s:bm}, relates the approach of Section~\ref{s:sdbg} to other studies of selfdual
Bianchi metrics~\cite{Dan:sfk,DaSt:cok,PePo:kzs,Tod:p6,Tod:com,Tod:p3}.
The second interlude, Section~\ref{s:svhe}, provides a~simple \emph{a priori} explanation for the conformal invariance
of Hitchin's selfduality equation on a~Riemann surface~\cite{Hit:sde} (and of course, reductions from Kleinian signature
also give conformally invariant equations: harmonic maps into a~Lie group, and the principal chiral model).
This demystif\/ies this conformal invariance: it is, after all, a~consequence of conformal invariance in four dimensions.
The third interlude, Section~\ref{s:hchk}, is a~unif\/ied treatment of various constructions (or at least interpretations)
of hypercomplex and hyperk\"ahler structures.
My aim here is three-fold:
\begin{enumerate}\itemsep=0pt
\item[(i)]
to provide geometrical descriptions of the well-known Mason--Newman~\cite{MaNe:eym},
Ashtekar--Jacobson--Smolin~\cite{AJS:hfe}, and Park--Ward~\cite{Par:sdg,Ward:suc} constructions of hyperk\"ahler
metrics from lower-dimensional gauge f\/ields;
\item[(ii)]
to give, at the same time, hypercomplex generalizations, following~\cite{Dun:tph,GrSt:his,Hit:hcm,Joy:esd};
\item[(iii)]
to prove that all hypercomplex and hyperk\"ahler structures are locally obtained from any of these constructions, in
a~way that is manifestly compatible with any reality conditions, and that clarif\/ies the extra choice that needs to be
made to reduce a~hypercomplex or hyperk\"ahler structure to a~solution of the relevant gauge f\/ield equation.
\end{enumerate}
I hope that the overview provided here is useful, at least to the reader who is not familiar with the treatments in the
physics literature.
In addition this work answers~-- and extends to the hypercomplex case~-- a question of Ward~\cite{Ward:suc}, who
conjectured that any hyperk\"ahler metric could be obtained from Hitchin's selfduality equation, with gauge group
$\SDiff(\Sigma^2)$.
I also show how this description gives a~Euclidean analogue of Plebanski's heavenly equations, which is well-adapted to
the study of hyperk\"ahler metrics on elliptic f\/ibrations.
The fourth interlude, Section~\ref{s:s1Hit}, is a~two-part analysis of hyperCR Einstein--Weyl spaces, the three-dimensional analogue of hypercomplex structures.
In the f\/irst part, following an approach of Tod~\cite{Tod:sew}, the hyperCR Einstein--Weyl equation is shown to be
equivalent to the $\Diff(S^1)$ Hitchin equation (revealing a~hidden $\SO(3)$ symmetry in the latter).
This can be viewed as a~three-dimensional version of the constructions of Section~\ref{s:hchk} although it is remarkable
that the f\/ibres are only one-dimensional, since such constructions are not suf\/f\/iciently general in four dimensions.
In the second part, Einstein--Weyl spaces admitting a~dimensional reduction with geodesic one-dimensional f\/ibres are
studied.
They are shown to be hyperCR, and the quotient spaces are trivial or spherical spinor-vortex geometries.
Conversely, any shear-free geodesic congruence on a~hyperCR Einstein--Weyl space def\/ines a~generalized dimensional
reduction.
The f\/inal interlude again consists of two-parts.
The f\/irst shows that hyperCR Einstein--Weyl spaces may also be constructed from the $\Diff(\Sigma^2)$ Nahm equations.
In the second part, I present a~proof that the well-known $\SU(\infty)$ Toda f\/ield equation $u_{xx}+u_{yy}+(e^u)_{zz}=0$
and the dKP equation $u_{yy}=(u_t-uu_x)_x$ are both equivalent to generalized Nahm equations, via a~hodograph
transformation (cf.~\cite{DuTo:p12} in the dKP case).
Even these results are to a~large extent self-contained, although the the work of Section~\ref{s:bgf} shows that the
construction is a~special case of the general theory, while Section~\ref{s:rs} shows that the backgrounds for these
generalized Nahm equations are the type (D) and (N) Riccati spaces respectively.
To the best of my knowledge, all examples of hodograph solutions to the $\SU(\infty)$ Toda f\/ield equation or dKP
equation arise in this way from explicit solutions of a~generalized Nahm equation.
I end the interlude by discussing these examples.
\subsection*{Addenda}
The majority of this paper was written in the period 1999--2001, and the present content is not
substantially dif\/ferent from a~January 2002 version which has been posted on my academic home page since that time.
The intervening 12 years have seen many advances in the f\/ield (for instance, by Dunajski and his
collaborators~\cite{Dun:hfcq,Dun:sitt,DGS:misd,DuKr:ewhv,DuSp:dis1}), and I have collected a~few of the most closely
related papers as ``Additional references'' at the end of the bibliography.
Some of these works develop ideas from the 2002 version of this paper, or discover related ideas independently.
There has also been much work on null reductions and geometries, which were only touched upon in the original version of
this paper.
A major driving force has been the introduction of a~global twistor theory of holomorphic discs by LeBrun and
Mason~\cite{LeMa:zmcs,LeMa:nghd,LeMa:zmbchd,Nak:ssdz}, both for Kleinian signature selfdual conformal structures and
$2$-dimensional projective structures.
Dunajski and West~\cite{DuWe:acs,DuWe:acns} established a~relationship between these structures by considering selfdual
conformal structures with a~null conformal Killing vector f\/ield.
As explained in~\cite{Cal:sdp}, the natural context for their construction is the null reduction of selfdual conformal
structures along a~$\beta$-surface foliation.
The quotient is a~surface with a~natural projective structure structure, and the gauge f\/ield equations are projectively
invariant on this background geometry.
In~\cite{Nak:sdza} the global Mason--LeBrun theories are related by this construction.
More recently, similar methods have been applied in Einstein--Weyl geometry~\cite{LeMa:ewhd,Nak:ewlm}.
One of the most intriguing recent developments has been the introduction, by Ferapontov and his coworkers, of the method
of hydrodynamic reductions to analyse integrability.
This was applied in~\cite{FHZ:cqim} to study the central quadric Ansatz and its relation to the Painlev\'e equations.
In doing so, they independently rediscovered equations equivalent to the $\SDiff(\Sigma^2)$ generalized Nahm equations
on Riccati spaces.
More recently, in~\cite{FeKr:disew}, Ferapontov and Kruglikov construct a~Weyl structure from the formal linearization
of a~second-order PDE in three dimensions, and show that it is Einstein--Weyl for all solutions if and only if the
system is integrable.
This and related conjectures for second-order PDEs in four dimensions suggest deep connections with the integrable
background geometry concept.
For these reasons, the main (odd-numbered) sections have been updated with addenda which place the above works in the
framework of this paper.
Section~\ref{section12} and Subsection~\ref{s:sdsn}
have also been updated to ref\/lect the exciting new directions that are currently unfolding.
\subsection*{Notation}
In order to manipulate, in a~tensorial way, the objects and structures entering into the equations and constructions of
this paper, it will be convenient to employ the formalism of densities.
If $V$ is a~real $n$-dimensional vector space and $w$ any real number, then the oriented one-dimensional linear space
$L^w=L^w_V$ carrying the representation $A\mapsto|\det A|^{w/n}$ of $\GL(V)$ is called the space of \emph{densities of
weight $w$} or \emph{$w$-densities}.
It can be constructed canonically as the space of maps $\rho\colon(\wedge^nV)\setminus0\to{\mathbb{R}}$ such that
$\rho(\lambda\omega)=|\lambda|^{-w/n}\rho(\omega)$ for all $\lambda\in{\mathbb{R}}^{\times}$ and
$\omega\in(\wedge^nV)\setminus0$.
For a~vector bundle ${\mathcal{V}}\to M$ this construction yields an oriented real line bundle $L^w_{\mathcal{V}}$,
a~\emph{density line bundle}.
If ${\mathcal{V}}$ is oriented and of rank $n$, then $L^{-n}_{\mathcal{V}}$ is canonically isomorphic to
$\wedge^n{\mathcal{V}}$; indeed an orientation may be def\/ined as an \emph{orientation form}
$*1\in\mathrm{C}^\infty(M,L^n_{\mathcal{V}} \wedge^n{\mathcal{V}}^{*})$.
(Here and elsewhere, when tensoring with a~density line bundle, I shall often omit the tensor product sign.) More
generally, the \emph{Hodge star operator} is the isomorphism
\begin{gather*}
{*}\colon \ L^{w-k}_{\mathcal{V}} \wedge^k{\mathcal{V}}\to L^{w+n-k}_{\mathcal{V}} \wedge^{n-k}{\mathcal{V}}^{*}
\end{gather*}
determined by the nondegenerate pairing $\wedge^k{\mathcal{V}}^{*}\otimes\wedge^{n-k}{\mathcal{V}}^{*}\to
\wedge^n{\mathcal{V}}^{*}\cong L^{-n}_{\mathcal{V}}$.
A \emph{conformal structure} on ${\mathcal{V}}$ is a~nondegenerate symmetric bilinear form on ${\mathcal{V}}$ with
va\-lues in~$L^2_{\mathcal{V}}$, or equivalently a~metric on $L^{-1}_{\mathcal{V}} {\mathcal{V}}$.
The conformal inner product of sections~$X$,~$Y$ is $\ip{X,Y}\in\mathrm{C}^\infty(M,L^2_{\mathcal{V}})$ and the conformal
structure itself may be viewed as a~section $\mathsf{c}\in\mathrm{C}^\infty(M,L^2_{\mathcal{V}}
S^2{\mathcal{V}}^{*})$.
I shall make free use of the isomorphism between $L^{w-k}_{\mathcal{V}}\wedge^k{\mathcal{V}}$ and
$L^{w+k}_{\mathcal{V}} \wedge^k{\mathcal{V}}^{*}$ given by a~conformal structure.
When ${\mathcal{V}}$ is the tangent bundle of $M$, $L^w_M=L^w_{TM}$ is called the bundle of $w$-densities of $M$,
denoted $L^w$ when $M$ is understood.
The line bundles $L^w$ are trivializable and a~nonvanishing (usually positive) section $\mu$ of $L=L^1$ will be called
a~\emph{length scale} or \emph{gauge}.
I shall also say that tensors in $L^w\otimes(TM)^j\otimes(T^{*} M)^k$ have \emph{weight} $w+j-k$.
A \emph{Weyl derivative} is a~covariant derivative $D$ on $L$.
It induces covariant derivatives on~$L^w$ for all $w$.
If $M$ is conformal, i.e., there is a~conformal structure $\mathsf{c}$ on~$TM$, then any Weyl derivative induces (via
the Koszul formula) a~\emph{Weyl connection}: a~torsion-free connection~$D$ on~$TM$ with $D\mathsf{c}=0$.
Compatible Riemannian metrics~$g$ correspond to length scales $\mu$, and the Weyl connection induced by the Weyl
derivative preserving $\mu$ is, of course, the Levi-Civita connection of $g=\mu^{-2}\mathsf{c}$.
The curvature $R^D$ of a~Weyl connection $D$, as a~$\mathop{\mathfrak{co}}(TM)$-valued $2$-form, decomposes as
\begin{gather*}
R^D_{X,Y}=W_{X,Y}-\abrack{r^D(X),Y}+ \abrack{r^D(Y),X}.
\end{gather*}
Here $W$ is the \emph{Weyl curvature} of the conformal structure, an $\mathop{{\rm so}}(TM)$-valued $2$-form
independent of the choice of $D$, and $r^D$ is a~covector valued $1$-form, the \emph{$($normalized$)$ Ricci
curvature} of~$D$.
For a~$1$-form $\gamma$ and tangent vector~$X$, $\abrack{\gamma,X}=\gamma(X)\iden+\gamma\mathinner{\vartriangle}X$, where
$(\gamma\mathinner{\vartriangle}X)(Y) =\gamma(Y)X-\ip{X,Y}\gamma$.
This bracket is part of a~Lie algebra structure on $TM\oplus\mathop{\mathfrak{co}}(TM)\oplus T^{*} M$ and the same
notation will be used for the commutator bracket in $\mathop{\mathfrak{co}}(TM)$.
The normalized Ricci curvature decomposes into a~symmetric traceless part $r^D_0$, a~scalar part $\scal^D$ (the scalar
curvature) and a~skew part, which is just a~multiple of the curvature of~$D$ on~$L$ called the \emph{Faraday curvature}~$F^D$.
In practice a~Weyl derivative is described by its connection $1$-form $\omega$ relative to a~length scale: $\omega$ is
called the \emph{Weyl $1$-form}, and $F^D={\rm d}\omega$.
If $F^D=0$ then $D$ is said to be \emph{closed}.
It follows that there are local length scales $\mu$ with $D\mu=0$.
If such a~length scale exists globally then $D$ is said to be \emph{exact}.
The above constructions can also be carried out locally on complex manifolds, except that~$L^1$ is now a~choice of local
$n$th root of $\wedge^nTM$.
Although confusion with the index $i$ is unlikely, as a~courtesy to the reader, I denote the (chosen) square root of~$-1$ by ${\boldsymbol i}$.
\subsection*{Twistors and Lax pairs} Several of the results in this paper were motivated by twistor or integrable
systems methods: in particular the idea of generalized dimensional reduction arises naturally when one considers
holomorphic foliations of twistor spaces.
However, I have deliberately suppressed discussion of twistor spaces and Lax pairs, for at least two reasons: f\/irst, to
make the paper accessible to the reader not familiar with these ideas; second, because I believe it is a~useful to
present all calculations and formulae in purely dif\/ferential geometric terms~-- it is often impossible to carry out
twistor constructions in practice.
An unfortunate consequence is that some of the results and formulae appear miraculous: the twistor point of view
provides a~quick way to see why such results are true, while the Lax pair formalism provides one way to carry out more
detailed calculations.
\section{Selfdual spaces and the background geometries}
\label{s:sdbg}
On a~conformal $4$-manifold, the Hodge star operator is an involution on $2$-forms, so there is a~decomposition
$\wedge^2 T^{*} M= \wedge^2_{\raise1pt\hbox{$\scriptscriptstyle +$}} T^{*} M\oplus \wedge^2_{\raise1pt\hbox{$\scriptscriptstyle -$}} T^{*} M$, and the eigenspaces $\wedge^2_{\raise1pt\hbox{$\scriptscriptstyle\pm$}} T^{*}
M$ are called the selfdual and antiselfdual $2$-forms.
This induces a~similar decomposition
$\mathop{{\rm so}}(TM)=\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle +$}}(TM)\oplus\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle -$}}(TM)$ of the skew
endomorphisms of $TM$.
The Weyl curvature $W$ splits as a~sum of selfdual and antiselfdual $2$-forms $W^{\raise1pt\hbox{$\scriptscriptstyle\pm$}}$ with values in
$\mathop{{\rm so}}(TM)$: in fact $W^{\raise1pt\hbox{$\scriptscriptstyle +$}}$ is $\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle +$}}(TM)$-valued and $W^{\raise1pt\hbox{$\scriptscriptstyle -$}}$ is
$\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle -$}}(TM)$-valued (essentially because $W$ is traceless).
A \emph{selfdual space} is a~conformal $4$-manifold with selfdual Weyl curvature, i.e., $W^{\raise1pt\hbox{$\scriptscriptstyle -$}}=0$.
In~\cite{Joy:esd}, Joyce studied selfdual spaces with a~surface-orthogonal action of the torus $T^2$ by conformal
transformations, and found new explicit selfdual conformal metrics on connected sums of complex projective planes.
The key idea in his approach is the use of conformal connections with torsion, and the following observation.
\begin{lem}[see Joyce~\cite{Joy:esd}]\label{lem:J}
Let $(M,\mathsf{c})$ be an oriented conformal $4$-manifold and ${\mathcal{D}}$ a~conformal connection $($i.e.,
${\mathcal{D}}\mathsf{c}=0$, but ${\mathcal{D}}$ may have torsion$)$.
Suppose that the antiselfdual part of the torsion of ${\mathcal{D}}$ is tracelike.
Then $(M,\mathsf{c})$ is selfdual if and only if the Weyl part of the curvature of~${\mathcal{D}}$ is selfdual.
\end{lem}
When the torsion is selfdual and traceless, this lemma follows easily, since ${\mathcal{D}}$ then dif\/fers from a~Weyl
connection by an $\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle +$}}(TM)$-valued $1$-form, so that the $\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle -$}}(TM)$-valued
part of the curvature of ${\mathcal{D}}$ agrees with that of the Weyl connection.
The general case is a~consequence of the fact that the trace parts of the torsion (which are $1$-forms) cannot
contribute to the Weyl part of the curvature.
I shall refer to this result as Joyce's lemma: although simple, and perhaps previously known, its application by
Joyce~\cite{Joy:esd} was one of the main motivations for the present work.
Indeed, for a~conformal manifold with a~surface-orthogonal $T^2$-action, Joyce constructed, on the open set where the
torus acts freely, a~conformal connection with torsion, and hence separated the selfduality equation for the conformal
structure into a~nonlinear equation for a~quotient geometry and a~linear equation def\/ined on this background.
He then showed that the quotient geometry in this case is the hyperbolic plane, and superposed known solutions of the
linear equation to f\/ind new explicit metrics.
In this section, Joyce's lemma will be applied to selfdual spaces with \emph{any} freely acting group of conformal
transformations, and a~large class of dimensional reductions of the selfduality equation will be obtained.
The restriction to freely acting groups of symmetries will also be relaxed later, leading to a~generalized version of
dimensional reduction.
Let $M$ be a~conformal manifold with a~free proper action of a~group $H$, so that $M$ is a~principal $H$-bundle over the
orbit space $Q=M/H$.
The generators of the action form a~Lie algebra $\lie{h}$ of vector f\/ields on $M$ and pointwise evaluation def\/ines an
isomorphism $M\mathbin{{\times}}\lie{h}\to VM$, where $VM$ is the vertical bundle of $M\to Q$.
Now suppose that $H$ acts conformally with nondegenerate orbits.
Then $Q$ is a~conformal manifold and the horizontal distribution $VM^\perp\leqslant TM$ def\/ines a~principal
$H$-connection $\alpha$ on $\pi\colon M\to Q$; furthermore, $VM$ is isomorphic to the pullback of a~conformal vector
bundle ${\mathcal{V}}\to Q$ with $L^1_{{\mathcal{V}}}=L^1_Q$.
Although $M\mathbin{{\times}}\lie{h}$ and $\pi^*{\mathcal{V}}$ are both isomorphic to~$VM$, it will be crucial in the
following to distinguish between them, since the trivialization of~$VM$ given by the $H$-action is not, in general,
compatible with the conformal structure.
However, the isomorphism $\pi^*{\mathcal{V}}\to M\mathbin{{\times}}\lie{h}$ is $H$-equivariant, so it may be viewed as
a~bundle isomorphism $\varphi\colon{\mathcal{V}}\to\lie{h}_Q$ over $Q$, where $\lie{h}_Q=M\mathbin{{\times}}_H\lie{h}$.
To summarize, the conformal geometry of $M$ is encoded by:
\begin{itemize}\itemsep=0pt
\item
a conformal structure on $Q$;
\item
a conformal vector bundle ${\mathcal{V}}\to Q$ with $L^1_{{\mathcal{V}}}=L^1_Q$;
\item
a principal $H$-connection $\alpha$ on $M\to Q$;
\item
a bundle isomorphism $\varphi\colon{\mathcal{V}}\to\lie{h}_Q$ over $Q$.
\end{itemize}
The data $(\alpha,\varphi)$ identify the tangent bundle $TM$ with the pullback of ${\mathcal{V}}\oplus TQ$: sections of~${\mathcal{V}}$ or~$TQ$ will be denoted $U$, $V$, $W$ or $X$, $Y$, $Z$ respectively, and identif\/ied with invariant vector f\/ields on~$M$.
The curvature of the principal connection $\alpha$ on $\pi\colon M\to Q$ is an $\lie{h}_Q$-valued $2$-form $F^\alpha$,
given by minus the Frobenius curvature of the horizontal distribution: $F^\alpha(X,Y)=-\varphi([X,Y])$, where~$\varphi$
extended by zero from ${\mathcal{V}}$ to ${\mathcal{V}}\oplus TQ$.
Choose a~conformal connection $D$ on ${\mathcal{V}}$ over $Q$.
This induces a~Weyl derivative on $L^1_{\mathcal{V}}=L^1_Q$, hence a~torsion-free conformal connection on $TQ$ and
a~direct sum connection on ${\mathcal{V}}\oplus TQ$.
These conformal connections will be denoted by $D$, as will the pullback connection on $TM=\pi^*({\mathcal{V}}\oplus
TQ)$, which is conformal, but not torsion-free in general: the f\/ibres of $\pi\colon M\to Q$ need not be umbilic unless
$Q$ is three-dimensional, the nonlinear connection on $M\to Q$ need not be f\/lat unless $Q$ is one-dimensional, and the
sections of ${\mathcal{V}}$ parallel along the f\/ibres of $\pi$ will have nontrivial Lie brackets unless $\lie{h}$ is
Abelian.
In order to apply Joyce's lemma, the torsion needs to be reduced.
To do this, introduce $\psi\colon\wedge^2TQ\to{\mathcal{V}}$ and ${\mathcal{C}}\colon S^2_0{\mathcal{V}}\to TQ$, and
def\/ine ${\mathcal{D}}=D+\widehat{\mathcal{C}}+\widehat\psi$, where $\widehat{\mathcal{C}}$ and $\widehat\psi$ are the
sections of $T^{*} M\otimes\mathop{{\rm so}}(TM)$ given by
\begin{gather*}
\widehat {\mathcal{C}}_{U+X} (V+Y)={\mathcal{C}}(U,V)-\ip{{\mathcal{C}}(U,\cdot),Y},
\\
2\widehat\psi_{U+X} (V+Y)= \ip{\psi(X,\cdot),V}+\ip{\psi(Y,\cdot),U}-\psi(X,Y).
\end{gather*}
The idea is that $\psi$ will compensate for the curvature of the horizontal distribution, while ${\mathcal{C}}$ will
of\/fset the traceless second fundamental form of the f\/ibres.
It will then be possible to make the torsion selfdual by the choice of $D$.
The torsion $T^{\mathcal{D}}$ of this modif\/ied conformal connection may be computed by applying it to invariant vector
f\/ields, as long as one is careful that the Lie bracket on invariant vertical vector f\/ields is minus the Lie bracket on
the Lie algebra $\lie{h}$ of generators of the action, hence minus the bracket on the associated Lie algebra bundle
$\lie{h}_Q$ over $Q$
\begin{gather}
\label{eq:t1}
T^{\mathcal{D}}(U,V)=\varphi^{-1}[\varphi(U),\varphi(V)]_{\lie{h}},
\\
T^{\mathcal{D}}(U,X)=\varphi^{-1}D^\alpha_X\varphi(U)-\ip{{\mathcal{C}}(U),X},
\\
T^{\mathcal{D}}(X,Y)=\varphi^{-1}F^\alpha(X,Y)-\psi(X,Y).
\label{eq:t3}
\end{gather}
Here $D^\alpha$ denotes the connection on $\pi^*({\mathcal{V}}\otimes\lie{h}_Q)$ induced by~$D$ and $\alpha$.
Note that the torsion is vertical-valued.
The computation of the curvature is a~little more complicated.
First of all the curvatures of~${\mathcal{D}}$ and~$D$ (on~$TM$) are related by
\begin{gather*}
\nonumber R^{\mathcal{D}}_{U+X,V+Y}=R^D_{U+X,V+Y}+{\rm d}^D(\widehat {\mathcal{C}}+\widehat\psi)_{U+X,V+Y}
+\Abrack{(\widehat{\mathcal{C}}+\widehat\psi)_{U+X}, (\widehat{\mathcal{C}}+\widehat\psi)_{V+Y}}.
\end{gather*}
The second term is the twisted exterior derivative of $\mathop{{\rm so}}(TM)$-valued $1$-forms and must be handled
carefully, since $D$ has torsion.
Relating this to the torsion of ${\mathcal{D}}$ gives
\begin{gather*}
{\rm d}^D(\widehat{\mathcal{C}}+\widehat\psi)_{U+X,V+Y}= D_{U+X}(\widehat{\mathcal{C}}+\widehat\psi)_{V+Y}
-D_{V+Y}(\widehat{\mathcal{C}}+\widehat\psi)_{U+X}
\\
\hphantom{{\rm d}^D(\widehat{\mathcal{C}}+\widehat\psi)_{U+X,V+Y}=}{}
+(\widehat{\mathcal{C}}+\widehat\psi)_{T^{\mathcal{D}}(U+X,V+Y)} +(\widehat{\mathcal{C}}+\widehat\psi)_{\psi(X,Y)}+
(\widehat{\mathcal{C}}+\widehat\psi)_{\ip{{\mathcal{C}}(U),Y}-\ip{{\mathcal{C}}(V),X}}.
\end{gather*}
Notice that the derivatives in vertical directions are zero, since the connection is a~pullback connection and
$\widehat{\mathcal{C}},\widehat\psi$ are invariant.
Also, $R^D$ is horizontal.
Hence putting everything together leads to the following formula for the ``torsion-adjusted'' curvature of
${\mathcal{D}}$
\begin{gather*}
R^{{\mathcal{D}},ta}_{U+X,V+Y}:=
R^{\mathcal{D}}_{U+X,V+Y}-(\widehat{\mathcal{C}}+\widehat\psi)_{T^{\mathcal{D}}(U+X,V+Y)}
\\
\phantom{R^{{\mathcal{D}},ta}_{U+X,V+Y}}~
=R^D_{X,Y}+D_X\widehat{\mathcal{C}}{}_V-D_Y\widehat{\mathcal{C}}{}_U +D_X\widehat\psi{}_V-D_Y\widehat\psi{}_U
\\
\phantom{R^{{\mathcal{D}},ta}_{U+X,V+Y}=}~
{}+\widehat{\mathcal{C}}_{\psi(X,Y)} +\widehat\psi_{\ip{{\mathcal{C}}(U),Y}}-\widehat\psi_{\ip{{\mathcal{C}}(V),X}}
+\abrack{\widehat\psi_{U+X},\widehat{\mathcal{C}}_V} +\abrack{\widehat{\mathcal{C}}_U,\widehat\psi_{V+Y}}
\\
\phantom{R^{{\mathcal{D}},ta}_{U+X,V+Y}=}~
{}+\widehat{\mathcal{C}}_{\ip{{\mathcal{C}}(U),Y}}
-\widehat{\mathcal{C}}_{\ip{{\mathcal{C}}(V),X}}+\abrack{\widehat{\mathcal{C}}_U,\widehat{\mathcal{C}}_V}
+\widehat\psi_{\psi(X,Y)} +\abrack{\widehat\psi_{U+X},\widehat\psi_{V+Y}}.
\end{gather*}
Now suppose that $T^{\mathcal{D}}$ is selfdual.
Then, by Joyce's lemma, the Weyl curvature $W$ of $M$ is selfdual if and only if the Weyl part of this formula is
selfdual.
This condition is a~nonlinear dif\/ferential equation on $Q$ def\/ining a~reduced background geometry.
A key feature of the formula is that the right hand side is manifestly independent of $(\alpha,\varphi)$ and so the
group structure of $H$ decouples from the reduced background geometry.
The details depend on the dimension; in particular, ensuring that the torsion is selfdual constrains $D$.
In the following subsections, I will explain these constraints and obtain the background geometries explicitly.
For later use I will also introduce individual notations for the geometries in each dimension.
As far as possible, the notation will be chosen to be consistent with existing usage.
The case of a~four-dimensional group acting on itself is left as an exercise: a~f\/ield
${\mathcal{Y}}\in\wedge^2_{\raise1pt\hbox{$\scriptscriptstyle -$}}{\mathcal{V}}^{*}\otimes{\mathcal{V}}$ is needed here to make the torsion selfdual; see
Remark~\ref{r:zdg}.
\subsection{Reduction to one dimension}
In this case $Q$ is a~oriented curve, with weightless unit tangent $\xi$, and $\psi=0$.
Write ${\mathcal{C}}(U,V)=\ip{{\mathcal{B}}(U),V}\xi$, so that ${\mathcal{B}}$ is a~symmetric traceless endomorphism of
weight $-1$.
I will denote the curve $Q$ by $C$ and the bundle ${\mathcal{V}}$ by ${\mathcal{E}}$.
Thus ${\mathcal{E}}$ is a~rank $3$ conformal vector bundle over $C$ with $L^1_{\mathcal{E}}=L^1_C=TC$, the last
identif\/ication being given by $\xi$.
The Hodge star operator is an isomorphism between $\wedge^2{\mathcal{E}}$ and $TC\otimes{\mathcal{E}}=L^1{\mathcal{E}}$.
The two components of the torsion of ${\mathcal{D}}$ are given by
\begin{gather*}
T^{\mathcal{D}}(U,V)=\varphi^{-1}[\varphi(U),\varphi(V)]_{\lie{h}},
\qquad
T^{\mathcal{D}}(U,\xi)=\varphi^{-1}D^\alpha_\xi\varphi(U)-{\mathcal{B}}(U).
\end{gather*}
This will be selfdual if and only if
\begin{gather}
\label{eq:riccdef}
\varphi^{-1}\bigl(D^\alpha_\xi\varphi-{*[\varphi,\varphi]_{\lie{h}}}\bigr) ={\mathcal{B}}.
\end{gather}
Conformal connections $D$ on ${\mathcal{E}}$ form an af\/f\/ine space modelled on $T^{*} C
\otimes\mathop{\mathfrak{co}}({\mathcal{E}})$.
Hence $D$ can be chosen uniquely so that the left hand side is symmetric and traceless and this in turn def\/ines
${\mathcal{B}}$.
The two components of the torsion-adjusted curvature of ${\mathcal{D}}$ are
\begin{gather*}
R^{{\mathcal{D}},ta}_{U,V} =\abrack{\widehat{\mathcal{B}}_U,\widehat{\mathcal{B}}_V},
\qquad
R^{{\mathcal{D}},ta}_{U,\xi} =-D_\xi\widehat{\mathcal{B}}_U+\widehat{\mathcal{B}}_{{\mathcal{B}}(U)}.
\end{gather*}
A straightforward calculation shows that the Weyl part is selfdual if and only if
$D_\xi{\mathcal{B}}=2({\mathcal{B}}^2)_0$, where the subscript zero denotes the traceless part.
Since $D$ is f\/lat, this is really just a~Riccati equation for a~$3\times 3$ symmetric traceless matrix.
On the other hand, the orientation of $C$ is not needed if ${\mathcal{B}}$ is viewed as an endomorphism-valued $1$-form
on $C$.
\begin{defn}
Suppose that ${\mathcal{E}}$ is a~rank $3$ conformal vector bundle over a~curve $C$ with $L^1_{\mathcal{E}}=L^1_C$.
Equip ${\mathcal{E}}$ with a~conformal connection $D$ and a~section ${\mathcal{B}}$ of $T^{*}
C\otimes\Sym_0{\mathcal{E}}$.
Then $(C,{\mathcal{E}})$ is said to be a~\emph{Riccati space} if $(D,{\mathcal{B}})$ satisfy the equation
\begin{gather*
D{\mathcal{B}}=2\big(\mathcal{B}^2\big)_0.
\end{gather*}
Equivalently, with respect to a~conformal trivialization of ${\mathcal{E}}$, the connection $D$ is given by a~section
$(a,\Theta)$ of $T^{*} C\otimes\mathop{\mathfrak{co}}({\mathcal{E}})=T^{*} C\oplus(T^{*}C\otimes\mathop{{\rm so}}({\mathcal{E}}))$ and
\begin{gather*}
\dot{\mathcal{B}}-a{\mathcal{B}}+\abrack{\Theta,{\mathcal{B}}}=2\big(\mathcal{B}^2\big)_0,
\end{gather*}
where the dot denotes dif\/ferentiation with respect to a~compatible coordinate.
In particular, using a~$D$-parallel trivialization of ${\mathcal{E}}$ and an af\/f\/ine coordinate $r$ (i.e.,
$D{\rm d} r=0$), ${\mathcal{B}}_r=2({\mathcal{B}}^2)_0$.
\end{defn}
Joyce's lemma now gives the following result.
\begin{thm}\label{th:ric}
Let $M$ be an oriented conformal manifold with a~$3$-dimensional Lie algebra $\lie{h}$ of linearly independent conformal
vector fields such that the projection $\pi$ onto the space of orbits is a~submersion over a~curve $C$.
Let ${\mathcal{E}}$ be a~rank $3$ conformal vector bundle on $C$ such that $\pi^*{\mathcal{E}}$ is the vertical bundle
of $M$ $($trivialized along the fibres by invariant vector fields$)$ and define $(D,{\mathcal{B}})$ by
equation~\eqref{eq:riccdef} as explained above.
Then $M$ is selfdual if and only if $(C,{\mathcal{E}})$ is a~Riccati space.
\end{thm}
\subsection{Reduction to two dimensions}
The two-dimensional geometry is perhaps the richest.
In the Euclidean case, $Q$ is a~Riemann surface, which will be denoted $N$, and the conformal vector bundle
${\mathcal{V}}$ has rank two, and so will be viewed as a~complex line bundle ${\mathcal{W}}\to N$.
It will be convenient to view $\varphi\colon{\mathcal{W}}\to\lie{h}_N$ as a~complex linear map
$\frac12(\varphi-{\boldsymbol i}\varphi\circ J)\colon{\mathcal{W}}\to\lie{h}_N\otimes{\mathbb{C}}$.
The constraint $L^1_{\mathcal{W}}=L^1_N$ means that there is a~Hermitian metric on
${\mathcal{W}}^{*}\otimes_{\mathbb{C}} TN$.
On f\/ixing orientations, this may also be interpreted as an identif\/ication
${\mathcal{W}}\otimes_{\mathbb{C}}\overline{\mathcal{W}}= TN\otimes_{\mathbb{C}}\overline{TN}$, or
$\wedge^{1,1}{\mathcal{W}}=\wedge^{1,1}TN$.
Here the orientations will be chosen so that the induced almost complex structure $J$ on $TM=\pi^*({\mathcal{W}}\oplus
TN)$ is selfdual, i.e., the corresponding weightless $2$-form $\Omega\in\mathrm{C}^\infty(M,L^2 \wedge^2T^{*} M)$ is selfdual.
The Hodge star operator on $\wedge^2TM$ interchanges the vertical and horizontal components using the identif\/ication
above, and acts on mixed bivectors by $*(U\wedge X)=-JU\wedge JX$.
In particular the torsion of the modif\/ied connection ${\mathcal{D}}$ is selfdual if and only if
\begin{gather}
\varphi^{-1}\bigl(F^\alpha-[\varphi,\overline\varphi]_{\lie{h}}\bigr) =\psi,
\nonumber
\\
\varphi^{-1}\bigl(D^\alpha_X\varphi(U)+D^\alpha_{JX}\varphi(JU)\bigr)
=\ip{{\mathcal{C}}(U),X}+\ip{{\mathcal{C}}(JU),JX}
\label{eq:svdef}
\end{gather}
for any vector f\/ield $X$ and section $U$ of ${\mathcal{W}}$.
The f\/irst equation def\/ines $\psi$ uniquely.
For the second equation, note that conformal connections $D$ on ${\mathcal{W}}$ form an af\/f\/ine space modelled on $T^{*}
N\otimes_{\mathbb{R}}\mathop{\mathfrak{co}}({\mathcal{W}})$, and therefore $D$ can be chosen uniquely such that
$\varphi^{-1}D^\alpha_X\varphi$ is symmetric and traceless for all $X$.
The remaining ambiguity in ${\mathcal{C}}$ is f\/ixed by supposing it is complex linear, i.e., a~section of
$S^2_0{\mathcal{W}}^{*}\otimes_{\mathbb{C}} TN$, so that the second equation determines it uniquely.
Note though, that the second equation does not use all of $D$: only the induced holomorphic structure of ${\mathcal{W}}$
is needed.
For the selfduality of the Weyl curvature, it now suf\/f\/ices to compute $R^{{\mathcal{D}},ta}_{U,Y}$ and one additional
component.
To see this, note f\/irst that $\smash{\widehat\psi}$ and $\smash{\widehat{\mathcal{C}}}$ are
$\mathop{{\rm so}}(TM)$-valued, and therefore no information is lost by considering $R^{\mathcal{D}}$ and $R^D$ to
be the curvatures on $L^{-1} TM$ rather than $TM$, so that the curvature equation is an identity
between $\mathop{{\rm so}}(TM)$-valued $2$-forms.
The selfduality condition is obtained by requiring that the Weyl part of $R^{{\mathcal{D}},ta}$ is selfdual.
This amounts to considering the traceless part of the induced bundle map from $\wedge^2_{\raise1pt\hbox{$\scriptscriptstyle -$}} T M$ to
$\mathop{{\rm so}}_{\raise1pt\hbox{$\scriptscriptstyle -$}}(TM)$.
Let $\overline\Omega$ and $\overline J$ be the sections of these bundles obtained by reversing the orientation of
$\Omega$ and $J$ on $TN$.
There are therefore three equations to f\/ind:
\begin{enumerate}\itemsep=0pt
\item[(i)]
The traceless part of the $\overline\Omega^\perp\otimes\overline J^\perp$ component of $R^{{\mathcal{D}},ta}$ should be
zero.
\item[(ii)]
The part of $R^{{\mathcal{D}},ta}$ in $\overline\Omega^\perp\otimes\langle\overline J\rangle$ should be zero.
\item[(iii)]
The multiple of $\overline\Omega\otimes\overline J$ should equal half the trace in
$\overline\Omega^\perp\otimes\overline J^\perp$.
\end{enumerate}
All of these except (iii) involve considering only the part of $R^{{\mathcal{D}},ta}$ in
$\overline\Omega^\perp\otimes\mathop{{\rm so}}(TM)$, which involves evaluating $R^{{\mathcal{D}},ta}$ on bivector
f\/ields of the form $U\wedge Y-{*(U\wedge Y)}=U\wedge Y+JU\wedge JY$.
An easy calculation gives
\begin{gather*}
R^{{\mathcal{D}},ta}_{U,Y}
=-D_Y{\mathcal{C}}(U,\cdot)+D_Y{\mathcal{C}}(U,\cdot)^{\scriptscriptstyle\mathrm T}-\tfrac14D_Y\psi(JU)(J-\overline J)
\\
\phantom{R^{{\mathcal{D}},ta}_{U,Y}=}
{}+{\mathcal{C}}\bigl(\ip{{\mathcal{C}}(U),Y},\cdot\bigr)
-{\mathcal{C}}\bigl(\ip{{\mathcal{C}}(U),Y},\cdot\bigr)^{\scriptscriptstyle\mathrm T}
-\tfrac14\ip{{\mathcal{C}}(U,J\psi),Y}(3J-\overline J)-\tfrac14J\psi(U)J\psi\mathinner{\vartriangle}Y,
\end{gather*}
where ${}^{\scriptscriptstyle\mathrm T}$ denotes the transpose, and I have contracted horizontal and vertical skew
endomorphisms using $X\mathinner{\vartriangle}Y=\frac12\Omega(X,Y)(J-\overline J)$ and $U\mathinner{\vartriangle}V
=\frac12\Omega(U,V)(J+\overline J)$.
Also, using $\Omega$, $\psi$ may be viewed as a~section of ${\mathcal{W}}^{*}$, so that $\psi(X,Y)=2\Omega(X,Y)J\psi$.
One readily obtains conditions (i) and (ii):{\samepage
\begin{gather*}
D_X{\mathcal{C}}(U,\cdot)+D_{JX}{\mathcal{C}}(JU,\cdot)=0,
\\
\tfrac12\bigl( D_X\psi(U)+D_{JX}\psi(JU)\bigr)=-3\ip{{\mathcal{C}}(\psi,U),X},
\end{gather*}
for any vector f\/ield $X$ and section $U$ of ${\mathcal{W}}$.}
It remains to compute condition (iii).
For this note that $R^D$ is given by $\frac14s_N(\Omega-\overline\Omega) \otimes(J-\overline
J)+\frac14s_{\mathcal{W}}(\Omega-\overline\Omega)\otimes(J+\overline J)$, where $s_N=\frac12\scal_N$ and
$s_{\mathcal{W}}=\frac12\scal_{\mathcal{W}}$ are the normalized scalar curvatures of $D$ on $TN$ and ${\mathcal{W}}$.
This yields, f\/inally,
\begin{gather*}
s_N-s_{\mathcal{W}}=|\psi|^2-2|{\mathcal{C}}|^2.
\end{gather*}
These equations, and the equations def\/ining ${\mathcal{C}}$ and $\psi$, depend only on $D$ through the induced
holomorphic structure $\overline\partial{}^a$ on ${\mathcal{W}}$.
The conformal connections on ${\mathcal{W}}$ and $TN$ may be viewed as Chern connections determined by the holomorphic
structures on these bundles together with the choice of a~Weyl derivative on $L^1$.
The dif\/ference $s_N-s_{\mathcal{W}}$ does not depend on the choice of Weyl derivative, since it is the normalized scalar
curvature of the Chern connection on the Hermitian line bundle ${\mathcal{W}}^{-1} TN$.
\begin{aside}
The construction of a~``Chern--Weyl'' connection does not seem to be known and relates to a~simple coordinate-free
description (also not well-known) of the usual Chern connection, so I will sketch it here.
Let $E$ be a~complex vector bundle (over a~complex manifold $M,J$) with holomorphic structure $\overline\partial{}^E$
and compatible conformal metric $\mathsf{c}\colon S^2_{\mathbb{R}} E\to L^2_E$ with respect to which the complex
structure on $E$ is orthogonal; the latter is equivalently a~conformal Hermitian structure $\smash{\overline
E}\otimes_{\mathbb{C}} E\to L^2_E\otimes{\mathbb{C}}$.
(Note that a~complex line bundle automatically has a~conformal Hermitian structure.) Now given any covariant derivative
$D$ on $L^1_E$, let $\overline\partial{}^D_X=\frac12(D_X+{\boldsymbol i} D_{JX})$ (for all vector f\/ields $X$) be the
induced almost holomorphic structure on $L^2_E\otimes{\mathbb{C}}$.
Then there is a~unique conformal Hermitian connection $D^E$ on $E$ inducing $D$ on $L^1_E$.
It is given by the formula
\begin{gather*}
\ip{D^E_Xs_1,s_2}=\overline\partial{}^D_X\ip{s_1,s_2}-\ip{s_1,\overline\partial{}^E_X s_2}+
\ip{\overline\partial{}^E_Xs_1,s_2},
\end{gather*}
where $s_1$ and $s_2$ are sections of $E$.
The proof is immediate (the idea behind the formula is simply that $D^E=\partial^E+\overline\partial{}^E$, where
$\partial^E$ is the complex-linear part of the derivative).
In the case that the covariant derivative on $L^1_E$ is just a~trivialization, so that
$\overline\partial{}^D=\overline\partial$, this is the usual Chern connection by uniqueness.
\end{aside}
\begin{defn}
Suppose that ${\mathcal{W}}$ is a~complex line bundle over a~Riemann surface $N$ such that $L^1_{\mathcal{W}}=L^1_N$
(i.e., there is a~Hermitian metric on ${\mathcal{W}}^{-1} TN$).
Equip ${\mathcal{W}}$ with a~holomorphic structure~$\overline\partial{}^a$, a~section ${\mathcal{C}}$ of
${\mathcal{W}}^{-2} TN$, and a~section $\psi$ of ${\mathcal{W}}^{-1}$.
Then $(N,{\mathcal{W}})$ is said to be a~\emph{spinor-vortex space} if $(\overline\partial{}^a,{\mathcal{C}},\psi)$
satisfy the equations
\begin{gather}
\overline\partial{}^a{\mathcal{C}}=0,
\label{eq:sv1}
\\
\overline\partial{}^a \psi=-3{\mathcal{C}}\overline\psi,
\label{eq:sv2}
\\
s_{\smash{{\mathcal{W}}^{-1} TN}} =\psi\overline\psi-2{\mathcal{C}}\overline{\mathcal{C}},
\label{eq:sv3}
\end{gather}
where $s_{\smash{{\mathcal{W}}^{-1} TN}}$ is the normalized scalar curvature of the Chern connection
on ${\mathcal{W}}^{-1} TN$.
\end{defn}
Joyce's lemma now gives the following result.
\begin{thm}
\label{th:sv}
Let $M$ be an oriented conformal manifold with a~$2$-dimensional Lie algebra $\lie{h}$ of linearly independent conformal
vector fields such that the projection $\pi$ onto the space of orbits is a~submersion over a~Riemann surface~$N$.
Let~${\mathcal{W}}$ be a~complex line bundle on $N$ such that~$\pi^*{\mathcal{W}}$ is the vertical bundle of~$M$
$($trivialized along the fibres by invariant vector fields$)$ and define
$(\overline\partial{}^a,{\mathcal{C}},\psi)$ by~\eqref{eq:svdef} as explained above.
Then $M$ is selfdual if and only if $(N,{\mathcal{E}})$ is a~spinor-vortex space.
\end{thm}
Of course there are only two possible $2$-dimensional Lie algebras.
In the case that $\lie{h}$ is Abelian and $\psi=0$, this result reduces to the original application of Lemma~\ref{lem:J}
by Joyce~\cite{Joy:esd}.
\begin{rem}
It is straightforward to adapt the calculations of this subsection to other signatures: in general $N$ is a~conformal
surface with two line bundles ${\mathcal{W}},\widetilde{\mathcal{W}}$ such that ${\mathcal{W}} \widetilde{\mathcal{W}}\cong T^{1,0}N T^{0,1}N$ where $T^{1,0}N$ and $T^{0,1}N$ are the null line
subbundles of $TN\otimes{\mathbb{C}}$.
The line bundles ${\mathcal{W}}$ and $\widetilde{\mathcal{W}}$ are equipped with ``(anti)holomorphic structures''
$\tilde\partial^a,\partial^a$ and there are sections $({\mathcal{C}},\widetilde{\mathcal{C}},\psi,\widetilde\psi)$
satisfying appropriate reality conditions: in the Euclidean case $\widetilde{\mathcal{W}}=\overline{\mathcal{W}}$,
$\widetilde{\mathcal{C}}=\overline{\mathcal{C}}$ and $\widetilde\psi=\overline\psi$, but there is also a~Lorentzian
case, when the f\/ields are all real, and a~Euclidean reduction from Kleinian signature $(2,2)$ when
$\widetilde\psi=-\overline\psi$.
\end{rem}
\subsection{Reduction to three dimensions}
The reduction to three dimensions is the Jones--Tod correspondence~\cite{JoTo:mew}, which was one of the motivations for
this work.
The proof using Joyce's lemma is outlined in~\cite{Joy:esd}, so I will only recall the main ideas and def\/initions.
In this case, $Q$ is an oriented $3$-dimensional conformal manifold, which will be denoted $B$, and ${\mathcal{V}}$ is
an oriented line bundle isomorphic to $L^1=L^1_B$.
Note that ${\mathcal{C}}$ automatically vanishes, and write $\psi=*\omega$, where $\omega$ is a~$1$-form on $B$.
Then the torsion $T^{\mathcal{D}}$ will be selfdual provided that
\begin{gather}
*(D\varphi-\omega\varphi)=F^\alpha.
\end{gather}
(The connection $\alpha$ does not act on $\varphi$ since $\lie{h}$ is Abelian.)
This equation determines $(D,\omega)$ up to the gauge freedom $D\mapsto D+\gamma,\omega\mapsto\omega-\gamma$ for any
$1$-form $\gamma$ on $B$ (note that $(D+\gamma)\varphi=D\varphi-\gamma\varphi$, since $\varphi$ has weight $-1$).
In other words, it determines uniquely the Weyl derivative $D+\omega$.
This is called the \emph{Jones--Tod} Weyl structure.
The gauge freedom can be used to set $D\varphi=0$, or to set $\omega=0$.
Taking the latter point of view, the Jones--Tod Weyl structure is just $D$, determined by $* D\varphi=F^\alpha$.
The curvature of ${\mathcal{D}}$ is simply the pullback of the curvature of $D$ on $B$.
In particular the Weyl part will be selfdual if\/f it vanishes and this is readily seen to be equivalent to the vanishing
of~$r^D_0$~\cite{Joy:esd}.
\begin{defn}
Suppose that $B$ is a~conformal $3$-manifold and let $D$ be a~Weyl connection on~$B$.
Then $B$ is said to be an \emph{Einstein--Weyl space} if\/f $r^D_0=0$, i.e., the symmetric traceless Ricci tensor of~$D$
vanishes.
\end{defn}
\begin{thm}
[\cite{JoTo:mew}]
\label{th:JnT}
Let $M$ be an oriented conformal $4$-manifold and $K$ a~nonvanishing conformal vector field such that the projection
$\pi$ of $M$ onto the space of trajectories is a~submersion over a~conformal manifold $B$.
Equip $B$ with the Jones--Tod Weyl structure.
Then $M$ is selfdual if and only if $B$ is Einstein--Weyl.
\end{thm}
All three background geometries are themselves def\/ined by geometric gauge theories: conformal local trivializations of
the bundle ${\mathcal{V}}$ over $Q$ are related by gauge transformations.
In the three-dimensional case, ${\mathcal{V}}$ is simply $L^1$, so conformal trivializations are length scales, and this
is Weyl's original gauge theory~\cite{Wey:stm}.
Analogously, in the one and two-dimensional case, ${\mathcal{E}}$~and~${\mathcal{W}}$ should be regarded as part of the
geometry of the space, \emph{not} auxiliary bundles, and these background equations, although gauge-theoretic, should
not be confused with the gauge f\/ield equations on auxiliary $G$-bundles which will be studied in Section~\ref{s:gfe}.
\subsection*{Addenda: null reductions}
The constructions of this section assume nondegeneracy of the conformal structure $\mathsf{c}$ on $M$ along the orbits
of the symmetry group $H$.
In (real) Euclidean signature, this is automatic, but this is not the case when $M$ has Kleinian signature, nor when
$M$ is a~holomorphic conformal manifold.
When the conformal structure degenerates on the orbits, the reduction is said to be \emph{null}.
For the generic local considerations of this paper, I assume that the radical (or kernel) $R_V:=VM\cap VM^\perp$ of the
conformal structure along the $H$ orbits has constant rank (as before, $VM$ denotes the tangent bundle to the $H$
orbits).
There are thus three possibilities:
\begin{enumerate}\itemsep=0pt
\item[(1)] $R_V$ has rank one;
\item[(2$+$)] $R_V=U_+$ has rank two and is selfdual;
\item[(2$-$)] $R_V=U_-$ has rank
two and is antiselfdual.
\end{enumerate}
In case (1), $R_V^\perp = VM+VM^\perp$ has rank three, and $R_V^\perp/R_V$ is the sum of two null subbundles with
a~nondegenerate pairing between them.
Their inverse images $U_\pm$ in $R_V^\perp$ are selfdual and antiselfdual null 2-plane distributions.
There are several subcases to consider here, depending on the rank of $VM$.
If either $U_\pm \subseteq VM$ then $VM$ must have rank $3$, in which case $VM^\perp\subset VM$, hence $VM^\perp=R_V$
and $R_V^\perp=VM$, so $VM=U_+ + U_-$.
If $VM$ has rank two, then $TM=VM+U_++U_-$, and clearly if $\rank V_M=1$, then $R_V=VM$.
Cases (2$\pm$) are simpler, at least when $\rank VM=2$, so $VM=R_V$ is a~bundle of selfdual or antiselfdual $2$-planes.
I concentrate on these two cases here.
One justif\/ication for such a~focus is that in the other cases, one may be able to prove that the distributions $U_+$ or
$U_-$ are integrable, and hence study the reduction in terms of their leaf spaces.
For instance, Dunajski--West~\cite{DuWe:acs} establish such an integrability result when $\rank VM=1$.
Suppose then that $\pi\colon M\to Q=M/H$ is a~principal $H$-bundle over a~manifold $Q$ such that $VM$ has rank two and
is totally null.
As $VM^\perp=VM$, neither $VM$ nor $TQ$ inherit conformal structures from $M$; instead there is a~nondegenerate pairing
$VM\times\pi^* TQ\to L^2$.
This pairing in $H$-invariant, and thus identif\/ies $VM=\pi^* (T^*Q\otimes {\mathcal{V}})$, for a~line bundle
${\mathcal{V}}\to Q$ (whose pullback to $M$ will be identif\/ied with $L^2$).
As in the nondegenerate setting the isomorphism of $VM$ with $M\times\lie{h}$ descends to a~bundle isomorphism
$\varphi\colon T^*Q\otimes {\mathcal{V}} \to \lie{h}_Q:= M\times_H\lie{h}$ over $Q$.
There is no canonical splitting of the short exact sequence $0\to \pi^*T^*Q\otimes {\mathcal{V}} \to TM \to \pi^*TQ\to
0$, but such a~splitting may be chosen so that the complementary subbundle to $VM$ is $H$-invariant and null.
This yields a~principal $H$-connection $\alpha$ on $\pi\colon M\to Q$.
A torsion-free connection on $TN$ and a~connection on ${\mathcal{V}}$ together induce a~conformal connection $D$ on $TM$
with vertical-valued torsion as before.
A conformal connection ${\mathcal{D}}$ with selfdual torsion may be obtained by adding correction terms to $D$.
Compared to the nondegenerate case, the correction term ${\mathcal{C}}$ may be absorbed into the choice of $D$, and
replaced by a~vertical correction $\chi\colon \wedge^2(T^*Q\otimes{\mathcal{V}})\to T^*Q\otimes {\mathcal{V}}$.
Thus ${\mathcal{D}}=D+\widehat\chi+\widehat\psi$, where $\psi\colon \wedge^2TQ\to T^*Q\otimes {\mathcal{V}}$ as
before, and
\begin{gather*}
2\widehat\chi_{U+X} (V+Y)= \ip{\chi(U,\cdot),Y}+\ip{\chi(V,\cdot),X}-\chi(U,V),
\\
2\widehat\psi_{U+X} (V+Y)= \ip{\psi(X,\cdot),Y}+\ip{\psi(Y,\cdot),X}-\psi(X,Y).
\end{gather*}
The torsion satisf\/ies
\begin{gather}
\label{eq:null-t1}
T^{\mathcal{D}}(U,V)=\varphi^{-1}[\varphi(U),\varphi(V)]_{\lie{h}}-\chi(U,V),
\\
T^{\mathcal{D}}(U,X)=\varphi^{-1}D^\alpha_X\varphi(U),
\\
T^{\mathcal{D}}(X,Y)=\varphi^{-1}F^\alpha(X,Y)-\psi(X,Y),
\label{eq:null-t3}
\end{gather}
while the curvature computes to
\begin{gather*}
R^{{\mathcal{D}}}_{U+X,V+Y}=R^D_{U+X,V+Y}+(\widehat\chi+\widehat\psi)_{T^{\mathcal{D}}(U+X,V+Y)}
\\
\phantom{R^{{\mathcal{D}}}_{U+X,V+Y}=}
{}+D_{U+X}(\widehat\chi+\widehat\psi)_{V+Y} -D_{V+Y}(\widehat\chi+\widehat\psi)_{U+X}
\\
\phantom{R^{{\mathcal{D}}}_{U+X,V+Y}=}
{}+(\widehat\chi+\widehat\psi)_{\psi(X,Y)}+ (\widehat\chi+\widehat\psi)_{\chi(U,V)}
+\Abrack{(\widehat\chi+\widehat\psi)_{U+X}, (\widehat\chi+\widehat\psi)_{V+Y}}.
\end{gather*}
Hence the torsion-adjusted curvature is
\begin{gather}
R^{{\mathcal{D}},ta}_{U+X,V+Y}=R^D_{X,Y}+\Abrack{(\widehat\chi+\widehat\psi)_X,
(\widehat\chi+\widehat\psi)_Y}+\widehat\chi_{\psi(X,Y)}
\nonumber
\\
\phantom{R^{{\mathcal{D}},ta}_{U+X,V+Y}=}
{}+D_X(\widehat\chi+\widehat\psi)_Y -D_Y(\widehat\chi+\widehat\psi)_X +D_X\widehat\chi{}_V-D_Y\widehat\chi{}_U
\nonumber
\\
\phantom{R^{{\mathcal{D}},ta}_{U+X,V+Y}=}
{}+\Abrack{(\widehat\chi+\widehat\psi)_X,\widehat\chi{}_V} +\Abrack{\widehat\chi{}_U,(\widehat\chi+\widehat\psi)_Y}
+\abrack{\widehat\chi_U,\widehat\chi_V} +\widehat\chi_{\chi(U,V)}.
\label{eq:null-c}
\end{gather}
\subsection*{Antiselfdual ($\boldsymbol{\alpha}$-surface) reduction}
When $VM$ is antiselfdual, the f\/ibres of $\pi\colon M\to Q$ are $\alpha$-surfaces (i.e., they correspond to points in
the twistor space of $M$).
Under the isomorphism of $TM$ with $\pi^*(T^*Q\otimes{\mathcal{V}} \oplus TQ)$, the antiselfdual bivectors decompose
into three rank one subbundles: $\wedge^2(T^*Q\otimes{\mathcal{V}})$, $\wedge^2TQ$, and the tracelike part of
$TQ\otimes T^*Q\otimes{\mathcal{V}}$.
The corresponding decomposition of $\mathop{{\rm so}}_-(TM)$ has summands $\Hom_-(TQ,T^*Q\otimes{\mathcal{V}})$,
$\Hom_-(T^*Q\otimes{\mathcal{V}},TQ)$ and the span of $\iden_{T^*Q\otimes{\mathcal{V}}} - \iden_{TQ}$; here $\Hom_-$
denotes the subbundle of skew symmetric operators.
In order to interpret equations~\eqref{eq:null-t1}--\eqref{eq:null-t3} and~\eqref{eq:null-c}, it is convenient to set
${\mathcal{V}}={\mathcal{L}}^2\otimes \wedge^2TQ$ for a~line bundle ${\mathcal{L}}$, so that $\varphi$, $\psi$ and
$\chi$ may be viewed as $1$-forms on $Q$, with values in ${\mathcal{L}}^2\otimes\lie{h}_Q$, ${\mathcal{L}}^{-2}$ and
${\mathcal{L}}^2$ respectively.
Then $T^{\mathcal{D}}$ is selfdual if\/f $\chi\wedge\varphi = \frac12 [\varphi\wedge\varphi]_{\lie{h}}$,
$\psi\wedge\varphi=F^\alpha$ and ${\rm d}^{D,\alpha}\varphi=0$.
The f\/irst two equations determine $\chi$ and $\psi$ uniquely (since $\varphi\colon TN\to {\mathcal{L}}\otimes\lie{h}_Q$
is injective), while the third depends only on (and essentially determines) the connection $a$ induced by $D$ on
${\mathcal{L}}$.
In the expression~\eqref{eq:null-c} for the torsion-adjusted curvature, very few terms contribute to the antiselfdual
Weyl part.
For instance, only the trace part of f\/irst and last lines contribute, and the latter trace vanishes identically.
After some tedious computations, the background equations for the $1$-forms $\psi$ and $\chi$ and the connection $a$ on
${\mathcal{L}}$ reduce to
\begin{gather*
{\rm d}^a\psi=0,
\qquad
{\rm d}^a\chi=0,
\qquad
F^a=\chi\wedge\psi,
\end{gather*}
which may be interpreted as the f\/latness of the connection ${\rm d}^a+\psi+\chi$ on ${\mathcal{L}}\oplus
{\mathcal{L}}^{-1}$ (where ${\rm d}^a$ is the direct sum connection, while $\psi$ and $\chi$ are viewed as
$1$-forms with values in $\Hom({\mathcal{L}}^{-1},{\mathcal{L}})$ and $\Hom({\mathcal{L}},{\mathcal{L}}^{-1})$).
\subsection*{Selfdual ($\boldsymbol{\beta}$-surface) reduction}
In the selfdual case, only the mixed part of $T^{\mathcal{D}}$ has a~antiselfdual component, so there is no loss in
setting $\psi=\chi=0$.
The selfduality of the torsion then reduces to $D^\alpha\varphi=\frac12{\rm d}^{D^\alpha}\varphi$, where
$\varphi$ is interpreted as a~$1$-form on $Q$ as before.
This equation determines $D$ up to projective transformation ($D_XY\mapsto D_XY +\gamma(X)Y+\gamma(Y)X$ for a~$1$-form
$\gamma$), and the background equations are vacuous.
Hence the background geometry is an arbitrary (torsion-free) projective surface $(Q,[D])$; this reduction was obtained
in~\cite{DuWe:acs} and~\cite{Cal:sdp}.
\section{Interlude: Bianchi metrics}
\label{s:bm}
Selfdual conformal manifolds with a~freely acting three-dimensional symmetry group have been studied in many
places~\cite{Dan:sfk,DaSt:cok,Hit:tem, Maz:csb,MMW:sbm,PePo:kzs,Tod:p6,Tod:com,Tod:p3}.
In this interlude, I will show brief\/ly how the direct approach to selfdual Bianchi metrics is related to the Riccati
space reduction of the previous section.
For simplicity, I focus on the case of diagonal Bianchi IX metrics.
Such a~metric may be written in the form
\begin{gather*}
g=w_1w_2w_3 {\rm d} t^2 +\frac{w_2w_3}{w_1}\sigma_1^{2}+\frac{w_3w_1}{w_2}\sigma_2^{2}
+\frac{w_1w_2}{w_3}\sigma_3^{2},
\end{gather*}
where $\sigma_1$, $\sigma_2$, $\sigma_3$ are the usual left-invariant $1$-forms on $\SU(2)$ and $w_1$, $w_2$, $w_3$ are functions of~$t$.
If $X_1$, $X_2$, $X_3$ are the dual vector f\/ields to $\sigma_1$, $\sigma_2$, $\sigma_3$, then the vector f\/ields
$\partial_t$, $\varphi_1=w_1 X_1$, $\varphi_2=w_2 X_2$ and $\varphi_3=w_3 X_3$ form a~conformal frame.
Notice that $\dot\varphi_1 -[\varphi_2,\varphi_3]$ is the multiple $\dot w_1-w_2w_3$ of $X_1$, where the dot denotes the
derivative with respect to $t$.
The other two components are similar.
Following~\cite{PePo:kzs,Tod:p6}, write
\begin{gather*}
\dot w_1=w_2w_3-w_1(A_2+A_3),
\\
\dot w_2=w_3w_1-w_2(A_3+A_1),
\\
\dot w_3=w_1w_2-w_3(A_1+A_2).
\end{gather*}
Comparing with equation~\eqref{eq:riccdef}, observe that the matrix ${\mathcal{B}}$ is diagonal, with eigenvalues
$-(A_2+A_3),-(A_3+A_1),-(A_1+A_2)$.
In the approach of the previous section, the conformal gauge freedom is used to set $A_1+A_2+A_3=0$, so that
${\mathcal{B}}$ is traceless with eigenvalues $A_1$, $A_2$, $A_3$.
This is not usually done in the literature, because by working with an arbitrary compatible metric, additional
(non-conformally-invariant) equations can be imposed.
In particular, vacuum metrics~-- and more generally, K\"ahler metrics (with antiselfdual complex structure)~-- are
scalar-f\/lat.
Hence scalar-f\/latness is often used as a~gauge condition, in which case the following well-known system, originating in
work of Brioschi, Chazy, Darboux, and Halphen, is obtained:
\begin{gather*}
\dot A_1=A_2A_3-A_1(A_2+A_3),
\\
\dot A_2=A_3A_1-A_2(A_3+A_1),
\\
\dot A_3=A_1A_2-A_3(A_1+A_2).
\end{gather*}
Joyce's lemma explains the remarkable fact that this system depends only on $w_1,w_2,w_3$ through the functions
$A_1,A_2,A_3$.
The trace of this system is the scalar-f\/lat gauge condition: $\dot A_1+\dot A_2+\dot A_3=-A_2A_3-A_3A_1-A_1A_2$, while
the traceless part is the selfduality equation for the Weyl curvature~-- the latter condition is independent of the
choice of conformal gauge, and may be rewritten as a~matrix Riccati equation in the following way:
\begin{gather*}
{\mathcal{B}} =\frac13\left[
\begin{matrix}
2A_1-A_2-A_3&0&0
\\
0&2A_2-A_3-A_1&0
\\
0&0&2A_3-A_1-A_2
\end{matrix}
\right],
\\
\dot{\mathcal{B}}-a{\mathcal{B}}=2\big(\mathcal{B}^2\big)_0,
\end{gather*}
{where}
\begin{gather*}
a=-\frac23(A_1+A_2+A_3).
\end{gather*}
In other words, the matrix Riccati equation is obtained by separating the diagonal matrix ${\mathcal{B}}+a\iden$, with
eigenvalues $-(A_2+A_3),-(A_3+A_1),-(A_1+A_2)$, into its trace (which def\/ines the connection $D=\partial_t+a$ on the one-dimensional quotient geometry), and its traceless part (which is ${\mathcal{B}}$).
One advantage of this geometric interpretation is that dif\/ferent gauge conditions can be easily compared.
Evidently the condition $a=0$ f\/ixes $t$ up to af\/f\/ine transformations.
In order to interpret the scalar-f\/lat gauge condition $\dot a=\frac23(A_2A_3+A_3A_1+A_1A_2)$, it is natural and
illuminating to express the right hand side in terms of $a$ and $\trace{\mathcal{B}}^2$.
The result is
\begin{gather*}
\dot a-\tfrac12a^2=-\tfrac13\trace{\mathcal{B}}^2.
\end{gather*}
Note that $(\partial_t-\frac12 a)(\partial_t+\frac12 a)=\partial_t^2+\frac12\dot a- \frac14 a^2$, so that on sections of
$L^{1/2}$, $D^2=\partial_t^2-\frac16\trace{\mathcal{B}}^2$.
Hence the scalar-f\/lat gauge condition may be interpreted as f\/ixing $t$ to be a~projective coordinate with respect to the
projective structure $D^2+\frac16\trace{\mathcal{B}}^2$; this determines $t$ up to a~projective transformation.
\section{The gauge f\/ield equations}
\label{s:gfe}
Let $\nabla$ be a~$G$-connection, on a~vector bundle $E$ over a~conformal manifold $M$, which is invariant under an
action of a~group $H$ of conformal transformations.
Inf\/initesimally, the generators form a~Lie algebra $\lie{h}$ of conformal vector f\/ields on $M$ and there is an action of
these vector f\/ields by Lie derivative on sections of $E$ such that ${\mathcal{L}}_\xi\nabla=0$ for all $\xi\in\lie{h}$.
Each $\xi\in\lie{h}$ therefore determines a~Higgs f\/ield $S_\xi$ in the associated Lie algebra bundle
$\lie{g}_M\leqslant\End(E)$ def\/ined by $S_\xi s={\mathcal{L}}_\xi s-\nabla_\xi s$.
(Note the unusual sign convention, which is necessary for consistency later.) Since ${\mathcal{L}}_\xi\nabla=0$ and
$[{\mathcal{L}}_\xi,{\mathcal{L}}_\chi]={\mathcal{L}}_{[\xi,\chi]}$, it follows that
\begin{gather*}
{\mathcal{L}}_\xi (S_\chi) = S_{[\xi,\chi]}
\end{gather*}
and so $S\colon M\mathbin{{\times}}\lie{h}\to\lie{g}_M$ is $H$-equivariant.
Part of the curvature $F^\nabla$ of $\nabla$ is determined by the Higgs f\/ields~-- one readily computes that for any
$\xi\in\lie{h}$ and any vector f\/ield $X$,
\begin{gather*}
F^\nabla(\xi,X)=\nabla_X(S_\xi).
\end{gather*}
In particular, for two vector f\/ields $\xi$, $\chi$ in $\lie{h}$,
\begin{gather*}
F^\nabla(\xi,\chi)=[S_\xi,S_\chi]_{\lie{g}}-S_{[\xi,\chi]},
\end{gather*}
where the f\/irst bracket is the Lie bracket in $\lie{g}_M$.
Now suppose that $H$ acts freely on $M$ with nondegenerate conformal metrics on the orbits, and def\/ine
$(\alpha,\varphi)$ as in Section~\ref{s:sdbg}.
The bundle $E$ is the pullback of a~bundle, also denoted~$E$, over~$Q$.
Since $\nabla$ is $H$-invariant, it descends to a~connection~$A$ over~$Q$.
The Higgs f\/ields are also $H$-invariant; hence setting $\Phi(U)=S_{\varphi(U)}$ def\/ines a~bundle map
$\Phi\colon{\mathcal{V}}\to\lie{g}_{\smash Q}$ over $Q$.
Next introduce a~conformal connection $D$ on ${\mathcal{V}}$.
Then $\nabla_X(S_\xi)=(D^A_X\Phi)\bigl(\varphi^{-1}(\xi)\bigr)
-\Phi\bigl(\varphi^{-1}(D^\alpha_X\varphi)\varphi^{-1}(\xi)\bigr)$ and a~simple computation of the curvature of $\nabla$
yields the following equations:
\begin{gather*}
F^\nabla(U,V)=[\Phi(U),\Phi(V)]_{\lie{g}} -\Phi\bigl(\varphi^{-1}[\varphi(U),\varphi(V)]_{\lie{h}}\bigr),
\\
F^\nabla(U,X)=(D^A_X\Phi)(U) -\Phi\bigl(\varphi^{-1}D^\alpha_X\varphi(U)\bigr),
\\
F^\nabla(X,Y)=F^A(X,Y)-\Phi\bigl(\varphi^{-1}F^\alpha(X,Y)\bigr).
\end{gather*}
This formulation makes manifest the analogy between $(A,\Phi)$ and $(\alpha,\varphi)$: the former is a~$G$-connection
and $\lie{g}_Q$-valued section of ${\mathcal{V}}^{*}$, while the latter is an $H$-connection and $\lie{h}_Q$-valued
section of ${\mathcal{V}}^{*}$.
Furthermore, these formulae for $F^\nabla$ are closely analogous to the formulae~\eqref{eq:t1}--\eqref{eq:t3} for
$T^{\mathcal{D}}$ obtained in the previous section: adding torsion terms to the above equations yields
\begin{gather}
\label{eq:gfe1}
\big(F^\nabla+\Phi\circ T^{\mathcal{D}}\big)(U,V)=[\Phi(U),\Phi(V)]_{\lie{g}},
\\
\big(F^\nabla+\Phi\circ T^{\mathcal{D}}\big)(U,X)=(D^A_X\Phi)(U) -\Phi\bigl(\ip{{\mathcal{C}}(U),X}\bigr),
\\
\big(F^\nabla+\Phi\circ T^{\mathcal{D}}\big)(X,Y)=F^A(X,Y)-\Phi\bigl(\psi(X,Y)\bigr).
\label{eq:gfe3}
\end{gather}
Assuming that $T^{\mathcal{D}}$ is selfdual (which can always be arranged, using the choice of $D$, by the work of
Section~\ref{s:sdbg}), the selfduality of $F^\nabla$ is now equivalent to the selfduality of the right hand sides
of~\eqref{eq:gfe1}--\eqref{eq:gfe3}.
This reduced gauge f\/ield equation is easy to compute in each dimension (and equivalent calculations have already been
used in Section~\ref{s:sdbg}).
Notice that no assumption of selfduality on~$M$ is needed for these computations: just as the selfdual Yang--Mills
equation makes sense on any oriented conformal $4$-manifold, so also the generalized Bogomolny equation is def\/ined on
any Weyl space, and the same is true on the one and two-dimensional geometries.
However, the principal dogma underlying this work is that the natural backgrounds for the gauge f\/ield equations are
selfdual spaces, Einstein--Weyl spaces, spinor-vortex spaces and Riccati spaces.
There are two reasons for this: f\/irst, the Ward correspondence predicts that the reduced gauge f\/ield equations on these
backgrounds will be integrable; second, it will soon be apparent, if not already, that the gauge f\/ield equations and
background equations are intimately related.
\subsection{Generalized Nahm equations on Riccati spaces}
The reduction to one dimension is straightforward, since there is no curvature.
Hence the connection $A$ on $\lie{g}_C$ can be assumed trivial, and the gauge f\/ield equation for
$\Phi\in\mathrm{C}^\infty(C,{\mathcal{E}}^{*}\otimes\lie{g}_C)$ is
\begin{gather}
\label{eq:nahm}
D\Phi-{*[\Phi,\Phi]_{\lie{g}}} ={\mathcal{B}}\mathinner{\cdot}\Phi,
\end{gather}
where ${\mathcal{B}}\mathinner{\cdot}\Phi=\Phi\circ{\mathcal{B}}$ and the Lie bracket pairing
$[\Phi,\Phi]_{\lie{g}}\in\mathrm{C}^\infty(C,\wedge^2{\mathcal{E}}^{*}\otimes\lie{g}_C)$ is interpreted as a~section
of $T^{*} C\otimes{\mathcal{E}}^{*}\otimes\lie{g}_C$ using the Hodge star and conformal structure on ${\mathcal{E}}$,
together with the identif\/ications $L^1_{\mathcal{E}}=L^1_C=TC$.
When ${\mathcal{B}}=0$, this is the Nahm equation.
\subsection{Generalized Hitchin equations on spinor-vortex spaces}
For the reduction to two dimensions, it is natural, as before, to reinterpret $\Phi$ as a~complex linear map from
${\mathcal{W}}$ to $\lie{g}_N\otimes{\mathbb{C}}$.
Then the gauge f\/ield equations for $(A,\Phi)$ are
\begin{gather*
F^A-[\Phi,\overline\Phi]_{\lie{g}} =\psi\wedge\overline\Phi+\overline\psi\wedge\Phi,
\qquad
\overline\partial{}^{a,A}\Phi={\mathcal{C}}\overline\Phi.
\end{gather*}
When ${\mathcal{C}}=\psi=0$ and ${\mathcal{W}}=TN$, with the induced holomorphic structure, these are Hitchin's
equations.
To adapt the gauge f\/ield equations to spinor-vortex spaces in general signature (when there are background f\/ields
${\mathcal{C}},\widetilde{\mathcal{C}},\psi,\widetilde\psi$), replace $\overline\Phi$ by an additional f\/ield
$\widetilde\Phi$, satisfying the analogous $\partial^{a,A}$-equation.
The Lorentzian reality condition ($\Phi,\widetilde\Phi$ real) provides a~generalized chiral model, while
a~generalization of the harmonic map equation is obtained by introducing the crucial sign change
$\widetilde\Phi=-\overline\Phi$ (recall also that $\widetilde\psi=-\overline\psi$ in this case).
\subsection{Generalized Bogomolny equations on Einstein--Weyl spaces}
The reduction to three dimensions gives the natural generalization to Weyl geometry of the Bogomolny equation for
magnetic monopoles
\begin{gather*
{* D^A\Phi}=F^A.
\end{gather*}
The Euclidean, hyperbolic or spherical Bogomolny equation arises when the Einstein--Weyl structure is given by a~metric
of constant curvature.
\subsection*{Addenda: twisted f\/lat pencils and projective pairs}
The same methodology as in the nondegenerate cases yields gauge f\/ield equations over the null reductions, using the
equations~\eqref{eq:null-t1}--\eqref{eq:null-t3} for the torsion $T^{\mathcal{D}}$.
In the background geometry $(a,\phi,\psi)$ on $(Q,{\mathcal{L}})$ obtained by reduction along an $\alpha$-surface
foliation, the gauge f\/ields consist of a~$G$-connection $A$ and a~$1$-form $\Phi$ with values in
${\mathcal{L}}^2\otimes\lie{g}_Q$, and the gauge f\/ield equations are
\begin{gather}
\label{eq:tfc}
F^A=\psi\wedge\Phi,
\qquad
{\rm d}^{a,A}\Phi=0,
\qquad
\tfrac12[\Phi\wedge\Phi]=\chi\wedge\Phi.
\end{gather}
If $\psi=\chi=0$ (so $a$ is f\/lat, and ${\mathcal{L}}$ may be trivialized) then these are the equations for a~pencil of
f\/lat connections ${\rm d}^A + \lambda\Phi$ on a~surface (also known as the topological chiral model).
Thus solutions of~\eqref{eq:tfc} may be called \emph{twisted flat pencils}.
The special case $\psi=0$ has been studied by Tafel and W{\'o}jcik in~\cite{TaWo:nkv}.
On a~projective surface $(Q,[D])$ obtained by reduction along a~$\beta$-surface foliation, the gauge f\/ields consist of
a~$G$-connection $A$ and a~$1$-form $\Phi$ with values in ${\mathcal{O}}_Q(2)\otimes\lie{g}_Q$ (where
${\mathcal{O}}_Q(3)=\wedge^2TQ$), and the gauge f\/ield equations are
\begin{gather*
D^A\Phi=\tfrac12 {\rm d}^{D,A}\Phi.
\end{gather*}
Solutions $(A,\Phi)$ were referred to (somewhat unimaginatively) as \emph{projective pairs} in~\cite{Cal:sdp}.
The special case of reductions of the selfdual Yang--Mills equation on ${\mathbb{R}}^{2,2}$ (or ${\mathbb{C}}^4$) by null
translations was considered already by Mason and Woodhouse~\cite{MaWo:ist}: the $\alpha$-plane reduction (yielding f\/lat
pencils) is denoted $H_{SD}$, while the $\beta$-plane reduction (yielding projective pairs) is denoted~$H_{ASD}$.
\section{Interlude: spinor-vortex spaces and Hitchin's equations}
\label{s:svhe}
In~\cite{Hit:sde}, Hitchin considered solutions of the selfdual Yang--Mills equation on ${\mathbb{R}}^4$ invariant under
two translations.
He observed that the Yang--Mills connection could be decomposed into a~connection over ${\mathbb{R}}^2$ and two Higgs
f\/ields.
He combined these Higgs f\/ields into a~complex Higgs f\/ield and then noticed that, remarkably, the reduced Yang--Mills
equation becomes conformally invariant provided this complex Higgs f\/ield is interpreted as a~$1$-form rather than
a~scalar.
This was a~surprise, because although the selfdual Yang--Mills equation is conformally invariant, the notion of
translation-invariance is not.
Furthermore, conformal invariance in two dimensions implies an inf\/inite-dimensional symmetry group.
The reduction process described here provides a~simple explanation of this phenomenon: $W^{\raise1pt\hbox{$\scriptscriptstyle -$}}=0$ is conformally
invariant, and so is the notion of torus symmetry.
Hence the equations for $(\overline\partial{}^a,\psi,{\mathcal{C}},\alpha,\varphi)$ are conformally invariant on a~f\/ixed
Riemann surface with a~complex line bundle~${\mathcal{W}}$ and a~Hermitian metric on ${\mathcal{W}}^{-1} TN$.
In particular, it is clear that the equations~\eqref{eq:sv1}--\eqref{eq:sv3} for
$(\overline\partial{}^a,\psi,{\mathcal{C}})$ are conformally invariant.
If ${\mathcal{C}}$ is not identically zero, then (on the open set where ${\mathcal{C}}$ is nonvanishing) the freedom in
the holomorphic structure on ${\mathcal{W}}$ can be f\/ixed by declaring that ${\mathcal{C}}$ is an identif\/ication of~${\mathcal{W}}^2$ with~$TN$.
The length of ${\mathcal{C}}$ now def\/ines a~gauge, breaking conformal invariance.
More precisely, given a~Weyl derivative $D$, the Chern connection on ${\mathcal{W}}$ only agrees with the connection
induced from $TN$ if $D=D^g$, where $g$ is the metric induced by $|{\mathcal{C}}|$.
Equation~\eqref{eq:sv3} becomes $\frac12s^g=-2+4|\psi|^2_g$; in particular, if $\psi$ is zero, then $g$ has constant
negative curvature, which is the case studied by Joyce~\cite{Joy:esd}, the \emph{hyperbolic spinor-vortex geometry}.
On the other hand if ${\mathcal{C}}$ is identically zero, then $\psi$ is holomorphic, so if $\psi$ is not identically
zero, then (on the open set where $\psi$ is nonvanishing), $\psi$ trivializes ${\mathcal{W}}$.
Again this breaks conformal invariance by introducing a~natural gauge, $|\psi|$; the corresponding metric has constant
positive curvature ($s^g=4$).
This is the \emph{spherical spinor-vortex geometry}.
Finally if ${\mathcal{C}}$ and $\psi$ both vanish, then the holomorphic structure can be f\/ixed by setting
${\mathcal{W}}=TN$, so that the Chern connection on ${\mathcal{W}}^{-1} TN$ is trivial.
This trivial spinor-vortex geometry does not break conformal invariance.
The Yang--Mills equation is reduced to two dimensions by interpreting the Higgs f\/ields as a~section $\Phi$ of
${\mathcal{W}}^{-1}\otimes_{\mathbb{R}}\End(V)$, using $\varphi$.
The resulting gauge f\/ield equations are independent of $(\alpha,\varphi)$, i.e., they are intrinsic to the spinor-vortex
space.
On a~trivial spinor-vortex space, ${\mathcal{W}}=TN$, so $\Phi$ becomes an endomorphism-valued $1$-form, and, as
remarked already, the gauge f\/ield equations for $(A,\Phi)$ are Hitchin's equations.
Thus it is the isomorphism $\varphi$, and the geometry of the trivial spinor-vortex space that are responsible for the
interpretation of the Higgs f\/ields as a~$1$-form, rather than as scalars.
\section{Selfdual spaces from gauge f\/ields}
\label{s:sgf}
In Sections~\ref{s:sdbg} and~\ref{s:gfe}, background geometries and gauge f\/ield equations were found by reducing the
selfduality equation for conformal structures and Yang--Mills f\/ields respectively.
In the approach taken, the symmetry group $H$, and the pair $(\alpha,\varphi)$ def\/ining the reduction, decoupled from
the construction, and the reduced background geometry and gauge f\/ield equations were found to be independent of these
data.
In some sense, this was a~sleight of hand, since $(\alpha,\varphi)$ were used to def\/ine the data on the quotient which
make up the background geometry (i.e., $D$, $\psi$ and ${\mathcal{C}}$).
However, the procedure may be turned around: starting with the background geometry, these def\/initions, originally
obtained by imposing selfduality on the torsion $T^{\mathcal{D}}$, may be viewed as equations for the pair
$(\alpha,\varphi)$.
Indeed, applying $\varphi$ to the formulae~\eqref{eq:t1}--\eqref{eq:t3} for $T^{\mathcal{D}}$ gives:
\begin{gather}
\label{eq:tg1}
\big(\varphi\circ T^{\mathcal{D}}\big)(U,V)=[\varphi(U),\varphi(V)]_{\lie{h}},
\\
\big(\varphi\circ T^{\mathcal{D}}\big)(U,X)=(D^\alpha_X\varphi)(U) -\varphi\bigl(\ip{{\mathcal{C}}(U),X}\bigr),
\\
\big(\varphi\circ T^{\mathcal{D}}\big)(X,Y)=F^\alpha(X,Y)-\varphi\bigl(\psi(X,Y)\bigr).
\label{eq:tg3}
\end{gather}
The right hand sides of these formulae correspond precisely to the right hand sides of~\eqref{eq:gfe1}--\eqref{eq:gfe3}.
Hence the selfduality equation for $T^{\mathcal{D}}$ coincides with the gauge f\/ield equation for $(\alpha,\varphi)$ on
the background geometry.
This leads immediately to an inverse construction of selfdual spaces with symmetry from gauge f\/ields with
$\ell$-dimensional gauge group on $k$-dimensional background geometries, where $k+\ell=4$.
However, the construction can be generalized further by noting that the selfduality of $T^{\mathcal{D}}$ is implied by
the gauge f\/ield equation as long as $\varphi\colon{\mathcal{V}}\to\lie{h}_Q$ is injective.
I will now explain this generalized construction.
Let $P\to Q$ be a~principal $H$-bundle with an $H$-connection $\alpha$ and a~Higgs f\/ield
$\varphi\colon{\mathcal{V}}\to\lie{h}_{\smash{Q}}$, where $\lie{h}_{\smash{Q}}=P\mathbin{{\times}}_H\lie{h}$.
Suppose that $H$ acts transitively on an $\ell$-manifold $\Sigma^\ell$, where $Q$ has dimension $k=4-\ell$, so that the
associated f\/ibre bundle $P\mathbin{{\times}}_H\Sigma^\ell$ is four-dimensional.
The basic example is the case that the action of $H$ is also free, in which case $\Sigma^\ell$ is a~principal
homogeneous space for $H$ and therefore there is a~commuting free transitive action of a~Lie group $\tilde H$ isomorphic
to $H$.
Choosing a~basepoint on $\Sigma^\ell$ identif\/ies $\tilde H$ and $\Sigma^\ell$ with $H$ and the two actions are the left
and right regular actions.
However, it can be useful to distinguish between the \emph{structure group} $H$, and the \emph{symmetry group} $\tilde
H$: if $P$ is a~principal $H$-bundle over $Q$, then $P\mathbin{{\times}}_H \Sigma^\ell$ is a~principal $\tilde
H$-bundle.
For general $\Sigma^\ell$, there is still an associated bundle $\pi\colon P\mathbin{{\times}}_H\Sigma^\ell\to Q$, but
this does not have any symmetries in general, since there is no longer a~commuting right action of $\tilde H$ on
$\Sigma^\ell$.
However, the f\/ibre of $\lie{h}_Q=P\mathbin{{\times}}_H\lie{h}$ is still a~Lie algebra of vertical vector f\/ields on
$P\mathbin{{\times}}_H\Sigma^\ell$, which will be called ``invariant'', but for consistency with the case
$\Sigma^\ell=H$, the Lie bracket of these vector f\/ields is opposite to the Lie bracket in $\lie{h}_Q$.
The map $\varphi\colon{\mathcal{V}}\to\lie{h}_Q$ therefore induces a~map~$\widehat\varphi$ from~$\pi^*{\mathcal{V}}$ to
the vertical bundle of $P\mathbin{{\times}}_H\Sigma^\ell$.
These bundles both have rank~$\ell$, so let~$M$ be an open subset of $P\mathbin{{\times}}_H\Sigma^\ell$ where
$\widehat\varphi$ is an isomorphism.
Note that $\widehat\varphi$ sends basic sections to ``invariant'' vector f\/ields.
Equip $TM\cong(\pi|_{M})^*({\mathcal{V}}\oplus TQ)$ with the direct sum conformal structure, so that a~conformal
connection $D$ on ${\mathcal{V}}$ induces a~conformal connection on $TM$.
As before, a~modif\/ied conformal connection ${\mathcal{D}}$ can be constructed, using the pullbacks of $\psi$ and
${\mathcal{C}}$: the torsion of this connection will be vertical-valued, and will
satisfy~\eqref{eq:tg1}--\eqref{eq:tg3}.
Since $\varphi$ is injective, the torsion will be selfdual if $(\alpha,\varphi)$ satisfy the gauge f\/ield equation on
$Q$.
The calculation of the curvature of ${\mathcal{D}}$ carries over immediately to this more general setting, hence Joyce's
Lemma can be applied to establish the following result.
\begin{thm}
\label{th:sdgf}
Let $H$ be a~transitive group of diffeomorphisms of an $\ell$-manifold $\Sigma^\ell$.
Suppose that $(\alpha,\varphi)$ is a~solution of the gauge field equation $($i.e., the Nahm, Hitchin or Bogomolny
equation$)$ on a~principal $H$-bundle $P\to Q$, where $Q$ is a~$(4-\ell)$-dimensional background geometry
$($i.e., an Einstein--Weyl, spinor-vortex or Riccati space$)$.
Then on the open subset $M$ of $\pi\colon P\mathbin{{\times}}_H\Sigma^\ell\to Q$ where $\widehat\varphi$ is an
isomorphism, $(\alpha,\varphi)$ identifies $TM$ with $\pi^*({\mathcal{V}}\oplus TQ)$, and the direct sum conformal
structure is selfdual.
\end{thm}
For later work, it will be useful to have a~more explicit description of the construction of this theorem.
Choose a~local conformal frame $e_i$ for $TQ\oplus{\mathcal{V}}$ over $Q$ compatible with the direct sum decomposition,
and a~local section of $P$.
Then, by identifying $M$ locally with $Q\mathbin{{\times}}\Sigma^\ell$ and viewing the connection $\alpha$ as
a~Vect$(\Sigma^\ell)$-valued $1$-form on $Q$, the components of $(\alpha,\varphi)$ with respect to~$e_i$ def\/ine four
vector f\/ields $X_i$ on $\Sigma^\ell$: $X_1,\ldots X_k$ are the components of the connection and $X_{k+1},\ldots X_4$ are
the components of $\varphi$.
Since $e_i$ is a~conformal frame on $TQ\oplus{\mathcal{V}}$, the conformal structure on~$M$ is clearly represented
contravariantly by the metric
\begin{gather}
\label{eq:invmetric}
(e_1-X_1)^2+\dots+(e_k-X_k)^2+{X_{k+1}}^2+\dots+{X_4}^2.
\end{gather}
This is a~metric on $T^{*} M$.
The covariant metric on $TM$ is dual to this, and will be discussed in Section~\ref{s:bgf}.
In fact it is sometimes more convenient to use contravariant metrics, since they push forward easily.
\begin{rems}\quad
\begin{enumerate}\itemsep=0pt
\item[(i)]
Note that the calculations leading to this theorem are entirely formal and so, at least for local considerations, $H$
need not be f\/inite-dimensional, but could be any subgroup of $\Diff(\Sigma^\ell)$.
Hence the conformal aspects of Ward's construction~\cite{Ward:suc} are included in the theorem when the background
geometry is trivial.
\item[(ii)]
Theorem~\ref{th:sdgf} provides a~new interpretation of the switch map~\cite{MaWo:ist, MMW:sbm}.
Let $M$ be a~selfdual conformal $4$-manifold with freely acting $\ell$-dimensional symmetry group $H$.
Then the local quotient $Q=M/H$ is an background geometry of dimension $4-\ell$.
An invariant selfdual Yang--Mills f\/ield (on a~principal bundle $P$) with $\ell$-dimensional gauge group $G$ descends to
a~solution of the gauge f\/ield equation on $Q$, from which a~new selfdual space $\tilde M$ may be constructed.
\centerline{\includegraphics{Calderbank-D1}}
Note the following features of this interpretation.
\begin{itemize}\itemsep=0pt
\item
The construction avoids considering the $(4+\ell)$-dimensional manifold $P$ explicitly.
\item
The procedure decomposes into two steps of independent interest: the construction of gauge f\/ields on $Q$ from
$H$-invariant selfdual Yang--Mills f\/ields on $M$, and the construction of selfdual spaces from gauge f\/ields on $Q$.
\item
The group $G$ need not be $\ell$-dimensional, so long as it acts transitively on an $\ell$-manifold.
Hence the constructions of~\cite{DMW:2dg} f\/it into the same framework.
\end{itemize}
\item[(iii)]
In the case that $\Sigma^\ell=H$, $M$ has symmetry group $H$, but enlarging the group $H$ acting on~$\Sigma^\ell$ gives
less symmetry, rather than more symmetry, since there are fewer dif\/feomorphisms of~$\Sigma^\ell$ commuting with the
$H$-action.
In general there will be none, and~$M$ will have no symmetry.
In other words, the group~$H$ is a~\emph{structure group} rather than a~\emph{symmetry group}: only in the case of~$H$
acting freely does it happen that the commuting symmetry group is (isomorphic to)~$H$.
\end{enumerate}
\end{rems}
Selfdual conformal manifolds with symmetry groups which do not act freely are rather special: for instance they are
foliated by the surfaces of equal isotropy group, and the curvature must be invariant under the isotropy representation.
Some examples arise as very special cases of the constructions of this paper.
More precisely, if~$G$ is a~(not necessarily free or transitive) group of dif\/feomorphisms of $\Sigma^\ell$, then~$G$
will be a~symmetry group of~$M$ provided there is a~representation of~$G$ on~${\mathcal{V}}$ such that~$\varphi$ and~$\alpha$ are $G$-invariant.
In the case that~$G$ acts trivially on~${\mathcal{V}}$, this means that the structure group~$H$ reduces to the group of
dif\/feomorphisms of~$\Sigma^\ell$ commuting with~$G$.
On the other hand, there are interesting examples where~$G$ acts nontrivially on~${\mathcal{V}}$, see for
instance~\cite{Hit:hcm}.
The use of structure groups rather than symmetry groups also turns out to be natural when partial reductions are
considered in view of the following remark.
If $H$ acts freely on~$M$ and~$K$ is a~subgroup of~$H$, then $K$ also acts freely on $M$ and the structure group of
$M/K$ is $H/K_0$ acting on $H/K$, where $K_0$ is the maximal normal subgroup of $H$ lying in~$K$.
For example, if $\SU(2)$ acts freely on $M$ then $\SO(3)$ acting on $S^2$ will be the induced structure group for~$M/\Un(1)$.
In Section~\ref{s:bgf}, it will be shown that such partial reductions~$M/K$ arise directly from gauge f\/ields on~$M/H$
with gauge group~$H/K_0$.
\subsection*{Addenda: generalized constructions from null reductions}
The same principles may be applied to null reductions to obtain constructions of selfdual $4$-manifolds from twisted
f\/lat pencils and projective pairs, where the gauge group is a~transitive group of dif\/feomorphisms of a~surface
$\Sigma^2$.
The only change needed, relative to the nondegenerate case, is that the conformal structure on~$M$ is obtained from the
natural pairing between~$VM$ and~$\pi^* TQ$, rather than conformal structures on each summand.
The construction of selfdual conformal structures from f\/lat pencils of connections on a~surface is not at all new: as
discussed in the following Interlude, it underpins Plebanski's heavenly equations and interpretations of hypercomplex
and hyperk\"ahler structures as topological chiral models.
On the other hand, the construction of selfdual conformal structures from \emph{twisted} f\/lat pencils with gauge group
$\Diff(\Sigma^2)$ has not been studied, as far as I am aware.
The analogous story for projective pairs on a~projective surface underpins the Dunajski--West construction of selfdual
conformal manifolds with a~null Killing vector~\cite{DuWe:acs}.
In~\cite{Cal:sdp}, their construction is shown to be a~reduction of gauge group from $\Diff(\Sigma^2)$ to the subgroup
commuting with a~nonvanishing vector f\/ield.
\section{Interlude: hypercomplex and hyperk\"ahler structures}
\label{s:hchk}
From the point of view of integrable systems, hypercomplex and hyperk\"ahler $4$-mani\-folds are considerably simpler
than the general selfdual space.
Recall that a~hypercomplex structure consists of a~triple $I$, $J$, $K$ of integrable complex structures, satisfying the
quaternionic relation $IJ=K$.
It is well known that a~hypercomplex manifold comes equipped with a~unique torsion-free connection~$D$ with
$DI=DJ=DK=0$, called the \emph{Obata connection}~\cite{Oba:cqh}.
A hypercomplex $4$-manifold possesses a~canonical conformal structure, def\/ined by requiring that $(X,IX,JX,KX)$ is
a~conformal frame for any nonzero tangent vector~$X$.
The Obata connection preserves this conformal structure, and is thus a~Weyl connection.
Since~$I$,~$J$,~$K$ are anticommuting orthogonal complex structures, their (weightless) K\"ahler forms are either all
selfdual or all antiselfdual: I f\/ix the orientation by requiring that they are antiselfdual.
Therefore~$D$ is f\/lat on $L^2 \wedge^2_{\raise1pt\hbox{$\scriptscriptstyle -$}} T^{*} M$, and in particular, $M$ is a~selfdual conformal
manifold.
The hypercomplex structure is hyperk\"ahler if and only if~$D$ is exact, i.e., the Obata connection preserves a~length
scale, and hence a~metric in the conformal class.
The simplicity of the hypercomplex condition manifests itself in the following local description of hypercomplex
$4$-manifolds.
\begin{thm}
\label{th:MN}
Let $V_0$, $V_1$, $V_2$, $V_3$ be linearly independent vector fields on a~$4$-manifold $M$ and let $\eta_0$, $\eta_1$, $\eta_2$, $\eta_3$
be the dual coframe of $1$-forms $($with $\eta_i(V_j)=\delta_{ij})$.
Define almost complex structures $I$, $J$, $K$ by
\begin{gather*}
I V_0=V_1,\qquad I V_2=V_3,
\qquad
J V_0=V_2,\qquad J V_3=V_1,
\qquad
K V_0=V_3,\qquad K V_1=V_2.
\end{gather*}
Then the following are equivalent.
\begin{enumerate}\itemsep=0pt
\item[$1.$]
The frame $V_i$ satisfies the equations
\begin{gather}
[V_0,V_1]+[V_2,V_3]=0,
\label{eq:MN1}
\\
[V_0,V_2]+[V_3,V_1]=0,
\label{eq:MN2}
\\
[V_0,V_3]+[V_1,V_2]=0.
\label{eq:MN3}
\end{gather}
\item[$2.$]
For each $i$, ${\rm d}\eta_i$ is selfdual with respect to the conformal structure represented by the metric
$g=\eta_0^2+\eta_1^2+\eta_2^2+\eta_3^2$ $($where $I$, $J$, $K$ are antiselfdual$)$.
\item[$3.$]
$(I,J,K)$ is hypercomplex with Obata connection $D$ and $\divg^D\eta_i=0$ for all $i$.
\end{enumerate}
Any hypercomplex $4$-manifold $M$ arises locally in this way, and is hyperk\"ahler if and only if the vector fields
$V_i$ all preserve a~volume form $\nu$.
\end{thm}
The hyperk\"ahler version of this theorem is due to Mason--Newman~\cite{MaNe:eym}, following a~construction of
Ashtekar--Jacobson--Smolin~\cite{AJS:hfe} which will be described below.
The hyperk\"ahler case is simpler, because for $D$ exact, the $\eta_i$ are divergence-free with respect to $D$ if and
only if the $V_i$ preserve a~volume form~-- see~\eqref{eq:divs}.
The general construction was f\/irst written explicitly by Joyce~\cite{Joy:esd}, but the fact that all four-dimensional
hypercomplex structures arise in this way is due to Dunajski~\cite{Dun:tph}.
The description I have given dif\/fers slightly from these references, and owes a~great deal to the approaches of
Hitchin~\cite{Hit:hcm} (see below) and Grant--Strachan~\cite{GrSt:his}.
Since the role of the divergence condition has perhaps not been fully elucidated before, and will be useful later, I
give a~complete proof.
\begin{proof}
Since
\begin{gather*}
\eta_i ( [V_j,V_k] )=-{\rm d}\eta_i(V_j,V_k)
\end{gather*}
for all $i$, $j$, $k$, it is manifest that~(i) and~(ii) are equivalent formulations of the same equations.
Also,~\eqref{eq:MN2} and~\eqref{eq:MN3} clearly imply that~$I$ is integrable since they may be rewritten as
\begin{gather*}
[V_0+{\boldsymbol i} V_1,V_2+{\boldsymbol i} V_3]=[V_0,V_2]-[V_1,V_3] +{\boldsymbol i} ( [V_0,V_3]+[V_1,V_2] )=0.
\end{gather*}
Similarly~\eqref{eq:MN3} and~\eqref{eq:MN1} imply that $J$ is integrable, and~\eqref{eq:MN1} and~\eqref{eq:MN2} imply
that $K$ is integrable.
Now note that for any $1$-form $\eta$ on a~hypercomplex manifold ${\rm d}\eta$ is selfdual if and only if it is
orthogonal to the weightless K\"ahler forms of $I$, $J$, $K$.
Since $DI=DJ=DK=0$, this is equivalent to $I\eta$, $J\eta$ and $K\eta$ being divergence-free with respect to $D$ (for
instance, $\divg^D(I\eta)=\sum\limits_i \ip{\varepsilon^i,I D_{e_i}\eta}$, and $I$ is skew).
Hence (iii) is equivalent to (i) and (ii).
On any hypercomplex manifold, the conditions $\divg^D\eta=0$ and ${\rm d}\eta^{\raise1pt\hbox{$\scriptscriptstyle -$}}=0$ form a~determined f\/irst-order linear system for a~$1$-form $\eta$, which therefore admits local (non-null) solutions: $(\eta,I\eta,J\eta,K\eta)$
is then a~divergence-free conformal coframe.
It remains to characterize the hyperk\"ahler case in terms of volume forms~-- or length scales.
Suppose that $\mu=e^{f}\mu_g$ is a~length scale; then $\mu^{-4} V_i=\mu^{-4} \mu_g^{2}\eta_i=e^{-2f}\mu^{-2}\eta_i$ and
so
\begin{gather}
\label{eq:divs}
\divg\big(\mu^{-4}V_i\big)=\divg^D\big(e^{-2f}\mu^{-2}\eta_i\big)=\Ip{D\big(e^{-2f}\mu^{-2}\big),\eta_i} +e^{-2f}\mu^{-2}\divg^D\eta_i.
\end{gather}
Since $\divg^D\eta_i=0$, $D$ preserves the length scale $e^f\mu=e^{2f}\mu_g$ if and only if the $V_i$ all preserve the
volume form $\nu=\mu^{-4}$ (note that $\wedge^4T^{*} M\cong L^{-4}$, using the orientation of $M$).
\end{proof}
There is an equivalent way to describe the divergence-free condition on the coframe, \mbox{using}
spinors~\cite{Dun:tph,MaWo:ist}.
Recall that any conformal $4$-manifold locally admits (weightless) \emph{spin bundles}~$\$^{\raise1pt\hbox{$\scriptscriptstyle\pm$}}$, which are
$\SL(2,{\mathbb{C}})$ bundles such that $\$^{\raise1pt\hbox{$\scriptscriptstyle +$}}\otimes\$^{\raise1pt\hbox{$\scriptscriptstyle -$}}$ is isomorphic to the complexif\/ied weightless cotangent
bundle $L T^{*} M\otimes{\mathbb{C}}$ with the metric induced by the two area forms.
The conventions are chosen so that $L^2 \wedge^2_{\raise1pt\hbox{$\scriptscriptstyle -$}} T^{*} M\otimes{\mathbb{C}}=S^2\$^{\raise1pt\hbox{$\scriptscriptstyle -$}}$.
On a~hypercomplex manifold $D$ induces a~f\/lat connection on $\$^{\raise1pt\hbox{$\scriptscriptstyle -$}}$ preserving the area form.
For Euclidean reality conditions, the real structure on the $L T^{*} M\otimes{\mathbb{C}}$ is induced
by (parallel) quaternionic structures on $\$^{\raise1pt\hbox{$\scriptscriptstyle\pm$}}$.
If frames $(\rho_0,\rho_1)$ for $L^{-1} \$^{\raise1pt\hbox{$\scriptscriptstyle +$}}$ and $(\sigma_0,\sigma_1)$ for $\$^{\raise1pt\hbox{$\scriptscriptstyle -$}}$ are chosen so
that the quaternionic structure sends $\rho_0$ to $\rho_1$ and $\sigma_0$ to $\sigma_1$, then
\begin{gather*}
\eta_0+{\boldsymbol i}\eta_1=\rho_0\otimes\sigma_0,
\qquad
\eta_2+{\boldsymbol i}\eta_3=\rho_0\otimes\sigma_1,
\\
\eta_0-{\boldsymbol i}\eta_1=\rho_1\otimes\sigma_1,
\qquad
\eta_2-{\boldsymbol i}\eta_3=-\rho_1\otimes\sigma_0
\end{gather*}
def\/ines a~real conformal coframe $(\eta_0,\eta_1,\eta_2,\eta_3)$.
\begin{prop}
Suppose that $M$ is hypercomplex with Obata connection $D$ and that $(\sigma_0,\sigma_1)$ is a~$D$-parallel frame for
$\$^{\raise1pt\hbox{$\scriptscriptstyle -$}}$.
\begin{enumerate}\itemsep=0pt
\item[$1.$]
Let $\rho$ be a~section of $L^{-1} \$^{\raise1pt\hbox{$\scriptscriptstyle +$}}$.
Then $\rho\otimes\sigma_0$ and $\rho\otimes\sigma_1$ are divergence-free with respect to $D$ if and only if $\rho$
satisfies the Dirac--Weyl equation $\sum\limits_i \varepsilon_i\mathinner{\cdot} D_{e_i}\rho=0$.
\item[$2.$]
Let $\eta$ be a~$1$-form.
Then $\eta\mathinner{\cdot}\sigma_0$ and $\eta\mathinner{\cdot}\sigma_1$ satisfy the Dirac--Weyl equation if and only if
$\divg^D\eta=0$ and ${\rm d}\eta^{\raise1pt\hbox{$\scriptscriptstyle -$}}=0$.
\end{enumerate}
Here the dot denotes the natural $($Clifford$)$ action $T^{*} M\otimes L^{w} \$^{\raise1pt\hbox{$\scriptscriptstyle\pm$}}\to
L^{w-1} \$^{\raise1pt\hbox{$\scriptscriptstyle\mp$}}$.
\end{prop}
\begin{proof}
These are direct calculations:
\begin{enumerate}\itemsep=0pt
\item[(i)]
For $A=0,1$, $\divg^D(\rho\otimes\sigma_A)=\sum\limits_i\varepsilon^i(D_{e_i}\rho\otimes\sigma_A)
=\omega^{\raise1pt\hbox{$\scriptscriptstyle -$}}\bigl(\sum\limits_i\varepsilon^i\mathinner{\cdot} D_{e_i}\rho,\sigma_A\bigr)$.
\item[(ii)]
For $A=0,1$, $\sum\limits_i\varepsilon^i\mathinner{\cdot} D_{e_i}(\eta\mathinner{\cdot}\sigma_A)=
\sum\limits_i\varepsilon^i\mathinner{\cdot}(D_{e_i}\eta)\mathinner{\cdot}\sigma_A
=(\divg^D\eta)\sigma_A+{\rm d}\eta\mathinner{\cdot}\sigma_A$.
\end{enumerate}
In (i) $\omega^{\raise1pt\hbox{$\scriptscriptstyle -$}}$ is the area form on $\$^{\raise1pt\hbox{$\scriptscriptstyle -$}}$, and in (ii) the Clif\/ford action of ${\rm d}\eta^{\raise1pt\hbox{$\scriptscriptstyle +$}}$ is
trivial.
\end{proof}
An important class of solutions to the equations $\divg^D\eta=0$ and ${\rm d}\eta^{\raise1pt\hbox{$\scriptscriptstyle -$}}=0$ is obtained by taking
$\eta={\rm d} r$ where $\Delta^Dr:=\divg^D {\rm d} r=0$.
Evidently the equation $\Delta^Dr=0$ admits local solutions on any hypercomplex $4$-manifold.
Hence the following result is obtained, which is the original construction of Ashtekar--Jacobson--Smolin~\cite{AJS:hfe}
in the hyperk\"ahler case, and is due to Hitchin~\cite{Hit:hcm} in general.
\begin{thm}
\label{th:s3Nahm}
Let $\Phi$ be a~solution of the Nahm equation $($on a~trivial Riccati space$)$ with gauge group
$\Diff(\Sigma^3)$ for some $3$-manifold $\Sigma^3$.
Then the selfdual space constructed from $\Phi$ is hypercomplex.
Any hypercomplex $4$-manifold arises locally in this way, and is hyperk\"ahler if and only if there is a~reduction to
the gauge group $\SDiff(\Sigma^3)$.
\end{thm}
\begin{proof}
Let $r$ be an af\/f\/ine coordinate on a~trivial Riccati space $C$, choose a~conformal trivialization of ${\mathcal{E}}$,
and write the components of $\Phi$, which are vector f\/ields on $\Sigma^3$, as $(V_1,V_2,V_3)$.
Then the vector f\/ields $(\partial_r,V_1,V_2,V_3)$ on $M\subset C\mathbin{{\times}}\Sigma^3$
satisfy~\eqref{eq:MN1}--\eqref{eq:MN3} as a~consequence of the Nahm equation, and so the dual frame $({\rm d}
r,\eta_1,\eta_2,\eta_3)$ has ${\rm d}\eta_i$ selfdual (note that I~use the opposite orientation
to~\cite{Hit:hcm}).
Conversely any hypercomplex manifold arises locally in this way by letting~$r$ be a~solution of~$\Delta^Dr=0$ and
setting $(\eta_1,\eta_2,\eta_3)=(I{\rm d} r,J{\rm d} r,K{\rm d} r)$.
Finally note that $\partial_r$ and $V_i$ preserve a~volume form $e^{-4f} {\rm d}
r\wedge\eta_1\wedge\eta_2\wedge\eta_3$ if and only if
\begin{gather*}
{\mathcal{L}}_{\partial_r}\big(e^{-4f}\eta_1\wedge\eta_2\wedge\eta_3\big)=0
\qquad
\text{and}
\qquad
{\rm d} r\wedge {\mathcal{L}}_{V_i}\big(e^{-4f}\eta_1\wedge\eta_2\wedge\eta_3\big)=0.
\end{gather*}
Here I use the fact that ${\rm d} r(V_i)=0$, ${\rm d} r(\partial_r)=1$ and that $\iota_{\partial_r}
(e^{-4f}\eta_1\wedge\eta_2\wedge\eta_3)=0$.
The f\/irst equation says that $e^{-4f}\eta_1\wedge\eta_2\wedge\eta_3$ is an $r$-independent volume element on $\Sigma^3$
(i.e., a~parallel volume form on $C\mathbin{{\times}}\Sigma^3\to C$), and the second equation says that the $V_i$ are
volume-preserving vector f\/ields for each f\/ixed $r$ (i.e., on each f\/ibre of $C\mathbin{{\times}}\Sigma^3\to C$).
This is exactly what it means to have a~reduction to $\SDiff(\Sigma^3)$.
\end{proof}
As this construction involves taking ${\rm d}\eta_0=0$, it is natural to ask if one can f\/ind divergence-free
coframes with ${\rm d}\eta_0=0={\rm d}\eta_1$ and hence formulate the hypercomplex equations as Hitchin
equations on a~trivial spinor-vortex space.
A number of constructions of four-dimensional hyperk\"ahler metrics from two-dimensional integrable models are known,
due to Park~\cite{Par:sdg}, Ward~\cite{Ward:suc} and (later) Husain~\cite{Hus:sdg}: see~\cite{Uen:ift} for a~review.
Unfortunately, it is not always clear in these constructions whether all hyperk\"ahler metrics are obtained, what
choices are needed to obtain the integrable model from a~hyperk\"ahler metric, and if they are compatible with
Euclidean reality conditions.
In particular, as far as I can tell, none of these works establish an equivalent formulation of the Euclidean
hyperk\"ahler condition as a~Euclidean two-dimensional integrable model.
Indeed, the usual approach is to use the equation $[V_0+{\boldsymbol i} V_1,V_2+{\boldsymbol i} V_3]=0$ to introduce
coordinates $(x,y,u,v)$ such that $V_0+{\boldsymbol i} V_1=\partial_x$ and $V_2+{\boldsymbol i} V_3=\partial_y$.
There are several variations on this theme, since the meaning of $\partial_x$ and $\partial_y$ in $(x,y,u,v)$
coordinates depends on $u$ and $v$, leading to dif\/ferent forms for $V_0-{\boldsymbol i} V_1$ and $V_2-{\boldsymbol i}
V_3$.
This procedure tends to obscure the nature of the choice made to obtain the frame, making it more dif\/f\/icult to argue
that any hyperk\"ahler metric admits such a~frame.
Fortunately there is an alternative approach, which clarif\/ies the choice of frame, is easily made compatible with any
reality conditions, and generalizes to the hypercomplex case.
The following elementary observation is very well known, at least in the hyperk\"ahler case.
\begin{prop}
Let $z$ be a~complex function on a~hypercomplex $4$-manifold $(M,D)$ which is holomorphic with respect to one of the
complex structures.
Then $\Delta^D z=0$, so that ${\rm d} z$ is a~complex null $1$-form which is divergence-free with respect to $D$.
\end{prop}
\begin{proof}
If $z$ is $I$-holomorphic, then $I{\rm d} z:=-{\rm d} z\circ I = -{\boldsymbol i}{\rm d} z$.
Now $I$ is skew and $DI=0$, so $\divg^D(I{\rm d} z)=\sum\limits_i\ip{\varepsilon^i,I D_{e_i}{\rm d} z}=0$
since ${\rm d}^2 z=0$.
Hence $\Delta^D z=\divg^D({\rm d} z-{\boldsymbol i} I{\rm d} z)=0$.
\end{proof}
\begin{rem}
I have presented this observation using language adapted to Euclidean signature manifolds.
In Kleinian signature, some of the complex structures are imaginary, so that ${\boldsymbol i} I$ (say) is a~real
involution, inducing a~decomposition $TM=T^{+}M\oplus T^{-}M$ into its $\pm1$ eigenspaces.
These distributions are integrable, and the analogue of a~holomorphic function is a~function constant on one of the
families of integral surfaces~-- such functions can of course be real valued.
Note that the orientation $2$-forms of these integral surfaces are the (weightless) K\"ahler forms of the null complex
structures $J\pm{\boldsymbol i} K$, which are decomposable (and up to rotation, $J$ and ${\boldsymbol i} K$ are real).
The Euclidean and Kleinian cases can be considered together by f\/irst working on a~complexif\/ied hypercomplex manifold,
then imposing reality conditions.
The above proposition applies equally in the complexif\/ied setting.
Other, more Kleinian, arguments are also available (cf.~\cite{CMN:cs4,Ple:sce}).
\end{rem}
Before discussing the non-null reduction to two dimensions, I will brief\/ly discuss the null reduction (closely related
to Plebanski's heavenly equations~\cite{Ple:sce}) which is used in the literature to relate the selfdual vacuum equation
to a~topological chiral model~\cite{Par:sdg,Ward:suc}.
I present the generalizations to the hypercomplex case, following~\cite{Dun:tph,GrSt:his}.
\subsection*{Topological models and heavenly equations}
Choose independent functions $(w,z)$, both holomorphic with respect to $I$; then ${\rm d} w,{\rm d} z$ are
null $1$-forms with $\ip{{\rm d} w,{\rm d} z}=0$, and one can take these to be $\eta_0+{\boldsymbol
i}\eta_1$ and $\eta_2+{\boldsymbol i}\eta_3$.
This can be done compatibly with Kleinian reality conditions by taking ${\boldsymbol i} I$ and $(w,z)$ real, but is
incompatible with Euclidean reality conditions.
Locally, $M$ is a~bundle of null surfaces over a~quotient surface $N$ with coordinates $w,z$.
Choosing f\/ibre coordinates amounts to choosing a~local trivialization of this bundle, and locally one can take
$M=N\mathbin{{\times}}\Sigma^2$.
Then $V_0-{\boldsymbol i} V_1=\partial_w-\alpha$, $V_2-{\boldsymbol i} V_3=\partial_z-\beta$, $V_0+{\boldsymbol i}
V_1=\phi$, $V_2+{\boldsymbol i} V_3=\psi$, where $\alpha$, $\beta$, $\phi$, $\psi$ are vector f\/ields tangent to the f\/ibres.
The equations~\eqref{eq:MN1}--\eqref{eq:MN3} now read:
\begin{gather*}
[\partial_w-\alpha,\partial_z-\beta]=0,
\qquad
[\phi,\psi]=0,
\qquad
[\partial_z-\alpha,\phi]+[\partial_w-\beta,\psi]=0,
\end{gather*}
which are the equations for a~pencil of f\/lat connections ${\rm d}+A+\lambda\Phi$ with gauge group
$\Diff(\Sigma^2)$, where $A=-\alpha {\rm d} w-\beta {\rm d} z$ and $\Phi=-\psi{\rm d}
z+\phi{\rm d} w$.
(This is also known as a~topological chiral or sigma model.)
Plebanski's f\/irst and second heavenly equations~\cite{Ple:sce}, and their generalizations to the hypercomplex case (due
to Grant--Strachan~\cite{GrSt:his} and Dunajski~\cite{Dun:tph} respectively) are obtained by f\/ixing the gauge freedom in
dif\/ferent ways.
$\bullet$ First, since ${\rm d}+A$ is f\/lat, one can set $A=0$ (i.e., $\alpha=\beta=0$),
then integrate the equation ${\rm d}\Phi=0$ (i.e., $\phi_z+\psi_w=0$) to get $V_0-{\boldsymbol i}
V_1=\partial_w$, $V_2-{\boldsymbol i} V_3=\partial_z$, $V_0+{\boldsymbol i} V_1=U_w$, $V_2+{\boldsymbol i} V_3=-U_z$ for
a~vector f\/ield $U$ tangent to the f\/ibres (with $U_z$ and $U_w$ linearly independent).
The remaining equation is $[U_w,U_z]=0$.
If $U$ is area preserving on the f\/ibres, with local hamiltonian $\Omega$, then Plebanski's f\/irst equation
$\{\Omega_w,\Omega_z\}=1$ is obtained, where $\{\cdot,\cdot\}$ denotes the Poisson bracket with respect to a~suitably
scaled area form on the f\/ibres.
$\bullet$ Second, since $[\phi,\psi]=0$, one can choose the f\/ibre coordinates $(x,y)$ so that
$\phi=\partial_x$ and $\psi=-\partial_y$, then integrate the equation $\alpha_x-\beta_y=0$ to give $V_0-{\boldsymbol i}
V_1=\partial_w-\gamma_y$, $V_2-{\boldsymbol i} V_3=\partial_z-\gamma_x$, $V_0+{\boldsymbol i} V_1=\partial_x$,
$V_2+{\boldsymbol i} V_3=-\partial_y$.
The remaining equation is $\gamma_{xw}-\gamma_{yz}=[\gamma_x,\gamma_y]$.
Again the area preserving condition reduces everything to a~single function $\Theta$, satisfying Plebanski's second
heavenly equation $\Theta_{xw}+\Theta_{yz}=\{\Theta_x,\Theta_y\}$.
All complexif\/ied hypercomplex and hyperk\"ahler metrics are obtained from these constructions, but information about
Euclidean real slices is lost.
\subsection*{Hypercomplex structures from the Hitchin equations}
In order to obtain a~formulation compatible with Euclidean reality conditions, take $z$ to be $I$-holomorphic and
$\tilde z$ to be $(-I)$-holomorphic.
Then ${\rm d} z$ and ${\rm d}\tilde z$ are null and so ${\rm d} z+{\rm d} \tilde z$ and
$I({\rm d} z+{\rm d}\tilde z)=-{\boldsymbol i}({\rm d} z-{\rm d}\tilde z)$ are orthogonal,
closed, divergence-free $1$-forms of the same length.
Generically, this length will be nonzero on a~dense open set, and one can take $\eta_0+{\boldsymbol
i}\eta_1={\rm d} z$ and $\eta_0-{\boldsymbol i}\eta_1={\rm d}\tilde z$.
Euclidean reality conditions are easily obtained by setting $\tilde z=\bar z$ for $I$ real.
\begin{thm}
\label{th:s2Hit}
Let $(A,\Phi)$ be a~solution to the Hitchin equations $($on a~trivial spinor-vortex space$)$, with gauge group
$\Diff(\Sigma^2)$ for some $2$-manifold $\Sigma^2$.
Then the selfdual space constructed from $(A,\Phi)$ is hypercomplex.
Any hypercomplex $4$-manifold arises locally in this way, and is hyperk\"ahler if and only if there is a~reduction to
the gauge group $\SDiff(\Sigma^2)$.
\end{thm}
\begin{proof}
Choose conformal coordinates $z=x+{\boldsymbol i} y$, $\tilde z=x-{\boldsymbol i} y$ on a~trivial spinor-vortex space
and write $A=\alpha {\rm d} z+\widetilde\alpha {\rm d}\tilde z$, $\Phi=\phi {\rm d}
z+\widetilde\phi {\rm d}\tilde z$, where $\alpha,\widetilde\alpha, \phi,\widetilde\phi$ are complex vector
f\/ields on $\Sigma^2$.
Then the Hitchin equations become
\begin{gather*}
[\partial_z-\alpha,\widetilde\phi]=0,
\qquad
[\partial_{\tilde z}-\widetilde\alpha,\phi]=0,
\qquad
[\partial_z-\alpha,\partial_{\tilde z}-\widetilde\alpha]-[\phi,\widetilde\phi]=0,
\end{gather*}
and so the vector f\/ields $V_0-{\boldsymbol i} V_1=\partial_z-\alpha$, $V_0+{\boldsymbol i} V_1=\partial_{\tilde
z}-\widetilde\alpha$, $V_2-{\boldsymbol i} V_3=\widetilde\phi$ and $V_2+{\boldsymbol i} V_3=\phi$ satisfy
equations~\eqref{eq:MN1}--\eqref{eq:MN3}.
Hence the selfdual space is hypercomplex, with a~divergence-free coframe $(\eta_0,\eta_1,\eta_2,\eta_3)$ such that
$\eta_0+{\boldsymbol i} \eta_1={\rm d} z$ and $\eta_0-{\boldsymbol i}\eta_1={\rm d}\tilde z$ so that
$\eta_0$ and $\eta_1$ are closed.
Conversely, any hypercomplex $4$-manifold admits such a~divergence-free coframe, so the distribution generated by $V_2$
and $V_3$ (i.e., annihilated by $\eta_0$ and $\eta_1$) is integrable.
Under this assumption the form of the vector f\/ields given above is entirely general, and so any hypercomplex structure
arises in this way.
The characterization of the hyperk\"ahler case is entirely analogous to Theorem~\ref{th:s3Nahm}: $V_i$ preserve
a~volume form $e^{-4f}{\rm d} z\wedge {\rm d}\tilde z\wedge\eta_2\wedge\eta_3$ if and only if the area
form $e^{-4f}\eta_2\wedge\eta_3$ is parallel with respect to the connection $A=\alpha {\rm d}
z+\widetilde\alpha {\rm d} \tilde z$, and the vector f\/ields $\phi$, $\widetilde\phi$ are area-preserving (on each
f\/ibre), which is exactly what it means to have a~reduction to $\SDiff(\Sigma^2)$.
\end{proof}
\begin{rem}
This result can also be interpreted in Kleinian signature, when the Hitchin equations are replaced by harmonic maps into
a~Lie group or the principal chiral model.
The latter is the context for Husain's formulation~\cite{Hus:sdg}.
The non-null reduction (in the hyperk\"ahler case) is also discussed brief\/ly by Ward~\cite{Ward:suc} and
Ueno~\cite{Uen:ift}.
\end{rem}
Although this discussion has been local, there are intriguing connections with the global geometry of elliptically
f\/ibred K3 surfaces.
Yau's solution of the Calabi problem shows that on any K3 surface there is a~unique hyperk\"ahler metric in each
K\"ahler class, but no explicit description is known.
Any such hyperk\"ahler metric will correspond to a~solution of the $\SDiff(\Sigma^2)$ Hitchin equations, once
a~holomorphic function is chosen on a~suitable open subset of the K3 surface to def\/ine the dimensional reduction.
Now there are K3 surfaces which admit f\/ibrations over $\CP1$ (meromorphic functions) with elliptic curves as f\/ibres,
and, generically, 24 singular f\/ibres.
On the complement of the singular f\/ibres, there is therefore a~dimensional reduction to the Hitchin equations (on $\CP1$
minus 24 points) with gauge group $\SDiff(T^2)$.
The work of Gross and Wilson~\cite{GrWi:lcsK3} shows that this solution is well approximated by an Abelian solution
(gauge group $T^2$) def\/ining a~`semi-f\/lat' metric.
\begin{rem}
Continuing the development of this section, it is natural to ask if the hypercomplex equations are equivalent to the
$\Diff(S^1)$ Bogomolny equation on ${\mathbb{R}}^3$.
In fact, it is shown in~\cite{Cal:sde} that the selfdual space constructed from a~solution of the $\Diff(S^1)$
generalized Bogomolny equation on an Einstein--Weyl space $B$ is hypercomplex if $B$ is ``hyperCR'' (see
Section~\ref{s:s1Hit}).
However, not all hypercomplex structures arise this way, since for the hyperk\"ahler case in particular,
$\SDiff(S^1)=\Un(1)$ and only metrics with symmetry are obtained.
\end{rem}
\begin{rem}\label{r:zdg}\sloppy
The Mason--Newman--Dunajski--Joyce construction of Theorem~\ref{th:MN} also has a~gau\-ge-theo\-retic interpretation, of
course: the $V_i$ satisfy gauge f\/ield equations on a~trivial zero-dimen\-sio\-nal geometry~(!) with gauge group
$\Diff(\Sigma^4)$ for some $4$-manifold $\Sigma^4$.
It has been observed in many places (in particular~\cite{MaNe:eym}) that these equations are the reduction of the
selfdual Yang--Mills equations on ${\mathbb{R}}^4$ by four translations, and it would therefore be natural to extend the
integrable background geometries programme to the zero-dimensional case.
The background geometry is a~four-dimensional conformal vector space ${\mathcal{M}}$ together with an element
${\mathcal{Y}}$ of $\wedge^2_{\raise1pt\hbox{$\scriptscriptstyle -$}}{\mathcal{M}}^{*}\otimes{\mathcal{M}}$ which acts as a~right hand side for
equations~\eqref{eq:MN1}--\eqref{eq:MN3}.
The background equation is an unpleasant quadratic condition on ${\mathcal{Y}}$, which I have left as an exercise for
the enthusiastic reader: including all the constructions involving this zero-dimensional geometry would have added
unnecessarily to the length of this paper.
\end{rem}
\section{Background geometries from gauge f\/ields}
\label{s:bgf}
Theorem~\ref{th:sdgf} admits the following generalization.
\begin{thm}
\label{th:bgf}
Let $H$ be a~transitive group of diffeomorphisms of an $\ell$-manifold $\Sigma^\ell$.
Suppose that $(\alpha,\varphi)$ is a~solution of the gauge field equation on a~principal $H$-bundle $P\to Q$, where $Q$
is a~$k$-dimensional background geometry and $k+\ell\leqslant 4$.
Then the open subset of $\pi\colon P\mathbin{{\times}}_H\Sigma^\ell\to Q$ where $\widehat\varphi$ is surjective carries
naturally the structure of a~$(k+\ell)$-dimensional background geometry.
\end{thm}
\begin{proof}
The idea is to apply Theorem~\ref{th:sdgf} using the group $H\mathbin{{\times}}{\mathbb{R}}^{4-k-\ell}$ acting on
$\Sigma^\ell\mathbin{{\times}}{\mathbb{R}}^{4-k-\ell}$.
Suppose that $\widehat\varphi$ is surjective at some point $x$ of the f\/ibre of $P\mathbin{{\times}}_H\Sigma^\ell$ over
$q\in Q$.
Let $K\leqslant{\mathcal{V}}$ be the kernel of $\widehat\varphi$ at $x$.
Then there exists a~solution $(\alpha_0,\varphi_0)$ of the gauge f\/ield equation, with gauge group
${\mathbb{R}}^{4-k-\ell}$, def\/ined on a~neighbourhood of $q$, such that $\kernel\varphi_0\cap K=\{0\}$ at $q$: the
linear gauge f\/ield equation can be solved with any initial condition.
The pair $(\widehat\varphi,\widehat\varphi_0)$ is therefore an isomorphism on a~neighbourhood
$M=U\mathbin{{\times}}{\mathbb{R}}^{4-k-\ell}$ of $\{x\}\mathbin{{\times}}{\mathbb{R}}^{4-k-\ell}$ in
$\bigl(P\mathbin{{\times}}_H\Sigma^\ell\bigr)\mathbin{{\times}}{\mathbb{R}}^{4-k-\ell}$.
The resulting selfdual conformal structure on $M$ clearly admits ${\mathbb{R}}^{4-k-\ell}$ as a~symmetry group.
Hence the quotient $U\subseteq P\mathbin{{\times}}_H\Sigma^\ell$ is a~$(k+\ell)$-dimensional background geometry by the
results of Section~\ref{s:sdbg}.
However, the conformal metric on this quotient is clearly independent of the choice of $(\alpha_0,\varphi_0)$, since the
pushforward of the inverse metric~\eqref{eq:invmetric} from $\Sigma^\ell\mathbin{{\times}}{\mathbb{R}}^{4-k-\ell}$ to
$\Sigma^\ell$ kills the components of the vector f\/ields $X_i$ in ${\mathbb{R}}^{4-k-\ell}$.
Hence the conformal metric is uniquely def\/ined wherever $\widehat\varphi$ is surjective.
One also sees that the other f\/ields def\/ining the background geometry are well def\/ined, but the details here depend on
the geometry and are rather complicated.
The general formulae are given in the following subsections: these will complete the proof, since they are manifestly
well def\/ined.
\end{proof}
I now give some explicit formulae, using the notation $(A,\Phi)$ for the gauge f\/ields, rather than $(\alpha,\varphi)$.
From the above description it is clear at least that the conformal metric on the $(k+\ell)$-dimensional background
geometry may be represented contravariantly by
\begin{gather*}
(e_1-X_1)^2+\cdots+(e_k-X_k)^2+{X_{k+1}}^2+\cdots+{X_4}^2,
\end{gather*}
but note that $X_{k+1},\ldots X_4$ are $4-k$ vector f\/ields on an $\ell$-manifold $\Sigma^\ell$ with $\ell\leqslant 4-k$,
so inverting this metric is only straightforward when $k+\ell=4$.
It will be convenient therefore to introduce a~volume form $\nu$ on $\Sigma^\ell$ and hence present the explicit
formulae in an `SDif\/f-gauge'.
This will also make it easy to understand the volume-preserving case.
\subsection{Riccati space constructions}
First suppose that $\Phi$ is a~generalized Nahm f\/ield with values in $\Vect(\Sigma^3)$ and that $\nu$ is a~volume form
on $\Sigma^3$.
Then the contravariant metric
\begin{gather*}
\partial_r^2+\ip{\Phi,\Phi}
\qquad
\text{is dual to}
\quad
{\rm d} r^2+\ip{\eta,\eta},
\end{gather*}
{where}
\begin{gather*}
\eta=\frac{\nu(\Phi\mathbin{{\times}}\Phi,\cdot)}{\nu(\Phi\mathbin{{\times}}\Phi\mathbin{{\times}}\Phi)}
\end{gather*}
is a~section of ${\mathcal{E}}^{*}$ with values in $\Omega^1(\Sigma^3)$ (the space of $1$-forms).
Here and in the following~$\mathbin{{\times}}$ denotes the cross product ${\mathcal{E}}^{*}\otimes{\mathcal{E}}^{*}\to
L^{-1} {\mathcal{E}}^{*}$ (given by the wedge product and Hodge star operator): the $\Vect(\Sigma^3)$
values of $\Phi$ are then contracted into the entries of the volume form~$\nu$.
${\rm d} r^2+\ip{\eta,\eta}$ is the covariant form of the selfdual conformal structure obtained from~$\Phi$.
Now suppose that $\Phi$ is a~generalized Nahm f\/ield with values in $\Vect(\Sigma^2)$ and that $\nu$ is an area form on
$\Sigma^2$.
Let $F$ be an Abelian Nahm f\/ield acting on ${\mathbb{R}}$ with coordinate $\theta$.
Then $\Phi+F\partial_\theta$ is a~generalized Nahm f\/ield with values in $\Vect(\Sigma^2\mathbin{{\times}}{\mathbb{R}})$
and $\nu\wedge {\rm d}\theta$ is a~volume form on $\Sigma^2\mathbin{{\times}}{\mathbb{R}}$.
The selfdual conformal structure is therefore represented by ${\rm d} r^2+\ip{\eta,\eta}$ where now
\begin{gather*}
\eta=\frac{\nu(\Phi\mathbin{{\times}}\Phi){\rm d}\theta+F\mathbin{{\times}}\nu(\Phi,\cdot)}
{\ip{F,\nu(\Phi\mathbin{{\times}}\Phi)}}.
\end{gather*}
Straightforward manipulations and triple cross product identities may be used to rediagonalize the conformal metric
\begin{gather*}
{\rm d} r^2+\ip{\eta,\eta}= {\rm d} r^2+\frac{|\nu(\Phi\mathbin{{\times}}\Phi)|^2{\rm d}\theta^2
+2\ip{\nu(\Phi\mathbin{{\times}}\Phi),F\mathbin{{\times}}\nu(\Phi,\cdot)}{\rm d}\theta
+|F\mathbin{{\times}}\nu(\Phi,\cdot)|^2}{\ip{F,\nu(\Phi\mathbin{{\times}}\Phi)}^2}
\\
\phantom{{\rm d} r^2+\ip{\eta,\eta}}
={\rm d} r^2+\frac{|\nu(\Phi\mathbin{{\times}}\Phi)|^2
|F\mathbin{{\times}}\nu(\Phi,\cdot)|^2-\ip{\nu(\Phi\mathbin{{\times}}\Phi),F\mathbin{{\times}}\nu(\Phi,\cdot)}^2}
{|\nu(\Phi\mathbin{{\times}}\Phi)|^2\ip{F,\nu(\Phi\mathbin{{\times}}\Phi)}^2}
\\
\phantom{{\rm d} r^2+\ip{\eta,\eta}=}
{}+\frac{|\nu(\Phi\mathbin{{\times}}\Phi)|^2}{\ip{F,\nu(\Phi\mathbin{{\times}}\Phi)}^2}
\biggl({\rm d}\theta+\frac{\ip{\nu(\Phi\mathbin{{\times}}\Phi),F\mathbin{{\times}}\nu(\Phi,\cdot)}}
{|\nu(\Phi\mathbin{{\times}}\Phi)|^2}\biggr)^2
\\
\phantom{{\rm d} r^2+\ip{\eta,\eta}}
={\rm d} r^2+\frac{\ip{\nu(\Phi,\cdot),\nu(\Phi,\cdot)}}{|\nu(\Phi\mathbin{{\times}}\Phi)|^2}
+\frac{|\nu(\Phi\mathbin{{\times}}\Phi)|^2}{\ip{F,\nu(\Phi\mathbin{{\times}}\Phi)}^2}
\biggl({\rm d}\theta+\frac{\ip{\nu(\Phi\mathbin{{\times}}\Phi),F\mathbin{{\times}}\nu(\Phi,\cdot)}}
{|\nu(\Phi\mathbin{{\times}}\Phi)|^2}\biggr)^2.
\end{gather*}
Hence the conformal structure on the quotient by $\partial_\theta$ is represented by the metric
\begin{gather*}
{\rm d} r^2+\frac{\ip{\nu(\Phi,\cdot),\nu(\Phi,\cdot)}}{|\nu(\Phi\mathbin{{\times}}\Phi)|^2},
\end{gather*}
which is, of course, inverse to $\partial_r^2+\ip{\Phi,\Phi}$: note in particular that
$\ip{\nu(\Phi\mathbin{{\times}}\Phi),\Phi}=0$, expressing the fact that neither the components of $\Phi$, nor the dual
$1$-form components of $\nu(\Phi,\cdot)$ are linearly independent (pointwise on $\Sigma^2$).
I want to give the Weyl structure in an SDif\/f-gauge, with representative metric
\begin{gather*}
|\nu(\Phi\mathbin{{\times}}\Phi)|^2{\rm d} r^2+\ip{\nu(\Phi,\cdot),\nu(\Phi,\cdot)}.
\end{gather*}
This metric can be conveniently diagonalized as
\begin{gather*}
g=|\nu(\Phi\mathbin{{\times}}\Phi){\rm d} r+\nu(\Phi,\cdot)|^2
\end{gather*}
using the fact that $\ip{\nu(\Phi\mathbin{{\times}}\Phi),\nu(\Phi,\cdot)}=0$.
It takes quite a~bit of calculation to compute the Jones--Tod Weyl structure $\omega$ in this gauge (i.e.,
$D=D^g+\omega$), but the result is
\begin{gather*}
\omega=\frac{\Ip{2{\mathcal{B}}\bigl(\nu(\Phi\mathbin{{\times}}\Phi)\bigr)
-\nu(\Phi\mathbin{{\times}}\Phi)\mathbin{{\times}}\divg_\nu\Phi, \nu(\Phi\mathbin{{\times}}\Phi){\rm d}
r+\nu(\Phi,\cdot)}}{|\nu(\Phi\mathbin{{\times}}\Phi)|^2}.
\end{gather*}
Writing $\nu={\rm d} p\wedge {\rm d} q$ and expanding the cross products in components gives a~fuller
expression
\begin{gather}
\label{eq:RicEW}
g= \eta_1^2+\eta_2^2+\eta_3^2,
\\
\nonumber
\omega= \frac{2\sum\limits_{j,k} {\mathcal{B}}_{jk} \nu_j\eta_k
-\sum\limits_{i,j,k}\varepsilon_{ijk}\nu_i(\phi^j_p+\psi^j_q)\eta_k} {\nu_1^2+\nu_2^2+\nu_3^2},
\end{gather}
where
\begin{gather*}
\eta_i= \nu_i {\rm d} r + \phi^i {\rm d} q - \psi^i {\rm d} p,
\\
\nu_1= \phi^2\psi^3-\phi^3\psi^2,
\qquad
\nu_2= \phi^3\psi^1-\phi^1\psi^3,
\qquad
\nu_3= \phi^1\psi^2-\phi^2\psi^1,
\end{gather*}
and
\begin{gather*}
\Phi=\big(\phi^1,\phi^2,\phi^3\big)\partial_p+\big(\psi^1,\psi^2,\psi^3\big)\partial_q.
\end{gather*}
Well, nobody said it was going to be easy!
The reward is the knowledge that this Weyl structure is Einstein--Weyl if
${\mathcal{B}}$ satisf\/ies the matrix Riccati equation and $\Phi$ is a~generalized Nahm f\/ield on this Riccati space.
The construction of spinor-vortex spaces from Riccati spaces is perhaps the most awkward to make explicit, because of
the gauge freedom in the bundle ${\mathcal{W}}$ on a~spinor-vortex space.
If $\Phi$ is a~generalized Nahm f\/ield on a~Riccati space with values in $\Vect(\Sigma^1)$ for a~$1$-manifold $\Sigma^1$
with coordinate $t$, then the only natural way to proceed is to take ${\mathcal{W}}$ to be the kernel of $\Phi$ in the
pullback of ${\mathcal{E}}$ to $C\mathbin{{\times}}\Sigma^1$.
This kernel is not preserved, in general, by the connection $D$ on ${\mathcal{E}}$, but is preserved by the conformal
connection
\begin{gather*}
\nabla=D+\frac{\Phi_r\mathinner{\vartriangle}\Phi{\rm d} r+\dot\Phi\mathinner{\vartriangle}\Phi {\rm d}
t}{|\Phi|^2},
\end{gather*}
where an af\/f\/ine coordinate $r$ and a~$D$-parallel trivialization of ${\mathcal{E}}$ have been introduced.
The complex structure on ${\mathcal{W}}$ is given by cross product with $\Phi/|\Phi|$, while the holomorphic structure
is def\/ined using the connection
\begin{gather*}
\nabla+\frac{\ip{{\mathcal{B}}\Phi,\Phi}\iden}{|\Phi|^2}{\rm d} r
\end{gather*}
on ${\mathcal{W}}$.
The other two f\/ields on the spinor-vortex space are
\begin{gather*}
{\mathcal{C}}={\mathcal{B}}-\frac{{\mathcal{B}}\Phi\otimes\Phi+\Phi\otimes{\mathcal{B}}\Phi}{|\Phi|^2}
+\frac{\ip{{\mathcal{B}}\Phi,\Phi}}{2|\Phi|^2}\left(\iden+\frac{\Phi\otimes\Phi} {|\Phi|^2}\right),
\\
\psi=\frac{\Phi\mathbin{{\times}}(2{\mathcal{B}}\Phi+\Phi\mathbin{{\times}}\dot\Phi)}{|\Phi|^2}.
\end{gather*}
These f\/ields satisfy the spinor-vortex equations if $\Phi$ is a~generalized Nahm f\/ield on a~Riccati space.
(Since $\Phi/|\Phi|$ is $D$-parallel, it is reasonably straightforward to check~\eqref{eq:sv1}--\eqref{eq:sv2} directly,
although~\eqref{eq:sv3} is harder.)
\subsection{Spinor-vortex space constructions}
The approach here is the same as in the previous subsection, and the details are slightly less complicated.
For explicitness, introduce conformal coordinates $z=x+{\boldsymbol i} y$, $\tilde z=x-{\boldsymbol i} y$ on~$N$ and let
$\nu$ be an area form on~$\Sigma^2$.
First suppose that $(\Phi,\widetilde\Phi, \alpha{\rm d} z+\widetilde\alpha{\rm d}\tilde z)$ is
a~generalized Hitchin f\/ield with values in~$\Vect(\Sigma^2)$.
Then the contravariant metric
\begin{gather*}
4(\partial_z-\alpha)(\partial_{\tilde z}-\widetilde\alpha)+4\ip{\Phi,\widetilde\Phi}
\qquad
\text{is dual to}
\quad
{\rm d} z {\rm d}\tilde z+\ip{\eta,\widetilde\eta},
\end{gather*}
{where}
\begin{gather*}
\eta=\frac{\nu(\Phi,\cdot)+\nu(\Phi,\alpha){\rm d} z+\nu(\Phi,\widetilde\alpha) {\rm d}\tilde
z} {\nu(\Phi,\widetilde\Phi)}
\qquad
\text{and}
\qquad
\widetilde\eta=\frac{\nu(\widetilde\Phi,\cdot)+\nu(\widetilde\Phi,\alpha){\rm d} z
+\nu(\widetilde\Phi,\widetilde\alpha){\rm d}\tilde z} {\nu(\widetilde\Phi,\Phi)}.
\end{gather*}
Now suppose that $(\Phi\otimes\partial_t,\widetilde\Phi\otimes\partial_t, (\alpha{\rm d} z+\widetilde\alpha
{\rm d}\tilde z)\otimes \partial_t)$ is a~generalized Hitchin f\/ield with values in $\Vect(\Sigma^1)$ and that
$(F\partial_\theta,\widetilde F\partial_\theta,(\beta{\rm d} z+\widetilde\beta{\rm d}\tilde
z)\otimes\partial_\theta)$ is an Abelian Hitchin f\/ield acting on~${\mathbb{R}}$ with coordinate $\theta$.
Adding these together produces a~generalized Hitchin f\/ield with values in
$\Vect(\Sigma^1\mathbin{{\times}}{\mathbb{R}})$ and ${\rm d} t\wedge{\rm d}\theta$ is an area form on
$\Sigma^1\mathbin{{\times}}{\mathbb{R}}$.
Direct substitution gives
\begin{gather*}
\eta=\frac{F({\rm d} t+\alpha{\rm d} z+\widetilde\alpha{\rm d}\tilde z)
-\Phi({\rm d}\theta+\beta{\rm d} z+\widetilde\beta{\rm d}\tilde z)}
{F\widetilde\Phi-\Phi\widetilde F},
\\
\widetilde\eta= \frac{\widetilde F({\rm d} t+\alpha{\rm d} z+\widetilde\alpha{\rm d}\tilde
z) -\widetilde\Phi({\rm d}\theta+\beta{\rm d} z+\widetilde\beta{\rm d}\tilde z)} {\widetilde
F\Phi-\widetilde\Phi F}.
\end{gather*}
It is straightforward to rediagonalize $\ip{\eta,\widetilde\eta}$ to obtain the metric
\begin{gather*}
{\rm d} z{\rm d}\tilde z+\frac{({\rm d} t + \alpha{\rm d} z + \widetilde\alpha
{\rm d}\tilde z)^2} {4\Phi\widetilde\Phi}
\\
\qquad
{}-\frac{\Phi\widetilde\Phi}{(F\widetilde\Phi-\Phi\widetilde F)^2} \biggl({\rm d}\theta+\beta{\rm d}
z+\widetilde\beta{\rm d}\tilde z-\frac{F \widetilde\Phi+\Phi\widetilde F}{2 \Phi\widetilde\Phi}({\rm d}
t+\alpha{\rm d} z+\widetilde\alpha{\rm d}\tilde z)\biggr)^2.
\end{gather*}
As in the Riccati space construction, the conformal structure is easy to obtain, and is dual to
$4(\partial_z-\alpha)(\partial_{\tilde z}-\widetilde\alpha) +4\Phi^2\partial_t^2$, while more work is required to
compute the Jones--Tod Weyl structure~$\omega$.
The result, again in an SDif\/f-gauge (i.e., the gauge given by ${\rm d} t$) is reasonably simple, however,
\begin{gather}
g= 4\Phi\widetilde\Phi {\rm d} z{\rm d}\tilde z +({\rm d} t + \alpha{\rm d} z +
\widetilde\alpha {\rm d}\tilde z)^2,
\nonumber
\\
\omega= \biggl( \dot\alpha-\frac{2\widetilde{\mathcal{C}}\Phi}{\widetilde\Phi}\biggr){\rm d} z +\biggl(
\dot{\widetilde\alpha}-\frac{2{\mathcal{C}}\widetilde\Phi}{\Phi}\biggr){\rm d}\tilde z
-\frac12\biggl(\frac{\psi+\dot\Phi}{\Phi}+ \frac{\widetilde\psi+\dot{\widetilde\Phi}}{\widetilde\Phi}\biggr)
({\rm d} t + \alpha{\rm d} z + \widetilde\alpha {\rm d}\tilde z).
\label{eq:SVtoEW}
\end{gather}
This Weyl structure is Einstein--Weyl if $(\Phi,\widetilde\Phi,\alpha{\rm d} z + \widetilde\alpha
{\rm d}\tilde z)$ is a~generalized Hitchin f\/ield on a~spinor-vortex space.
\subsection{Einstein--Weyl constructions}
For completeness, I record here the explicit form of the generalized Jones--Tod construction of selfdual spaces from
Einstein--Weyl spaces~\cite{JoTo:mew,Cal:sde}.
The conformal structure on $M$ is obtained from the $\Diff(\Sigma^1)$ monopole $(A,\Phi)\partial_t$ on $B$ by the
formula
\begin{gather*}
\mathsf{c}=\mathsf{c}_B + \Phi^{-2}({\rm d} t+A)^2,
\end{gather*}
where $t$ is a~coordinate on $\Sigma^1$.
Compatible metrics for $\mathsf{c}$ are easily obtained by introducing a~compatible metric $g_B=\mu^{-2}\mathsf{c}_B$ on
$B$ and writing $\Phi=V\mu^{-1}$.
Then $g_B+V^{-2}({\rm d} t+A)^2$, $Vg_B+V^{-1}({\rm d} t+A)^2$ and $V^2g_B+({\rm d} t+A)^2$ are all
possibilities.
The latter may be written more invariantly as $\Phi^2\mathsf{c}_B +({\rm d} t+A)^2$: it is the SDif\/f-gauge
determined by ${\rm d} t$.
\subsection*{Addendum: null and non-null reductions}
Unlike the preceding addenda, the main observation here is a~negative one: the methods of this section do not extend
readily to relate nondegenerate background geometries to the~$\alpha$ and~$\beta$ surface reductions.
One might hope to obtain closer links by considering intermediate null reductions, in which the radical of~$VM$ is both
proper and nontrivial.
However, this is beyond the scope of this paper.
\section{Interlude: the Dif\/f(1) Hitchin equation}
\label{s:s1Hit}
\subsection*{HyperCR Einstein--Weyl spaces}
An important class of Einstein--Weyl spaces are the \emph{hyperCR} Einstein--Weyl spaces~\cite{GaTo:hms}.
In~\cite{Tod:sew}, Paul Tod presented a~way of reducing the hyperCR Einstein--Weyl equation to a~single second-order
dif\/ferential equation for a~complex function of three variables.
Since the equation is expected to be integrable, he posed the problem of identifying it.
In this section, I will follow a~similar approach to Tod and identify the hyperCR Einstein--Weyl equation with the
$\Diff(S^1)$ Hitchin equation on a~trivial spinor-vortex space.
An Einstein--Weyl structure $(\mathsf{c},D)$ on $B$ is said to be~\emph{hyperCR} if it admits an orthonormal frame
$\chi_1$, $\chi_2$, $\chi_3$ for the weightless (co)tangent bundle $L T^{*} B\cong L^{-1} TB$ such that $D\chi_i=\kappa {*\chi_i}$ for some section $\kappa$ of $L^{-1}$ and each~$i$~-- indeed any~$\chi$ in
the unit sphere generated by $\chi_1$, $\chi_2$, $\chi_3$ satisf\/ies the same equation; these $\chi$'s are called the
\emph{hyperCR congruences} of $B$.
Since the Weyl connection is torsion-free and conformal, it is easy to see that these equations are implied by their
skew parts
\begin{gather}
\label{eq:hcrEW1}
{\rm d}^D\chi_1 = 2\kappa \chi_2\wedge\chi_3,
\\
{\rm d}^D\chi_2 = 2\kappa \chi_3\wedge\chi_1,
\\
{\rm d}^D\chi_3 = 2\kappa \chi_1\wedge\chi_2.
\label{eq:hcrEW3}
\end{gather}
Tod noticed that $\chi_i$ satisfying~\eqref{eq:hcrEW1}--\eqref{eq:hcrEW3} determine the Einstein--Weyl space: the
conformal metric is $\chi_1^2+\chi_2^2+\chi_3^2$ and the Einstein--Weyl equation follows
from~\eqref{eq:hcrEW1}--\eqref{eq:hcrEW3}~-- see also~\mbox{\cite{CaPe:sdc,GaTo:hms}}.
Now introduce a~gauge $\mu$ with $D\mu=\omega\mu$ and def\/ine $1$-forms $\alpha_i=2\mu^{-1}\chi_i$.
Then the equations~\eqref{eq:hcrEW1}--\eqref{eq:hcrEW3} may be rewritten in the form given by Tod~\cite{Tod:sew}:
\begin{gather}
\label{eq:hcrg1}
{\rm d}\alpha_1=-\omega\wedge\alpha_1+\kappa\alpha_2\wedge\alpha_3,
\\
{\rm d}\alpha_2=-\omega\wedge\alpha_2+\kappa\alpha_3\wedge\alpha_1,
\\
{\rm d}\alpha_3=-\omega\wedge\alpha_3+\kappa\alpha_1\wedge\alpha_2.
\label{eq:hcrg3}
\end{gather}
These equations are easier to interpret after complexif\/ication, so that the conformal structure is determined by its
null lines, which form a~bundle of conics in $P(T^{*} B)$.
This bundle is trivial, since $T^{*} B$ is trivialized by $\alpha_1$, $\alpha_2$, $\alpha_3$.
The pullback of the tautological $1$-form on $P(T^{*} B)$ by a~constant section is a~constant linear combination of
$\alpha_1$, $\alpha_2$ and $\alpha_3$, which is null for a~section of the bundle of conics.
Then~\eqref{eq:hcrg1}--\eqref{eq:hcrg3} are equivalent to the integrability of the distributions def\/ined by these null
$1$-forms, i.e., to the integrability of a~rank~$2$ distribution~${\mathcal{H}}$ on the bundle of conics.
The integral surfaces are the null surfaces which motivated Cartan~\cite{Car:cew} to study $3$-dimensional
Einstein--Weyl geometry, and the quotient of the bundle of conics by ${\mathcal{H}}$ is the \emph{minitwistor space}
${\mathcal{S}}$ of $B$~\cite{Hit:cme}.
On a~general Einstein--Weyl space, ${\mathcal{H}}$ is def\/ined by the Weyl connection: the hyperCR case is special in
that there is a~preferred trivialization of the bundle of conics with respect to which the distribution is horizontal.
Using this trivialization, the bundle of conics is $B\mathbin{{\times}}{\mathbb P}^1$, and after choosing a~projective
coordinate $\zeta$ on ${\mathbb P}^1$, the null $1$-forms are
\begin{gather*}
\alpha_\zeta = \alpha_1+{\boldsymbol i}\alpha_2 + 2\zeta \alpha_3-\zeta^2(\alpha_1-{\boldsymbol i}\alpha_2).
\end{gather*}
The system~\eqref{eq:hcrg1}--\eqref{eq:hcrg3} is equivalent to $\alpha_\zeta\wedge {\rm d}\alpha_\zeta=0$ for all
$\zeta$.
If $X_1$, $X_2$, $X_3$ is the dual frame, this means that the vector f\/ields $X_1+{\boldsymbol i} X_2+\zeta X_3$ and
$X_3-\zeta(X_1-{\boldsymbol i} X_2)$ span an integrable distribution (tangent to the null surfaces) for each f\/ixed~$\zeta$.
This is the hyperCR analogue of the Mason--Newman--Dunajski--Joyce description of hypercomplex structures.
To see explicitly what this means, put $\varphi=\alpha_1+{\boldsymbol i}\alpha_2$ and $\omega=-\tau\alpha_3+\gamma$,
with $\ip{\gamma,\alpha_3}=0$, so that~\eqref{eq:hcrg1}--\eqref{eq:hcrg3} become
\begin{gather}
\label{eq:hcrh}
{\rm d}\varphi=\bigl((\tau+{\boldsymbol i}\kappa)\alpha_3-\gamma\bigr)\wedge\varphi,
\\
{\rm d}\alpha_3=-\gamma\wedge\alpha_3+\tfrac{{\boldsymbol i}}2\kappa \varphi\wedge\overline\varphi.
\label{eq:hcrc}
\end{gather}
The f\/irst equation implies the integrability of the distribution def\/ined by $\varphi$.
Tod~\cite{Tod:sew} uses~\eqref{eq:hcrh} to put $\varphi=w{\rm d} z$.
I will not repeat this here.
Instead I want to analyse these equations from the point of view of integrable background geometries.
The idea is that the foliation determined by~$\chi_3$ is a~generalized dimensional reduction.
To see this, break the equations into horizontal and vertical parts by writing $\alpha_3={\rm d} t+A$ for some
f\/ibre coordinate~$t$ so that~$A$ is horizontal (i.e., in the span of~$\alpha_1$ and~$\alpha_2$).
Then for any $1$-form~$\beta$,
\begin{gather*}
{\rm d}\beta=d_N\beta+{\rm d} t\wedge\dot\beta=d_N\beta-A\wedge\dot\beta +\alpha_3\wedge\dot\beta,
\end{gather*}
where $d_N\beta$ is a~multiple of $\alpha_1\wedge\alpha_2$, $\dot\beta=\partial_t\beta$,
$\alpha_1(\partial_t)=0=\alpha_2(\partial_t)$, and ${\rm d} t(\partial_t)=1$.
Hence~\eqref{eq:hcrh}--\eqref{eq:hcrc} become
\begin{gather*}
d_N\varphi-A\wedge\dot\varphi=-\gamma\wedge\varphi,
\qquad
\dot\varphi=(\tau+{\boldsymbol i}\kappa)\varphi,
\\
d_NA -A\wedge\dot A=\tfrac{{\boldsymbol i}}2\kappa\varphi\wedge\overline\varphi,
\qquad
\dot A=\gamma.
\end{gather*}
The equations on the right simply def\/ine $\tau+{\boldsymbol i}\kappa$ and $\gamma$, so after computing that
$\dot\varphi\wedge\overline\varphi -\varphi\wedge\dot{\overline\varphi} =2{\boldsymbol
i}\kappa\varphi\wedge\overline\varphi$, the equations on the left reduce to
\begin{gather*}
d_N\varphi+\dot A\wedge\varphi-A\wedge\dot\varphi=0,
\\
d_NA+\dot A\wedge A=\tfrac14\bigl(\dot\varphi\wedge\overline\varphi -\varphi\wedge\dot{\overline\varphi}\bigr).
\end{gather*}
The conformal structure on $N$ has representative metric $\varphi\overline\varphi$ so the orientation can be chosen so
that $\varphi$ has type $(1,0)$.
It is now easy to see that these equations are Hitchin's equations with gauge group $\Diff(S^1)$: $\varphi$ and $A$ are
$1$-forms on $N$ with values in $\lie{g}=\Vect(S^1)$; $\varphi$ is the Higgs f\/ield and $A$ is the connection $1$-form,
satisfying
\begin{gather*}
F^A=[\varphi,\overline\varphi]_{\lie{g}}
\qquad
\overline\partial{}^{A}\varphi=0.
\end{gather*}
These are equivalent to the~\eqref{eq:hcrh}--\eqref{eq:hcrc} and hence to~\eqref{eq:hcrg1}--\eqref{eq:hcrg3}: since
$\dot\varphi$ has type $(1,0)$, one can locally write $\dot\varphi=(\tau+{\boldsymbol i}\kappa)\varphi$ and thus def\/ine
$\alpha_1$, $\alpha_2$, $\alpha_3$, $\omega$.
\begin{thm}
Let $(A,\varphi)$ be a~solution of the $\Diff(S^1)$ Hitchin equations $($on a~trivial spinor-vortex space
$N)$.
Then the Einstein--Weyl space defined by $(A,\varphi)$ is hyperCR, and one of its hyperCR congruences defines the
foliation over $N$.
Any hyperCR Einstein--Weyl space arises in this way.
\end{thm}
Tod's simplif\/ication of the $\Diff(S^1)$ Hitchin equations amounts to a~f\/ixing the $\Diff(S^1)$ gauge via
$e^{2{\boldsymbol i} t}=\varphi/\overline\varphi$.
{\sloppy In order to obtain new examples of hyperCR Einstein--Weyl spaces, consider Hitchin f\/ields where the gauge group is
a~f\/inite-dimensional subgroup of $\Diff(S^1)$.
The Abelian gauge group~$\Un(1)$ yields only f\/lat Einstein--Weyl spaces, but the af\/f\/ine and projective groups,
Af\/f$({\mathbb{R}})$ and~PSL$(2,{\mathbb{R}})$ are more interesting.
The former is more tractable, since Af\/f$({\mathbb{R}})$ is a~solvable group, meaning that the nonlinear Hitchin
equations can be solved by integrating a~sequence of linear equations.
Indeed, writing $\varphi=\varphi_0+\varphi_1 t$ and $A=A_0+A_1 t$ gives
\begin{gather*}
\begin{split}
& d \varphi_1=0,
\qquad
d \varphi_0+A_1\wedge \varphi_0-A_0\wedge \varphi_1=0,
\\
& d A_1=0,
\qquad
d A_0+A_1\wedge A_0=\tfrac14(\varphi_1\wedge\overline\varphi_0 -\varphi_0\wedge \overline\varphi_1).
\end{split}
\end{gather*}
Locally, the af\/f\/ine gauge freedom can be used to eliminate the linear term $A_1$ of the connection~$A$, while the
conformal gauge freedom can be used to make the linear term~$\varphi_1$ of the Higgs f\/ield~$\varphi$ equal to~$\lambda{\rm d} z$ with $\lambda$ constant (without loss of generality $\lambda=1$ unless it vanishes, which is
the Abelian case).
Let $\varphi_0= {\boldsymbol i} f {\rm d} z$ so that the equations reduce to
\begin{gather*}
{\boldsymbol i} f_{\bar z} {\rm d}\bar z\wedge {\rm d} z = \lambda A_0\wedge {\rm d} z,
\qquad
d A_0=-\tfrac12 {\boldsymbol i} f{\rm d} z\wedge {\rm d}\bar z.
\end{gather*}
For $\lambda\neq 0$, the general solution up to gauge transformation is therefore determined by a~real function
$f(z,\bar z)$ satisfying $\Delta f+2\lambda^2 f=0$: $\varphi=(t+{\boldsymbol i} f){\rm d} z$ and
$A=-{*{\rm d} f}/\lambda$.
Hence (local) eigenfunctions of the Laplacian on ${\mathbb{R}}^2$ give rise to hyperCR Einstein--Weyl spaces.
}
\subsection*{Einstein--Weyl spaces with a~geodesic generalized symmetry}
A \emph{shear-free geodesic congruence} on an Einstein--Weyl space is a~weightless unit vector f\/ield
$\chi\in\mathrm{C}^\infty(B,L^{-1} TB)$ such that
\begin{gather*}
D\chi=\tau(\iden-\chi\otimes\chi)+\kappa {*\chi}
\end{gather*}
and for sections $\tau$, $\kappa$ of $L^{-1}$ called the \emph{divergence} and \emph{twist} of $\chi$.
On a~hyperCR Einstein--Weyl space, the hyperCR congruences are examples: they are also divergence-free and in fact this
characterizes them.
In the previous subsection it was found that the foliation def\/ined by a~hyperCR congruence is a~generalized symmetry, so
it is natural to ask, more generally, \textit{when does a~shear-free geodesic congruence define a~generalized symmetry?}
Note that the weightless unit vector f\/ield tangent to a~generalized symmetry over a~spinor-vortex space is always
shear-free, so this question can be rephrased: \textit{when is a~generalized symmetry geodesic?} Since the
Einstein--Weyl space $B$ is completely explicit~\eqref{eq:SVtoEW} in terms of the Hitchin f\/ield on the spinor-vortex
space $N$, this question is easily answered: the generalized symmetry is geodesic if and only if ${\mathcal{C}}$ (and
$\widetilde{\mathcal{C}}$, which is the complex conjugate in the Euclidean case) vanishes.
The Einstein--Weyl structure is then given by
\begin{gather}
g= 4\Phi\overline\Phi {\rm d} z{\rm d}\bar z +({\rm d} t + \alpha{\rm d} z +
\overline\alpha {\rm d}\bar z)^2,
\nonumber
\\
\omega=\dot\alpha{\rm d} z+\dot{\overline\alpha}{\rm d}\bar z
-\frac12\biggl(\frac{\psi+\dot\Phi}{\Phi}+ \frac{\overline\psi+\dot{\overline\Phi}}{\overline\Phi}\biggr)
({\rm d} t + \alpha{\rm d} z + \overline\alpha {\rm d}\bar z).
\label{eq:ewggs}
\end{gather}
If $\psi$ vanishes, then the spinor-vortex space is (locally) trivial and this is the case of the previous subsection.
Otherwise, $\psi$ is a~holomorphic trivialization of ${\mathcal{W}}$ on the open set where it is nonzero, and the
spinor-vortex space is given by a~spherical metric on $N$.
Two special cases of this construction have already been studied: the case that $\partial_t$ is a~genuine
symmetry~\cite{CaPe:sdc}, and the case that the congruence is also twist-free~\cite{CaTo:emh}.
The f\/irst class arises by supposing that the gauge group reduces to $\Un(1)$ or ${\mathbb{R}}$ and $B$ is said to be
Einstein--Weyl \emph{with a~geodesic symmetry}.
The Abelian Hitchin equations are easily solved on the spherical spinor-vortex space yielding the explicit formula
\begin{gather}
g=|h|^{2} g_{S^2}+\beta^2,
\qquad
\omega=-\frac{{\boldsymbol i}(h-\overline h)}{2|h|^2}\beta,
\qquad
{\rm d}\beta=\tfrac12(h+\overline h)\vol_{S^2},
\label{eq:ewgs}
\end{gather}
where $h$ is a~holomorphic function on an open subset of $S^2$.
The second class, the \emph{hyperCR-Toda spaces}, are obtained by from some explicit solutions of the af\/f\/ine Hitchin
equations: the twist-free condition reduces the gauge group to Af\/f$({\mathbb{R}})$; the connection is f\/lat, the linear
part of the Higgs f\/ield is constant, while the translational part of the Higgs f\/ield is given by a~holomorphic function~$h$.
The resulting Einstein--Weyl structure is{\samepage
\begin{gather}
g=(t+h)(t+\overline h)g_{S^2}+{\rm d} t^2,
\qquad
\omega=-\frac{2t+h+\overline h}{(t+h)(t+\overline h)}{\rm d} t.
\label{eq:hcrtoda}
\end{gather}
In this case the twist of the geodesic generalized symmetry $\partial_t$ vanishes.}
Note that the general theory of this paper justif\/ies the f\/inal remarks of~\cite{CaTo:emh}, by explaining the sense in
which the spherical metric is the natural quotient geometry of both structures~\eqref{eq:ewgs} and~\eqref{eq:hcrtoda}.
In both cases, the Einstein--Weyl space is hyperCR, although the congruence generated by~$\partial_t$ is no longer one
of the hyperCR congruences.
The same holds in general.
\begin{thm}
An Einstein--Weyl space with a~geodesic generalized symmetry is hyperCR.
\end{thm}
\begin{proof}
Gauduchon and Tod~\cite{GaTo:hms} show that an Einstein--Weyl space $(B,\mathsf{c},D)$ is hyperCR with twist
$\hat\kappa$ if and only if $*D\hat\kappa=\frac12F^D$ and $\hat\kappa^2=\frac16\scal^D$.
On an Einstein--Weyl space with a~geodesic generalized symmetry, direct computation of ${\rm d}^D\chi$ in the
gauge $(g,\omega)$ for the Einstein--Weyl structure~\eqref{eq:ewggs} yields
\begin{gather*}
\kappa=\frac1{4{\boldsymbol i}}\biggl(\frac{\dot\Phi-\psi}{\Phi}-
\frac{\dot{\overline\Phi}-\overline\psi}{\overline\Phi}\biggr).
\end{gather*}
Now put
\begin{gather*}
\hat\kappa=\frac1{2{\boldsymbol i}}\biggl(\frac{\dot\Phi}{\Phi}- \frac{\dot{\overline\Phi}}{\overline\Phi}\biggr)-\kappa
=\frac1{4{\boldsymbol i}}\biggl(\frac{\dot\Phi+\psi}{\Phi}-
\frac{\dot{\overline\Phi}+\overline\psi}{\overline\Phi}\biggr).
\end{gather*}
In the gauge $(g,\omega)$, the equation $*({\rm d}\hat\kappa-\omega\hat\kappa)=\frac12{\rm d}\omega$ is
a~straightforward though tedious computation.
The equation $\hat\kappa^2=\frac16\scal^D$ also follows by direct computation, although the calculation is greatly
simplif\/ied by using the general formula $-\frac16\scal^D=D_\chi\tau+\tau^2-\kappa^2$ for the scalar curvature of an
Einstein--Weyl space with a~shear-free geodesic congruence $\chi$~\cite{CaPe:sdc, PeTo:3ew}.
In this case $-2\tau=\psi/\Phi+\overline\psi/\overline\Phi$: the verif\/ication that
$\dot\tau-\omega(\partial_t)\tau+\tau^2 =\kappa^2-\hat\kappa^2$ is now easy.
\end{proof}
Conversely, twistor methods show that the foliation def\/ined by any shear-free geodesic cong\-ruen\-ce on any hyperCR
Einstein--Weyl space is a~generalized dimensional reduction over a~trivial or spherical spinor-vortex geometry, although
this is not easy to see by direct computation.
\section{Riccati spaces}
\label{s:rs}
Riccati spaces form a~foundation on which higher-dimensional geometries can be built, so although the matrix Riccati
equation is easy to solve, it is invaluable to understand the solutions carefully.
For this reason, I will begin by tackling the Riccati equation in an invariant way, without choosing a~conformal
trivialization of ${\mathcal{E}}$ or a~coordinate on $C$.
Recall that $\wedge^3{\mathcal{E}}=(TC)^3$, and it will be convenient to write $TC=L$, although the choice of
orientation implicit in this identif\/ication is not essential.
The matrix ${\mathcal{B}}$ is a~section of $L^{-1} \Sym_0{\mathcal{E}}$ and so, at each point of $C$,
it has two obvious invariants, of weight $-2$ and $-3$ respectively: $x=\frac23\trace({\mathcal{B}}^2)$ and
$y=4\det{\mathcal{B}}$, normalized so that the characteristic polynomial of ${\mathcal{B}}$ is $4\lambda^3-3x\lambda-y$.
The discriminant of this polynomial is the section $y^2-x^3$ of $L^{-6}$: more precisely, writing $4(y^2-x^3)=-27c^2$
yields $c=\psi_1 \psi_2 \psi_3$, where $\psi_1=\frac23(\lambda_2-\lambda_3)$, $\psi_2=\frac23(\lambda_3-\lambda_1)$ and
$\psi_3=\frac23(\lambda_1-\lambda_2)$, $\{\lambda_i\}$ being the eigenvalues of ${\mathcal{B}}$; the sign of $c$ depends
on the ordering of the eigenvalues.
Note that
\begin{gather*}
\psi_1+\psi_2+\psi_3=0,
\qquad
\psi_1^2+\psi_2^2+\psi_3^2=2x
\end{gather*}
{and}
\begin{gather*}
(\psi_1-\psi_2)(\psi_2-\psi_3)(\psi_3-\psi_1)=-8\lambda_1\lambda_2\lambda_3=-2y.
\end{gather*}
\subsection*{Pencils of conics}
In order to interpret this geometrically, complexify $C$ and ${\mathcal{E}}$ so that the
conformal structure $\mathsf{c}$ on ${\mathcal{E}}$ is determined by its null lines, which form a~bundle of conics
${\mathcal{S}}({\mathcal{E}})$ in the bundle $P({\mathcal{E}})$ of projective planes over $C$.
The role of ${\mathcal{B}}$ is to determine a~pencil of conics in each f\/ibre, associated to the two-dimensional family
of bilinear forms $\mathsf{c}\circ(s\iden+t{\mathcal{B}})$: $\mathsf{c}\circ{\mathcal{B}}$ is distinguished by being
traceless with respect to the f\/ixed bilinear form~$\mathsf{c}$.
Now, two distinct conics meet in four points, counted with multiplicity, so there are six possible conf\/igurations: the
generic case (I), where the four points are distinct; the four degenerations (II, III, D, N),
when two, three, two pairs, or four points come together; and the trivial case~(0), when ${\mathcal{B}}=0$ and the `pencil' is constant.
$$
\includegraphics{Calderbank-D2}
$$
The notation here follows the well-known application of this classif\/ication to Weyl tensors in four dimensions.
Only types (I, D, 0) are compatible with Euclidean reality conditions: in this case ${\mathcal{B}}$ must be
diagonalizable, and the eigenvalues are either distinct $(\lambda_1,\lambda_2,\lambda_3)$ with
$\lambda_1+\lambda_2+\lambda_3=0$, or of the form $(\lambda,\lambda,-2\lambda)$, or all zero.
A natural way to analyse a~pencil of conics is to identify one of the conics with a~projective line~$\mathbb{P}^1$.
In the present situation, this is done by introducing a~bundle of spinors ${\mathcal{U}}$ for ${\mathcal{E}}$, i.e.,
with ${\mathcal{E}}=S^2{\mathcal{U}}$, so that ${\mathcal{S}}({\mathcal{E}})=P({\mathcal{U}})$ and ${\mathcal{U}}$
inherits a~connection from ${\mathcal{E}}$.
Trivializing ${\mathcal{U}}$ using this connection identif\/ies ${\mathcal{S}}({\mathcal{E}})$ locally with
$C\mathbin{{\times}}\mathbb{P}^1$, on which an af\/f\/ine coordinate $r$ for $C$ and a~projective coordinate $\zeta$ for
$\mathbb{P}^1$ may be introduced.
The isomorphism between sections of the inner product bundle $L^{-1} {\mathcal{E}}$ and vertical
vector f\/ields on ${\mathcal{S}}({\mathcal{E}})$ may be described concretely using the induced parallel orthonormal frame
for $L^{-1} {\mathcal{E}}$: sections induce vector f\/ields via their inner product with the
tautological null vector f\/ield $e_\zeta\otimes\partial_\zeta$ where
\begin{gather}
\label{eq:tnvf}
e_\zeta=\bigl(\tfrac12\big(\zeta^2+1\big),{\boldsymbol i}\zeta,\tfrac{{\boldsymbol i}}2\big(\zeta^2-1\big)\bigr).
\end{gather}
The base locus of the pencil of conics (at each point of $C$) consists of the four zeros of the quartic polynomial
$\ip{{\mathcal{B}}(e_\zeta),e_\zeta}\otimes\partial_\zeta^2$, which is a~section of $L^{-1}\otimes{\mathcal{O}}(4)$.
It is now straightforward and entirely classical~\cite{Todd:pag} to analyse the zeros of this quartic in the various
cases, and hence relate the six types of pencil to properties of the matrix ${\mathcal{B}}$.
In particular, the generic case is given by $c\neq0$, when the four points are distinct.
\begin{rem}
Four distinct points on $\mathbb{P}^1$ are determined up to projective transformation by their cross-ratio, and this
freedom is often used to identify the points with $0$, $1$, $\infty$, $t$, with $t$ being the cross-ratio.
However, as remarked by Yoshida in his wonderful book~\cite{Yos:hml}, ``it is not fair that only the fourth point is
allowed to move freely''.
The democratic cross-ratio used there is the point $[\psi_1,\psi_2,\psi_3]$ on the line $\psi_1+\psi_2+\psi_3=0$ in~$\mathbb{P}^2$.
\end{rem}
\subsection*{Solution of the Riccati equation}
I shall now f\/ind the solutions of the Riccati equation $D{\mathcal{B}}=2({\mathcal{B}}^2)_0$.
Here and elsewhere $D$ denotes dif\/ferentiation with respect to the af\/f\/ine structure on $C$, and takes a~section of
a~natural bundle~$F$ to a~section of $L^{-1} F$.
In naive terms, $D$ is dif\/ferentiation with respect to an af\/f\/ine coordinate~$r$ (i.e., ${\rm d} r$ is
a~$D$-parallel section of $L^{-1}=T^{*} C$, and $L^{-1}$ is trivialized using this section).
The Riccati equation shows that $D{\mathcal{B}}$ commutes with ${\mathcal{B}}$, and so dif\/ferentiating the
Cayley--Hamilton equation $4{\mathcal{B}}^3-3x{\mathcal{B}}-y\iden=0$, gives
\begin{gather*}
\begin{split}
& 0=12 (D{\mathcal{B}}){\mathcal{B}}^2-3xD{\mathcal{B}}-3Dx{\mathcal{B}}-Dy\iden
=24{\mathcal{B}}^4-18x{\mathcal{B}}^2+3x^2\iden-3Dx{\mathcal{B}}-Dy\iden
\\
& \phantom{0}
=3(2y-Dx){\mathcal{B}}+(3x^2-Dy)\iden.
\end{split}
\end{gather*}
Hence $Dx=2y$, $Dy=3x^2$, $D^2x=6x^2$ and so $D(y^2-x^3)=0$.
In other words, the discriminant~$c^2$ is constant ($D$-parallel).
The constancy of $c$ means that if~${\mathcal{B}}$ has distinct eigenvalues at a~point, it has distinct eigenvalues
everywhere, in which case it is diagonalizable, and can be assumed diagonal, since $D{\mathcal{B}}$ commutes with~${\mathcal{B}}$.
I consider this f\/irst.
\begin{itemize}\itemsep=0pt
\item[I.] \textit{${\mathcal{B}}$ has distinct eigenvalues}.
\end{itemize}
Taking ${\mathcal{B}}$ diagonal, it is straightforward to compute the equation for the eigenvalues:
$D\lambda_1=2\lambda_1^2-\frac23(\lambda_1^2+\lambda_2^2+\lambda_3^2)$ and similarly for $\lambda_2$ and $\lambda_3$.
Hence $D\psi_1=\psi_1(\psi_2-\psi_3)=-2\lambda_1\psi_1$ and similarly for $\psi_2$ and $\psi_3$.
Now let $\chi_1=-\psi_2\psi_3$ etc., so that $\chi_1\chi_2\chi_3=-c^2$ and
$D\chi_1=-\chi_1(\psi_2-\psi_3)=2\lambda_1\chi_1$.
Squaring this gives $(D\chi_1)^2=\chi_1^2 (\psi_2^2+\psi_3^2-2\psi_2\psi_3)=\chi_1^2(\psi_1^2-4\psi_2\psi_3)
=4\chi_1^3+c^2$, since $\psi_1+\psi_2+\psi_3=0$.
Hence the $\chi_i$ all satisfy the equation
\begin{gather*}
(D\chi)^2=4\chi^3+c^2,
\\
\intertext{which is closely related to the equation for $x$, namely} (Dx)^2=4x^3-27c^2.
\end{gather*}
Thus $x$ and the $\chi$'s are equianharmonic Weierstra\ss\;elliptic functions of an af\/f\/ine coordinate, the period
lattice for $x$ being the barycentric subdivision of the lattice for the $\chi$'s.
\begin{rem}
The advantage of an invariant description is the f\/lexibility in the choice of coordinates: one does not have to use the
af\/f\/ine coordinate $r$.
Indeed, in Section~\ref{s:bm}, a~dif\/ferent gauge choice was motivated: there is a~natural projective structure
$D^2+\frac16\trace{\mathcal{B}}^2=D^2+\frac14 x$ and with respect to a~projective coordinate $t$, $D=\partial_t+a$,
where $\dot a-\frac12 a^2=-\frac12 x$.
Now $\dot x=2(ax+y)$ and $\dot y=3(ay+x^2)$, so that $\ddot a=a\dot a-\frac12 \dot x=a\dot a-ax-y$ and
\begin{gather*}
\dddot a= a \ddot a+\dot a^2-\dot a x-\tfrac52a\dot x+\bigl(\tfrac32a\dot x-\dot y\bigr)
= a \ddot a+\dot a^2-\dot a (a^2-2\dot a)-5a(a\dot a-\ddot a)+3(a^2-x)x
\\
\phantom{\dddot a}
=6a\ddot a+3\dot a^2-6a^2\dot a+6\dot a(a^2-2\dot a)
=6a\ddot a-9\dot a^2,
\end{gather*}
which is the \emph{Chazy equation}.
Of course it is well-known that the Chazy equation arises in the study of selfdual metrics in a~scalar-f\/lat gauge.
The generic solution, given by elliptic functions of the af\/f\/ine coordinate, may instead be presented in terms of modular
functions of the projective coordinate~\cite{AbCl:sis,Dub:g2d}.
\end{rem}
I turn next to the degenerate cases, when $c=0$, i.e., $y^2=x^3=64\lambda^6$, where the eigenvalues of ${\mathcal{B}}$
are $\lambda,\lambda$ and $-2\lambda$.
Assume f\/irst that $\lambda$ is not identically zero.
Then, on an open set at least, the (generalized) eigenspaces of ${\mathcal{B}}$ are constant (again using the fact that
$D{\mathcal{B}}$ commutes with ${\mathcal{B}}$) and hence ${\mathcal{B}}$ can be assumed to take the the form
\begin{gather*}
{\mathcal{B}}=\left[
\begin{matrix}
\lambda+\mu&{\boldsymbol i}\mu& 0
\\
{\boldsymbol i}\mu&\lambda-\mu&0
\\
0&0&-2\lambda
\end{matrix}
\right].
\end{gather*}
The matrix Riccati equation now yields
\begin{gather*}
D\lambda=-2\lambda^2,
\qquad
D\mu=4\lambda\mu.
\end{gather*}
Since $\lambda$ is not identically zero, it is given (up to translation) by $\lambda=1/2r$.
This is def\/ined for $r$ nonzero, and $\lambda$ has no zeros.
In this coordinate $\mu=b r^2$ for constant $b$, which is either identically zero, or nonzero for all nonzero $r$.
There are thus two cases.{\samepage
\begin{itemize}\itemsep=0pt
\item[II.] \textit{$y^2=x^3$ is nowhere zero, and ${\mathcal{B}}$ is nowhere diagonalizable}.
\item[D.] \textit{$y^2=x^3$ is nowhere zero, and ${\mathcal{B}}$ is everywhere diagonalizable}.
\end{itemize}
The notation is justif\/ied by noting that $\ip{{\mathcal{B}}(e_\zeta),e_\zeta}$ is the quartic
$\frac14(\zeta-1)^2(\mu(\zeta-1)^2+3\lambda(\zeta+1)^2)$: $\zeta=1$ is a~repeated root, as is $\zeta=-1$ when $\mu=0$.}
\begin{rem}
\label{r:Dproj}
The type (D) solution is even simpler with respect to a~projective coordinate.
Since $2{\mathcal{B}}^2_0=-2\lambda{\mathcal{B}}$, setting $D=\partial_t+2\lambda$ gives $\partial_t{\mathcal{B}}=0$, so
that $\lambda$ is constant in this gauge and $a=2\lambda$ satisf\/ies $\dot a-\frac12
a^2=-2\lambda^2=-\frac13\trace{\mathcal{B}}^2=-\frac12 x$; thus $t$ is a~projective coordinate with respect to the
natural projective structure, and ${\mathcal{B}}$ and $a$ are constant.
\end{rem}
It remains to consider the case that $y^2=x^3$ is identically zero, i.e., the eigenvalues of ${\mathcal{B}}$ are all
zero, so that ${\mathcal{B}}^3=0$.
If ${\mathcal{B}}^2=0$ then ${\mathcal{B}}$ is constant, so that in any case the kernel and image of ${\mathcal{B}}$ are
constant; ${\mathcal{B}}$ may thus be assumed to take the form
\begin{gather*}
{\mathcal{B}}=\left[
\begin{matrix}
\mu& {\boldsymbol i}\mu& \beta
\\
{\boldsymbol i}\mu&-\mu&{\boldsymbol i}\beta
\\
\beta&{\boldsymbol i} \beta&0
\end{matrix}
\right]
\end{gather*}
and the matrix Riccati equation yields
\begin{gather*}
D\beta=0,
\qquad
D\mu=2\beta^2.
\end{gather*}
Hence $\beta$ is a~constant, zero if and only if ${\mathcal{B}}$ is constant, and if $\beta$ is nonzero, then in
a~suitably translated af\/f\/ine coordinate $\mu=2\beta^2r$.
Thus there are three more cases.
\begin{itemize}\itemsep=0pt
\item[III.] \textit{${\mathcal{B}}^3=0$ and ${\mathcal{B}}^2$ is nowhere zero}.
\item[N.] \textit{${\mathcal{B}}^2=0$ and ${\mathcal{B}}$ is constant and nonzero}.
\item[0.] ${\mathcal{B}}=0$.
\end{itemize}
Again the notation is justif\/ied by computing $\ip{{\mathcal{B}}(e_\zeta),e_\zeta}=
\frac14(\zeta-1)^3(\mu(\zeta-1)+2{\boldsymbol i} \beta(\zeta+1))$.
\subsection*{Isomonodromic deformations} Recall that, for
$\Phi\in\mathrm{C}^\infty(C,{\mathcal{E}}^{*}\otimes\lie{g}_C)$, the generalized Nahm equation~\eqref{eq:nahm} is
\begin{gather*}
D\Phi-{*[\Phi,\Phi]_{\lie{g}}}={\mathcal{B}}\mathinner{\cdot}\Phi.
\end{gather*}
I claim that this equation, for ${\mathcal{B}}$ not identically zero, is equivalent to the fact that
\begin{gather*}
{\rm d}+\frac{\Phi}{{\mathcal{B}}}={\rm d}+\frac{\Phi(e_\zeta)
{\rm d}\zeta}{\ip{{\mathcal{B}}(e_\zeta),e_\zeta}}
\end{gather*}
is an isomonodromic family of connections on $\CP1$ parameterized by $C$, with four poles along the base locus of the
pencil of conics, where I use the tautological null vector f\/ield $e_\zeta$~\eqref{eq:tnvf} to clarify the meaning of the
connection $1$-form.
More precisely, this follows from the fact that the meromorphic connection
\begin{gather*
{\rm d}+\frac{\Phi+\ip{*{\mathcal{B}},\Phi}}{{\mathcal{B}}} ={\rm d}+\frac{\Phi(e_\zeta)
{\rm d}\zeta+\ip{*e_\zeta\wedge{\mathcal{B}}(e_\zeta),\Phi}{\rm d} r}
{\ip{{\mathcal{B}}(e_\zeta),e_\zeta}},
\end{gather*}
def\/ined on the pullback of $\lie{g}_C$ to the bundle of conics ${\mathcal{S}}({\mathcal{E}})$ over the Riccati space
$C$, is f\/lat if and only if $\Phi$ satisf\/ies the generalized Nahm equation.
This is a~straightforward verif\/ication.
Equivalently, if the gauge algebra $\lie{g}_C$ is represented as a~Lie algebra of vector f\/ields, then this observation
can be reformulated as the integrability of a~rank two distribution on a~bundle over ${\mathcal{S}}({\mathcal{E}})$, and
the poles of the connection can be viewed as points of ${\mathcal{S}}({\mathcal{E}})$ over which the distribution is
tangent to the f\/ibres.
Explicitly, in the above coordinates, two vector f\/ields generating this distribution are
\begin{gather}
\ip{{\mathcal{B}}(e_\zeta),e_\zeta}\partial_\zeta+\Phi(e_\zeta),
\nonumber
\\
\partial_r+\ip{{\mathcal{B}}(e_\zeta),e_\zeta'}\partial_\zeta+\Phi(e_\zeta'),
\label{eq:NLP}
\end{gather}
where $e_\zeta'=(\zeta,{\boldsymbol i},{\boldsymbol i}\zeta)$.
One can easily check that the distribution is closed under Lie bracket.
An advantage of this vector f\/ield interpretation is that it also makes sense on the trivial Riccati space
${\mathcal{B}}=0$, when~\eqref{eq:NLP} reduces to the standard Lax pair for the Nahm equation.
\subsection*{Addendum: Riccati spaces and integrability\\ by the method of hydrodynamic reductions}
In~\cite{FHZ:cqim}, Ferapontov et al.~independently discover the integrability of a~system of equations equivalent to
the generalized Nahm equations on Riccati spaces.
Their study concerns equations of the form
\begin{gather*}
\frac{\partial^2 (A^{\alpha\beta}(u))}{\partial x^\alpha \partial x^\beta}=0,
\end{gather*}
where $u=u(x^1,x^2,x^3)$ is a~function of three variables, and $A^{\alpha\beta}$ is a~$3\times 3$ symmetric matrix of
functions of one variable.
(Here I use the summation convention on Greek indices; this may also be viewed as abstract index notation.) Expanding
one derivative yields the equivalent formulation
\begin{gather}
\label{eq:fhz1}
\frac{\partial}{\partial x^\alpha} \left( V^{\alpha\beta}(u) \frac{\partial u}{\partial x^\beta} \right) =0,
\end{gather}
where $V^{\alpha\beta}$ is the derivative of $A^{\alpha\beta}$ (i.e., $V=A'$).
The approach in~\cite{FHZ:cqim} characterizes integrability using the method of hydrodynamic reductions; this tests for
the existence of suf\/f\/iciently many multi-phase solutions $u=U(R^1,R^2,\ldots R^N)$, where the $R^j(x^1,x^2,x^3)$ are
solutions to arbitrarily many commuting $(1+1)$-dimensional systems of hydrodynamic type.
The result of this analysis is that~\eqref{eq:fhz1} is integrable in this sense if and
only if there is a~scalar function $k(u)$ of one variable such that
\begin{gather}
\label{eq:fhz2}
V''(u) = (\log\det V )'(u) V'(u) + k(u) V(u)
\end{gather}
or equivalently $(V'/\det(V))' = k V$.
The relation to the Riccati space equation is subtle, and involves introducing a~$u$-dependent triple of vectors
$\theta^i_\alpha(u)$ ($i=1,2,3$) such that
\begin{gather}
\label{eq:V}
V=(\det\theta)\theta^{-1}(\theta^{\scriptscriptstyle\mathrm T})^{-1},
\qquad
\text{i.e.,}
\qquad
V^{\alpha\beta}=\sum\limits_{i=1}^3(\det\theta)\big(\theta^{-1}\big)_i^\alpha\big(\theta^{-1}\big)_i^\beta,
\end{gather}
where $\theta=(\theta^i_\alpha)$ is assumed invertible.
Now write ${\mathcal{C}}_{ij}:=(\theta^{-1})_i^\alpha(\theta^j_{\alpha})'$ so that $\theta^i_\alpha =\sum\limits_{j=1}^3
{\mathcal{C}}_{ij}\theta^j_\alpha$.
There is a~gauge freedom in the choice of $\theta$, which may be used to suppose ${\mathcal{C}}$ is symmetric.
Substituting~\eqref{eq:V} into~\eqref{eq:fhz2}, straightforward computation, using the symmetry of ${\mathcal{C}}$
together with standard matrix identities such as $(A^{-1})'=-A^{-1}A'A^{-1}$ and $(\det A)^{-1}(\det A)'=\trace(A^{-1}
A')$, then yields
\begin{gather*}
{\mathcal{C}}' = 2 {\mathcal{C}}^2 - (\trace {\mathcal{C}}) {\mathcal{C}} + c I,
\end{gather*}
where $c$ is an unknown scalar function of $u$ (related to $k$).
Now decompose ${\mathcal{C}}={\mathcal{B}}+a I$ with ${\mathcal{B}}$ tracefree; then the trace part of the above
equation determines $c$, while the trace-free part is the Riccati equation for ${\mathcal{B}}$.
Thus, remarkably, the hydrodynamic integrability condition agrees with the twistor-theoretic Riccati equation.
After classifying solutions of~\eqref{eq:fhz2} (equivalently, solutions of the Riccati equation), Ferapontov et al.
compute the form of the equation~\eqref{eq:fhz1} for $u$ arising from each solution.
They then observe that the central quadric Ansatz
\begin{gather*}
x^\alpha M_{\alpha\beta}(u) x^\beta =1
\end{gather*}
for $u$ yields all Painlev\'e equations~\cite{FHZ:cqim}.
It is this geometry that originally led to the identif\/ication of~\eqref{eq:fhz2} with the Riccati equation.
The key observation is that~\eqref{eq:fhz1} is a~divergence form equation, meaning that $u$ is in the kernel of the
Laplace--Beltrami operator of the metric $g_{\alpha\beta}(u)$ with $g={\rm adj}(V)=(\det V)V^{-1}$ so that
$V=\sqrt{\det g} g^{-1}$.
Hence the equations in this class have the form
\begin{gather}
\label{eq:ibg}
{\rm d}{*_g {\rm d} u}=0,
\end{gather}
where $g=g(u)$, and $*_g$ is the associated Hodge star operator.
Now $\theta^i_\alpha(u)$ has an interpretation, as a~\emph{framing} of the metric $g$:
\begin{gather*}
g_{\alpha\beta}(u)=\sum\limits_{i=1}^3 \theta^i_{\alpha}(u)\theta^i_{\beta}(u).
\end{gather*}
In terms of the geometry of $g$, $\theta$ is thus an ${\mathbb{R}}^3$-valued $1$-form: on vector f\/ields $X,Y$, the
metric is $g(u)(X,Y)=\ip{\theta(u)X,\theta(u)Y}$.
Thus there is a~gauge freedom to rotate $\theta$ by an $\SO(3)$-valued function of $u$: this is the freedom used to make
${\mathcal{C}}_{ij}$ symmetric above (which in turn f\/ixes $\theta$ up to a~rigid rotation).
As will be explained in~Section~\ref{s:sdsn}
(the last part of the next and f\/inal interlude),~\eqref{eq:fhz1} is a~hodograph
transformation of the generalized Nahm equation with gauge group $\SDiff(\Sigma^2)$.
The central quadric Ansatz corresponds to a~reduction of gauge group from $\SDiff(\Sigma^2)$ to $\SU(2)$ (which acts by
area preserving dif\/feomorphisms on the $2$-sphere).
This in turn explains the appearance of the Painlev\'e equations: generalized Nahm equations with gauge group $G$
describe isomonodromic deformation problems for $G$-connections on ${\mathbb{C}} P^1$ with four poles, and, when
$G=\SU(2)$ (rank~$2$ bundles over ${\mathbb{C}} P^1$), the connection between such isomonodromy problems and the
Painlev\'e equations is well known~\cite{JMU:mpd}.
\section{Interlude: the Dif\/f(2) generalized Nahm equation}
This interlude is devoted to Weyl structures of the form~\eqref{eq:RicEW}.
It follows from the work of Section~\ref{s:bgf}, that these Weyl structures are Einstein--Weyl provided that
${\mathcal{B}}$ satisf\/ies the matrix Riccati equation and $\Phi^i=\phi^i\partial_p+\psi^i\partial_q$ ($i=1,2,3$) def\/ines
a~generalized Nahm f\/ield on this Riccati space, the gauge group being a~subgroup of $\Diff(\Sigma^2)$ where $\Sigma^2$
is a~surface with coordinates $p$, $q$.
However, as details were omitted in Section~\ref{s:bgf}, I present a~self-contained study of these Weyl structures in
the following two cases:
\begin{enumerate}\itemsep=0pt
\item[(i)]
${\mathcal{B}}=0$ (so the Riccati space is trivial);
\item[(ii)]
the $\Phi^i$ are area preserving vector f\/ields.
\end{enumerate}
These examples have an interest that stretches beyond the proscenium of Einstein--Weyl geo\-met\-ry, since they are closely
related both to well-known integrable systems, and also to hypercomplex, hyperk\"ahler and scalar-f\/lat K\"ahler
structures in four dimensions.
\subsection{HyperCR Einstein--Weyl spaces and the Dif\/f(2) Nahm equation}
In Section~\ref{s:s1Hit}, hyperCR Einstein--Weyl spaces were related to the $\Diff(S^1)$ Hitchin equation.
Now it is known~\cite{Cal:sde, GaTo:hms} that any solution of the $\Diff(S^1)$ Einstein--Weyl Bogomolny equation on
a~hyperCR Einstein--Weyl space gives rise to a~hypercomplex $4$-manifold (which is sometimes hyperk\"ahler).
This two step construction ties in with the construction of hypercomplex and hyperk\"ahler structures from the
$\Diff(\Sigma^2)$ Hitchin equation: see Section~\ref{s:hchk}.
On the other hand, also in Sec\-tion~\ref{s:hchk}, hypercomplex and hyperk\"ahler structures were related to the
$\Diff(\Sigma^3)$ Nahm equation, so it is natural to expect that hyperCR Einstein--Weyl spaces may be constructed from
the $\Diff(\Sigma^2)$ Nahm equation.
In fact this is not hard to see.
Suppose that $D=D^g+\omega$ is a~Weyl structure given by
\begin{gather*}
g= \eta_1^2+\eta_2^2+\eta_3^2,
\qquad
\omega= -\frac{\sum\limits_{i,j,k}\varepsilon_{ijk}\nu_i(\phi^j_p+\psi^j_q)\eta_k} {\nu_1^2+\nu_2^2+\nu_3^2},
\end{gather*}
{where}
\begin{gather*}
\eta_i= \nu_i {\rm d} r + \phi^i {\rm d} q - \psi^i {\rm d} p,
\\
\nu_1= \phi^2\psi^3-\phi^3\psi^2,
\qquad
\nu_2= \phi^3\psi^1-\phi^1\psi^3,
\qquad
\nu_3= \phi^1\psi^2-\phi^2\psi^1,
\end{gather*}
{and}
\begin{gather*}
\Phi=\big(\phi^1,\phi^2,\phi^3\big)\partial_p+\big(\psi^1,\psi^2,\psi^3\big)\partial_q.
\end{gather*}
The Nahm equation expands to give
\begin{gather*}
\phi^1_r=\phi^2\phi^3_p-\phi^3\phi^2_p+\psi^2\phi^3_q-\psi^3\phi^2_q,
\qquad
\psi^1_r=\phi^2\psi^3_p-\phi^3\psi^2_p+\psi^2\psi^3_q-\psi^3\psi^2_q
\end{gather*}
and cyclic permutations.
From this it follows easily that ${\rm d}\eta_1+\omega\wedge\eta_1=\kappa \eta_2\wedge\eta_3$ (and similarly
for the cyclic permutations), where
\begin{gather*}
\kappa=\frac{\nu_1\big(\phi^1_p+\psi^1_q\big)+\nu_2\big(\phi^2_p+\psi^2_q\big) +\nu_3\big(\phi^3_p+\psi^3_q\big)}{\nu_1^2+\nu_2^2+\nu_3^2}.
\end{gather*}
Hence the Weyl structure is hyperCR Einstein--Weyl.
Note that if the Nahm f\/ield preserves the (\textit{a priori} arbitrary) area form ${\rm d} p\wedge{\rm d}
q$ then $\kappa=0$ and the Einstein--Weyl space is f\/lat.
Ward~\cite{Ward:sut} uses this fact to linearize the Nahm equation in this case.
In fact any hyperCR Einstein--Weyl space $B$ arises locally from this construction.
One way to see this is to choose an Abelian monopole on $B$.
This def\/ines a~hypercomplex structure with a~triholomorphic vector f\/ield $K$ tangent to the f\/ibres over $B$.
For example, $\kappa$ is an Abelian monopole on any hyperCR Einstein--Weyl space and the corresponding hypercomplex
structure is hyperk\"ahler with a~triholomorphic homothetic vector f\/ield~\cite{GaTo:hms}.
Now let $r$ be a~solution of $\Delta r=0$, which is constant on the f\/ibres over $B$, where $\Delta$ is the Laplacian of
the Obata connection.
This def\/ines a~divergence-free coframe ${\rm d} r$, $I{\rm d} r$, $J{\rm d} r$, $K{\rm d} r$ and
a~dual frame of vector f\/ields $V_0$, $V_1$, $V_2$, $V_3$.
Since $r$ is constant on the f\/ibres over $B$, $K$ is tangent to the level surfaces of $r$, so it is in the span of
$V_1$, $V_2$, $V_3$ and commutes with them all (since it is triholomorphic).
Therefore the gauge group of the Nahm f\/ield def\/ined by~$V_1$, $V_2$, $V_3$ reduces to the group of dif\/feomorphisms of~$\Sigma^3$ commuting with the f\/low of~$K$.
Taking the local quotient by~$K$ gives a~Nahm f\/ield with gauge group~$\Diff(\Sigma^2)$ for some surface~$\Sigma^2$, and
this is clearly a~Nahm f\/ield giving rise to the Einstein--Weyl space~$B$.
\begin{thm}
Let $\Phi$ be a~solution of the Nahm equation with gauge group $\Diff(\Sigma^2)$ for some surface~$\Sigma^2$.
Then the Einstein--Weyl space defined by $\Phi$ is hyperCR, and any hyperCR Einstein--Weyl space arises locally in this way.
If there is a~reduction to~$\SDiff(\Sigma^2)$, the Einstein--Weyl space is flat.
\end{thm}
There is a~great deal more freedom in this construction than in the $\Diff(S^1)$ Hitchin construction of hyperCR
Einstein--Weyl spaces, which can be an advantage or a~drawback, since the same hyperCR Einstein--Weyl space will arise
in many ways.
There are some interesting special cases one could consider, such as the group $\SL(2,{\mathbb{C}})$ acting on~$\CP1$
(which does not preserve an area form).
The Nahm equations in this case reduce to a~(complexif\/ied) Euler top equation, solvable in terms of elliptic functions.
\subsection{The SDif\/f(2) generalized Nahm equation}
\label{s:sdsn}
In addition to hyperCR Einstein--Weyl spaces, another important class consists of the Einstein--Weyl spaces arising from
solutions of the $\SU(\infty)$ Toda f\/ield equation \mbox{$u_{xx}+u_{yy}+(e^u)_{zz}=0$} \cite{LeBr:cp2,Tod:p3,Ward:sut}.
The terminology originates by regarding the equation as a~dispersionless limit of the $\SU(N)$ Toda equation as
$N\to\infty$: the Toda equation is a~two-dimensional system with independent variables $(x,y)$; the $z$ variable is
normally discrete, but becomes continuous in the limit.
The large $N$ limit of $\SU(N)$ may be interpreted as the group $\SDiff(S^1\mathbin{{\times}} S^1)$ of area preserving
dif\/feomorphisms of a~torus, with its Lie algebra ${\mathbb{Z}}$-graded by the Fourier components in one of the circles.
There is a~potential source of confusion here, however, since the $\SU(\infty)$ Toda equation is related to area
preserving dif\/feomorphisms in another way.
Namely, each solution gives rise to a~hyperk\"ahler metric with a~Killing vector $K$~\cite{BoFi:kve}, and hyperk\"ahler
metrics are in turn obtained from the $\SDiff(\Sigma^2)$ Hitchin equations on ${\mathbb{R}}^2$, as shown in
Theorem~\ref{th:s2Hit} (cf.~\cite{Ward:suc}).
Hence one expects the $\SU(\infty)$ Toda equation to be equivalent to a~symmetry reduction of the $\SDiff(\Sigma^2)$
Hitchin equations to one dimension.
$$
\includegraphics{Calderbank-D3}
$$
The Einstein--Weyl structure on a~Toda Einstein--Weyl space $B^3$ is given by the metric and Weyl $1$-form
\begin{gather*}
h=e^u\big({\rm d} x^2+{\rm d} y^2\big)+{\rm d} z^2,
\qquad
\omega_h=-u_z{\rm d} z.
\end{gather*}
In these geometric terms, the $\SU(\infty)$ Toda equation may be written ${\rm d}{*_h{\rm d} u}=0$, and so
$*_h{\rm d} u$ is a~closed $2$-form, which is therefore locally of the form ${\rm d} p\wedge{\rm d}
q$.
Now $(p,q,u)$ will be functionally independent unless $0={\rm d} u\wedge{*_h {\rm d}
u}={*_h|{\rm d} u|^2}$.
Hence (in Euclidean signature), $(p,q,u)$ may be used as coordinates, except in the trivial case ($u$ constant).
The solution of the Toda equation is now given implicitly by the functions $(x,y,z)$ of $(p,q,u)$.
These functions will turn out to satisfy a~generalized Nahm equation on the type (D) Riccati space with gauge group
$\SDiff(\Sigma^2)$.
In order to see this, and relate it to the work of Ferapontov et al.~\cite{FHZ:cqim}, consider the generalized Nahm
equation on any Riccati space with af\/f\/ine coordinate $r$, where the Higgs f\/ields
$\Phi^i=\phi^i\partial_p+\psi^i\partial_q$ preserve the area form ${\rm d} p\wedge{\rm d} q$ on a~surface
$\Sigma^2$ with coordinates $(p,q)$.
Motivated by the general construction of Section~\ref{s:bgf}, introduce the Weyl structure
\begin{gather*}
g= \eta_1^2+\eta_2^2+\eta_3^2,
\qquad
\omega= 2\frac{\lambda_1\nu_1\eta_1+\lambda_2\nu_2\eta_2+\lambda_3\nu_3\eta_3} {\nu_1^2+\nu_2^2+\nu_3^2},
\end{gather*}
{where}
\begin{gather*}
\eta_i= \nu_i {\rm d} r + \phi^i {\rm d} q - \psi^i {\rm d} p,
\\
\nu_1= \phi^2\psi^3-\phi^3\psi^2,
\qquad
\nu_2= \phi^3\psi^1-\phi^1\psi^3,
\qquad
\nu_3= \phi^1\psi^2-\phi^2\psi^1.
\end{gather*}
The metric $g$ can be simplif\/ied considerably using the fact that area preserving vector f\/ields are locally hamiltonian:
this means one can write $\phi^i={F^i}_q$ and $\psi^i=-{F^i}_p$ for some functions $F^1$, $F^2$, $F^3$ of $(p,q,r)$.
The ($r$-dependent) constants of integration for the hamiltonians may be chosen so that the $F^i$ satisfy the
generalized Nahm equation in the Lie algebra of functions under Poisson bracket: ${F^1}_r-\{F^2,F^3\}=\sum\limits_j
{\mathcal{B}}_{1j} F^j$ and so on.
(The system ${p^i}_r-\sum\limits_j {\mathcal{B}}_{ij} p^i=q^i$ has local solutions for any $q^i(r)$, and these can be
added to $F^i$.
However, if the $\Phi^i$ take values in a~Lie subalgebra of the area preserving vector f\/ields, then the $F^i$ may take
values in a~central extension of this Lie subalgebra.) The metric now reduces to
\begin{gather*}
\nonumber g=\sum\limits_i \left({\rm d} F^i-\sum\limits_{j}{\mathcal{B}}_{ij} F^j {\rm d} r\right)^2
=\sum\limits_i \theta^i_{\alpha}(r) \theta^i_\beta(r) {\rm d} x^\alpha {\rm d} x^\beta,
\end{gather*}
where $\theta^i_\alpha$ is a~basis of solutions to the linear system ${\theta^i}_r=\sum\limits_j {\mathcal{B}}_{ij}
\theta^j$ and, using the summation convention on Greek indices, $F^i=\theta^i_\alpha x^\alpha$, for some functions
$x^\alpha(p,q,r)$ ($\alpha=1,2,3$).
The equation ${F^i}_r-\frac12\sum\limits_{j,k}\varepsilon_{ijk}\{F^j,F^k\} =\sum\limits_j {\mathcal{B}}_{ij} F^j$ is
immediately equivalent to
\begin{gather}
\label{eq:gnH}
\theta^i_{\alpha} {x^\alpha}_r =\sum\limits_{j,k} \varepsilon_{ijk} \theta^j_\alpha \theta^k_\beta\big\{x^\alpha,x^\beta\big\}.
\end{gather}
Now for \emph{any} functions $x^\alpha(p,q,r)$ direct calculations yield the general jacobian identities
\begin{gather*}
\{x^2,x^3\}{\rm d} x^1+\{x^3,x^1\}{\rm d} x^2+\{x^1,x^2\}{\rm d} x^3=J(x){\rm d} r,
\\
{x^1}_r{\rm d} x^2\wedge {\rm d} x^3+{x^2}_r{\rm d} x^3\wedge {\rm d} x^1
+{x^3}_r{\rm d} x^1\wedge {\rm d} x^2 =J(x){\rm d} p\wedge{\rm d} q,
\end{gather*}
where $J(x)$ is the determinant of the jacobian of $(x^1,x^2,x^3)$ with respect to $(p,q,r)$.
Hence~\eqref{eq:gnH} holds if and only if $*_g {\rm d} r={\rm d} p\wedge {\rm d} q$.
This argument was carried out using an af\/f\/ine coordinate $r$.
However, any coordinate $t$ can be used, simply by writing $\partial_t=\partial_r+a$.
The following result summarizes the general construction.
\begin{prop}
Consider the $3$-dimensional metric
\begin{gather*}
g=\sum\limits_{i} \bigl(\theta^i_{\alpha}(t){\rm d} x^\alpha\bigr)^2,
\end{gather*}
where $\theta^i_\alpha$ is a~basis of solutions of the linear system ${\theta^i}_t-a \theta^i =\sum\limits_j
{\mathcal{B}}_{ij} \theta^j$ for some symmetric traceless matrix ${\mathcal{B}}_{ij}(t)$, and $x^\alpha(p,q,t)$ are
arbitrary functions.
Then $*_g {\rm d} t={\rm d} p\wedge{\rm d} q$ if and only if the functions $F^i=\theta^i_\alpha
x^\alpha$ satisfy
\begin{gather*}
F^i_t-aF^i-\frac12\sum\limits_{j,k}\varepsilon_{ijk}\big\{F^j,F^k\big\}=\sum\limits_j {\mathcal{B}}_{ij} F^j.
\end{gather*}
\end{prop}
\begin{rem}
We pause brief\/ly to reiterate (conversely) the link between this analysis and the work of Ferapontov et
al.~\cite{FHZ:cqim}.
The framing $\theta^i_\alpha$ of the metric $g$ appearing in ${\rm d} *_g{\rm d} u =0$, i.e., the
formulation~\eqref{eq:ibg} of~\eqref{eq:fhz1}, provides a~diagonalization
\begin{gather*}
g_{\alpha\beta} {\rm d} x^\alpha {\rm d} x^\beta= \sum\limits_{i=1}^3 \bigl(\theta^i_{\alpha}
{\rm d} x^\alpha\bigr)^2 =\sum\limits_{i=1}^3 \bigl({\rm d} F^i - ({\mathcal{C}} F)^i {\rm d} u)^2,
\end{gather*}
where $F^i=\theta^i_{\alpha} x^\alpha$, $({\mathcal{C}} F)^i=\sum\limits_{i=1}^3 {\mathcal{C}}_{ij}F^j$.
Writing $*_g {\rm d} u = {\rm d} p \wedge {\rm d} q$ yields
\begin{gather*}
(F^i)'-\frac12 \sum\limits_{i,j,k} \varepsilon_{ijk}\big\{F^j,F^k\big\} =\sum\limits_j {\mathcal{C}}_{ij} F^j,
\end{gather*}
which reduces to the generalized Nahm equation after splitting ${\mathcal{C}}$ into its tracefree and tracelike parts.
\end{rem}
Returning to the $\SU(\infty)$ Toda equation, $(\theta^i_\alpha)$ is now a~diagonal matrix with eigenvalues
$(e^{u/2},e^{u/2},1)$.
The case that $u$ is constant (so that ${\mathcal{B}}=0$) is the overlap with the previous subsection, and this
construction recovers Ward's linearization of the $\SU(\infty)$ Nahm equation~\cite{Ward:sun}.
On the other hand if~$u$ is not constant, $t=u$ can be used as a~coordinate, so that~$a=1/3$ and~${\mathcal{B}}_{ij}$ is
diagonal with constant eigenvalues $1/6$, $1/6$, $-1/3$.
This is the type~(D) solution of the matrix Riccati equation with $\lambda=1/6$ (see Remark~\ref{r:Dproj}).
There is another class of Einstein--Weyl spaces, governed by the dKP equation~\cite{DMT:dKP}.
The Weyl structure in this case is Lorentzian:
\begin{gather*}
h={\rm d} y^2-4{\rm d} x {\rm d} t-4u {\rm d} t^2,
\qquad
\omega_h=2u_x{\rm d} t.
\end{gather*}
The Einstein--Weyl condition may again be written ${\rm d}{*_h{\rm d} u}=0$, which now reduces to the dKP
equation $u_{yy}=(u_t-uu_x)_x$.
If the solutions $(\theta^i_\alpha)$ are taken to be $(1+u,2{\boldsymbol i} u,0)$, $(2{\boldsymbol i} u,1-u,0)$ and
$(0,0,1)$, then $h= \sum\limits_{i,\alpha,\beta} \theta^i_{\alpha}(r)\theta^i_\beta(r) {\rm d} x^\alpha
{\rm d} x^\beta$ with $-\sqrt2t=x^1+{\boldsymbol i} x^2$, $2\sqrt2 x=x^1-{\boldsymbol i} x^2$, $y=x^3$, whereas
${\mathcal{B}}$ is the type (N) solution of the Riccati equation.
The equivalence works as long as $|{\rm d} u|_h\neq0$.
\begin{thm}
The $\SU(\infty)$ Toda and dKP equations are generically locally equivalent to $\SDiff(\Sigma^2)$ generalized Nahm
equations, on the type {\rm (D)} and {\rm (N)} Riccati space respectively.
\end{thm}
The equivalence in each case is obtained by a~``hodograph transformation'', i.e., dependent and independent variables
are exchanged.
The beauty of such transformations is that they interchange coordinate-invariance and gauge-invariance, which is why
they are potentially useful for studying geometric dif\/ferential equations, where the independent variables are often not
well def\/ined.
This result was f\/irst obtained in for the $\SU(\infty)$ Toda equation and type (D) Nahm equation.
The extension to more general Nahm equations resulted from discussions with Maciej Dunajski.
Together with Paul Tod, he has given a~detailed discussion of the dKP case~\cite{DuTo:p12}.
Solutions of the $\SU(\infty)$ Toda equation may be divided into two classes according to whether they are most easily
presented explicitly or implicitly: in the former class, there are very few examples~-- to the best of my knowledge, the
only examples are the separable solutions~\cite{BoFi:kve,GeDa:sst} and the hyperCR-Toda solutions~\cite{CaTo:emh} (the
latter were discussed in Section~\ref{s:s1Hit}).
The class of implicit solutions is much richer and all of them (that I know of) are obtained by the above hodograph
transformation.
There are two general classes.
\subsection*{Solutions with a~Killing vector~\cite{CaPe:sdc, Ward:sut}}
Suppose that the generalized Nahm f\/ield is invariant under a~one-dimensional symmetry group of~$\Sigma^2$, generated by
a~hamiltonian vector f\/ield $X$, with momentum map $\eta$, and let $\psi$ be a~function on~$\Sigma^2$ with
${\rm d}\psi(X)=1$.
Then $\psi,\eta$ can be used as coordinates on $\Sigma^2$, $X=\partial_\psi$ and the area form is
${\rm d}\psi\wedge{\rm d}\eta$.
The Lie algebra of hamiltonian vector f\/ields commuting with $\partial_\psi$ consists of vector f\/ields of the form
$b\partial_\eta+W(\eta)\partial_\psi$, where $b$ is constant (on $\Sigma^2$), and the hamiltonian is $-b\psi+V(\eta)$
where $V_\eta=W$.
The solutions found by Ward~\cite{Ward:sut} are given by the metric
\begin{gather*}
h=\rho^2\bigl({\rm d}(U_\eta)^2+{\rm d}\psi^2\bigr)+{\rm d}(\rho U_\rho)^2,
\end{gather*}
where $U(\rho,\eta)$ satisf\/ies $(\rho U_\rho)_\rho+\rho U_{\eta\eta}=0$, the equation for axisymmetric harmonic
functions on~${\mathbb{R}}^3$.
In this case one of the hamiltonians is simply~$\psi$.
The general solutions were found in~\cite{CaPe:sdc} following similar ideas.
\subsection*{Solutions constant on central ellipsoids~\cite{Tod:p3}}
Motivated by the Pedersen--Poon Ansatz for scalar-f\/lat K\"ahler metrics~\cite{PePo:kzs}, Tod~\cite{Tod:p3} observed
that solutions of the Toda equation which are constant on central ellipsoids may be obtained from an ordinary
dif\/ferential equation, namely Painlev{\'e}'s third equation.
Painlev{\'e} III is equivalent to the isomonodromic deformation problem for an $\SU(2)$ connection with two double
poles, and hence to the generalized Nahm equation with gauge group $\SU(2)$ on the type (D) Riccati space.
Tod's solutions are obtained by reducing to the gauge algebra $\lie{so}(3)$ of inf\/initesimal isometries of $S^2$.
Solutions constant on central ellipsoids arise in this way because ellipsoids are precisely the quotients of left
invariant metrics on $S^3=\SU(2)$ by a~$\Un(1)$ subgroup.
Similar reductions are known in the dKP case~-- see~\cite{DMT:dKP,DuTo:p12}~-- and the latter construction is the context
for the central quadric Ansatz and Painlev\'e equations of~\cite{FHZ:cqim}.
\section{Further speculations}\label{section12}
The integrable background geometries discussed in this paper are those arising from the selfduality equation for
conformal structures in four dimensions.
Roughly speaking, they are characte\-ri\-zed as geometries associated to a~twistor space containing rational curves with
degree $2$ normal bundle, although this requires some interpretation for the one or two-dimensional geometries, where
the twistor space is not Hausdorf\/f and has dimension zero or one respectively.
Null reductions lead to twistor spaces with lower degree normal bundles.
For instance, a~surface with a~projective structure may be viewed as a~moduli space of rational curves in a~complex
surface with normal bundle ${\mathcal{O}}(1)$ (see Hitchin~\cite{Hit:gse} and LeBrun~\cite{LeB:phd}).
These considerations lead to the idea of extending the integrable background geometry concept to higher degree normal
bundles.
This would encompass the Einstein--Weyl and selfdual hierarchies (where the normal bundle is ${\mathcal{O}}(n)$ or
${\mathcal{O}}(n)\otimes{\mathbb{C}}^2$ respectively), as well as quaternionic geometry (normal bundle
${\mathcal{O}}(1)\otimes{\mathbb{C}}^{2k}$).
In the language of integrable systems, the concept of a~background geometry appears to be related to Lax systems
involving a~derivative with respect to the spectral parameter $\zeta$.
The zeros of the coef\/f\/icient of $\partial_\zeta$ are associated with poles of the Lax system: the prototype here is the
one-dimensional case (isomonodromic deformations).
In its most general form, the background geometry idea might be regarded as a~study of these extra $\partial_\zeta$
terms: general principles for the introduction of such terms have been developed by
Burtsev--Zakharov--Mikhailov~\cite{BMZ:isv}.
At the very least, it is desirable to know when these terms can be eliminated.
For the geometries studied in this paper, this question has an answer: they can be eliminated provided that the
background geometry is a~trivial Riccati space, a~trivial or spherical spinor-vortex space, a~hyperCR Einstein--Weyl
space, or a~hypercomplex selfdual space.
Returning to the general context of this paper, recent work of Ferapontov and Kruglikov~\cite{FeKr:disew} suggest deep
links between selfduality (hence twistor theory) and integrability in low dimensions, going beyond the inspirational
analysis by Mason and Woodhouse.
For example, in the particular case of equations of the form~\eqref{eq:ibg}, the Einstein--Weyl structure may be
recovered from the linearization ${\rm d}{*_{g(u)} {\rm d} v}+{\rm d}(v {*_g}'(u){\rm d}
u)=0$.
The leading term in $v$ is
\begin{gather*}
g^{\alpha\beta}(u)\frac{\partial^2 v}{\partial x^\alpha \partial x^\beta},
\end{gather*}
so the symbol of the linearized equation is the inverse metric $g^{-1}$ (up to a~conformal factor) of the Einstein--Weyl
structure.
Ferapontov and Kruglikov also show how to recover (the $1$-form of) the Weyl structure.
They obtain a~similar result for several classes of PDEs, relating the integrability of such equations (by the method of
hydrodynamic reductions) to Einstein--Weyl or selfdual conformal structures on their moduli spaces.
|
1,116,691,500,472 | arxiv | \section{Introduction}
Supersymmetric models with extra dimensions
has attracted interest.
Since extra dimensions and supersymmetry have not been discovered,
these must be invisible
at low scales.
One of the simple ways to compactify extra dimensions and
break supersymmetry simultaneously
is the Scherk-Schwarz mechanism.
It is known that supersymmetry breaking by the Scherk-Schwarz mechanism
\cite{Scherk:1978ta}\cite{
Scherk:1979zr}
is equivalent
to the supersymmetry
breaking by
constant superpotentials
in flat bulk space \cite{Marti:2001iw}--%
\cite{Biggio:2002rb}.
These two scenarios generate the same mass spectrum.
The equivalence between supersymmetry breakings by
the Scherk-Schwarz mechanism and constant
superpotentials has been discussed also in warped space
\cite{Altendorfer:2000rr}--%
\cite{Correia:2006pj},
particularly in the Randall-Sundrum background \cite{Randall:1999ee}.
Here the more fundamental question has been examined,
i.e., whether
supersymmetry can be
broken by the Scherk-Schwarz mechanism in Randall-Sundrum
background.
From the viewpoint of supergravity,
the answer seems to be negative.
This statement means that
the Scherk-Schwarz mechanism cannot be only the source of
supersymmetry breaking in Randall-Sundrum background.
On the other hand, the degrees of freedom of the Scherk-Schwarz twist
may exist in Randall-Sundrum background
if other supersymmetry-breaking sources are taken into
account \cite{Abe:2004ar}.
Thus it would be important to clarify the effects of
Scherk-Schwarz twists or constant superpotentials
in systems with additional sources.
In the previous papers \cite{Maru:2006id}\cite{Maru:2006ji},
we have shown that
a warped space model with a constant boundary superpotential
is an efficient model both to break supersymmetry
and to stabilize the radius
when a hypermultiplet, a compensator and a radion multiplet are
taken into account.
We presented possible additional supersymmetry-breaking sources of
$F$-term and $D$-term
to cancel the cosmological constant.
The resulting
soft scalar mass, gravitino mass and radion mass as well as
zero
cosmological constant
all gave evidence that this model is phenomenologically viable.
In this model,
the sectors of constant superpotentials and
additional supersymmetry breaking
are decoupled.
It would be worth to work with
systems where these sectors are coupled
because if constant superpotentials are allowed only
in the case with supersymmetry breaking in additional sectors,
they may be mixing each other.
In this Letter
we study supersymmetry breaking in a warped space model with
constant boundary superpotentials, a hypermultiplet,
a compensator, a radion multiplet and boundary chiral supermultiplets.
Equations of motion are solved together for these fields.
We take into account
the mass parameter $c$ for the hypermultiplet and
a superpotential of
O'Raifeartaigh model \cite{O'Raifeartaigh:1975pr}
for the boundary chiral supermultiplets.
If the hypermultiplet is decoupled, the model reduces to
ordinary O'Raifeartaigh model.
There is a flat direction of the chiral supermultiplets.
In the presence of the hypermultiplet,
it is shown that the flat direction
is lifted due to the mixing
of the equations of motion.
Then we show that
a modulus of the hypermultiplet
remains unfixed for zero constant superpotentials
and that it is stabilized
in the case with nonzero constant superpotentials for large negative
$c$.
In other words, the additional stabilization
of moduli can be developed when the sector of
constant superpotentials is coupled to a system
with spontaneous supersymmetry breaking.
\section{Model}\label{sc:model}
We consider a five-dimensional supersymmetric model of
a single hypermultiplet
and three chiral supermultiplets
on the Randall-Sundrum background \cite{Randall:1999ee}
whose metric is
\begin{eqnarray}
ds^2 = e^{-2R\sigma}\eta_{\mu\nu}dx^\mu dx^\nu +R^2 dy^2,
\quad ~~
\sigma(y)\equiv k|y|,
\end{eqnarray}
where $R$ is the radius of $S^1$ of the orbifold $S^1/Z_2$,
$k$ is the
curvature of the five-dimensional Anti-de-Sitter (AdS$_5$) space,
and the angle of $S^1$
is denoted by $y\,(0 \le y \le \pi)$.
In terms of superfields for
four-dimensional $N=1$ supersymmetry,
our Lagrangian
is \cite{Marti:2001iw}\cite{Maru:2006id}
\begin{eqnarray}
\!\!\!
{\cal L} &\!\!\!=\!\!\!& \int
d^2 \theta d^2\bar{\theta}~
\frac{1}{2}\, \varphi^\dag \varphi (T+T^\dag)
e^{-(T+T^\dag)\sigma}
(\Phi^\dag \Phi + \Phi^c \Phi^{c\dag} - 6M_5^3)
\nonumber \\
&& + \int d^2 \theta
\left[
\varphi^3 e^{-3T \sigma} \left\{
\Phi^c \left[
\partial_y - \left( \frac{3}{2} - c \right)T \sigma'
\right] \Phi + W_c
\right\} + \textrm{H.c.}
\right]
\nonumber
\\
&&
+\delta(y) \left[
\int
d^2\theta d^2\bar{\theta} ~
\varphi^\dagger \varphi
(\Phi_1^\dagger \Phi_1 +\Phi_2^\dagger \Phi_2 + \Phi_3^\dagger \Phi_3)
+\left\lbrace
\int d^2\theta ~
\varphi^3 W(\Phi_1,\Phi_2,\Phi_3)+ \textrm{H.c.}\right\rbrace
\right] ,
\nonumber
\\
\label{lag}
\end{eqnarray}
where the compensator chiral supermultiplet $\varphi$,
and the radion chiral supermultiplet $T$ are denoted as
\begin{eqnarray}
\varphi = 1 + \theta^2 F_{\varphi}, ~~~~~~
T=R + \theta^2 F_T ,
\end{eqnarray}
respectively and the chiral supermultiplets representing
the hypermultiplet are denoted as
\begin{eqnarray}
\Phi =\phi +\theta^2 F , ~~~~~~ \Phi^c =\phi^c +\theta^2 F^c .
\end{eqnarray}
The $Z_2$ parity is assigned to be even for
$\Phi$ and odd for $\Phi^c$.
The derivative with respect to $y$ is denoted by $'$,
such as $\sigma'\equiv d\sigma/dy$.
The five-dimensional Planck mass is denoted as $M_5$.
We assume the constant (field independent)
superpotential localized at the fixed points $y=0$ and $y= \pi$,
\begin{eqnarray}
W_c \equiv 2M_5^3 (w_0 \delta(y) + w_\pi \delta(y-\pi)),
\label{eq:boundary_pot}
\end{eqnarray}
where $w_{0}$ and $w_{\pi}$ are dimensionless constants.
In the Lagrangian (\ref{lag}), the last line is
the O'Raifeartaigh model
coupled to the compensator.
The three chiral supermultiplets are denoted as
\begin{eqnarray}
\Phi_i = \phi_i +\theta^2 F_i , ~~~~i=1,2,3 .
\end{eqnarray}
which are confined at $y=0$.
The superpotential $W$ is given by
\begin{eqnarray}
W(\Phi_i)=\lambda(\Phi_1^2-\mu^2)\Phi_2+m\Phi_1\Phi_3
\end{eqnarray}
where $\lambda,\mu,m$ are real parameters.
As the part of the Lagrangian (\ref{lag}) containing auxiliary
components is relevant to extra dimensions,
we extract the part
\begin{eqnarray}
\!\!\!
{\cal L}_{\textrm{\scriptsize aux}} &\!\!\!=\!\!\!&
\left[
\frac{1}{2}e^{-2R\sigma}(2RF^\dag F + F_T F^\dag \phi
+ F_T^\dag F \phi^\dag)
\right. \nonumber \\
&& \left. + \left\{
\frac{1}{2}e^{-2R\sigma}(2R \phi^\dag F
+ F_T(\phi^\dag \phi -3M_5^3))
(F_{\varphi}^\dag - F_T^\dag \sigma)
+ {\rm h.c.} \right\}
+(\phi \leftrightarrow \phi^c) \right] \nonumber \\
&&+e^{-2R\sigma}R(\phi^\dag \phi + \phi^c \phi^{c\dag} - 6M_5^3)
(F_{\varphi}^\dag - F_T^\dag \sigma)
(F_{\varphi} - F_T \sigma) \nonumber \\
&& + \left[
3e^{-3R\sigma}(F_{\varphi} - F_T \sigma)
\left\{
\phi^c \left[ \partial_y
-\left( \frac{3}{2} - c \right) R \sigma' \right]
\phi + W_c
\right\} \right. \nonumber \\
&&\left. +e^{-3R\sigma}\left\{
F^c \left[ \partial_y
-\left( \frac{3}{2} - c \right) R \sigma' \right]\phi
+ \phi^c \left[ \partial_y
-\left( \frac{3}{2} - c \right) R \sigma' \right]F
\right. \right. \nonumber \\
&& \left. \left.
-\phi^c \left( \frac{3}{2} - c \right) F_T \sigma' \phi
\right\} +{\rm H.c.}
\right]
\nonumber
\\
&&
+
\delta (y)
\left[|F_\varphi|^2|\phi_i|^2
+|F_i|^2
+
\left\lbrace
F_\varphi \phi_iF_i^\dagger
+3F_\varphi W
+W_i F_i
+\textrm{H.c.}
\right\rbrace\right]
\label{aux}
\end{eqnarray}
where the summation over $i$ is taken.
The derivatives of the superpotential are denoted
as $W_i\equiv \partial W/\partial\phi_i$, $i=1,2,3$.
The Lagrangian (\ref{aux}) gives the following equations of
motion for auxiliary fields:
\begin{eqnarray}
F &\!\!\!=&\!\!\! -\frac{e^{-R\sigma}}{R}
\left[
-\partial_y \phi^{c\dag}
+ \left( \frac{3}{2} + c \right)R \sigma' \phi^{c\dag}
+\frac{\phi}{2M_5^3}W_c^\dagger
+\frac{\phi}{6M_5^3}(3W^\dagger-\phi_i^\dagger W_i^\dagger)\delta(y)
\right. \nonumber \\
&\!\!\!&\!\!\! \left. +\frac{1}{6M_5^3}\phi^\dag \phi
\partial_y \phi^{c\dag}
+\frac{1}{3M_5^3}\phi^{c\dag} \phi
\partial_y \phi^{\dag}
-\frac{1}{6M_5^3}\phi^\dag \phi
\phi^{c\dag}\left( \frac{9}{2} -c \right)
R \sigma'
\right],
\label{Feom} \\
F^c &\!\!\!=&\!\!\! -\frac{e^{-R\sigma}}{R}
\left[
\partial_y \phi^{\dag}
- \left( \frac{3}{2} - c \right)R \sigma' \phi^{\dag}
+\frac{\phi^c}{2M_5^3}W_c^\dagger
+\frac{\phi^c}{6M_5^3}(3W^\dagger-\phi_i^\dagger W_i^\dagger)\delta(y) \right. \nonumber \\
&\!\!\!&\!\!\! \left. +\frac{1}{6M_5^3}\phi^c \phi^\dag
\partial_y \phi^{c\dag}
+\frac{1}{3M_5^3}\phi^{c\dag} \phi^c
\partial_y \phi^{\dag}
-\frac{1}{6M_5^3}\phi^c \phi^\dag
\phi^{c\dag} \left( \frac{9}{2} -c \right)
R \sigma'
\right],
\label{Fceom} \\
F_{\varphi} &\!\!\!=&\!\!\! -\frac{e^{-R\sigma}}{R}
\left[
-\frac{1}{6M_5^3}\phi^\dag \partial_y \phi^{c\dag}
-\frac{1}{3M_5^3}\phi^{c\dag}\partial_y \phi^\dag
+\frac{1}{6M_5^3} \phi^\dag \phi^{c\dag}
\left( \frac{9}{2} -c \right)R\sigma'
-\frac{1}{2M_5^3}W_c^\dagger
\right. \nonumber \\
&\!\!\!&\!\!\! \left.
-\frac{3(1-2R\sigma)}{r}\phi^{c\dag}
\partial_y \phi^\dag -\frac{3(1-2R\sigma)}{r}W_c^\dagger
+\frac{1-2R\sigma}{r}
\phi^{c\dag}\phi^\dag \left(\frac{3}{2} -c \right)R\sigma'
\right.
\nonumber
\\
&\!\!\!&\!\!\!\left.
+\left(-{1\over 6M_5^3}
-{(1-2R\sigma)^2\over r}\right)
(3W^\dagger-\phi_i^\dagger W_i^\dagger)
\delta(y)
\right], \label{Fpeom} \\
F_T &\!\!\!=&\!\!\! -\frac{e^{-R\sigma}}{r}
\left[
6\phi^{c\dag} \partial_y \phi^\dag -2\phi^{c\dag}\phi^\dag
\left(\frac{3}{2} -c \right)R\sigma' +6W_c^\dagger
\right.
\nonumber
\\
&\!\!\!&\!\!\! \left.
+2(1-2R\sigma)(3W^\dagger-\phi_i^\dagger W_i^\dagger)\delta(y)
\right], \label{FTeom} \\
F_i &\!\!\!=&\!\!\! \frac{e^{-R\sigma}}{R}
\phi_i\left[
-\frac{1}{6M_5^3}\phi^\dag \partial_y \phi^{c\dag}
-\frac{1}{3M_5^3}\phi^{c\dag}\partial_y \phi^\dag
+\frac{1}{6M_5^3} \phi^\dag \phi^{c\dag}
\left( \frac{9}{2} -c \right)R\sigma'
-\frac{1}{2M_5^3}W_c^\dagger
\right. \nonumber \\
&\!\!\!&\!\!\! \left.
-\frac{3(1-2R\sigma)}{r}\phi^{c\dag}
\partial_y \phi^\dag -\frac{3(1-2R\sigma)}{r}W_c^\dagger
+\frac{1-2R\sigma}{r}
\phi^{c\dag}\phi^\dag \left(\frac{3}{2} -c \right)R\sigma'
\right.
\nonumber
\\
&\!\!\!&\!\!\!\left.
+\left(-{1\over 6M_5^3}
-{(1-2R\sigma)^2\over r}\right)
(3W^\dagger-\phi_i^\dagger W_i^\dagger)
\delta(y)
\right]
-W_i^\dagger
\label{f1}
\end{eqnarray}
where we have defined
\begin{eqnarray*}
r \equiv \phi^\dag \phi + \phi^{c\dag} \phi^c - 6M_5^3 .
\end{eqnarray*}
In Eq.(\ref{Feom})
a partial integration has been performed.
The Lagrangian (\ref{aux}) are written as
\begin{eqnarray}
\!\!\!\!\!\!\!
{\cal L}_{\textrm{\scriptsize aux}}
&\!\!\!=\!\!\!& e^{-3R\sigma}\left[
(-\partial_y \phi^c+\left({3\over 2}+ c\right)R\sigma'\phi^c)F
\right.
+(\partial_y\phi-\left(\frac{3}{2}-c\right)R\sigma'\phi)F^c
\nonumber
\\
&&\!\!\!\!\!
+\left(3\phi^c(\partial_y\phi-\left(\frac{3}{2} -c\right)
R\sigma'\phi) +3W_c +(3W-\phi_i W_i) \delta(y)\right) F_\varphi
\nonumber
\\
&&\!\!\!\!\! \left.
-\left(3\sigma\phi^c(\partial_y\phi-\left(\frac{3}{2} -c\right)
R\sigma'\phi)+3\sigma W_c
+\phi^c\left({3\over 2}-c\right)\sigma'\phi\right)F_T
\right]
-|W_i|^2 \delta(y).
\label{auxap}
\end{eqnarray}
with $F,F^c,F_\varphi,F_T$
given in Eqs.(\ref{Feom})--(\ref{FTeom}).
\section{Moduli and potential
}\label{sc:oo}
\subsection{The O'Raifeartaigh sector (no hypermultiplet)}
We begin with examining moduli
in the part of the O'Raifeartaigh model coupled to the compensator.
If the hypermultiplet is absent,
the Lagrangian (\ref{auxap}) becomes
\begin{eqnarray}
{\cal L}_{\textrm{\scriptsize aux}}
&\!\!\!=\!\!\!&{e^{-R\sigma}\over 6M_5^3}\, 4\sigma(1-R\sigma)
\left|3W-\phi_i W_i\right|^2(\delta(y))^2
-\left|W_i\right|^2\delta(y)
=
-\left|W_i\right|^2\delta(y)
\label{nos}
\end{eqnarray}
where we used $\sigma(\delta(y))^2=0$.
The Lagrangian (\ref{nos}) is the same as in
the O'Raifeartaigh model without the compensator.
The solution of $\phi_1$ is
\begin{eqnarray}
\phi_1=\left\{\begin{array}{l}
0 ~\textrm{or}~ \pm\sqrt{\mu^2-{m^2/ (2\lambda^2)}}
\qquad \textrm{for}~ \mu^2 > {m^2/ (2\lambda^2)}
\\
0 \qquad\qquad\qquad\qquad\qquad\qquad
\textrm{for}~ \mu^2 < {m^2/ (2\lambda^2)}
\\
\end{array}\right.
\equiv \underline{\phi_1} .
\label{xo}
\end{eqnarray}
The other fields $\phi_2,\phi_3$ only need to satisfy a single equation
\begin{eqnarray}
2\lambda \phi_1\phi_2+ m\phi_3=0 ,
\label{phi23}
\end{eqnarray}
and one (or two) of $\phi_2$ and $\phi_3$
is undetermined.
From Eqs.(\ref{xo}), (\ref{phi23}) and (\ref{nos}),
the potential is obtained as
\begin{eqnarray}
V =-\int_0^\pi dy \,{\cal L}_{\textrm{\scriptsize aux}}
=|\lambda(\underline{\phi_1}^2-\mu^2)|^2 +|m\underline{\phi_1}|^2
\ge 0
\label{po}
\end{eqnarray}
where $\underline{\phi_1}$ is given in (\ref{xo}).
In order to be consistent with the Randall-Sundrum background,
the parameters $\lambda$ and $m$ must be zero.
In this pure boundary chiral supermultiplet
case,
the compensator has no
effects on the
potential and the background solution.
\subsection{Mixing of the two sectors ($W_c=0$
}\label{sc:mix}
We next examine the background, potential and moduli
in the case with nonzero hypermultiplet and chiral supermultiplets.
In the model without chiral supermultiplets,
we found that the hypermultiplet solution for $W_c=0$
is \cite{Maru:2006id}
\begin{eqnarray}
&&\phi=N_2\exp\left[\left({3\over 2}-c\right)R\sigma\right]
\equiv
\underline{\phi},
\label{phibar}
\\
&&\phi^c=0
\end{eqnarray}
where $N_2$ is an overall complex constant for the
flat direction $\phi$.
From this situation, it is one possibility that
a simplest nontrivial solution may exist for $W_c=\phi^c=0$
even with chiral supermultiplets.
In this section, we consider the case $W_c=\phi^c=0$.
From the Lagrangian (\ref{auxap}),
the equations of motion ($W_c=\phi^c=0$) are
\begin{eqnarray*}
&&\!\!\!\!\!
(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
\left[-\left({3\over 2}-c\right)R\sigma'\right]
-e^{4R\sigma}\partial_y\{(\partial_y\phi
-\left({3\over 2}-c\right)R\sigma'\phi)
e^{-4R\sigma}\}
\nonumber
\\
&& \qquad\qquad
+|3W-\phi_i W_i|^2 (\delta(y))^2
{(1-2R\sigma)^2\over r^2}\phi=0
\qquad\qquad\qquad \textrm{for}~ \phi^\dagger ,
\label{phieom}
\\
&&\!\!\!\!\!
(3W-\phi_i W_i)\delta(y)
\left[
-{1\over 3M_5^3}\partial_y\phi^\dagger
+{1\over 6M_5^3}\phi^\dagger\left({9\over 2}-c\right)R\sigma'
-{3\over r}\partial_y\phi^\dagger
+{1\over r}\phi^\dagger\left({3\over 2}-c\right)R\sigma'\right]
=0
\nonumber
\\ &&
\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad\quad
\textrm{for}~ \phi^c{}^\dagger ,
\\
&&\!\!\!\!\!
(3W-\phi_i W_i)(\delta(y))^2 \left(-{1\over R}\right)
(-{1\over 6M_5^3}-{1\over r})(m\phi_3)^\dagger
\nonumber
\\
&& \qquad\qquad
-((2\lambda \phi_1\phi_2+m\phi_3)(2\lambda \phi_2)^\dagger
+\lambda(\phi_1^2-\mu^2)(2\lambda \phi_1)^\dagger
+m\phi_1 m^\dagger)\delta(y)=0
\nonumber
\\ &&
\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad\quad
\textrm{for}~ \phi_1^\dagger,
\\
&& \!\!\!\!\!
(3W-\phi_i W_i)(\delta(y))^2\left(-{1\over R}\right)
(-{1\over 6M_5^3}-{1\over r})(-2\lambda\mu^2)^\dagger
-(2\lambda \phi_1\phi_2+ m\phi_3)(2\lambda \phi_1)^\dagger \delta(y)
=0
\nonumber
\\
&&
\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad\quad
\textrm{for} ~ \phi_2^\dagger ,
\\
&&\!\!\!\!\!
(3W-\phi_i W_i)(\delta(y))^2\left(-{1\over R}\right)
(-{1\over 6M_5^3}-{1\over r})(m\phi_1)^\dagger
-(2\lambda \phi_1\phi_2+m\phi_3)m^\dagger \delta(y)=0
\nonumber
\\
&&
\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad\quad
\textrm{for} ~\phi_3^\dagger .
\end{eqnarray*}
These equations of motion reduce to
\begin{eqnarray*}
&&2\lambda \phi_1\phi_2+m\phi_3=0,
~~~~~\qquad\qquad\qquad
-2\lambda\mu^2\phi_2+m\phi_1\phi_3=0, \\
&&\lambda(\phi_1^2-\mu^2)(2\lambda \phi_1)^\dagger
+m\phi_1 m^\dagger=0,
~~~~~~
\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi=0 ,
\end{eqnarray*}
where $3W-\phi_i W_i=-2\lambda\mu^2\phi_2+m\phi_1\phi_3$.
The first equation above is the same as Eq.(\ref{phi23}).
The second equation gives an additional constraint to
$\phi_i$.
As a result,
there are four equations to determine
the four variables $\phi_i$ and $\phi$.
We find the solution (for $W_c=0$)
\begin{eqnarray}
\phi= \underline{\phi},~~
\phi^c=0 ,~~
\phi_1=\underline{\phi_1},~~
\phi_2=0,~~
\phi_3=0 .
\label{solwo}
\end{eqnarray}
Thus the fields $\phi_i$ are all determined
unlike no hypermultiplet case
in the previous section.
Still $\underline{\phi}$ includes unfixed $N_2$ and $R$.
From the solution (\ref{solwo}) and the Lagrangian (\ref{lag}),
the potential itself is seen to be the same as
in the O'Raifeartaigh model
which is given in Eq.(\ref{po}).
In this potential, the moduli $N_2$ and the radius $R$
are not stabilized.
In other words, even if the two sectors have been mixed,
there still exist moduli for the $W_c=0$ case.
\subsection{Moduli stabilization with $W_c\neq 0$
}\label{sc:w}
Now we study the case with a
nonzero constant superpotential $W_c$.
We assume $|w_0|\sim |w_\pi|\equiv w\ll 1$
and work out perturbative solutions of the equations of motion
for $\phi$, $\phi^c$ and $\phi_i$ similarly to analysis in \cite{Maru:2006id}.
To allow possible discontinuities of the $Z_2$-odd field
$\phi^c$ across the fixed points $y=0$ and $y= \pi$, we define
\begin{equation}
\phi^c(x,y) \equiv \hat \epsilon(y) \chi^c(x,y),
\qquad
\hat \epsilon(y)\equiv
\left\{
\begin{array}{cc}
+1, & 0<y<\pi \\
-1, & -\pi<y<0
\end{array}
\right.
,
\label{eq:odd_field}
\end{equation}
where $\chi^c(x,y)$ is a parity even function with possibly
nonvanishing value at $y=0, \pi$.
Up to ${\cal O}(w)$,
the solution for $\phi$ is found to be $\phi=\underline{\phi}$
which is given in Eq.(\ref{phibar}).
Using this solution $\phi=\underline{\phi}$
and
examining $(\delta(y))^2$ terms
in the equation of motion
for $\phi^\dagger$ derived from the Lagrangian (\ref{auxap}),
we find that
\begin{eqnarray*}
3W-\phi_i W_i \propto w_0 ,
\end{eqnarray*}
which is of order of ${\cal O}(w)$.
As for singular terms,
we use the following
identity valid as a result of a properly regularized
calculation:
\begin{equation}
\delta(y) (\hat \epsilon(y))^2 = \frac{1}{3} \delta(y),
\qquad
\delta(y-\pi) (\hat \epsilon(y))^2 = \frac{1}{3} \delta(y-\pi).
\label{eq:delta_epsilon2}
\end{equation}
This
respects
the relation $2\delta(y)=d\epsilon(y)/dy$.
From the Lagrangian (\ref{auxap}),
the equation of motion for $\phi^\dagger$ is identical to
Eq.(\ref{phieom})
up to the first order of $w$.
The equation of motion for $\phi^c{}^\dagger$ up to ${\cal O}(w)$ is
\begin{eqnarray}
&&
(-\partial_y\phi^c+\left({3\over 2}+c\right)R\sigma'\phi^c)
\left[\left({3\over 2}+c\right)R\sigma'
+{1\over 3M_5^3}\phi\partial_y\phi^\dagger
-{1\over 6M_5^3}\phi^\dagger\phi\left({9\over
2}-c\right)R\sigma'\right]
\nonumber
\\
&&
-e^{4R\sigma}\partial_y\{(-\partial_y\phi^c+\left({3\over 2}+c\right)
R\sigma'\phi^c)e^{-4R\sigma}\left[
-1+{1\over 6M_5^3}\phi^\dagger\phi\right]\}
\nonumber
\\
&&
+(\partial_y\phi -\left({3\over 2}-c\right)R\sigma'\phi)
\left[{1\over 3M_5^3}\phi^c\partial_y\phi^\dagger
-{1\over 6M_5^3}\phi^c\phi^\dagger
\left({9\over 2}-c\right)R\sigma'\right]
\nonumber
\\
&&-e^{4R\sigma}\partial_y\{
(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)e^{-4R\sigma}
{1\over 6M_5^3}\phi^c\phi^\dagger\}
\nonumber
\\
&&
+(3\phi^c(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi))
\left[
-{1\over 3M_5^3}\partial_y\phi^\dagger
+{1\over 6M_5^3}\phi^\dagger\left({9\over 2}-c\right)R\sigma'
\right.
\nonumber
\\
&& \left.
-{3(1-2R\sigma)\over r}\partial_y\phi^\dagger
+{1-2R\sigma\over r}
\phi^\dagger \left({3\over 2}-c\right)R\sigma'\right]
\nonumber
\\
&&-e^{4R\sigma}\partial_y\{
(3\phi^c(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
+3W_c+(3W-\phi_i W_i)\delta(y))e^{-4R\sigma}
\left[-{1\over 6M_5^3}\phi^\dagger\right]\}
\nonumber
\\
&&-(3\sigma\phi^c(\partial_y\phi-\left({3\over 2}-c\right)
R\sigma'\phi)+\phi^c\left({3\over 2}-c\right)R\sigma'\phi)
{1\over r}\left[
6\partial_y\phi^\dagger
-2\phi^\dagger\left({3\over 2}-c\right)R\sigma'\right]
\nonumber
\\
&&
=0
\end{eqnarray}
where $\hat{\epsilon}\delta(y)=0$ is used
and $r=\phi^\dagger\phi-6M_5^3+{\cal O}(w^2)$ should be taken.
The equation of motion for $\phi_1^\dagger$
up to ${\cal O}(w)$ is
\begin{eqnarray}
&&(-\partial_y\phi^c+\left({3\over 2}+c\right)R\sigma'\phi^c)
{\phi \over 6M_5^3}(m\phi_3)^\dagger \delta(y)
+(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
{\phi^c\over 6M_5^3}(m\phi_3)^\dagger \delta(y)
\nonumber
\\
&&
+(3\phi^c(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
+3W_c+(3W-\phi_i W_i)\delta(y))
(-{1\over 6M_5^3}-{1\over r})(m\phi_3)^\dagger\delta(y)
\nonumber
\\
&&
-\phi\phi^c\left({3\over 2}-c\right)R\sigma'\,{1\over r}\,(m\phi_3)^\dagger
\,2\delta(y)
\nonumber
\\
&&-((2\lambda \phi_1\phi_2+m\phi_3)(2\lambda \phi_2)^\dagger
+\lambda(\phi_1^2-\mu^2)(2\lambda \phi_1)^\dagger
+m\phi_1m^\dagger)\delta(y) =0 .
\end{eqnarray}
The equation of motion for $\phi_2^\dagger$ up to ${\cal O}(w)$ is
\begin{eqnarray}
&&(-\partial_y\phi^c+\left({3\over 2}+c\right)R\sigma'\phi^c)
{\phi \over 6M_5^3}(-2\lambda\mu^2)^\dagger \delta(y)
\nonumber
\\
&&
+(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
{\phi^c\over 6M_5^3}(-2\lambda\mu^2)^\dagger \delta(y)
\nonumber
\\
&&
+(3\phi^c(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
+3W_c+(3W-\phi_i W_i)\delta(y))
(-{1\over 6M_5^3}-{1\over r})(-2\lambda\mu^2)^\dagger\delta(y)
\nonumber
\\
&&
-\phi\phi^c\left({3\over 2}-c\right)
R\sigma'\,{1\over r}\,(-2\lambda\mu^2)^\dagger
\,2\delta(y)
-(2\lambda \phi_1\phi_2+m\phi_3)(2\lambda \phi_1)^\dagger\delta(y) =0 .
\end{eqnarray}
The equation of motion for $\phi_3^\dagger$ up to ${\cal O}(w)$ is
\begin{eqnarray}
&&(-\partial_y\phi^c+\left({3\over 2}+c\right)R\sigma'\phi^c)
{\phi \over 6M_5^3}(m\phi_1)^\dagger \delta(y)
+(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
{\phi^c\over 6M_5^3}(m\phi_1)^\dagger \delta(y)
\nonumber
\\
&&
+(3\phi^c(\partial_y\phi-\left({3\over 2}-c\right)R\sigma'\phi)
+3W_c+(3W-\phi_i W_i)\delta(y))
(-{1\over 6M_5^3}-{1\over r})(m\phi_1)^\dagger\delta(y)
\nonumber
\\
&&
-\phi\phi^c\left({3\over 2}-c\right)
R\sigma'\,{1\over r}\,(m\phi_1)^\dagger
\,2\delta(y)
-(2\lambda \phi_1\phi_2+m\phi_3)m^\dagger\delta(y) =0 .
\end{eqnarray}
From the equations of motion for $\phi^\dagger$ and $\phi^c{}^\dagger$,
it is seen that the hypermultiplet has
the same bulk solutions as in the model without
the boundary chiral supermultiplets.
The solutions are given for generic values of the bulk
mass parameter $c$ ($\not=3/2, 1/2$) as \cite{Maru:2006id}
\begin{eqnarray}
\phi&\!\!\!=\!\!\!&\underline{\phi} ,
\label{solchi}
\\
\phi^c&\!\!\!=\!\!\!&
\hat{\epsilon}\,
{(X+1)^{(5/2-c)/(3-2c)}\over X}
\left[c_1 +c_2 (X+1)^{-(1-2c)/(3-2c)}\left(X+{3-2c\over
1-2c}\right)\right]
\label{solchic}
\end{eqnarray}
where $c_1$ and $c_2$ are constants of integration.
We have changed a variable from $y$ to a dimensionless
variable
$X\equiv \underline{\phi}^\dagger\underline{\phi}/(6M_5^3)-1$.
The remaining parts of the equations of motion give boundary conditions.
The $\partial_y\delta(y)$, $\partial_y\delta(y-\pi)$ terms
of the equation of motion for $\phi^c{}^\dagger$ gives rise to
the boundary conditions
\begin{eqnarray}
\chi^c\bigg|_{y=0}
&\!\!\!=\!\!\!&-\left({1\over 2X}\,
\underline{\phi}^\dagger\left(w_0+{3W-\phi_i W_i\over
6M_5^3}\right)\right)\bigg|_{y=0} ,
\label{bc1}
\\
\chi^c\bigg|_{y=\pi}
&\!\!\!=\!\!\!&\left({1\over
2X}\,\underline{\phi}^\dagger
w_\pi\right)\bigg|_{y=\pi} .
\label{bc2}
\end{eqnarray}
From the equations of motion for $\phi_2^\dagger$ and $\phi_3^\dagger$,
the boundary conditions are
\begin{eqnarray}
&&2\lambda \phi_1\phi_2+m\phi_3=0 ,
\label{xym}
\\
&&(-\partial_y\phi^c+\left({3\over 2}+c\right)R\sigma'\phi^c)
\underline{\phi}\delta(y)
+(6M_5^3 w_0+3W-\phi_i W_i)(\delta(y))^2
\left(-{\underline{\phi}^\dagger\underline{\phi}\over r}\right)
\nonumber
\\
&&\quad
-\underline{\phi}\hat{\epsilon}\chi^c
\left({3\over 2}-c\right)R\sigma'{6M_5^3\over r}
2\delta(y)=0 .
\label{yz}
\end{eqnarray}
In Eq.(\ref{yz}), the $(\delta(y))^2$ terms give the same boundary
condition as Eq.(\ref{bc1}).
The other terms lead to
\begin{eqnarray}
\delta(y)\hat{\epsilon}
\left[-\partial_y\chi^c+\left({3\over 2}+c\right)R\sigma'\chi^c
-\chi^c\left({3\over 2}-c\right)R\sigma'
{12M_5^3\over r} \right] =0 .
\label{bc3}
\end{eqnarray}
This equation gives the boundary condition for $\chi^c$ at $y=0$.
Eq.(\ref{bc1}) gives the boundary condition for $(3W-\phi_i W_i)$ rather than
for $\chi^c$ at $y=0$.
The boundary condition for $\chi^c$ at $y=\pi$ is given
by Eq.(\ref{bc2}).
Finally, the equation of motion for $\phi_1^\dagger$ becomes
\begin{eqnarray}
\phi_1=\underline{\phi_1}
\label{bc4}
\end{eqnarray}
subject to the boundary conditions (\ref{xym}) and (\ref{yz}).
The boundary conditions given above are solved in the following.
Firstly we calculate the constants of integration $c_1$ and $c_2$.
Substituting the bulk solution (\ref{solchic})
into the boundary conditions (\ref{bc2}) and (\ref{bc3}) one obtains
\begin{eqnarray*}
&&c_1+c_2 \hat{N}^{(2c-1)/(3-2c)}{1-2c\over 3-2c}=0 ,
\\
&&c_1+c_2(\hat{N}e^{(3-2c)Rk\pi})^{(2c-1)/(3-2c)}
(\hat{N}e^{(3-2c)Rk\pi}-{2\over 3-2c})
={N_2^\dagger e^{(3/2-c)Rk\pi}w_\pi
\over 2(\hat{N}e^{(3-2c)Rk\pi})^{(5/2-c)/(3-2c)}} ,
\end{eqnarray*}
where we defined a dimensionless parameter
$\hat{N}\equiv |N_2|^2/(6M_5^3)$.
These equations are solved as
\begin{eqnarray}
c_1&\!\!\!=\!\!\!&
{(2c-1)N_2^\dagger \hat{N}^{-(5/2-c)/(3-2c)}e^{-Rk\pi}
\over e^{(2c-1)Rk\pi}((3-2c)\hat{N}e^{(3-2c)Rk\pi}-2)
+2c-1}{w_\pi\over 2} ,
\label{c1}
\\
c_2&\!\!\!=\!\!\!&
{(3-2c)N_2^\dagger \hat{N}^{-(3/2+c)/(3-2c)}e^{-Rk\pi}
\over e^{(2c-1)Rk\pi}((3-2c)\hat{N}e^{(3-2c)Rk\pi}-2)
+2c-1}{w_\pi\over 2} .
\label{c2}
\end{eqnarray}
The coefficients $c_1$ and $c_2$ are independent of $w_0$.
Lastly, from Eqs.(\ref{bc1}), (\ref{xym}) and (\ref{bc4}),
$\phi_i$ are solved as
\begin{eqnarray}
\phi_1&\!\!\!=\!\!\!&\underline{\phi_1} ,
\\
\phi_2&\!\!\!=\!\!\!&-{6M_5^3\over 2\lambda (\mu^2+\underline{\phi_1}^2)}
\left({(3-2c)(1-\hat{N})e^{-Rk\pi}w_\pi
\over e^{(2c-1)Rk\pi}((3-2c)\hat{N}e^{(3-2c)Rk\pi}-2)
+2c-1}
-w_0\right) ,
\\
\phi_3&\!\!\!=\!\!\!& {6M_5^3 \underline{\phi_1}\over m (\mu^2+\underline{\phi_1}^2)}
\left({(3-2c)(1-\hat{N})e^{-Rk\pi}w_\pi
\over e^{(2c-1)Rk\pi}((3-2c)\hat{N}e^{(3-2c)Rk\pi}-2)
+2c-1}
-w_0\right) ,
\label{zsol}
\end{eqnarray}
where $\underline{\phi_1}$ is given in Eq.(\ref{xo}).
Obviously, the solutions
(\ref{solchi}),(\ref{solchic})
and (\ref{c1})--(\ref{zsol})
include the result in the previous section where $w_0=w_\pi=0$.
As in the previous section,
$\phi_i$ are determined unambiguously.
Here we would like to stress that
dependence of the above solutions on $w_0,w_\pi$ are
different from that of the case
decoupled to
boundary chiral supermultiplets.
When boundary chiral supermultiplets are absent,
the boundary conditions for $\chi^c$
are given in Eq.(\ref{bc2}) and
Eq.(\ref{bc1}) with zero ($3W-\phi_i W_i$).
At $y=0$ the boundary condition includes $w_0$.
Then $c_1$ and $c_2$ depend on $w_0$ and $w_\pi$.
Even for zero $w_\pi$, there exists a nontrivial solution for $\phi^c$.
On the other hand, when the boundary chiral supermultiplets are coupled,
Eq.(\ref{bc1}) is interpreted as a boundary condition
for $\phi_i$ or more concretely for ($3W-\phi_i W_i$).
The three fields $\phi_i$ are
solved for the three equations
(\ref{bc1}), (\ref{xym}) and (\ref{bc4}).
The boundary conditions for $\chi^c$ are Eqs.(\ref{bc2})
and (\ref{bc3}). They do not
include $w_0$.
Thus the coefficients $c_1$ and $c_2$ are independent of $w_0$.
In obtaining a nontrivial solution for $\phi^c$,
it is required that $w_\pi$ is at least nonzero.
We have
solved the equations of motion.
We can now calculate the potential.
By inserting the solutions into the Lagrangian (\ref{aux}) and
integrating over the extra dimension $y$,
we obtain the potential as a function of the radius $R$
and the complex normalization parameter $N_2$
\begin{eqnarray}
V&\!\!\!=\!\!\!&{k\over 2M_5^3}\int_0^{\pi}
dy \bigg\{
-2c_2^\dagger\hat{N}^{5/2-2c+2/(3-2c)}
e^{((3-2c)(5/2-2c)+2)R\sigma}
\nonumber
\\
&& +\left({3\over 2}+c+(3-2c)
\left(-{5\over 2}+2c-\left[3(\hat{N}
e^{(3-2c)R\sigma}-1)\right]^{-1}\right)\right)
\chi^c{}^\dagger \bigg\}
{\underline{\phi}^\dagger \widetilde{W}\over 2}e^{-4R\sigma}
\nonumber
\\
&& +|\lambda(\underline{\phi_1}^2-\mu^2)|^2+|m\underline{\phi_1}|^2
\end{eqnarray}
where $\widetilde{W}\equiv W_c+(W-\phi_i W_i/3)\delta(y)$
and $c_2$ is given in Eq.(\ref{c2})
for generic values of $c$.
This form of the potential is similar to the
decoupled model
\cite{Maru:2006id}.
Performing integration and
using the boundary conditions (\ref{bc2}) and
(\ref{bc3}) lead to the potential
\begin{eqnarray}
V&=& -N_2^\dagger k
c_2^\dagger\hat{N}^{5/2-2c+2/(3-2c)}
\left(\widetilde{w}_0+w_\pi e^{((3-2c)^2-2)Rk\pi}\right)
\nonumber\\
&&+{|N_2|^2 k\widetilde{w}_0^2\over 4(1-\hat{N})}
\left(-4c^2+12c-6+{3-2c\over 3(1-\hat{N})}\right)
\nonumber\\
&&+{|N_2|^2 k w_\pi^2 e^{-(1+2c)Rk\pi}
\over 4(\hat{N}e^{(3-2c)Rk\pi}-1)}
\left(-4c^2+12c-6-{3-2c\over 3(\hat{N}e^{(3-2c)Rk\pi}-1)}\right)
\nonumber
\\
&& +|\lambda(\underline{\phi_1}^2-\mu^2)|^2+|m\underline{\phi_1}|^2
\end{eqnarray}
where we defined $\widetilde{w}_0\equiv w_0+(3W-\phi_i W_i)/(6M_5^3)$.
Using Eqs.(\ref{c1})--(\ref{zsol}) for
$c_1,c_2,\phi_i$, we find the potential
\begin{eqnarray}
V&\!\!\!=\!\!\!&{(6M_5^3)kw_\pi^2\over 4}
\bigg\{
-{2(3-2c)\hat{N}^{7/2-2c+(1/2-c)/(3-2c)}
e^{((3-2c)^2-3)Rk\pi}
\over e^{(2c-1)Rk\pi}((3-2c)\hat{N}e^{(3-2c)Rk\pi}-2)+2c-1}
\nonumber
\\
&&
+\hat{N}(1-\hat{N})
\left({(3-2c)e^{-Rk\pi}\over
e^{(2c-1)Rk\pi}((3-2c)\hat{N}e^{(3-2c)Rk\pi}-2)+2c-1}\right)^2
\nonumber
\\
&&\quad \times
\left(-4c^2+12c-6+{3-2c\over 3(1-\hat{N})}
-2\hat{N}^{5/2-2c+(1/2-c)/(3-2c)}\right)
\nonumber
\\
&&
+{e^{-(1+2c)Rk\pi}\over \hat{N}e^{(3-2c)Rk\pi}-1}
\left(-4c^2+12c-6-{3-2c\over 3(\hat{N}e^{(3-2c)Rk\pi}-1)}\right)
\bigg\}
\nonumber
\\
&& +|\lambda(\underline{\phi_1}^2-\mu^2)|^2+|m\underline{\phi_1}|^2 .
\label{nowo}
\end{eqnarray}
This potential is independent of $w_0$
as it is seen from the fact that $\widetilde{w}_0$ is proportional to
$w_\pi$ subject to Eq.(\ref{bc1}).
We move on
the stabilization of the radius $R$
and the modulus $N_2$.
For simplicity, we consider the case where $-c\gg 1$ and the
constant $N_2$ is real.
Then the potential becomes
\begin{eqnarray}
\!\!\!\!\!
V\approx
-(6M_5^3)kw_\pi^2 c^2 ~
{(-\hat{N}^2-\hat{N}^{-2c}/(2c)^2)e^{-2Rk\pi}+\hat{N}-1
\over (\hat{N}-1)(\hat{N}-e^{2cRk\pi})}
+|\lambda(\underline{\phi_1}^2-\mu^2)|^2+|m\underline{\phi_1}|^2 .
\label{vw}
\end{eqnarray}
We need to require the stationary condition for both modes
$R$ and $N_2$
\begin{eqnarray}
{\partial V\over \partial R}=0
\textrm{~~and~~}
{\partial V\over \partial \hat{N}}=0 .
\end{eqnarray}
The former condition $\partial V/\partial R=0$ leads to
\begin{eqnarray}
e^{-2Rk\pi}\approx {(\hat{N}-1)^2\over \hat{N}^2}
\label{vv1}
\end{eqnarray}
whereas the latter condition gives
\begin{eqnarray}
0\approx \left(\hat{N}^3+{\hat{N}^{-2c}\over 2c^2}\right)
{(\hat{N}-1)^2\over \hat{N}^2}
+c(-\hat{N}^2+\hat{N}-1)e^{2cRk\pi} .
\label{vv2}
\end{eqnarray}
From Eqs.(\ref{vv1}) and (\ref{vv2}),
we find that the stationary condition is satisfied for infinite
radius
and that the modulus $N_2$ is stabilized at
\begin{eqnarray}
N_2\approx \sqrt{6M_5^3} .
\end{eqnarray}
It is important to notice that
this stabilization originates from the terms proportional to $w_\pi^2$
in the potential (\ref{vw}).
Only when
a constant superpotential at $y=\pi$ is nonzero,
the modulus of the hypermultiplet is stabilized.
At the stationary point, the potential is
\begin{eqnarray}
V\approx
-(6M_5^3)kw_\pi^2 c^2
+|\lambda(\underline{\phi_1}^2-\mu^2)|^2+|m\underline{\phi_1}|^2
\end{eqnarray}
which can be zero
dependently on the parameters.
\section{Conclusion}\label{sc:conclusion}
We have studied supersymmetry breaking in
a warped space model with constant boundary superpotentials,
hypermultiplet, boundary chiral supermultiplets,
compensator and radion multiplet.
We have presented the classical background solution
and have shown that all of the fields are determined unambiguously.
Dependence of the potential on $w_0$ and $w_\pi$
is significantly different from that of the potential in
the model without the mixing between bulk and brane
field equations \cite{Maru:2006id}.
The potential (\ref{nowo}) is independent of $w_0$.
In the situation we have considered where
the boundary chiral supermultiplets are only at $y=0$,
the constant superpotential at $y=\pi$ is required to be nonzero
to stabilize
the modulus of the hypermultiplet.
For large negative $c$,
we have shown that the modulus of the hypermultiplet
is stabilized at a finite value and that the radius is infinite.
It would be worth mentioning that
large $|c|$ is closely related to
flat space limit.
The bulk mass parameter $c$ should have large magnitude in
order to take a proper flat space limit $k\to 0$ as
seen from the Lagrangian (\ref{lag}).
In Ref.\cite{Maru:2006id}, we found that
there is
a similarity of hypermultiplet mass spectrum
between flat space case and $k\to 0$ limit of warped space case
with fixed $ck$.
Infinite radius that we have obtained for large negative $c$
might be analogous to disappearance of the potential over $R$
in flat space case.
The radius stabilization has been studied also in the
AdS$_4$ background where Scherk-Schwarz supersymmetry
breaking is formulated.
In models with nonzero superpotentials
\cite{Katz:2005wp}\cite{Katz:2006mv},
it has been found that hypermultiplets give positive
contributions to the radion potential, contrary to the
negative contributions from the gravity multiplet.
This provides various patterns of radion potential.
It would be interesting to study such a model with
mixing of equations of motion for bulk and brane fields.
\subsubsection*{Acknowledgments}
I thank Masud Chaichian, Nobuhito Maru and Norisuke Sakai
for useful discussions.
This work is supported by Bilateral exchange program between
Japan Society for the Promotion of Science and the Academy of Finland.
\vspace*{10mm}
|
1,116,691,500,473 | arxiv | \section{Classical Lagrange and Wolfe \\dual programs}
In this Section, we re-discuss the well-known programs with
holonomic constraints insisting on the following issues \cite{B}, \cite{H}, \cite{R}:
(i) the Lagrange-dual problem with weak respectively strong duality;
(ii) the Wolfe-dual problem; (iii) pertinent examples.
\subsection{The Lagrange dual problem}
Let $D$ be a domain in $R^n$, let $x=(x^1,...,x^n)$ be a point in D and $f:D\to R$ and $g_\alpha: D\to R$ be convex functions.
Denote $g=(g_\alpha)$ and we introduce the set
$$\Omega = \{x\in D \,|\,g_\alpha(x)\leq 0,\,\alpha=1,...,m\}= \{x\in D \,|\,g(x)\preceq 0\}.$$
A complete notation for this set is $(\Omega, g, \preceq)$, but for short the sign $\preceq$
or the pair $(g,\preceq)$ are suppressed in the notation.
Let us consider the {\it convex program}
$$\min_x\{f(x)\,|\, x \in \Omega\}.\leqno(P)$$
The {\it Lagrange function (or Lagrangian)} of (P)
$$L(x, \lambda) = f(x)+ \sum_{\alpha=1}^m \lambda_\alpha g_\alpha(x)
= f(x)+<\lambda,g>, x\in D, \lambda\succeq 0$$
is convex in $x$ and linear in $\lambda$.
{\bf Remark} {\it We can create an original Riemannian geometry on the set of critical points,
using similar ideas we shall develop in a further 3.2.1}.
For all $\lambda\succeq 0$, the inequality
$$\sum_{\alpha =1}^m\lambda_\alpha g_\alpha(x)\leq 0, \,\,\forall x \in \Omega$$
holds. Consequently
$$L(x,\lambda)\leq f(x), \,\,\forall x \in \Omega,\,\, \forall \lambda\succeq 0.\leqno(1)$$
The equality holds iff (complementarity conditions)
$$\lambda_\alpha g_\alpha(x)=0 \,(\hbox{for each}\,\, \alpha =1,...,m).$$
Let us introduce the {\it Lagrange dual function}
$$\psi(\lambda)=\inf_{x\in D}\{f(x)+ \sum_{\alpha =1}^m \lambda_\alpha g_\alpha(x),\,\, x\in D,\,\, \lambda\succeq 0\}.$$
This function $\psi(\lambda)$ is concave, because it is a point-wise infimum of affine functions. Indeed,
using the linearity of $L(x,\lambda)$ with respect to $\lambda$, and introducing $\lambda^1\succeq 0$, $\lambda^2\succeq 0$,
and $0\leq t\leq 1$, we have
$$\psi(t\lambda^1 +(1-t)\lambda^2)=\inf_{x\in D}L(x,t\lambda^1 +(1-t)\lambda^2)$$
$$=\inf_{x\in D}(tL(x,\lambda^1) +(1-t)L(x,\lambda^2))\geq \inf_{x\in D}(tL(x,\lambda^1)) + \inf_{x\in D}((1-t)L(x,\lambda^2))$$
$$= t\inf_{x\in D}L(x,\lambda^1) + (1-t)\inf_{x\in D}L(x,\lambda^2)=t\psi(\lambda^1)+ (1-t)\psi(\lambda^2).$$
\begin{definition}
The problem
$$\sup_\lambda \,\{\psi(\lambda)\,|\, \lambda\succeq 0\}$$
is the so-called {\it Lagrange dual problem} of (P).
\end{definition}
The Lagrange dual problem can be called convex
because it is equivalent to the convex problem
$$\inf_\lambda\,\{-\psi(\lambda)\,|\, \lambda\succeq 0\}.$$
The Lagrange-dual problem is also defined
in this way if (P) is not convex. The following theorem holds also in that case.
\begin{theorem} ({\bf weak duality}) The dual function yields lower bounds of the initial optimal value $f_*$, i.e., for any
$\lambda$, we have $\psi(\lambda)\leq f_*$. In other words,
$$\sup_\lambda\,\{\psi(\lambda)\,|\, \lambda\succeq 0\}\leq \inf_{x\in \Omega}\{f(x)\}.$$
\end{theorem}
{\bf Proof} In the foregoing statements, we have the relation (1).
Since $\Omega\subset D$, for each $\lambda\succeq 0$, we find
$$\psi(\lambda)=\inf_{x\in D}L(x,\lambda)\leq \inf_{x\in \Omega}L(x,\lambda)\leq \inf_{x\in \Omega}f(x).$$
Thus the statement in the theorem is true.
The problem of finding the best lower bound on $f_*$ obtained from the Lagrange dual function is called the {\it
Lagrange dual problem} for the original or primal problem.
The optimal values may be different. However, they are equal if (P) satisfies the Slater
condition and has finite optimal value. This is the next result.
\begin{theorem} {\bf (strong duality)} If the program (P) satisfies the Slater condition and has finite optimal
value, then
$$\sup_\lambda\, \{\psi(\lambda)\,|\, \lambda\succeq 0\}= \inf_{x\in D}\{f(x)\,|\,\,g(x)\preceq 0\}.$$
Moreover, then the dual optimal value is attained.
\end{theorem}
{\bf Proof} Denote by $f_*$ the optimal value of (P). Taking $a = f_*$ in the Convex Farkas Lemma,
it follows that there exists a vector $\lambda_*=(\lambda_{1*},...,\lambda_{m*})\geq 0$ such that
$$L(x, \lambda_*) = f(x)+ \sum_{\alpha =1}^m \lambda_{\alpha *}\, g_\alpha(x)\geq f_*,\, \forall x \in D.$$
Using the definition of $\psi(\lambda_*)$ this implies $\psi(\lambda_*)\geq f_*$. By the weak duality theorem, it
follows that $\psi(\lambda_*)= f_*$. This not only proves that the optimal values are equal, but also
that $\lambda_*$ is an optimal solution of the dual problem.
{\bf Remark} Unlike in Linear Programming theory, the strong duality theorem cannot always be established for
general optimization problems.
\subsection{Topology of Lagrange multipliers}
Let $f:D\subseteq R^{n}\rightarrow R$ and $g:D\subseteq R^{n}\rightarrow
R^{p}$, $p<n,$ of class $C^{2}$ with $rank\,J_g = p$ in $D$. Let
$L\left( x,\lambda \right) =f\left( x\right) +\lambda \cdot g\left( x\right),$ with $%
\lambda \in R^{p}.$ We recall that
\begin{equation*}
H\left( x,\lambda \right)=\nabla f\left( x\right) +\lambda\cdot \nabla g\left( x\right) =0
\end{equation*}%
is the equation of critical points with respect to $x$ of the Lagrange function.
Let $A=\left\{ x\ |\ \exists \lambda \text{ cu }H\left( x,\lambda \right)
=0\right\} $ and $B=\left\{ \lambda \ |\ \exists x\text{ cu }H\left(
x,\lambda \right) =0\right\} .$ Introduce $h:A\rightarrow B$ such that
$H\left( x,h\left( x\right) \right) =0.$ The function $h$ is well defined
since the equation $H\left( x,\lambda \right) =0$ is linear in $\lambda$ (system with unique solution).
Hence, for any $\lambda \in B$, the set $h^{-1}\left( \lambda \right)
$ is non-void, and it consists of all critical points corresponding to $\lambda$
(set in which the nondegenerate critical points are isolated).
\begin{proposition}
Let $\lambda _{0}\in B$ such that there exists $x_{0}\in
h^{-1}\left( \lambda _{0}\right) $ with the property that $x_{0}$ is
nondegenerate, i.e., the Hessian $d^{2}f\left( x_{0}\right) +\lambda _{0}\cdot d^{2}g\left(
x_{0}\right) $ is nondegenerate. Then $h$ admits a differentiable section $s_{\lambda _{0}}:I_{\lambda _{0}}\rightarrow A.$
\end{proposition}
\begin{proof}
Since $\frac{\partial H}{\partial x}\left( x_{0},\lambda _{0}\right)
=d^{2}f\left( x_{0}\right) +\lambda _{0}\cdot d^{2}g\left( x_{0}\right)$ is non-degenerate, by hypothesis,
there exists a neighborhood $I_{\lambda _{0}}$ of $%
\lambda _{0}$ and a differentiable function $s_{\lambda
_{0}}:I_{\lambda _{0}}\rightarrow A$ such that $H\left(
s_{\lambda _{0}}\left( \lambda \right) ,\lambda \right) =0,$ $\forall
\lambda \in I_{\lambda _{0}}$ and $s_{\lambda _{0}}\left( \lambda
_{0}\right) =\lambda _{0}.$ Moreover, the function $s_{\lambda _{0}}$ is unique,
with these properties.
\end{proof}
For any $\lambda \in B$, let $S_{\lambda }$ be the set of all sections of $h$ defined
in a neighborhood of $\lambda,$ set which is eventually void.
\begin{remark}
(i) If $h^{-1}\left( \lambda \right) $ contains at least one nondgenerate critical point,
then $S_{\lambda }$ is non-void. If $%
h^{-1}\left( \lambda \right) $ does not contain degenerate critical points,
then the sets $h^{-1}\left( \lambda \right) $ and $S_{\lambda }$
have the same cardinal and are discrete sets.
(ii) The set $C=\left\{ \lambda \in B\ |\text{ }S_{\lambda }\neq
\emptyset \right\} $ is open.
\end{remark}
In the following, we suppose that the set $S_{\lambda }$
is finite, for any $\lambda \in B.$ We can define $f^{\ast
}:B\rightarrow R$ by $f^{\ast }\left( \lambda \right) =\max\nolimits_{s\in
S_{\lambda }}f\left( s\left( \lambda \right) \right) ,$ if $S_{\lambda
}\neq \emptyset $ and $f^{\ast }\left( \lambda \right) =-\infty ,$ if $S_{\lambda }=\emptyset .$
\begin{proposition}
(i) For any $\lambda \in B$, we have
\begin{equation*}
f^{\ast }\left( \lambda \right) \leq \sup\nolimits_{x\in h^{-1}\left(
\lambda \right) }f\left( x\right) .
\end{equation*}
(ii) If $h^{-1}\left( \lambda \right) $ does not contain degenerate critical points, then
\begin{equation*}
f^{\ast }\left( \lambda \right) =\sup\nolimits_{x\in h^{-1}\left( \lambda
\right) }f\left( x\right) =\max\nolimits_{x\in h^{-1}\left( \lambda \right)
}f\left( x\right) .
\end{equation*}
\end{proposition}
\begin{proof}
(i) Let $s_{0}\in S_{\lambda }$ cu $f\left( s_{0}\left( \lambda \right)
\right) =\max\nolimits_{s\in S_{\lambda }}f\left( s\left( \lambda \right)
\right) =f^{\ast }\left( \lambda \right) .$ Since $s_{0}\left( \lambda
\right) \in h^{-1}\left( \lambda \right) ,$ it follows that $f^{\ast
}\left( \lambda \right) \leq \sup\nolimits_{x\in h^{-1}\left( \lambda
\right) }f\left( x\right) .$
(2) By hypothesis, the sets
$S_{\lambda }$ \c{s}i $h^{-1}\left( \lambda \right) $ have the same cardinal, hence
$h^{-1}\left( \lambda \right) $ is finite. Let $y_{0}\in h^{-1}\left( \lambda \right) $ with $f\left( y_{0}\right) =\max_{x\in
h^{-1}\left( \lambda \right) }f\left( x\right) .$ Since $\left(y_{0},\lambda \right) $ is a nondegenerate critical point, there exists $%
s_{1}\in S_{\lambda }$ with $s_{1}\left( \lambda \right) =y_{0}.$ Then, it follows
that $$\max\limits_{s\in S_{\lambda }}f\left( s\left( \lambda
\right) \right) \geq f\left( y_{0}\right) =\max_{x\in h^{-1}}\left( \lambda
\right)f\left( x\right) .$$
\end{proof}
\begin{proposition}
Let $\lambda _{0}\in B$ such that $h^{-1}\left( \lambda
_{0}\right) $ does not contain degenerate critical points. Suppose, also,
that $\ f|_{h^{-1}\left( \lambda _{0}\right) }$ is injective. Then there exists $s_{0}\in S_{\lambda _{0}},$ $s_{0}:I_{0}%
\rightarrow A$ such that $f^{\ast }\left( \lambda \right)
=f\left( s_{0}\left( \lambda \right) \right) ,$ for any $\lambda \in
I_{0}.$
\end{proposition}
\begin{proof}
Let $s_{0}\in S_{\lambda _{0}},$ $s_{0}:I_{0}\rightarrow A$ such that
$f^{\ast }\left( \lambda _{0}\right) =f\left( s_{0}\left( \lambda
_{0}\right) \right).$ Then $f^{\ast }\left( \lambda _{0}\right) =f\left(
s_{0}\left( \lambda _{0}\right) \right) >f\left( s\left( \lambda _{0}\right)
\right) ,$ $\forall s\in S_{\lambda _{0}},\ s:I_{s}\rightarrow A.$ Since
$f$ is continuous and the set $S_{\lambda _{0}}$ is finite,
it follows that we can restrict the neighborhood $I_{0}$ such that
$f\left( s_{0}\left( \lambda \right) \right) >f\left( s\left(
\lambda \right) \right) ,$ $\forall \lambda \in I_{0},\forall s\in
S_{\lambda _{0}},$ i.e., $f^{\ast }\left( \lambda \right) =f\left(
s_{0}\left( \lambda \right) \right) ,\forall \lambda \in S_{\lambda _{0}}.$
\end{proof}
\subsection {The meaning of Lagrange multiplier}
In our mostly geometrical discussion, $\lambda$ is just an artificial variable that lets us compare the
directions of the gradients without worrying about their magnitudes. To express mathematically the meaning of the multiplier,
write the constraint in the form $g(x) = c$ for some constant $c$.
This is mathematically equivalent to our usual $g(x)=0$, but allows us to easily describe a whole family of constraints.
For any given value of $c$, we can use Lagrange multipliers to find the optimal value of $f(x)$ and the point where it occurs.
Call that optimal value $f_*$, occurring at coordinates $x_0$ and with Lagrange multiplier $\lambda _0$.
The answers we get will all depend on what value we used for $c$ in the constraint, so we can think of these as functions of $c$ :
$ f_*(c), x_0(c), \lambda_0(c)$ .
Of course, $ f(x)$ only depends on $c$ because the optimal coordinates $x_0$ depend on $c$: we could write it as $f_*(c)$.
To find how the optimal value changes when we change the constraint, just take the derivative
$$\frac {df_*}{dc} = \frac {\partial f_*}{\partial x^i_0}\, \frac {dx^i_0}{dc} = \nabla f_* \,\cdot \, \frac {dx_0}{dc}.$$
Use the equation of critical points to substitute $\nabla f_* = - \lambda_0 \, \nabla g_0$ and obtain
$$\frac {df_*}{dc} = - \lambda_0 \, \nabla g_0 \,\cdot \, \frac {dx_0}{dc} = - \lambda_0 \,\frac {dg_0}{dc}\,.$$
But the constraint function $g_0 = g(x_0(c))$ is {\it always} equal to $c$, so $dg_0/dc = 1$. Thus, $df_*/dc = - \lambda _0$.
That is, the Lagrange multiplier is the {\it rate of change of the optimal value with respect to changes in the constraint}.
Of course, $f_*$ depends on $c$ through of $\lambda$, and then ${\displaystyle \frac {df_*}{dc} = \frac {df_*}{d \lambda}\,
\frac {d\lambda}{dc} }$. We can define $c(\lambda ) $ by Cauchy problem
$$ \frac {dc}{d \lambda } = - \frac {1}{\lambda }\, \frac {df_*}{d \lambda}\,,\,\,\,\,\,\, c(\lambda _0) = 0\,. \leqno (EC)$$
Then another Lagrange dual function may be
$$ \varphi (\lambda ) = f_*( x_0(\lambda ) ) + \lambda \, c(\lambda )\,. \leqno (LDF)$$
{\bf Proposition} {\it If optimum points are critical points, both Lagrange dual functions give the same solution. Hence strong duality holds.}
{\bf Proof} Indeed, using (EC) we have
$$ \varphi ^\prime (\lambda ) = \frac {df_*}{d \lambda} + c(\lambda ) + \lambda \, \frac {dc}{d \lambda } = c(\lambda )\, $$
and $ \varphi ^\prime (\lambda ) = 0$ implies $ c(\lambda ) =0$, that is for $\lambda _0$.
Often the Lagrange multiplier have
an interpretation as some quantity of interest:
(i) $\lambda$ is the rate of change of the quantity being optimized as a function of the constraint variable
since $\frac{\partial L}{\partial c}=\lambda$;
(ii) by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation
as the marginal effect of the corresponding constraint constant upon the optimal attainable
value of the original objective function: if we denote values at the optimum with an asterisk, then it can be shown that
$$ \frac{d}{dc}\, f_*= \frac{d}{dc} f(x(c)) = \lambda_*.$$
For details regarding classical theory of programs see \cite{B}, \cite{H}, \cite{R}.
If we have more constraints $g_\alpha (x) = c_\alpha \,, \alpha = 1, ..., m$, then the Lagrange function is $L(x, \lambda) = f(x) +
\sum_{\alpha=1}^m \lambda _\alpha (g_\alpha (x) -c_\alpha)$ and the system of critical points is
$$ \frac {\partial f}{\partial x^i} + \sum_{\alpha=1}^m \lambda_\alpha \,\frac {\partial g_\alpha}{\partial x^i} = 0\,.$$
Because the optimal coordinates $x_0$ and the optimal value $f_*$ depend on vector $c$, taking the derivatives we have
$$\frac { \partial f_*}{\partial c_\alpha} = \frac {\partial f_*}{\partial x^i_0}\, \frac {\partial x^i_0}{\partial c_\alpha} =
- \sum_{\beta=1}^m \lambda_\beta^0 \, \frac {\partial g_\beta}{\partial x^i_0}\, \frac {\partial x^i_0}{\partial c_\alpha} =
- \sum_{\beta=1}^m \lambda_\beta^0\, \frac {\partial g_\beta}{\partial c_\alpha} = - \sum_{\beta=1}^m \lambda_\beta^0\, \delta_{\alpha\beta} = - \lambda_\alpha^0\,.$$
Then we can define $c(\lambda ) $ by the partial differential system, with initial condition, written in matrix language as
$$[\lambda_1 \,\,...\,\, \lambda_m]\,
\left[\begin{array}{ccc}\frac{\partial c_1}{\partial \lambda_1}&...&\frac{\partial c_1}{\partial \lambda_m}\\ \
...&...&...\\ \
\frac{\partial c_{m}}{\partial \lambda_1}&...&\frac{\partial c_{m}}{\partial \lambda_m}
\end{array}\right] =
- \left [ \frac {\partial f_*}{\partial \lambda_1}\,\, ...\,\, \frac {\partial f_*}{\partial \lambda_m} \right ] \,,\,\,\,\,\,\, c(\lambda _0) = 0\,. \leqno (EC)$$
Then another Lagrange dual function may be
$$ \varphi (\lambda ) = f_*( x_0(\lambda ) ) + \sum_{\alpha=1}^m \lambda_\alpha\, c_\alpha (\lambda )\,. \leqno (LDF)$$
Deriving the $\beta$-th equation with respect to $\lambda_\alpha$ and the $\alpha$-th equation with respect to $\lambda_\beta$ , in the previous
system, we obtain the complete integrability conditions ${\displaystyle \frac {\partial c_\alpha}{\partial \lambda_\beta} =
\frac {\partial c_\beta}{\partial \lambda_\alpha}}\,$ (symmetric Jacobian matrix); consequently $c(\lambda )$ is
the gradient of a scalar function, namely the Lagrange dual function
$\varphi (\lambda )$.
Moreover, the previous square matrix being a symmetrical one, we can write the equation $(EC)$ as
$$[\lambda ^1 \,\,...\,\, \lambda ^m]\,
\left[\begin{array}{ccc}\displaystyle\frac{\partial c_1}{\partial \lambda^1}&...&\displaystyle\frac{\partial c_m}{\partial \lambda^1}\\ \
...&...&...\\ \
\displaystyle\frac{\partial c_1}{\partial \lambda^m}&...&\displaystyle\frac{\partial c_{m}}{\partial \lambda^m}
\end{array}\right] =
- \left [ \frac {\partial f_*}{\partial \lambda ^1}\,\, ...\,\, \frac {\partial f_*}{\partial \lambda ^m} \right ] \,.$$
Consequently, in a regular case, we have the following situation: Solving a constrained optimum problem we obtain the optimal value as
$f_* = f(c_1, ... , c_m)$. For the dual problem we use a $f_* = f( \lambda^1, ... , \lambda^m)$. If the correspondence between $(c_1, ... , c_m)$ and
$( \lambda^1, ... , \lambda^m)$ is like a change of variables there hold the relations:
$$grad_c\, f_* = - \lambda\,; \,\,\,\,\, c ^\prime (\lambda )\, \lambda = - grad_\lambda \, f_* \,,\,\, c ^\prime (\lambda ) \in L(R^m,\, R^m)\,.$$
\subsection{Examples and counter-examples}
(1) Let us consider the functions $f(x,y)=x^2+y^2$ and $g(x,y)=x^2+y^2-2x$ and the problem
$$\min f(x,y) \,\,\hbox{constrained by}\,\, g(x,y)=c, \,c\geq -1.$$
The Lagrange function of this problem is
$$L(x,y,\lambda)= x^2+y^2+\lambda (x^2+y^2-2x - c).$$
The critical points of the partial function $(x,y)\to L(x,y,\lambda)$ are the solutions of the system
$$\frac{1}{2}\frac{\partial L}{\partial x}= x+\lambda x -\lambda=0,\,\frac{1}{2}\frac{\partial L}{\partial y}= y+\lambda y=0.$$
It follows $x=\frac{\lambda}{\lambda+1},\,y=0$ and hence $f_* = \left(\frac{\lambda}{\lambda +1}\right)^2$.
On the other hand, by restriction, in critical points, we have the relation
$$\left(\frac{\lambda}{\lambda +1}\right)^2 - 2 \frac{\lambda}{\lambda +1}=c.$$
It follows
$$\frac{df_*}{dc}= \frac{df_*}{d\lambda}\,\,\frac{1}{\frac{dc}{d\lambda}}$$
and finally, we obtain the geometrical interpretation $\frac{df_*}{dc}= -\lambda$.
The dual function is
$$\psi(\lambda)= L(x(\lambda),y(\lambda),\lambda)= - \frac{\lambda^2}{\lambda+1}-\lambda c.$$
The value $\psi(\lambda)$ is a minimum for $\lambda >-1$ and a maximum for $\lambda<-1$, in the initial problem.
The condition of extremum (critical point), $\psi^\prime(\lambda)=0$, is equivalent to
$(c+1)(\lambda+1)^2=1$ and the dual problem has the same solution as the primal one.
\begin{figure}
\centering
\includegraphics[width={10cm}]{1.jpg}\\
\caption{Geometry of Lagrange duality}\label{Fig.1}
\end{figure}
On the other hand the equation $(EC)$ for this problem is
$$ \frac {d\bf{c}}{d \lambda } = - \frac {1}{\lambda }\, \frac {d}{d \lambda}\,\left (\frac {\lambda}{\lambda + 1} \right )^2,\,\,\,\,\,\,
{\bf c} (\lambda _0) = 0\,,\,\, \hbox {where}\,\, \lambda _0 +1 = \pm \frac {1}{ \sqrt {c + 1}}\,.$$
We find $\displaystyle {\bf c}(\lambda ) = \frac {1}{(\lambda + 1)^2} - 1 - c $ and the dual Lagrange function
$$\varphi (\lambda ) = \left ( \frac {\lambda }{\lambda + 1} \right )^2 + \lambda \,\left (\frac {1}{(\lambda + 1)^2} - 1 - c \right ) = \psi(\lambda) $$
as the above one.
(2) {\bf A problem with two constraints} Solve the following constrained optimum problem: $$f(x,y,z) = xyz = extremum $$
{\it constrained by}
$$g_1(x,y,z)=x+y -a=0,\,\,g_2(x,y,z)=xz+yz-b=0\,.$$
The Lagrange function is
$$L(x,y,z, \lambda , \mu ) = xyz + \lambda (x+y-a) + \mu (xz+yz-b) $$
and the feasible solution of the problem is, only,
$$x=y=\frac {a}{2}= -2 \mu \,,\, z=\frac {b}{a}= \frac {\lambda}{\mu}\,,\, \lambda = - \frac {b}{4}\,,\, \mu = - \frac {a}{4}\,, $$
$$f_* = \frac {ab}{4} = 4 \lambda \mu \,.$$
The Lagrange dual function is $\psi(\lambda , \mu) = -4 \lambda \mu -a \lambda - b \mu \,.$
The partial differential system which defines $c_1(\lambda , \mu )$ and $c_2(\lambda , \mu )$ becomes, in this case,
$$\left [\lambda \,\,\,\, \mu \right ]\,
\left[\begin{array}{cc} \displaystyle {\frac{\partial c_1}{\partial \lambda}}& \displaystyle { \frac{\partial c_1}{\partial \mu}}\\
\displaystyle {\frac{\partial c_{2}}{\partial \lambda}} & \displaystyle {\frac{\partial c_{2}}{\partial \mu }}
\end{array}\right] = - [ 4 \mu \,\,\,\, 4 \lambda ] \,,\,\,\, c_1\left (- \frac {b}{4}, - \frac {a}{4} \right ) =
c_2\left (- \frac {b}{4}, - \frac {a}{4} \right ) = 0\,. \leqno (EC)$$
Taking into account that $\displaystyle \frac{\partial c_1}{\partial \mu} = \frac{\partial c_{2}}{\partial \lambda} $,
we obtain two quasilinear PDEs
$$\lambda \frac{\partial c_1}{\partial \lambda } + \mu \frac{\partial c_1}{\partial \mu} = -4 \mu,\,\,
\lambda \frac{\partial c_2}{\partial \lambda } + \mu \frac{\partial c_2}{\partial \mu} = -4 \lambda \,, $$
with solutions, respectively
$$c_1(\lambda, \mu) = -4 \mu + \alpha \left (\frac {\lambda}{\mu} \right ), \, \, c_2(\lambda, \mu) = -4 \lambda + \beta \left (\frac {\lambda}{\mu} \right ),\,$$
$\alpha \,, \beta $ arbitrary functions.
The condition $\displaystyle \frac{\partial c_1}{\partial \mu} = \frac{\partial c_{2}}{\partial \lambda} $
is verified, for instance, if $\alpha $ and $\beta $ are constant functions.
Using the initial conditions, we find finally
$$c_1(\lambda, \mu) = -4 \mu - a \,, \, c_2(\lambda, \mu) = -4 \lambda - b\,,$$
$$\varphi (\lambda , \mu ) = 4 \lambda \mu + \lambda (-4 \mu - a) + \mu (-4 \lambda - b) = \psi(\lambda , \mu)\,. $$
The Geometry of Lagrange duality is represented in Figure 1.
(3) {\bf A strange problem} Solve the following constrained optimum problem: $$f(x,y,z) = xyz = extremum $$
{\it constrained by}
$$g_1(x,y,z)=x+y+z-a=0\,,$$ $$g_2(x,y,z)=xy+xz+yz-b=0\,.$$
So the Lagrange function is
$$L(x,y,z, \lambda , \mu ) = xyz + \lambda (x+y+z-a) + \mu (xy+xz+yz-b) $$
and one from the solutions of the problem is, for instance,
$$\mu = \frac {-a- \sqrt {a^2-3b}}{3}\,,\,\,\, \lambda = \frac {2a^2-3b+2a \sqrt {a^2-3b}}{9} = \mu ^2\,,$$
$$x=y= \frac {a+ \sqrt {a^2-3b}}{3} = - \mu \,,\,\,\,\, z= \frac {a-2 \sqrt {a^2-3b}}{3} = a + 2 \mu \,,$$
with the extremum value
$$f_*= \frac {1}{27} \left ( -2a^3 + 9ab - 2(a^2 -3b)^{3/2} \right ) = \lambda (a + 2 \mu )\,,$$
only if $a^2-3b \geq 0$.
{\bf Remark} Another solution of the problem is $$\mu = \frac {-a + \sqrt {a^2-3b}}{3}\,, ... \,\hbox {and so on} $$
with the extremum value
$$f^*= \frac {1}{27} \left ( -2a^3 + 9ab + 2(a^2 -3b)^{3/2} \right ) = \lambda (a + 2 \mu )\,.$$
The interval $[f_*\,, f^*]$ solves the following algebraic problem: {\it Find the real numbers $m$ such that the equation
$t^3 - at^2 + bt - m = 0, \,\, a, b \in R$, has tree real roots.}
It is easily to verify that
$$\frac {\partial }{\partial a} f_*(a,b) = - \lambda \,\, \hbox {and} \,\, \frac {\partial }{\partial b} f_*(a,b) = - \mu \,.$$
On the other hand, $\displaystyle \frac {\partial (\lambda , \mu )}{\partial (a, b)} = 0 $ and $f_*$ cannot be expressed as function
of $\lambda $ and $\mu $ only. Then
we have to consider $f_* = f_*(a,b, \lambda (a,b), \mu (a,b))$ and the following relations
$$ - \lambda = D_a f_* = \frac {\partial f_*}{\partial a} + \frac {\partial f_*}{\partial \lambda} \frac {\partial \lambda}{\partial a} +
\frac {\partial f_*}{\partial \mu} \frac {\partial \mu}{\partial a}$$
$$ - \mu = D_b f_* = \frac {\partial f_*}{\partial b} + \frac {\partial f_*}{\partial \lambda} \frac {\partial \lambda}{\partial b} +
\frac {\partial f_*}{\partial \mu} \frac {\partial \mu}{\partial b}$$
which is easily to verify also (here $D_.$ is an operator of total derivative.)
{\bf Question } Which is the dual Lagrange function $\psi (\lambda, \mu )$ in this case?
Solving the system of the critical points with respect to $x, y$ and $z$ we find, for instance, $x=y= -\mu $, $z$ undeterminate and
$\lambda = \mu ^2$. With these, one obtains the dual Lagrange function
$$\psi (\lambda, \mu )=\chi(\mu) = - \mu ^3 - a \mu ^2 - b \mu \,.$$
{\bf Remark} Although $z$ is undeterminate, the dual Lagrange function does not depend upon $z$, because with the above solutions
the derivative $\displaystyle {\frac {\partial L}{\partial z}}$ vanishes identically. The critical points condition for the dual Lagrange function
$$\frac {d \chi }{d \mu } = - (3 \mu ^2 + 2a \mu + b) = 0$$
gives us the same solutions as in primal problem.
{\bf Open problem} How it means and how we find the functions $c_1$ and $c_2$ in the situation, like this, when
$\displaystyle \frac {\partial (\lambda , \mu )}{\partial (a, b)} = 0 $ ?
(4) Let us consider the functions $f(x,y)=x^2+y^2$ and $g(x,y)=x + y$, with $(x,y)\in R^2$, and the problem
$$\min f(x,y) \,\,\hbox{constrained by}\,\, g(x,y)\geq 1.$$
The Lagrange function is
$$L(x,y,\lambda)= x^2+y^2+\lambda (1- x - y),\,\,(x,y)\in R^2,\, \lambda \geq 0.$$
The function $(x,y)\to L(x,y,\lambda)$ is convex. Consequently, it is minimal iff
$$\frac{\partial L}{\partial x}=0, \frac{\partial L}{\partial y}=0.$$
This holds if $x = \frac{\lambda}{2},\,y = \frac{\lambda}{2}$.
Substitution gives
$$\psi(\lambda)=\lambda - \frac{\lambda^2}{2},\, \lambda \geq 0.$$
The dual problem $\max\psi(\lambda)$ has the optimal point $\lambda =1$.
Consequently, $x = y = \frac{1}{2}$ is the optimal solution of the original (primal)
problem. In both cases the optimal value equals $\frac{1}{2}$, i.e.,
at optimality the duality gap is zero!
(5) Let us solve the program
$$\min x \,\,\hbox{subject to}\,\,x^2\leq 0,\, x \in R.$$
This program is not Slater regular. On the other hand, we have
$$\psi(\lambda)=\inf_{x\in R}(x+ \lambda x^2)=\left\{\begin{array}{ccc} -\frac{1}{2\lambda} & for & \lambda >0\\ \
-\infty& for & \lambda =0.\end{array}\right.$$
Obviously, $\psi(\lambda)<0$ for all $\lambda \geq 0$. Consequently,
$\sup \{\psi(\lambda)\,|\,\lambda \geq 0\} = 0$.
So the Lagrange-dual has the same optimal value as the primal problem. In spite of the lack
of Slater regularity there is no duality gap.
(6) {\bf (Example with positive duality gap)}
We consider the program
$$\min e^{-y}\, \,\,\hbox{subject to}\,\,\,\sqrt{x^2+y^2}- x \leq 0,\, (x,y) \in R^2.$$
Here the feasible region is $\Omega =\{(x,y)\in R^2\,|\, x\geq 0, y=0\}$.
Consequently this program is not Slater regular. The optimal value of the objective function is $1$.
The Lagrange function is
$$L(x,y,\lambda)= e^{-y} + \lambda(\sqrt{x^2+y^2}- x).$$
The Lagrange dual program can be written in the form
$$\sup \psi(\lambda)\, \,\,\hbox{subject to}\,\,\,\lambda \geq 0.$$
Note that $L(x, y,\lambda) > 0$ implies $\psi(\lambda)\geq 0$. Now let $\epsilon >0$. Fixing $y = - \ln \epsilon$ and
$x= \frac{y^2-\epsilon^2}{2\epsilon}$, we find $\sqrt{x^2+y^2}- x=\epsilon$
and
$$L(x,y,\lambda)= (1+\lambda)\epsilon.$$
In this way,
$$\psi(\lambda)=\inf_{(x,y)\in R^2}L(x,y,\lambda)\leq \inf_{\epsilon>0}(1+\lambda)\epsilon=0.$$
On the other hand, we also have $\psi(\lambda)\geq 0$, and consequently the optimal value of the Lagrange dual
program is $0$, and hence the minimal duality gap equals $1$! ( No strong duality here).
\subsection{The Wolfe-dual problem}
The {\it Lagrange dual program} can be written in the form
$$\sup_{\lambda \geq 0}\,\,\{\,\inf_{x \in D}\{f(x) + \sum_{\alpha=1}^m \lambda_\alpha g_\alpha(x)\}\}.$$
Assume that $D = R^n$ and the functions $f, g_1, ... , g_m$ are continuously differentiable and
convex. For a given $\lambda \geq 0$ the inner minimization problem is convex, and we can use the
fact that the infimum is attained if and only if the gradient with respect to $x$ is zero.
\begin{definition}
The problem
$$\sup_{x,\lambda}\,\,\{f(x) + \sum_{\alpha=1}^m \lambda_\alpha g_\alpha(x)\} \leqno(WP)$$
subject to
$$\frac{\partial f}{\partial x^i}(x) + \sum_{\alpha=1}^m \lambda_\alpha \frac{\partial g_\alpha}{\partial x^i}(x)=0,\,\, \lambda \geq 0$$
is called the {\it Wolfe dual} of the program (P).
\end{definition}
Obviously, the constraints in Wolfe dual are usually nonlinear. In such cases the
Wolfe-dual is not a convex program.
The Wolfe dual has the weak duality property.
\begin{theorem} {\bf (weak duality property)}
Suppose that $D= R^n$ and the functions $f, g_1, ... , g_m$ are continuously differentiable and
convex. If $\hat x$ is a feasible solution of (P) and $(\bar x, \bar \lambda)$ is a feasible solution for (WP), then
$$L(\bar x, \bar \lambda) \leq f(\hat x).$$
In other words, weak duality holds for (P) and (WP).
\end{theorem}
\subsection{Example}
(1) Let us consider the convex program
$$\min_{x,y}\, x+ e^{y}\, \,\,\hbox{subject to}\,\,\,3x-2 e^y \geq 10,\, y\geq 0,\,\,(x,y) \in R^2.$$
Then the optimal value is $5$ with $x = 4, y=0$. The Wolfe dual of this program is
$$\sup_{x,y,\lambda}\{x+ e^{y} + \lambda_1 (10- 3x + 2 e^y)-\lambda_2 y\}$$
subject to
$$1-3\lambda_1=0,\,e^{y} + 2 e^{y}\lambda_1-\lambda_2=0,\,\,(x,y)\in R^2,\, \lambda \geq 0.$$
Obviously, the Wolfe dual program is not convex. It follows
$\lambda_1=\frac{1}{3}$ and the second constraint becomes $\frac{5}{3}e^{y}-\lambda_2=0$.
Eliminating $\lambda_1$, $\lambda_2$ from the objective function, we find
$$g(y)=\frac{5}{3}\,e^{y}- \frac{5}{3}\,y\,e^{y}+\frac{10}{3}.$$
This function has a maximum when $g^\prime(y)=0$, i.e., $y=0$ and $f(0)=5$.
Hence the optimal value of (WP) is $5$
and then $(x, y,\lambda_1,\lambda_2) = (4, 0, \frac{1}{3}, \frac{5}{3})$.
{\bf Remark} The substitution $z = e^y \geq 1$ makes the problem linear.
\section{Minimax inequality}
For any function $\phi$ of two vector variables $x\in X,\, y\in Y$,
the {\it minimax inequality}
$$\max_{y\in Y}\, \min_{x\in X}\, \phi(x,y)\leq \min_{x\in X}\, \max_{y\in Y} \,\phi(x,y)$$
is true. Indeed, start from
$$\forall x, y: \min_{x^\prime\in X}\phi(x^\prime, y)\leq \max_{y^\prime\in Y} \phi(x, y^\prime)$$
and take the minimum over $x \in X$ on the right-hand side, then the maximum over $y \in Y$
on the left-hand side.
Weak duality is a direct consequence of the minimax inequality. To see this, start from the
unconstrained formulation of Lagrange, and apply the above inequality, with $\phi = L$ the Lagrangian
of the original problem, and $y = \lambda$ the Lagrange vector multiplier.
\section{Nonholonomic Lagrange and Wolfe \\dual programs}
The Pfaff nonholonomic constraints in optimal programs were introduced by the
mathematical school coordinated by Prof. Dr. Constantin Udri\c ste at University
Politehnica of Bucharest (see \cite{DD}, \cite{DT}, \cite{RUU}-\cite{UFO}).
In this section we address the following original issues:
(i) difference between a program constrained by an integral submanifold of a Pfaff equation
and a program constrained by a Pfaff equation;
(ii) the non-holonomic Lagrange-dual problem with weak respectively strong duality;
(iii) the non-holonomic Wolfe-dual problem; (iv) pertinent examples.
Let $$\omega= \omega_i(x)dx^i= 0,\,i=1,...,n,\,\,x \in R^n,\,\, n\geq 3,$$ be a
non-completely integrable Pfaff equation (see also, \cite{D}, \cite{E}, \cite{M}, \cite{S}). The condition $n\geq 3$ is imposed by the nonholonomy theory.
For theoretical reasons, we understand that the co-vector field $\omega =(\omega_i(x))$ is $C^1$ on $R^n$, and has
no critical point in $R^n$. To this Pfaff equation we attach the $(n-1)$-hyperplane
$$H_x= \{\omega_i(x)dx^i= 0\}=\{y=(y^1,...,y^n)\in R^n\,|\, \omega_i(x)y^i=0\}$$
in the $n$-space $R^n$.
\begin{definition}
(i) A submersion $g=(g_\alpha),\,\alpha=1,...,n-p,$ is called solution of the Pfaff equation $\omega= \omega_i(x)dx^i= 0$
if $g_1(x)=0,...,g_{n-p}(x)=0$ and $dg_1(x)=0,...,dg_{n-p}(x)=0$ imply $\omega= \omega_i(x)dx^i= 0$.
(ii) A $p$-dimensional submersed submanifold $$(M,\omega, g, =)$$ of $R^n$ is called an integral manifold of the Pfaff equation on $R^n$ if
$$dg(M_x)\subseteq H_x,\,\, \hbox{for each point}\, x\, \hbox{in}\, M.$$
\end{definition}
\begin{definition}
The Pfaff equation is said to be {\it completely integrable} if there is one and only one integral
manifold of maximum possible dimension $n-1$ through each point of $R^n$.
\end{definition}
\begin{theorem} {\bf (Frobenius theorem)} A necessary and sufficient condition for
the Pfaff equation $\omega=0$ to be completely integrable is $\omega \wedge d\omega=0$.
\end{theorem}
Here $d\omega$ is the differential form of degree $2$ obtained from $\omega$ by exterior differentiation,
and $\wedge$ is the exterior product.
\subsection{Integral curves}
Let $(\Gamma,g=(g_1,...,g_{n-1}), =)$ be an integral curve of the non-completely
integrable Pfaff equation $\omega=\omega_i(x)dx^i=0$ can be written in the Cartesian implicit form
$$\Gamma: g_1(x)=0,..., g_{n-1}(x)=0,$$
i.e., the equations
$g_1(x)=0,..., g_{n-1}(x)=0$ and $dg_1(x)=0,..., dg_{n-1}=0$ imply
$\omega_i(x)dx^i=0$, for any $dx$. This means that
$$\det\left(\begin{array}{ccc}\frac{\partial g_1}{\partial x^1}&...&\frac{\partial g_1}{\partial x^n}\\ \
...&...&...\\ \
\frac{\partial g_{n-1}}{\partial x^1}&...&\frac{\partial g_{n-1}}{\partial x^n}\\ \
\omega_1&...&\omega_n \end{array}\right) =0.$$
It follows that there exist the functions $\nu_1(x),...,\nu_{n-1}(x)$, and the constant $\mu$, such that
$$\sum_{\alpha=1}^{n-1}\nu_\alpha(x) \frac{\partial g_\alpha}{\partial x^i}(x)= \mu \omega_i(x),\,\, i=1,...,n,$$
for any $x$.
\subsection{The connection between critical points on \\an integral submanifold and critical points \\with Pfaff non-holonomic constraint}
Let $f:R^n\to R$ be a $C^2$ function. Let us consider a non-holonomic program:
$$\min f(x)\,\, \hbox{subject to}\,\, \omega = \omega_i(x)dx^i=0,\,\, i=1,...,n.\leqno(NP)$$
Suppose that a $p$-dimensional integral submanifold of the Pfaff equation $\omega = \omega_i(x)dx^i=0$ is
$(M,\omega, g=(g_1,...,g_{n-p}), =)$. We consider the attached program
$$\min_x f(x)\,\,\hbox{subject to}\,\, x \in M$$
or, equivalently,
$$\min_x f(x)\,\,\hbox{subject to}\,\, g_1(x)=0,...,g_{n-p}(x)=0.$$
\begin{theorem}
A point in $R^n$ is a nonholonomic constrained critical point if and only if it is
critical point constrained by each integral submanifold containing it.
\end{theorem}
{\bf Proof} ({\bf if}) For a given integral submanifold, the associated Lagrange function is
$$L(x,\lambda)= f(x)+\lambda_1 g_1(x)+...+ \lambda_{n-p} g_{n-p}(x).$$
The critical point conditions are
$$df(x)+ \lambda_1 dg_1(x)+...+ \lambda_{n-p} dg_{n-p}(x)=0,\,\,\forall dx; g_1(x)=0,...,g_{n-p}(x)=0.$$
Let $X$ be a vector field tangent to the integral submanifold (and hence from distribution), i.e.,
$$dg_1(X)=0,...,dg_{n-p}(X)=0,\,\omega(X)=0.$$
It follows that at a critical point we must have $df(X)=0$.
If the integral submanifold is arbitrary (both as dimension and as way of description), i.e., $g$ is arbitrary, i.e., $X$ is an arbitrary
vector field tangent to $M$, then from the relations
$\omega(X)=0, df(X)=0, \forall X$, we obtain the existence of a constant $\mu$ such that,
at a critical point, which is independent on $g$, we must have
$$df(x)+ \mu \omega(x)=0,\,\,\forall dx.\leqno(NCP)$$
({\bf only if}) Each nonholonomic constrained critical point belongs to each critical point set
associated to a constrained integral submanifold. Indeed, if $X$ is a vector field tangent to an integral submanifold,
then
$$dg_1(X)=0,...,dg_{n-p}(X)=0,\,\omega(X)=0.$$
It follows the existence of multipliers $\lambda_1,...,\lambda_{n-p}$ such that
$$\omega = \lambda_1 dg_1(x)+...+ \lambda_{n-p} dg_{n-p}(x).$$
But $df= -\mu \omega$.
\subsubsection{Attached Riemannian geometry}
Let us consider the system "NCP = nonholonomic critical point", where the parameter $\mu$ is arbitrary.
According to the implicit function theorem, if the matrix
$$\left(a_{ij}\right)=\left(\frac{\partial^2 f}{\partial x^i \partial x^j}+\mu \frac{\partial \omega_i}{\partial x^j}\right)$$
is non-degenerate at a fixed critical point, then the system define a curve $x=x(\mu)$. On the other hand, the matrix of elements
$g_{jk}=\delta^{il} a_{ij}a_{kl}$ is a Riemannian metric. Symbolically, $a= (a_{ij})$, $g= {}^ta\,a$, $g^{-1} = a^{-1}\,{}^t(a^{-1})$
and the geometry induced by $g$ follows by usual rules.
By differentiation with respect to $\mu$, we obtain
$$a_{ij}\frac{dx^j}{d\mu} +\omega_i=0.$$
Let $\eta_k= \delta^{il} a_{kl}\omega_i$. Then $g_{kj}\frac{dx^j}{d\mu}+ \eta_k=0$
and hence
$$g_{kj}\frac{dx^k}{d\mu}\frac{dx^j}{d\mu}+ \eta_k \frac{dx^k}{d\mu}=0$$
{\bf Proposition} {\it The angle between the vectors $\eta_k$ and $\frac{dx^j}{d\mu}$ is always obtuse}.
{\bf Completion} It follows
$$\left(\frac{\partial^2 f}{\partial x^i \partial x^j}+\frac{\mu}{2} \left(\frac{\partial \omega_i}{\partial x^j}+ \frac{\partial \omega_j}{\partial x^i}\right)\right)\frac{dx^i}{d\mu} \frac{dx^j}{d\mu} +\omega_i \frac{dx^i}{d\mu}=0.$$
We reinterpret this equality, introducing the {\it fundamental tensor}
$$h_{ij}=\frac{\partial^2 f}{\partial x^i \partial x^j}+\frac{\mu}{2} \left(\frac{\partial \omega_i}{\partial x^j}+ \frac{\partial \omega_j}{\partial x^i}\right),$$
and writing
$$h_{ij}\frac{dx^i}{d\mu} \frac{dx^j}{d\mu} +\omega_i \frac{dx^i}{d\mu}=0.$$
\subsection{Eliminating ambiguities by \\geometric interpretation}
The Pfaff equations theory is a source of misunderstanding for beginners.
We can eliminate such problems thinking in terms of differential geometry.
\subsubsection{Language of Vranceanu}
Let $\gamma_{x_0}$ be the image of an integral curve of the
Pfaff equation $\omega_i(x)dx^i= 0$ through the point $x_0$. Denote by $\Sigma_{x_0}=\{\gamma_{x_0}\}$
the family of all images of integral curves through the point $x_0$. The pair
$$(D,\Sigma),\,\, \Sigma=\{\Sigma_{x_0}\,|\, x_0\in D\}$$
is called {\it nonholonomic hypersurface} on $D$ attached to the Pfaff equation $\omega_i(x)dx^i= 0$ (see also, \cite{Vr}).
To the Pfaff equation $\omega_i(x)dx^i= 0$ and to the point $x_0\in D$, we attach the unique $(n-1)$-hyperplane
$$H_{x_0}= \{x \in R^n\,|\, \,\omega_i(x_0)(x^i - x_0^i)=0\}.$$
Since all straight lines tangent to integral curves which pass through $x_0$ are included in $H_{x_0}$,
the hyperplane $H_{x_0}$ is called the {\it tangent hyperplane} at $x_0$ of $(D,\Sigma)$.
\subsubsection{Language of distributions}
Let us introduce the $(n-1)$-hyperplane
$$H_x= \{\omega_i(x)dx^i= 0\}=\{y=(y^1,...,y^n)\in R^n\,|\, \omega_i(x)y^i=0\}$$
in the $n$-space $R^n$. The rule $x\to H_x$ gives a field $H$ of hyperplanes in $R^n$,
or what we call {\it $(n-1)$-dimensional distribution: a linear subbundle of the tangent bundle}.
In short $H=\cup_{x\in R^n}H_x\subset TR^n$.
In subsequent explanations, we shall prefer the distributions language being more suggestive.
Similarly, we can introduce the half-hyperplane
$$H_x^-= \{\omega_i(x)dx^i\leq 0\}=\{y=(y^1,...,y^n)\in R^n\,|\, \omega_i(x)y^i\leq 0\}.$$
The rule $x\to H^-_x$ gives a field $H^-$ called {\it $(-)$-distribution}.
\begin{definition}
A $p$-dimensional submersed submanifold $(M,H, g, =)$ of $R^n$ is called an integral manifold of the distribution $H$
on $R^n$ if
$$dg(M_x)\subseteq H(g(x))=H_x,\,\, \hbox{for each point}\, x\, \hbox{in}\, M.$$
\end{definition}
The field $H^-$ is used when the program refers to {\it manifolds whose boundary contains an integral manifold of a Pfaff equation}.
A manifold with boundary is a manifold with an edge. The boundary of a $(p+1)$-manifold with boundary is a $p$-manifold.
In technical language, a manifold with boundary is a space containing both interior points and boundary points.
\subsection{The Lagrange dual problem}
Let $(M,\omega, g=(g_1,...,g_{n-p}), =)$ be a $p$-dimensional integral submanifold
of the Pfaff equation $\omega = \omega_i(x)dx^i=0$ and $f:R^n\to R$ be a $C^2$ function.
Denote $g=(g_\alpha),\,\alpha=1,...,n-p,$ and we introduce the set
$$\Omega = \{x\in R^n \,|\,g_\alpha(x)= 0,\,\alpha=1,...,n-p\}= \{x\in R^n \,|\,g(x)= 0\}.$$
For each program
$$\min_x f(x)\,\,\hbox{subject to}\,\, x \in M,$$
equivalently
$$\min_x\{f(x)\,|\, x \in \Omega\},$$
we can repeat the theory of dual programs.
The {\it Lagrange function (or Lagrangian)} of this program is
$$L(x, \lambda) = f(x)+ \sum_{\alpha=1}^{n-p} \lambda_\alpha g_\alpha(x)
= f(x)+<\lambda,g>, x\in R^n, \lambda\in R^{n-p}.$$
The critical points with respect to the variable $x$ are given by the system
$$df(x)+ <\lambda,dg(x)>=0, \forall dx.\leqno(1)$$
It follows $x=x(\lambda)$, the dual function $\psi(\lambda)= L(x(\lambda), \lambda)$ and the {\it Lagrange dual problem}
$$\max_\lambda \psi(\lambda).$$
But, what we understand by the dual theory
for the nonholonomic program (NP)? Of course, we must ask an arbitrary integral submanifold.
That is why, the system (1) must be replace with the system
$$df(x)+ \mu \omega(x) =0, \forall dx.$$
The solution of this system is of the form $x=x(\mu)$, an arbitrary dual function is
$\psi(\mu,\lambda)= L(x(\mu), \lambda)$ and the {\it Lagrange dual problem} can be written
$$\max_{\mu,\,\lambda} \psi(\mu,\lambda).$$
Each Lagrange function $L(x, \lambda) $ of (NP)
is linear in $\lambda$.
If the distribution $H$ is described by $q$ Pfaff equations, then $\lambda$ has $(n-p)q$ components and
$\mu$ has $q$ components.
\subsubsection{Dual nonholonomic Lagrange function}
In the non-holonomic context, the constraint function $g$ does not exists, but it should be built
at least on the critical point set $\{x(\mu)\}$.
By analogy with the holonomic equality $g( \bar x(\mu )) = c$,
from the relation ${\displaystyle \frac {df_*}{d \mu} \left ( \frac {dc}{d \mu} \right )^{-1} = - \mu} $,
we can define $c(\mu ) $ by Cauchy problem
$$ \frac {dc}{d \mu } = - \frac {1}{\mu }\, \frac {df_*}{d \mu}\,,\,\,\,\,\,\, c(\mu_0) = 0\,, \leqno (EC)$$
and then a Lagrange dual function will be
$$ \theta (\mu ) = f_*(\bar x(\mu ) ) + \mu \, c(\mu )\,. \leqno (LDF)$$
This function has the derivative $\theta^\prime (\mu )= c(\mu )$.
{\bf Example 1.} Let the objective function be $f(x,y,z) = x^2 + y^2 + z^2$ and the constraint Pfaff form $\omega = x dy + dz = 0$.
Then the critical points condition $df + \mu \omega = 0$ gives us
$$ \bar x(\mu ) =0\,,\,\, \bar y(\mu ) = 0\,,\,\, \bar z(\mu ) = - \frac {\mu }{2}\,,\,\, f_*(\bar x,\bar y, \bar z) = \frac {\mu^2}{4}\,.$$
If, for instance, we take $\mu_0 = 2$, the solution for the {\it primal problem} will be $\bar x=0, \bar y =0, \bar z = -1$ and $ f_*= 1$.
For the {\it Lagrange dual problem} the equation (EC) gives us $$\frac {dc}{d \mu } = - \frac {1}{\mu }\, \frac { \mu}{2} = - \frac {1}{2}\,.$$
Consequently ${\displaystyle c(\mu) = - \frac {\mu}{2} + \alpha }$ . If, as instance, $c_0 = 0$ for $\mu_0 = 2$, then $\alpha = 1$ and the
Lagrange dual function is $$\theta (\mu ) = \frac {\mu^2}{4} + \mu \left ( - \frac {\mu}{2} + 1 \right ) = - \frac {\mu^2}{4} + \mu\,.$$
It follows ${\displaystyle \theta ^\prime (\mu ) = - \frac {\mu}{2} + 1} = 0$. Hence $\mu_0 = 2$ and we obtain the same solution as in primal
problem.
{\bf Example 2.} Let consider the function $f(x,y,z)=x^2+y^2-z$ and the Pfaff form $\omega =x dy - z dz$. Solve the {\it primal problem}
$$f(x,y,z)= extremum\,\,\, \hbox {with constraint}\,\,\, \omega = 0.$$
The differential Lagrange form of the problem is
$$dL = 2x dx + 2y dy - dz + \mu (x dy - z dz)\,. $$
The condition $dL = 0$, (as differential form) leads to the equations $2x = 0,\, 2y + \mu x = 0,\, -1 - \mu z = 0$, whose solutions,
the critical points, are $x = 0,\, y = 0,\, z = -1/ \mu \, $. For $\mu < 0$ the critical points are points of constrained minimum and
the corresponding minimum values are $f_* = 1/ \mu \,.$ For $\mu \geq 0$ the critical points are not constrained extremum points.
For construct the {\it dual problem}, let us use the above described method. From equation (EC) we have $dc/ d\mu = 1/ \mu ^3$. Then
$\displaystyle c(\mu ) = - \frac {1}{2} \left ( \frac {1}{\mu ^2} - \frac {1}{\mu ^2_0} \right )$ and the Lagrange dual function will be
$$\theta (\mu) = \frac {1}{\mu } - \frac {\mu }{2} \left ( \frac {1}{\mu ^2} - \frac {1}{\mu ^2_0} \right )\,.$$
The critical points are given by the equation
$$\theta ^\prime (\mu) = - \frac {1}{2} \left ( \frac {1}{\mu ^2} - \frac {1}{\mu ^2_0} \right ) = c(\mu ) = 0$$
and a solution is $\mu _0$, i.e. the strong duality holds.
{\it Another way } Remind that a point $(x_0,y_0,z_0)$ is a minimum (maximum) point for the function $f(x,y,z)$,
constrained by the Pfaff equation $\omega = 0$, if this point is a minimum (maximum) point for $f$ restricted at any line solution of $\omega = 0$,
passing through $(x_0,y_0,z_0)\,$. So we reformulate the {\it primal problem} for a suitable line passing through critical points, in our case $(0,0,1/c)\,$.
Such a line has the cartesian implicit equations
$$\left \{ \begin{array}{c} y^2 - z - 1/c = 0 \\ 2yz - x = 0\,. \end{array} \right .$$
Then the Lagrange function of the problem is
$$L(x,y,z, \lambda , \mu ) = x^2+y^2-z + \lambda (y^2 - z - 1/c) + \mu (2yz - x)\,. $$
We obtain the following system of the critical points:
$$\frac {\partial L}{\partial x}=2x- \mu =0 $$
$$\frac {\partial L}{\partial y}=2y+2\lambda y + 2\mu z =0 $$
$$\frac {\partial L}{\partial z}=-1- \lambda + 2 \mu y=0\,. $$
This system has the solution $(0,0,1/c)\,$ for $\mu = 0 $ and $\lambda = 1\,.$
\subsection{Case of nonholonomic inequalities}
Let $(R^n,\omega, g=(g_1,...,g_{n-p}),\leq, \preceq)$ be a subset of $R^n$,
attached to the Pfaff inequation $\omega = \omega_i(x)dx^i\leq 0$, whose boundary
contains the $p$-dimensional integral submanifold $(M,\omega, g=(g_1,...,g_{n-p}),=, =)$ of the Pfaff equation.
Let $f:R^n\to R$ be a $C^2$ function.
Denote $g=(g_\alpha),\,\alpha=1,...,n-p,$ and we introduce the set
$$\Omega = \{x\in R^n \,|\,g_\alpha(x)\leq 0,\,\alpha=1,...,n-p\}= \{x\in R^n \,|\,g(x)\preceq 0\}.$$
\begin{theorem} ({\bf weak duality}) The dual function yields lower bounds of the initial optimal value $f_*$, i.e., for any
$\lambda$, we have $\varphi(\lambda)\leq f_*$. In other words,
$$\sup_\lambda\,\{\varphi(\lambda)\,|\, \lambda \succeq 0\}\leq \min_{x\in \Omega}\,\,\{f(x)+ <\lambda,g(x)>,\,\, x\in \Omega,\,\, \lambda\succeq 0\}.$$
\end{theorem}
\begin{theorem} {\bf (strong duality)} If the program (P) satisfies the Slater condition and has finite optimal
value, then
$$\sup_\lambda\,\{\varphi(\lambda)\,|\, \lambda \succeq 0\}= \min_{x\in \Omega}\{f(x)\}.$$
Moreover, then the dual optimal value is attained.
\end{theorem}
\subsection{Nonholonomic Wolfe dual}
The problem
$$\max_{x,\,\mu}\,\,\{f(x)\}$$
subject to
$$\frac{\partial f}{\partial x^i}(x) + \mu\, \omega_i(x)=0,\,\, \mu \geq 0$$
is called the {\it nonholonomic Wolfe dual} (WDNP) of the nonholonomic program (NP).
It follows $x=x(\mu)$ and $f(x(\mu))$. That is why, solving the dual problem
is equivalent to find extrema of the function $\mu \to f(x(\mu))$.
\subsection{Examples}
(1) see \cite{UFO}, p.190-191 {(\bf Consumer theory with nonholonomic constraint)}
Suppose that
$$u(x)=x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_n^{\alpha_n},\,\,\alpha_i\geq 0,\,\,\sum_{i=1}^n\, \alpha_i<1,\,\, x=(x_1,...,x_n) \in R_+^n$$
is the utility function defined over $n$ goods. Denote by $p_i(x)>0,\, i=1,...,n,$ the prices of the goods.
Let us determine what is the proportion of income that the associated consumer will spend on each good, if the
budget constraint is the Pfaff inequality $\sum_{i=1}^n\, p_i(x)dx_i\geq 0$.
The utility function is concave. We must look for critical points of the
utility function $u$ subject to the given nonholonomic constraint. To determine the
constrained critical points, we use the Lagrange $1$-form
$$\eta = du(x) - \mu \sum_{i=1}^n\, p_i(x)dx_i= \sum_{i=1}^n\left(\frac{\alpha_i}{x_i}\,u(x) - \mu p_i(x)\right)dx_i,$$
and we write the system
$$\alpha_i u(x)- \mu p_i(x)x_i=0,\,i=1,...,n.$$
Suppose we have a critical point (solution) $x^* = x^*(\mu),\, \mu>0$.
Each component $X_i^*$ of the critical point is the quantity consumed of the $i^{th}$ good.
Moreover, $p_i(x^*)x_i^*$ is the income spend on the $i^{th}$ good and
$\sum_{i=1}^n p_i(x^*)x_i^*$ is the total income. Since
$$u(x)\sum_{i=1}^n \alpha_i - \mu \sum_{i=1}^n p_i(x^*)x_i^*=0,$$
we find the proportion of the income spent on the $i^{th}$ good,
$$\frac{p_i(x^*)x_i^*}{\sum_{i=1}^n p_i(x^*)x_i^*}=\frac{\alpha_i}{\sum_{i=1}^n\alpha_i},$$
which is independent on consumed quantities and of prices ({\it economic law}).
Suppose we are interested in the maximum of the utility function $u$ subject to the nonholonomic constraint.
To solve this problem, as usual we look for a critical point $x^* = x^*(\mu),\, \mu>0$,
which verify the equality in the budget constraint (hyperplane) $\sum_{i=1}^n p_i(x^*(\mu))dx_i=0$ and
the negative definiteness of the restriction of the quadratic form
$$d^2u(x^*) - \frac{\mu}{2}\,\,\sum_{i,j=1}^n \left(\frac{\partial p_i}{\partial x_j}+\frac{\partial p_j}{\partial x_i}\right)(x^*)\,dx_i\,dx_j$$
to the budget hyperplane.
(2) {\bf Nonholonomic initial program} Find extremum points of the function $f(x,y,z)=2xy + z^2$ subject to $zdx-dy \geq 0,\, xdy+dz \geq 0$\,\, (see \cite{UD6}).
The constrained critical points are solutions of the system
$$2y-\mu_1 z=0,\,\, 2x + \mu_1 -\mu_2 x =0,\,\, 2z-\mu_2=0,\,\,\mu_1 \leq 0,\,\,\mu_2\leq 0.$$
It follows the family of critical points
$$x= \frac{\mu_1}{\mu_2-2},\,\, y=\frac{1}{4}\mu_1 \mu_2, \,\,z=\frac{1}{2}\mu_2,\,\,\mu_1 \leq 0,\,\,\mu_2\leq 0.$$
The nature of each critical point is fixed by the signature of the quadratic form $(4-\mu_2)dxdy-\mu_1 dxdz+2dz^2$
restricted to $\frac{\mu_2}{2}dx-dy=0,\,\frac{\mu_1}{\mu_2-2}dy + dz =0$. It follows the restriction
$q= \mu_2\left(\frac{\mu_1^2 }{(\mu_2 -2)^2}+2 -\frac{\mu_2}{2}\right)\,dx^2$, which is negative definite.
All critical points are maximum points. The manifold of critical points has the implicit Cartesian equation
$$xz(z-1)=y,\,x\geq 0, y\geq 0,\, z\leq 0.$$
The maximum value of the function $f$ is
$$f(x(\mu_1,\mu_2), y(\mu_1,\mu_2), z(\mu_1,\mu_2))= \frac{\mu_1^2 \mu_2}{2(\mu_2-2)}+ \frac{1}{4}\mu_2^2.$$
If we change the constraints into
$$zdx-dy\leq 0,\, xdy+dz \leq 0,$$
then it will be sufficient to have positive multipliers and a positive
definite quadratic form $q$ in order that each critical point becomes a minimum point.
The PDEs system which gives us $c_1(\mu_1 , \mu ), c_2(\mu_1 , \mu_2 )$ is (see 1.2 )
$$\mu_1 \, \frac{\partial c_1}{\partial \mu_1 } + \mu_2 \, \frac{\partial c_1}{\partial \mu_2} = - \frac {\mu_1 \mu_2 }{\mu_2 - 2 }$$
$$\mu_1 \, \frac{\partial c_2}{\partial \mu_1 } + \mu_2 \, \frac{\partial c_2}{\partial \mu_2} = - \frac{1}{2} \, \mu_2 + \left ( \frac {\mu_1 }{\mu_2 - 2} \right )^2 \,, $$
with solutions, respectively,
$$c_1 (\mu_1 , \mu_2 ) = - \mu_1 - \frac {2\, \mu_1 }{\mu_2 } \, \ln |\mu_2 -2| + \alpha _1 \left ( \frac {\mu_1}{ \mu_2} \right ) \,,$$
$$c_2 (\mu_1 , \mu_2 ) = - \frac {1}{2} \, \mu_2 - \frac {\mu_1^2}{\mu_2^2} \, \left ( \frac {2}{\mu_2 -2} - \ln |\mu_2 -2| \right ) +
\alpha _2 \left ( \frac {\mu_1}{ \mu_2} \right )\,,$$
where $\alpha _1\,, \alpha _2 $ are arbitrary functions. The condition $\displaystyle \frac{\partial c_1}{\partial \mu_2} =
\frac{\partial c_{2}}{\partial \mu_1} $ is verified, for instance, if $\displaystyle \alpha_1 \left ( \frac {\mu_1}{\mu_2} \right ) = ct. = \alpha _1 $
and $\displaystyle \alpha _2 \left ( \frac {\mu_1}{\mu_2 } \right ) = - \frac {\mu_1^2}{\mu_2^2} + \alpha _2\,, \alpha _2 = ct. $
Finally, the Lagrange dual function $\theta (\mu_1, \mu_2) = f_* + \mu_1 c_1 + \mu_2 c_2 $ is
$$\theta (\mu_1, \mu_2) = - \frac {\mu_1^2}{2} - \frac {1}{4} \mu_2^2 - \frac { \mu_1^2}{\mu_2} \ln |\mu_2 -2| +
\alpha _1 \mu_1 + \alpha _2 \mu_2 \,.$$
{\bf Nonholonomic Wolfe dual program} Find the extrema of the function
$$(\mu_1,\mu_2)\to \varphi(\mu_1,\mu_2)=f(x(\mu_1,\mu_2), y(\mu_1,\mu_2), z(\mu_1,\mu_2)),\,\,\mu_1 \leq 0,\,\,\mu_2 \leq 0.$$
Since
$$\varphi(\mu_1,\mu_2)= \frac{\mu_1^2 \mu_2}{2(\mu_2-2)}+ \frac{1}{4}\mu_2^2\geq 0,$$
in the conditions of the problem, the critical point $\mu_1=0,\mu_2=0$ is a minimum point.
Also, all the points of the form $(\mu_1 <0,\mu_2=0)$
are minimum points.
\section{Extrema in canonical coordinates}
The next Theorem, which gives the canonical Pfaff forms (and hence canonical nonholonomic constraints),
is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem (see \cite{D}, \cite{S}, \cite{Vr}).
\begin{theorem} ({\bf Darboux Theorem})
Suppose that $\omega$ is a differential $1$-form on an $n$ dimensional manifold, such that $d\omega$ has constant rank $p$. If
$\omega \wedge (d\omega)^p=0$ everywhere,
then there is a local system of coordinates $x^1,...,x^p, y^1, ..., y^p$ in which
$$\omega = \sum_{i=1}^p x^i dy^i.$$
If, on the other hand, $\omega \wedge (d\omega)^p \neq 0$ everywhere,
then there is a local system of coordinates $x^1,...,x^p$, $y^1, ..., y^p, z$ in which
$$\omega = \sum_{i=1}^p x^i dy^i + dz\,\, \hbox{(contact form)}$$
or
$$\omega = \frac{1}{2}\left(\sum_{i=1}^p x^i dy^i- \sum_{i=1}^p y^i dx^i\right)+dz\,\, \hbox{(symmetric normal form)}.$$
\end{theorem}
From the normal form of Darboux, we see that the maximal integral manifolds are of
dimension $p$. For the contact form equation
$$\omega = \sum_{i=1}^p x^i dy^i + dz=0,$$
they are given by
$$z=f(y^1,...,y^{p}), \,\,x_1=- \frac{\partial f}{\partial y^1},...,x_p=- \frac{\partial f}{\partial y^p},$$
where $f$ is a $C^2$ arbitrary function.
\subsection{Case of even number of variables}
Let us find the extrema of a function $f(x,y), x=(x^i), y=(y^i), i=1,...,n$, subject to a nonholonomic constraint
written as Pfaff equation $\omega = x^1 dy^1 + ... + x^n dy^n =0$.
The constrained critical points are solutions of the system
$$\frac{\partial f}{\partial x^i} =0,\, \frac{\partial f}{\partial y^i}+\mu x^i =0.$$
\subsection{Case of odd number of variables}
Let us find the extrema of a function
$f(x,y,z), x=(x^i), y=(y^i), i=1,...,n$ subject to a nonholonomic constraint
$\omega = x^1 dy^1 + ... + x^n dy^n + dz=0$. The constrained
critical points are solutions of the system
$$\frac{\partial f}{\partial x^i} =0,\, \frac{\partial f}{\partial y^i} + \mu x^i=0,\,\frac{\partial f}{\partial z} + \mu =0.$$
To decide the type of a critical point $(x_0,y_0,z_0)$, we use the restriction of the quadratic form
$Q= d^2f(x_0,y_0,z_0)+ \mu \delta_{ij}dx^idy^j$ to the hyperplane $\delta_{ij} x^{i}_0 dy^j+dz=0$
and its signature. Since
$$d^2f= \frac{\partial^2 f}{\partial x^i\partial x^j}dx^i dx^j+ \frac{\partial^2 f}{\partial y^i\partial y^j}dy^i dy^j + 2 \frac{\partial^2 f}{\partial x^i\partial y^j}dx^i dy^j$$
$$+ 2\left(\frac{\partial^2 f}{\partial x^i\partial z}dx^i +\frac{\partial^2 f}{\partial y^i\partial z}dy^i\right)dz + \frac{\partial^2f}{\partial z^2}dz^2,$$ the restriction of $Q$ is
$$q= \frac{\partial^2 f}{\partial x^i\partial x^j}dx^i dx^j+\frac{\partial^2 f}{\partial y^i\partial y^j}dy^i dy^j + 2\frac{\partial^2 f}{\partial x^i\partial y^j}dx^i dy^j + \mu \delta_{ij}dx^idy^j$$
$$- 2\left(\frac{\partial^2 f}{\partial x^i\partial z}dx^i + \frac{\partial^2 f}{\partial y^i\partial z}dy^i\right) \delta_{kl} x^{k}_0 dy^l + \frac{\partial^2f}{\partial z^2}(\delta_{kl} x^{k}_0 dy^l )^2.$$
\subsubsection{Another point of view}
The general solution of the Pfaff equation $\omega = x^1 dy^1 + ... + x^n dy^n + dz=0$ is
$$z=\varphi(y),\,\,x=- \frac{\partial \varphi}{\partial y}(y),\,\, \hbox{where}\,\, \varphi\,\, \hbox{is arbitrary}.$$
Consequently the previous nonholonomic program can be written
$$\min f(x,y,z)\,\,\hbox{subject to}\,\, z=\varphi(y),\,\, x=- \frac{\partial \varphi}{\partial y}(y).$$
In this form, it is similar to a classical program, but the function $\varphi$ is arbitrary. For each $\varphi$,
we attach a Lagrangian
$$L(x,y,z,\lambda_1,\lambda_2)= f(x,y,z) +\lambda_1(z-\varphi(y))+\lambda_2\left(x+ \frac{\partial \varphi}{\partial y}(y)\right).$$
It follows the system which describes the critical points
$$\frac{\partial L}{\partial x}= \frac{\partial f}{\partial x} +\lambda_2=0,\,\, \frac{\partial L}{\partial z}= \frac{\partial f}{\partial z} +\lambda_1=0$$
$$\frac{\partial L}{\partial y}= \frac{\partial f}{\partial y} -\lambda_1 \frac{\partial \varphi}{\partial y}+\lambda_2 \frac{\partial^2 \varphi}{\partial y \partial y}=0$$
$$\frac{\partial L}{\partial \lambda_1}= z-\varphi(y)=0,\,\,\frac{\partial L}{\partial \lambda_2}= x+ \frac{\partial \varphi}{\partial y}(y)=0.$$
This means that the critical points with respect to $x$, $y$ and $z$ must verifies the constraints of the initial program.
In this context, it is very clear what means a Lagrange dual program. The dual function is
$$\psi(\lambda_1,\lambda_2)= L(x(\lambda_1,\lambda_2),y(\lambda_1,\lambda_2),z(\lambda_1,\lambda_2),\lambda_1,\lambda_2).$$
{\bf Example} Find extremum points of the function
$f(x,y,z)=x+y+z+\frac{1}{2}(x^2+y^2+z^2)$ subject to $dz-xdy=0$.
The constrained critical points are solutions of the system
$$\frac{\partial f}{\partial x}=0, \frac{\partial f}{\partial y}-\mu x=0, \frac{\partial f}{\partial z}+\mu=0,$$
i.e., $x=1, y=1-\mu, z=1+\mu$.
The nature of critical points is determined by the signature of the restriction $q$ of
the quadratic form $Q= -(dx^2+dy^2+dz^2)-\mu dx dy$ to the plane $dz=dy$. It follows
$q= -(dx^2+2dy^2) - \mu dx dy$. This quadratic form is negative definite for $\mu^2 <8$, i.e.,
$\mu \in (-2\sqrt{2},2\sqrt{2})$. In this case, all critical points are maximum points.
The function $\varphi(\mu)=f(x(\mu),y(\mu),z(\mu))= \frac{3}{2}-\mu^2$ is increasing on $(-2\sqrt{2},0)$
and decreasing on $(0,2\sqrt{2})$. Also, $\inf \varphi(\mu)= \frac{3}{2}-8$.
\subsubsection{Passing to an even number of variables}
The point $(x,y,z)$ belong to the contact manifold defined by $\omega =xdy +dz=0$. Let $t$ be the number to be multiplied with
$1$-form $\omega$ to obtain a point of a symplectic $(2n+2)$-manifold described by the $1$-form $\eta=t\omega = tx dy + t dz$.
If we pass to the coordinates $P= (P^I)=(p,p_0), p=tx,p_0=t$, $Q= (Q^I)=(q,q_0), q=y, q_0=z$, then $\eta = PdQ$ and hence $d\eta=dP \wedge dQ$.
Having in mind the changing of the variables, the function $f(x,y,z)$ becomes $f\left(\frac{p}{p_0}, q, q_0\right)=\varphi (P,Q)$.
The constrained critical points are solutions of the system
$$\frac{\partial \varphi}{\partial P^I} =0,\, \frac{\partial \varphi}{\partial Q^i}+\mu x^i =0,\,\frac{\partial \varphi}{\partial z}+ \mu=0.$$
\subsubsection{Contact Hamiltonian}
Let $X= \dot x^i\frac{\partial}{\partial x^i}+\dot y^i\frac{\partial}{\partial y^i} + \dot z\frac{\partial}{\partial z},i=1,...,n$ be the contact vector field.
Let $K(x,y,z)$ be the contact Hamiltonian, which is defined by
$$K(x,y,z)= \omega(X),\,\, X\rfloor d\omega|_{\omega=0} =dK.$$
In case of $\omega = xdy +dz$, we have $d\omega|_{\omega=0}=-dx\wedge dy$. It follows the Hamiltonian
$K=x\dot y + \dot z$ and the contact flow
$$\dot x= -\frac{\partial K}{\partial y} + x \frac{\partial K}{\partial z},\,\,\dot y= \frac{\partial K}{\partial x},\,\, \dot z = K-x \frac{\partial K}{\partial x}.$$
{\bf Open problem} Find extrema of the contact Hamiltonian $K(x,y,z)$
constrained by $\omega = xdy +dz=0$.
|
1,116,691,500,474 | arxiv | \section{Introduction}
A long time has been passed since a straight double helical
structure had been proposed by Watson and Crick as relaxed
configuration of a DNA, but today, we know it
can almost never be found in this form in nature. The macromolecule
is able to play its important and essential role in the life, when
it is interacting with different proteins which force it to bend,
twist, melt and/or pack~\cite{matthews92,schleif92}.
To understand the macromolecule's functionality, DNA has been
examined in different length scales. Single molecule force-extension
studies on $10-100\mu$~m DNAs~\cite{Smith92}, cyclization
experiments on $100-1000$~nm DNAs~\cite{han97} and recent
experiments on sharply bent DNAs in loops with the lengths less
than $50$~nm, persistence length of
DNA~\cite{widom04,widom05,saiz05,Vologodskii05} . All these
experiments are of great biological interest. Depending on relevant
length scale of the problem, experimental results are usually
analyzed in two types of theoretical models: variants of the
continuum elastic worm-like chain (WLC) and base-pair steps.
The worm-like chain models are based on the physics of elastic rod
where the energy has a quadratic form in deformations. There are two
approaches to this model. The first one is based on Kirchhoff-like
equations of balance of forces and torques for every segment of the
rod~\cite{mahadevan96,westcott97} and the other one is based on
energy minimization~\cite{moroz98,zhang03}. In these models
statistical mechanics of the rod at non-zero temperature is also
encountered~\cite{yamakawa78}. A force constant or
equivalently a deformation module is defined for bending, twisting,
stretching and their coupling terms.
In the classical WLC model, all cross terms as well as stretching energy
are neglected and the rod is isotropic in bending in different
directions~\cite{shimada84}. So the model is identified with just
two persistence lengths assigned to each of bending or twisting
degrees of freedom. Modifications to this model have been made
by encountering twist-stretch coupling, twist-bend coupling and
stretch-bend
coupling~\cite{moroz98,marko94,kamien97,O'Hern98,FarshidPRL05} and
estimating the free parameters by fitting the theory to
force-extension experimental data. WLC model
has also been used to evaluate loop formation
probability, $J$-factor~\cite{zhang03,shimada84}.
Looking at microscopic structure of DNA macromolecule suggests that
bending toward the groove is easier than bending toward the
backbone. Though, there is no direct experiment to measure the
anisotropy, there are some theoretical postulates for double
stranded DNA~\cite{sclellman74,zhurkin79,matsumoto02}.
Monte Carlo simulations have also confirmed that B-DNA bends more
easily in the groove direction (roll) than in the backbone direction
(tilt)~\cite{zhurkin91}. It has been encountered in some models for
describing force-extension and DNA cyclization experiments and found
to have no significant improvement to the
results~\cite{moroz98,zhang03,levene86}, though existence
of bending anisotropy affects the existence of twist-stretch
coupling~\cite{O'Hern98}. Balaeff
\textit{et al.} have claimed that bending anisotropy can reduce the
energy of loops in length scale of $76$~bp by a factor of one
third~\cite{balaeff04}. In a recent experiment an asymmetry in the
periodic behavior of free energy of loop formation as a function of
loop length for $60-100$~bp loops has been detected~\cite{saiz05}.
Fourier analysis of free energy shows two main frequencies. One with
a period of $\sim10.5$~bp due to helical shape of the double strand
and one with a period of $\sim5.6$~bp that might be a result of
bending anisotropy.
Sequence dependence and anisotropy of bending persistence length has
been widely noticed in base-pair steps approaches, in which relative
rotation and displacement of every two segments are defined through
six parameters slide, shift, rise, tilt, roll and
twist~\cite{olson93}. These sequence dependent parameters
for individual base-pair steps have been determined from their
standard deviation in crystal complexes~\cite{olson98}. Theoretical
work has also been done on extracting these parameters from atomic
level parameters via the analysis of Molecular Dynamics (MD)
trajectories~\cite{Gonzales00}. Different simulation studies give
estimations for roll and tilt values. Munteanu \textit{et al.} have
shown that for $3-11$~bp DNAs bending rigidity oscillates with
bending direction and the values of roll is $\sim8-10$ times greater
than the values of tilt~\cite{munteanu98}. Olson \textit{et al.}
estimate the ratio of the hard bending rigidity, $A_1$, to the soft
bending rigidity $A_{2}$ to be between $4$
and $16$~\cite{olson93} and between $1$ and $5$ in a more
recent study~\cite{olson04}. MD simulations of $17$~bp dsDNAs
gives $A_{1}$ almost twice as $A_{2}$~\cite{Lankas00}. In a stack
of plates simulations, Mergell \textit{et al.} state that spacial
constraints make roll twice as favorable as tilt
\cite{mergell03}. Effect of sequence dependency on persistence
length and also intrinsic curvature on loop formation problem have
been studied analytically by Popov \textit{et al.} which result
in a wide distribution of cyclization probabilities~\cite{popov}.
An effective persistence length, $A$, can be assigned to a rod with
hard and soft
anisotropic bending rigidities $A_{1}$ and $A_{2}$ (corresponding to
tilt and roll). In an analytical
stack of plates study, done by Olson {\it et al.}, a complicated
dependence of $A$ on detailed couplings, anisotropic constants and
sequence alphabet has been shown for a free DNA~\cite{olson04}.
For a free, long and highly twisted DNA, the effective persistence
length exactly equals the harmonic mean of soft and hard
rigidities,
\begin{equation}
\label{Eq:harmonic} A=2(\frac{1}{A_{1}}+\frac{1}{A_{2}})^{-1}.
\end{equation}
This can be easily deduced from the equipartition principal.
The equipartition principle states that the total energy of a rod is
equal to $\frac{1}{2}k_BT$ times number of its degrees of freedom.
Bending of a segment of a rod around a principal axis with length
equal to its persistence length can be considered as one degree of
freedom.
Therefore number of degrees of freedom in hard and soft directions is
simply counted as $\frac{L}{A_1}$ and $\frac{L}{A_2}$
and the total energy is found to be $\frac{1}{2}k_BT (\frac{L}{A_1} +
\frac{L}{A_2})$.
To find the efficient persistence length, the anisotropic rod is
considered as an isotropic one.
Since the isotropic rod is able to be bent around two similar
directions, its energy is found to be $2(\frac{1}{2}k_BT \frac{L}{A})$ in
which $A$ is the efficient persistence length.
Having the two estimations equal to each other results
equation~\ref{Eq:harmonic}. This is more
accurately derived by Kehrbaum using an averaging
theory for a non-isotropic elastic rod with high intrinsic
twist~\cite{kehrbaumthesis}. Adding geometrical constraints on a free
DNA might
affect the above relation.
Here we are going to give an expression for effective persistence
length of in-plane DNA loops, by energy
minimization, when the entropic effects are neglected. Although an isotropic
WLC loop with length smaller enough than its persistence length is
planar, it is not obvious for the case of anisotropic DNA. In fact
experiments~\cite{amzallag06} and MD simulations~\cite{Lankas06}
show that DNA minicircles are not completely, but are almost planar.
The rest of paper is organized as follows: Section 2 describes the
model that is used in studying the short DNA loops, followed by the
presentation of the results in Section 3. Finally, Section 4
concludes the paper, while the overestimation of deviation of DNA
local twist from its mean value appears in Appendix.
\section{The Model}
The anisotropic elastic model of DNA represents the macromolecule as
an elastic rod of length $L$, parameterized by arclength $s$. As the
double-strand is ribbon-like, it is \ anisotropic in bending around
two different directions. On the other hand, DNA is twisted, so
while bending around a fixed axis, it should bend around each of the
two directions, periodically. It is energetically preferable for the
DNA to bend more around the ``soft'' axis and less around the
``hard'' one. To reduce elastic energy, the DNA also may modify its
twist to have more soft bending along its planar path.
For a planar DNA loop, the tangent unit vector $\hat{t}$ and the
twist angle $\psi$ at each point contain enough information to
parameterize DNA conformation. Actually an anisotropic bent DNA has
a higher tendency to go out of the bending plane which depends on
the value of anisotropy $(A_{1} - A_{2})/(A_{1} + A_{2})$. For short
loops, this
tendency decreases as the anisotropy becomes smaller~\cite{farshid03}.
Elastic energy of anisotropic rod is
\begin{equation}
\label{eq:E_A} E=\frac{1}{2}k_{B}T \int_{0}^{L}\left[
A(s)|\dot{\hat{t}}\,|^2+C(\dot{\psi} - \omega_0)^{2} \right]ds,
\end{equation}
where $A(s)$ is the local bending rigidity, $C$ is the twist
rigidity, $\omega_{0}$ is the spontaneous twist of the helix, and
the dots indicate derivatives with respect to $s$. The first term is
the usual elastic term, with $s$ dependent persistence length $A(s)$
and the second term is the energy needed to over-(under-)twist the
DNA, which may be implied by the boundary conditions on $\psi$ or
the anisotropic effects. It should be noted that in this model $s$
dependence of $A$ is not because of sequence dependence, but it is
due to rotation of soft and hard axes. As we will show later, this
is a source of twist-bend coupling, although there is no explicit
coupling term in the Hamiltonian.
DNA bending could be decomposed into two principal axes $\hat{e}_1$
and $\hat{e}_2$, attached to the DNA. The hard one, $\hat{e}_1$ is
perpendicular to double strand's local plane, and the soft one,
$\hat{e}_2$, is defined to lie in the local plane of the double
strand, and to be perpendicular to both strands. Due to the helical
structure of DNA, $\hat{e}_1$ and $\hat{e}_2$ rotate with the helix.
Since $\dot{\hat{t}}$ is perpendicular to $\hat{t}$ axis and lies
in the $\hat{e}_1-\hat{e}_2$ plane, by its decomposition in our
coordinate system we obtain
\begin{eqnarray}
\label{eq:E_A1A2} E &=& \frac{1}{2} k_{B}T \int_{0}^{L} \left[(A_1
\sin^{2}\psi +A_2 \cos^{2}\psi ) \, \dot{\theta}^{2} \right.
\nonumber
\\
&& ~~~~~~~~~~~~~ \left. + ~ C(\dot{\psi} - \omega_0)^{2}\right]ds ,
\end{eqnarray}
where $A_1$ and $A_2$ are constant bending
rigidities about the rotating axes, $\hat{e}_1$ and $\hat{e}_2$, and
$\theta$ is the angle between $\hat{t}$ and a fixed, arbitrary
direction in loop's plane. Using Euler-Lagrange equation and applying
corresponding boundary conditions, we are able to find $\theta(s)$ and
$\psi(s)$ as well as DNA's shape and bending energy of the loop. The
integral form of closed loop or ``ends meeting'' condition is
\begin{equation}
\label{eq:EndMeet} \int_{0}^{L}\sin\theta(s)\,\mathrm{d}s =
\int_{0}^{L}\cos\theta(s)\, \mathrm{d}s = 0.
\end{equation}
Also because the DNA strands are antiparallel they can not switch in
the ends and they should bind in phase, then we have
$\psi(L)-\psi(0)=2k\pi$ with $k$ being an integer that is the number
of full turns of the helix. On the other hand to avoid any
singularity in tangent vector changes we consider a constraint on
$\theta$ values in the ends by $\theta(L)-\theta(0)=2\pi$ (no self
crossing).
\section{Results}
Integrating Euler-Lagrange Equations of (\ref{eq:E_A1A2})
results following
equations of motions:
\begin{equation}
\label{eq:EL1} \dot{\theta}=\frac{\gamma}{(A_1+A_2) - (A_1-A_2)\cos 2\psi},
\end{equation}
\begin{equation}
\label{eq:EL3} \dot{\psi}^{2}=-\frac{2\gamma^2}{C} \,
\frac{1}{(A_1+A_2) - (A_1-A_2)\cos 2\psi}+\beta,
\end{equation}
where integral constants of $\gamma$ and $\beta$ should be
determined from the boundary conditions. As it is seen in above
equations, $\psi$ depends on $s$, as on the DNA's local
bending in a
complicated form. This twist-bend coupling is a direct result of
anisotropy of the model and vanishes in the case of isotropic rods.
Even if $A_1$ and $A_2$ differ by one order of magnitude,
which is the case of our double stranded DNA, the coupling is very
weak and $s$ dependence of $\dot{\psi}$ is negligible. Indeed,
numerical studies show that the relative variations of $\dot{\psi}$
is less than one percent, even if two bending rigidities differ by
two orders of magnitude~\cite{farshid03}. This leads us to consider
homogeneous twist along the DNA and set $\psi(s)= \omega s$, in
which $\omega=2 k \pi /L$ is found by applying the ``ends in phase''
boundary condition.
This is more discussed in Appendix. The above approximation
decouples torsional part of energy from its bending part.
Integrating~(\ref{eq:EL1}) with considering above approximation gives
\begin{eqnarray}
\theta(s)=\theta(0)&+&\frac{\gamma}{\omega\sqrt{A_1
A_2}} \\
&\times&\left( \tan^{-1}\left(\sqrt{\frac{A_1}{A_2}}\tan(\omega
s)\right) +\pi\left[\frac{1}{2}+\frac{\omega
s}{\pi}\right]\right),\nonumber
\end{eqnarray}
where the bracket means ``integer part''. This term is added to
get rid of discontinuity in the $\tan^{-1}$ function. Without lack
of generality, we set $\theta(0)=0$ and therefore the condition on
total bending results $\theta(L)=2\pi$. Applying this condition simply
lets us to fix $ \gamma = (2\pi/L) \sqrt{A_1
A_2} $. These would yield the functional form of $\theta(s)$ and
loop's shape as
\begin{equation}
\theta(s)=\frac{1}{k}\left(\tan^{-1}
\left(\sqrt{\frac{A_1}{A_2}}\tan(\omega s) \right) +\pi
\left[\frac{1}{2}+\frac{\omega s}{\pi} \right]\right).
\end{equation}
The above solution automatically satisfies the ``ends meeting''
condition~(\ref{eq:EndMeet}). $k$ in above equation counts the
number of turns of helix along the loop. To have minimum torsional
energy it should be fixed to the closest integer value to
$\omega_{0}L/2\pi$, so $k=round(\omega_{0}L/2\pi)$. Thus the DNA is
undertwisted in case
$\Delta\psi = \omega_{0} L - 2 k \pi < 0$
and is overtwisted otherwise.
In the case $A_1 >> A_2$ for a closed DNA loop, bending is not
homogeneous and it is localized in ``soft" parts. Here the loop
looks more like a polygon rather than a circle. This is the direct
effect of bending anisotropy. The
shape of looped DNA for anisotropic model is given in
figure~\ref{fig:conformation}. Increasing the ratio of $A_1/A_2$,
the polygonal shape is more visible (e.g. for $A_1/A_2>50$, the DNA
loop will have sharp edges at the soft points). In a full helix turn
the DNA meets the soft axis of
rotation two times when it bends in plane, thus the number of
polygon edges equals $2k$. As the number of edges ($\sim$ helical
turns) increases by length of the loop, the polygonal shape of the
loop is less visible for larger DNAs.
\begin{figure}
\includegraphics[width=1\columnwidth]{L.eps}
\includegraphics[width=1\columnwidth]{aniso.eps}
\caption{\textbf{Above}, Loop shapes for different DNA lengths. An
exaggerated ratio of $\frac{A_{1}}{A_{2}}=100$ for all illustrations has been
used. \textbf{Below}, Effect of
anisotropy on sharpness of polygonal edges. A loop length of
$L=9.6$~nm for illustrations has been used. In all figures $\omega=1.8$~
nm$^{-1}$ (see text).} \label{fig:conformation}
\end{figure}
To find the energy of the loop, we read $(A_1\sin^{2}\psi +
A_2\cos^{2}\psi)\, \dot{\theta}=\gamma$ from~(\ref{eq:EL1}) and
substitute it in~(\ref{eq:E_A1A2}),
\begin{equation}
\label{eq:E_gamma} E=\frac{1}{2}k_{B}T \int_{0}^{L} \left(\gamma
\dot{\theta} + C(\dot{\psi} - \omega_0)^{2} \right) ds.
\end{equation}
As $\int_{0}^{L} \dot\theta ds = 2\pi$ and $\dot\psi$ and $\gamma$
are constant, the elastic energy is
\begin{equation}
E=\frac{2\pi^{2}}{L}k_{B}T \sqrt{A_1 A_2} + \frac{k_{\rm B}T}{2L} C
\Delta\psi^2.
\end{equation}
The second term is the twist energy which is due to the
over-(under-)twist implied by the ``ends in phase'' condition on
DNA loop. Because $k$ is a step function of L, this term leads to a
well known oscillatory behavior which is damping by an $L^{-1}$
factor.
In analogy with the bending energy of a circular loop,
$(2\pi^{2}k_{B}T A/L)$, we can read the effective persistence length
of in-plane small loops as $\sqrt{A_1 A_2}$. This is different from
the effective persistence length of anisotropic DNA in larger
scales (equation~(\ref{Eq:harmonic})).
\section{Conclusion}
We studied the effect of bending anisotropy on planar DNA loops
using energy minimization and neglecting entropic effects under
constraints of parallel and in phase ends. Bending anisotropy causes
anisotropy in curvature and twist. However, anisotropy induced a
twist-bend coupling to the model even in the lack of explicit
appearance of such coupling in the Hamiltonian, though, it is small
enough to be neglected in the calculations. The bending anisotropy
results in polygonal shape of the loop. Increasing the anisotropy
makes the edges sharper where the number of helical turns and hence
the number of edges of the polygon grows with loop length.
Energy of such a loop includes an oscillating term for twist energy
and a bending term similar to that of anisotropic loop with an
effective persistence length $A=\sqrt{A_{1}A_{2}}$, which is
different from harmonic mean of $A_{1}$ and $A_{2}$
(equation~\ref{Eq:harmonic}) for an unconstrained long DNA. As the
geometric mean of $A_1$ and $A_2$ for $A_1 \neq A_2$ is always larger
than their harmonic mean, the results shows that considering
anisotropy in DNA bending rigidities is not able to explain large
loop formation probability of DNA minicircles. Thus it seems other
efforts, as like as bubble formation~\cite{YanMarkoPRL04}, effect
of sequence dependence~\cite{popov}, or generalized semiflexible
model~\cite{Wiggins2006} are more successful in this direction.
|
1,116,691,500,475 | arxiv | \section{Introduction
The main purpose of this paper is to deepen the study of qualitative theory of differential equations induced by differentiable vector fields of the plane which are not necessarily of class $C^1$. For instance, problems addressed in \cite{MR2096702, MR2287882,MR2266382} include global injectivity, uniqueness of solutions and asymptotic stability of the point at infinity of the Riemann sphere $\Real\cup \{\infty\}$. These works, like many others in the literature, extend to the differentiable case previous results concerning $C^1-$vector fields. Thus, the present paper pretends to extend to the differentiable case the work in \cite{MR2257430}. The authors of \cite{MR2257430} guarantee the change of stability at infinity of a one--parameter family of $C^1-$vector fields as the parameter goes through zero.
This behaviour was presented as Hopf bifurcation at infinity.
\par
Before \cite{MR2257430} the study of Hopf bifurcation at infinity had been approached only from two points of view. One deals with families which have finitely many parameters and are made up of continuous perturbations of linear systems. The other point of view deals with polynomial families. The former case uses the strong domination imposed by the linear part of the vector field on the continuous nonlinear term, which can have sub--linear growth \cite{MR1028236,MR1124985,MR1364308,MR1446100,MR1780452}. The later point of view involves standard Poincar\'e compactification \cite{MR967475,MR1213941,MR1231467,MR1421060}.
\par
Both points of view are related to the existence of limit cycles of arbitrarily large amplitudes. Particularly, the case of polynomial systems can be associated to the sixteenth Hilbert problem, see for instance \cite{MR934515,MR909943,MR2093918,MR3130553}. In contrast, a different point of view was given in \cite{MR2257430}, where the authors are only interested in the change of stability at infinity. This point of view has been strongly articulated in other works related to planar systems induced by differentiable vector fields \cite{MR1339178,MR2040002,MR2287882,MR2257430,MR3062761,MR3223368}. The main motivation of this approach was the definition of Hopf bifurcation at infinity given in \cite{MR909943} (see also \cite{MR934515,MR2142367}). Concretely, it is said in \cite{MR909943} that the polynomial family of vector fields $\{Z_{\mu}:-\epsilon<\mu<\epsilon\}$ presents a Hopf bifurcation at infinity when $\mu$ cross $0$ if\, \lq the infinity changes its stability\rq.
\par
The paper \cite{MR2257430} deals with a class of one--parameter family ${\{Z_{\mu}\; ;\; -\epsilon<\mu<\epsilon\}}$ of planar $C^1$ vector fields defined on the complement of an open ball centered at origin which are also free of singularities. The authors find the change of stability at infinity when the sign of $\displaystyle{\int_{\Real^2}\; div(\hat{Z_\mu})}$ varies as the parameter crosses zero. In the context of \cite{MR2257430}, $\hat{Z_\mu}: \Real^2 \to \Real^2$ is any $C^1-$ extension of $Z_\mu$ with divergence Lebesgue almost--integrable (see Section \ref{sec_prelim} for details). The families considered in \cite{MR2257430} are neither polynomial nor a perturbation of linear systems, consequently the work in \cite{MR2257430} contains the two point of view described above.
\par
We consider a one--parameter family $\{X_{\mu}(z)=X(z)+\mu z, \;\mu\in \Real\}$, where $X$ has isolated singularities and belongs to a class of differentiable (not necessarily of class $C^1$) planar vector fields defined on the complement of some compact ball centered at the origin. The main motivation to address the existence of Hopf bifurcation at infinity for this family is the work given in \cite{MR2096702} and \cite{MR2287882}. In those papers the authors prove the uniqueness of the positive trajectory starting at every point $p\in \Real^2$ due some conditions on the eigenvalues of the Jacobian matrix of the differentiable vector field. In addition, results in \cite{MR2287882} allow us to focus on the integral $\displaystyle{\int_{\Real^2}\; div(\hat{X_\mu})}$ in order to find a change of stability at infinity. In this context, since the qualitative change at infinity of the family $\{X_{\mu}(z)=X(z)+\mu z, \;\mu\in \Real\}$ may be exhibit from global extensions $\hat{X_\mu}: \Real^2 \to \Real^2$, the main obstruction to describe the bifurcation is the existence of unbounded sequences of singular points. In order to avoid this phenomenon, we impose some extra condition on the extension $\hat{X_\mu}$ which imply that every vector field of the family induces an injective map. Consequently, every vector field of the family has at most one singular point in the plane.
\par
Observe that the one--parameter family considered in this paper is not necessarily a perturbation of linear systems in the sense of \cite{MR1028236,MR1124985,MR1364308,MR1446100,MR1780452}.
Moreover, \mbox{Corollary \ref{cor:1}} complements the results presented in~\cite{MR2257430} for $C^1-$vector fields because in our work the existence of isolated singularities of $X$ is allowed.
\par
\section{Notation and definitions} \label{sec_prelim}
Let $\overline{D}_\sigma=\{z\in\mathbb{R}^2:||z||\leq\sigma\}$ be the compact ball bounded by
$\partial\overline{D}_\sigma=\{z\in\mathbb{R}^2:||z||=\sigma\}$ with $\sigma>0$, and suppose that $X:\mathbb{R}^2\setminus\overline{D}_{\sigma}\to\mathbb{R}^2$ is a differentiable vector field, defined on the complement of $\overline{D}_{\sigma}$ on $\mathbb{R}^2$.
Here, the term differentiable means \textit{Frechet differentiable} at each point $z\in\mathbb{R}^2\setminus\overline{D}_{\sigma}$ and $X^{\prime}(z):\mathbb{R}^2\to\mathbb{R}^2$ is the respective derivative. This derivative is defined as the bounded linear operator induced by the standard Jacobian matrix at $z$. This matrix is denoted by $DX(z)$, so $\mbox{det}(DX(z))$ and $\mbox{Trace}(DX(z))$ are the typical Jacobian determinant and divergence, respectively.
On the other hand, if the point $q\in\mathbb{R}^2\setminus\overline{D}_{\sigma}$ is kept fixed, then a \textit{trajectory} of $X$ starting at $q$ is defined as the integral curve $I_q\ni t\mapsto\gamma_q(t)\in \mathbb{R}^2\setminus\overline{D}_{\sigma}$, determined by a maximal solution of the Initial Value Problem $\dot{z}=X(z),~ z(0)=q$.
Of course, it means that $\frac{d }{dt}\gamma_q(t)=X\big(\gamma_q(t)\big),\forall t\in I_q$ and $\gamma_q(0)=q$. In this context, it is useful to have a term that refers to the image of the solution.
Hence, we define the \textit{orbit} of $\gamma_q$ to be the set $\{\gamma_q(t):t\in I_q\}$, and
we identify the trajectory with its orbit.
Similarly, ${\gamma^+_q}=\{\gamma_q(t):t\in I_q\cap[0,+\infty)\}$ and ${\gamma^-_q}=\{\gamma_q(t):t\in (-\infty,0]\cap I_q\}$ are the \textit{semi--trajectories} of $X$, and they are called \textit{positive} and \textit{negative}, respectively.
In consequence, $\gamma^-_q\cup \gamma^+_q=\gamma_q$, and each trajectory has its two limit sets, $\alpha(\gamma_q^-)$ and $\omega(\gamma_q^+)$.
These limit sets are well defined in the sense that they only depend on the respective solution.
And lastly, such a vector field induces a well defined \textit{positive semi--flow} (respectively \textit{negative semi--flow}), if the condition $q\in\mathbb{R}^2\setminus\overline{D}_{\sigma}$ implies the existence and uniqueness of $\gamma_q^{+}$ (respectively $\gamma_q^{-}$).
\par
The trajectories of a vector field may be unbounded.
One way to obtain some information about the behavior of such solutions is to compactify the plane, so that the vector field is extended to a new manifold that contains the \lq points at infinity\rq.\,
In this context, the so called Alexandroff compactification has been most successful in the study of planar systems induced by $C^1$-vector fields, not necessarily polynomial (see for example \cite{MR0176180,MR934515,MR2287882,MR2257430,MR3062761}).
To describe the results, $\mathbb{R}^2$ is embedded in the Riemann sphere $\mathbb{R}^2\cup\{\infty\}$.
Consequently, $(\mathbb{R}^2\setminus\overline{D}_\sigma)\cup\{\infty\}$ is the subspace of $\mathbb{R}^2\cup\{\infty\}$ with the induced topology, and \lq infinity\rq ~refers to the point $\infty$ of $\mathbb{R}^2\cup\{\infty\}$.
Moreover, a vector field $X:\mathbb{R}^2\setminus\overline{D}_\sigma\to\mathbb{R}^2\setminus\{0\}$ (without singularities) can be extended to a map
\[
\hat{X}:\big(({\mathbb{R}^2\setminus\overline{D}_\sigma})\cup\{\infty\},\infty\big)\longrightarrow(\mathbb{R}^2,0)
\]
(which takes $\infty$ to $0$).
In this manner, all questions concerning the local theory of isolated singularities of planar vector fields can be formulated and examined in the case of the extended vector field $\hat{X}$, which coincides with $X$ on $\mathbb{R}^2\setminus\overline{D}_\sigma$.
For instance, if $\gamma_p^+\subset\mathbb{R}^2\setminus\overline{D}_\sigma$ is an unbounded semi--trajectory of $X:\mathbb{R}^2\setminus\overline{D}_\sigma\to\mathbb{R}^2$ with empty $\omega-$limit set, then we declare that ${\gamma_p^+}$ \textit{goes to infinity}, and we write $\omega({\gamma_p^+})=\infty$.
Similarly, $\alpha({\gamma_p^-})=\infty$ denotes that $\gamma_p^-$ \textit{comes from infinity}, and it means that $\gamma_p^-\subset\mathbb{R}^2\setminus\overline{D}_\sigma$ is an unbounded semi--trajectory whose $\alpha-$limit set is empty.
Therefore, it is also possible to talk about the phase portrait of $X$ in a neighborhood of $\infty$, as shown \cite[\mbox{Proposition 29}]{MR2287882}.
Throughout this paper, given $C\subset\mathbb{R}^2$, a closed (compact, no boundary) curve ($1-$manifold), $\overline{D}(C)$ (respectively $D(C)$) denotes the compact disc (respectively open disc) bounded by $C$.
Thus, the boundaries $\partial \overline{D}(C)$ and $\partial D(C)$ are equal to $C$ besides homeomorphic to the circle $\partial D_1=\{z\in\mathbb{R}^2:||z||=1\}$.
\subsection{Index at infinity}
The index at infinity was defined firstly for $C^1-$vector fields in \cite{MR1339178} and lately in \cite{MR2287882} for differentiable vector fields (not necessarily of class $C^1$). The relationship between this index and the stability of the point at infinity of the vector field is also stated in those works in both cases.
\begin{defn}[\cite{MR2287882}] Consider $X:\mathbb{R}^2\setminus\overline{D}_\sigma\to\mathbb{R}^2$ be a differentiable vector field. The \textit{index of $X$ at infinity}, denoted by $\mathcal{I}(X)$, is the number of the extended line $[-\infty,+\infty]$ given by
\[
\mathcal{I}(X)={\int_{\mathbb{R}^2}}\mbox{\rm
Trace}(D\widehat{X})dx\wedge dy,
\]
where $\widehat{X}:\mathbb{R}^2\to\mathbb{R}^2$ is a global differentiable vector field such that:
\begin{itemize}
\item In some $\mathbb{R}^2\setminus\overline{D}_s$ with $s\geq\sigma$, both $X$ and $\widehat{X}$ coincide.
\item $z\mapsto\mbox{\rm Trace}(D\widehat{X}(z))$ is Lebesgue almost--integrable on $\mathbb{R}^2,$ in the sense {of \cite{MR2287882}}.
\end{itemize}
\end{defn}
This index is a well--defined number in $[-\infty,+\infty]$, and it does not depend on the pair $(\widehat{X},s)$ as shown in \cite[\mbox{Lemma 12}]{MR2287882} (see also \cite{MR2257430} for the $C^1$ case).
\subsection{Hopf bifurcation at infinity} \label{def:hopfbif
The definition of Hopf bifurcation at infinity for differentiable vector fields presented in this paper is based on the definition of Hopf bifurcation at infinity for $C^1$ vector fields given in \cite{MR1339178} and the definition of attractor/repellor given in \cite{MR2287882} for differentiable vector fields.
In \cite{MR2287882}, the authors relate the index at infinity of a differentiable vector field with the stability of the point at $\infty$ of the Riemann sphere.
\begin{defn}[\cite{MR2287882,MR3062761}]\label{def:atractor repellor}
The point at infinity $\infty$ of the Riemann sphere $\mathbb{R}^2\cup\{\infty\}$ is an \textit{attractor} (res\-pec\-ti\-vely, a \textit{repellor}) for
$X:\mathbb{R}^2\setminus\overline{D}_\sigma\to\mathbb{R}^2$ if:
\begin{itemize}
\item There is a sequence of closed curves, transversal to $X$ and tending to infinity.
It means that for every $r\geq\sigma$ there exists a closed curve $C_r$ such that $D(C_r)$ contains $D_r$ and $C_r$ has transversal contact to each small local integral curve of $X$ at any $p\in C_r$.
\item For some $C_s$ with $s\geq\sigma,$ all the trajectories $\gamma_p$ starting at a point $p\in
\mathbb{R}^2\setminus\overline{D}(C_s)$ satisfy $\omega({\gamma^+_p})=\infty$, that is $\gamma_p^+$ goes to infinity (respectively, $\alpha({\gamma^-_p})=\infty$, that is $\gamma_p^-$ comes from infinity).
\end{itemize}
\end{defn}
Notice that in the $C^1$ case, \mbox{Definition \ref{def:atractor repellor}} is equivalent to saying that the vector field $\hat{X}$ induced by $X$ on the Riemann sphere is locally topologically equivalent in an open neighborhood of the infinity to $z\mapsto -z$ (respectively, $z\mapsto z$) at the origin \cite{MR1339178,MR2040002,MR2257430}.
\par
The authors of \cite{MR2287882} relate the index at infinity of a differentiable vector field with the stability of the point at $\infty$ of the Riemann sphere.
\begin{defn}[\cite{MR2257430}]\label{def:hopf
We will say that the family of differentiable vector fields $\big\{X_{\mu}:\mathbb{R}^2\setminus{\overline{D}_{\sigma}}\to\mathbb{R}^2\; :\;-\varepsilon<\mu<\varepsilon\big\}$ has at $\mu=0$ a \textit{Hopf bifurcation at $\infty$} if the following two conditions are satisfied:
\begin{itemize}
\item For $\mu<0$ (resp. $\mu>0$), the vector field $X_{\mu}$ has a repellor at $\infty$, and for $\mu>0$ (resp. $\mu<0$), the vector field $X_{\mu}$ has an attractor at $\infty$.
\item The vector field $X_{\mu}$ has no singularities in $\mathbb{R}^2\setminus{\overline{D}_{s}}$, for $-\varepsilon<\mu<\varepsilon$ and some $s\geq\sigma$.
\end{itemize}
\end{defn}
In order to capture the essential features of \mbox{Definition \ref{def:hopf}}, we remark that the element $X_0$ might be unstable in the sense that the vector field $\hat{X_0}$ induced by $X_0$ on the Riemann sphere might be locally topologically equivalent in an open neighborhood of the infinity to a center at the origin.
In this context, $X_0$ has first degree of instability \cite{MR0344606} in the sense that it is unstable whereas any vector field in $\{X_{\mu}\}$, with $\mu$ sufficiently close to $0$, is either stable (i.e $\infty\in\{attractor, repellor\}$) or topologically equivalent to $X_0$ (see Figure \ref{fig:mainf proposition}). In Section \ref{sec:free eigen}, we also impose some condition on $X_0$ in order to force a specific family $\{X_{\mu}\}$ to present a Hopf bifurcation at $\infty$ as a topological characterization of the bifurcation when $\infty$ reverses its stability.
\par
The authors of \cite{MR2257430} relate the existence of a \emph{Hopf bifurcation at infinity} for a family of $C^1-$vector fields with the \emph{index at infinity} of each vector field of the family.
\begin{thm}[\cite{MR2257430}] \label{teoAGG07} Let $\, \{ Z_{\mu} :\Real^2\setminus D_{r}\; ;\; -\varepsilon < \mu
< \varepsilon\} \,$ be a family of $C^1-$vector fields such that:
\begin{enumerate}
\item[(1)] $z\mapsto\mbox{\rm Trace}\big(DZ_{\mu}(z)\big)$ is Lebesgue almost--integrable on $\mathbb{R}^2\setminus{D_{r}}$,
\item[(2)] $\mathcal{I}(Z_{\mu})$ is well defined, this index belongs to $[-\infty,+\infty]$, and
\[
{\int_{\sigma}^{+\infty}}\Upsilon_{\mu}(r) dr=+\infty,
\]
where $\Upsilon_{\mu}(r)=\inf\big\{||Z_{\mu}(z)||:||z||=r\big\}$,
\item[(3)] $\mu\neq0$ implies that the product $\mu \cdot\mathcal{I}(Z_{\mu}) >0$,
\item[(4)] $Z_{\mu}$ has no singularities, and the \textit{Poincar\'e index} at $\infty$ of the vector field
\[\hat{Z}_{\mu}:\big(({\mathbb{R}^2\setminus{D_r}})\cup\{\infty\},\infty\big)\longrightarrow(\mathbb{R}^2,0)\]
(which extends $Z_{\mu}$ at $\infty$) is less than or equal to $1$.\\
\end{enumerate}
Then $ \mu = 0 $ is a Hopf bifurcation at $ \infty $ of the family $ \{ X_{\mu}\} $.
\end{thm}
\mbox{Theorem \ref{teoAGG07}} is the one we extend to the differentiable case in \mbox{Theorem \ref{main 1}}.
\section{Hopf bifurcation at infinity for dissipative vector fields}\label{sec:3
A differentiable planar vector field $X=f\frac{\partial}{\partial x}+g\frac{\partial}{\partial y}$ is called \emph{dissipative} on a region $D$ if $div(X)=\frac{\partial f}{\partial x}+\frac{\partial g}{\partial y}< 0$ on $D$ and the equality only holds on a set with Lebesgue measure zero. See \cite{MR1094380} for more details.
Let $\mathcal{H}(2,\sigma)$ be the set of differentiable vector fields $X:\mathbb{R}^2\setminus{\overline{D}_\sigma}\to\mathbb{R}^2$ which are dissipative and have strictly positive Jacobian determinant. This means that all vector field $X$ in $\mathcal{H}(2,\sigma)$ verifies
\begin{equation}\label{eq:spectrum dissipative}
\mbox{\rm Spc}(X)\subset\Big\{z\in\mathbb{C}:\Re(z)\leq0\Big\}\setminus\big\{(0,0)\big\},
\end{equation} where $Spc(X)\subset \mathbb{C}$ is the set of eigenvalues of $DX(z)$, for all $z\in \mathbb{R}^2\setminus{\overline{D}_\sigma}$.
\par
In this section we study the case of dissipative vector fields in $\mathcal{H}(2,\sigma)$. This is complemented in the next section, where we focus on vector fields that are free of real eigenvalues. Observe that the divergence of every vector field verifying \eqref{eq:spectrum dissipative} is Lebesgue almost--integrable on $\mathbb{R}^2\setminus{\overline{D}_\sigma}$, in the sense of \cite{MR2287882}.
\par
Recall that, a map $\widetilde{X}:\mathbb{R}^2\to \mathbb{R}^2$ is a local homeomorphism (respectively a local diffeomorphism) if for every $z\in\mathbb{R}^2$ there exist open neighborhoods $U\subset \mathbb{R}^2$ of $z$ y $V\subset \mathbb{R}^2$ of $\widetilde{X}(z)$ such that the mapping
U\ni p\mapsto\widetilde{X}(p)\in V
$
is a continuous (resp. differentiable) bijection whose inverse is also continuous (resp. differentiable).
\subsection{On extensions of dissipative vector fields}\label{sec:extensions
Consider the family of vector fields $$\{X_{\mu}(z)= X(z)+ \mu z \; : \; \mu \in \Real \},$$ where $X\in \mathcal{H}(2,\sigma)$ and $X$ has some singularity in $\mathbb{R}^2\setminus{\overline{D}_\sigma}$. In this section, we also prove the existence of globally injective extensions of $X_{\mu}: \Real^2\setminus{\overline{D}_{\sigma}}\to \Real^2$ to the whole plane whose divergence is Lebesgue almost--integrable in $\mathbb{R}^2$.
\par
Consider $X=(f,g) \in \mathcal{H}(2,\sigma)$. Here, $\mathcal{F}(h)$ with $h\in\{f,g,\tilde{f},\tilde{g}\}$ denotes the continuous foliation given by the level sets $\{h=\mbox{constant}\}$ and then the leaves of the foliations are differentiable.
The concept of half-Reeb component is a natural generalization of the description of the planar foliations given in \cite{MR0089412}.
More precisely, $\mathcal{A}$ is a \textit{half--Reeb component} of $\mathcal{F}(h)$ if there is a homeomorphism
\[
H: B=\Big\{(x,y)\in [0,2]\times[0,2]:0<x+y\leq 2 \Big\}\to\mathcal{A},
\]
which is a topological equivalence between $\mathcal{F}(h) \vert_{\mathcal{A}}$ and ${\mathcal{F}}(h_0) \vert_B$ with $h_0(x,y)=xy$ such that:
\begin{itemize}
\item The segment $\{(x,y)\in B : x+y=2\}$ is sent by $H$ onto a transversal section for the foliation $\mathcal{F}(h)$ in the complement of the point $H(1,1)$.
\item Both segments $\{(x,y)\in B : x=0 \}$ and $\{(x,y)\in B : y=0\}$ are sent by $H$ onto full half--leaves of $\mathcal{F}(h)$.
\end{itemize}
\begin{lem}\label{lem: existence of the index} If $X\in\mathcal{H}(2,\sigma)$ has some singularity, the following hold:
\begin{itemize}
\item[(a)] $\exists s_0\geq\sigma$ and a globally injective local homeomorphism $\widetilde{X}:\mathbb{R}^2\to\mathbb{R}^2$ such that $\widetilde{X}(0)=0$ and $\widetilde{X}$ and $X$ coincide on $\mathbb{R}^2\setminus{\overline{D}_{s_0}}$;
\item[(b)] the index $\mathcal{I}(X)$ at infinity is a well defined number in $[-\infty,+\infty)$.
\end{itemize}
\end{lem}
\begin{proof}
Observe that every $X=(f,g)\in\mathcal{H}(2,\sigma)$ satisfies $\mbox{\rm Spc}(X)\cap[0,+\infty)=\emptyset$ and $\mbox{\rm Spc}(X)\subset\{z\in\mathbb{C}:\Re(z)\leq0\}$. Thus, the methods of \cite[Proposition 2]{MR3062761} lead us to obtain that:
\begin{itemize}
\item[(a.1)] Any half--Reeb component of either $\mathcal{F}(f)$ or $\mathcal{F}(g)$ is bounded.
\end{itemize}
Under conditions (a.1) and $\mbox{\rm Spc}(X)\cap[0,+\infty)=\emptyset$, the means developed in the last section of \cite{MR2266382} (see Proposition 5.1) can be applied. Consequently, the proof of \cite[Theorem 3]{MR3062761} gives the existence of a closed curve $C$, surrounding the singularity of $X$ and the point at the origin, so that $C$ is embedded in the plane, and it admits an exterior collar neighborhood $U\subset\mathbb{R}^2\setminus D(C)$ such that:
\begin{itemize}
\item [(a.2)] $X(C)$ is a non--trivial closed curve surrounding the origin, $X(U)$ is an exterior collar
neighborhood of $X(C)$ and the restriction $X|_{U}:U\to X(U)$ is a homeomorphism.
\item[(a.3)] The foliation $\mathcal{F}(f)$, restricted to $\mathbb{R}^2\setminus D(C)$ is topologically equivalent to the foliation made up by all the vertical straight lines on $\mathbb{R}^2\setminus D_1$.
\end{itemize}
By Schoenflies Theorem, \cite{MR1535106,MR728227,MR2190924}, the map $X|_{C}:C\to X(C)$ can be extended to a homeomorphism ${X_1}:{\overline{D}(C)}\to{\overline{D}(X(C))}$ with ${X_1}(0)=0$.
We also extend $X|_{\mathbb{R}^2\setminus D(C)}:\mathbb{R}^2\setminus D(C)\to \mathbb{R}^2$ to $\widetilde{X}=(\tilde{f},\tilde{g}):\mathbb{R}^2 \to \mathbb{R}^2$ by defining $\widetilde{X}{|_{\overline{D}(C)}}=X_1.$
Thus, $\widetilde{X}|_{U}:U\to X(U)$ is a homeomorphism, and $U$ (resp. $X(U)$) is a exterior collar neighborhood of $C$ (resp. $X(C)$).
\par
Furthermore, (a.3) implies that $\widetilde{X}$ is a local homeomorphism whose foliation $\mathcal{F}(\tilde{f})$ is trivial.
Therefore, $\tilde{X}$ is injective by \cite[Proposition 1.4]{MR2096702}. Observe that $\tilde{X}(0)=0$ by construction. Hence $(a)$ holds considering $s_0\geq\sigma$ with $D(C)\subset\overline{D}_{s_0}$.
\par
Under these conditions, \cite[\mbox{Theorem 11}]{MR2287882} gives the existence of some $r>s_0$ such that the restriction $X|_{\mathbb{R}^2\setminus \overline{D}_{r}}:{{\mathbb{R}^2\setminus \overline{D}_{r}}}\to \mathbb{R}^2$ admits a global differentiable extension $\widehat{X}$ which divergence is Lebesgue almost--integrable on $\mathbb{R}^2$ and $\widehat{X}(0)=0$. Consequently, the index of $X$ at infinity $\mathcal{I}(X)$ is a well defined number of the extended real line $\in[-\infty,+\infty)$. This concludes the proof.
\end{proof}
\begin{cor}\label{cor:3.2} If $X\in\mathcal{H}(2,\sigma)$ has some singularity and $\mu\leq 0$, then $\exists s_{\mu}>\sigma$ such that:
\begin{itemize}
\item[(a)] the restriction $X|_{{\mathbb{R}^2\setminus \overline{D}_{s_{\mu}}}}:{{\mathbb{R}^2\setminus \overline{D}_{s_{\mu}}}}\to \mathbb{R}^2$ satisfies the statement of \mbox{Lemma \ref{lem: existence of the index}};
\item[(b)] the vector field $X_{\mu}$ has a well--defined index at infinity and
\[\mu<0\Rightarrow\mathcal{I}(X_{\mu})<0.\]
\end{itemize}
\end{cor}
\begin{proof} Observe that a direct computation of the eigenvalues in $X_{\mu}(z)=X(z)+\mu z$ shows that
\begin{equation}\label{eq: translation spectrum}
\mbox{Spc}(X_{\mu})=\mu+\mbox{Spc}(X).
\end{equation}
Thus, as a consequence of \eqref{eq: translation spectrum}, we can obtain that:
\begin{equation}\label{eq:strong dissipative}
\mbox{Spc}(X_{\mu})\subset\{z\in\mathbb{C}:\Re(z)\leq\mu\}.
\end{equation}
Therefore, for every $\mu\leq0$, (a) follows by \mbox{Lemma \ref{lem: existence of the index}}.
Item (b) holds by \eqref{eq:strong dissipative} and definition of index at infinity of $X_{\mu}$, provided by $$\mbox{Trace}(DX_{\mu})\leq\mu<0.$$
\end{proof}
\begin{cor}\label{cor:3.2BIS} Let $X\in\mathcal{H}(2,\sigma)$ with some singularity. Suppose in addition that
\begin{equation} \nonumber
\mbox{det}(DX_{\mu})>0,\mbox{ for all $\mu$ in some open interval } (0,\epsilon_0).
\end{equation}
Then, $\exists \varepsilon >0$ such that for every $\mu\in [-\varepsilon, \varepsilon]$ there exists $s_{\mu}>\sigma$ for which the restriction $X|_{{\mathbb{R}^2\setminus \overline{D}_{s_{\mu}}}}:{{\mathbb{R}^2\setminus \overline{D}_{s_{\mu}}}}\to \mathbb{R}^2$ satisfies the statement of Lemma \ref{lem: existence of the index}.
\end{cor}
\begin{proof}
By hypothesis, $\mbox{det}(DX_{\mu})>0$ for all $\mu\in(0,\epsilon_0)$, then \eqref{eq: translation spectrum} implies that
\[
\mbox{Spc}(X_{\mu}) \cap\Big(-\frac{\epsilon_0}{4},+\infty\Big)=\emptyset, \quad\mbox{when} \quad \mu\in\Big[-\frac{\epsilon_0}{4},\frac{\epsilon_0}{4}\Big].
\]
Consider $\varepsilon=\frac{\epsilon_0}{4}>0$. Thus, Proposition 2.7 in \cite{MR2266382} implies that, for all $\mu\in[-\varepsilon,\varepsilon]$, $X_{\mu}=(f_{\mu},g_{\mu})$ has the following property: \lq\lq Any half--Reeb component of either $\mathcal{F}(f_{\mu})$ or $\mathcal{F}(g_{\mu})$ is bounded\rq\rq.
Therefore, the proof of Lemma \ref{lem: existence of the index} holds verbatim.
\end{proof}
\begin{rem}\label{rem:main} Under conditions of Corollary \ref{cor:3.2BIS}, the set $\big\{s_{\mu}\geq\sigma:\mu\in[-\epsilon,\epsilon]\big\}$ is a well defined set. In addition, this set is bounded when $\mu\mapsto s_{\mu}$ is continuous.
\end{rem}
When the extension given by Lemma \ref{lem: existence of the index}, Corollary \ref{cor:3.2} and Corollary \ref{cor:3.2BIS} is also a local diffeomorphism, we say that the vector field $\widetilde{X}_{\mu}:\Real^2 \to \Real^2$ is a \textit{strong extension} of the vector field $X_{\mu}:\Real^2\setminus \overline{D}_{s_\mu}\to \Real^2$.
\subsection{Global vector fields}\label{sec: global case} In this section we study the stability at infinity of the family of global vector fields $$\displaystyle{\{Y_{\mu}(z)= Y(z)+ \mu z \; : \;\mu\in \Real\}},$$ where $Y: \Real^2\to \Real^2$ is a vector field such that $Y(0)=0$ and the restriction $Y|_{\mathbb{R}^2\setminus{\overline{D}_{\sigma}}}:\mathbb{R}^2\setminus{\overline{D}_{\sigma}}\to\mathbb{R}^2$ belongs to $\mathcal{H}(2,\sigma)$.
\begin{prop}\label{prop:without sing
If there exists $\varepsilon_0>0$ such that the map $Y_{\mu}(z)=Y(z)+\mu z$ is a local diffeomorphism, for all $\mu\in(-\varepsilon_0,\varepsilon_0)$. Then, $\exists \varepsilon>0$ such that the restriction $Y_{\mu}|_{\mathbb{R}^2\setminus{\overline{D}_{\sigma}}}:\mathbb{R}^2\setminus{\overline{D}_{\sigma}}\to\mathbb{R}^2$ has no singularities in $\mathbb{R}^2\setminus{\overline{D}_{\sigma}}$, for every $\mu\in[-\varepsilon,\varepsilon]$.
\end{prop}
\begin{proof}
As a consequence of \eqref{eq: translation spectrum}, the following assumptions are equivalent
\begin{itemize}
\item[(i)] $\displaystyle{0\not\in\mbox{Spc}(Y_{\mu}) \quad \text{for all} \; \mu\in[0,\varepsilon_0) \quad (\text{resp. \; for all} \; \mu\in(-\varepsilon_0,0]),}$
\item[(ii)] $\displaystyle{ \mbox{Spc}(Y)\cap(-\varepsilon_0,0]=\emptyset \quad (\text{resp.} \quad \mbox{Spc}(Y)\cap[0,\varepsilon_0)=\emptyset).}$
\end{itemize}
Since the restriction $Y|_{\mathbb{R}^2\setminus{\overline{D}_{\sigma}}}:\mathbb{R}^2\setminus{\overline{D}_{\sigma}}\to\mathbb{R}^2$ belongs to $\mathcal{H}(2,\sigma)$, condition \eqref{eq: translation spectrum} implies that
$$\mbox{Spc}(Y_\mu|_{\Real^2\setminus \overline{D}_{\sigma}})\bigcap\left(-\frac{\varepsilon_0}{4},\frac{\varepsilon_0}{4}\right)=\emptyset, \; \forall \mu \in [-\frac{\varepsilon_0}{2},\frac{\varepsilon_0}{2}].$$
Hence, by \cite[Theorem 2.1]{MR2266382}, the following holds:
\begin{equation}\label{eq:Thm C gut}
Y_{\mu}:\mathbb{R}^2\to\mathbb{R}^2 \quad\mbox{is globally injective, for every }\mu\in\Big[\displaystyle-\frac{\varepsilon_0}{2},\frac{\varepsilon_0}{2}\Big],
\end{equation}
(see also \cite[Theorem 3]{MR2609212} and \cite{MR1045818}).
\par
Consequently, since $Y_{\mu}(0)=0$, \eqref{eq:Thm C gut} implies that there exists $\varepsilon=\frac{\varepsilon_0}{2}>0$ such that the restriction $Y_{\mu}|_{\mathbb{R}^2\setminus{\overline{D}_{\sigma}}}:\mathbb{R}^2\setminus{\overline{D}_{\sigma}}\to\mathbb{R}^2$ has no singularities in $\mathbb{R}^2\setminus{\overline{D}_{\sigma}}$, for every $\mu\in[-\varepsilon,\varepsilon]$. Thus, the proposition holds.
\end{proof}
\begin{prop}\label{prop:main global
Let $\varepsilon >0$ be as in Proposition \ref{prop:without sing}. Suppose in addition that $\exists s>\sigma$ such that, for each $0<\mu\leq\varepsilon$,
\begin{itemize}
\item[(1)] $Y_{\mu}$ induces a well--defined negative semi--flow on $\mathbb{R}^2\setminus\overline{D}_s$,
\item[(2)] the index $\mathcal{I}(Y_\mu)\in[-\infty,+\infty)$ satisfies $\mu\cdot\mathcal{I}(Y_\mu)>0$.
\end{itemize}
Then, for each $\mu\in[-\varepsilon,\varepsilon]$, $\infty$ is an attractor (respectively a repellor) for the vector field $Y_{\mu}|_{\mathbb{R}^2\setminus{\overline{D}_{s}}}:\mathbb{R}^2\setminus{\overline{D}_{s}}\to\mathbb{R}^2$ provided by $\mathcal{I}(Y_\mu)>0$ (respectively $\mathcal{I}(Y_\mu)<0$).
\end{prop}
\begin{proof}
Observe that, for all $-\varepsilon\leq \mu<0$
$$\mbox{Spc}(Y_{\mu}|_{\mathbb{R}^2\setminus\overline{D}_{\sigma}})\subset \{z\in \mathbb{C} : \Re(z)\leq \mu<0\}\setminus (-\frac{\varepsilon}{2},0]$$
Hence, for all $s_0 \geq \sigma$ the following hold:
\begin{itemize}
\item[(a.1)] $\mbox{\rm Trace}\big(DY_{\mu}\big)<0$ and so $\;z\mapsto\mbox{\rm Trace}\big(DY_{\mu}(z)\big)$ is Lebesgue almost--integrable in $\mathbb{R}^2\setminus\overline{D}_{s_0}$(\cite[\mbox{Lemma 7}]{MR2287882});
\item[(a.2)] $\mathcal{I}(Y_{\mu})$ is a well-defined number in $[-\infty,+\infty)$ (\cite[\mbox{Corollary 13}]{MR2287882}).
\item[(a.3)] for every $z\in \mathbb{R}^2\setminus\overline{D}_{s_0}$, there is only one
positive semi--trajectory of $Y_{\mu}$ passing through $z$.
(\cite[\mbox{Teorema 4}]{MR2287882}).
\end{itemize}
Therefore, $\mathcal{I}(Y_{\mu})$ is a well--defined number and $\mathcal{I}(Y_{\mu})\neq 0$, for all $\mu \in [-\varepsilon, \varepsilon]$ .
\begin{itemize}
\item[(a.4)] We claim that for every $\mu\in[-\varepsilon,\varepsilon]$, the condition $\mathcal{I}(Y_\mu)\neq0$ implies that $Y_{\mu}$ has a bounded set of periodic trajectories
\end{itemize}
Since $Y_{\mu}(0)=0$ and $Y_{\mu}$ is injective by \eqref{eq:Thm C gut}, we proceed by contradiction:
\begin{itemize}
\item [(*)] We suppose that $\{\Gamma_1,\Gamma_2,\dots,\Gamma_n,\dots\}$ is an unbounded set of periodic trajectories of $Y_{\mu}$ such that
\[
\overline{D}(\Gamma_1)\subset\overline{D}(\Gamma_2)\subset\cdots\subset\overline{D}(\Gamma_n)\subset\cdots
\]
\end{itemize}
Under these conditions, by using the Green Theorem in $\overline{D}(\Gamma_n)$ and the arc length element $ds,$ we obtain that
\[
{\int_{\overline{D}(\Gamma_n)}}\mbox{\rm
Trace}(DY_{\mu})dx\wedge dy=\oint_{\Gamma_n}\langle Y_{\mu}(s),\eta_n^{e}(s)\rangle ds,
\]
where $\eta_n^{e}(p)$ is the unitary outer normal vector to $\Gamma_n$ and $\langle Y_{\mu}(s),\eta_n^{e}(s)\rangle$ is the inner product of $Y_{\mu}(s)$ with $\eta_n^{e}(s)$.
Thus,
\[
\mathcal{I}(Y_{\mu})=\lim_{n\to\infty}{\int_{\overline{D}(\Gamma_n)}}\mbox{\rm
Trace}(DY_{\mu})dx\wedge dy=0,
\]
provided by $\langle Y_{\mu}(s),\eta_n^{e}(s)\rangle=0$, for all $\eta_n^{e}(s)$. This contradiction with $\mathcal{I}(Y_\mu)\neq0$ shows that (*) never happens, and gives the proof of (a.4).
Furthermore, since $[-\varepsilon,\varepsilon]$ is a compact set, the following is directly obtained applying (a.4):
\begin{itemize}
\item[(a.5)] $\exists s>s_0>\sigma$ such that $Y_{\mu}$ has no periodic trajectories in $\mathbb{R}^2\setminus\overline{D}_s$ as long as $\mu\in(-\varepsilon,\varepsilon)$ and $\mathcal{I}(Y_\mu)\neq0$.
\end{itemize}
Hence, $Y_\mu$ generates a positive semi--flow on $\mathbb{R}^2\setminus\overline{D}_s$, for all $-\varepsilon<\mu<0$. In addition, for $s>\sigma$ as in (a.5) we have that for all $\mu\in(-\varepsilon,\varepsilon)$ the following hold:
\begin{itemize}
\item[(a.6)] $Y_\mu(0)=0$;
\item[(a.7)] $\mbox{det}(DY_{\mu}(z))>0$, $\forall z\in\mathbb{R}^2$, so $Y_\mu$ preserves orientation;
\item[(a.8)] there exists $c>0$ such that $||Y_\mu(z)||>c$, for any $z\in \mathbb{R}^2\setminus{\overline{D}_{s}}$, by the openness of $Y_\mu$.
\end{itemize}
Under these properties: \eqref{eq:Thm C gut} and (a.1) to (a.8), the vector field $Y_{\mu}$ satisfies all the conditions of \cite[Theorem 26]{MR2287882}. Consequently:
\begin{itemize}
\item[(a.9)]
For every $r\geq s$ there exist a closed curve $C_r$ transversal to $Y_{\mu}$ contained in the regular set $\mathbb{R}^2\setminus \overline{D}_r$. In particular, $D(C_r)$ contains $D_r$ and $C_r$ has transversal contact to each small local integral curve of $Y$ at any $p\in C_r$.
\end{itemize}
Moreover, for $r\geq s$ large enough the closed curve $C_r\subset\mathbb{R}^2$ is transversal to $Y_{\mu}$, near $\infty$ and it is such that $\mu\in(-\varepsilon,\varepsilon)$, the index $\mathcal{I}(Y_{\mu})$ and
\[
{\int_{\overline{D}(C_r)}}\mbox{\rm Trace}(DY_{\mu})dx\wedge dy=\oint_{C_r}\langle Y_{\mu}(s),\eta_n^{e}(s)\rangle ds
\]
have the same sign. Thus, the point at infinity of the Riemann sphere $\mathbb{R}^2\cup\{\infty\}$ is either an attractor or a repellor of $Y_{\mu}:\mathbb{R}^2\setminus\overline{D}_s\to\mathbb{R}^2$.
Actually, by (a.9) and \cite[Theorem 28]{MR2287882} we obtain that:
\begin{enumerate}
\item[(a.10)] If $\mathcal{I}(Y_{\mu})<0$ (respectively $\mathcal{I}(Y_{\mu})>0$), then $\infty$ is a repellor (respectively an attractor) of the vector field $Y_{\mu}:\mathbb{R}^2\setminus\overline{D}_s\to\mathbb{R}^2$.
\end{enumerate}
It concludes the proof, and the proposition holds.
\end{proof}
\subsection{Hopf bifurcation at infinity for dissipative vector fields} \label{subsec: hopf bif
We are now ready to state our main theorem.
\begin{thm}\label{main 1
Consider the family $\displaystyle{\{X_{\mu}(z)=X(z)+\mu z \; : \; \mu\in\mathbb{R}\}}$ where $X$ is a differentiable vector field in $\mathcal{H}(2,\sigma)$ with some singularity and verifying
\begin{equation} \label{eq:det}
\mbox{det}(DX_{\mu})>0,\mbox{ for all $\mu$ in some open interval } (0,\epsilon_0).
\end{equation}
Suppose in addition that
\begin{enumerate}
\item $\exists \varepsilon_0>0$ such that the set $\big\{s_{\mu}\geq\sigma:\mu\in[-\varepsilon_0,\varepsilon_0]\big\}$ induced by strong extensions is bounded;
\item for each $\mu>0$, $X_{\mu}$ induces a well defined negative semi--flow and the index $\mathcal{I}(X_{\mu})\in[-\infty,+\infty]$ satisfies $\mu \cdot\mathcal{I}(X_{\mu}) >0$.
\end{enumerate}
Then there are $\varepsilon>0$ and $s> \sigma$ such that the family of vector fields
\[\Big\{X_{\mu}:\mathbb{R}^2\setminus{\overline{D}_s}\to\mathbb{R}^2\; ;\; -\varepsilon<\mu<\varepsilon\Big\}\]
has at $\mu=0$ a Hopf bifurcation at $\infty$.
\end{thm}
\begin{proof}
By the assumption (1) there exist a number $\tilde{s}=\sup\{s_{\mu}\geq\sigma:-\epsilon_0\leq\mu\leq\epsilon_0\}$ and a local diffeomorphism $\widehat{X}_0:\mathbb{R}^2\to\mathbb{R}^2$ with $\widehat{X}_0(0)=0$ such that
\begin{equation}\nonumber
\widehat{X}_0(z)+\mu z=X_{\mu}(z),\quad\mbox{for all} \; z\in\mathbb{R}^2\setminus\overline{D}_{\tilde{s}} \quad\mbox{and } \; \mu\in(-\epsilon_0,\epsilon_0).
\end{equation}
In addition, the map $\widehat{X}_\mu:\Real^2\to \Real^2$ such that $\widehat{X}_\mu(z)=\widehat{X}_0(z)+\mu z$ is also a local diffeomorphism. Therefore, there exist $s\geq\tilde{s}\geq\sigma$ and $0<\varepsilon<\epsilon_0$ such that for every $\mu\in[-\varepsilon,\varepsilon]$:
\begin{itemize}
\item The restriction $X_{\mu}|_{\mathbb{R}^2\setminus{\overline{D}_{s}}}:\mathbb{R}^2\setminus{\overline{D}_{s}}\to\mathbb{R}^2$ has no singularities in $\mathbb{R}^2\setminus{\overline{D}_{s}}$. By Proposition \ref{prop:without sing}.
\item $\mu \cdot I(X)>0$, for all $\mu<0$. By Corollary \ref{cor:3.2}.
\item For $\mu<0$, the restriction $X_{\mu}|_{\mathbb{R}^2\setminus{\overline{D}_{s}}}$ has a repellor at $\infty$, and for $\mu>0$, the restriction $X_{\mu}|_{\mathbb{R}^2\setminus{\overline{D}_{s}}}$ has an attractor at $\infty$. By hypotheses (2) and Proposition \ref{prop:main global}.
\end{itemize}
Thus,
$\big\{X_{\mu}:\mathbb{R}^2\setminus{\overline{D}_s}\to\mathbb{R}^2;-\varepsilon<\mu<\varepsilon\big\}$
has at $\mu=0$ a Hopf bifurcation at $\infty$.
\end{proof}
Observe that the vector fields $X_{\mu}(z)=X(z)+\mu z$ as in \mbox{Theorem \ref{main 1}} satisfy
\[
X_{\mu}(z)-X_{\tilde{\mu}}(z)=(\mu-\tilde{\mu})z, \quad \forall z\in \mathbb{R}^2\setminus{\overline{D}_\sigma}.
\]
Therefore, if $\mathfrak{X}(\mathbb{R}^2\setminus{\overline{D}_\sigma})$ is the space of the continuous vector fields of $\mathbb{R}^2\setminus{\overline{D}_\sigma}$ endowed with the topology induced by the uniform convergence on compact sets, the functional
\[
\mathbb{R}\ni\mu \mapsto X_{\mu}\in\mathfrak{X}(\mathbb{R}^2\setminus{\overline{D}_\sigma})
\]
is continuous.
\begin{rem}\label{rem:discontinuity}
If the maps $h_{\mu}:\mathbb{R}^2\setminus{\overline{D}_\sigma}\to(0,+\infty)$ and $\tilde{h}_{\mu}:\mathbb{R}^2\setminus{\overline{D}_\sigma}\to(-\infty,0)$ are differentiable and
the family $\big\{X_{\mu}:\mathbb{R}^2\setminus{\overline{D}_s}\to\mathbb{R}^2;-\varepsilon<\mu<\varepsilon\big\}$ is given by \mbox{Theorem \ref{main 1}}, then the families
\[
h_{\mu}(z)X_{\mu}(z)
\quad
\mbox{and}
\quad
\tilde{h}_{\mu}(z)X_{\mu}(z)
\]
also have at $\mu=0$ a Hopf bifurcation at $\infty$. For instance, the map
\[
h_{\mu}(z)=\left\{
\begin{array}{ll}
\displaystyle\frac{1}{\mu}, & \hbox{ if } \mu\neq0; \\
1, & \hbox{ if } \mu=0
\end{array}
\right.
\]
Notice that $h_{\mu}$ induces a well--defined map $\mu \mapsto h_{\mu} X_{\mu}$ which is discontinuous at $\mu=0$.
\end{rem}
In some sense, \mbox{Remark \ref{rem:discontinuity}} shows that \mbox{Theorem \ref{main 1}} includes the results where the strong domination imposed by the linear part is used.
\par
Furthermore, we do not assume the existence of some open neighborhood of infinity free of singularities, as in Theorem \ref{teoAGG07} item (4). Recall that the results in this work are obtained from the weak hypothesis on the Jacobian determinant like \eqref{eq:det}, which is natural in order to work with isolated singularities. In addition, the assumption (2) in Theorem \ref{main 1} is necessary, since the theorem considers the differentiable vector fields, not necessarily of class $C^1$.
In the particular case of $C^1-$vector fields, Theorem \ref{main 1} directly gives the next illustrative result.
\begin{cor}\label{cor:1
Consider $Z\in\mathcal{H}(2,\sigma)$ of class $C^1$. Suppose $Z$ has a singularity and the maps $\mathbb{R}^2\setminus{\overline{D}_\sigma}\ni z\mapsto Z_{\mu}(z)=Z(z)+\mu z$ are orientation preserving local diffeomorphisms, for all $\mu$ in some interval $(0,\epsilon_0)$.
If assumption (1) in Theorem \ref{main 1} holds, then the condition
\begin{equation}\label{eq:index}
\mu\neq0\Rightarrow\mu \cdot\mathcal{I}(Z_{\mu}) >0
\end{equation}
implies the existence of $\varepsilon>0$ and $s> \sigma$ such that
$\Big\{Z_{\mu}:\mathbb{R}^2\setminus{\overline{D}_s}\to\mathbb{R}^2;-\varepsilon<\mu<\varepsilon\Big\}$
has at $\mu=0$ a Hopf bifurcation at $\infty$.
\end{cor}
Observe that condition \eqref{eq:index} extends the case where the spectrum of $X_{\mu}$ crosses the imaginary axis transversally when $\mu$ moves from negative to positive values.
\section{Vector fields free of real eigenvalues} \label{sec:free eigen
This subsection addresses the special case of differentiable vector fields whose Jacobian matrix is free of real eigenvalues. These vector fields induce local diffeomorphisms with zero divergence.
\begin{prop} \label{cor:3.3}
Let $X:\mathbb{R}^2\setminus{\overline{D}_\sigma}\to\mathbb{R}^2$ be a differentiable vector field with some singularity. Suppose that the following holds
\begin{equation} \label{eq:spectrum free divergence}
\mbox{\rm Spc}(X)\subset\Big\{z\in\mathbb{C}:\Re(z)=0\Big\}\setminus\big\{(0,0)\big\}.
\end{equation}
Then, $\forall \mu\in \Real$ $\exists s_{\mu}\geq \sigma$ such that
\begin{itemize}
\item[(a)] the restriction $X|_{{\mathbb{R}^2\setminus \overline{D}_{s_\mu}}}:{{\mathbb{R}^2\setminus \overline{D}_{s_\mu}}}\to \mathbb{R}^2$ is injective and admits a global differentiable extension $\widehat{X}:\Real^2\to \Real^2$ with $\widehat{X}(0)=0$ and $$\widehat{X}(z)=X(z), \;\forall z\in \mathbb{R}^2\setminus \overline{D}_{s_\mu};$$
\item[(b)] the index $\mathcal{I}(X)$ at infinity is a well defined number in $[-\infty,+\infty)$;
\item [(c)] $X_{\mu}$ induces a well defined positive (respectively negative) semi--flow as long as $\mu<0$ (respectively $\mu>0$). Moreover, the trajectories of $X$ are unique in the sense that only depend of the initial condition.
\end{itemize}
\end{prop}
\begin{proof}
Since $X_{\mu}(z)=X(z)+\mu z$ implies that
$\mbox{\rm Spc}(X_\mu)\cap\mathbb{R}=\emptyset$ for all $\mu\in\mathbb{R}$. Proposition 2.7 in \cite{MR2266382}, implies that $X_{\mu}=(f_{\mu},g_{\mu})$ has the following property:
\begin{itemize}
\item[(a.1)] Any half-Reeb component of either $\mathcal{F}(f_{\mu})$ or $\mathcal{F}(g_{\mu})$ is a bounded set.
\end{itemize}
Therefore, the proof of Lemma \ref{lem: existence of the index} holds verbatim and statements (a) and (b) remain true.
In addition, \eqref{eq: translation spectrum} implies that:
\[
\begin{array}{ccl}
\mu<0 & \Rightarrow & \mbox{Spc}(X_{\mu}) \subset \big\{z\in\mathbb{C}:\Re(z)\leq\mu<0\big\},\\
\mu>0 & \Rightarrow & \mbox{Spc}(-X_{\mu}) \subset \big\{z\in\mathbb{C}:\Re(z)\leq-\mu<0\big\}.
\end{array}
\]
Therefore, \cite[Lemma 3.3]{MR2096702} implies that the vector fields $X_{\mu}$ with $\mu<0$, and $-X_{\mu}$ with $\mu>0$, induce a well defined positive semi--flow.
Similarly, \cite[Lemma 2.1]{MR2448587} gives the uniqueness of the trajectories induced by $X$.
Thus, statement (c) holds.
\end{proof}
\begin{prop}\label{prop:div 0}
Consider the family of differentiable vector fields $\{X_{\mu}(z)=X(z)+\mu z : \mu\in \Real\}$, where $X$ verifies hypotheses of Proposition \ref{cor:3.3}.
If $\exists \varepsilon_0>0$ such that the set $\big\{s_{\mu}\geq\sigma:\mu\in[-\varepsilon_0,\varepsilon_0]\big\}$ induced by strong extensions is bounded, then $\exists s>\sigma$ and $\varepsilon>0$ such that the family
\[\big\{X_{\mu}:\mathbb{R}^2\setminus{\overline{D}_s}\to\mathbb{R}^2;-\varepsilon<\mu<\varepsilon\big\}\]
has at $\mu=0$ a Hopf bifurcation at $\infty$.
\end{prop}
\begin{proof}Follows by Proposition \ref{cor:3.3} and Proposition \ref{prop:main global}.
\end{proof}
Observe that a similar result as described in Remark \ref{rem:discontinuity} is obtained in the case of families given by \mbox{Proposition \ref{prop:div 0}}.
\begin{figure}[htb]
\centering
\psfrag{D}{$\overline{D}_s$}
\psfrag{In}{$\infty$}
\psfrag{c}{$\mu=0$}
\psfrag{m}{$\mu<0$}
\psfrag{M}{$\mu>0$}
\includegraphics[width=12cm]{fig_hopf.eps}\\
\caption{\footnotesize The bifurcation in \mbox{Proposition \ref{prop:div 0}}.}\label{fig:mainf proposition}
\end{figure}
\begin{rem}\label{rem:center}
Consider now the special case of planar $C^1-$vector fields $Y:\mathbb{R}^2\to\mathbb{R}^2$ such that $Y(0)=0$ and
\[
\mbox{\rm Spc}(Y)\subset\Big\{z\in\mathbb{C}:\Re(z)=0\Big\}\setminus\big\{(0,0)\big\}.
\]
The trajectories in $\mathbb{R}^2\setminus\{0\}$ induced by $Y$ are periodic orbits surrounding the origin \cite{MR2448587,MR3223368}.
Moreover, a direct application of \mbox{Proposition \ref{prop:div 0}} gives the existence of some $\varepsilon>0$, for which the family $\big\{Y_{\mu}:\mathbb{R}^2\setminus{\overline{D}_s}\to\mathbb{R}^2;-\varepsilon<\mu<\varepsilon\big\}$ has at $\mu=0$ a Hopf bifurcation at $\infty$. See Figure \ref{fig:mainf proposition}.
\end{rem}
\section*{Acknowledgements}
The first author was partially supported by grants \textsc{micinn-12-mtm2011-22956 } from Spain and CNPq 474406/2013-0 from Brazil.
The second author was partially supported by Pontif\'{\i}cia Universidad Cat\'{o}lica del Per\'{u} (\textsc{dgi}:70242.0056), by Instituto de Ci\^encias Matem\'aticas e de Computa\c{c}\~ao (\textsc{icmc--usp}: 2013/16226-8).
This paper was written while the second author served as an Associate Fellow at the Abdus Salam \textsc{ictp} in Italy. He also acknowledges the hospitality of \textsc{icmc--usp} in Brazil during the preparation of part of this work.
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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|
1,116,691,500,476 | arxiv | \section{Introduction}
In 1914, Ramanujan \cite{Ramanujan} gave a number of rapidly convergent series of $1/\pi$. Although the following series, due to Bauer \cite{Bauer},
is not listed in \cite{Ramanujan}, it gives an example of this kind:
\begin{align}
\sum^{\infty}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} =\frac{2}{\pi}, \label{eq:Ramanujan}
\end{align}
where $(a)_k=a(a+1)\cdots (a+k-1)$ denotes the rising factorial.
Ramanujan's formulas for $1/\pi$ got widely admired in 1980's
when they were discovered to offer efficient algorithms for
calculating decimal digits of $\pi$. See the monograph \cite{BB} of the Borwein brothers.
For a recent proof of Ramanujan's series, see Guillera \cite{Guillera}.
In 1997, Van Hamme \cite{Hamme} developed interesting $p$-adic analogues of Ramanujan-type series. In particular, he conjectured the following supercongruence
corresponding to \eqref{eq:Ramanujan}:
\begin{align}
\sum^{(p-1)/2}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv p(-1)^{(p-1)/2} \pmod{p^3},\label{eq:VAN}
\end{align}
where $p$ is an odd prime. Note that we may calculate the sum in \eqref{eq:VAN} for $k$ up to $p-1$,
since the $p$-adic order of $(\frac{1}{2})_k/k!$ is $1$ for $k$ in the range $(p+1)/2\leqslant k\leqslant p-1$.
Congruences of this kind are called Ramanujan-type supercongruences. The congruence \eqref{eq:VAN} was first proved by Mortenson \cite{Mortenson} in 2008
using a $_6F_5$ transformation and the $p$-adic Gamma function, and received a WZ (Wilf--Zeilberger \cite{WZ1,WZ2}) proof by Zudilin \cite{Zudilin} shortly afterwards.
In 2012, also employing the WZ method, Sun \cite{Sun}
gave the following refinement of \eqref{eq:VAN}: for any prime $p>3$,
\begin{align}
\sum^{m}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4},\label{eq:Sun}
\end{align}
where $m=p-1$ or $(p-1)/2$, and $E_{p-3}$ is the $(p-3)$-th Euler number.
In recent years, $q$-analogues (or rational function generalizations) of congruences and supercongruences have
aroused the interest of many researchers (see \cite{Guo2018,Guo-t,Guo-a2,Guo-div,Guo-mod4,GPZ,GS3,GS,GZ14,GuoZu,LW,Liu,LP,NP,Straub2,WY0,Zu19}).
For instance, the author \cite{Guo2018} gave the following $q$-analogue of \eqref{eq:VAN}: for any odd integer $n>1$,
\begin{align*}
\sum_{k=0}^{(n-1)/2}(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}
\equiv [n]q^{(n-1)^2/4} (-1)^{(n-1)/2}\pmod{[n]\Phi_n(q)^2},
\end{align*}
Here we need to be familiar with the standard $q$-notation. The {\it $q$-integer} is defined by $[n]=1+q+\cdots+q^{n-1}$,
and the {\it $q$-shifted factorial} is defined as $(a;q)_n=(1-a)(1-aq)\cdots (1-aq^{n-1})$ for $n\geqslant 1$ and $(a;q)_0=1$.
Moreover, the $n$-th {\it cyclotomic polynomial} $\Phi_n(q)$ is given by
\begin{align*}
\Phi_n(q)=\prod_{\substack{1\leqslant k\leqslant n\\ \gcd(k,n)=1}}(q-\zeta^k),
\end{align*}
where $\zeta$ is an $n$-th primitive root of unity. It is well known that $\Phi_n(q)$ is an irreducible polynomial in $\mathbb{Z}[q]$.
For two rational functions $A(q)$ and $B(q)$ in $q$ and a polynomial $P(q)$ in $q$ with integer coefficients, we say that $A(q)$ is congruent to $B(q)$
modulo $P(q)$, denoted by $A(q)\equiv B(q)\pmod{P(q)}$,
if $P(q)$ divides the numerator of the reduced form of $A(q)-B(q)$
in the polynomial ring $\mathbb{Z}[q]$.
In this paper, we shall give a $q$-analogue of \eqref{eq:Sun}.
\begin{theorem}\label{thm:1}
Let $n$ be a positive odd integer. Then
\begin{align}
\sum_{k=0}^{N}
(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}
&\equiv
(-1)^{(n-1)/2}q^{(1-n^2)/4}
\left(q^{n\choose 2}[n]+\frac{(n^2-1)(1-q)^2}{24}[n]^3\right) \notag \\
&\quad+[n]^3\sum_{k=1}^{(n-1)/2}\frac{q^k (q^2;q^2)_{k}}{[2k][2k-1](q;q^2)_{k}} \pmod{[n]\Phi_n(q)^3}, \label{eq:main-1}
\end{align}
where $N=(n-1)/2$ or $n-1$.
\end{theorem}
Note that Sun \cite[Equation (3.1)]{Sun2011} proved that, for any odd prime $p$,
$$
\sum_{k=1}^{(p-1)/2}\frac{4^k}{k(2k-1){2k\choose k}}\equiv 2 E_{p-3} \pmod{p}.
$$
Letting $n=p$ be a prime greater than $3$ and taking $q\to 1$ in \eqref{eq:main-1}, we immediately get \eqref{eq:Sun}.
Still using the WZ method, Guillera and Zudilin \cite{GuZu} established the following supercongruence: for odd primes $p$,
\begin{align}
\sum_{k=0}^{(p-1)/2}(-1)^k (3k+1)\frac{(\frac{1}{2})_k^3}{k!^3} 2^{3k} \equiv p(-1)^{(p-1)/2}\pmod{p^3}. \label{eq:div-3}
\end{align}
Moreover, in 2016, Chen, Xie, and He \cite{CXH} gave the following refinement of \eqref{eq:div-3}:
\begin{align}
\sum_{k=0}^{p-1}(-1)^k (3k+1)\frac{(\frac{1}{2})_k^3}{k!^3} 2^{3k} \equiv p(-1)^{(p-1)/2}+p^3E_{p-3} \pmod{p^4}, \label{eq:div-3-2}
\end{align}
which was originally conjectured by Sun \cite[Conjecture 5.1]{Sun0}. The author \cite{Guo-div} established a $q$-analogue of \eqref{eq:div-3}:
\begin{align}
\sum_{k=0}^{n-1}(-1)^k [3k+1]\frac{(q;q^2)_k^3}{(q;q)_k^3}
\equiv [n]q^{(n-1)^2/4} (-1)^{(n-1)/2} \pmod{[n]\Phi_n(q)^2}. \label{eq:q-Zudilin-3}
\end{align}
We point out that a supercongruence for the left-hand side of \eqref{eq:div-3} modulo $p^4$,
also conjectured by Sun \cite[Conjecture 5.1]{Sun0}, was recently confirmed by Mao \cite{Mao}.
In this paper, we shall also establish a $q$-analogue of \eqref{eq:div-3-2}.
\begin{theorem}\label{thm:2}
Let $n$ be a positive odd integer. Then
\begin{align}
\sum_{k=0}^{n-1} (-1)^k [3k+1]\frac{(q;q^2)_k^3}{(q;q)_k^3}
&\equiv
(-1)^{(n-1)/2}q^{(1-n^2)/4}
\left(q^{n\choose 2}[n]+\frac{(n^2-1)(1-q)^2}{24}[n]^3\right) \notag \\
&\quad+[n]^3\sum_{k=1}^{(n-1)/2}\frac{q^k (q^2;q^2)_{k}}{[2k][2k-1](q;q^2)_{k}} \pmod{[n]\Phi_n(q)^3}. \label{eq:main-2}
\end{align}
\end{theorem}
The remainder of the paper proceeds as follows. We prove Theorems \ref{thm:1} and \ref{thm:2}
in Sections 2 and 3, respectively, by using the $q$-WZ method, together with a $q$-analogue of Wolstenholme's congruence and a $q$-analogue of Morley's congruence.
In Section 4, we give some concluding remarks and two open
problems. Particularly in Corollary \ref{cor:one}, using a recent result of Wei \cite{Wei}, we shall deduce a $q$-analogue of another supercongruence related to
Euler numbers of Sun from Theorem \ref{thm:1}.
\section{Proof of Theorem \ref{thm:1}}
Recall that the {\it $q$-binomial coefficients} ${M\brack N}$
are defined by
$$
{M\brack N}={M\brack N}_q
=\begin{cases}\displaystyle\frac{(q;q)_M}{(q;q)_N(q;q)_{M-N}} &\text{if $0\leqslant N\leqslant M$,} \\[10pt]
0 &\text{otherwise.}
\end{cases}
$$
We need the following $q$-analogue of Wolstenholme's congruence (see \cite[Lemma 3.1]{GW}).
\begin{lemma}\label{lem:first}
Let $n$ be a positive integer. Then
\begin{align*}
{2n-1\brack n-1} \equiv (-1)^{n-1}q^{n\choose 2}+\frac{(n^2-1)(1-q)^2}{12}[n]^2
\pmod{\Phi_n(q)^3}.
\end{align*}
\end{lemma}
Moreover, a $q$-analogue of Morley's congruence (see \cite[(1.5)]{LPZ})
and a $q$-analogue of Fermat's little theorem (see \cite[Lemma 3.2]{GW}) will also be used in our proof.
\begin{lemma}\label{lem:second}
Let $n$ be a positive odd integer. Then, modulo $\Phi_n(q)^3$,
\begin{align*}
{n-1\brack \frac{n-1}{2}}_{q^2}
&\equiv (-1)^{(n-1)/2}q^{(1-n^2)/4}\left((-q;q)_{n-1}^2-\frac{(n^2-1)(1-q)^2}{24}[n]^2\right).
\end{align*}
\end{lemma}
\begin{lemma}\label{lem:third}
Let $n$ be a positive odd integer. Then
\begin{align}
(-q;q)_{n-1}\equiv 1\pmod{\Phi_n(q)}. \label{eq:phi}
\end{align}
\end{lemma}
\begin{proof}[Proof of Theorem {\rm\ref{thm:1}}]
By \cite[Theorem 6.1]{Guo-mod4}, modulo $[n]\Phi_n(q)(1-aq^n)(a-q^n)$,
\begin{align}
&\sum_{k=0}^{(n-1)/2}[4k+1]\frac{(aq;q^2)_k (q/a;q^2)_k (q/b;q^2)_k (q;q^2)_k}
{(aq^2;q^2)_k(q^2/a;q^2)_k (bq^2;q^2)_k (q^2;q^2)_k}b^k \notag\\[5pt]
&\quad\equiv \sum_{k=0}^{n-1}[4k+1]\frac{(aq;q^2)_k (q/a;q^2)_k (q/b;q^2)_k (q;q^2)_k}
{(aq^2;q^2)_k(q^2/a;q^2)_k (bq^2;q^2)_k (q^2;q^2)_k}b^k, \label{eq:equiv}
\end{align}
where $a$ and $b$ are indeterminates.
Letting $b\to\infty$ and $a=1$ in \eqref{eq:equiv}, we get
\begin{align}
\sum_{k=0}^{(n-1)/2}(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}
\equiv \sum_{k=0}^{n-1}(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3} \pmod{\Phi_n(q)^4}. \label{eq:equiv-2}
\end{align}
By \cite[Theorem 4.1]{GuoZu}, both sides of \eqref{eq:equiv-2} are congruent to $0$ modulo $[n]$, and so
it is also true modulo $[n]\Phi_n(q)^3$. Thus, to prove Theorem \ref{thm:1}, it suffices to prove
the $N=(n-1)/2$ case.
We introduce two rational functions in $q$:
\begin{align*}
F(m,k) &=(-1)^{m+k}q^{(m-k)^2}\frac{[4m+1](q;q^2)_{m}^2(q;q^2)_{m+k}}{(q^2;q^2)_{m}^2(q^2;q^2)_{m-k}(q;q^2)_{k}^2}, \\[5pt]
G(m,k) &=\frac{(-1)^{m+k}q^{(m-k)^2}(q;q^2)_{m}^2(q;q^2)_{m+k-1}}{(1-q)(q^2;q^2)_{m-1}^2(q^2;q^2)_{m-k}(q;q^2)_{k}^2},
\end{align*}
where we assume that $1/(q^2;q^2)_{M}=0$ for negative integers $M$. As mentioned in \cite{Guo2018}, the functions $F(m,k)$ and $G(m,k)$ form a $q$-WZ pair.
Namely, they satisfy the following equality
\begin{align}
F(m,k-1)-F(m,k)=G(m+1,k)-G(m,k). \label{eq:fnk-gnk}
\end{align}
Summing \eqref{eq:fnk-gnk} over $m$ from $0$ to $(n-1)/2$, we get
\begin{align}
\sum_{m=0}^{(n-1)/2}F(m,k-1)-\sum_{m=0}^{(n-1)/2}F(m,k)=G\left(\frac{n+1}{2},k\right)-G(0,k)=G\left(\frac{n+1}{2},k\right). \label{eq:fnk-gn0-00}
\end{align}
Summing \eqref{eq:fnk-gn0-00} further over $k$ from $1$ to $(n-1)/2$ and noticing that $F(m,(n-1)/2)=0$ for $m<(n-1)/2$, we obtain
\begin{align}
\sum_{m=0}^{(n-1)/2}F(m,0)-F\left(\frac{n-1}{2},\frac{n-1}{2}\right)=\sum_{k=1}^{(n-1)/2}G\left(\frac{n+1}{2},k\right). \label{eq:fnk-gn-new}
\end{align}
Note that, for $k=1,2,\ldots,(n-1)/2$, we have
\begin{align}
G\left(\frac{n+1}{2},k\right)
&=\frac{(-1)^{(n+1)/2+k}q^{((n+1)/2-k)^2}(q;q^2)_{(n+1)/2}^2(q;q^2)_{(n-1)/2+k}}{(1-q)(q^2;q^2)_{(n-1)/2}^2(q^2;q^2)_{(n+1)/2-k}(q;q^2)_{k}^2} \notag \\[5pt]
&=(-1)^{(n+1)/2+k}q^{((n+1)/2-k)^2}\frac{(1-q^n)^3(q;q^2)_{(n-1)/2}^3 (q^{n+2};q^2)_{k-1}}{(1-q)(q^2;q^2)_{(n-1)/2}^2(q^2;q^2)_{(n+1)/2-k}(q;q^2)_{k}^2}.
\label{eq:gppk}
\end{align}
Since $q^n\equiv 1\pmod{\Phi_n(q)}$, there hold
\begin{align*}
(q^2;q^2)_{(n+1)/2-k}&=\frac{(q^2;q^2)_{(n-1)/2}}{(q^{n+3-2k};q^2)_{k-1}} \\[5pt]
&\equiv \frac{(q^2;q^2)_{(n-1)/2}}{(q^{3-2k};q^2)_{k-1}}
=(-1)^{k-1}q^{(k-1)^2}\frac{(q^2;q^2)_{(n-1)/2}}{(q;q^2)_{k-1}} \pmod{\Phi_n(q)},
\end{align*}
and
\begin{align*}
\frac{(q;q^2)_{(n-1)/2}}{(q^2;q^2)_{(n-1)/2}}&=\prod_{j=1}^{(n-1)/2}\frac{1-q^{2j-1}}{1-q^{n-2j+1}} \\[5pt]
&\equiv \prod_{j=1}^{(n-1)/2}\frac{1-q^{2j-1}}{1-q^{1-2j}}=(-1)^{(n-1)/2}q^{(n-1)^2/4} \pmod{\Phi_n(q)}.
\end{align*}
Employing the above two $q$-congruences, we deduce from \eqref{eq:gppk} that, for $1\leqslant k\leqslant (n-1)/2$,
\begin{align}
G\left(\frac{n+1}{2},k\right)
\equiv \frac{q^k(1-q^n)^3 (q^{2};q^2)_{k-1}}{(1-q)(1-q^{2k-1})(q;q^2)_{k}}
=\frac{q^k[n]^3 (q^{2};q^2)_{k}}{[2k][2k-1](q;q^2)_{k}} \pmod{\Phi_n(q)^4}.
\label{eq:gppk-new}
\end{align}
Since $(q;q^2)_k/(q^2;q^2)_k={2k\brack k}/(-q;q)_k^2$ and the $q$-binomial coefficient can be written as a product of different cyclotomic polynomials
(see \cite[Lemma 1]{CH}), we see that the right-hand side of \eqref{eq:gppk-new} is congruent to $0$ modulo $[n]$, and so is \eqref{eq:gppk}.
Namely, the $q$-congruence \eqref{eq:gppk-new} holds modulo $[n]\Phi_n(q)^3$.
In addition, by Lemmas \ref{lem:first}--\ref{lem:third},
\begin{align}
&F\left(\frac{n-1}{2},\frac{n-1}{2}\right) \notag\\[5pt]
&\quad=\frac{[n]}{(-q;q)_{n-1}^2}{2n-1\brack n-1}{n-1\brack \frac{n-1}{2}}_{q^2} \notag \\[5pt]
&\quad\equiv (-1)^{(n-1)/2}q^{(1-n^2)/4}
\left\{q^{n\choose 2}[n]+\frac{(n^2-1)(1-q)^2}{24}\left(2-\frac{q^{n\choose 2}}{(-q;q)_{n-1}^2}\right)[n]^3\right\} \notag\\[5pt]
&\quad\equiv (-1)^{(n-1)/2}q^{(1-n^2)/4}
\left(q^{n\choose 2}[n]+\frac{(n^2-1)(1-q)^2}{24}[n]^3\right) \pmod{[n]\Phi_n(q)^3}, \label{eq:abcd}
\end{align}
where we have used $q^{n\choose 2}\equiv 1\pmod{\Phi_n(q)}$ for odd $n$ in the last step.
Substituting \eqref{eq:abcd} and the modulus $[n]\Phi_n(q)^3$ case of \eqref{eq:gppk-new} into \eqref{eq:fnk-gn-new},
we are led to \eqref{eq:main-1} in the case where $N$ is equal to $(n-1)/2$. This completes the proof of the theorem.
\end{proof}
\section{Proof of Theorem \ref{thm:2}}
The author \cite{Guo-div} employed the following functions
\begin{align*}
F(m,k) &=(-1)^{m}[3m-2k+1]{2m-2k\brack m}\frac{(q;q^2)_{m}(q;q^2)_{m-k} }{(q;q)_{m} (q^2;q^2)_{m-k}}, \\[5pt]
G(m,k) &=(-1)^{m+1}[m]{2m-2k\brack m-1}\frac{(q;q^2)_{m}(q;q^2)_{m-k} q^{m+1-2k} }{(q;q)_{m} (q^2;q^2)_{m-k}}
\end{align*}
to establish \eqref{eq:q-Zudilin-3}.
It is not difficult to verify that $F(m,k)$ and $G(m,k)$ satisfy the relation
\begin{align}
F(m,k-1)-F(m,k)=G(m+1,k)-G(m,k). \label{eq:fnk-gnk-new}
\end{align}
That is, they form a $q$-WZ pair.
Since \eqref{eq:main-2} is clearly true for $n=1$, we now assume that $n\geqslant 3$.
Summing \eqref{eq:fnk-gnk-new} over $m=0,1,\ldots,n-1$, we obtain
\begin{align}
\sum_{m=0}^{n-1}F(m,k-1)-\sum_{m=0}^{n-1}F(m,k)=G(n,k). \label{eq:fnk-gn0-0011}
\end{align}
Summing \eqref{eq:fnk-gn0-0011} further over $k=1,\ldots,n-1$ and noticing that $F(m,n-1)=0$ for $m\leqslant n-1$, we arrive at
\begin{align}
\sum_{m=0}^{n-1}F(m,0)=\sum_{k=1}^{n-1}G(n,k)=\sum_{k=1}^{(n+1)/2}G(n,k). \label{eq:fm0-gmk}
\end{align}
In view of
$$
{2m\brack m}
=\frac{(q;q^2)_m (-q;q)_m^2}{(q^2;q^2)_m},
$$
the identity \eqref{eq:fm0-gmk} can be written as
\begin{align}
\sum_{m=0}^{n-1}(-1)^n [3m+1]\frac{(q;q^2)_m^3}{(q;q)_m^3}
=\frac{[n]{2n-1\brack n-1}}{(-q;q)_{n-1}}\sum_{k=1}^{(n+1)/2}{2n-2k\brack n-1}\frac{(q;q^2)_{n-k} q^{n+1-2k} }{(q^2;q^2)_{n-k}}.
\label{eq:fn03n+1}
\end{align}
For $1\leqslant k\leqslant (n-1)/2$, we have
\begin{align*}
\frac{(q;q^2)_{n-k}}{(q^2;q^2)_{n-k}}
&=\frac{(1-q^n)(q;q^2)_{(n-1)/2}(q^{n+2};q^2)_{(n-1)/2-k}}{(1-q^{n-1})(q^2;q^2)_{(n-3)/2}(q^{n+1};q^2)_{(n+1)/2-k}} \\[5pt]
&\equiv \frac{(1-q^n)(q;q^2)_{(n-1)/2}(q^2;q^2)_{(n-1)/2-k}}{(1-q^{n-1})(q^2;q^2)_{(n-3)/2}(q;q^2)_{(n+1)/2-k}} \\
&=\frac{(1-q^n)(q^{n+2-2k};q^2)_{k-1}}{(1-q^{n-1})(q^{n+1-2k};q^2)_{k-1}} \\
&\equiv \frac{(1-q^n)(q^{2-2k};q^2)_{k-1}}{(1-q^{-1})(q^{1-2k};q^2)_{k-1}}
=-\frac{q^k(1-q^n)(q^2;q^2)_{k-1}}{(q;q^2)_{k}} \pmod{\Phi_n(q)^2},
\end{align*}
and
\begin{align*}
{2n-2k+1\brack n}=\prod_{k=1}^{n-2k+1}\frac{1-q^{n+j}}{1-q^{j}}\equiv 1\pmod{\Phi_n(q)}.
\end{align*}
It follows that
\begin{align}
&\sum_{k=1}^{(n+1)/2}{2n-2k\brack n-1}\frac{(q;q^2)_{n-k} q^{n+1-2k} }{(q^2;q^2)_{n-k}} \notag\\[5pt]
&\quad=\frac{(q;q^2)_{(n-1)/2}}{(q^2;q^2)_{(n-1)/2}}+\sum_{k=1}^{(n-1)/2}{2n-2k\brack n-1}\frac{(q;q^2)_{n-k} q^{n+1-2k} }{(q^2;q^2)_{n-k}} \notag\\[5pt]
&\quad={n-1\brack \frac{n-1}{2}}_{q^2}\frac{1}{(-q;q)_{n-1}}
+\sum_{k=1}^{(n-1)/2}{2n-2k+1\brack n}\frac{(1-q^n)(q;q^2)_{n-k} q^{n+1-2k} }{(1-q^{2n-2k+1})(q^2;q^2)_{n-k}} \notag\\[5pt]
&\quad\equiv (-1)^{(n-1)/2}q^{(1-n^2)/4}\left((-q;q)_{n-1}-\frac{(n^2-1)(1-q)^2}{24}[n]^2\right) \notag\\[5pt]
&\quad\quad+[n]^2\sum_{k=1}^{(n-1)/2}\frac{q^k (q^2;q^2)_k}{[2k][2k-1] (q;q^2)_k} \pmod{\Phi_n(q)^3}, \label{eq:2nk}
\end{align}
where we have used Lemmas \ref{lem:second} and \ref{lem:third} in the last step.
By Lemmas \ref{lem:first} and \ref{lem:third}, we have
\begin{align}
\frac{{2n-1\brack n-1}}{(-q;q)_{n-1}} \equiv \frac{(-1)^{n-1}q^{n\choose 2}}{(-q;q)_{n-1}}+\frac{(n^2-1)(1-q)^2}{12}[n]^2
\pmod{\Phi_n(q)^3}. \label{eq:2nk-2}
\end{align}
Substituting \eqref{eq:2nk} and \eqref{eq:2nk-2} into \eqref{eq:fn03n+1} and making some simplifications, we immediately obtain
\eqref{eq:main-2}.
\section{Concluding remarks and open problems}
From \eqref{eq:Sun} and \eqref{eq:div-3-2} one sees that, for any prime $p>3$,
\begin{align}
\sum^{m}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3}
\equiv \sum_{k=0}^{p-1}(-1)^k (3k+1)\frac{(\frac{1}{2})_k^3}{k!^3} 2^{3k}\pmod{p^4}, \label{eq:sun-combin}
\end{align}
where $m=p-1$ or $(p-1)/2$. Further, combining \eqref{eq:main-1} and \eqref{eq:main-2}, we have the following $q$-analogue of \eqref{eq:sun-combin}:
for any positive odd integer $n$,
\begin{equation}
\sum_{k=0}^{N}
(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}
\equiv
\sum_{k=0}^{n-1}
(-1)^k [3k+1]\frac{(q;q^2)_k^3}{(q;q)_k^3} \pmod{[n]\Phi_n(q)^3}, \label{eq:q4k-3k}
\end{equation}
where $N=(n-1)/2$ or $n-1$.
We now provide a conjectural parametric generalization of \eqref{eq:q4k-3k}.
\begin{conjecture}Let $n$ be a positive odd integer. Then, modulo $[n]\Phi_n(q)(1-aq^n)(a-q^n)$,
\begin{align}
&\sum_{k=0}^{N}
(-1)^k q^{k^2} [4k+1]\frac{(aq;q^2)_k (q/a;q^2)_k (q;q^2)_k }{(aq^2;q^2)_k (q^2/a;q^2)_k (q^2;q^2)_k } \notag\\[5pt]
&\quad\equiv
\sum_{k=0}^{n-1}
(-1)^k [3k+1]\frac{(aq;q^2)_k (q/a;q^2)_k (q;q^2)_k }{(aq;q)_k (q/a;q)_k (q;q)_k }, \label{eq:q4k-3k-a}
\end{align}
where $N=(n-1)/2$ or $n-1$.
\end{conjecture}
Using the `creative microscoping' method introduced in \cite{GuoZu} and the Chinese remainder theorem for relatively prime polynomials,
the author \cite[Theorem 5.3]{Guo-mod4} has shown that the left-hand side of \eqref{eq:q4k-3k-a}
is congruent to
\begin{align*}
&(-1)^{(n-1)/2}q^{(n-1)^2/4}[n]+(-1)^{(n-1)/2}q^{(n-1)^2/4}\frac{(1-aq^n)(a-q^n)}{(1-a)^2}[n] \\[5pt]
&\quad{}-\frac{(1-aq^n)(a-q^n)}{(1-a)^2} [n]\sum_{k=0}^{(n-1)/2}\frac{(q;q^2)_k^2}{(aq^2;q^2)_k(q^2/a;q^2)_k}
\end{align*}
modulo $[n]\Phi_n(q)(1-aq^n)(a-q^n)$. But it seems rather difficult to prove that the right-hand side of
\eqref{eq:q4k-3k-a} is also congruent to the above expression, though a three-parametric generalization of
\eqref{eq:q-Zudilin-3} was already proved by the author and Schlosser \cite[Theorem 6.1]{GS} (see also \cite[Conjecture 4.6]{GuoZu}).
In 2011, Sun \cite{Sun0} studied many interesting supercongruences related to Euler numbers. In particular, Sun \cite[Theorems 1.1 and 1.2]{Sun0} proved that, for any prime $p>3$,
\begin{align}
\sum_{k=0}^{p-1}\frac{1}{2^k}{2k\choose k}
&\equiv (-1)^{(p-1)/2}-p^2 E_{p-3} \pmod{p^3}, \label{eq:Sun-2}\\[5pt]
\sum_{k=0}^{(p-1)/2}\frac{1}{16^k}{2k\choose k}^2
&\equiv (-1)^{(p-1)/2}+p^2 E_{p-3} \pmod{p^3}, \label{eq:Sun-3}\\[5pt]
\sum_{k=0}^{p-1}\frac{1}{16^k}{2k\choose k}^2
&\equiv (-1)^{(p-1)/2}-p^2 E_{p-3} \pmod{p^3}. \label{eq:Sun-4}
\end{align}
Recently, Wei \cite[Theorem 1.1 with $c=q$, $d\to\infty$]{Wei} gave the following result: for any positive odd integer $n$,
and $N=(n-1)/2$ or $n-1$,
\begin{align}
\sum_{k=0}^{N}
(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3}
&\equiv
q^{(1-n)/2}\left([n]+\frac{(n^2-1)(1-q)^2}{24}[n]^3\right) \notag\\[5pt]
&\quad\times\sum_{k=0}^{(n-1)/2}
\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2} q^{2k} \pmod{[n]\Phi_n(q)^3}, \label{eq:q4k-3k}
\end{align}
which is clearly a $q$-analogue of the relation between \eqref{eq:Sun} and \eqref{eq:Sun-3}: for any prime $p>3$,
\begin{align*}
\sum^{m}_{k=0}(-1)^k(4k+1)\frac{(\frac{1}{2})_k^3}{k!^3}
\equiv p\sum_{k=0}^{(p-1)/2}\frac{1}{16^k}{2k\choose k}^2\pmod{p^4}.
\end{align*}
where $m=p-1$ or $(p-1)/2$.
Note that the author, Pan and Zhang \cite{GPZ} gave the following $q$-supercongruence:
\begin{align}
\sum_{k=0}^{(n-1)/2}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{2k}\equiv (-1)^{(n-1)/2}q^{(n^2-1)/4} \pmod{\Phi_n(q)^2} \label{eq:guozeng}
\end{align}
(see \cite{GZ14} for $n$ being a prime). Hence, the $q$-supercongruence \eqref{eq:q4k-3k} may be written as
\begin{align}
& q^{(n-1)/2}\sum_{k=0}^{N}
(-1)^k q^{k^2} [4k+1]\frac{(q;q^2)_k^3}{(q^2;q^2)_k^3} \notag\\[5pt]
&\equiv
[n]\sum_{k=0}^{(n-1)/2}
\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2} q^{2k}
+\frac{(n^2-1)(1-q)^2}{24}[n]^3 (-1)^{(n-1)/2}q^{(n^2-1)/4} \pmod{[n]\Phi_n(q)^3}, \label{eq:q4k-3k-2}
\end{align}
Combining \eqref{eq:main-1} and \eqref{eq:q4k-3k-2} we immediately obtain a $q$-analogue of \eqref{eq:Sun-3}.
\begin{corollary} \label{cor:one}
Let $n$ be a positive odd integer. Then
\begin{align*}
\sum_{k=0}^{(n-1)/2}
\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2} q^{2k}
&\equiv
(-1)^{(n-1)/2}q^{(n^2-1)/4} \\[5pt]
&\quad+(-1)^{(n-1)/2}\frac{(n^2-1)(1-q)^2}{24}[n]^2 \left(q^{-(n-1)^2/4}-q^{(n^2-1)/4}\right) \\[5pt]
&\quad+ q^{(n-1)/2}[n]^2\sum_{k=1}^{(n-1)/2}\frac{q^k (q^2;q^2)_{k}}{[2k][2k-1](q;q^2)_{k}} \pmod{\Phi_n(q)^3}.
\end{align*}
\end{corollary}
However, to the best of the author's knowledge,
no $q$-analogues of \eqref{eq:Sun-2} and \eqref{eq:Sun-4}, even conjectural, are known in the literature.
It follows from \eqref{eq:Sun-2} and \eqref{eq:Sun-4} that, for any odd prime $p$,
\begin{equation}
\sum_{k=0}^{p-1}\frac{1}{2^k}{2k\choose k}
\equiv
\sum_{k=0}^{p-1}\frac{1}{16^k}{2k\choose k}^2 \pmod{p^3}. \label{eq:Sun2-4}
\end{equation}
On the other hand, for odd $n$, the author \cite{Guo-t} proved that
\begin{align}
\sum_{k=0}^{n-1}\frac{q^k}{(-q;q)_{k}}{2k\brack k}\equiv (-1)^{(n-1)/2}q^{(n^2-1)/4} \pmod{\Phi_n(q)^2}, \label{eq:q-tauraso}
\end{align}
confirming a conjecture of Tauraso \cite{Tauraso}.
In light of \eqref{eq:guozeng} and \eqref{eq:q-tauraso}, we believe that the following $q$-analogue of \eqref{eq:Sun2-4} is true.
\begin{conjecture}
Let $n$ be a positive odd integer. Then
\begin{align*}
\sum_{k=0}^{n-1}\frac{q^k}{(-q;q)_k}{2k\brack k} \equiv
\sum_{k=0}^{n-1}\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{2k}
\pmod{\Phi_n(q)^3}.
\end{align*}
\end{conjecture}
|
1,116,691,500,477 | arxiv | \section{Introduction}
Recently, deep neural networks is used to model implicit representations of 3D shapes has been widely applied for reconstruction \cite{duggal2021secrets}, generation \cite{chen2019learning}, compression \cite{tang2020deep} and rendering \cite{remelli2020meshsdf, takikawa2021neural}.
As one of the most popular methods, DeepSDF~\cite{park2019deepsdf} represents the zero-level surface of the whole 3D shape by regressing its continuous signed distance function (SDF).
However, the effectiveness of such models depends on the complexity of 3D shapes and the capacity of neural networks.
In Fig~\ref{fig:compare2degeneration}~(d) compared to (c) we demonstrate an example, where the capacity of the model is insufficient. Consequently, the reconstruction of one complex 3D shape is of poor quality.
To alleviate this problem, we propose to learn a set of local SDFs to represent the whole surface. In this case, each local SDF is responsible for a part of the reconstructed shape.
Learning such local SDFs is much easier. Here we use the assumption that the complexity of one local part of the 3D shape is much simpler than the whole and usually similar to other local parts.
However, learning a set of local SDFs instead of one global SDF makes the training process of the model more difficult, since the distribution of training data is not uniform, especially when more SDFs are considered in a local region.
One of the possible solutions is to make these local SDFs learnable using both the database and the local latent code. Here we utilize the reasonable assumption that similarity of the neighbor parts means similarity in the local code of the shape.
Recently, graph neural networks (GNNs) have demonstrated high effectiveness in geometric learning field, especially in 3D reconstruction problem~\cite{ranjan2018generating,bouritsas2019neural,zhou2020fully,hanocka2019meshcnn}.
The discussed works show that it is possible to decode latent representation of the shape geometry with highly accurate reconstruction, even for finer details.
The GNNs provide geometric constraints to facilitate the information smoothed over the graph as the original purpose for clustering vertices of a graph~\cite{kipf2016semi, defferrard2016convolutional}.
This property leads to a message exchange mechanism among the vertices and results in their local similarity.
Consequently, we introduce Global-to-Local~(G2L) network based on GNNs to learn local latent codes that came from one global code. Thus, our G2L combines the advantages of the GNNs and SDF approaches.
We assume that geometrical locality is learnable in the space of latent code as in the original geometric space of the shape.
In addition, we also include one geometric similarity loss function based on the geometric structure to enhance the effectiveness of the GNN in local latent code learning.
Remind our assumption is that the neighboring local regions should have similar latent codes in general. Our experimental results confirm the mentioned assumption with the local latent codes learning.
Several methods~\cite{chabra2020deep, jiang2020local, genova2020local, tretschk2020patchnets, hao2020dualsdf} learn the SDF of 3D shape locally and have shown promising results.
However, all of them need either the voxel representation of 3D shape to align the local latent codes or explicit parametric models (for example, a sphere) to fit.
Both approaches face problems of the volumetric representation that in many cases highly difficult to solve or even is insolvable.
In contrast, our work leverages geometric learning techniques to model local SDFs of 3D shapes directly in the latent code space.
Contributions of our work are:
\begin{itemize}
\item We propose the Local Geometry Code Learning method, where we learn the shape as zero-surfaces with local latent codes.
\item We use graph neural networks to generate local latent codes and distribute them on the 3D shape, which does not request voxelization of the 3D shape as it is in locally modeling methods.
\item We introduce a geometric similarity in the loss function that helps to learn and reduce the fluctuation of the reconstructed surface.
\item Our experimental validation shows that the proposed approach could keep more details of the reconstructed shape in comparison with the original SDF decoder.
\end{itemize}
\begin{figure*}
\begin{center}
\includegraphics[width=0.95\linewidth]{figs/pipeline.pdf}
\end{center}
\caption{(a) Overview of the LGCL model; (b) Local region separation, where one 3D shape sample is represented with mesh data and projected to a 2D plane.}
\label{fig:geo_sdf_decoder}
\end{figure*}
\begin{figure}
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/gcn.pdf}
\end{center}
\caption{Architecture of the G2L network.}
\label{fig:g2l}
\end{figure}
\section{Related work}
\subsection{SDF learning methods}
Learning a signed distance function to represent the 3D shape as a set of iso-surfaces recently receives extensive attention in the field.
Chen and Zhang~\cite{chen2019learning} proposed to assign a value to each point in 3D space and use a binary classifier to extract an iso-surface.
Mescheder~\etal~\cite{mescheder2019occupancy} utilize a truncated SDF to decide the continuous boundary of 3D shapes. In contrast with~\cite{chen2019learning}, they predict the probability of occupancy in voxels, which could be used in a progressive multi-resolution procedure to get refined output.
Park \etal~\cite{park2019deepsdf} learn a continuous field of 3D shape instead of the discrete representation of SDF in the grid and is understandable as a learned shape-conditioned classifier for the decision boundary of 3D shape surface.
Atzmon and Lipman~\cite{atzmon2020sal} leveraged sign agnostic learning (SAL) to learn from unsigned distance data, such as the distance directly from points to triangle soups.
Gropp \etal~\cite{gropp2020implicit} suggested using a geometric regularization paradigm to approximate the signed distance function, which can be achieved without 3D supervision and/or a direct loss on the surface of the shape.
Inspired by incorporating derivatives in a regression loss leads to a lower sample complexity, Atzmon and Lipman~\cite{atzmon2020sald} generalized SAL to include derivatives and show a significant improvement in the quality of 3D shape reconstruction.
Note that it is extremely hard to obtain the ground-truth data about the signed distance from sampling points to the surface of 3D shape for training.
This is the main motivation for us to propose a method that can avoid such a voxalization used in algorithms that we discuss in the above paragraphs.
\subsection{Modelling local SDFs}
Instead of learning a single SDF for representing a whole shape, Jiang \etal~\cite{jiang2020local} designed an embedding of local crops of 3D shapes during training, and optimize a set of latent codes on a regular grid of overlapping crops with one single shared decoder when run on inference.
Inspired by DeepSDF~\cite{park2019deepsdf}, Chabra \etal~\cite{chabra2020deep} replaced the dense volumetric SDF representation used in traditional surface reconstruction with a set of locally learned continuous SDFs defined by a single parameterized neural network.
In contrast with the voxels(grid)-based representation of SDFs, Genova \etal~\cite{genova2019learning} proposed a network to encode shapes into structured implicit functions (SIF) as a composition of local shape elements.
Tretschk \etal~\cite{tretschk2020patchnets} designed an explicit parametric surface model, which fits an implicit SDF in each local patch separated from one sphere.
Hao \etal~\cite{hao2020dualsdf} represent 3D shapes as two levels of granularity with SDF, which provides interpretability for latent space of SDF in local parts of the shape.
They introduced a novel shape manipulation method by editing the primitives of local SDFs.
In the most recent works, Genova \etal~\cite{genova2020local} developed the SIF to learn a set of local SDFs that are arranged and blended according to a SIF template.
The method associates a latent vector with each local region that can be decoded with the SDF to produce finer geometric detail.
\subsection{Geometric learning on 3D shapes}
Generalize neural networks to data with the non-Euclidean structure are known as Graph Neural Networks (GNNs) in the domain of geometric learning.
Ranjan \etal~\cite{ranjan2018generating} proposed to learn a non-linear representation of human faces by spectral convolutions with Chebychev biasis~\cite{kipf2016semi, defferrard2016convolutional} as filters.
Bouritsas \etal~\cite{bouritsas2019neural} replaced the convolution kernel with operators applied along a spiral path around the graph vertices.
Hanocka \etal~\cite{hanocka2019meshcnn} leveraged the intrinsic geodesic connections of edges to define convolution operators, which inherited the direction invariant property as in 3D points convolution methods~\cite{qi2017pointnet, qi2017pointnet++}.
Zhou \etal~\cite{zhou2020fully} further improved the reconstruction precision by using locally adaptive convolution operators for registered mesh data.
We incorporate several ideas of this and the previous subsection in our method, but do it in different ways. More details are in the next section below.
\section{Local geometry code learning method}
\subsection{Modeling SDF locally}
A shape $\mathcal{S}$ can be represented as the zero level-set of $f_{\theta}(\mathbf{x}, \mathbf{z})$ as:
\begin{equation}
\mathcal{S} = \{ \mathbf{x} \in \mathbb{R}^3 \mid f_{\theta}(\mathbf{x,z})=0 \},
\end{equation}
where $f_{\theta}(\mathbf{x},\mathbf{z}):\mathbb{R}^3 \times \mathbb{R}^m \rightarrow \mathbb{R}$ is a signed surface distance function implemented as a neural network (usually as multilayer perceptron network) with learnable parameters $\theta$. The latent code $\mathbf{z}$ decides output shape of $f_{\theta}$ along with the sampling coordinates $\mathbf{x}$.
Similar with DeepLS~\cite{chabra2020deep}, we want to make $f_{\theta}$ model the whole shape as a composition of its local parts:
\begin{equation}
\mathcal{S} = \{ \mathbf{x} \in \mathbb{R}^3 \mid \bigcup\nolimits_{i} \mathbbm{1}_{\mathbf{x} \in \mathcal{R}_i} f_{\theta}(T_i(\mathbf{x}),\mathbf{z}_{i}^{L})=0 \},
\end{equation}
where $T_i(\cdot)$ supposes to transfer global location $\mathbf{x}$ to the local coordinate system $\mathbf{x}_{i}^{L}$ of a local region $\mathcal{R}_i$, and $\mathbf{z}_i^{L}$ indicates its related local latent code, as illustrated in Fig~\ref{fig:geo_sdf_decoder}~(a).
Different from splitting the 3D space into volumes~\cite{jiang2020local,chabra2020deep} or explicitly parametric surface model~\cite{tretschk2020patchnets, hao2020dualsdf}. we define the local region $\mathcal{R}_i$ with a key point $\mathrm{p}_i$ as:
\begin{equation}
\mathcal{R}_i = \{ \mathbf{x} \in \mathbb{R}^3 \mid \argmin\nolimits_{\mathbf{p} \in \mathcal{P}} d(\mathbf{x}, \mathbf{p}) = \mathbf{p}_i\},
\end{equation}
where $\mathcal{P}$ is a set of key points and $d(\cdot)$ is a distance function, \eg Euclidean distance.
Note that key points $\mathcal{P}$ are only used for aligning the sampling points to their corresponding local latent code $\mathbf{z}_i^L$. Thus,these key points do not necessary in the training or inference.
One simple illustration for our region division method is shown Fig~\ref{fig:geo_sdf_decoder}~(b). One patch with different color from their neighbours indicates a local region, which owns a corresponding latent code.
\subsection{Geometric leaning on local latent codes}
Different from the volume representation in DeepLS~\cite{chabra2020deep} and LDIF~\cite{genova2020local}, we do not know which local region includes the 3D shape, and it will lead to defining some unnecessary local latent codes.
Another problem is that these local latent codes are learned independently, which will lead to inconsistent surface estimates at the local region boundaries as mentioned in~\cite{chabra2020deep}.
Inspired by the geometric learning of COMA~\cite{ranjan2018generating}, Neural3DMM~\cite{bouritsas2019neural}, FCM~\cite{zhou2020fully}, we introduce the mesh structure of 3D shape as a "scaffold" to put key points and propagate information between local latent codes.
Then we can get two benefits that it allows to keep each local region has an intersection with the 3D shape and construct communications among them by representing the mesh as a graph.
Let us assume that a 3D surface mesh can be defined as a set of vertices $\mathcal{V}$ and edges $\mathcal{E}$, where $\mathcal{V}$ is replaced by $\mathcal{P}$ to define the local regions as shown in Fig \ref{fig:geo_sdf_decoder} (b).
Cooperate with several graph convolution layers to construct a graph network $\mathrm{G2L}(\mathcal{E}, \mathbf{z}^G): \mathbf{z}^G \rightarrow \{ \mathbf{z}^L_i \}$, we could predict the local SDF with the local latent codes aligned to these local regions, as shown in Fig \ref{fig:geo_sdf_decoder} (a).
Such graph neural network provides geometric deformations on each local latent code.
Consequently, each local latent code can contribute to the shape representation since each key point is on the shape.
Since our method gets benefits from modeling SDF in \textbf{Local} with latent codes and learning these latent codes through \textbf{Geometric Learning} with graph neural networks, thus we name it Local Geometry Code Learning (LGCL).
\subsection{Loss function}
\noindent {\bf Sign agnostic learning.}
Due to the advanced works of SAL~\cite{atzmon2020sal} and IGR~\cite{gropp2020implicit}, we do not request the true distance of sampling points to the surface of shape during training.
Instead of getting the true distance in a rendering way, directly calculating the distance from a point to a triangle soup is more convenient and efficient, and also without the requirement of watertight structures.
Thus, we construct basic loss function as:
\begin{equation}
\mathcal{L}_{\mathrm{basic}} = \mathcal{L}_{\mathrm{sal}} + \mathcal{L}_{\mathrm{igr}} ,
\end{equation}
where $\mathcal{L}_{\mathrm{sal}}$ just needs the unsigned distance $d_{\mathrm{u}}$ from point $\mathbf{x}$ that sampled from whole space $\Omega$ to the shape $\mathcal{S}$, and it is defined as:
\begin{equation}
\mathcal{L}_{\mathrm{sal}} = \mathbb{E}_{\mathbf{x}\in\Omega} \big| \left|f_{\theta}(T_i(\mathbf{x}), \mathbf{z}_i^L) \right| - d_{\mathrm{u}}(\mathbf{x}, \mathcal{S}) \big|.
\end{equation}
For the $\mathcal{L}_{\mathrm{igr}}$, we use its variant type from Siren~\cite{sitzmann2020implicit} as:
\begin{equation}
\begin{aligned}
\mathcal{L}_{\mathrm{igr}} &= \lambda_{\mathrm{grad}} \mathbb{E}_{\mathbf{x}\in\Omega}(\| \nabla_{\mathbf{x}}f_{\theta} \|_2 -1)^2 +\\
&\quad \quad
\mathbb{E}_{\mathbf{x}\in\Omega_0} \| \nabla_{\mathbf{x}}f_{\theta} - n(\mathbf{x}) \|_2,
\end{aligned}
\end{equation}
where $\Omega_0$ means the domain of zero-iso surface of the shape, $\|\cdot\|_2$ is the euclidean 2-norm.
\noindent {\bf Geometric similarity loss.}
There is a contradiction between the distributions of key points and sampling points.
Consider a complex part of the shape, it needs more key points to get more local latent codes for better modeling.
However, the more key points are allocated, the harder the optimization of local latent codes is.
Since each local region would be smaller and get fewer sampling points for training.
Even if increasing the number of sampling points, it is still difficult to ensure assigning enough sampling points to each local latent code.
To alleviate this problem, we propose a loss $\mathcal{L}_{\mathrm{sim}}$ to make these local latent codes not only learn from the sampling points but also learn from each other.
The assumption here is the difference between the adjacent local latent codes is smaller than the ones that are far away from each other.
On the other hand, it provides a kind of regularization effect on the local latent codes that cannot get sufficient training, which forces them to be similar to their neighbors.
Again, we use the geometric structure as a graph to calculate $\mathcal{L}_{\mathrm{sim}}$ as:
\begin{equation}
\mathcal{L}_{\mathrm{sim}} = \frac{1}{N_{\mathrm{v}}} \sum^{N_{\mathrm{v}}}_{i} \left| \mathbf{z}_i^L - \sum^{K}_{k} G_{\mathrm{l}}(\mathbf{z}_i^L, \mathcal{N}_k(i)) \right|
\end{equation}
where $G(x_i, \mathcal{N}(i)): x = \frac{1}{\left| \mathcal{N}(i) \right|} \sum_{j\in \mathcal{N}(i)}(x_j)$ means to update the value of $x_i$ by the average value of its neighbours.
Here $\mathcal{N}_k(i)$ means the neighbours of vertex $i$ in the $k$ layer.
And for better learning, we increase the neighbor region of one local latent code by $K$ layers.
We use $K=3$ layers for our experiments.
\noindent {\bf Total loss}
Our final loss function consists of above losses and a regular term $\| \mathrm{z}^G\|$ as:
\begin{equation}
\mathcal{L}_{\mathrm{total}} = \lambda_{\mathrm{sim}} \mathcal{L}_{\mathrm{sim}} + \mathcal{L}_{\mathrm{basic}} + \lambda_{reg} \|\mathrm{z}^G\|_2
\end{equation}
In our experiments, we use the setting of $\lambda_{\mathrm{grad}}=0.1$, $\lambda_{\mathrm{sim}}=1.0$ and $\lambda_{\mathrm{reg}}=0.001$ if without extra explanation.
\section{Experiments}
In our experiments, all of the used models are trained and evaluated mainly on a subset of the D-Faust dataset~\cite{bogo2017dynamic}, which is the No.50002 subset of mesh registrations for 14 different actions about the human body, such as leg and arm raises, jumps, etc.
Due to the low variation between adjacent scans, we sample the used dataset at a ratio of 1:10 and then split them randomly with 90\% for training and 10\% for the test.
Our data pre-processing method inherits from both IGR~\cite{gropp2020implicit} and SAL~\cite{atzmon2020sal}, which will generate 600K sampling points from each object, 300K are sampled on the object surface with their normals and the other 300K are sampled from two Gaussian distributions centered at points on the object surface.
\subsection{Reconstruction}
As one of the main baselines, we train the DeepSDF with the same setting in ~\cite{park2019deepsdf} on the completed No.50002 sub dataset of the D-FAUST dataset.
In addition, we also train other two different sizes of DeepSDFs with the loss function $\mathcal{L}_{\mathrm{basic}}$: SDF-8, which is similar to the original DeepSDF, but with 8 fully connected layers and 512 neurons in each hidden layer. The dimension of its latent code is 256. One skip connection is also used at the fourth layer with the latent code. SDF-4, has 4 fully connected layers, 128 neurons in each one and none skip connection, and the length of latent code is 64. Each fully connected layer in both SDF-8 and SDF-4 except the last one is followed by Softplus activation and initialized as in ~\cite{atzmon2020sal}.
In our LGCL-based method, we use a 4-layers G2L (as shown in Fig.~\ref{fig:g2l}) network followed by an SDF-4 network. The graph convolution kernels in the G2L are from~\cite{ranjan2018generating} as chebConv or~\cite{zhou2020fully} as vcConv, we gave the difference of their performance in the following results.
we re-implemented the kernels in a more compact form, and the parameters of one graph convolution layer can be represented as $(\mathrm{n},\mathrm{m},\mathrm{k})$, where $\mathrm{n}$ means the size of input channel and $\mathrm{m}$ is the size of the output channel. $\mathrm{k}$ stands for the size of the Chebyshev filter when using chebConv and the number of weight basis when using vcConv.
More details about the architecture of our models can be viewed in Table~\ref{tab:arch}.
The local latent code for each vertex of the G2L is set to an 8 length vector.
\begin{table}
\begin{center}
\begin{tabular}{l|c|c}
\hline
& LGCL-VC & LGCL-Cheb \\
\hline
\multirow{4}{*}{G2L} & vcConv(8,8,8) & chebConv(8,8,6) \\
& vcConv(8,16,16) & chebConv(8,16,6) \\
& vcConv(16,32,32) & chebConv(16,32,6) \\
& vcConv(32,64,64) & chebConv(32,64,6) \\
\hline
\multirow{4}{*}{SDF-4}
& \multicolumn{2}{c}{Linear(67,128)} \\
& \multicolumn{2}{c}{Linear(128,128)} \\
& \multicolumn{2}{c}{Linear(128,128)} \\
& \multicolumn{2}{c}{Linear(128,1)} \\
\hline
\end{tabular}
\end{center}
\caption{Setting of the architecture in the LGCL method. LGCL-VC means using the graph convolution kernels from~\cite{zhou2020fully} and LGCL-Cheb is from~\cite{ranjan2018generating}.}
\label{tab:arch}
\end{table}
We train all the models with 300 epochs at a learning rate of 5e-4 for the parameters of neural networks and 1e-3 for latent codes optimization.
Both learning rates are decayed to half after 200 epochs.
We evaluate all the methods on the split test dataset.
As same as in DeepSDF~\cite{park2019deepsdf}, the latent code will be estimated with the frozen neural network before the inference.
\begin{table}
\begin{center}
\setlength{\tabcolsep}{2pt}{
\begin{tabular}{l|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} & Net & Latent & \multirow{2}{*}{CD} & \multirow{2}{*}{HD} & \multicolumn{2}{c}{ED} \\
& Params & Params & & & Mean & Std \\
\hline
DeepSDF & 1.84 M & 0.26 K & 0.28 & 68.11 & 12.51 & 16.37 \\
SDF-8 & 1.58 M & 0.26 K & 0.22 & 59.86 & 6.94 & 10.30 \\
SDF-4 & 41.86 K & 0.06 K & 2.20 & 119.56 & 24.45 & 31.19 \\
LGCL-Cheb & 58.42 K & 55.12 K & 2.56 & 108.58 & 2.51 & 1.87 \\
LGCL-VC & 0.19 M & 55.12 K & 1.55 & 71.67 & 3.35 & 2.44 \\
\hline
\end{tabular}}
\end{center}
\caption{Quantitative evaluation. CD means Chamfer Distance, HD means Hausdorff Distance and ED stands for Euclidean Distance of the point to surface.
All the distances are represented in millimeters.
We directly run the DeepSDF code as the baseline.}
\label{tab:qe}
\end{table}
\begin{table}
\begin{center}
\setlength{\tabcolsep}{1pt}{
\begin{tabular}{l|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} & \multicolumn{2}{|c}{Error(mm)} & \multicolumn{3}{|c}{Percentage($\%$)}\\
& $<50\%$ & $<90\%$ & $>5$ mm & $>10$ mm & $>20$ mm\\
\hline
DeepSDF & 6.19 & 37.79 & 57.42 & 33.27 & 17.44 \\
SDF-8 & 3.55 & 18.59 & 34.79 & 14.14 & 6.39 \\
SDF-4 & 11.91 & 73.42 & 34.80 & 43.07 & 25.37 \\
LGCL-Cheb & 2.13 & 5.21 & 11.44 & 0.03 & 0.00 \\
LGCL-VC & 2.92 & 6.85 & 24.47 & 0.98 & 0.00 \\
\hline
\end{tabular}}
\end{center}
\caption{Statistics of reconstruction errors.}
\label{tab:statis}
\end{table}
To evaluate the performance of reconstruction, we measure the Euclidean Distance (ED) from the vertices of ground truth to the surface of reconstruction generated from different methods.
We also report our results under the metrics of Chamfer Distance (CD) and Hausdorff Distance (HD).
Due to CD and HD are applied on the point cloud, then we sample 30000 points on both surfaces of ground truths and reconstruction for it.
For a more fair comparison, we also list the size of network parameters and latent codes of different methods, while both are necessary to represent 3D shapes.
All of these quantitative results are shown in Table~\ref{tab:qe}.
As one can see that SDF-only-based methods need much more parameters of network to achieve comparable performance with our method.
We can see that there is a positive correlation between the size of the DeepSDF network and its performance on reconstruction, as all the quantitative results of SDF-4 are worse than SDF-8's.
By introducing local latent codes, our LGCL-based model outperforms SDF-4 by approximately one order of magnitude under the metric of Euclidean distance.
Even compared to the DeepSDF-8 which has a huge size of parameters, our results still has competitive advantages and obtain the smallest Euclidean error as $2.51 \pm 1.87$ mm with chebConv kernels.
More details about the Euclidean errors of different methods can be found in Table~\ref{tab:statis}.
Consequently, LGCL-Vc decreases the errors of CD and HD of SDF-4 by about 30\% and 40\% respectively.
We visualize two examples about the Euclidean error of each vertex shown in Fig~\ref{fig:compare2degeneration}.
It obviously shows that the small size of the DeepSDF network struggles to reconstruct the details, note that it almost loses the whole hands of the human.
In contrast, our LGCL model could keep more information in local regions though it causes more fluctuation.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/compareerror-00115-00170.pdf}
\end{center}
\caption{Visualization of the per-vertex Euclidean error of the reconstructions.
GT means the ground truth shape, the model of Ours here used the LGCL-VC.}
\label{fig:compare2degeneration}
\end{figure}
\subsection{Ablation study}
\label{sec:ablation}
We perform ablative analysis experiments to evaluate the influence of our proposed geometric similarity loss $\mathcal{L}_{\mathrm{sim}}$.
It is controlled by adjusting its coefficient $\lambda_{\mathrm{sim}}$ to constrain the similarity between local latent codes with their neighbours.
As shown in Table~\ref{tab:lambda_influence},
the geometric similarity loss takes different influences on LGCL-VC and LGCL-Cheb.
Specifically, it tends to get better ED results with less constraint on similarities of local latent codes of LGCL-VC.
In contrast, one needs more similar local latent codes to decrease the errors of CD and HD since the large freedom of graph convolution kernels that used in LGCL-VC.
And for LGCL-Cheb, it implicitly has a stronger geometric constraint set by its ChebConv kernels.
Thus the extra geometric similar loss takes a little impact on the errors of CD and HD, but it should be patient to pick the adopted when you consider the ED errors.
\begin{table}
\begin{center}
\setlength{\tabcolsep}{3pt}{
\begin{tabular}{l|c|c|c|c|c|c}
\hline
\multirow{2}{*}{Model} &
\multirow{2}{*}{$\lambda_{\mathrm{sim}}$} &
\multirow{2}{*}{CD} &
\multirow{2}{*}{HD} & \multicolumn{3}{c}{ED} \\
& & & & Mean & Std & Median \\
\hline
\multirow{3}{*}{LGCL-VC}
& 0.1 & 2.66 & 105.59 & 2.45 & 1.72 & 2.16 \\
& 1.0 & 1.55 & 71.67 & 3.35 & 2.44 & 2.92 \\
& 10.0 & 1.45 & 66.05 & 4.07 & 2.97 & 3.56 \\
\hline
\multirow{3}{*}{LGCL-Cheb}
& 0.1 & 2.43 & 108.65 & 2.72 & 1.91 & 2.39 \\
& 1.0 & 2.56 & 108.58 & 2.51 & 1.87 & 2.13 \\
& 10.0 & 2.74 & 108.69 & 4.44 & 3.11 & 3.94 \\
\hline
\end{tabular}}
\end{center}
\caption{Influence of geometric similarity loss, all results are shown in millimetres.}
\label{tab:lambda_influence}
\end{table}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth]{figs/failed_results.pdf}
\end{center}
\caption{Failed results of different methods for getting local latent codes.
(a) is the ground truth~(GT);
(b) uses G2L but with the same initial input of global latent code for each graph vertex;
(c) uses a pyramid G2L to get local latent codes;
(d) does not include $\mathcal{L}_{\mathrm{sim}}$.
Please check more details in Sec~\ref{sec:ablation}.}
\label{fig:failed_results}
\end{figure}
We found some interesting explorations on the influence of different methods for getting local latent codes, as shown in Fig~\ref{fig:failed_results}.
In our LGCL-based methods, we split the global code $\mathrm{z}^G$ evenly into parts, which is equal to the number of the graph vertices, and then align these parts to different vertices.
However, for the result in Fig~\ref{fig:failed_results}~(b), we directly align the same initial input, which is the global latent code, to each vertex of the G2L.
Since each local latent code has the same initial value, it provides a similarity constraint implicitly between them.
Then we do not introduce the $\mathcal{L}_{\mathrm{sim}}$.
In this case, each local region of reconstruction tends to shrink to the same type of mini polyhedron.
We attribute this degeneration of modelling local SDFs to the over-constraint on the similarity among the local latent codes, which introduces a limitation to geometric deformation of them.
We also show the results without the usage of $\mathcal{L}_{\mathrm{sim}}$ in our LGCL-based methods in Fig~\ref{fig:failed_results}~(d).
It is obvious to see the dramatic vibrations in some local regions.
We argue that it is caused by insufficient training of some local latent codes in corresponding local regions.
Furthermore, as shown in Fig~\ref{fig:failed_results}~(c), the result looks like fall in between (b) and (d), which has both similar polyhedron and vibrations that exist in local regions.
We got this result by modifying the G2L network as a pyramid structure as in the COMA~\cite{ranjan2018generating} decoder.
The modification changes the graph structure of each graph convolution layer with pooling layer.
The pyramid structure provides a cluster mapping from top to bottom in this G2L variant.
In other words, the mentioned modification adds an extra similarity constraint on local latent codes within a more large geometric range.
However, the approach is also limited to deform the local latent codes and leads to a compromise result compared to (b), (d) and our major results in Fig~\ref{fig:compare2degeneration}.
\section{Conclusions}
In this work, we propose the LGCL method which is based on a new architecture for the local geometry learning handling.
The idea is to perform the learning regression process directly in the latent code space. Consequently, our approach makes general GNNs more flexible, compact, and simple in realization.
The experimental results show that our method considerably outperforms baselines DeepSDFs both in accuracy and model size.
We think that our architecture is novel, promising, and can be further improved in future work.
\section{Acknowledgements}
This work has been supported by the Spanish project TIN2015-65464-R, TIN2016-79717-R (MINECO/FEDER) and the COST Action IC1307 iV\&L Net (European Network on Integrating Vision and Language), supported by COST (European Cooperation in Science and Technology).
We acknowledge the CERCA Programme of Generalitat de Catalunya.
Shun Yao acknowledges the Chinese Scholarship Council (CSC) grant No.201906290108.
We also acknowledge the generous GPU support from Nvidia.
{\small
\bibliographystyle{ieee_fullname}
|
1,116,691,500,478 | arxiv | \section*{Appendix}
\subsection*{A. Transference Formalism and Proofs}
\begin{example}[Freeze Quantification]
Suppose we want to express that whenever the event $Q$ occurs, it is followed
later by $R$, and then by $S$, such that the time difference between occurrences
of $Q$ and $R$ is at most $5$, and also the time difference between occurrences
of $Q$ and $S$ is at most $10$.
This can be expressed in \mTLTL{\small$(\TFun)$} as
\[\Box\Big( x.Q \rightarrow \Diamond \big(y.\big[ R\wedge(y\leq x+5) \wedge
\Diamond \left(z. \left(S \wedge z\leq x+10\right)\right)\big]\big) \Big).\]
Thus, freeze quantification, by giving a mechanism to bind times to variables,
allows us to relate,
with several constraints, far apart events.
\qed
\end{example}
\begin{example}[Freeze Quantification Functions]
Suppose we want to express that whenever the event $Q$ occurs,
it must be followed by a response $R$ within time $\lambda^{t_Q}$ for some
$\lambda > 1$ where $t_Q$ is the time at which $Q$ occurred;
thus, the later
$Q$ occurs the more delay we can tolerate in the response time.
The requirement can be expressed as
$x. \left( Q \rightarrow \Diamond
\big(y. \left(R \wedge 0 \leq y\leq \lambda^x\right)\big)\right)$.
\qed
\end{example}
\begin{example}[$\delta$-relaxation for Bounded Temporal Operators -- \MTL]
\label{example:RelaxationMTL}
We demonstrate how $\delta$-relaxation operates on bounded time constraints
through an example.
Consider an \MTL formula $\phi= Q \until_{[a,b]} R$.
The $\delta$-relaxation of this formula over the closed interval ${ I_{\TFun}} = \reals_+$
is $Q \until_{[a-2\cdot\delta\,,\, b+2\cdot \delta]} R$.
This can be seen as follows.
The formula $\phi$ can be written in \TLTL syntax as:
\[
x. Q \until y. \left( (y\leq x+b) \wedge (y \geq x+b) \wedge R\right).\]
The $\delta$-relaxation of this formula according to
Definition~\ref{definition:Relaxation} is:
\begin{align*}
\myrelax_{\reals_+}^{\delta}
\left(x. Q \until y. \left( (y\leq x+b) \wedge (y \geq x+a) \wedge R\right) \right) \ & =\\
&\hspace{-30mm} = \myrelax_{\reals_+}^{\delta}
\left(x. Q \until y. \left( (y-x-b\leq 0) \wedge (y -x-a\geq 0) \wedge R\right) \right) \\
& \hspace{-30mm} =
x. Q \until y. \left(
\begin{array}{l}
(y-x-b - 2\vdot \delta\leq0 ) \ \wedge\\
(y -x -a +2\vdot \delta\geq 0) \wedge R
\end{array}\right)
\\
&\hspace{-20mm}
\begin{array}{l}
\text{ since the Lipschitz constant of } y-x-c \text{ is } 2 \\
\text{ for any constant } c
\end{array}\\
&\hspace{-30mm} = x. Q \until y. \left( (y \leq x+b+2\vdot\delta ) \wedge (y \geq x +a -2\vdot \delta) \wedge R\right) \\
& \hspace{-30mm} = Q \until_{[a-2\cdot\delta,b+2\cdot \delta]} R.
\end{align*}
Thus, the time constraint interval boundaries are relaxed by $2\vdot \delta$.
The factor of $2$ arises because there are two
contributing factors: the starting time of $Q$ can be ``pulled back'' by
$\delta$, and the time of $R$ can be postponed by $\delta$; thus, the time
duration
in between $Q$ and $R$ increases by $2\vdot \delta$.
\qed
\end{example}
\smallskip\noindent\textbf{Removing Negation using the $\awaits$ Operator.}
The following identities hold relating the $\awaits$ operator to the
$\until$ operator
\begin{compactenum}
\item
$ \phi_1 \until \phi_2\, \equiv\, \neg\left(\neg(\phi_2) \awaits (\neg\phi_1 \wedge \neg\phi_2)\, \right)$; and
\item
$ \phi_1 \awaits \phi_2 \, \equiv \, \neg\left(\neg(\phi_2) \until (\neg\phi_1 \wedge \neg\phi_2)\, \right)$.
\end{compactenum}
Informally, the first identity states that $\neg (\phi_1 \until \phi_2)$ holds iff either
(i)~$\phi_2$ never holds; or
(ii)~there is a point where $\phi_1$ is false, and at that point and all points before it, $\phi_2$ has remained false.
The second identity is similar.
The first identity above allows us to ``push'' the negations down using the $\awaits$ operator.
The mechanism for the three interesting cases is below.
\begin{align*}
\neg\left(f_{\mytime}(x_1, \dots, x_l) \, \sim\, 0 \right) &
\ \equiv\ f_{\mytime}(x_1, \dots, x_l) \, \myneg(\sim)\, 0,\\
& \hspace{2.5cm}
\text{where, for } \sim \,\in \set{\leq, <, \geq, >} \text{ we have}\\
&\hspace{2.5cm} \begin{aligned}
\myneg(\leq) & \text{ to be } >; \hspace{5mm} &
\myneg(<) & \text{ to be } \geq;\\
\myneg(\geq) & \text{ to be } <; \hspace{5mm} &
\myneg(>) & \text{ to be } \leq
\end{aligned}\\
\neg(x.\psi) \, & \equiv\, x.\neg(\psi)\\
\neg\left(\phi_1 \until \phi_2 \right) \, & \equiv\, \neg(\phi_2) \awaits (\neg\phi_1 \wedge \neg\phi_2)
\end{align*}
\begin{proposition}
\label{proposition:PropositionalRelaxation}
The function $\myrelax$ is a relaxation on \mTLTL{\small$(\TFun)$} formulae,
\emph{i.e.} if a timed propositional trace $\pi\models \phi$ for
a \mTLTL{\small$(\TFun)$} formula $\phi$,
then $\pi\models \myrelax_{ I_{\TFun}}^{\delta}(\phi)$.
\end{proposition}
\begin{proof}
Observe that, over the predicates $f_{\mytime}(x_1, \dots, x_l) \sim 0$, the
function $\myrelax$ is indeed a relaxation, \emph{i.e},
if $f_{\mytime}(t_1, \dots, t_l) \sim 0$ for
values $t_1, \dots, t_l$, then
$\myrelax_{ I_{\TFun}}^{\delta}\left(f_{\mytime}(t_1, \dots, t_l) \right) \sim 0) $ also holds.
The result follows by a straightforward induction argument.\qedhere
\end{proof}
\begin{proof}[\textbf{Proof of
Theorem~\ref{theorem:PropositonalRobustness}}]
Let $\untime(\phi)$ be the formula where all freeze variable constraints are replaced
by $\true$ (\emph{e.g.} $\untime(x. (Q \wedge x<5)) $ is $x. (Q\wedge \true)$).
Since $\dist(\pi, \pi') < \delta$, we have that there exists a retiming
$r: \tdom(\pi) \mapsto \tdom(\pi')$ such that
\begin{equation}
\label{equation:SameSequence}
\pi(t) = \pi'(\retime(t)).
\end{equation}
This implies that
both $\pi$ and $\pi'$ satisfy $\untime(\phi)$, which can be
shown by an induction argument.
The interesting cases are for the $\until$ and $\awaits$ operators.
We sketch the argument for the $\until$ case (the argument for
$\awaits$ is similar).
The time environment $\env'$ for $\pi'$ assigns the time $\retime(t_x) $
to the freeze variable $x$ where the witnessing freeze variable environment $\env$ for
$\pi\models \phi$ assigns $t_x$ to $x$.
Let $\pi \models_{\env} \phi_1 \until \phi_2$, and let
$t$ be the time value which demonstrates this satisfaction (as in
Definition~\ref{definition:PropositionalSemantics}), with the corresponding
freeze variable environment $\env$.
To show $\pi' \models_{\env'} \phi_1 \until \phi_2$, we pick the time
$\retime(t)$, with the environment $\env'$ for $\pi'$ which
assigns the time $\retime(t_x) $
to the freeze variable $x$ where $\env(x) = t_x$.
It can be checked that, due to Equation~\ref{equation:SameSequence},
we have
(i)~$\retime(t) \geq \env'(x)$, for a freeze variable $x$ in
$\phi_1 \until \phi_2$
(which was previously bound);
(i)~${\pi'}^{\retime(t)} \models_{\env'} \phi_2$; and
(ii)~for all $t_0' \leq t^{\dagger} < \retime(t)$, we have
${\pi'}^{t^{\dagger}} \models_{\env'} \phi_1\vee \phi_2$.
Thus, $\retime(t)$, and $\env'$ demonstrate that
$\pi'\models_{\env'} \phi_1 \until \phi_2$.
We now check what is the relaxation needed on the original freeze variable constraints
so that $\pi'$ satisfies the relaxed constraints.
Without loss of generality, assume that each freeze variable $x$ is only quantified once
in $\phi$, \emph{i.e.} once it is bound to a value by ``$x.$'', it is not ``re-bound'' with
another application of ``$x.$''.
Let $\kappa_{\pi}$ denote an assignment of time values (from $I$)
to the freeze variables
such that all the freeze variable constraints in $\phi$ are satisfied, \emph{i.e.} $\kappa_{\pi}$
is an time environment witness to the satisfaction of $\phi$ by $\pi$.
Consider a free variable assignment $\kappa_{\pi'}$ corresponding to
$\kappa_{\pi}$, where $\kappa_{\pi'}(x) = \retime\left(\kappa_{\pi}(x)\right)$.
This is a legal variable assignment compatible with some $\until, \awaits$ time
witnesses which demonstrate that
$\pi'$ satisfies
$\untime(\phi)$, as shown previously.
Observe that by the existence of a retiming function, for all freeze variables $x$ occurring in $\phi$, we have
that $\abs{\kappa_{\pi'}(x) -\kappa_{\pi}(x)} < \delta$.
Since the time \emph{values} of variables are different in $\kappa_{\pi}$
and $\kappa_{\pi'}$, the original freeze constraints (\emph{e.g.} $x<5$) in $\phi$
might not be satisfied with the assignment $\kappa_{\pi'}$.
Consider a freeze variable constraint $f_{\mytime}(x_1, \dots, x_l) \sim 0$ in $\phi$.
We know that $f_{\mytime}(\kappa_{\pi}(x_1), \dots, \kappa_{\pi}(x_l)) \sim 0$ is true.
As $\abs{\kappa_{\pi'}(x) -\kappa_{\pi}(x)} \leq \delta$ for all
freeze variables $x$ occurring in $\phi$,
by the definition of relaxation, we have that
\begin{compactenum}
\item
$f_{\mytime}(\kappa_{\pi}(x_1), \dots, \kappa_{\pi}(x_l)) + K_{\mytime} ( \delta) \sim 0$
if $\sim\, \in \set{>, \geq}$; and
\item
$f_{\mytime}(\kappa_{\pi}(x_1), \dots, \kappa_{\pi}(x_l)) - K_{\mytime} (\delta) \sim 0$
if $\sim\, \in \set{<, \leq}$.
\end{compactenum}
This implies that $\kappa_{\pi'}$ is also a witness to the satisfaction of
$\myrelax_{I_{\pi,\pi'}}^{\delta}(\phi)$ by $\pi'$.
Thus, $\pi'\models \myrelax_{I_{\pi,\pi'}}^{\delta}(\phi)$.
\qedhere
\end{proof}
\begin{proof}[\textbf{Example~\ref{example:Transference-one} details}]
Since $\pi$ satisfies $\phi$, we must have time-stamps $t_x, t_y, t_z$ bound to
$x,y,z$ respectively so that with these assignments, the formula $\phi$
is satisfied.
Since $\pi'$ is $\delta$ close to $\pi$, for every $\epsilon > 0$, there is a
retiming from $\pi$ to $\pi'$ such that the times $t_x, t_y, t_z$ in $\pi$
are mapped to $t_x', t_y', t_z'$ in $\pi'$ such that
(a)~$|t_x-t_x'| \leq \delta+\epsilon$; and
(b)~$|t_y-t_y'| \leq \delta+\epsilon$; and
(c)~$|t_z-t_z'| \leq \delta+\epsilon$.
Let $\delta'= \delta+\epsilon$.
The sum $(t_x'-t_y')^2 + (t_y'-t_z') ^2+(t_z'-t_x')^2$ is
\begin{align*}
& =\, \left( (t_x'-t_x) + (t_x-t_y) + (t_y-t_y')\right)^2 \, +\,
\left( (t_y'-t_y) + (t_y-t_z) + (t_z-t_z')\right)^2 \, +\\
& \hspace*{63mm} \left( (t_z'-t_z) + (t_z-t_x) + (t_x-t_x')\right)^2\\
& =\,
2 \left((t_x'-t_x)^2 + (t_y'-t_y)^2 + (t_z'-t_z)^2\right) \, +\,
(t_x-t_y)^2 + (t_y-t_z) ^2+(t_z-t_x)^2\, +\,\\
& \hspace*{25mm}
2\left( (t_x'-t_x)(t_x-t_y) + (t_y-t_y')(t_x-t_y)+ (t_x'-t_x)(t_y-t_y')\right)
\, +\\
&\hspace*{25mm}
2\left( (t_y'-t_y)(t_y-t_z) + (t_z-t_z')(t_y-t_z)+ (t_y'-t_y)(t_z-t_z')\right)
\, +\\
& \hspace*{25mm}
2\left( (t_z'-t_z)(t_z-t_x) + (t_x-t_x')(t_z-t_x)+ (t_z'-t_z)(t_x-t_x')\right)\\
& \leq \,
6\delta'^2 + d+
2\delta'\abs{t_x-t_y} + 2\delta'^2
+2\delta'\abs{t_y-t_z} + 2\delta'^2
+ 2\delta'\abs{t_z-t_x} + 2\delta'^2\\
& = \,
d + 12 \delta'^2 + 4\delta'\left(\abs{t_x-t_y} + \abs{t_y-t_z} + \abs{t_z-t_x} \right)\\
& \leq \,
d + 12 \cdot\delta'^2 + 4\sqrt{3}\cdot\delta'\cdot \sqrt{d}
\end{align*}
In the last step above, we use the inequality:
$\abs{a} + \abs{b} + \abs{c} \leq \sqrt{3}\cdot\sqrt{ a^2 + b^2+c^2}$
This inequality is obtained by applying the Cauchy-Schwarz inequality to
the tuples $(\abs{a}, \abs{b}, \abs{c})$ and $(1,1,1)$.
Thus, by Theorem~\ref{theorem:PropositonalRobustness},
for every $\epsilon >0$, we have
\[\pi'\models
x. \left( Q \rightarrow \Diamond
\big(y. \left(R \wedge
\Diamond\left[z. \left(S \wedge
\left( (y-x)^2 + (z-y)^2 + (z-x)^2 \leq d^{\dagger} \right) \right)
\right] \right)
\big)\right) \]
where
$d^{\dagger} = d + 12 \cdot\delta'^2 + 4\sqrt{3}\cdot\delta'\cdot \sqrt{d}$.
\qedhere
\end{proof}
\begin{definition}[$\delta$-relaxation of \mTLTL{\small$(\TFun, \SFun)$} formulae]
\label{definition:Signal Relaxation}
Let $\phi$ be a \mTLTL{\small$(\TFun, \SFun)$} formula in which negations appear only on the
prepositional symbols .
The $\delta$ relaxation of $\phi$ (for $\delta \geq 0$), denoted
$\myrelax_{I_{\TFun}, \imap}^{\delta}(\phi)$ is defined as follows,
where $I_{\TFun}$, a closed subset of
$reals_+$, is the domain of the variables in $V_{\mytime}$;
and $ \imap$ is a mapping from $V_{\mysig}$ to closed intervals
of $\reals$ such that $\imap(z)$ denotes the domain of $z$.
\begin{align*}
\myrelax_{I_{\TFun}, \imap}^{\delta}(\true) & = \true;
\hspace{30mm} \myrelax_{I_{\TFun}, \imap}^{\delta}(\false) = \false;\\
\myrelax_{I_{\TFun}, \imap}^{\delta}( \phi_1 \wedge \phi_2 ) & =
\myrelax_{I_{\TFun}, \imap}^{\delta}( \phi_1) \wedge \myrelax_{\delta}(\phi_2 );\\
\myrelax_{I_{\TFun}, \imap}^{\delta}( \phi_1 \vee \phi_2 ) & =
\myrelax_{I_{\TFun}, \imap}^{\delta}( \phi_1) \vee \myrelax_{I_{\TFun}, \imap}^{\delta}(\phi_2 );\\
\myrelax_{I_{\TFun}, \imap}^{\delta}( x.\psi ) & = x.\myrelax_{I_{\TFun}, \imap}^{\delta}(\psi);\\
\myrelax_{I_{\TFun}, \imap}^{\delta}( \phi_1 \until \phi_2 )& =
\myrelax_{I_{\TFun}, \imap}^{\delta}(\phi_1) \until \myrelax_{I_{\TFun}, \imap}^{\delta}(\phi_2);\\
\myrelax_{I_{\TFun}, \imap}^{\delta}( \phi_1 \awaits \phi_2 )& =
\myrelax_{I_{\TFun}, \imap}^{\delta}(\phi_1) \awaits \myrelax_{I_{\TFun}, \imap}^{\delta}(\phi_2)
\\
\myrelax_{I_{\TFun}, \imap}^{\delta}\left(f_{U}(z_1, \dots, z_l) \right) \sim 0) & =
\begin{cases}
f_{U}(z_1, \dots, z_l) \, +\, K_{f_{U}}(\delta) \ \sim\, 0 & \text{ if }
\sim\, \in\set{>, \geq};\\
f_{U}(z_1, \dots, z_l) \, -\, K_{f_{U}}(\delta) \ \sim \, 0 & \text{ if }
\sim \,\in\set{<, \leq};\\
\end{cases}\\
& \qquad \text{ where } U \in \set{ \mytime, \mysig} \text{ with } K_{f_{U}}
\text{ being as in Definition~\ref{definition:Relaxation};}\\
&\qquad\text{ and }
K_{f_{\mysig}}: \big[0,\ \max_{z\in V_{\mysig}} |\max \imap(z) \, - \, \min\imap(z)| \big]
\mapsto \reals_+\\
&\qquad \text{ is a function such that:}\\
&\hspace{-40mm}
K_{f_{\mysig}}(\delta) =
\sup_{
\begin{array}{c}
z_i\in \imap(z_i); \, z'_i\in \imap(z'_i)\\
\text{ for all } i
\end{array}
}
\left\{
\left\arrowvert
\begin{array}{c}
f_{\mysig}(z_1,\dots, z_l) \\
-\\
f_{\mysig}(z_1',\dots,z_l')
\end{array}\right\arrowvert
\text{ s.t. }
|z_i-z_i'| \leq \delta \text{ for all } i
\right\}\qedhere
\\
\end{align*}
\qed
\end{definition}
The functions $K_{f_{\mysig}}(\delta)$ define the maximal change in the value
of $f_{\mysig}$ that can occur when the input variables can vary by $\delta$.
The role of $\imap$ in the above definition is to restrict the domain of the signal variables
in order to obtain the least possible bounds relaxation bounds on the signal
constraints; as was done in Definition~\ref{definition:Relaxation} for the
freeze variables.
\begin{proposition}
\label{proposition:SignalTLTLRelaxation}
The function $\myrelax_{I_{\TFun}, \imap}^{\delta}$ is a relaxation on \mTLTL{\small$(\TFun, \SFun)$} formulae,
\emph{i.e.} if a timed $\reals^n$-valued trace $\pi\models \phi$ for
a \mTLTL{\small$(\TFun, \SFun)$} formula $\phi$,
then $\pi\models \myrelax_{I_{\TFun}, \imap}^{\delta}(\phi)$.
\end{proposition}
\begin{proof}
The proof is similar to the proof of Proposition~\ref{proposition:PropositionalRelaxation}.
\qedhere
\end{proof}
\begin{proof}[\textbf{Proof of
Theorem~\ref{theorem:SignalTLTLRobustness}}]
The proof use the result for the propositional case, Theorem~\ref{theorem:PropositonalRobustness}.
We construct the propositions $p_{f_{\mysig}}$ defined to be
$\myrelax_{I_{\TFun}, \imap}^{\delta}\left(f_{\mysig}(\overline{y}) \right) \sim 0) $
for the constraints over $V_{\mysig}$
in the formula $\phi$; and define the \mTLTL{\small$(\TFun)$} formula
$\phi_{\prop}$ as that obtained from $\phi$ by syntactically replacing
each constraint $ f_{\mysig}(\overline{y}) \sim 0$ in $\phi$ by $p_{f_{\mysig}}$.
Let $\prop_{\mysig}$ denote all such predicates for $\phi$.
We obtain the timed $\prop_{\mysig}$ propositional traces
$\pi_{P_{\mysig}}, \pi_{P_{\mysig}}'$
from $\pi, \pi'$ by mapping to propositions.
By the definition of the skorokhod distance, the distance between
$\pi_{\prop_{\mysig}}$ and $\pi_{\prop_{\mysig}}'$ is less than $\delta$.
By Theorem~\ref{theorem:PropositonalRobustness},
$ \pi_{\prop_{\mysig}}' \models \phi_{\prop}$.
This implies $\pi' \models \myrelax_{I_{\TFun}, \imap}^{\delta}(\phi)$.
\qedhere
\end{proof}
\subsection*{B. Details on Case Studies}
\paragraph{LQR-based pitch controller.} The aircraft pitch controller
system has $3$ state variables, and the state vector $\mathbf{x}$ =
$[\alpha\ q\ \theta]$, where $\alpha$ is the angle of attack, $q$ is the
pitch rate, and $\theta$ is the pitch angle. The system has a single input
$\delta$ (the elevator deflection angle). In deriving the control law, the
designers use the state feedback law to substitute $\delta$ =
$\theta_{des}-K\mathbf{x}$, where $\theta_{des}$ is the desired pitch
angle. The resulting dynamical equations of the system are of the form
$\dot{\mathbf{x}} = (A-BK)\mathbf{x} + B\theta_{des}$, and the output of
the system is the state variable $\theta$. Note that the $K$ matrix is the
gain matrix resulting from the LQR control design technique. The values of
the $A$, $B$ and $K$ matrices are as given below:
\[
\begin{array}{ll}
\vspace{1em}
A = \left[\begin{array}{l@{\hspace{1em}}l@{\hspace{1em}}l}
-0.313 & 56.7 & 0 \\
-0.0139 & -0.426 & 0 \\
0 & 56.7 & 0 \\
\end{array}
\right] &
B = \left[\begin{array}{l}
0.232 \\
0.0203 \\
0 \\
\end{array}
\right] \\
K = [-0.6435\ \ 169.6950\ \ 7.0711] & \\
\end{array}
\]
\paragraph{Air-Fuel Ratio Controller.} The Air-Fuel (A/F) ratio control
systems that we consider are simplified versions of industrial-scale
models. Both versions have $2$ exogenous inputs, and $4$ continuous states.
The inputs are engine speed (measured in rpm) and the throttle angle (in
degrees). The throttle angle is a user input, and it is common to assume a
series of pulses or steps as throttle angle inputs. The engine speed is
considered an input to avoid modeling parts of the powertrain dynamics. In
our experiments, we typically hold the engine speed constant. This is to
mimic a common engine testing scenario involving a dynamometer, which is a
device to provide external torque to the engine to maintain it at a
constant speed. Of the $4$ continuous states, we assume that $2$ of these
states are from the plant model (that encapsulates physical processes
within the engine), while $2$ states belong to the controller. The plant
states $p$ and $\lambda$ denote intake manifold pressure and the A/F ratio
respectively. The controller states $p_e$ denotes the estimated manifold
pressure (with the use of an observer) used in the feed-forward control,
and the state $i$ denotes the integrator state in the P+I feedback control.
We check conformance with respect to the system output $\lambda$. For the
dynamical system equations, please refer to
\cite{hscc14benchmark,hscc14lyapunov}.
\section{Skorokhod Distance based Conformance Testing}
\label{sec:conformance}
\vspace{-3mm}
In conformance testing, we test for the variance in behavior of two given
systems $\system_1$ and $\system_2$ under the same input\footnote{It is
also possible to extend our approach to allow inputs that are within some
bounded Skorokhod distance.}. Given the same input, the two systems
produce potentially differing output traces; the goal is to quantify this
difference, and to determine an input signal that causes the corresponding
output signals to exceed a user provided bound on the maximum tolerable
output trace distance.
\begin{comment}
In some cases, given systems $M_1$ and $M_2$ such that $M_1$ satisfies the
temporal requirement $\varphi_1$, then a bound $\delta$ could be provided
by the user based on the weakest $\delta$-relaxation of $\varphi_1$ that
the user tolerates for the behaviors of $M_2$ to satisfy. We begin by
discussing a simulation-guided (sound but incomplete) testing algorithm. In
rest of the section, we discuss several case studies, providing rationale
for choosing the appropriate $\delta$ and presenting results on the
computation time.
\end{comment}
\newcommand{\leftarrow}{\leftarrow}
\newcommand{\mathit{cost}}{\mathit{cost}}
\newcommand{\mathit{maxCost}}{\mathit{maxCost}}
\newcommand{\mathit{maxIterations}}{\mathit{maxIterations}}
{\small
\begin{algorithm}
\DontPrintSemicolon
\caption{Algorithm to test if $\displaystyle\max_{y_1 , y_2} \dist_\skoro(y_1,y_2) \, <\delta$}
\label{algo:simuskoro}
\KwIn{Systems $\system_1$, $\system_2$, Bound $\delta$, Input Parameterization
$(P,F,B)$, Time Horizon $T$}
\KwOut{$u(t)$ s.t. $y_1 = \system_1(u)$, $y_2 = \system_2(u)$, and $\dist_{\skoro}(y_1,y_2) > \delta$}
$u \leftarrow \mathtt{random}(P,F,B)$ \;
$\mathit{maxCost} \leftarrow -\infty$, $m \leftarrow 0$ \;
\While{($maxCost < \delta$) \texttt{or} ($m < \mathit{maxIterations}$)}{
$y_1 \leftarrow \mathtt{simulate}(M_1, u, T)$ \;
$y_2 \leftarrow \mathtt{simulate}(M_2, u, T)$ \;
$\mathit{cost} \leftarrow \dist_{\skoro}(y_1,y_2)$ \;
\lIf{$\mathit{cost} > \mathit{maxCost}$}{
$\mathit{maxCost} \leftarrow \mathit{cost}$
}
$u \leftarrow \mathtt{pickNewInputs}(\mathit{cost})$ \;
$m \leftarrow m+1$\;
}
\end{algorithm}
}
Algorithm~\ref{algo:simuskoro} is a standard optimization-guided testing
algorithm in which we have used the Skorokhod distance between two output
traces as the cost function. In such algorithms, it is common to define a
finite parameterization of the input space, represented by the tuple
$(P,F,B)$, where $P = \{p_1,\ldots,p_k\}$ represents a set of parameters,
$F = \{f_1,\ldots,f_k\}$ represents a finite set of basis functions from
$[0,T]$ to $\reals^n$, where $T$ is some finite time-horizon, and for each
$p_i \in P$, there is a $b_i \in B$ that is a closed interval in $\reals$
over which $p_i$ is assumed to take values. An input signal $u$ is defined
such that, for all $t$, $u(t) = \sum_i p_i\cdot f_i(t)$. A valid input
signal has the property that for all $i$, $p_i \in b_i$.
In each step, the algorithm picks an input signal $u$ and computes the
Skorokhod distance between the corresponding outputs $y_1 = \system_1(u)$
and $y_2 = \system_2(u)$. Based on heuristics that rely on the current
cost, and a possibly bounded history of costs, the procedure then picks a
new value for $u$. For instance, in a gradient-ascent based procedure, the
new value of $u$ is chosen by estimating the local gradient in each
direction in the input-parameter space, and then picking the direction that
has the largest (positive) gradient. In our implementation, we use the
Nelder-Mead (or nonlinear simplex) algorithm.
The algorithm terminates when a violation is found (i.e., a pair of inputs
that exceed the user-provided Skorokhod distance bound), or when the number
of iterations is exhausted. The Skorokhod distance bound $\delta$ is
chosen based on engineering requirements, \emph{e.g.}, based on the maximum
allowed weakening of the temporal logical properties that have been
verified/tested on one system.
\smallskip\noindent\textbf{Sampling and Polygonal Approximations.}
In practice, the output behaviors of the systems are observed with
a sampling process, thus in implementations of Algorithm~\ref{algo:simuskoro},
entities $y_1$ and $y_2$ on lines $4,5$ are time-sampled output trace
\emph{sequences}, from which
the Skorokhod distance algorithm of Theorem~\ref{theorem:SkoroFinal}
constructs (continuous time) signals using linear
interpolation.
Given a timed trace sequence $\tseq$, let $\symb{\tseq}_{\LI}$ denote the
continuous time trace obtained from $\tseq$ by linear interpolation.
Let $\tseq_{\pi}, \tseq_{\pi'}$ be two corresponding samplings
of the traces $\pi, \pi'$.
Since the Skorokhod distance is a metric, we have that
\vspace{-1mm}
\[\dist_{\skoro}(\pi, \pi') \leq
\dist_{\skoro}\left(\symb{\tseq_{\pi}}_{\LI}, \symb{\tseq_{\pi'}}_{\LI}\right)\, +\,
\dist_{\skoro}\left(\symb{\tseq_{\pi}}_{\LI}, \pi\right) +
\dist_{\skoro}\left(\symb{\tseq_{\pi'}}_{\LI}, \pi'\right). \vspace{-1mm}\]
If $\Delta_{\sampleerr}$ is a bound on the distance between a trace, and an
interpolated completion of its sampling, we have that
$
\dist_{\skoro}(\pi, \pi') \leq \dist_{\skoro}\left(\symb{\tseq_{\pi}}_{\LI}, \symb{\tseq_{\pi'}}_{\LI}\right)\, +\,
2\vdot \Delta_{\sampleerr}$.
Thus, in a sampling framework, a value of $2\vdot \Delta_{\sampleerr}$
needs to be added to the Skorokhod distance between the polygonal approximations.
Section~\ref{section:Logic} presents a theory of (quantifiable)
transference of logical properties.
Section~\ref{section:Experiment} presents results on our implementation
of Algorithm~\ref{algo:simuskoro}.
We also discuss several case studies,
providing rationale for choosing the appropriate $\delta$ value,
and present results on the
computation time and the conformance distance found.
\section{Conclusion}
\vspace{-2mm}
Metrics for comparing behaviors of dynamical systems
which quantify both time and value distortions
have heretofore been
an object of mathematical inquiry, without enough attention being paid to
computational aspects and connections to logical requirements. We argue
that the Skorokhod metric provides a robust definition of conformance by
proving transference of a rich class of temporal logic properties.
We also demonstrate the computationally tractability of the metric for practical use
by constructing a conformance testing tool
in a simulation and optimization guided
approach for finding and quantifying non-conformant behavior of
dynamical systems.
Pinpointing the source of trace deviations is
necessary in many engineering applications;
our tool allows for independent weighing of time and value-dimension distortions
in order to achieve this objective.
\section{Experimental Evaluation}
\label{section:Experiment}
\mypara{Skorokhod Distance Computation Benchmark} The Skorokhod
distance is computed with the help of a streaming, sliding window
monitoring routine which checks for a fixed $\delta$ whether the
linear interpolations of two time-sampled traces are at most $\delta$
away from each other. The least such $\delta$ value is computed by
binary search over the monitoring routine. The upper limit of the
search range is set to the pointwise metric (\emph{i.e} assuming the
identity retiming) between the two traces. The traces to the
Skorokhod procedure are pre-scaled, each dimension (and the
time-stamp) is scaled by a different constant. The constants are
chosen so that after scaling, one unit of deviation in one dimension
is as undesirable as one unit of jitter in other dimensions. We next
present a benchmark on the distance computing routine.
Consider the hybrid dynamical system $\system_1$ shown in
Fig.~\ref{fig:watertank}. The system consists of two water tanks,
each with an outlet from which water drains at a constant rate $d_j$.
Both tanks share a single inlet pipe that is switched between the
tanks, filling only one tank at any given time at a constant inflow
rate of $i$. When the water-level in tank $j$ falls below level
$\ell_j$, the pipe switches to fill it. The drain and inflow rates
$d_1$, $d_2$ and $i$ are assumed to be inputs to the system. Now
consider a version $\system_2$ that incorporates an actuation delay
that is a function of the inflow rate. This means that after the level
drops to $\ell_j$ for tank $j$, the inlet pipe starts filling it only
after a finite time. $\system_1$ and $\system_2$ have the same
initial water level. We perform a fixed number of simulations by
systematically choosing drain and inflow rates $d_1$, $d_2$, $i$ to
generate traces (water-level vs. time) of both systems and compute
their Skorokhod distance. We summarize the results in
Table~\ref{tab:distance_evaluation}.
\vspace{-1mm}
\begin{figure}[t]
\centering
\begin{tikzpicture}
\tikzstyle{smalltext}=[font=\fontsize{8}{8}\selectfont]
\tikzstyle{state}=[draw,rectangle,rounded corners,minimum width=4.5em,smalltext]
\node[state] (s1) {%
$\left[\begin{array}{l}
\dot{h_1} \\
\dot{h_2} \\
\end{array}\right]\!\! = \!\!
\left[\begin{array}{l}
i - d_1 \\
-d_2 \\
\end{array}\right]$};
\node[state,right of=s1,node distance=40mm] (s2) {%
$\left[\begin{array}{l}
\dot{h_1} \\
\dot{h_2} \\
\end{array}\right]\!\! = \!\!
\left[\begin{array}{l}
- d_1 \\
i -d_2 \\
\end{array}\right]$};
\draw[->,>=latex'] (s1) to[out=30,in=150] node[smalltext,above] {$h_2 < \ell_2$} (s2);
\draw[->,>=latex'] (s2) to[out=210,in=330] node[smalltext,below] {$h_1 < \ell_1$} (s1);
\end{tikzpicture}
\vspace{-2mm}
\caption{System $\system_1$ used for benchmarking Skorokhod Distance computation.
Inflow rate $i$, Drain rate $d_1$ for tank $1$ and $d_2$ for tank $2$ are
all inputs to the system.\label{fig:watertank}}
\vspace{-2mm}
\end{figure}
\newcommand{\muc}[1]{\multicolumn{2}{c}{#1}}
\begin{table}[t]
\caption{Benchmarking the computation of $\dist_{\skoro}(\pi_1,\pi_2)$, where
$\pi_1$ is a trace of system $\system_1$ described in Fig.~\ref{fig:watertank}, and
$\pi_2$ is a trace of system $\system_2$, which is $\system_1$ with an actuation delay.
$\dist_2$ is the naive pointwise distance.
Both
$\pi_1$ and $\pi_2$ contain equally spaced $2001$ time points over a simulation
horizon of $100$ seconds. \label{tab:distance_evaluation}}
\centering
\begin{tabular*}{0.99\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Window size & Avg. $\dist_{\skoro}$ & \muc{Avg. Time taken (secs)} & $\max\frac{\dist_2 - \dist_\skoro}{\dist_2}$ \\
\cline{3-4}
& & Computation & Monitoring & \\
\midrule
20 & 8.58 & 0.81 & 0.13 & 0.09 \\
40 & 8.35 & 1.55 & 0.26 & 0.18 \\
60 & 8.09 & 2.31 & 0.39 & 0.26 \\
80 & 7.88 & 3.05 & 0.52 & 0.33 \\
100 & 7.72 & 3.77 & 0.64 & 0.38 \\
\midrule
\bottomrule
\end{tabular*}
\vspace{2mm}
\end{table}
Recall that $\dist_\skoro$ (the Skorokhod distance) computation involves a
sequence of monitoring calls with different $\delta$ values picked by a
bisection-search procedure. Thus, the total time to compute $\dist_\skoro$
is the sum over the computation times for individual monitoring calls plus
some bookkeeping. In Table~\ref{tab:distance_evaluation}, we make a
distinction between the average time to monitor traces (given a $\delta$
value), and the average time to compute $\dist_\skoro$. There are an
average of $6$ monitoring calls per $\dist_\skoro$ computation. We ran
$64$ simulations by choosing different input values, and then computing
$\dist_\skoro$ for increasing window sizes. As the window size increases,
the average $\dist_\skoro$ is seen to decrease; this is expected as a
better match may be achieved in a larger window. The computation time is
also seen to increase linearly, as postulated by
Theorem~\ref{theorem:SkoroFinal}. Finally, we see that the Skorokhod
distance is less aggressive at classifying traces as distant (as shown by
its lower overall numbers) than a simpler metric $\dist_2$ (defined as as
the maximum of the pointwise $L_2$ norm\footnote{Even though the difference
is only $38\%$ with respect to the pointwise metric, this difference is
amplified in the original state value domain, as in the experiment, the
input state values to the Skorokhod routine were scaled by $0.1$.}). We can
see this discrepancy becomes more prominent with increased window size
(because of better matches being available).
\mypara{Case Study: LQR-based Controller} The first case study is an
example of an aircraft pitch control application taken from the openly
accessible control tutorials for Matlab and Simulink \cite{ctms}. The
authors describe a linear dynamical system of the form:
$\dot{\mathbf{x}} = (A-BK)\mathbf{x} + B\theta_{des}$. Here,
$\mathbf{x}$ describes the vector of continuous state variables and
$\theta_{des}$ is the desired reference provided as an external input.
One of the states in the $\mathbf{x}$ vector is the pitch angle
$\theta$, which is also the system output. The controller gain matrix
$K$ is computed using the linear quadratic regulator method
\cite{lqr}, a standard technique from optimal control. We are
interested in studying a digital implementation of the continuous-time
controller obtained using the LQR method. To do so, we consider
sampled-data control where the controller samples the plant output,
computes, and provides the control input to the plant every $\Delta$
seconds. To model sensor delay, we add a fixed delay element to the
system; thus, the overall system now represents a delay-differential
equation.
Control engineers are typically interested in the step response of a
system. In particular, quantities such as the overshoot/undershoot of
the output signal (maximum positive/negative deviation from a
reference value) and the settling time (time it takes for transient
behaviors to converge to some small region around the reference value)
are of interest. Given a settling time and overshoot for the first
system, we would like the second system to display similar
characteristics. We remark that both of these properties can be
expressed in \STL, see \cite{hscc14benchmark} for details. We
quantify system conformance (and thereby adherence to requirements) in
terms of the Skorokhod distance, or, in other words, maximum permitted
time/space-jitter value $\delta$. For this system, we know that at
nominal conditions, the settling time is approximately $2.5$ seconds,
and that we can tolerate an increase in settling time of about $0.5$
seconds. Thus, we chose a time-saling factor of $2 = \frac{1}{0.5}$.
We observe that the range of $\theta$ is about $0.4$ radians, and
specify an overshoot of $20\%$ of this range as being permissible.
Thus, we pick a scaling factor of $0.08$ for the signal domain. In
other words, Skorokhod distance $\delta = 1$ corresponds to either a
time-jitter of $0.5$ seconds, or a space-discrepancy of $0.08$
radians.
We summarize the results of conformance testing for different values
of sampling time $\Delta$ in Table~\ref{table:example1}. It is clear
that the conformance of the systems decreases with increasing $\Delta$
(which is to be expected). The time taken to compute the Skorokhod
distance decreases with increasing $\Delta$, as the number of
time-points in the two traces decreases.
\begin{table}[t] \centering
\caption{Variation in Skorokhod Distance with changing sampling time for an
aircraft pitch control system with an LQR-based controller. Time taken
indicates the total time spent in computing the upper bound on the
Skorokhod distance across all simulations. We scale the signals such that
a time-jitter of $0.5$ seconds, is treated the same as a value-difference
of $0.08$ radians, and the window size chosen is $150$. The system is
simulated for $5$ seconds, with a variable-step solver.
\label{table:example1}}
\begin{tabular*}{.8\textwidth}{@{\extracolsep{\fill}}lrrr}
\toprule
Controller & Skorokhod & Time taken (seconds) & Number of \\
Sample-Time & distance & to compute $\dist_{\skoro}$ & simulations \\
(seconds) & & & \\
\midrule
0.01 & 0.012 & 232 & 104\\
0.05 & 0.049 & 96 & 104\\
0.1 & 0.11 & 70 & 106\\
0.3 & 0.39 & 45 & 104\\
0.5 & 1.51 & 40 & 101\\
\bottomrule
\end{tabular*}
\end{table}
\mypara{Case Study: Air-Fuel Ratio Controller} In
\cite{hscc14benchmark}, the authors present three systems representing
an air-fuel ratio ($\lambda$) controller for a gasoline engine, that
regulate $\lambda$ to a given reference value of $\lambda_{\text{ref}} =
14.7$. Of interest to us are the second and the third systems. The
former has a continuous-time plant model with highly nonlinear
dynamics, and a discrete-time controller model. In
\cite{hscc14lyapunov}, the authors present a version of this system
where the controller is also continuous. We take this to be
$\system_1$. The third system in \cite{hscc14benchmark} is a
continuous-time closed-loop system where all the system differential
equations have right-hand-sides that are polynomial approximations of
the nonlinear dynamics in $\system_1$. We call this polynomial
dynamical system $\system_2$. The rationale for these system versions
is as follows: existing formal methods tools cannot reason about
highly nonlinear dynamical systems, but tools such as Flow*
\cite{flowstar}, C2E2 \cite{c2e2}, and CORA \cite{althoff} demonstrate
good capabilities for polynomial dynamical systems. Thus, the hope is
to analyze the simpler systems instead. In \cite{hscc14benchmark}, the
authors comment that the system transformations are not accompanied by
formal guarantees. By quantifying the difference in the system
behaviors, we hope to show that if the system $\system_2$ satisfies
the temporal requirements $\varphi$ presented in
\cite{hscc14benchmark}, then $\system_1$ satisfies a moderate
relaxation of $\varphi$. We pick a scaling factor of $2$ for the time
domain, as a time-jitter of $0.5$ seconds is the maximum deviation we
wish to tolerate in the settling time, and pick $0.68 =
\frac{1}{0.1*\lambda_{\text{ref}}}$ as the scaling factor for $\lambda$
(which corresponds to the worst case tolerated discrepancy in the
overshoot).
The results of conformance testing for these systems are summarized in
Table~\ref{table:example2}. In \cite{arch14benchmark}, the authors posed a
challenge problem for conformance testing. In it, the authors
reported that the original nonlinear system and the approximate polynomial
system both satisfy the \STL requirements specifying overshoot/undershoot
and settling time. We, however, found an input that causes the outputs of
the two systems to have a high Skorokhod distance. Thus, comparing the two
systems by considering equi-satisfaction of a given set of \STL
requirements such as overshoot/undershoot and settling time may not always
be sufficient, and our experiment indicates that the more nuanced Skorokhod
metric may be a better measure of conformance.
\begin{table}[t]
\centering
\caption{Conformance testing for closed-loop A/F ratio controller at
different engine speeds. We scale the signals such that 0.5 seconds of
time-jitter is treated equivalent to 10\% of the steady-state value (14.7)
of the A/F ratio signal. The simulation traces correspond to a time horizon
of $10$ seconds, and the window size is $300$. \label{table:example2}}
\begin{tabular*}{.8\textwidth}{@{\extracolsep{\fill}}lrrrr}
\toprule
Engine & Skorokhod & Computation & Total Time & Number of \\
speed (rpm) & distance & Time (secs) & Taken (secs) & simulations \\
\midrule
1000 & 0.31 & 218 & 544 & 700 \\
1500 & 0.20 & 240 & 553 & 700 \\
2000 & 0.27 & 223 & 532 & 700 \\
\bottomrule
\end{tabular*}
\end{table}
\mypara{Case Study: Engine Timing Model} The Simulink demo palette
presented by the Mathworks \cite{mathworks_simulink_demo} contains a
system representing a four-cylinder spark ignition internal combustion
engine based on a model by Crossley and Cook
\cite{crossley1991nonlinear}. This system is then enhanced by adding a
proportional plus integral (P+I) control law. The integrator is used
to adjust the steady-state throttle as the desired engine speed
set-point changes, and the proportional term compensates for phase lag
introduced by the integrator. In an actual implementation of such a
system, such a P+I controller is implemented using a discrete-time
integrator. Such integrator blocks are typically associated with a
particular numerical integration technique, {\em e.g.}, forward-Euler,
backward-Euler, trapezoidal, {\em etc}. It is expected for different
numerical techniques to produce slight variation in the results, and
we wish to quantify the effect of using different numerical
integrators in a closed-loop setting. We try to check if the
user-provided bound of $\delta = 1.0$ is satisfied by systems
$\system_1$ and $\system_2$, where $\system_1$ is the original system
provided at \cite{mathworks_simulink_demo}, while $\system_2$ is a
modified system that uses the backward Euler method to compute the
discrete-time integral in the controller. We try to determine the
input signal that leads to a violation of this $\delta$ bound, using a
simulation-guided approach as described before. We scale the outputs
in such a way that a value discrepancy of $1\%$ of the the output
range (~$1000$) is equivalent to a time discrepancy of $0.1$ seconds.
These values are chosen to bias the search towards finding signals
that have a small time jitter. This is an interesting scenario for
this case study where the two systems are exactly equivalent except
for the underlying numerical integration solver. We find the signal
shown in Fig.~\ref{fig:violation}, for which we find output traces
with Skorokhod distance $1.04$. The experiment uses $296$ simulations
and the total time taken to find the counterexample is $677$ seconds.
\begin{figure}[t]
\centering
\scalebox{0.9}{
\includegraphics[trim=15mm 87mm 0mm 82mm,clip,width=100mm]{./figures/cex1.pdf}}
\vspace{-2mm}
\caption{Example of non-conformant behavior found using a simulation-guided
optimization algorithm with the Skorokhod distance between system output
trajectories as the cost function.}
\label{fig:violation}
\vspace{2mm}
\end{figure}
\section{Introduction}
\vspace{-2mm}
A fundamental question in model-based design is {\em conformance testing}: whether two models of a system
are equivalent.
For discrete systems, this question is
well-studied~\cite{Milner80,HennessyM85,HHK95,BisimBook2011},
and there is a rich theory of process equivalences based on similarity and bisimilarity.
For continuous and hybrid systems, however, the state of the art is somewhat unsatisfactory.
While there is a straightforward generalization of process equivalences to the continuous case,
in practice, equivalence notions such as bisimilarity are always too strong and most systems are not
bisimilar.
Since equivalence is a Boolean notion, one gets no additional information about the systems other than they are ``not bisimilar,''
and even if two dynamical systems are bisimilar, they may still differ in many properties that are of control-theoretic
interest.
Thus, classical notions for equivalence and conformance have been of limited use in industrial practice.
In recent years, the notion of bisimulation has therefore been generalized to \emph{metrics} on systems,
which quantify the distance between them.
For example, one approach is that of $\epsilon$-bisimulation,
which requires that the states of the two systems remain ``close'' forever (within an $\epsilon$-ball), rather than coincide exactly.
Under suitable stability assumptions on the dynamics, one can prove results about $\epsilon$-bisimulation \cite{GirardPT10,HaghverdiTP05}.
Unfortunately, proving the pre-requisites for the existence of $\epsilon$-bisimulations for complex dynamical
models, or coming up with suitable and practically tractable bisimulation functions, is extremely difficult in practice.
In addition, establishing $\epsilon$-bisimulation
requires full knowledge of the system dynamics making the scheme
inapplicable where one system is an actual physical component
with unknown mathmatical dynamics.
Bisimulation notions have hence been of limited practical use.
Instead, a more pragmatic semi-formal approach has gained prominence in industrial practice.
In this approach, the two systems are executed on the same input sequences and a metric on finite trajectories
is used to evaluate the closeness of these trajectories.
The key to this methodology is the selection of a {\em good} metric, with the following properties:
\begin{compactitem}
\item \emph{Transference.}
Closeness in the metric must translate to preserving
interesting classes of logical and functional specifications between systems, and
\item \emph{Tractability.}
The metric should be efficiently computable.
\end{compactitem}
In addition, there is the more informal requirement of \emph{applicability}: the metric should classify systems, that
the engineers consider close, as being close, and conversely.
A number of metrics have been proposed recently.
The simplest is a
{\em pointwise} metric that computes the maximum pointwise difference between two trajectories,
sometimes generalized to apply a constant time-shift
to one trajectory~\cite{DonzeM10}.
Unfortunately, for many practical models, two trajectories may be close only under variable time-shifts.
This is the case, for example, for two dynamical models that may use different numerical integration techniques
(e.g., fixed step versus adaptive step) or when some component in the implementation has some jitter.
Thus, the pointwise metric spuriously report large distances for ``close'' models.
More complicated hybrid distances have been proposed \cite{GeorgiosMemo14}.
The transference properties of these metrics w.r.t.\
common temporal logics for dynamical systems are not yet clear.
In this work we present a methodology for quantifying conformance between
real-valued dynamical systems based on the \emph{Skorokhod}
metric~\cite{Davoren09}.
The Skorokhod metric allows for mismatches in both the trace values \emph{and} in the
timeline, and quantifies temporal and spatial variation of the system dynamics under
a unifying framework.
The distortion of the timeline is specified by a \emph{retiming} function $\retime$ which
is a continuous bijective strictly increasing function from $\reals_+$ to $\reals_+$.
Using the retiming function, we obtain the \emph{retimed trace} $x\left(\retime(t)\right)$ from the
original trace $x(t)$.
Intuitively, in the retimed trace $x\left(\retime(t)\right)$,
we see exactly the same values as before, in
exactly the same order, but the
time duration between two values might now be different than
the corresponding duration in the original trace.
The amount of distortion for the retiming $\retime$ is given by $\sup_{t\geq 0} \abs{\retime(t)-t}$.
Using retiming functions, the Skorokhod distance between two traces $x$ and $y$ is defined to be
the least value over all possible retimings $\retime$ of:
\vspace{-1mm}
\[\max\left(\sup_{t\in[0,T]} \abs{\retime(t)-t},\, \sup_{t\in [0,T]}\dist\big(x\left(\retime(t)\right), y(t)\big)
\right),\]
\vspace{-1mm}
where $\dist$ is a pointwise metric on values.
The Skorokhod distance thus
incorporates two components: the first component quantifies
the {\em timing discrepancy} of the timing distortion required to ``match'' two traces,
and the second quantifies the \emph{value mismatch} (in the metric space ${\myO}$)
of the values under the timing distortion.
The Skorokhod metric was introduced as a theoretical basis for defining the semantics of hybrid systems by providing
an appropriate hybrid topology \cite{CaspiB02,Broucke98}.
We now demonstrate its usefulness in the context of conformance testing.
\smallskip
\noindent
\textbf{Transference.}
We show that the Skorokhod metric gives a robust quantification of system conformance
by relating the metric to \TLTL (timed \LTL) enriched with
(i)~predicates of the form $f(x_1,\dots, x_n) \geq 0$, as in Signal Temporal Logic,
for specifying constraints on trace values; and
(ii) \emph{freeze quantifiers}, as in \TPTL~\cite{AlurH94}, for specifying temporal constraints
(freeze quantifiers can express more complex timing constraints than bounded
timing constraints, \emph{e.g} of \MTL).
This logic subsumes the \MITL-based logic \STL~\cite{DonzeM10}.
We prove a \emph{transference theorem}: flows (and propositional traces) which are close under
the Skorokhod metric satisfy ``close'' \TLTL formulae for a rich class of temporal
and spatial predicates; where the untimed structure of the formulae
remains unchanged, only the predicates are enlarged.
\noindent\textbf{Tractability.}
We improve on recent polynomial-time algorithms for the Skorokhod
metric~\cite{MajumdarPHSCC15}
by taking advantage of the fact that, in practice, only retimings
that map the times in one trace to ``close'' times in the other are of interest.
This enables us to obtain a streaming
sliding-window based monitoring procedure which takes only
$O(W)$ time per sample, where $W$ is the window size
(assuming the dimension $n$ of the system to be a constant).
\smallskip\noindent\textbf{Usability.}
Using the Skorokhod distance checking procedure as a subroutine,
we have implemented a Simulink toolbox for conformance testing.
Our tool integrates with Simulink's model-based design flow for
control systems, and provides a stochastic search-based approach to
find inputs which maximize the Skorokhod distance between
systems under these inputs.
We present three case studies from the control domain, including industrial challenge problems;
our empirical evaluation
shows that our tool computes sharp estimates of the conformance distance reasonably
fast on each of them.
Our input models were complex enough that more theoretically appealing techniques such as
$\epsilon$-bisimulation function generation could not be applied.
In particular, we demonstrate how two models that only differ in the underlying ODE solver
can nevertheless deviate enough to invalidate system requirements on settling time.
We conclude that the Skorokhod metric can be an effective foundation for semi-formal conformance
testing for complex dynamical models.
Proofs of the theorems are given in the accompanying technical report [REF].
\smallskip\noindent\textbf{Related Work.}
The work of~\cite{GeorgiosMemo14,GeorgiosHFDKU14} is closely related
to ours.
In it, robustness properties of hybrid state \emph{sequences} are derived
with respect to a trace metric which also
quantifies temporal and spatial variations.
Our work differs in the following ways.
First, we guarantee robustness properties over \emph{flows} rather than
only over (discrete) sequences.
Second, the Skorokhod metric is a stronger form of the
$(T,J,(\tau, \epsilon))$-closeness degree\footnote{Instead of
having two separate parameters $\tau$ and $\epsilon$ for
time and state variation, we pre-scale time and the $n$ state components
with $n+1$ constants, and have a single value quantifying closeness of the
scaled traces.}\textsuperscript{,}\footnote{Informally, two signals
$x,y$ are $(T,J,(\tau, \epsilon))$-close if for each point
$x(t)$, there is a point $y(t')$ with $|t-t'| < \tau$ such that
$\dist(x(t), y(t')) <\epsilon$; and similarly for $y(t)$.}(for systems
which do not have hybrid time); and
allows us to give stronger robustness transference guarantees.
The Skorokhod metric requires order preservation of the timeline, which
the $(T,J,(\tau, \epsilon))$-closeness function does not.
Preservation of the timeline order allows us to
(i)~keep the untimed structure of the formulae the same (unlike
in the transference theorem of~\cite{GeorgiosMemo14});
(ii)~show transference of a rich class of global timing constraints
using freeze quantifiers (rather than only for the standard bounded time
quantifiers of \MTL/\MITL).
However, for implementations where the timeline order is not preserved,
we have to settle for the less stronger guarantees provided
by~\cite{GeorgiosMemo14}.
The work of~\cite{DonzeM10}, in terms of robustness,
deals mainly with spatial robustness of \STL;
the only temporal disturbances considered are constant time-shifts for the entire signal
where the entire signal is moved to the past, or to the future by the same amount.
The Skorokhod metric incorporates time-shifts which are variable along the timeline.
\section{Transference of Logical Properties}
\label{section:Logic}
\vspace{-2mm}
In this section, we demonstrate transference of logical properties.
If two traces are at a distance of $\delta$, and one trace satisfies a logical
specification $\phi$, we derive the ``relaxation'' needed (if any) in $\phi$ so that
the other trace also satisfies this relaxed logical specification.
The logic we use is a version of the timed linear time logic
\TLTL~\cite{AlurH94} (a timed version of the logic \LTL).
We show that the Skorokhod distance provides robust transference of
specifications in this logic: if the Skorokhod distance between
two traces is small, they satisfy close \TLTL formulae.
We first present the results in a propositional framework, and then extend
to $\reals^n$-valued spaces.
\vspace{-3mm}
\subsection{The Logic \TLTL}
\vspace{-1mm}
Let $\prop$ be a set of propositions.
A \emph{propositional trace} $\pi$ over $\prop$ is a
trace where the topological space is $2^{\prop}$, with the associated metric:
$\dist_{\prop}(\sigma, \sigma') = \infty$ if $\sigma \neq \sigma'$, and $0$ otherwise
for $\sigma,\sigma'\in 2^{\prop}$.
We restrict our attention to propositional
traces with finite variability: we require
that there exists a finite partition of $\tdom(\pi)$ into disjoint subintervals
$I_0, I_1, \dots, I_m$ such that $\pi$ is constant on each subinterval.
The set of all timed propositional traces over $\prop$ is denoted by $\Pi(\prop)$.
\vspace{-1mm}
\begin{definition}[\mTLTL{\small$(\TFun)$} Syntax]
Given a set of propositions $\prop$, a set of (time) variables $V_{\mytime}$, and a set $\TFun$ of functions
from $\reals_+^l$ to $\reals$,
the formulae of \mTLTL($\TFun$)
are defined by the following grammar.
\[
\phi := p \mid \true \mid f_{\mytime}(\overline{x}) \sim 0 \mid \neg\phi \mid \phi_1 \wedge \phi_2 \mid \phi_1 \vee \phi_2 \mid \phi_1 \until \phi_2 \mid x.\phi
\quad \text{ where}
\]
\begin{compactitem}
\item
$p\in \prop$ and $x\in V_{\mytime}$, and $\overline{x} = (x_1, \dots, x_l)$ with $x_i\in V_{\mytime}$ for all $1\leq i \leq l$;
\item $f_{\mytime} \in \TFun$ is a real-valued function,
and $\sim$ is one of $ \set{\leq, <, \geq, >}$.\qed
\end{compactitem}
\end{definition}
We say that the variable $x$ is \emph{bound} in $\phi$ if $\phi$ is $ x.\Psi$, otherwise it is \emph{free}.
The quantifier ``$x.$'' is known as the \emph{freeze quantifier}, and binds the variable $x$ to the current
time.
A formula is \emph{closed} if it has no free variables.
\begin{definition}[\mTLTL{\small$(\TFun)$} Semantics]
\label{definition:PropositionalSemantics}
Let $\pi: I \mapsto 2^{\prop}$ be a timed propositional trace,
$t_0 = \min(I)$,
and let $\env: V \mapsto I$ be the time environment
mapping the variables in $V$ to time values in $I$.
The satisfaction of the timed sequence $\pi$ with respect to the \mTLTL($\TFun)$ formula $\phi$ in the time environment $\env$ is
written as $\pi \models_{\env} \phi$, and is
defined inductively as follows (denoting $t_0 = \min \tdom(\pi)$).
\begin{align*}
\pi & \models_{\env} p \text{ for } p\in \prop \text{ iff } p\in \pi(t_0);
\qquad \pi \models_{\env} \true;
\qquad \pi \models_{\env} \neg\Psi \text{ iff } \pi \not\models_{\env} \Psi;\\
\pi & \models_{\env} \phi_1 \wedge \phi_2 \text{ iff } \pi \models_{\env} \phi_1 \text{ and } \pi \models_{\env} \phi_2;
\quad \pi \models_{\env} \phi_1 \vee \phi_2 \text{ iff } \pi \models_{\env} \phi_1 \text{ or } \pi \models_{\env} \phi_2;\\
\pi & \models_{\env} f_{\mytime}(x_1, \dots, x_l) \sim 0 \text{ iff } f_{\mytime}(\env(x_1), \dots, \env(x_l)) \sim 0
\text{ for } \sim \,\in \set{\leq, <, \geq, >};\\
\pi & \models_{\env}\! x.\psi \text{ iff } \pi\! \models_{\env[x\!:=\!t_0]}\! \psi \text{ where } \env[x\!:=\!t_0] \text{ agrees}\, \text{with }
\env\! \text{ for}\, \text{all } z\!\neq\! x, \!\text{ and}\,\text{maps}\, x\, \text{to } t_0; \\
\pi & \models_{\env} \phi_1 \until \phi_2 \text{ iff } \pi^t \models_{\env}
\phi_2 \text{ for some } t\in I
\text{ and } \pi^{t'} \models_{\env} \phi_1\vee \phi_2 \text{ for all } t_0\leq t' < t.
\end{align*}
A timed trace $\pi$ is said to satisfy the closed formula $\phi$ (written
as $ \pi \models \phi$) if there is some environment $\env$ such that
$ \pi \models_{\env} \phi$.
\qed
\end{definition}
\vspace{-1mm}
The definition of additional temporal operators in terms of these base operators is standard:
the ``eventually'' operator $\Diamond \phi$ stands for $\true \until \phi$; and the
``always'' operator $\Box \phi$ stands for $\neg \Diamond \neg \phi$.
\mTLTL{\small$(\TFun)$} provides a richer framework than \MTL~\cite{Koymans90}
for expressing timing
constraints as:
(i)~freeze quantifiers
allow specification of constraints between distant contexts, which the bounded temporal operators in \MTL
cannot do; and
(ii)~the predicates $f_{\mytime}() \sim 0$ for $f_{\mytime}\in \TFun$ allow the
specification of complex timing requirements not expressible in \MTL.
\vspace{-3mm}
\begin{example}[Freeze quantifiers; \mTLTL{\small$(\TFun)$} subsumes \MTL]
Let $\TFun$ be the set of two variable functions of the form $f(x,y) = x-y+c$ where $c$ is a rational constant.
Then \mTLTL{\small$(\TFun)$} subsumes \MTL.
The \MTL formula $Q \until_{[a,b]} R$ can be written as
\vspace{-1mm}
\[
x. \Big(Q \until y. \big( \,(y\leq x+b) \wedge (y\geq x+a) \wedge R\big)\Big).\]
\vspace{-1mm}
We explain the formula as follows.
We assign the ``current'' time $t_x$ to the variable $x$, and some future time $t_y$ to the variable $y$.
The values $t_x$ and $t_y$ are such that
at time $t_y$, we have $R$ to be true, and moreover, at all times between $t_x$ and $t_y$,
we have $Q\vee R$ to be true.
Furthermore, $t_y$ must be such that $t_y\in [t_x+a, t_x+b]$, which is specified by the term
$(y\leq x+b) \wedge (y\geq x+a) $.
\qed
\end{example}
\vspace{-4mm}
\begin{example}[Temporal Constraints]
\label{example:TemporalConstraints}
Suppose we want to express that whenever the event $Q$ occurs,
it must be followed by a response $R$, and then
by $S$.
In addition, we have the following timing requirement:
if $\varepsilon_{QR}, \varepsilon_{RS}, \varepsilon_{QS}$ are
the time delays between $Q$ and $R$, between $R,S$, and between
$Q$ and $S$ respectively, then:
we must have
$\varepsilon^2_{QR}+ \varepsilon^2_{RS}+\varepsilon^2_{QS} \leq d$ for
a given positive constant $d$.
This can be written using freeze quantifiers as the \TLTL formula $\phi$:
\[
x. \left( Q \rightarrow \Diamond
\big(y. \left(R \wedge
\Diamond\left[z. \left(S \wedge
\left( (y-x)^2 + (z-y)^2 + (z-x)^2 \leq d \right) \right)
\right] \right)
\big)\right).\qed\]
\end{example}
\vspace{-3mm}
\subsection{Transference of \TLTL Properties for Propositional Traces}
\vspace{-1mm}
We show in this section that if a timed propositional trace
$\pi$ satisfies a \mTLTL{\small$(\TFun)$} formula $\phi$, then any timed trace $\pi'$ that is at most
$\delta$ distance away from $\pi$ satisfies a slightly relaxed version of the formula $\phi$,
the degree of relaxation being governed by $\delta$; and
the variance of the functions in $\TFun$ over the time interval
containing the time domains of $\pi$ and $\pi'$.
Recall that the distance between two sets of propositions $\sigma, \sigma'$ is
$\infty$ if $\sigma\neq \sigma'$, and $0$ if $\sigma = \sigma'$.
The distance between two propositional traces is defined
to be the Skorokhod distance with the aforementioned metric on $2^{\prop}$.
Next, we define relaxations of \mTLTL{\small$(\TFun)$}formulae.
The relaxations are defined as a syntactic transformation
on formulae which do not have negations, except on the propositions.
Every \mTLTL{\small$(\TFun)$}formula can be expressed in this negation-normal form.
To remove negations from the until operator, we use the waiting for operator, $\awaits$, defined as:
\begin{verse}
\vspace{-2mm}
$\pi\models_{\env} \phi_1 \awaits \phi_2$ iff either (1)~$\pi^t \models_{\env}\phi_1$ for all
$ t\in I $;
or (2)~$\pi^t \models_{\env} \phi_2$ for some $ t\in I $;
and $\pi^{t'} \models_{\env} \phi_1\vee \phi_2$ for all $t_0\leq t' < t$.
\vspace{-2mm}
\end{verse}
It can be showed that every \mTLTL{\small$(\TFun)$} formula can be rewritten
using the $\awaits$ operator such that negations appear only over the propositions
(the procedure is given in the Appendix).
\vspace{-1mm}
\begin{definition}[$\delta$-relaxation of \mTLTL{\small$(\TFun)$} formulae]
\label{definition:Relaxation}
Let $\phi$ be a \mTLTL{\small$(\TFun)$} formula in which negations appear only on the
propositional symbols.
The $\delta$ relaxation of $\phi$ (for $\delta\! \geq\! 0$) over a closed interval $J$,
denoted
$\myrelax_{J}^{\delta}(\phi)$, is defined as:
\vspace{-1mm}
\[
\begin{array}{l|l}
\begin{array}{lll}
\myrelax_{ J}^{ \delta}(p) & = & p \\
\myrelax_{ J}^{\delta}(\neg p) & = & \neg p \\
\myrelax_{ J}^{\delta}( \phi_1 \wedge \phi_2 ) & = &
\myrelax_{ J}^{\delta}( \phi_1) \wedge \myrelax_{J}^{\delta}(\phi_2 ) \\
\myrelax_{ J}^{\delta}( x.\psi ) & = & x.\myrelax_{ J}^{\delta}(\psi)\\
\myrelax_{ J}^{\delta}( \phi_1 \until \phi_2 ) & = &
\myrelax_{ J}^{\delta}(\phi_1) \until \myrelax_{J}^{\delta}(\phi_2) \\
\end{array} &
\begin{array}{lll}
\myrelax_{ J}^{\delta}(\true) & = & \true \\
\myrelax_{ J}^{\delta}(\false) & = & \false \\
\myrelax_{ J}^{\delta}( \phi_1 \vee \phi_2 ) & = &
\myrelax_{ J}^{\delta}( \phi_1) \vee \myrelax_{ J}^{\delta}(\phi_2 )\\
& & \\
\myrelax_{ J}^{\delta}( \phi_1 \awaits \phi_2 ) & = &
\myrelax_{ J}^{\delta}(\phi_1) \awaits \myrelax_{ J}^{\delta}(\phi_2) \\
\end{array}
\end{array}
\]
\newcommand{\mathrm{def}}{\mathrm{def}}
\newcommand{\ensuremath{\mathop{\overset{\MyDef}{=}}}}{\ensuremath{\mathop{\overset{\mathrm{def}}{=}}}}
\newcommand{\mathop{\overset{\MyDef}{\resizebox{\widthof{\eqdefU}}{\heightof{=}}{=}}}}{\mathop{\overset{\mathrm{def}}{\resizebox{\widthof{\ensuremath{\mathop{\overset{\MyDef}{=}}}}}{\heightof{=}}{=}}}}
\vspace{-0.4em}
\begin{equation}
\label{equation:RelaxProp}
\begin{array}{l}
\vspace{0.3em}
\myrelax_{ J}^{\delta}\left(f_{\mytime}(x_1, \dots, x_l) \right) \sim 0) =
\left\{
\begin{array}{ll}
f_{\mytime}(x_1, \dots, x_l) \, +\, K_{f_{\mytime}}^I(\delta) \ \sim\, 0 & \text{ if }
\sim\, \in\set{>, \geq}\\
f_{\mytime}(x_1, \dots, x_l) \, -\, K_{f_{\mytime}}^I(\delta) \ \sim \, 0 & \text{ if }
\sim \,\in\set{<, \leq},\\
\end{array}
\right. \\
\vspace{0.3em}
\text{where } K_{f_{\mytime}}^I:[0, \max \tdom( J)\, - \, \min \tdom(
J)]\mapsto \reals_+ \text{, and} \\
K_{f_{\mytime}}^I(\delta) \mathop{\overset{\MyDef}{\resizebox{\widthof{\eqdefU}}{\heightof{=}}{=}}}
\displaystyle\sup_{
\begin{array}{l}
{t_1,\dots,t_l\in J}\\
{t_1',\dots,t_l'\in J}
\end{array}
} \left\{
\left\arrowvert
\begin{array}{c}
f_{\mytime}(t_1,\dots, t_l) \\
-\\
f_{\mytime}(t_1',\dots,t_l')
\end{array}\right\arrowvert \
\text{ s.t. }
|t_i-t_i'| \leq \delta \text{ for all } i
\right\}
\end{array}
\hspace{-3em}
\end{equation}
\end{definition}
\vspace{-2mm}
Thus, instead of comparing the $f_{\mytime}()$ values to $0$, we relax by comparing
instead to $\pm K_{f_{\mytime}}^J(\delta)$.
The other cases recursively relax the subformulae.
The functions $K_{f_{\mytime}}^J(\delta)$ define the maximal change in the value
of $f_{\mytime}$ that can occur when the input variables can vary by $\delta$.
The role of $J$ is the above definition is to restrict the domain of the freeze
quantifier variables to the time interval $J$ (from $\reals_+$)
in order to obtain the least possible relaxation on a given trace
$\pi$ (\emph{e.g.} we do not care about the values of a function in ${\TFun}$ outside
of the domain $\tdom(\pi)$ of the trace).
\vspace{-1mm}
\begin{example}[$\delta$-relaxation for Bounded Temporal Operators -- \MTL]
We demonstrate how
$\delta$-relaxation operates on bounded time constraints
through an example.
Consider an \MTL formula $\phi= Q \until_{[a,b]} R$.
This can be written as a \TLTL formula, and relaxed using the
$ \myrelax_{\reals_+}^{\delta}$ function.
The relaxed \TLTL formula is again equivalent to an \MTL
formula, namely $Q \until_{[a-2\cdot\delta\,,\, b+2\cdot \delta]} R$.
The details are explained in Example~\ref{example:RelaxationMTL}
in the Appendix.
\qed
\end{example}
\begin{theorem}[Transference for Propositional Traces]
\label{theorem:PropositonalRobustness}
Let $\pi, \pi'$ be two timed propositional traces such that
$\dist(\pi, \pi') < \delta$ for some finite $\delta$.
Let
$\phi$ be a closed \mTLTL{\small$(\TFun)$} formula in negation-normal form.
If $\pi\models \phi$, then
$\pi'\models \myrelax_{I_{\pi,\pi'}}^{\delta}(\phi)$
where
${I_{\pi,\pi'}}$ is the convex hull of $\tdom(\pi) \cup \tdom(\pi')$.
\qed
\end{theorem}
Theorem~\ref{theorem:PropositonalRobustness} relaxes the freeze variables
over the entire signal time-range ${I_{\pi,\pi'}}$;
it can be strengthened by relaxing over a smaller range:
if $\pi\models \phi$, and $t_1, \dots, t_k$ are time-stamp assignments to the
freeze variables $x_1, \dots, x_k$ which witness $\pi$ satisfying $\phi$,
then $x_i$ only needs to be relaxed over $[t_i-\delta, t_i+\delta]$ rather
than the larger interval ${I_{\pi,\pi'}}$.
These smaller relaxation intervals for the freeze variables can be incorporated
in Equation~\ref{equation:RelaxProp}.
We omit the details for ease of presentation.
\begin{example}
\label{example:Transference-one}
Recall Example~\ref{example:TemporalConstraints}, and the
formula $\phi$ presented in it.
Suppose a flow $\pi$ satisfies $\phi$; and let $\pi'$ be $\delta$ close to $\pi$
under the Skorokhod metric (for propositional traces).
Our robustness theorem ensures that
(i)~$\pi'$ will satisfy the same untimed formula
$Q\rightarrow \, \Diamond \left(R \wedge \Diamond S\right) $; and
(ii) it gives a bound on how much the timing constraints need to be relaxed in $\phi$
in order
to ensure satisfaction by $\pi'$; it states that $\pi'$ satisfies the following relaxed
formula $\phi'$ for every $\epsilon >0$:
\[\pi'\models
x. \left( Q \rightarrow \Diamond
\big(y. \left(R \wedge
\Diamond\left[z. \left(S \wedge
\left( (y-x)^2 + (z-y)^2 + (z-x)^2 \leq d^{\dagger} \right) \right)
\right] \right)
\big)\right)\]
where
$d^{\dagger} = d + 12 \cdot(\delta+\epsilon)^2 + 4\sqrt{3}\cdot(\delta+\epsilon)
\cdot \sqrt{d}$.
The constant $d^{\dagger} $ is derived in the appendix.
\qed
\end{example}
\vspace{-5mm}
\subsection{Transference of \TLTL properties for $\reals^n$-valued Signals}
\vspace{-1mm}
A \emph{timed $\reals^n$-valued trace}
$\pi$ is a function from a closed interval $I$ of $\reals_+$ to $\reals^n$.
For $ \ol{\alpha} = (\alpha^0, \dots, \alpha^n)\in \reals^n$, we denote the
$k$-th dimensional value $\alpha^k$ as $\ol{\alpha}[k]$.
The $\pi$ projected function onto the $k$-th $\reals$ dimension is
denoted by $\pi_{k}: I \mapsto \reals$.
In order to define the satisfaction of \TLTL formulae over timed $\reals^n$-valued
sequences,
we use booleanizing predicates $\mu: \reals^n \mapsto \bool$, as in
\STL~\cite{DonzeM10}, to transform
$\reals^n$-valued sequences in to timed propositional sequences.
These predicates are part of the logical specification.
In this work, we restrict our attention to traces and predicates such that each predicate
varies only finitely often on the finite time traces under consideration.
\vspace{-1mm}
\begin{definition}[\mTLTL{\small$(\TFun, \SFun)$} Syntax]
Given
a set of variables $V_{\mytime}$ (the freeze
variables), a set of \emph{ordered} variables
$V_{\mysig}$ (the signal variables),
and two sets $\TFun, \SFun$ of
functions,
the formulae of \mTLTL{\small$(\TFun, \SFun)$}
are defined by the grammar:
\[
\phi := \true \mid f_{\mytime}(\overline{x}) \sim 0 \mid
f_{\mysig}(\overline{y}) \sim 0 \mid
\neg\phi \mid \phi_1 \wedge \phi_2 \mid \phi_1\vee \phi_2\mid \phi_1 \until \phi_2 \mid x.\phi
\quad \text{ where}
\]
\begin{compactitem}
\item $x\in V_{\mytime}$, and $\overline{x} = (x_1, \dots, x_l)$ with $x_i\in V_{\mytime}$ for all $1\leq i\leq l$;
\item $\overline{y} = (y_1, \dots, y_d)$ with $y_j\in V_{\mysig}$ for all $1\leq j \leq d$;
\item $V_{\mytime} $ and $V_{\mysig}$ are disjoint;
\item $f_{\mytime} \in \TFun$ and $f_{\mysig} \in \SFun$ are real-valued functions,
and $\sim $ is $ \leq, <, \geq, $ or $>$.\qed
\end{compactitem}
\end{definition}
\vspace{-1mm}
The semantics of \mTLTL{\small$(\TFun, \SFun)$} is straightforward and similar to the propositional case (Definition~\ref{definition:PropositionalSemantics}).
The only new ingredients are the booleanizing predicates
$f_{\mysig}(\overline{y}) \sim 0 $: we define
$\pi \models_{\env} f_{\mysig}(y_1, \dots, y_d) \sim 0$
iff $ f_{\mysig}( \pi_{j_1}[t_0], \dots, \pi_{j_d}[t_0]) \sim 0$ for any freeze variable
environment $\env$, where $t_0 = \min \tdom(\pi)$, and $y_i$ is the $j_i$-th
variable in $V_{\mysig}$ (\emph{i.e.}, $y_i$ refers to the $j_i$-th dimension
in the signal trace).
We require that for a timed $\reals^n$-valued trace
$\pi$ to satisfy $\phi$, the arity of the functions in $ \SFun$
occurring in $\phi$ should not be more than $n$, that is, functions should not refer
to dimensions greater than $n$ for an $\reals^n$ trace.
\smallskip\noindent\textbf{$\delta$ relaxation of \mTLTL{\small$(\TFun, \SFun)$}.}
Let $ \jmap$ be a mapping from $V_{\mysig}$ to closed intervals
of $\reals$, thus $\jmap(z)$ denotes a sub-domain of $z\in V_{\mysig}$.
The relaxation function $\myrelax_{{J}, \jmap}^{\delta}$ which
operates on \mTLTL{\small$(\TFun, \SFun)$} formulae is defined
analogous to the relaxation function $\myrelax_{J}^{\delta}$
in Definition~\ref{definition:Relaxation}.
We omit the similar cases, and only present the new case for the predicates
formed from $ \SFun$ (the full definition can be found in the appendix).
\vspace{-3mm}
\[
\myrelax_{J, \jmap}^{\delta}\left(f_{\mysig}(z_1, \dots, z_l) \right) \sim 0)
\ =
\begin{cases}
f_{\mysig}(z_1, \dots, z_l) \, +\, K_{f_{\mysig}}(\delta) \ \sim\, 0 & \text{ if }
\sim\, \in\set{>, \geq};\\
f_{\mysig}(z_1, \dots, z_l) \, -\, K_{f_{\mysig}}(\delta) \ \sim \, 0 & \text{ if }
\sim \,\in\set{<, \leq}
\end{cases}
\]
where
$K_{f_{\mysig}}: \big[0,\ \max_{z\in V_{\mysig}} |\max \jmap(z) \, - \, \min\jmap(z)| \big]
\mapsto \reals_+$
is a function s.t.
\vspace{-1mm}
\[
K_{f_{\mysig}}(\delta) =
\sup_{
\begin{array}{c}
z_i\in \jmap(z_i); \, z'_i\in \jmap(z'_i)\\
\text{ for all } i
\end{array}
}
\left\{
\left\arrowvert
\begin{array}{c}
f_{\mysig}(z_1,\dots, z_l) \\
-\\
f_{\mysig}(z_1',\dots,z_l')
\end{array}\right\arrowvert
\text{ s.t. }
|z_i-z_i'| \leq \delta \text{ for all } i
\right\}.
\]
The functions $K_{f_{\mysig}}(\delta)$ define the maximal change in the value
of $f_{\mysig}$ that can occur when the input variables can vary by $\delta$ over the
intervals in $\jmap(z)$ and $J$.
The role of $\jmap$ in the above definition is to restrict the domain of the signal variables
in order to obtain the least possible relaxation bounds on the signal
constraints; as was done in Definition~\ref{definition:Relaxation} for the
freeze variables.
\begin{theorem}[Transference for $\reals^n$-valued Traces]
\label{theorem:SignalTLTLRobustness}
Let $\pi, \pi'$ be two $\reals^n$-valued traces such
the Skorokhod distance between them is less than $\delta$ for some finite $\delta$.
Let $\phi$ be a closed \mTLTL{\small$(\TFun, \SFun)$} formula in negation-normal form.
If $\pi \models \phi$, then $\pi'\models \myrelax_{I_{\pi, \pi'}, \imap}^{\delta}(\phi)$, where
\begin{compactitem}
\item ${I_{\pi,\pi'}}$ is the convex hull of $\tdom(\pi) \cup \tdom(\pi')$; and
\item $\imap(z)$
is the convex hull of
$\set{\pi(t)[k] \mid t\in \tdom(\pi)} \cup \set{\pi'(t)[k] \mid t\in \tdom(\pi')} $;
where $z$ is the $k$-th variable in the ordered
set $V_{\mysig}$.\qed
\end{compactitem}
\end{theorem}
\vspace{-2mm}
Theorem~\ref{theorem:SignalTLTLRobustness} can be strengthened similar to
the strengthening mentioned for Theorem~\ref{theorem:PropositonalRobustness}
by relaxing the variables over smaller intervals obtained from
assignments to variables which witness $\pi\models \phi$.
\vspace{-1mm}
\begin{example}[Spatial Constraints and Transference]
Recall Example~\ref{example:TemporalConstraints},
suppose that the events $Q,R,S$ are defined by the
following predicates over real variables $\alpha_1$ and $\alpha_2$.
Let $Q \equiv \alpha_1 + 10\vdot\alpha_2 \geq 3$; the
predicate $R \equiv |\alpha_1| + |\alpha_2| \leq 20$; and
$S \equiv |\alpha_1| + |\alpha_2| \leq 15$.
Let $\pi$ satisfy this formula with these predicates, and let
$\pi'$ be $\delta$ close to $\pi$,
for a finite $\delta$ under the Skorokhod metric for $\reals^2$.
Our robustness theorem ensures that
$\pi'$ will satisfy the relaxed formula
\[
x. \left( Q^{\delta} \rightarrow \Diamond
\big(y. \left(R^{\delta} \wedge
\Diamond\left[z. \left(S^{\delta} \wedge
\left( (y-x)^2 + (z-y)^2 + (z-x)^2 \leq d+12\vdot \delta^2 \right) \right)
\right] \right)
\big)\right). \]
where the relaxed predicates $Q^{\delta},R^{\delta},S^{\delta}$
are defined as follows:
$Q^{\delta} \equiv \alpha_1 + 10\vdot\alpha_2 \geq 3- 22\vdot\delta$;
and $R^{\delta} \equiv |\alpha_1| + |\alpha_2| \leq 20+4\vdot\delta$;
and $S^{\delta} \equiv |\alpha_1| + |\alpha_2| \leq 15+4\vdot\delta$.
\qed
\end{example}
\begin{comment}
\subsection{Quantitative Semantics of \TLTL on Real-Valued Signals}
\begin{defintion}[Quantitative Semantics]
Let $\phi$ be a \mTLTL{\small$(\TFun, \SFun)$} formula.
The function $\symb{\cdot}$ i assigns values in
$\reals \cup\set{-\infty, +\infty}$ to timed traces as follows.
For a timed trace
$\pi: I \maptso \reals^n$,
\begin{align*}
\end{align*}
\end{defintion}
\end{comment}
\section{Preliminaries}
\vspace{-3mm}
\noindent\textbf{Traces.}
A (finite) \emph{trace} or a \emph{signal}
$\pi: [T_i,T_e] \mapsto {\myO} $ is a mapping from a
finite closed interval $[T_i,T_e]$ of $\reals_+$,
with $0 \leq T_i < T_e$,
to some topological space ${\myO}$.
If $\myO$ is a metric space, we refer to the associated metric as $\dist_{\myO}$.
The time-domain of $\pi$, denoted $\tdom(\pi)$ is the time domain $[T_i,T_e]$
over which it is defined.
The time-duration of $\pi$, denoted as $\tlen(\pi)$, is $\sup\left( \tdom(\pi) \right)$.
The $t$-suffix of $\pi$ for $t\in \tdom(\pi)$, denoted by $\pi^t$, is the
trace $\pi$ restricted to the interval $( \tdom(\pi) \cap [t, \tlen(\pi)]$.
We denote by $\pi_{\downarrow T'_e}$ the prefix trace obtained from
$\pi$ by restricting the domain to $[T_i,T'_e]\subseteq \tdom(\pi)$.
\noindent\textbf{Systems.}
A (continuous-time) \emph{system}
$\system: \left( \reals_+^{\scalebox{0.6}{[\ ]}}\mapsto {\myO}_{\myinput}\right)\, \mapsto\,
\left( \reals_+^{\scalebox{0.6}{[\ ]}}\mapsto {\myO}_{\myoutput}\right)$,
where $ \reals_+^{\scalebox{0.6}{[\ ]}}$ is the set of finite closed intervals of $\reals_+$,
transforms input traces
$\pi_{\myinput}: [T_i,T_e] \mapsto {\myO}_{\myinput} $ into output traces
$\pi_{\myoutput}: [T_i,T_e] \mapsto {\myO}_{\myoutput} $ (over the same time domain).
We require that if $\system(\pi_{\myinput}) \mapsto \pi_{\myoutput}$,
then for every $\min \tdom(\pi) \leq T_e' < \max \tdom(\pi)$, the system $\system$ maps
${\pi_{\myinput}}_{\downarrow T'_e} $ to ${\pi_{\myoutput}}_{\downarrow T'_e} $.
Thus, we only consider causal systems.
Common examples of such systems are (causal) dynamical, and hybrid dynamical
systems~\cite{Branicky1995PhD,tabuadabook}.
\noindent\textbf{Conformance.}
A system $\system'$ conforms to the system $\system$ over
an input trace $\pi_{\myinput}$ if
$\system'(\pi_{\myinput}) = \system(\pi_{\myinput}) $,
\emph{i.e.} if the behavior of $\system'$ on the input trace $\pi_{\myinput}$
is the same as that of $\system$.
The system $\system'$ conforms to the system $\system$ over
the input trace set $\Pi_{\myinput}$ if conformance holds for each input trace
in $\Pi_{\myinput}$.
Given a metric $\dist$ over input traces, and an input trace set $\Pi_{\myinput}$,
the \emph{quantitative conformance} between $\system'$ and
$\system$ over $\Pi_{\myinput}$ is defined as the quantity
$
\sup_{\pi_{\myinput}\in \Pi_{\myinput}}
\dist\left(\system'\left(\pi_{\myinput}\right),
\system\left(\pi_{\myinput}\right) \right).
$
If $\Pi_{\myinput}$ is the set of all input traces, this quantity is the
distance between the two systems.
\smallskip\noindent\textbf{Retimings.}
A \emph{retiming} $\retime: I \mapsto I' $, for closed
intervals $I, I'$ of $\reals_+$ is an order-preserving (\emph{i.e.} monotone)
continuous
bijective function from
$I$ to $I'$;
thus if $t<t'$ then $\retime(t) < \retime(t')$.
Let the class of retiming functions from $I$ to $ I' $
be denoted as $\retimeclass_{I \mapsto I' }$, and let
$\iden$ be the identity retiming.
Intuitively, retiming can be thought of as follows: imagine a stretchable and compressible
timeline; a retiming of the original timeline gives a new timeline
where some parts have been
stretched, and some compressed, without the timeline having been broken.
Given a trace $\pi: I_{\pi} \rightarrow {\myO} $, and a retiming
$\retime: I \mapsto I_{\pi} $; the function
$\pi\circ \retime$ is another trace from $I$ to ${\myO}$.
\vspace{-1mm}
\begin{definition}[Skorokhod Metric]
Given a retiming $\retime: I \mapsto I' $, let $||\retime-\iden||_{\sup}$ be defined as
$
||\retime-\iden||_{\sup} =\sup_{t\in I }|\retime(t)-t|$.
Given two traces $\pi: I_{\pi}\mapsto {\myO} $ and $\pi': I_{\pi'} \mapsto {\myO} $,
where ${\myO}$ is a metric space with the associated
metric $ \dist_{{\myO}}$,
and a retiming $\retime: I_{\pi} \mapsto I_{\pi'}$, let
$\norm{\pi\,-\, \pi'\circ \retime}_{\sup}$ be defined as:
\vspace{-2mm}
\[
\norm{\pi\,-\, \pi'\circ \retime}_{\sup}
=
\sup\nolimits_{t\in I_{\pi}} \dist_{{\myO}}\big(\, \pi(t)\ ,\ \pi'\left(\retime(t)\right)\, \big).\]
The \emph{Skorokhod distance}\footnote{
The two components
of the Skorokhod distance (the retiming, and the value difference components) can
be weighed with different weights -- this simply corresponds to a change of scale.}
between the traces $\pi()$ and $\pi'()$ is defined to be:
\begin{equation}
\label{equation:Skoro}
\dist_{\skoro}(\pi,\pi') = \inf_{r\in \retimeclass_{ I_{\pi} \mapsto I_{\pi'}}}
\max(\norm{\retime-\iden}_{\sup} \, ,\, \norm{\pi\,-\, \pi'\circ \retime}_{\sup}).\qed
\end{equation}
\end{definition}
\vspace{-2mm}
Intuitively, the Skorokhod distance
incorporates two components: the first component quantifies
the {\em timing discrepancy} of the timing distortion required to ``match'' two traces,
and the second quantifies the \emph{value mismatch} (in the metric space ${\myO}$)
of the values under the timing distortion.
In the retimed trace $ \pi\circ \retime$, we see exactly the same values as in $\pi$, in
exactly the same order, but the times at which the value are seen can be different.
\smallskip\noindent\textbf{Polygonal Traces.}
A
polygonal trace $\pi: I_{\pi} \mapsto \myO$ where $\myO$ is a vector space
with the scalar field $\reals$ is a continuous trace such that
there exists a finite sequence $\min I_{\pi}= t_0 < t_1 < \dots < t_m = \max I_{\pi}$
of time-points such that
the trace segment between $t_k$ and $t_{k+1}$ is affine for all $0\leq k < m$,
\emph{i.e.}, for $t_k \leq t \leq t_{k+1}$ we have
$\pi(t) = \pi(t_k) + \frac{t- t_k}{t_{k+1}-t_k}\vdot \left(\pi( t_{k+1}) - \pi(t_k)\right)$.
Polygonal traces are obtained when discrete-time traces are completed by
linear interpolation.
We remark that after retiming, the retimed trace $\pi \circ \retime$ \emph{need not}
be piecewise linear (see \emph{e.g.}~\cite{MajumdarP14}).
\vspace{-2mm}
\begin{theorem}[Computing the Distance between Polygonal
Traces~\cite{MajumdarPHSCC15}]
\label{theorem:SkoroFinal}
Let $\pi : I_{\pi} \mapsto \reals^n$ and $\pi': I_{\pi'}\mapsto \reals^n$ be two
polygonal traces with $m_{\pi}$ and $m_{\pi'}$ affine segments respectively.
Let the Skorokhod distance between them (for the $L_2$ norm on $\reals^n$) be
denoted as $\dist_{\skoro} (\pi, \pi') $.
\begin{compactenum}
\item Given $\delta \geq 0$, it can be checked whether
$\dist_{\skoro} (\pi, \pi') \leq \delta$ in time $O\left(m_{\pi}\vdot m_{\pi'}\vdot n\right)$.
\item
Suppose we restrict retimings to be such that the $i$-th affine segment of $\pi$
can only be matched to $\pi'$ affine segments $i-W$ through $i+W$ for all $i$,
where $W\geq 1$.
Under this retiming restriction, we can determine, with a streaming algorithm,
whether
$\dist_{\skoro} (\pi, \pi') \leq \delta$ in time $O\left(\left(m_{\pi}+ m_{\pi'}\right)\vdot n\vdot W\right)$.\qed
\end{compactenum}
\end{theorem}
\vspace{-2mm}
Let us denote by $\dist_{\skoro}^W (\pi, \pi') $ the Skorokhod difference
between $\pi, \pi'$ under the retiming restriction of the second part of
Theorem~\ref{theorem:SkoroFinal}, \emph{i.e.}, the value obtained by
restricting the retimings in Equation~\ref{equation:Skoro}\footnote{$\dist_{\skoro}^W$
is not a metric over traces (the triangle inequality fails).}.
The value $ \dist_{\skoro}^W (\pi, \pi') $
is an upper bound on $ \dist_{\skoro} (\pi, \pi') $.
In addition, for $W' < W$, we have $ \dist_{\skoro}^W (\pi, \pi') \leq
\dist_{\skoro}^{W'} (\pi, \pi') $.
\begin{comment}
\noindent\textbf{Estimating Quantitative Conformance using
Polygonal Approximations.}
An output trace $\system(\pi_{\myinput})$ is usually observed by sampling the trace
at discrete intervals, and assuming a linear (or piecewise constant) interpolation between
the sample points.
Sampling a trace $\pi$ gives us a timed trace sequence
$(o_1, t_1), \dots, (o_m, t_m)$ where $\pi(t_m) = o_m$, and where the
time-stamps depend on the sampling.
Given a timed timed trace sequence $\tseq$, let $\symb{\tseq}_{\LI}$ denote the
continuous time trace obtained from $\tseq$ by linear interpolation.
Let $\tseq_{\pi}, \tseq_{\pi'}$ be two corresponding samplings
of the traces $\pi, \pi'$.
Since the Skorokhod distance is a metric, we have that
\[\dist_{\skoro}(\pi, \pi') \leq
\dist_{\skoro}\left(\symb{\tseq_{\pi}}_{\LI}, \symb{\tseq_{\pi'}}_{\LI}\right)\, +\,
\dist_{\skoro}\left(\symb{\tseq_{\pi}}_{\LI}, \pi\right) +
\dist_{\skoro}\left(\symb{\tseq_{\pi'}}_{\LI}, \pi'\right). \]
If $\Delta_{\sampleerr}$ is a bound on the distance between a trace, and an
interpolated completion of its sampling, we have that
\[
\dist_{\skoro}(\pi, \pi') \leq \dist_{\skoro}\left(\symb{\tseq_{\pi}}_{\LI}, \symb{\tseq_{\pi'}}_{\LI}\right)\, +\,
2\vdot \Delta_{\sampleerr}.\]
Using this fact, given the error bound $\Delta_{\sampleerr}$,
a procedure for computing the Skorokhod distance between polygonal traces in
$\reals^n$
can be used to estimate the Skorokhod distance between $\reals^n$-valued traces.
This, in turn, can be used to estimate the quantitative conformance between
two systems with the output space $\reals^n$.
\end{comment}
|
1,116,691,500,479 | arxiv | \section{Introduction}
\hspace*{0.5 cm}Let $\Omega\subset {\bf R}^N$ be a bounded domain
with a $C^2$-smooth boundary $\partial\Omega$ and let
$\omega\subset\Omega$ be a subdomain. Write $Q=\Omega \times (0,T)$
with $T>0$ and write $\Sigma=\partial\Omega\times (0,T) $. Consider
the following parabolic equation:
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y}{\partial t}(x,t)}+L_0
y(x,t)+e(x,t)y(x,t)=f(x,t), \;&
\mbox{in }\;Q=\Omega \times (0,T),\\
y(x,t)=0, & \mbox{on }\;\Sigma=\partial \Omega \times (0,T),\\
\end{array}\right.
\eqno{(1.1)}
$$
where
$$
L_0y(x,t) = - \sum ^N _{i,j=1} \frac{\partial }{\partial x_j}
(a^{ij}(x)\frac{\partial }{\partial x_i} y(x,t)) +c(x)y(x,t)
$$
is considered as the system operator. Here and in all that follows,
we make the following regularity assumptions for the coefficients of
$L_0$:
\noindent (I):
$$\begin{array}{ll}
a^{ij}(x) \in Lip(\overline{\Omega}),\;a^{ij}(x)=a^{ji}(x),\;
\mbox{and}\; \lambda ^*|\xi|^2 \leq \displaystyle{\sum_{i,j=1}^{N}} a^{ij}(x)
\xi _i \xi_j & \leq \displaystyle{\frac{1}{\lambda^*}} |\xi|^2 ,\;
\mbox{for}\; \xi
\in {\bf R}^N \\
\end{array}
\eqno{(1.2)}
$$
with $\lambda^*$ a certain positive constant;
\noindent (II): $$
\begin{array}{ll}
c(x) \in
L^\infty(\Omega),\;
e(x,t) \in L^\infty (0,T;L^q(\Omega))\ \hbox{with }\ q
>\max\{N,2\},\ \hbox{and}\ f(x,t)\in L^2(Q).
\end{array}
\eqno{(1.3)}
$$
In such a system, we regard $e(x,t)$ as a perturbation in the system
conductivity. Suppose in the ideal case, namely, in the case when
the perturbation $e(x,t)\equiv 0$, (1.1) has a periodic solution
$y_0(x,t)$:
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y_0}{\partial t}(x,t)}+L_0
y_0(x,t)=f(x,t), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&
\mbox{in }\;\;Q,\\
y_0(x,t)=0, & \mbox{on }\;\; \Sigma,\\
y_0(x,0)=y_0(x,T),&\mbox{in }\Omega.
\end{array}\right.
\eqno{(1.4)}
$$
Then the presence of the error term $e(x,t)$ may well destroy the
periodicity of the system. Indeed, (1.1) may no longer have any
periodic solution. (See Section 3.) The problem that we are
interested in in this paper is to understand if there is a finite
(constructible) dimensional subspace ${\bf U}\subset L^2(Q)$, such
that, after imposing a control $u_e\in \bf{U}$, we can restore the
periodic solution $y_e$. Moreover, we would like to know if $y_e$ is
close to $y_0$ and if the energy of $u_e$ is small, when $e(x,t)$ is
small. Our main purpose of this paper is to show that we can indeed
achieve this goal in the small perturbation case, even if the
control is only imposed over a subregion $\omega$ of $\Omega$. The
basic tool for this study is the existence and energy estimate for
the approximate periodic solutions obtained in the author's previous
paper \cite{kn:[1]}. \medskip
To state our results, we first recall the definition of approximate
periodic solutions with respect to the elliptic operator $L_0$.
Notice that $L_0$ is a symmetric operator. Consider the eigenvalue
problem of $L_0$:
$$
\left\{
\begin{array}{ll}
L_0 X(x)=\lambda X(x),\\
X(x)|_{\partial \Omega} =0.
\end{array}\right.
\eqno{(1.5)}
$$
Making use of the regularity assumptions of the coefficients of
$L_0$, we know (see, \cite{kn:[2]} \cite{kn:[3]}, for example) that
(1.5) has a complete set of eigenvalues
$\{\lambda_j\}_{j=1}^{\infty}$ with the associated eigenvectors
$\{X_j(x)\}_{j=1}^{\infty}$ such that $$L_0 X_j(x)=\lambda_j X_j
(x),$$ $$-\infty<\lambda_1\leq\lambda_2
\leq\cdots\leq\lambda_j\leq\cdots<\infty,\ \lim_{j\rightarrow
\infty}\lambda_j=\infty,\;X_j(x)\in H^1_0(\Omega)\cap
C(\overline{\Omega}).$$ Choose $\{X_j(x)\}_{j=1}^{\infty}$ such that
it forms an orthonormal basis of $L^2(\Omega)$. Therefore, for any
$y(x,t)\in L^2(Q)$, we have $
y(x,t)=\displaystyle\sum^{\infty}_{j=1}y_j(t)X_j(x)$, where
$$y_j(t)=\langle y(x,t),X_j(x)\rangle= \displaystyle\int_\Omega
y(x,t)X_j(x)dx\in L^2(0,T).$$\medskip
{\bf Definition 1.1.} {\it We call $y(x,t)$ is a K-approximate
periodic solution of (1.1) with respect
to $L_0$ if \\
(a): $y \in C([0,T];L^2(\Omega))\cap L^2(0,T;H^1_0 (\Omega ))$ is
a weak solution of (1.1);\\
(b): $ y\in {\bf S_{K}} $, where ${\bf S_{K}}$ is the space of the
following functions: $${\bf S_{K}} = \{y(x,t)\in
L^2(Q);\;y_j(0)=y_j(T),\;\mbox{for}\;j \ge K+1,\;
y_j(t)=\displaystyle{\int_\Omega } y(x,t)X_j (x)dx \}.$$}
\medskip
When $K= 0$, we will always regard $\sum ^{0} _{j=1} = 0$. Hence, a
0-approximate periodic solution of (1.1) is a regular periodic
solution. In what follows, we write $\langle y(\cdot,t),y(\cdot
,t)\rangle=\displaystyle{\int_\Omega} y^2(x,t)dx=\|y(\cdot,t)\|^2$,
and we denote $y_t$ for the derivative of $y(x,t)$ with respect to
$t$.
\medskip
Our first result of this paper can be stated as follows:
\bigskip
{\bf Theorem 1.1} {\it Consider the system (1.1), where $e(x,t)$ is
regarded as a perturbation in the system conductivity. Suppose that
(1.1) has a periodic solution $y_0(x,t)$ at the ideal case with
$e(x,t)\equiv 0$. Assume that
$\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}=ess\displaystyle{\sup_{t\in(0,T)}}\|e(\cdot,t)\|_{L^q(\Omega)}<\varepsilon$,
where $\varepsilon<1$ is a small constant which depends only on
$L_0,\Omega, N, q, T$ with $q>\max\{N,2\}$. Then there are a
non-negative integer $K_0$, depending only on $L_0,\Omega, N, q, T$
(but not $f$), and a unique outside force of the form
$$u_e(x):=\sum_{j=1}^{K_0}u_jX_j(x)\in {\bf U}=span_{{\mathbf
R}}\{X_1(x),X_2(x),\cdots,X_{K_0}(x)\},$$ where $u_j\in {\mathbf
R}$, such that the following has a unique periodic solution $y$
satisfying:
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y(x,t)}{\partial
t}}+L_0y(x,t)+e(x,t)y(x,t)=f(x,t)+u_e(x),
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&
\mbox{in }\;\;Q,\\
y(x,t)=0, & \mbox{on }\;\; \Sigma,\\
\langle y(x,0),X_j(x)\rangle=\langle y_0(x,0),X_j(x)\rangle, & \mbox{for }\;j\leq K_0,\\
y(x,0)=y(x,T),&\mbox{in }\;\;\Omega.
\end{array}\right.
\eqno{(1.6)}
$$
Moreover, we have the following energy estimate:
$$\begin{array}{ll}
&\displaystyle\sup_{t\in[0,T]}\|(y-y_0)(\cdot,t)\|^2+\displaystyle{\int^T_0}\|\nabla(y-y_0)(\cdot,t)\|^2dt\\
&\leq C(system,K_0)\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}(1
+|\vec{a}|^2+\displaystyle{\int_Q} f^2dxdt),\end{array}\eqno{(1.7)}
$$
and
$$
\|u_e\|^2_{L^2(\Omega)}\leq
C(system,K_0)\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}(1+|\vec{a}|^2+\displaystyle{\int_Q}
f^2dxdt), \eqno{(1.8)}
$$
where $\vec{a}=(a_1,a_2,\cdots,a_{K_0})=(\langle
y_0(x,0),X_1(x)\rangle,\langle y_0(x,0),X_2(x)\rangle,\cdots,\langle
y_0(x,0),X_{K_0}(x)\rangle)$.
Here and in what follows, $C(system,K_0)$
denotes a constant depending only on $L_0,\Omega, N,q,T$, which may
be different in different contexts.}\bigskip
\bigskip
In Section 3 of this paper, we will construct an example, showing
that without outside controls, (1.1) has no periodic solutions in
general. This is one of the main features in our Theorem 1.1: The
control can always be taken from a certain fixed constructible {\it
finite dimensional subspace} to regain the periodicity, while
the perturbation space for $e(x,t)$, which destroys the periodicity, is
{\it of infinite dimension}. We also notice that our system
operator $L_0$ is not assumed to be positive.
\medskip
The second part of this work is to consider the same problem as
studied in the first part, but with the control only imposed over a
subregion $\omega\subset\Omega$ and time interval $E\subset [0,T]$,
$m(E)>0$. We will similarly obtain the following:\bigskip
{\bf Theorem 1.2.} {\it Suppose that the system (1.1) has a periodic
solution $y_0(x,t)$ at the ideal case with $e(x,t)\equiv 0$. Then
there are a positive integer $K_0$, a small constant
$\varepsilon>0$, depending only on $L_0,\Omega, N,q,T$
$(q>\max\{N,2\})$, such that, when
$$\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}=ess\displaystyle{\sup_{t\in(0,T)}}\|e(x,t)\|_{L^q(\Omega)}<\varepsilon,$$
the following has a unique periodic solution:
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y(x,t)}{\partial
t}}+L_0y(x,t)+e(x,t)y(x,t)=f(x,t)+\displaystyle{\sum_{j=1}^{K_0}}\chi_\omega(x)\chi_E(t)u_jX_j(x),
\;\;\;\;\;\;\;\;&
\mbox{in }\;\;Q,\\
y(x,t)=0, & \mbox{on }\;\; \Sigma,\\
\langle y(x,0),X_j(x)\rangle=a_j, & \mbox{for }\;j\leq K_0,\\
y\in {\bf S_{K_0}},
\end{array}\right.
\eqno{(1.9)}
$$
where $(a_1,a_2,\cdots,a_{K_0})=(\langle
y_0(x,0),X_1(x)\rangle,\langle y_0(x,0),X_2(x)\rangle,\cdots,\langle
y_0(x,0),X_{K_0}(x)\rangle)=\vec{a}$,
$(u_1,u_2,\cdots,u_{K_0})=\vec{u}\in {\mathbf R}^{K_0}$. Moreover,
$$
|\vec{u}|^2\leq
C(system,K_0,\omega)\displaystyle\frac{\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}}{(m(E))^2}(1+|\vec{a}|^2+\displaystyle{\int_Q}
f^2dxdt),\eqno{(1.10)}
$$
and
$$\begin{array}{ll}
&\displaystyle\sup_{t\in[0,T]}\|(y-y_0)(\cdot,t)\|^2+\displaystyle{\int^T_0}\|\nabla(y-y_0)(\cdot,t)\|^2dt\\
&\leq
C(system,K_0,\omega)\displaystyle\frac{\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}}{(m(E))^2}(1+|\vec{a}|^2+\displaystyle{\int_Q}
f^2dxdt).\end{array}\eqno{(1.11)}
$$
Here, $$\chi_\omega(x),\;\chi_E(t)$$ are the characteristic
functions for $\omega$ and $E$, respectively; and $C(system,K_0,
\omega)$ is a constant depending only on $\omega,L_0,\Omega, N,q,T$.
}
\bigskip
Theorem 1.1 and Theorem 1.2 give stabilization results for the
periodic solutions of a linear parabolic system under small
perturbation of the system conductivity, modifying a control from a
fixed finite dimensional subspace. We do not know if similar
results as in Theorem 1.1 hold under the large perturbation case.
\medskip
The paper is organized as follows. In Section 2, we prove Theorem
1.1. In Section 3, we give an example to show that with a small
perturbation $e(x,t)$, (1.1) has no periodic solution in general. In
section 4, we give the proof of Theorem 1.2.
\section{Small perturbation}
\hspace*{0.5 cm}In this Section, we give a proof of Theorem
1.1, based on the author's previous paper \cite{kn:[1]}. For convenience of the reader, we first recall the following result of \cite{kn:[1]}, which will be used here.
\medskip
{\bf Theorem 2.1}. {\it Assume (1.2) and (1.3). Let $e(x,t)\in {\mathcal{M}}(q,M)$, where, for any positive
number $M$ and $q> \frac{N}{2}$,
$${\mathcal{M}}(q,M):=
\{e(x,t) \in L^{\infty}(0,T;L^q (\Omega)); \mbox{ess sup}_{t\in
(0,T)} \|e(x,t)\|_{L^q (\Omega)}\le M\}.$$ Then, there exists an
integer $K_0(L_0,M,\Omega, q,N,T)$ $\geq 0$, depending only on
$(L_0, M,\Omega, q,N, T)$ (but not $f(x,t)$), such that for any
$K\geq K_0(L_0,M,\Omega,q,N, T)$ and any initial value
$\vec{a}=(a_1,a_2,\cdots,a_K)\in {\bf R^K}$, we have a unique
solution to the following equation:
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y(x,t)}{\partial
t}}+L_0y(x,t)+e(x,t)y(x,t)=f(x,t),
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&
\mbox{in }\;\;Q,\\
y(x,t)=0, & \mbox{on }\;\; \Sigma,\\
\langle y(x,0),X_j(x)\rangle=a_j, & \mbox{for }\;j\leq K,\\
y\in {\bf S_{K}}.
\end{array}\right.
\eqno{(2.1)}
$$
Moreover, for such a solution $y(x,t)$, we have the following energy estimate:
$$
\begin{array}{ll}
\displaystyle{\sup_{t\in[0,T]}}\|y(\cdot,t)\|^2
+\displaystyle{\int^T_0}\|\nabla y(\cdot,t)\|^2 dt \leq
C(L_0,M,\Omega,q,N,T) (|\vec{a}|^2 +\displaystyle{\int_Q}f^2dxdt).
\end{array}
\eqno (2.2) $$}
\bigskip
Now, suppose $y_0$ is a periodic solution of (1.1) with $e(x,t)=0$,
namely,
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y_0}{\partial t}(x,t)}+L_0
y_0(x,t)=f(x,t), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&
\mbox{in }\;\;Q,\\
y_0(x,t)=0, & \mbox{on }\;\; \Sigma,\\
y_0(x,0)=y_0(x,T),&\mbox{in }\Omega.
\end{array}\right.
\eqno{(2.3)}
$$
Let $$ a_j=(y_0)_j(0)=\langle y_0(x,0),X_j(x)\rangle,\mbox{ for }
j=1,2,\cdots.
$$
In all that follows, we assume that $e(x,t)\in {\mathcal{M}}(q,M)$
with $M=1$. By Theorem 2.1, there exists an integer
$K_0(L_0,M,\Omega, q,N,T)$ $\geq 0$, such that for the initial value
$\vec{a}=(a_1,a_2,\cdots,a_{K_0})\in {\mathbf R}^{K_0}$, we have a
unique solution $y(x,t)$ satisfying the following equations:
$$
\left\{\begin{array}{ll}
\displaystyle{\frac{\partial y(x,t)}{\partial
t}}+L_0y(x,t)+e(x,t)y(x,t)=f(x,t)+\displaystyle{\sum_{j=1}^{K_0}}u_jX_j(x),
\;\;\;\;\;\;\;\;&
\mbox{in }\;\;Q,\\
y(x,t)=0, & \mbox{on }\;\; \Sigma,\\
\langle y(x,0),X_j(x)\rangle=a_j, & \mbox{for }\;j\leq K_0,\\
y\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(2.4)}
$$Here, $\vec{u}=(u_1,u_2,\cdots,u_{K_0})\in {\mathbf R}^{K_0}$.
Subtracting (2.3) from (2.4), we get the following equation:
$$
\left\{\begin{array}{ll}
(y-y_0)_t
+L_0(y-y_0)+e(x,t)(y-y_0)=\displaystyle{\sum_{j=1}^{K_0}}u_jX_j(x)-e(x,t)y_0,
\;\;\;&
\mbox{in }\;\;Q,\\
(y(x,t)-y_0(x,t))=0, & \mbox{on }\;\; \Sigma,\\
(y-y_0)_j(0)=\langle y(x,0)-y_0(x,0),X_j(x)\rangle=0, & \mbox{for }\;j\leq K_0,\\
(y-y_0)\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(2.5)}
$$
We define a map$$J:\;{\mathbf R}^{K_0}\longmapsto {\mathbf
R}^{K_0}$$by
$$J(u_1,u_2,\cdots,u_{K_0})=((y-y_0)_1(T),(y-y_0)_2(T),\cdots,(y-y_0)_{K_0}(T)).$$Write $v=y-y_0=v_0+v_u$. Here,
$v_0$ and $v_u$ are the solution of the following equations, respectively,
$$
\left\{\begin{array}{ll}
(v_0)_t +L_0v_0+e(x,t)v_0=-e(x,t)y_0, \;\;\;&
\mbox{in }\;\;Q,\\
v_0=0, & \mbox{on }\;\; \Sigma,\\
(v_0)_{j}(0)=0, & \mbox{for }\;j\leq K_0,\\
v_0\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(2.6)}
$$and
$$
\left\{\begin{array}{ll} (v_u)_t
+L_0v_u+e(x,t)v_u=\displaystyle{\sum_{j=1}^{K_0}}u_jX_j(x), \;\;\;&
\mbox{in }\;\;Q,\\
v_u=0, & \mbox{on }\;\; \Sigma,\\
(v_u)_{j}(0)=0, & \mbox{for }\;j\leq K_0,\\
v_u\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(2.7)}
$$
We are led to the question to find out if there is a vector $\vec{u}=(u_1,u_2,\cdots,u_{K_0})\in {\bf R^{K_0}}$
such that
$$
J(\vec{u})=((y-y_0)_1(T),(y-y_0)_2(T),\cdots,(y-y_0)_{K_0}(T))=(0,0,\cdots,0).
$$
Indeed, if this is the case, then $y$ is a periodic solution with
the required estimate as we will see later.
For this purpose, we write
$J_0=((v_0)_1(T),(v_0)_2(T),\cdots,(v_0)_{K_0}(T))$ and
$$J^*(\vec{u})=((v_u)_1(T),(v_u)_2(T),\cdots,(v_u)_{K_0}(T)).$$Then $$J(\vec{u})=J_0+J^*(\vec{u}).$$
Now, it is easy to see that $J^*$ is linear in
$(u_1,u_2,\cdots,u_{K_0})$. We next claim that $J^*$ is invertible
under the small perturbation case. If not, we can find a vector
$\vec{\xi}=(\xi_1,\xi_2,\cdots,\xi_{K_0})\in {\mathbf R}^{K_0}$ with
$|\vec{\xi}|=\sqrt{\xi^2_1+\xi^2_2+\cdots+\xi^2_{K_0}}=1$ such that
$J^*(\vec{\xi})=0$. Hence, we have a unique solution to the
following problem:
$$
\left\{\begin{array}{ll} w_t
+L_0w+e(x,t)w=\displaystyle{\sum_{j=1}^{K_0}}\xi_jX_j(x), \;\;\;&
\mbox{in }\;\;Q,\\
w=0, & \mbox{on }\;\; \Sigma,\\
w_{j}(0)=w_{j}(T)=0, & \mbox{for }\;j\leq K_0,\\
w\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(2.8)}
$$
First, by the energy estimate in Theorem 2.1, we have for $w(x,t)$,
$$\begin{array}{ll}
\displaystyle{\sup_{t\in[0,T]}}\|w(\cdot,t)\|^2
+\displaystyle{\int^T_0}\|\nabla w(\cdot,t)\|^2 dt &\leq
C(system)\cdot T\cdot |\vec{\xi}|^2\\ &\leq C(system,K_0).
\end{array}
\eqno (2.9)$$As mentioned before, we use $C(system, K_0)$ to denote
a constant depending only on $L_0,M,\Omega,q,N,T$, which may be
different in different contexts.
\medskip
Write $w=\displaystyle{\sum^{\infty}_{j=1}}w_j(t)X_j(x)$ as before. Then we have
$$
\displaystyle{\frac{dw_j(t)}{dt}}+\lambda_j w_j(t)+\int_\Omega
e(x,t)w(x,t)X_j(x)dx =\xi_j,\;\mbox{for
}j=1,2,\cdots,K_0.\eqno{(2.10)}
$$
Next, by the H$\ddot{o}$lder inequality (see Claim 2.2 of
\cite{kn:[1]}), we have
$$\begin{array}{ll}
\displaystyle{\int_\Omega}|e(x,t)w(x,t)X_j(x)|dx&\leq
C(\Omega,N,q)\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}[\|w(\cdot,t)\|^2_{L^2(\Omega)}\\
&+\|X_j(x)\|^2_{L^2(\Omega)}+ \|\nabla w(\cdot,t)\|^2_{L^2(\Omega)}+\|\nabla X_j(x)\|^2_{L^2(\Omega)}]\\
&\leq C(\Omega,N,q)\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}[1+\lambda^2_j\\
&+ \|w(\cdot,t)\|^2_{L^2(\Omega)}+
\|\nabla w(\cdot,t)\|^2_{L^2(\Omega)}].
\end{array}
$$
By (2.9), we have
$$
\displaystyle{\int^T_0\int_\Omega}|e(x,t)w(x,t)X_j(x)|dxdt\leq
C(system,K_0)\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}.\eqno{(2.11)}
$$
Next, from (2.10), we get
$$(e^{\lambda_j t}w_j(t))'_t+\int_\Omega e(x,t)w(x,t)X_j(x)e^{\lambda_j t}dx
=e^{\lambda_j t}\xi_j,\;\mbox{for }j=1,2,\cdots,K_0.$$ Integrating the above over [0,T], we get, for
$j=1,2,\cdots,K_0$,
$$0+\displaystyle{\int^T_0\int_\Omega}e(x,t)w(x,t)X_j(x)e^{\lambda_j t}dxdt=\xi_j\displaystyle{\int^T_0}e^{\lambda_j t}dt.$$
Namely,
$$
\xi_j=\left\{\begin{array}{ll} \displaystyle{\frac{\displaystyle{\int^T_0\int_\Omega}e(x,t)w(x,t)X_j(x)e^{\lambda_j
t}dxdt}{\frac{1}{\lambda_j}(e^{\lambda_j T}-1)}},\;\;\;\;&\mbox{for }\lambda_j\neq 0,\\
\displaystyle{\frac{\displaystyle{\int^T_0\int_\Omega}e(x,t)w(x,t)X_j(x)dxdt}{T}},&\mbox{for }\lambda_j=0.
\end{array}\right.
$$
Hence, we get, for $j=1,2,\cdots,K_0$,
$$\begin{array}{ll}
|\xi_j|&\leq C(system,K_0)\displaystyle{\int^T_0\int_\Omega}|e(x,t)w(x,t)X_j(x)|dxdt\\
&\leq C(system,K_0)\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}.
\end{array}\eqno{(2.12)}
$$
We get
$$1=|\vec{\xi}|\leq C(system,K_0)\sqrt{K_0}\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}.$$
This gives a contradiction when
$$\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}<
\displaystyle{\frac{1}{C(system,K_0)\sqrt{K_0}}}.$$ Therefore, we showed that
$J^*$ is invertible when
$\|e(x,t)\|_{L^\infty(0,T;L^q(\Omega))}<\epsilon $ with a certain
$\epsilon$ depending only on $L_0,\Omega, N,q, T$.
Hence, for any given
$\vec{b}=(b_1,b_2,\cdots,b_{K_0})\in {\mathbf R}^{K_0}$, there
exists a unique $$\vec{u}=(u_1,u_2,\cdots,u_{K_0})\in {\mathbf
R}^{K_0}$$ such that
$$J^*(\vec{u})=J^*(u_1,u_2,\cdots,u_{K_0})=(b_1,b_2,\cdots,b_{K_0}).$$
Back to the equation (2.7), we have
$$
\displaystyle{\frac{d(v_u)_j(t)}{dt}}+\lambda_j (v_u)_j(t)+\int_\Omega e(x,t)v_u(x,t)X_j(x)dx =u_j,\;\mbox{for
}j=1,2,\cdots,K_0.
$$
Then
$$
\displaystyle{\frac{d[e^{\lambda_j t}(v_u)_j(t)]}{dt}}+\int_\Omega e(x,t)v_u(x,t)X_j(x)e^{\lambda_j t}dx=u_j
e^{\lambda_j t},\;\mbox{for }j=1,2,\cdots,K_0.
$$
Integrating the above over [0,T], by the definition of $J^*$, we have
$$
b_j e^{\lambda_j T}-0+\displaystyle{\int^T_0\int_\Omega} e(x,t)v_u(x,t)X_j(x)e^{\lambda_j t}dxdt=u_j \int^T_0e^{\lambda_j
t}dt,\;\mbox{for }j=1,2,\cdots,K_0.
$$
We then get
$$
u_j=\left\{\begin{array}{ll} \displaystyle{\frac{b_j e^{\lambda_j
T}+\displaystyle{\int^T_0\int_\Omega}e(x,t)v_u(x,t)X_j(x)e^{\lambda_j
t}dxdt}{\frac{1}{\lambda_j}(e^{\lambda_j T}-1)}},\;\;\;\;&\mbox{for }\lambda_j\neq 0,\\
\displaystyle{\frac{\displaystyle{\int^T_0\int_\Omega}e(x,t)v_u(x,t)X_j(x)dxdt}{T}},&\mbox{for }\lambda_j=0.
\end{array}\right.
$$
$$\begin{array}{ll}
|u_j|^2 &\leq 2e^{2\lambda_{K_0}T}|b_j|^2
+2e^{2\lambda_{K_0}T}[\displaystyle{\int^T_0\int_\Omega}e(x,t)v_u(x,t)X_j(x)dxdt]^2\\
&\leq 2e^{2\lambda_{K_0}T}|b_j|^2 +2e^{2\lambda_{K_0}T}\cdot
\hbox{sup}_{\Omega}|X_j|^2[\displaystyle{\int^T_0}\|e(\cdot,t)\|_{L^q(\Omega)}\|v_u(\cdot,t)\|_{L^{q'}(\Omega)}dt]^2
\end{array}
$$
Here $1/q+1/q'=1$. Since $\Omega$ is bounded and
$q'=\frac{q}{q-1}\le 2$, by the H\"older inequality, we have
$\|v_u\|_{L^{q'}(\Omega)}\le C(\Omega, q)\|v_u\|_{L^{2}(\Omega)}.$
Hence,
$$[\displaystyle{\int^T_0}\|e(\cdot,t)\|_{L^q(\Omega)}\|v_u(\cdot,t)\|_{L^{q'}(\Omega)}dt]^2\le
C(\Omega,T,q)\|e\|^2_{L^\infty(0,T;L^q(\Omega))}\|v_u\|^2_{L^{2}(Q)}.$$
By the energy estimate in Theorem 2.1, we have
$\|v_u\|^2_{L^{2}(Q)}\le C(system, K_0) |u|^2.$ Hence, as argument
before, when $\|e\|^2_{L^\infty(0,T;L^q(\Omega))}$ is small, we can
solve the above to obtain the following:
$$|\vec{u}|^2 \leq C(system,K_0)|\vec{b}|^2.\eqno{(2.13)}$$
Back to (2.5), we need to find $\vec{u}=(u_1,u_2,\cdots,u_{K_0})$
such that the solution in (2.5) has the property $(y-y_0)_j(T)=0$
for $j=1,2,\cdots,K_0$. As mentioned before, $v=y-y_0$ is then a
periodic solution. Thus $y=v+y_0$ is a periodic solution of (2.4)
after applying the control force
$\displaystyle{\sum_{j=1}^{K_0}}u_jX_j(x)$. To this aim, we need
only to find $\vec{u}$ such that
$$J(\vec{u})=0\;\mbox{or }\;J^*(\vec{u})=-J_0.$$ By the definition
of $J_0$, $J_0=-\vec{b}=(-b_1,-b_2,\cdots,-b_{K_0})$ is given by
$$
\left\{\begin{array}{ll}
(v_0)_t +Lv_0+e(x,t)v_0=-e(x,t)y_0, \;\;\;&
\mbox{in }\;\;Q,\\
v_0=0, & \mbox{on }\;\; \Sigma,\\
(v_0)_{j}(0)=0,\;(v_0)_j(T)=-b_j, & \mbox{for }\;j\leq K_0,\\
v_0\in {\bf S_{K_0}}.
\end{array}\right.
$$
By the energy estimate of Theorem 2.1, we have
$$\begin{array}{ll}
|\vec{b}|^2&\leq \|v_0(\cdot,T)\|^2_{L^2(\Omega)}\\
&\leq C(system,K_0)\displaystyle{\int^T_0\int_\Omega (-ey_0)^2dxdt}\\
&\leq C(system,K_0)\displaystyle{\int^T_0}\{\|e(x,t)\|^2_{L^q(\Omega))}\|y_0(\cdot,t)\|^2_{L^{\frac{2q}{q-2}}(\Omega)}\}dt\\
&\leq C(system,K_0)\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}\|\nabla y_0\|^2_{L^2(Q)}\\
&\leq
C(system,K_0)\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}(|\vec{a}|^2+\displaystyle{\int_Q}
f^2dxdt),
\end{array}\eqno{(2.14)}
$$
where $\vec{a}=(a_1,a_2,\cdots,a_{K_0})=(\langle y_0(x,0),X_1(x)\rangle,\langle
y_0(x,0),X_2(x)\rangle,\cdots,\langle y_0(x,0),X_{K_0}(x)\rangle)$.
Thus, by (2.13), we get
$$|\vec{u}|^2\leq C(system,K_0)\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}
(1+|\vec{a}|^2+\displaystyle{\int_Q} f^2dxdt).\eqno{(2.15)}
$$
By (2.2), (2.14) and (2.15), we obtain
$$\begin{array}{ll}
&\displaystyle\sup_{t\in[0,T]}\|(y-y_0)(\cdot,t)\|^2+\displaystyle{\int^T_0}\|\nabla(y-y_0)(\cdot,t)\|^2dt\\
&\leq C(system,K_0)\|e(x,t)\|^2_{L^\infty(0,T;L^q(\Omega))}(1
+|\vec{a}|^2+\displaystyle{\int_Q} f^2dxdt),\end{array}
$$
Summarizing the above, we complete the proof of Theorem 1.1.
$\hbox{\vrule height1.5ex width.5em}$
\bigskip
\section{An example}
\hspace*{0.5 cm}In this section, we present an example, showing that
with a small perturbation $e(x,t)$, (1.1) has no periodic solution
in general. This demonstrates the importance of an outside control
to gain back the periodicity as in Theorem 1.1.\medskip
We consider the following one dimensional parabolic equation:
$$
\left\{\begin{array}{ll} y_t-y_{xx}-y-e(x)y=f(x),\;\;\;\;\;\;&0\leq x\leq \pi,\;0\leq t\leq T,\\
y(0,t)=y(\pi,t)=0,&0\leq t\leq T.
\end{array}\right.\eqno{(3.1)}
$$
Let $L_e y=-y_{xx}-y-e(x)y$ with $e(x)\in C^0[0,\pi]$. Suppose $0$
is an eigenvalue of $L_e$ with eigenvectors $\{X_j(x)\}^m_{j=1}$.
Then (3.1) has a periodic solution if and only if
$$\displaystyle{\int^\pi_0}f(x)X_j(x)dx=0,\;\mbox{for}\;j=1,2,\cdots,m.$$Now, when $e(x)=0$, then $0$ is the first
eigenvalue of $L_0$ with $\sin x$ as a basis of the $0$-eigenspace.
Hence, (3.1) has a periodic solution if and only if
$$\displaystyle{\int^\pi_0}f(x)\sin xdx=0\;\mbox{or
}f(x)=\displaystyle{\sum^\infty_{j=2}}a_j\sin jx,\
\sum_{j=2}^{\infty}|a_j|^2<\infty.$$ Now suppose $e(x)\approx 0$.
The first eigenvalue $\lambda_e$ of $L_e$ is given by
$$\lambda_e=\displaystyle{\min_{\varphi\in
H^1_0(0,\pi),\|\varphi\|_{L^2(0,\pi)}=1}}J_e(\varphi,\varphi),$$where
$$J_e(\varphi,\varphi)=\displaystyle{\int^\pi_0}(\varphi^2_x-\varphi^2-e(x)\varphi^2)dx.$$
(See \cite{kn:[3]}). Hence,
$$\begin{array}{ll}
\lambda_e&\leq \displaystyle{\min_{\varphi\in
H^1_0(0,\pi),\|\varphi\|_{L^2(0,\pi)}=1}}\displaystyle{\int^\pi_0}(\varphi^2_x-\varphi^2)dx+\max|e(x)|\displaystyle{\int^\pi_0}\varphi^2dx\\
&\leq 0+\max|e(x)|\\
& \leq \max|e(x)|.
\end{array}\eqno{(3.2)}
$$
$$\lambda_e=J_e(\varphi_e,\varphi_e)=\displaystyle{\int^\pi_0}(\varphi_e)^2_xdx-\displaystyle{\int^\pi_0}(1+e(x))\varphi_e^2dx$$with
$\varphi_e$ the eigenvector corresponding to $\lambda_e$ and $\|\varphi_e\|_{L^2(0,\pi)}=1$.\medskip
Since $0$ is the first eigenvalue of $L_0$, we have
$$\begin{array}{ll}
\lambda_e&=\displaystyle{\int^\pi_0}((\varphi_e)^2_x-(\varphi_e)^2)dx-\displaystyle{\int^\pi_0}e(x)\varphi_e^2dx\\
&\geq -\max|e(x)|
\end{array}\eqno{(3.3)}
$$
By (3.2) and (3.3), we get $$|\lambda_e|\leq \max|e(x)|,\;\mbox{and
}\lambda_e\rightarrow 0 \;\mbox{as }e(x)\rightarrow 0.$$ Next,
consider the system with $e(x)+\lambda_e$ as the perturbation in
the system conductivity:
$$\left\{\begin{array}{ll} y_t-y_{xx}-y-(e(x)+\lambda_e)y=f(x),\;\;\;\;\;\;&0\leq x\leq \pi,\;0\leq t\leq T,\\
y(0,t)=y(\pi,t)=0,&0\leq t\leq T.
\end{array}\right.\eqno{(3.4)}
$$
Then when $e(x)\approx 0$, we have $(e(x)+\lambda_e)\approx 0$. However, if (3.4) still has a periodic solution,
we have
$$
\displaystyle{\int^\pi_0}f(x)\varphi_edx=0.
$$
If this is the case for any given $f$, we then have
$$
\displaystyle{\int^\pi_0}\sin jx\varphi_e dx=0,\;\mbox{for }j=2,3,\cdots.
$$
This implies that $\varphi_e=C \sin x$ and thus
$$
-e(x)\sin x=\lambda_e \sin x,\;\mbox{or }e(x)=-\lambda_e.
$$
This is a contradiction unless $e(x)\equiv const.$. This shows that
for any non-constant small perturbation in $e(x)$, for most a priori
given $f$, the periodicity of the system will get lost.\bigskip
\section{Local stabilization}
\hspace*{0.5 cm}In this section, we consider the same problem as
studied in Section 2, but with the control only imposed over a
subregion $\omega\subset\Omega$ and time interval $E\subset [0,T]$
with $m(E)>0$.
For the proof of Theorem 1.2, we need the following lemma, whose
quantitative version in the Laplacian case can be found in
\cite{kn:[4]} and \cite{kn:[5]}:
\medskip
{\bf Lemma 4.1} {\it Let $X_{ij}(\omega)=\displaystyle\int_\omega
X_i(x)X_j(x)dx$. Then the symmetric matrix
$X(\omega,k)=(X_{ij}(\omega))_{1\leq i,j\leq k}$ is positive
definite for any $k\geq 1$. In particular, it is
invertible.}\medskip
{\it Proof of Lemma 4.1:} Let $a=(a_1,a_2,\cdots,a_k)\in {\bf R^k}$
and let
$$
I(a,a)=\displaystyle\int_\omega|\sum^k_{j=1}a_jX_j(x)|^2dx.
$$
Then $$I(a,a)=a\cdot X(\omega,k)\cdot a^\tau,\mbox{ where
}a^\tau=\left(
\begin{array}{ccc}
a_1
\\
a_2
\\
\vdots\\
a_k
\end{array}
\right).
$$
Apparently, $I(a,a)\geq 0$. If $X(\omega,k)$ is not positive
definite, then there is a vector $a'=(a'_1,a'_2,\cdots,a'_k)\neq 0$
such that $I(a',a')=0$. Without loss of generality, assume that
$a'_k\not =0$. Hence,
$$\displaystyle\sum^k_{j=1}a'_jX_j(x)|_{\omega}=0.\eqno{(4.1)}$$ We thus get over $\omega$:
$$X_k(x)=\displaystyle\sum_{j<k}b_jX_j(x),\;\mbox{with }b_j=-\displaystyle\frac{a'_j}{a'_k}.\eqno{(4.2)}$$
Applying $(L_0)^m$ to (4.2) over $\omega$, we have
$$\lambda^m_k X_k(x)=\displaystyle\sum_{j<k}b_j\lambda^m_jX_j(x).$$
We get
$$X_k(x)=\displaystyle\sum_{j<k}b_j(\frac{\lambda_j}{\lambda_k})^mX_j(x)\mbox{ over }\omega.$$
Letting $m\rightarrow\infty$, we get over $\omega$
$$
X_k(x)=\displaystyle\sum_{k'\leq j<k}b_jX_j(x),\eqno{(4.3)}
$$
where
$$
\left\{\begin{array}{ll}
\lambda_j=\lambda_k,\;\;\;&\mbox{for }j\geq k',\\
\lambda_j<\lambda_k,&\mbox{for }j<k'.
\end{array}\right.\eqno{(4.4)}
$$
By (4.4), we get over $\Omega$,
$$
L_0(X_k(x)-\displaystyle\sum_{k'\leq
j<k}b_jX_j(x))=\lambda_kX_k(x)-\displaystyle\sum_{k'\leq
j<l}b_j\lambda_jX_j(x)=\lambda_k[X_k(x)-\displaystyle\sum_{k'\leq
j<k}b_jX_j(x)].
$$
By (4.3) and the unique continuation for solutions of elliptic
equations, we get
$$
X_k(x)-\displaystyle\sum_{k'\leq j<k}b_jX_j(x)\equiv 0\;\mbox{over }\Omega.
$$
This contradicts the linear independence of the system
$\{X_j\}$.\hbox{\vrule height1.5ex width.5em}
\medskip
{\bf Proof of Theorem 1.2.}: Similar to the proof of Theorem 1.2, we
need only to find a vector $\vec{u}=(u_1,u_2,\cdots,u_{K_0})\in
{\mathbf R}^{K_0}$ such that
$$J^*_\omega(\vec{u})=-J_{0,\omega},$$
where
$$J^*_\omega(\vec{u})=(\langle v(x,T),X_1(x)\rangle,\langle v(x,T),X_2(x)\rangle,\cdots,\langle
v(x,T),X_{K_0}(x)\rangle)=(v_1(T),v_2(T),\cdots,v_{K_0}(T))
$$
with $v$ the solution of the following equation:
$$
\left\{\begin{array}{ll} v_t
+L_0v+e(x,t)v=\displaystyle{\sum_{j=1}^{K_0}}\chi_\omega(x)\chi_E(t)u_jX_j(x),
\;\;\;&
\mbox{in }\;\;Q,\\
v=0, & \mbox{on }\;\; \Sigma,\\
v_j(0)=0, & \mbox{for }\;j\leq K_0,\\
v\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(4.5)}
$$
and
$$
J_{0,\omega}=((v_0)_1(T),(v_0)_2(T),\cdots,(v_0)_{K_0}(T))
$$
with $v_0$ the solution of the following system
$$
\left\{\begin{array}{ll}
(v_0)_t +L_0v_0+e(x,t)v_0=-e(x,t)y_0, \;\;\;&
\mbox{in }\;\;Q,\\
v_0=0, & \mbox{on }\;\; \Sigma,\\
(v_0)_{j}(0)=0, & \mbox{for }\;j\leq K_0,\\
v_0\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{(4.6)}
$$
In the same way, if $J^*_\omega$ is not invertible, then for a
vector $\vec{\xi}=(\xi_1,\xi_2,\cdots,\xi_{K_0})$ with
$|\vec{\xi}|=1$, we have a solution to the following system:
$$
\left\{\begin{array}{ll} v_t
+L_0v+e(x,t)v=\displaystyle{\sum_{j=1}^{K_0}}\chi_\omega(x)\chi_E(t)\xi_jX_j(x),
\;\;\;&
\mbox{in }\;\;Q,\\
v=0, & \mbox{on }\;\; \Sigma,\\
v_j(0)=0=v_j(T), & \mbox{for }\;j\leq K_0,\\
v\in {\bf S_{K_0}}.
\end{array}\right.
\eqno{}
$$
We then get
$$
v_j(t)'+\lambda_jv_j(t)+\displaystyle\int_\Omega
e(x,t)v(x,t)X_j(x)dx=\displaystyle\sum^{K_0}_{l=1}\xi_l\chi_E(t)X_{lj}(\omega),\;\mbox{for }j=1,2,\cdots,K_0.
$$
We similarly get
$$
(e^{\lambda_j t}v_j(t))'_t+e^{\lambda_j t}\displaystyle\int_\Omega e(x,t)v(x,t)X_j(x)dx=e^{\lambda_j
t}\displaystyle\sum^{K_0}_{l=1}\xi_l\chi_E(t)X_{lj}(\omega),\;\mbox{for }j=1,2,\cdots,K_0.
$$
$$
0+\displaystyle\int^T_0\int_\Omega e^{\lambda_j t}e(x,t)v(x,t)X_j(x)dxdt=\displaystyle\int^T_0e^{\lambda_j
t}\displaystyle\sum^{K_0}_{l=1}\xi_l\chi_E(t)X_{lj}(\omega)dt.
$$
We then get
$$
\left(
\begin{array}{ccc}
\displaystyle\int^T_0e^{\lambda_1 t}\chi_E(t)dt&\;&\;
\\
\;&\displaystyle\int^T_0e^{\lambda_2 t}\chi_E(t)dt&\;
\\
\;&\ddots&\;\\
\;&\;&\displaystyle\int^T_0e^{\lambda_{K_0} t}\chi_E(t)dt
\end{array}
\right)
X(\omega,K_0)\left(
\begin{array}{ccc}
\xi_1
\\
\xi_2
\\
\vdots\\
\xi_{K_0}
\end{array}
\right)$$
$$
=\left(
\begin{array}{ccc}
\displaystyle\int^T_0\int_\Omega e^{\lambda_1 t}e(x,t)v(x,t)X_1(x)dxdt\\
\displaystyle\int^T_0\int_\Omega e^{\lambda_2 t}e(x,t)v(x,t)X_2(x)dxdt\\
\vdots\\
\displaystyle\int^T_0\int_\Omega e^{\lambda_{K_0} t}e(x,t)v(x,t)X_{K_0}(x)dxdt
\end{array}
\right)
$$
$$
\left(
\begin{array}{ccc}
\xi_1
\\
\xi_2
\\
\vdots\\
\xi_{K_0}
\end{array}
\right)=X(\omega,K_0)^{-1}
\left(
\begin{array}{ccc}
(\displaystyle\int^T_0e^{\lambda_1 t}\chi_E(t)dt)^{-1} \displaystyle\int_Q e^{\lambda_1 t}evX_1dxdt\\
(\displaystyle\int^T_0e^{\lambda_2 t}\chi_E(t)dt)^{-1} \displaystyle\int_Q e^{\lambda_2 t}evX_2dxdt\\
\vdots\\
(\displaystyle\int^T_0e^{\lambda_{K_0} t}\chi_E(t)dt)^{-1} \displaystyle\int_Q e^{\lambda_{K_0} t}evX_{K_0}dxdt
\end{array}
\right)\eqno{(4.7)}
$$
By Lemma 4.1, we know $X(\omega,K_0)^{-1}$ is a bounded linear
operator from ${\mathbf R}^{K_0}$ to ${\mathbf R}^{K_0}$.
By the energy estimate in Theorem 2.1, we have for $v(x,t)$,
$$\begin{array}{ll}
\displaystyle\sup_{t\in [0,T]}\|v(\cdot,t)\|^2+\int^T_0\|\nabla
v(\cdot,t)\|^2dt&\leq C(system,K_0)\displaystyle\int_Q(\sum_{j=1}^{K_0}\chi_\omega(x)\chi_E(t)\xi_jX_j(x))^2dxdt\\
&\leq C(system,K_0)T|\vec{\xi}|^2\\
&\leq C(system,K_0).
\end{array}\eqno{(4.8)}$$
By the H$\ddot{o}$lder inequality, we have,
$$\begin{array}{ll}
\displaystyle\int_\Omega|evX_j|dx\leq
C(\Omega,N,q)\|e\|_{L^\infty(0,T;L^q(\Omega))}[1+\lambda_j^2+\|v(\cdot,t)\|^2+\|\nabla
v(\cdot,t)\|^2].
\end{array}\eqno{(4.9)}$$
Together with (4.8), we thus have
$$\begin{array}{ll}
\displaystyle\int^T_0\int_\Omega|evX_j|dxdt\leq
C(system,K_0)\|e\|_{L^\infty(0,T;L^q(\Omega))}.
\end{array}\eqno{(4.10)}$$
Back to (4.7), we have
$$\begin{array}{ll} |\vec{\xi}|^2&\leq
C(system,\omega,K_0)\frac{1}{(m(E))^2}\|X(\omega,K_0)^{-1}\|^2\|e\|^2_{L^\infty(0,T;L^q(\Omega))}\\
&\leq
C(system,\omega,K_0)\frac{1}{(m(E))^2}\|e\|^2_{L^\infty(0,T;L^q(\Omega))}.
\end{array}$$
Hence, when $\|e\|^2_{L^\infty(0,T;L^q(\Omega))}$ is sufficient
small, we get $|\vec{\xi}|^2<1$. This gives a contradiction.
Therefore, we showed that $J^*_\omega$ is invertible under small
perturbation. By the same arguments as those in the proof of Theorem
1.1, we can also show the energy estimates as stated in Theorem 1.2.
This completes the proof of Theorem 1.2. $\hbox{\vrule height1.5ex width.5em}$
|
1,116,691,500,480 | arxiv | \section{Introduction}\label{section Introduction}
Geometric group theorists traditionally restrict their attention to finitely
generated groups equipped with a word metric. A typical proof of \v Svarc-Milnor Lemma
(see \cite{Roe lectures} or \cite{Bridson-HaefligerBook}, p.140) involves such metrics. Recently, the study of large scale geometry of groups was expanded to all countable groups by usage of proper, left-invariant metrics:
in \cite{Smith} such metrics were constructed and it was shown that they all induce the same coarse structure on a group (see also \cite{DS}). The point of this note is that a
proper action of a group $G$ on a space ought to be viewed as a geometric way of creating a coarse structure on $G$. That structure is not given by a proper metric but by something very similar; a pseudo-metric where only a finite set of points may be at mutual distance
$0$. From that point of view the proof of \v Svarc-Milnor Lemma is automatic and the Lemma can be summarized as follows. There are two ways of creating coarse structures on countable groups: algebraic (via word or proper metrics) and geometric (via group actions), and both ways are equivalent.
\begin{Def}\label{LSMetricDef}
A pseudo-metric $d_X$ on a set $X$ is called a {\it large-scale metric}
(or ls-metric)
if for each $x\in X$ the set $\{y\in X \mid d_X(x,y)=0\}$ is finite.
$(X,d_X)$ is called a {\it large-scale metric space}
(or an ls-metric space)
if $d_X$ is an ls-metric.
\end{Def}
\begin{Def}\label{LSProperMetricOnGroups}
An ls-metric $d_G$ on a group $G$ is {\it proper} and {\it left-invariant}
if $d_G(g,h)=d_G(f\cdot g,f\cdot h)$ for all $f,g,h\in G$
and $\{h \mid d_G(g,h) < r\}$ is finite for all $r > 0$ and all $g\in G$.
\end{Def}
Notice $G$ must be countable if it admits a proper ls-metric.
One aspect of \v Svarc-Milnor Lemma is $G$ being finitely generated.
That corresponds to $(G,d_G)$ being metrically connected, i.e. there is $M > 0$
such that any two points in $G$ can be connected by a chain of points
separated by at most $M$.
\begin{Lem} \label{MetricConnectedness}
Suppose $d_G$ is a proper and left-invariant ls-metric on $G$.
$(G,d_G)$ is metrically connected if and only if $G$ is finitely generated.
\end{Lem}
{\bf Proof. } If $G$ is generated by a finite set $F$, put $M=\max\{d_G(1_G,f)\mid f\in F\}$.
If $(G,d_G)$ is $M$-connected, put $F=B(1_G,M+1)$.
\hfill $\blacksquare$
\begin{Def}\label{LSUniformDef}
A function $f\colon (X,d_X)\to (Y,d_Y)$ of ls-metric spaces is called {\it large-scale uniform}
(or ls-uniform)
if for each $r > 0$ there is $s > 0$ such that $d_X(x,y)\leq r$ implies
$d_Y(f(x),f(y))\leq s$.
\par $f$ is a {\it large-scale uniform equivalence} if there is
an ls-uniform $g:Y\to X$ such that both $g\circ f$ and $f\circ g$ are within
a finite distance from the corresponding identities.
\end{Def}
\begin{Lem} \label{LSUniformForGroups}
Suppose $(G,d_G)$ and $(H,d_H)$ are two groups equipped
with proper and left-invariant ls-metrics.
A function $f\colon (G,d_G)\to (H,d_H)$ is ls-uniform if and only
if for each finite subset $F$ of $G$ there is a finite subset $E$ of $H$
such that $x^{-1}\cdot y\in F$ implies $f(x)^{-1}\cdot f(y)\in E$
for all $x,y\in G$.
\end{Lem}
{\bf Proof. } Suppose $f$ is ls-uniform and $F$ is a finite subset of $G$.
Let $r$ be larger that all $d_X(1_G,g)$, $g\in F$.
Pick $s> 0$ such that $d_G(g,h) < r$ implies $d_H(f(g),f(h)) < s$
and put $E=\{x\in H \mid d_H(1_H,x) < s\}$.
If $x^{-1}\cdot y\in F$, then $d_G(x,y) < r$. Therefore
$s > d_H(f(x),f(y))=d_H(1_H,f(x)^{-1}\cdot f(y))$ and
$f(x)^{-1}\cdot f(y)\in E$.
Conversely, if $r > 0$ put $F=\{x\in G \mid d_G(1_G,x) < r\}$
and consider $E$ so that $x^{-1}\cdot y\in F$ implies $f(x)^{-1}\cdot f(y)\in E$.
If $s$ is bigger that all $d_H(1_H,g)$, $g\in E$, then
$d_G(x,y) < r$ implies $f(x)^{-1}\cdot f(y)\in E$ and $d_H(f(x),f(y)) < s$.
\hfill $\blacksquare$
\begin{Cor} \label{LSMetricsForGroupsAreUnique}
Given two proper and left-invariant ls-metrics $d_1$ and $d_2$ on the same group $G$,
the identity $id_G:(G,d_1)\to (G,d_2)$ is a coarse equivalence.
\end{Cor}
{\bf Proof. } The choice of $E=F$ always works for $id_G$.
\hfill $\blacksquare$
We are interested in creating proper left-invariant ls-metrics on groups $G$
using actions on metric spaces $X$ via the formula $d_G(g,h)=d_X(g\cdot x_0,h\cdot x_0)$
for some $x_0\in X$. To make $d_G$ left-invariant, a practical requirement is the action occurs
via isometries. Let's characterize the situation in which $d_G$ is a proper ls-metric.
\begin{Lem} \label{CharOflsMetrics}
Suppose $G$ acts via isometries on $X$ and $x_0\in X$. If $d_G$ is defined by
$d_G(g,h)=d_X(g\cdot x_0,h\cdot x_0)$, then $d_G$ is a proper left-invariant ls-metric on $G$ if and only if the following conditions are satisfied:
\begin{itemize}
\item[1.] The stabilizer $\{g\in G\mid g\cdot x_0=x_0\}$ of $x_0$ is finite.
\item[2.] $G\cdot x_0$ is topologically discrete.
\item[3.] Every bounded subset of $G\cdot x_0$ that is metrically discrete is finite.
\end{itemize}
\end{Lem}
{\bf Proof. } Recall that $A$ is metrically discrete if there is $s > 0$ such that $d_X(a,b) > s$
for all $a,b\in A$, $a\ne b$. Clearly, if one of Conditions 1-3 is not valid, then there is $r > 0$
such that $B(1_G,r)$ is infinite and $d_G$ is not proper. Thus, assume 1-3 hold.
Suppose $B(1_G,r)$ is infinite for some $r > 0$ and pick $g_1$ in that set.
Suppose $\{g_n\}_{n=1}^k\subset B(1_G,2r)$ is constructed so that
$d_X(g_i\cdot x_0,x_0) < \frac{1}{i}$. Put $A=B(1_G,r)\setminus \{g_n\}_{n=1}^k$
and notice $A\cdot x_0$ is infinite (otherwise the stabilizer of $x_0$ is infinite). Hence there are two different
elements $g,h\in A$ such that $g\cdot x_0\ne h\cdot x_0$ and $d_X(g\cdot x_0,h\cdot x_0) < \frac{1}{k+1}$. Put $g_{k+1}=g^{-1}\cdot h$.
However, $g_n\cdot x_0\to x_0$, a contradiction.
\hfill $\blacksquare$
It turns out, for nice spaces $X$, $d_G$ being a proper ls-metric is equivalent to the action being
proper.
\begin{Cor} \label{PreSvarc-Milnor}
Suppose $(X,d_X)$ is a metric space so that all infinite bounded subsets of $X$
contain an infinite Cauchy sequence.
If a group $G$ acts via isometries on $X$ and $x_0\in X$, then
$d_G(g,h)=d_X(g\cdot x_0,h\cdot x_0)$ defines a proper left-invariant ls-metric on $G$ if and only if there is a neighborhood $U$ of $x_0$ such that the set $\{g\in G\mid g\cdot U\cap U\ne\emptyset\}$ is finite.
\end{Cor}
{\bf Proof. } Suppose there is a neighborhood $U$ of $x_0$ such that the set $\{g\in G\mid g\cdot U\cap U\ne\emptyset\}$ is finite. Notice there is no converging sequence $g_n\cdot x_0\to x_0$
with $g_n$'s being all different.
\par If $d_G$ is proper, then choose any ball $U=B(x_0,r)$ around $x_0$.
Now, $g\cdot U\cap U\ne\emptyset$ means there is $x_g\in U$ so that $d_X(g\cdot x_g,x_0) < r$.
Therefore $d_G(g,1_G)=d_X(g\cdot x_0,x_0)\leq d_X(g\cdot x_0,g\cdot x_g)+
d_X(g\cdot x_g,x_g)+d_X(x_g,x_0)\leq r+2r+r=4r$ and there are only finitely many
such $g$'s.
\hfill $\blacksquare$
\begin{Cor} \label{BasicSvarc-Milnor}
If a group $G$ acts cocompactly and properly via isometries on a proper metric space $X$, then $g\to g\cdot x_0$ induces a coarse equivalence between $G$ and $X$
for all $x_0\in X$.
\end{Cor}
{\bf Proof. } Define $d_G(g,h)=d_X(g\cdot x_0,h\cdot x_0)$ for all $g,h\in G$.
Clearly, $d_G$ is left-invariant. Since action is proper, $d_G$ is
a proper ls-metric. Since action is cocompact, $X$ is within bounded distance from $G\cdot x_0$.
\hfill $\blacksquare$
\begin{Cor}[\v Svarc-Milnor] \label{Svarc-Milnor}
A group
$G$ acting properly and cocompactly via isometries on a length space $X$
is finitely generated and
induces a quasi-isometry equivalence $g\to g\cdot x_0$ for any $x_0\in X$.
\end{Cor}
{\bf Proof. } Consider the proper left-invariant metric $d_G$ induced on $G$ by the action.
The cocompactness of the action implies $G\cdot x_0$ is metrically connected.
So is $(G,d_G)$ and $G$ must be finitely generated.
Both $X$ and a Cayley graph of $G$ are proper geodesic spaces.
Therefore any coarse equivalence between them is a quasi-isometric equivalence.
\hfill $\blacksquare$
\centerline{\bf Final comments.}
Let us point out that \v Svarc-Milnor Lemma \ref{BasicSvarc-Milnor} for non-finitely generated groups
is useful when considering spaces of asymptotic
dimension $0$.
A large scale analog ${\mathcal M}^0$ of 0-dimensional Cantor set is
introduced in~\cite{Dran-Zar}: it is the set of all positive
integers with ternary expression containing 0's and 2's only (with
the metric from ${\mathbb R}_+$):
$$ {\mathcal M}^0=\{\sum\limits_{i=-\infty}^\infty a_i3^i\mid a_i=0,2\} .$$
\begin{Prop}\cite[Theorem 3.11]{Dran-Zar}\label{M0 is universal}
The space ${\mathcal M}^0$ is universal for proper metric spaces of
bounded geometry and of asymptotic dimension zero.
\end{Prop}
\begin{Prop}\label{M0 is equivalent to Z2}
The space ${\mathcal M}^0$ is coarsely equivalent to
$\bigoplus\limits_{i=1}^\infty {\mathbb{Z}}_{2}$.
\end{Prop}
{\bf Proof. } Consider the subset $A=\{\sum\limits_{i=0}^\infty a_i3^i\mid a_i=0,2\}$
of ${\mathcal M}^0$. Notice $ {\mathcal M}^0$ is within bounded distance from $A$, so $A\to {\mathcal M}^0$ is a coarse
equivalence. Also, there is an obvious action of $\bigoplus\limits_{i=1}^\infty {\mathbb{Z}}_{2}$ on $A$ (flipping $a_i=0$ to $2$ or $a_i=2$ to $0$ if the corresponding term
in $\bigoplus\limits_{i=1}^\infty {\mathbb{Z}}_{2}$ is not zero) that is proper
and cocompact.
\hfill $\blacksquare$
Notice any infinite countable group $G$ of asymptotic dimension $0$
is locally finite (see \cite{Smith}). Thus it can be expressed as the union of
a strictly increasing sequence of its finite subgroups $G_1\subset G_2\subset\ldots$
Put $n_{1}=|G_1|$, $n_i=|G_i/G_{i-1}|$ for $i > 1$, and observe (using
\ref{LSUniformForGroups}) that $G$ is coarsely equivalent to $\bigoplus\limits_{i=1}^\infty {\mathbb{Z}}_{n_i}$. We do not know if any two infinite countable groups of asymptotic dimension $0$ are coarsely equivalent.
|
1,116,691,500,481 | arxiv |
\section{Introduction}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\hsize]{imgs/ResultsOfSiamFC}
\caption{Examples of NIR partial face images and detection results. First row shows input images. Left image was captured with iris on the move (IOTM) system, and right image is from public iris dataset CASIA-Iris-M1-S2 \cite{CASIAIrisM1}. Second row shows results of OpenFace2.0 \cite{baltrusaitis2018openface} (face detection + landmark detection). Third row shows results of proposed method. Last two rows show heat maps indicates right and left eye similarity of reference features generated from SiamEDP.}
\label{fig:introExamples}
\end{figure}
Iris and periocular recognition \cite{Daugman}, pupillometry \cite{wilhelm2003clinical,couret2016reliability}, and gaze tracking \cite{cheng2021appearance} are used for identifying individuals and human state estimation from near infrared (NIR) images. These methods require high resolution to capture the fine texture of irises from moving subject. In iris capturing systems, the field of view is limited to the both eye area only due to the constraint of resolution. CASIA-Iris-M1 \cite{CASIAIrisM1} and CASIA-Iris-Distance \cite{casiairisdistance} (examples in Fig. \ref{fig:introExamples}, and Fig. \ref{fig:DLibs}), public datasets for iris recognition, are good examples. These datasets were created using mobile devices and iris imaging systems at a distance (IAAD \cite{nguyen2017long}). These images are partial face images, which include both eyes but not an entire face. The iris-recognition process using these partial face images first detects and classifies a right eye and a left eye respectively, then extract iris regions by segmentation on single eye images, and finally extract features from them for authentication.
In particular, Iris On the Move (IOTM) \cite{matey2006iris} system, an iris authentication system for a walking individual, requires a high frame rate. High frame rate increases the chances of capturing focused irises from a walking individual passing through the narrow depth of field reduced by the constraints of high resolution. Recent IOTM system \cite{zhang2020all} captures in 30 fps with 12M pixels. For real time processing of iris recognition in this system, the processing time for eye detection is required faster than 33 msec. The IOTM system requires a wide horizontal angle of view to expand width of walking pathway. The position of the face in the captured image is greatly shifted due to individual differences in gait and walking position, so the eyes do not always appear in the same position in images (examples in Fig. \ref{fig:introExamples}). A technique to detect each eyes with high speed and positional accuracy from NIR partial face images is expected.
In iris recognition, and certain gaze-tracking methods, precise eye landmark detectors \cite{choi2019accurate,ablavatski2020real,ahmed2021real,lee2020deep} have been proposed for estimating gaze direction and segmentation of the pupil and iris regions. These methods first detect the rectangle face region (bounding box) from the input image and crop the single eye regions from the bounding box using landmark detection. Landmark detection is a method of detecting a set of landmark points representing facial parts such as eyes, nose, and mouth. Finally, iris-landmark estimations and iris segmentations are executed for a single eye region.
Current face-detection \cite{dalal2005histograms,king2015max,deng2020retinaface,bazarevsky2019blazeface,xiang2017joint} and facial-landmark-detection methods \cite{kazemi2014one,guo2019pfld,kartynnik2019real,zadeh2017convolutional} are fast and accurate. These detection methods improved performance of occlusions. These occlusions indicate that a face is shielded by objects such as a mask, or another person's face. In other words, the target image contains objects on the face, which is different from the partial face in which some parts of the face is out of the angle of view. Landmark detection methods does not assume the partial face image, since the input is a fixed resized face region obtained by face detection. Moreover, these detectors are trained using images captured under visible light and including whole face, so the pre-trained model cannot be used for NIR partial face images. Annotation of many landmark points such as facial bounding box, eyes, contours, eyebrows and nose, for a NIR partial-face dataset requires a great deal of effort.
Direct eye-detection methods, such as object-detection methods, have also been proposed \cite{ren2015faster,yolov4}. Generic object detection methods \cite{liu2020deep} extract features from the input image and regress object classes and bounding boxes from the features to detect objects in different classes and different scales. These systems are not fast enough to meet the requirement for the real-time performance of the iris recognition system.
We proposes a fast eye detection method for NIR partial face images that is based on a Siamese network (SiamEDP) and directly detects right and left eye centers. We focused on a fully-convolutional Siamese network \cite{SiamFC} (SiamFC) to accurately obtain the eye center with a lightweight model. The Siamese network extracts features from two kind of images. One is a NIR partial face image as a search image and the other is a single eye image prepared in advance as a reference image. SiamEDP outputs a coarse similarity heat map between the reference feature and the search feature. Classification by similarity using the reference features is expected to reduce training parameters, improve discriminative performance between left and right eyes, and stabilize training. We further extended the two-dimensional convolutional similarity to cosine-margin-based loss \cite{wang2018cosface} to improve the performance. SiamEDP regresses a search feature vector with a highest similarity, and obtain the local fine position of the eye center. Therefore, SiamEDP can detect coarse to fine eye-center positions with high speed and accuracy. We evaluated the accuracy of SiamEDP, and demonstrated the effectiveness of the Siamese network and cosine-margin-based loss. We also compared it with current facial-landmark-detection methods by using public iris-recognition datasets. The results indicate that SiamEDP is faster and more accurate than these two types of methods.
Our main contributions are as follows:
\vspace{-5pt}
\begin{itemize}
\setlength{\itemsep}{3pt}
\setlength{\parskip}{0pt}
\item We propose an eye detector as a pre-process of iris segmentation for partially cropped NIR face image.
\item We apply SiamFC\cite{SiamFC} for object tracking to eye detection with a light weight network.
\item Using cosine-margin-based loss (CosFace\cite{wang2018cosface}) on training improves accuracy of detection.
\item Coarse to fine approach improves positional accuracy.
\item Our method reduces the cost of annotation less than facial-landmark detection. Only two landmarks of eye center are required for a single face.
\end{itemize}
\section{Related Work}
\subsection{Face and Landmark Based Eye Detection}
Eye-center or pupil-center detection methods have been proposed for gaze tracking \cite{cheng2021appearance}. These methods first extract the face region using face detection, then resize the region to a fixed size, extract the single eye regions using facial-landmark detection, and execute high-precision position estimation, or directly detect the single eye region from the face region. Therefore, the accuracy of these methods depends on the accuracy of the underlying face region and facial-landmark detection.
Face detection methods predict the facial bounding box. Early methods were mainly based on the classifiers using hand-crafted features extracted from an image \cite{violajones2001}. After the breakthrough of the CNN, CNN based models were proposed, such as Cascade-CNN, Faster-RCNN, and single-Shot Detection \cite{Li_2015_CVPR,Sun_2017,Zhang_2017,bazarevsky2019blazeface,deng2020retinaface}. To improve detection accuracy, several studies focused on the loss function or multi-task learning \cite{Ranjan2016HyperFace,deng2020retinaface}. Dent et al. \cite{deng2020retinaface} proposed RetinaFace which predicts facial bounding box by leveraging extra-supervised and self-supervised multi-task learning and showed significant improvement in accuracy. One of the challenges in face detection is occlusion, i.e, the lack of facial information due to obstacles or masks. Chen et al. \cite{Chen2018AOFD} proposed the Occlusion-aware Face Detector (AOFD) which detects faces with few exposed facial landmarks using adversarial training strategy.
The above face-detection methods use annotated facial-image datasets that include images captured under visible light. Several visible-light face datasets are publicly available. For example, WIDER FACE \cite{Yang2016WIDERFACE} includes more than 30,000 images and about 4 million labeled faces. There are several other datasets containing hundreds to tens of thousands of labeled faces. The majority of images are wide-angle shots of the face \cite{fddbTech,YAN2014790,Yang2015,Nada2018,Cao2018VGGFace2}.
Several of eye-detector and eye-center estimation methods detect eyes from resized facial bounding boxes. Some methods \cite{ahmed2021real,leo2013unsupervised} use statistical facial-landmark information for cropping out single eyes before an eye segmentation process in real time. The other method \cite{putro2020fast} uses a face region resized to $128 \times 128$ pixels before a bounding-box estimation of eyes.
Facial-landmark-detection methods detect key points that represent facial landmarks from facial bounding boxes. Early landmark-detection methods were mainly based on fitting a deformable face mesh by using statistical methods \cite{WANG201850}. V. Kazemi et al, proposed ensemble of regression trees which is based on gradient boosting initialized with the mean shape of landmarks \cite{kazemi2014one}. They achieved high speed and high accuracy in detecting 68 points from frontal-face images with less occlusion. CNN based landmark detectors are also proposed, showing significant improvement in in-the-wild facial-landmark detection \cite{sun2013deep,zhou2013extensive,Chandran2020CVPR,Zhang2016joint,Feng2018CVPR}.
The models of these methods are typically evaluated with 68 points using annotated visible-light-image datasets. Several datasets \cite{Sagonas2013_300Face,AFLW2011,Burgos2013COFW,Wu2018WFLW} are available. Each dataset includes several thousand of annotated images.
For example, the 300W \cite{Sagonas2013_300Face} dataset contains 4437 images with 68 landmark annotations. AFLW \cite{AFLW2011} contains 24386 images with 21 landmark annotations, COFW \cite{Burgos2013COFW} contains 1852 images with 29 landmark annotations, and WFLW \cite{Wu2018WFLW} have 10000 images with 98 landmark annotations.
Several iris-landmark-detection methods \cite{choi2019accurate,lee2020deep,ablavatski2020real} uses cropped single eye regions from facial-landmark-detection results. Choi et al. \cite{choi2019accurate} proposed segmentation based eye center estimation. They cut out a rectangle region using the landmarks of the eye socket and eye corner from 68 points of landmarks \cite{kazemi2014one} before pupil segmentation. Ablabatski et al. \cite{ablavatski2020real} detects 5 points of iris landmarks from a $64 \times 64$ single eye region from facial-landmark detection \cite{bazarevsky2019blazeface} results.
Our assumption of NIR partial face image data is images under intense NIR illumination, such as CASIA-Iris-Distance \cite{casiairisdistance} and CASIA-Iris-M1 dataset \cite{CASIAIrisM1}. Since the modality of these images is different from the visible light data set, the pre-trained models of the above detection methods are insufficient to provide accuracy. In addition, there are currently hardly any public datasets with annotations for near-infrared light face images. Therefore, it is necessary to annotate facial bounding box and landmark annotations on existing datasets. However, these annotations are very costly for the task of eye detection.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\hsize]{imgs/SFCDreg2022}
\caption{
Framework of SiamEDP. Reference and search images are fed into same feature extractor. Heat map is generated by calculating spatial cosine similarity between extracted features. Networks are optimized by the binary cross entropy (BCE) loss with CosFace as shown in Eq. \ref{eq:loss}. Eye feature vector is extracted from similarity heat map, and regressor estimates fine position in eye feature. Blue region shows testing framework. Detector computes spatial cosine similarity between extracted search and pre-extracted reference features. Reference feature is saved as trained parameter.
}
\label{fig:flow}
\end{figure*}
\subsection{Direct Eye Detection Methods}
Methods have been proposed to detect eyes directly from input images using CNN-based object detection \cite{ren2015faster,redmon2018yolov3} without face and landmark detection. These methods, called generic object-detection methods \cite{liu2020deep}, detect and classify objects of different scales and classes.Faster R-CNN \cite{ren2015faster} is one such method. It generates a feature map from an input image by using convolutional layers and estimates regions of interest (ROIs) with high objectness using a region proposal network. Then, a fully connected layer outputs the object class and bounding box of the ROIs. Nasaif et al. \cite{nsaif2021frcnn} proposed FRCNN-GNB for eye detection. It uses Faster R-CNN \cite{ren2015faster} to detect the initial eye regions then applies Gabor filters and a naïve Bayes model for the final eye detection. Generic object-detection methods tend to have weak discriminability against similar classes such as right and left eye class, and tend to increase processing time due to the large size of the backbone to handle multiple classes and multiple scales.
We focused on object tracking methods for fast detection. Methods based on correlation filters or Siamese networks have been proposed and are often used in the Visual Object Tracking (VOT) challenge \cite{VOT_TPAMI}. Certain eye detectors using correlation filters have been proposed \cite{bolme2009average,araujo2014fast}. Araujo et al. \cite{araujo2014fast} proposed a correlation-filter-based method in the pixel domain. They use cosine similarity during training to avoid the values outside the interval $[0, 1]$. On the other hand, there is no eye detection method using Siamese network. The well-known object tracking method, SiamFC\cite{SiamFC} has achieved high performance and speed in the VOT challenge. It extracts features from input reference and search images by using common networks. A similarity score map is then generated using the correlation of the extracted reference and search features. SiamFC detects the position of a reference object in a search image by thresholding the similarity heat map.
\section{Proposed Method}
\subsection{Basic Idea}
We consider SiamFC \cite{SiamFC} as two-class classifier that determines whether a subregion of a search feature is a tracking target. We considered applying cosine-margin-based loss, which is recently used in metric learning. It is a method for showing higher classification performance by embedding features into a hypersphere and providing a margin for the same class on training. We apply CosFace \cite{wang2018cosface} to 2D convolution to improve classification performance. The position of eye in the heat map is rough because the resolution of heat map is reduced by the stride setting of feature extraction CNN architecture. Therefore, we considered a method to obtain a fine eye position using the coarsely obtained eye positions. We designed the spatial size of the search feature to be equal to the size of the heat map, and there is a feature vector corresponding to one pixel in the heat map. We assume a feature vector with the highest similarity includes the information of eye, and directly obtain the eye center coordinates by regressing on the feature vector. This enables high-speed and highly accurate eye detection.
\input{Tables/CNNNetworkArchitecture}
\subsection{Framework}
We define reference image as $I_{ref}$ and search image as $I_{srch}$. The same CNN feature extractors $\phi$ calculate features from these two images. The features of $I_{ref}$ and $I_{srch}$ are defined as $f_{ref} := \phi(I_{ref}) \in \mathbb{R}^{m \times n \times c}$ and $F_{srch} := \phi(I_{srch}) \in \mathbb{R}^{M \times N \times c}$, respectively. The numbers of channels $c$ in $F_{srch}$ and $f_{ref}$ are equal, but each has a different spatial size. Let $F_{srch}[u] \in \mathbb{R}^{m \times n \times c}$ denote the partial region of size $m \times n \times c$ at spatial position $u \in U$ of feature $F_{srch}$.
Figure \ref{fig:flow} shows the flow of SiamEDP. In training, $I_{ref}$ is fixed to a single average image of eyes, and the same image is flipped between the left and right eyes. We assume that $I_{srch}$ always includes both eyes and detection tasks for the left and right eyes are learned, respectively. In testing, the same $I_{ref}$ as in training is used to obtain both features $f_{refR}$ and $f_{refL}$ in advance. We define Q as the heat map generated by the cosine similarity between $F_{srch}$ and $f_{ref}$. We define $\mathbf{x'}$ as the spatial position where $argmax{Q}$, and extract the feature vector $f_{eye}$ such that $F_{srch}[\mathbf{x'}]\in \mathbb{R}^{1 \times 1 \times c}$. The local position $d\mathbf{x}$ of the eye center in $f_{eye}$ is obtained by regression on $f_{eye}$ and the final position is estimated using $\mathbf{x'}$ and $d\mathbf{x}$.
\subsection{Coarse to Fine Eye Center Estimation}
SiamEDP calculates the kernel-wise cosine similarity between $F_{srch}[u]$ and $f_{ref}$, and obtains $Q$.
The edge of $F_{srch}$ is replicated before calculating the cosine similarity so that $Q$ has the same spatial size ($M, N$) as $F_{srch}$. The cosine similarity at spatial position $u$ on $Q$ can be calculated as
\begin{equation}
\label{eq:map}
Q[u] = \cos{\theta_u}= \frac{f_{ref} \cdot F_{srch}[u]}{||f_{ref}||_2||F_{srch}[u]||_2} .
\end{equation}
The spatial position$\mathbf{x}'$ is highest in intensity in $Q$ as $\mathbf{x}' = [x',y']^T = argmax{Q}$. The feature $f_{eye}$ representing the eye is denoted as
\begin{equation}
\label{eq:fea}
f_{eye} = F_{srch}[\mathbf{x}'] \in \mathbb{R}^{c}.
\end{equation}
A local eye center position $d\mathbf{x} = [dx, dy]$ in the feature block $f_{eye}$ is obtained by linear regression using weight parameters $\mathbf{w} \in \mathbb{R}^{2 \times c}$ as
\begin{equation}
\label{eq:reg}
d\mathbf{x} =
\begin{bmatrix}
dx \\
dy
\end{bmatrix}
= \mathbf{w}f.
\end{equation}
Using $\mathbf{x}'$, $\mathbf{x}$ and the size ratio of the input image to the output similarity map $\alpha$, the final eye position coordinates $\mathbf{x}$ for the input image are obtained by $\mathbf{x} = \alpha(\mathbf{x}' + d\mathbf{x})$.
\subsection{Loss Function}
We define two types of loss functions. One is the loss $L_s$ for the similarity map, and the other is the loss $L_p$ for the eye center coordinates. We design a loss function on the basis of binary cross entropy (BCE) with CosFace \cite{wang2018cosface} to accurately classify one side of the eye (two classes) from others. CosFace has a margin parameter $m$ applied only to the positive label locations and a scale parameter $s$. These parameters enables cosine decision margin between classes. CosFace is expected to separate the feature distribution of the right-eye class from that of the left-eye class.
The loss function $L_s$ based on the BCE is given by
\begin{eqnarray}
\label{eq:loss}
L_s = - \frac{1}{|U|} \sum_{u \in U} \{y_u {\rm log}\frac{e^{s(Q_u - m)}}{ e^{s(Q_u - m)} + \Sigma_{t \neq u}{e^{s Q_t} }} \nonumber \\
+ (1-y_u) {\rm log}\frac{e^{s Q_u}}{ e^{s(Q_u - m)} + \Sigma_{t \neq u}{ e^{s Q_t} }} \}
.
\end{eqnarray}
The $L_p$ for fine eye center position is the L1 norm expressed as
\begin{equation}
\label{eq:lp}
L_p = \Sigma_{p \in {\mathbf x}}b\cdot|p-\hat{p}|\cdot \frac{1}{\alpha},
\end{equation}
where $\alpha$ denotes the scale ratio between the input image and similarity heat map. $b$ denotes a mask to avoid calculating regression losses for features without eye information due to incorrect heat-map predictions and defined as follows.
\begin{equation}
\label{eq:mask}
b =
\left \{
\begin{aligned}
1, \ && if \ ||\mathbf{x}'-\mathbf{\hat{x}}'||_2<2
\\
0, \ && otherwize\
\end{aligned}
\right.
,
\end{equation}
The above $L_s$ and $L_p$ are combined into a loss function $L$ using the weights $\beta,\gamma$ as follows. We use a sum of loss calculated from the right and left eyes.
\begin{equation}
L = \beta L_s + \gamma L_p.
\end{equation}
\input{Tables/DataSet}
\input{Tables/abrationStudyTable.tex}
\input{Tables/landmarkDetectionMethods.tex}
\section{Experiments and Results}
We present three experiments we conducted evaluate the performance of SiamEDP. The first was an ablation study to confirm the contributions of SiamEDP (SiamFC, cosine similarity and CosFace). The second was a comparison between SiamEDP and current facial-landmark-detection methods. The third was a comparison between SiamEDP and generic object-detection methods. We considered SiamEDP is preprocessing for iris and pupil segmentations, so we did not compare segmentation methods.
We used a single network architecture for all three experiments. The base network for SiamFC was modified from ResNet \cite{he2016deep} as shown in Table \ref{tb:network}. The differences from the original ResNet are single channel input and the number of layers. Each layer contained convolution, batch normalization and rectified-linear-unit (ReLU) activation. For model training, we used stochastic gradient descent (SGD) as the optimizer. The learning rate was 0.1 on the first epoch and switched to 0.01 from the second epoch. The weight decay was 0.0001. The batch size was 16 for each iteration, and the total number of iterations was 60,000.
We used four iris datasets, CASIA-Iris-Distance \cite{casiairisdistance}, CASIA-Iris-M1-S1\cite{CASIAIrisM1}, CASIA-Iris-M1-S2\cite{CASIAIrisM1}, and CASIA-Iris-M1-S3 \cite{CASIAIrisM1} described in Table. \ref{tb:Dataset}. We manually annotated eye center points on all images. The input image was scaled down from the original size, with the resolution of the iris diameter at about 10 pixels. Therefore, the images of CASIA-Iris-M1-S1 were rescaled to 1/10, the others were rescaled to 1/16. Parameter $\alpha$ is the scale ratio between the resized input image and similarity heat map and is $8$ because of the network stride in Table \ref{tb:network}. The ground truth heat map is a binary map labelled on the eye-center pixel and its four neighboring pixels.
The evaluation metric was the root-mean-square error (RMSE) of the eye-center position or the relative error considering both eyes and expressed as
\begin{equation}
\label{eq:eyeError}
E = \frac{max{(d_l,d_r)}}{d},
\end{equation}
where $d$ is the Euclidean distance between the left- and right-eye centers, where $d_l$ and $d_r$ are the RMSEs of the right- and left-eye centers, respectively.
\input{Tables/landmarkrmseTable.tex}
\subsection{Ablation Study}
\label{sec:AS}
In this experiment, we evaluated the performance of selecting reference images, effectiveness of CosFace, and that of the Siamese network. We selected CASIA-Iris-Distance and CASIA-M1-S1 as training datasets and used 60\% of each training data set for training.
We first evaluated the effectiveness of the CosFace parameters $(s,m)$ and feature normalization and estimated suitable parameters. For training without reference images, we set the best parameters in the evaluation of CosFace. We trained all models three times, each with the same parameters and same training data, then averaged the results. The evaluation was done using CASIA-M1-S2 and CASIA-M1-S3 datasets, which were not used for training.
We then evaluated the following three methods for selecting reference images:
\begin{enumerate}
\item {\bf random}. 10\% of the images in the training dataset is selected as reference image data, and eyes were randomly selected from them and used as reference images during training. In the evaluation, we used the average of the reference features in a batch in the end of training as a reference feature.
\item {\bf fixed avg image}. The histogram-stretched image averaged over 128 reference images is used as the reference image. The reference average image is fixed during the training and testing.
\item {\bf without ref image}. A heat map is learned directly from a search feature without using the reference image. Channel 2 and 1×1 convolution is executed on the search features to directly output the right- and left-eye heat maps. We applied CosFace to this method and selected parameters from the other methods.
\end{enumerate}
The results are listed in Table \ref{tb:abStudy}. There was no significant difference between random reference and avg reference, and both methods significantly decreased in accuracy when the features were not normalized. We also found significant performance differences depending on the presence or absence of $s$ (same as NormFace \cite{wang2017normface}). Without reference images, the learning was not stable, resulting in lower accuracy.
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\hsize]{imgs/SiamAndDLib}
\caption{Results of Dlib (use CNN based detection and 68 points detection) and SiamEDP. The left three columns are the results using the images from CASIA-Iris-M1-S3 \cite{CASIAIrisM1}. Right three columns are results using the images from CASIA-Iris-M1-S2 \cite{CASIAIrisM1}.}
\label{fig:DLibs}
\end{figure*}
\begin{figure}[h]
\centering
\includegraphics[width=0.85\hsize]{imgs/ObjF1scores}
\caption{Comparisons with generic object detection methods. We evaluate F1 scores of each eyes per RMSE.}
\label{fig:objF1Scores}
\end{figure}
\subsection{Landmark Detection}
\label{sec:LD}
In the next experiment, we evaluated pre-trained face-detection and facial-landmark-detection models, which are publicly available as software development kits (SDKs) for face detection and recognition, and compared them using relative eye error and RMSE using eye-center points or the average of landmarks around the eye. We evaluated possible combinations of the methods in each of the four SDKs: Dlib \cite{king2009dlib}, FaceXZoo \cite{wang2021facex}, Mediapipe \cite{lugaresi2019mediapipe}, and OpenFace \cite{baltrusaitis2018openface}. The specifications of most SDKs first requires calculating the bounding box by face detection, then inputing the image and bounding box region to landmark detection. Since many SDKs did not detect faces in the CASIA datasets and did not output bounding boxes from detection modules, we recorded the success rate of detecting at least one bounding box and carried out landmark detection when a bounding box was detected. As an exception, when the evaluation using Dlib failed to detect faces, the bounding box was set as the entire image area and input to landmark detection because we assumed could be optimized from the initial points on the partial face image. Several facial-landmark-detection methods output only landmarks around the eyes, so the average value of the points around the eyes is output as the eye center. The combinations of each method and calculate eye-center positions are listed in Table \ref{tb:SDK}.
The results of CASIA-Iris-M1-S2 are listed in Table \ref{tb:landmarkss2} and those of CASIA-Iris-M1-S3 are listed in Table \ref{tb:landmarkss3}. The accuracy of face detection for CASIA-Iris-M1-S2 is decreased because most of the images are partial images of the face (from the nose up). Since landmark detection with Dlib involves optimization by placing initial points, it can achieve 90\% of images with RMSE less than 15 to some extent even if applied directly to the image but requires a large margin when cropped to a single eye. CASIA-Iris-M1-S3 includes the entire face, which has shown higher performance on SDKs than CASIA-Iris-M1-S2, but it is less accurate than SiamEDP in Eye relative Error and RMSE. Fig. \ref{fig:DLibs} shows detection examples with Dlib.
We evaluated processing times averaged 10 times in a CPU environment on CASIA-Iris-M1-S3 scaled down to 120 $\times$ 120 pixels. SiamEDP was 14 msec, FaceXZoo was 32.5 msec/33 msec depending if detection engine for mask was used, mediapipe facial-landmark detection was 14 msec and 35 msec with iris-landmark detection, Dlib with CNN face detection is 219 msec with 68-point detection, 218 msec with 5-point detection, Dlib with not-CNN-based detection is 10 msec with 68-point detection and 9 msec with 10 msec 5-point detection. We were unable to measure exact execution times when using OpenFace because we evaluated using a built executable file. Therefore, SiamEDP fast and the most accurate.
\subsection{Object Detection}
\label{sec:OD}
We compared SiamEDP with the major generic object detection methods FRCNN \cite{ren2015faster} and YOLOv5 (a PyTorch implementation of YOLOv4 \cite{yolov4}). We re-trained the YOLO and FRCNN models to detect both eyes from partial face images. The YOLO models were trained in 70 epochs, while FRCNN was trained in 5 epochs because the training time with FRCNN is much longer than the other methods.
The training datasets included CASIA-Iris-Distance and CASIA-Iris-M1-S1. We trained 50\% of the images from the training domain datasets. The other 50\% of the training domain datasets was used in the evaluation (trained domain). The un-training domain datasets included CASIA-Iris-M1-S2 and CASIA-Iris-M1-S3. All un-training domain datasets (untrained domain) were used for evaluation. We labeled a 16 $\times$ 32 pixels bounding box centered on the eye center with two classes, i.e., right eye and left eye, for YOLO and FRCNN.
We compared the three methods using the F1 score per RMSE to evaluate the discriminative performance between right- and left-eye classes. Since it assumed with SiamEDP that both eyes are always included in an input image, a false positive (FP) is always equal to a false negative (FN). This assumption makes precision equal to the recall for SiamEDP. In generic object detection, the number of detection targets is unlimited, so YOLO and FRCNN may detect more targets than expected (the FN may differ from the FP). Thus, we needed to evaluate them on the basis of the F1 score instead of accuracy.
The results are shown in Figure \ref{fig:objF1Scores}.
The results of YOLO indicate that the F1 scores converged around 0.7, indicating low discriminability between the right and left eyes. FRCNN had the highest accuracy in regions where RMSE was small. The results of SiamEDP are converged to the highest F1 score. The execution speed of SiamEDP was 11m sec, that of FRCNN was 1970 msec, and that of YOLO was 87 msec on average of 10 CPU runs for a $123 \times 96$ pixels of image, with SiamEDP having the best results.
\section{Conclusion}
We proposed a fast eye-detection method that is based on a Siamese network for NIR partial face images. By using the Siamese network and cosine-margin-based loss function, a shallow network was able to detect the left- and right-eye centers with high accuracy. Experimental results indicate that the Siamese network and CosFace is effective in achieving high-speed and high-accuracy detecion in a CPU environment compared with current facial-landmark-detection methods and generic object-detection methods. |
1,116,691,500,482 | arxiv | \section{Introduction}
The resolution limit of conventional microscopes is determined by Rayleigh's criterion \cite{Rayleigh}. In the last few decades, various techniques have been invented to circumvent Rayleigh's limit by changing the imaging conditions. Such techniques utilize nonlinear optical properties of the object \cite{2Photon, STED}, near-field optics \cite{SNOM, durig} or work with photo-switchable samples \cite{STORM, PALM}. However, recently it was found that sub-Rayleigh resolution can be achieved for certain microscopy-related tasks without resorting to nonlinear optics or near-field interactions. Such is the case, for example, for estimating the distance separating two point sources \cite{Tsang, Sliver, Ranjith, Lupo}.
The idea of this new approach was to count photons in the Hermite-Gaussian or transverse-electromagnetic (TEM) modes $\{\rm{TEM}_{0q}; q = 0,1, \ldots\}$ in the image plane \cite{Tsang}. The precision of this estimation has been calculated as the inverse of the Fisher information (FI) in accordance with the Cram\'er-Rao bound of classical statistics \cite{Bos, HarryL}. Remarkably, this FI is independent of the separation distance, in contrast to direct imaging in which the FI tends to zero in the limit of low separations. Moreover, it was shown that this method is quantum optimal, i.e. it permits extracting the maximum possible FI from each photon available to the observer \cite{Tsang}. Inspired by this analysis, a number of groups around the world demonstrated proof-of-principle experiments to achieve super-resolution \cite{Sheng, Tham, Paur, Yang}.
However, direct implementation of the scheme of Ref.~\cite{Tsang} requires a setup for spatial mode filtering in the Hermite-Gaussian basis, which is a challenge \cite{Lam2010,Vamivakas2016}. It is thus tempting to use a homodyne or heterodyne detector instead of a mode filter, taking advantage of such a detector's sensitivity to the optical signal only in the mode that matches that of the local oscillator, which, in turn, can be readily prepared in any TEM mode by using spatial light modulators or optical cavities. Ref. \cite{Yang} demonstrated the viability of this method for achieving sub-Rayleigh resolution. The first-order mode $\rm{TEM}_{01}$ was chosen as the local oscillator in the experiments of Ref. \cite{Yang}. As shown in Ref. \cite{Ranjith}, this mode contains most of the information on the source separation in the sub-Rayleigh regime. Consequently, we focus on dyne measurements of this mode in this paper.
Because homodyne and heterodyne detection are physically different from direct photon counting, the FI associated with these measurements needs to be evaluated independently. Ref.~\cite{tsanghomo} argues that homodyne detection offers no advantage with respect to direct imaging for weak thermal light because of the shot noise. However, there has been no similar analysis for arbitrary thermal sources. Here we show that homodyne and heterodyne detection do possess an advantage over direct imaging for estimating separations well below the Rayleigh limit when the average received photon number per-source of the thermal state exceeds two and four respectively.
\section{Concepts}
\subsection{Displacement and TEM$_{01}$ mode}
To illustrate the mode transformation and detection process, we begin with a brief description of homodyne detection in TEM$_{01}$ using classical optics. A complete quantum optical derivation that includes the effect of shot noise is given in the later sections and appendices. Heterodyne detection is closely related, see below. We work in a single transverse dimension and assume quasi-monochromatic light in the paraxial approximation. We also assume a translationally invariant imaging system with a Gaussian point spread function. With such assumptions, a pointlike light source located at the optical axis of the objective lens is imaged in the TEM$_{00}$ mode. When the light source is displaced by $\pm d$, the beam amplitude in the image plane is
\begin{equation}
\alpha E_0(x\pm d)=\alpha\left(\frac{1}{2 \pi \sigma^2}\right)^{1/4}e^{-\left(\frac{x\pm d}{2\sigma}\right)^2},\label{profile}
\end{equation}
where $\alpha$ is the amplitude, $E_0(x)$ is the normalized amplitude profile of TEM$_{00}$ and $\sigma$ is the beam width.
For small displacement, this can be approximated by Taylor expansion
\begin{equation}
\alpha E_0(x\pm d)\approx\alpha E_0(x)\pm\alpha d\cdot E_0'(x)=\alpha E_0(x)\mp \frac{d}{2\sigma}\alpha E_1(x),\label{amplitude}
\end{equation}
where $E_0'(x)$ is the derivative of $E_0(x)$ with respect to $x$ and $E_1(x)$ is the normalized amplitude profile of TEM$_{01}$. This means that, when the source becomes displaced, TEM$_{01}$ acquires a nonzero amplitude that is $\mp \frac d{2\sigma}$ of the amplitude of TEM$_{00}$, and a nonzero power corresponding to $d^2/(4\sigma^2)$ of that in TEM$_{00}$. We detect the image by homodyne detection with the local oscillator prepared in TEM$_{01}$, resulting in a photocurrent proportional to the displacement $d$ (with added shot noise, which is included later in our quantum formalism of analysis). A null measurement of the displacement is thereby achieved, in contrast to direct imaging, in which a signal in the form of a certain intensity distribution is present for all displacements.
\subsection{Fisher information (FI)}
As is standard in astronomy \cite{EricD, JonasZ}, single-molecule microscopy \cite{JerryChao}, asymptotic statistics \cite{AW}, and engineering statistics \cite{HarryL}, we adopt here the FI as the sensitivity measure. Given a probability distribution pr$(Y|\lambda)$ of measurement outcome $Y$ as a function of parameter $\lambda$, the FI is defined as
\begin{equation}
F_\lambda=\left<\left(\frac\partial{\partial\lambda}\log \text{pr}(Y|\lambda)\right)^2\right>
\end{equation}
where $\avg\cdot$ represents statistical average.
The inverse of FI gives the Cram\'er-Rao bound, which is a lower bound on the mean-square error of any unbiased estimator. The bound can be attained in the asymptotic limit of infinite repetitions by the maximum-likelihood estimator \cite{AW,HarryL}. Although a biased estimator can violate the Cram\'er-Rao bound for limited repetitions \cite{Tham,MT}, one can generalize the bound for any biased or unbiased estimator by adopting a modified error criterion from the Bayesian or minimax perspective \cite{MT}. In particular, the Bayesian Cram\'er-Rao bound by Sch\"utzenberger \cite{MP} and Van Trees \cite{HarryL,MT,RichardD} and the local asymptotic minimax theorem by H\'ajek and Le Cam \cite{AW,RichardD} are valid for any biased or unbiased estimator, and both depend on the FI.
For the imaging problem, the probability distribution and therefore the FI depend on the optical measurement method. In this paper, we compare the FI of three measurements---homodyne detection, heterodyne detection and direct imaging---of the light on the image plane from two thermal point sources. Let us note that the quantum Fisher information computed in Refs.\cite{Tsang,Ranjith,Lupo} is an upper bound on the FI for any measurement allowed by quantum mechanics, but otherwise outside the scope of this paper.
\section{Measuring the displacement of a single source}
\subsection{Coherent source}
The noise properties of homodyne and heterodyne detection of coherent or thermal fields can be discussed using either a semiclassical or quantum-optical formalism. Since both approaches give exactly the same quantitative results\cite{Shapiro}, the choice of formalism is a matter of taste and familiarity. Here, we use the quantum-optical formalism to make explicitly sure that our measurement models agree with quantum mechanics.
In order to introduce our approach for calculating the per-photon FI, we first consider a single coherent source. As is evident from Eq. (\ref{amplitude}), a coherent state $\ket{\alpha}$ in TEM$_{00}$ displaced by $\pm d$ is approximately equivalent to the direct product
\begin{equation}\label{alphapm}
\ket{\alpha}_\pm=\ket{\alpha}_0\otimes\ket{\mp\frac{d}{2\sigma}\alpha}_1,
\end{equation}
where the subscripts 0 and 1 label the two lowest order TEM modes centered on the optical axis of the lens. A full quantum optical analysis leading to \eeqref{alphapm} is given in Appendix A. Importantly, displacements in opposite directions give rise to opposite amplitudes of the TEM$_{01}$ component because of the antisymmetric shape of that mode.
Without loss of generality, we assume $\alpha$ to be real. The homodyne detector will measure the probability distribution of the quadrature $X$ in the state $\ket{\mp\frac{d}{2\sigma}\alpha}$, which is given by \cite{Leonhardt}
\begin{equation}
\pr_\alpha(X|d)=\bigg|\bigg<X\bigg|\mp\frac{d}{2\sigma}\alpha\bigg>\bigg|^2=\frac{1}{\sqrt\pi}\text{exp}\Bigg[{-\left(X\pm\frac{d}{2\sigma}\sqrt{2}\alpha\right)^2}\Bigg].
\end{equation}
A single quadrature measurement yields a sample of this distribution, from which the displacement $d$ can be estimated. The FI is given by
\begin{equation}
\begin{split}
F^{(\alpha)}(d)&=\left<\left(\frac{\partial}{\partial d} \log{\pr_\alpha(X|d)}\right)^2\right>
\\&=\intinf\left(\frac{\partial}{\partial d} \log{\pr_\alpha(X|d)}\right)^2\pr_\alpha(X|d)\dd X=\frac{\alpha^2}{\sigma^2}.
\end{split}
\end{equation}
For the coherent state, the average photon number $N=\alpha^2$, so the per-photon FI is $F^{(\alpha)}_1(d)=1/\sigma^2$. We notice that Ref. \cite{Hsu} also analyzes the performance of homodyne detection with TEM$_{01}$ mode to estimate the displacement of a single coherent light source by calculating the quantum noise limited sensitivity.
\subsection{Thermal source}
Let us now consider a single thermal source. If the average photon number of the original thermal state is $N$, then we have a thermal state with average photon number $\frac{d^2}{4\sigma^2}N$ in TEM$_{01}$. This follows from the fact that linear mode transformations (beam splitters) map thermal states into (in general correlated) states, each single mode of which is in a thermal state. It can also be verified more formally using the Sudarshan-Glauber $P$-representation, as is shown in Appendix B.
A thermal state with average photon number $N$ can be described by the Wigner function \cite{Leonhardt}
\begin{equation}
W(X,P)=\frac{1}{\pi (2N+1)}\exp\left[-\frac{X^2+P^2}{2N+1}\right].
\end{equation}
For a single thermal light source displaced by $d$, the Wigner function of the thermal state in TEM$_{01}$ is therefore
\begin{equation}
W^{(1)}_{01}(X,P)=\frac{1}{\pi \left(\frac{d^2}{2\sigma^2}N+1\right)}\text{exp}\left[-{\frac{X^2+P^2}{\frac{d^2}{2\sigma^2}N+1}}\right].
\end{equation}
Using homodyne detection, we obtain the distribution of quadrature $X$ in TEM$_{01}$\cite{Leonhardt}
\begin{equation}
\pr_{01}^{(1)}(X|d)=\frac{1}{\sqrt{\pi\left(\frac{d^2}{2\sigma^2}N+1\right)} }\exp\left[-\frac{X^2}{\frac{d^2}{2\sigma^2}N+1}\right].
\end{equation}
The width of this Gaussian distribution depends on $d$, and hence a single sample thereof permits inferring this parameter. We find the FI for this inference to be
\begin{equation}
F^{(1)}_{N}(d)=\left<\left(\frac{\partial}{\partial d} \log{\pr_{01}^{(1)}(X|d)}\right)^2\right>=\frac{2d^2N^2}{(d^2 N+2\sigma^2)^2},
\label{homFI}
\end{equation}
and the per-photon FI is
\begin{equation}
F^{(1)}_{1}(d)=\frac{1}{N} F^{(1)}_{N}(d)=\frac{2d^2N}{(d^2 N+2\sigma^2)^2}.
\end{equation}
An important observation we can make here is that $F^{(1)}_1(d)$ depends on $N$, i.e.~the FI is not additive with respect to the number of incoming photons. For example, for low $N$, $F^{(1)}_1(d)\approx d^2N/2\sigma^4$, which means that performing a single measurement of $d$ on a mode with $N$ photons gives a higher precision than two separate measurements on a mode with $N/2$ photons.
In practice it is often advantageous to use heterodyne rather than homodyne detection (i.e. a local oscillator with a slightly different frequency) in order to reduce flicker noise \cite{Yang}. Heterodyne detection is formally equivalent to mixing the input light with vacuum on a 50/50 beam splitter, followed by a homodyne detection of orthogonal quadratures in the two output modes \cite{Wiseman}. The overall FI for heterodyne detection can be obtained from that for homodyne detection by the following steps: replace $N$ by $N/2$, and multiply by 2 to take into account the fact that two quadratures with independent statistics are measured. The per-photon FI is still obtained by dividing by $N$. Since the numerator in Eq. (\ref{homFI}) is quadratic in $N$, this means that the per-photon FI for heterodyne detection can be obtained from that for homodyne detection simply by replacing $N$ with $N/2$.
\section{Measuring the separation of two thermal sources}
\subsection{Homodyne and heterodyne detection}
In practice, we are most interested in measuring the distance between two point light sources (e.g. stars) separated below the Rayleigh limit, for which the direct imaging approach offers reduced precision \cite{Tsang}. For two thermal sources each with average photon number $N$ and displaced by $\pm d$, the average photon number detected in TEM$_{01}$ is the sum of the photon numbers from each state, because the random phase between the sources does not lead to intereference when we sum up the photon numbers. From the results of Sec. III.B, the $\rm{TEM}_{01}$ mode is in a thermal state of average photon number $\frac{d^2}{2\sigma^2}N$.
We can then perform the same calculation as in the previous section. In this case, the Wigner function for the light in TEM$_{01}$ is
\begin{equation}
W^{(2)}_{01}(X,P)=\frac{1}{\pi \left(\frac{\theta^2}{4\sigma^2}N+1\right)}\text{exp}\left[{-(X^2+P^2)\bigg/\left(\frac{\theta^2}{4\sigma^2}N+1\right)}\right],
\end{equation}
where we choose to work with $\theta=2 d$, the separation of the light sources. This corresponds to the distribution of the $X$ quadrature
\begin{equation}\label{pr2S}
\pr_{01}^{(2)}(X|\theta)=\frac{1}{\sqrt{\pi\left(\frac{\theta^2}{4\sigma^2}N+1\right)} }\text{exp}\left[{-X^2\bigg/\left(\frac{\theta^2}{4\sigma^2}N+1\right)}\right],
\end{equation}
and the per-photon FI
\begin{equation}\label{F12S}
F^{(2)}_1(\theta)=\frac{\theta ^2 N }{(\theta^2 N+4\sigma^2)^2}.
\end{equation}
Following the same arguments as in the previous section, the per-photon FI for heterodyne detection can be obtained by replacing $N$ with $N/2$ in Eq. (\ref{F12S}).
As a side remark, since none of the calculations depend on the two sources being of equal strength, the total FI under either detection method for two sources of unequal strengths $N_1$ and $N_2$ can be obtained by replacing $N$ with $(N_1+N_2)/2$.
\subsection{Direct imaging}
We now evaluate the FI for spatially-resolved direct imaging of two incoherent thermal sources. An exact expression for this quantity for arbitrary source strengths is unknown and appears to be difficult to obtain. We can, however, approximate it by noting that, in practice, the photon counts on each pixel are integrated over a large number of temporal modes, and their statistics can be approximated as Gaussian by virtue of the central limit theorem. The FI then becomes simple to evaluate, as shown in Appendix C. We also find that this approximate FI for any $N$, when evaluated on the per-photon basis, is upper-bounded by the per-photon FI in the $N\ll 1$ limit. Thus we simply show the calculation of the FI for $N\ll 1$ here and use it as an upper bound on the approximate FI for arbitrary $N$.
In the $N\ll 1$ limit, information comes from one-photon events only, and the per-photon FI can be computed from the probability distribution of each photon. For direct imaging, the measurement outcome is the position of arrival $x$ of the photon in the image plane whose probability density is
\begin{equation}\label{prDI}
\pr^{(DI)}(x|\theta)=\frac{1}{2\sqrt{2\pi} \sigma}\left[e^{-\frac{(x-\theta/2)^2}{2\sigma^2}}+e^{-\frac{(x+\theta/2)^2}{2\sigma^2}}\right]
\end{equation}
The per-photon information is hence
\begin{equation}
\begin{split}
F_1^{(DI)}(\theta)
&=\int_{-\infty}^\infty \dd x\, \bigg[\frac{(2x-2xe^{\frac{x\theta}{\sigma^2}}+\theta+\theta e^{\frac{x\theta}{\sigma^2}})^2}{32\sqrt{2\pi}(1+e^{\frac{x\theta}{\sigma^2}})\sigma^5}e^{-\frac{(x+\theta/2)^2}{2\sigma^2}}\bigg]\\
&=\frac1{4\sigma^2}-\frac1{2\sqrt{2\pi}\sigma^5}\int_{-\infty}^\infty\dd x\,\frac{x^2 e^{-(x+\theta/2)^2/2\sigma^2}}{1+e^{-x\theta/\sigma^2}}
\end{split}
\label{directintegral}
\end{equation}
which can be evaluated numerically.
\begin{figure}\label{compgraph}
\centering
\includegraphics[width=\columnwidth]{incoh.pdf}
\caption{Per-photon FI of homodyne and direct imaging for two incoherent thermal sources. The (black) direct imaging curve is for weak thermal states with $N\ll1$, which gives the approximate FI for direct imaging (see main text and appendix C). Inset: per-photon FI for $\theta\ll\sigma$. In this limit, the FI for direct imaging is the same as homodyne detection when $N=2$. Black dots in the inset represent the result for direct imaging. Note that when $\theta/\sigma$ becomes close to 1, the population in TEM$_{01}$ starts to decrease because higher-order modes become important. This higher-order mode effect is not included in the figure. Our results for the FI are accurate in the regime of small $\theta/\sigma$. The per-photon FI for heterodyne detection can be obtained from that for homodyne detection by replacing $N$ with $N/2$, see text.}
\end{figure}
\subsection{Comparison}
In Fig.~1, we plot the per-photon FI for direct imaging and homodyne detection. One sees that homodyne detection is advantageous for small separations as long as the average photon number $N>2$, which is consistent with the conclusion in Ref. \cite{tsanghomo} that there is no advantage for small $N$. This also means that heterodyne detection is advantageous for $N>4$. This advantage can be understood by noting that, for small separations $\theta\ll\sigma$, the FI in Eq. (\ref{F12S}) for homodyne detection scales as $\theta^2 N/16\sigma^4$, whereas for direct imaging it scales like $\theta^2/8\sigma^4$ [see Eq.~(\ref{directintegral}) and Appendix C].
The maximum per-photon FI with homodyne detection is achieved for $\theta^2 N=4\sigma^2$ (i.e. when there are $\frac{d^2}{4\sigma^2}N=\frac 14$ photons in TEM$_{01}$ mode per source) and equals $1/16\sigma^2$. This corresponds to $1/4$ of the per-photon FI obtained in the quantum optimal measurement, which is achieved by means of a photon number measurement in TEM$_{01}$ \cite{Tsang}.
Let us compare the three methods (direct imaging, homodyne detection in TEM$_{01}$ and photon number measurement in TEM$_{01}$) in order to better understand the difference in their precision. For a direct image, the probability distribution Eq. (\ref{prDI}) of a photon's position of arrival in the image plane is a sum of two Gaussians, which, for $\theta\ll\sigma$, are almost indistinguishable from a single Gaussian centered $x=0$, and hence the inference on the source separation $\theta$ is very poor. The photon number measurement, on the other hand, is a null measurement: the signal power is proportional to $\theta^2$, so there is no signal whatsoever at $\theta=0$, leading to a FI that is independent of $\theta$. The homodyne measurement lies in between. It is not an ideal null measurement because of the shot noise, but it can still give a substantial advantage relative to direct imaging for sufficiently large $N$. Eq. (\ref{F12S}) shows that maximum sensitivity for the homodyne measurement is achieved when $N \theta^2=4\sigma^2$.
The requirement of $N>2$ (or $N>4$) means that the homodyne (or heterodyne) approach is most promising for measurements of distances between objects with rough surfaces that scatter laser light. Such measurements can occur, for example, in LIDARs that are used to operate autonomous vehicles. Since the laser can have a very large number of photons in a single mode, the scattered light is likely to contain speckles with multiple photons per mode. For astronomical applications, e.g.~measurements of distances between binary star components, the advantages and disadvantages of this method require further analysis to account not only for fundamental noise sources, but also for technical issues, such as atmospheric turbulences.
\setcounter{figure}{0}
\setcounter{equation}{0}
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1,116,691,500,483 | arxiv |
\section{Future Applications}
We have analyzed the individual components of this dataset and presented experiments with baseline results for tasks such as attribute classification, relationship classification, description generation, and question answering. There are, however, more applications and experiments for which our dataset can be used. In this section, we note a few potential applications that our dataset can enable.
\paragraph{Dense image captioning.} We have seen numerous image captioning papers~\cite{kiros2014multimodal, mao2014explain, karpathy2014deep, vinyals2014show} that attempt to describe an entire image with a single caption. However, these captions do not exhaustively describe every part of the scene. An natural extension to this application, which the Visual Genome dataset enables, is the ability to create dense captioning models that describe parts of the scene.
\paragraph{Visual question answering.} While visual question answering has been studied as a standalone task~\cite{VisualMadlibs,ren2015image,antol2015vqa,gao2015you}, we introduce a dataset that combines all of our question answers with descriptions and scene graphs. Future work can build supervised models that utilize various components of Visual Genome to tackle question answering.
\paragraph{Image understanding.} While we have seen a surge of image captioning~\cite{kiros2014multimodal} and question answering~\cite{antol2015vqa} models, there has been little work on creating more comprehensive evaluation metrics to measure how well these models are performing. Such models are usually evaluated using BLEU, CIDEr, or METEOR and other similar metrics that do not effectively measure how well these models understand the image~\cite{chen2015microsoft}. The Visual Genome scene graphs can be used as a measurement for image understanding. Generated descriptions and answers can be matched against the ground truth scene graph of an image to evaluate its corresponding model.
\paragraph{Relationship extraction.} Relationship extraction has been extensively studied in information retrieval and natural language processing~\cite{zhou122007tree, guodong2005exploring, culotta2004dependency, socher2012semantic}. Visual Genome is the first large-scale visual relationship dataset. This dataset can be used to study the extraction of visual relationships\cite{sadeghi2015viske} from images, and its interactions between objects can also be used to study action recognition~\cite{yao2010modeling, ramanathan2015learning} and spatial orientation between objects~\cite{gupta2009observing, prest2012weakly}.
\paragraph{Semantic image retrieval.} Previous work has already shown that scene graphs can be used to improve semantic image search~\cite{Johnson2015CVPR, schustergenerating}. Further methods can be explored using our region descriptions combined with region graphs. Attention-based search methods can also be explored where the area of interest specified by a query is also localized in the retrieved images.
\section{Conclusion}
\label{sec:conlusion}
Visual Genome provides a multi-layered understanding of pictures. It allows for a multi-perspective study of an image, from pixel-level information like objects, to relationships that require further inference, and to even deeper cognitive tasks like question answering. It is a comprehensive dataset for training and benchmarking the next generation of computer vision models. With Visual Genome, we expect these models to develop a broader understanding of our visual world, complementing computers' capacities to detect objects with abilities to describe those objects and explain their interactions and relationships. Visual Genome is a large \textit{formalized knowledge representation} for visual understanding and a more \textit{complete set of descriptions and question answers} that \textit{grounds visual concepts to language}.
\section{Crowdsourcing Strategies}
\label{sec:crowdsourcing_pipeline}
Visual Genome was collected and verified entirely by crowd workers from Amazon Mechanical Turk. In this section, we outline the pipeline employed in creating all the components of the dataset. Each component (region descriptions, objects, attributes, relationships, region graphs, scene graphs, questions and answers) involved multiple task stages. We mention the different strategies used to make our data accurate and to enforce diversity in each component. We also provide background information about the workers who helped make Visual Genome possible.
\subsection{Crowd Workers}
We used Amazon Mechanical Turk (AMT) as our primary source of annotations. Overall, a total of over $33,000$ unique workers contributed to the dataset. The dataset was collected over the course of $6$ months after $15$ months of experimentation and iteration on the data representation. Approximately $800,000$ Human Intelligence Tasks (HITs) were launched on AMT, where each HIT involved creating descriptions, questions and answers, or region graphs. Each HIT was designed such that workers manage to earn anywhere between \$$6$-\$$8$ per hour if they work continuously, in line with ethical research standards on Mechanical Turk~\cite{salehi2015we}. Visual Genome HITs achieved a $94.1$\% retention rate, meaning that $94.1$\% of workers who completed one of our tasks went ahead to do more. Table~\ref{table:worker_nation} outlines the percentage distribution of the locations of the workers. $93.02$\% of workers contributed from the United States.
\begin{table}[h]
\centering
\begin{tabular}{l r}
\textbf{Country} & \textbf{Distribution} \\
\midrule
United States & 93.02\% \\
Philippines & 1.29\% \\
Kenya & 1.13\% \\
India & 0.94\% \\
Russia & 0.50\% \\
Canada & 0.47\% \\
(Others) & 2.65\% \\
\end{tabular}
\caption{Geographic distribution of countries from where crowd workers contributed to Visual Genome.}
\label{table:worker_nation}
\end{table}
Figures~\ref{fig:worker_demographic} (a) and (b) outline the demographic distribution of our crowd workers. The majority of our workers were between the ages of $25$ and $34$ years old. Our youngest contributor was $18$ years old and the oldest was $68$ years old. We also had a near-balanced split of $54.15$\% male and $45.85$\% female workers.
\begin{figure*}[t]%
\centering
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/worker_age.png} }}%
\qquad
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/worker_gender.png} }}%
\caption{(a) Age and (b) gender distribution of Visual Genome's crowd workers.}%
\label{fig:worker_demographic}%
\end{figure*}
\subsection{Region Descriptions}
\label{sec:crowdsourcing_region_description}
Visual Genome's main goal is to enable the study of cognitive computer vision tasks. The next step towards understanding images requires studying relationships between objects in scene graph representations of images. However, we observed that collecting scene graphs directly from an image leads to workers annotating easy, frequently-occurring relationships like \relationship{man}{wearing}{shirt} instead of focusing on salient parts of the image. This is evident from previous datasets~\cite{Johnson2015CVPR, luvisualrelationship} that contain a large number of such relationships. After experimentation, we observed that when asked to describe an image using natural language, crowd workers naturally start with the most salient part of the image and then move to describing other parts of the image one by one. Inspired by this finding, we focused our attention towards collecting a dataset of region descriptions that is diverse in content.
When a new image is added to the crowdsourcing pipeline with no annotations, it is sent to a worker who is asked to draw three bounding boxes and write three descriptions for the region enclosed by each box. Next, the image is sent to another worker along with the previously written descriptions. Workers are explicitly encouraged to write descriptions that have not been written before. This process is repeated until we have collect $50$ region descriptions for each image. To prevent workers from having to skim through a long list of previously written descriptions, we only show them the top seven most similar descriptions. We calculate these most similar descriptions using BLEU~\cite{papineni2002bleu} (n-gram) scores between pairs of sentences. We define the BLEU score between a description $d_i$ and a previous description $d_j$ to be:
\begin{equation}
BLEU_N(d_i, d_j) = b(d_i, d_j) \exp(\frac{1}{N}\sum_{n=1}^{N} \log p_n(d_i, d_j))
\end{equation}
\vspace{0.3cm}
where we enforce a brevity penalty using:
\vspace{0.3cm}
\begin{equation}
b(d_i, d_j) = \left\{ \begin{array}{lr} 1 & \mathrm{if} \; len(d_i) > len(d_j) \\ e^{1-\frac{len(d_j)}{len(d_i)}} & \mathrm{otherwise} \end{array} \right.
\end{equation}
\vspace{0.3cm}
and $p_n$ calculates the percentage of n-grams in $d_i$ that match n-grams in $d_j$.
When a worker writes a new description, we programmatically enforce that it has not been repeated by using BLEU score thresholds set to $0.7$ to ensure that it is dissimilar to descriptions from both of the following two lists:
\begin{enumerate}
\item \textbf{Image-specific descriptions.} A list of all previously written descriptions for that image.
\item \textbf{Global image descriptions.} A list of the top 100 most common written descriptions of all images in the dataset. This prevents very common phrases like ``sky is blue'' from dominating the set of region descriptions.
\end{enumerate}
Finally, we ask workers to draw bounding boxes that satisfy one requirement: \textbf{coverage}. The bounding box must cover all objects mentioned in the description. Figure~\ref{fig:bbox_coverage} shows an example of a good box that covers both the \object{street} as well the \object{car} mentioned in the description, as well as an example of a bad box.
\begin{figure}[t]%
\centering
\iftoggle{smallfigs}{
\includegraphics[width=0.95\columnwidth]{png_graphics/bbox_coverage_small.png}
}{
\includegraphics[width=\columnwidth]{png_graphics/bbox_coverage.png}
}
\caption{Good (left) and bad (right) bounding boxes for the phrase ``a street with a red car parked on the side,'' judged on \textbf{coverage}.}%
\label{fig:bbox_coverage}%
\end{figure}
\subsection{Objects}
Once $50$ region descriptions are collected for an image, we extract the visual objects from each description. Each description is sent to one crowd worker, who extracts all the objects from the description and grounds each object as a bounding box in the image. For example, from Figure~\ref{fig:data_representation}, let's consider the description ``woman in shorts is standing behind the man.'' A worker would extract three objects: \object{woman}, \object{shorts}, and \object{man}. They would then draw a box around each of the objects. We require each bounding box to be drawn to satisfy two requirements: \textbf{coverage} and \textbf{quality}. Coverage has the same definition as described above in Section~\ref{sec:crowdsourcing_region_description}, where we ask workers to make sure that the bounding box covers the object completely (Figure~\ref{fig:bbox_quality}). Quality requires that each bounding box be as tight as possible around its object such that if the box's length or height were decreased by one pixel, it would no longer satisfy the coverage requirement. Since a one pixel error can be physically impossible for most workers, we relax the definition of quality to four pixels.
\begin{figure}[t]%
\centering
\iftoggle{smallfigs}{
\includegraphics[width=\columnwidth]{png_graphics/bbox_quality_small.png}
}{
\includegraphics[width=\columnwidth]{png_graphics/bbox_quality.png}
}
\caption{Good (left) and bad (right) bounding boxes for the object \object{fox}, judged on both \textbf{coverage} as well as \textbf{quality}.}%
\label{fig:bbox_quality}.%
\end{figure}
Multiple descriptions for an image might refer to the same object, sometimes with different words. For example, a \object{man} in one description might be referred to as \object{person} in another description. We can thus use this crowdsourcing stage to build these co-reference chains. With each region description given to a worker to process, we include a list of previously extracted objects as suggestions. This allows a worker to choose a previously drawn box annotated as \object{man} instead of redrawing a new box for \object{person}.
Finally, to increase the speed with which workers complete this task, we also use Stanford's dependency parser~\cite{manning-EtAl:2014:P14-5} to extract nouns automatically and send them to the workers as suggestions. While the parser manages to find most of the nouns, it sometimes misses compound nouns, so we avoided completely depending on this automated method. By combining the parser with crowdsourcing tasks, we were able to speed up our object extraction process without losing accuracy.
\subsection{Attributes, Relationships, and Region Graphs}
Once all objects have been extracted from each region description, we can extract the attributes and relationships described in the region. We present each worker with a region description along with its extracted objects and ask them to add attributes to objects or to connect pairs of objects with relationships, based on the text of the description. From the description ``woman in shorts is standing behind the man'', workers will extract the attribute \attribute{standing} for the \object{woman} and the relationships \relationship{woman}{in}{shorts} and \relationship{woman}{behind}{man}. Together, objects, attributes, and relationships form the region graph for a region description. Some descriptions like ``it is a sunny day'' do not contain any objects and therefore have no region graphs associated with them. Workers are asked to not generate any graphs for such descriptions. We create scene graphs by combining all the region graphs for an image by combining all the co-referenced objects from different region graphs.
\subsection{Scene Graphs}
The scene graph is the union of all region graphs extracted from region descriptions. We merge nodes from region graphs that correspond to the same object; for example, \object{man} and \object{person} in two different region graphs might refer to the same object in the image. We say that objects from different graphs refer to the same object if their bounding boxes have an overlap over union of $0.8$. However, this heuristic might contain false positives. So, before merging two objects, we ask workers to confirm that a pair of objects with significant overlap are indeed the same object. For example, in Figure~\ref{fig:bbox_combined} (right), the \object{fox} might be extracted from two different region descriptions. These boxes are then combined together (Figure~\ref{fig:bbox_combined} (left)) when constructing the scene graph. Two region graphs are combined together by merging objects that are co-referenced by both the graphs.
\begin{figure}[t]%
\centering
\iftoggle{smallfigs}{
\includegraphics[width=\columnwidth]{png_graphics/bbox_combined_small.png}
}{
\includegraphics[width=\columnwidth]{png_graphics/bbox_combined.png}
}
\caption{Each object (\object{fox}) has only one bounding box referring to it (left). Multiple boxes drawn for the same object (right) are combined together if they have a minimum threshold of $0.9$ intersection over union.}%
\label{fig:bbox_combined}.%
\end{figure}
\subsection{Questions and Answers}
To create question answer (QA) pairs, we ask the AMT workers to write pairs of questions and answers about an image. To ensure quality, we instruct the workers to follow three rules: 1) start the questions with one of the ``seven Ws'' (\qa{who}, \qa{what}, \qa{where}, \qa{when}, \qa{why}, \qa{how} and \qa{which}); 2) avoid ambiguous and speculative questions; 3) be precise and unique, and relate the question to the image such that it is clearly answerable if and only if the image is shown.
We collected two separate types of QAs: freeform QAs and region-based QAs. In freeform QA, we ask a worker to look at an image and write eight QA pairs about it. To encourage diversity, we enforce that workers write at least three different Ws out of the seven in their eight pairs. In region-based QA, we ask the workers to write a pair based on a given region. We select the regions that have large areas (more than 5k pixels) and long phrases (more than 4 words). This enables us to collect around twenty region-based pairs at the same cost of the eight freeform QAs. In general, freeform QA tends to yield more diverse QA pairs that enrich the question distribution; region-based QA tends to produce more factual QA pairs at a lower cost.
\subsection{Verification}
All Visual Genome data go through a verification stage as soon as they are annotated. This stage helps eliminate incorrectly labeled objects, attributes, and relationships. It also helps remove region descriptions and questions and answers that might be correct but are vague (``This person seems to enjoy the sun.''), subjective (``room looks dirty''), or opinionated (``Being exposed to hot sun like this may cause cancer'').
Verification is conducted using two separate strategies: majority voting~\cite{snow2008cheap} and rapid judgments~\cite{krishnaembracing}. All components of the dataset except objects are verified using majority voting. Majority voting\cite{snow2008cheap} involves three unique workers looking at each annotation and voting on whether it is factually correct. An annotation is added to our dataset if at least two (a majority) out of the three workers verify that it is correct.
We only use rapid judgments to speed up the verification of the objects in our dataset. Meanwhile, rapid judgments~\cite{krishnaembracing} use an interface inspired by rapid serial visual processing that enable verification of objects with an order of magnitude increase in speed than majority voting.
\subsection{Canonicalization}
All the descriptions and QAs that we collect are freeform worker-generated texts. They are not constrained by any limitations. For example, we do not force workers to refer to a man in the image as a \object{man}. We allow them to choose to refer to the man as \object{person}, \object{boy}, \object{man}, etc. This ambiguity makes it difficult to collect all instances of \object{man} from our dataset. In order to reduce the ambiguity in the concepts of our dataset and connect it to other resources used by the research community, we map all objects, attributes, relationships, and noun phrases in region descriptions and QAs to synsets in WordNet~\cite{miller1995WordNet}. In the example above, \object{person}, \object{boy}, and \object{man} would map to the synsets: \synset{person.n.01 (a human being)}, \synset{male\_child.n.01 (a youthful male person)} and \synset{man.n.03 (the generic use of the word to refer to any human being)} respectively. Thanks to the WordNet hierarchy it is now possible to fuse those three expressions of the same concept into \synset{person.n.01 (a human being)} since this is the lowest common ancestor node of all aforementioned synsets.
We use the Stanford NLP tools~\cite{manning-EtAl:2014:P14-5} to extract the noun phrases from the region descriptions and QAs. Next, we map them to their most frequent matching synset in WordNet according to WordNet lexeme counts. We then refine this simple heuristic by hand-crafting mapping rules for the 30 most common failure cases. For example according to WordNet's lexeme counts the most common semantic for ``table'' is \synset{table.n.01 (a set of data arranged in rows and columns)}. However in our data it is more likely to see pieces of furniture and therefore bias the mapping towards \synset{table.n.02 (a piece of furniture having a smooth flat top that is usually supported by one or more vertical legs)}. The objects in our scene graphs are already noun phrases and are mapped to WordNet in the same way.
We normalize each attribute based on morphology (so called ``stemming'') and map them to the WordNet adjectives. We include 15 hand-crafted rules to address common failure cases, which typically occur when the concrete or spatial sense of the word seen in an image is not the most common overall sense. For example, the synset \synset{long.a.02 (of relatively great or greater than average spatial extension)} is less common in WordNet than \synset{long.a.01 (indicating a relatively great or greater than average duration of time)}, even though instances of the word ``long'' in our images are much more likely to refer to that spatial sense.
For relationships, we ignore all prepositions as they are not recognized by WordNet. Since the meanings of verbs are highly dependent upon their morphology and syntactic placement (e.g.\ passive cases, prepositional phrases), we try to find WordNet synsets whose sentence frames match with the context of the relationship. Sentence frames in WordNet are formalized syntactic frames in which a certain sense of a word might appear; for example, \synset{play.v.01: participate in games or sport} occurs in the sentence frames ``Somebody [play]s'' and ``Somebody [play]s something.'' For each verb-synset pair, we then consider the root hypernym of that synset to reduce potential noise from WordNet's fine-grained sense distinctions. The WordNet hierarchy for verbs is segmented and originates from over $100$ root verbs. For example, \synset{draw.v.01: cause to move by pulling} traces back to the root hypernym \synset{move.v.02: cause to move or shift into a new position}, while \synset{draw.v.02: get or derive} traces to the root \synset{get.v.01: come into the possession of something concrete or abstract}. We also include $20$ hand-mapped rules, again to correct for WordNet's lower representation of concrete or spatial senses.
These mappings are not perfect and still contain some ambiguity. Therefore, we send all our mappings along with the top four alternative synsets for each term to Amazon Mechanical Turk. We ask workers to verify that our mapping was accurate and change the mapping to an alternative one if it was a better fit. We present workers with the concept we want to canonicalize along with our proposed corresponding synset with 4 additional options. To prevent workers from always defaulting to the our proposed synset, we do not explicitly specify which one of the 5 synsets presented is our proposed synset. Section~\ref{sec:canonicalization_stats} provides experimental precision and recall scores for our canonicalization strategy.
\section{Visual Genome Data Representation}
\label{sec:data_representation}
The Visual Genome dataset consists of seven main components: \textit{region descriptions}, \textit{objects}, \textit{attributes}, \textit{relationships}, \textit{region graphs}, \textit{scene graphs}, and \textit{question-answer pairs}. Figure~\ref{fig:data_representation} shows examples of each component for one image. To enable research on comprehensive understanding of images, we begin by collecting descriptions and question answers. These are raw texts without any restrictions on length or vocabulary. Next, we extract objects, attributes and relationships from our descriptions. Together, objects, attributes and relationships fabricate our scene graphs that represent a formal representation of an image. In this section, we break down Figure~\ref{fig:data_representation} and explain each of the seven components. In Section~\ref{sec:crowdsourcing_pipeline}, we will describe in more detail how data from each component is collected through a crowdsourcing platform.
\subsection{Multiple regions and their descriptions}
In a real-world image, one simple summary sentence is often insufficient to describe all the contents of and interactions in an image. Instead, one natural way to extend this might be a collection of descriptions based on different regions of a scene. In Visual Genome, we collect human-generated image region descriptions, with each region localized by a bounding box. In Figure~\ref{fig:data_representation_regions}, we show three examples of region descriptions. Regions are allowed to have a high degree of overlap with each other when the descriptions differ. For example, ``yellow fire hydrant'' and ``woman in shorts is standing behind the man'' have very little overlap, while ``man jumping over fire hydrant'' has a very high overlap with the other two regions. Our dataset contains on average a total of $42$ region descriptions per image. Each description is a phrase ranging from $1$ to $16$ words in length describing that region.
\begin{figure}[t]%
\centering
\includegraphics[width=0.46\textwidth]{eps_graphics/data_representation_regions.pdf}
\caption{To describe all the contents of and interactions in an image, the Visual Genome dataset includes multiple human-generated image regions descriptions, with each region localized by a bounding box. Here, we show three regions descriptions: ``man jumping over a fire hydrant,'' ``yellow fire hydrant,'' and ``woman in shorts is standing beghind the man.''}
\label{fig:data_representation_regions}
\end{figure}
\begin{figure}[ht]%
\centering
\includegraphics[width=0.45\textwidth]{eps_graphics/data_representation_objects.pdf}
\caption{From all of the region descriptions, we extract all objects mentioned. For example, from the region description ``man jumping over a fire hydrant,'' we extract \object{man} and \object{fire hydrant}.}%
\label{fig:data_representation_objects}%
\end{figure}
\subsection{Multiple objects and their bounding boxes}
Each image in our dataset consists of an avarege of $21$ objects, each delineated by a tight bounding box (Figure~\ref{fig:data_representation_objects}). Furthermore, each object is canonicalized to a synset ID in WordNet~\cite{miller1995WordNet}. For example, \object{man} and \object{person} would get mapped to \synset{man.n.03 (the generic use of the word to refer to any human being)}. Similarly, \object{person} gets mapped to \synset{person.n.01 (a human being)}. Afterwards, these two concepts can be joined to \synset{person.n.01} since this is a hypernym of \synset{man.n.03}. This is an important standardization step to avoid multiple names for one object (e.g.\ man, person, human), and to connect information across images.
\begin{figure}[t]%
\centering
\includegraphics[width=0.4\textwidth]{eps_graphics/data_representation_attributes.pdf}
\caption{Some descriptions also provide attributes for objects. For example, the region description ``yellow fire hydrant'' adds that the \object{fire hydrant} is \attribute{yellow}. Here we show two attributes: \attribute{yellow} and \attribute{standing}.}
\label{fig:data_representation_attributes}
\end{figure}
\subsection{A set of attributes}
Each image in Visual Genome has an average of 16 attributes. Objects can have zero or more attributes associated with them. Attributes can be color (\attribute{yellow}), states (\attribute{standing}), etc. (Figure~\ref{fig:data_representation_attributes}). Just like we extract objects from region descriptions, we also extract the attributes attached to these objects. In Figure~\ref{fig:data_representation_attributes}, from the phrase ``yellow fire hydrant,'' we extract the attribute \attribute{yellow} for the \object{fire hydrant}. As with objects, we canonicalize all attributes to WordNet~\cite{miller1995WordNet}; for example, \attribute{yellow} is mapped to \synset{yellow.s.01 (of the color intermediate between green and orange in the color spectrum; of something resembling the color of an egg yolk)}.
\subsection{A set of relationships}
Relationships connect two objects together. These relationships can be actions (\predicate{jumping over}), spatial (\predicate{is behind}), verbs (\predicate{wear}), prepositions (\predicate{with}), comparative (\predicate{taller than}), or prepositional phrases (\predicate{drive on}). For example, from the region description ``man jumping over fire hydrant,'' we extract the relationship \predicate{jumping over} between the objects \object{man} and \object{fire hydrant} (Figure~\ref{fig:data_representation_relationships}). These relationships are directed from one object, called the subject, to another, called the object. In this case, the subject is the \object{man}, who is performing the relationship \predicate{jumping over} on the object \object{fire hydrant}. Each relationship is canonicalized to a WordNet~\cite{miller1995WordNet} synset ID; i.e.\ \predicate{jumping} is canonicalized to \synset{jump.a.1 (move forward by leaps and bounds)}. On average, each image in our dataset contains 18 relationships.
\begin{figure}[t]%
\centering
\includegraphics[width=0.42\textwidth]{eps_graphics/data_representation_relationships.pdf} \caption{Our dataset also captures the relationships and interactions between objects in our images. In this example, we show the relationship \predicate{jumping over} between the objects \object{man} and \object{fire hydrant}.}%
\label{fig:data_representation_relationships}%
\end{figure}
\subsection{A set of region graphs}
Combining the objects, attributes, and relationships extracted from region descriptions, we create a directed graph representation for each of the $42$ regions. Examples of region graphs are shown in Figure~\ref{fig:data_representation}. Each region graph is a structured representation of a part of the image. The nodes in the graph represent objects, attributes, and relationships. Objects are linked to their respective attributes while relationships link one object to another. The links connecting two objects in Figure~\ref{fig:data_representation} point from the subject to the relationship and from the relationship to the other object.
\subsection{One scene graph}
While region graphs are localized representations of an image, we also combine them into a single scene graph representing the entire image (Figure~\ref{fig:scene_graph_2}).
The scene graph is the \emph{union} of all region graphs and contains all objects, attributes, and relationships from each region description. By doing so, we are able to combine multiple levels of scene information in a more coherent way. For example in Figure~\ref{fig:data_representation}, the leftmost region description tells us that the ``fire hydrant is yellow,'' while the middle region description tells us that the ``man is jumping over the fire hydrant.'' Together, the two descriptions tell us that the ``man is jumping over a yellow fire hydrant.''
\subsection{A set of question answer pairs}
We have two types of QA pairs associated with each image in our dataset: \textit{freeform QAs}, based on the entire image, and \textit{region-based QAs}, based on selected regions of the image. We collect 6 different types of questions per image: \qa{what}, \qa{where}, \qa{how}, \qa{when}, \qa{who}, and \qa{why}. In Figure~\ref{fig:data_representation}, ``Q. What is the woman standing next to?; A. Her belongings'' is a freeform QA\@. Each image has at least one question of each type listed above. Region-based QAs are collected by prompting workers with region descriptions. For example, we use the region ``yellow fire hydrant'' to collect the region-based QA: ``Q. What color is the fire hydrant?; A. Yellow.'' Region based QAs allow us to independently study methods that use NLP and vision priors to answer questions.
\section{Dataset Statistics and Analysis}
\label{sec:dataset_statistics}
In this section, we provide statistical insights and analysis for each component of Visual Genome. Specifically, we examine the distribution of \textit{images} (Section~\ref{sec:image_stats}) and the collected data for \textit{region descriptions} (Section~\ref{sec:region_stats}) and \textit{questions and answers} (Section~\ref{sec:qa_stats}). We analyze \textit{region graphs} and \textit{scene graphs} together in one section (Section~\ref{sec:graph_stats}), but we also break up these graph structures into their three constituent parts---\textit{objects} (Section~\ref{sec:object_stats}), \textit{attributes} (Section~\ref{sec:attribute_stats}), and \textit{relationships} (Section~\ref{sec:relationship_stats})---and study each part individually. Finally, we describe our canonicalization pipeline and results (Section~\ref{sec:canonicalization_stats}).
\begin{figure}[t]
\centering
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.5\textwidth]{pdf_graphics/region_drawing.pdf} }}%
}{
\subfloat[]{{\includegraphics[width=0.5\textwidth]{png_graphics/region_drawing.png} }}%
}
\qquad
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.35\textwidth]{png_graphics/region_drawing_boxes_small.png} }}%
}{
\subfloat[]{{\includegraphics[width=0.35\textwidth]{png_graphics/region_drawing_boxes.png} }}%
}
\caption{(a) An example image from the dataset with its region descriptions. We only display localizations for $6$ of the $42$ descriptions to avoid clutter; all 50 descriptions do have corresponding bounding boxes. (b) All $42$ region bounding boxes visualized on the image.}
\label{fig:region_drawing}
\end{figure}
\label{sec:region_stats}
\begin{figure*}[t]%
\centering
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/region_width_distribution.png} }}%
\qquad
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/region_height_distribution.png} }}%
\caption{(a) A distribution of the width of the bounding box of a region description normalized by the image width. (b) A distribution of the height of the bounding box of a region description normalized by the image height.}%
\label{fig:region_distributions}%
\end{figure*}
\subsection{Image Selection}
\label{sec:image_stats}
The Visual Genome dataset consists of all $108,249$ images from the intersection of MS-COCO's~\cite{lin2014microsoft} $328,000$ images and YFCC's~\cite{thomee2015yfcc100m} $100$ million images. These images are real-world, non-iconic images that were uploaded onto Flickr by users. The images range from as small as $72$ pixels wide to as large as $1280$ pixels wide, with an average width of $500$ pixels. We collected the WordNet synsets into which our $108,249$ images can be categorized using the same method as ImageNet~\cite{deng2009imagenet}. Visual Genome images cover $972$ synsets. Figure~\ref{fig:synsets} shows the top synsets to which our images belong. ``ski'' is the most common synset, with $2612$ images; it is followed by ``ballplayer'' and ``racket,'' with all three synsets referring to images of people playing sports. Our dataset is somewhat biased towards images of people, as Figure~\ref{fig:synsets} shows; however, they are quite diverse overall, as the top $25$ synsets each have over $800$ images, while the top $50$ synsets each have over 500 examples.
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{png_graphics/region_length_distribution.png}
\caption{A distribution of the number of words in a region description. The average number of words in a region description is $5$, with shortest descriptions of $1$ word and longest descriptions of $16$ words.}
\label{fig:region_length}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{png_graphics/region_clustering_method.png}
\caption{The process used to convert a region description into a 300-dimensional vectorized representation.}
\label{fig:region_clustering_pipeline}
\end{figure}
\subsection{Region Description Statistics}
One of the primary components of Visual Genome is its region descriptions. Every image includes an average of $42$ regions with a bounding box and a descriptive phrase. Figure~\ref{fig:region_drawing} shows an example image from our dataset with its $50$ region descriptions. We display bounding boxes for only $6$ out of the $50$ descriptions in the figure to avoid clutter. These descriptions tend to be highly diverse and can focus on a single object, like in ``A bag,'' or on multiple objects, like in ``Man taking a photo of the elephants.'' They encompass the most salient parts of the image, as in ``An elephant taking food from a woman,'' while also capturing the background, as in ``Small buildings surrounded by trees.''
MS-COCO~\cite{lin2014microsoft} dataset is good at generating variations on a single scene-level descriptor. Consider three sentences from MS-COCO dataset on a similar image: ``there is a person petting a very large elephant,'' ``a person touching an elephant in front of a wall,'' and ``a man in white shirt petting the cheek of an elephant.'' These three sentences are single scene-level descriptions. In comparison, Visual Genome descriptions emphasize different regions in the image and thus are less semantically similar. To ensure diversity in the descriptions, we use BLEU score~\cite{papineni2002bleu} thresholds between new descriptions and all previously written descriptions. More information about crowdsourcing can be found in Section~\ref{sec:crowdsourcing_pipeline}.
Region descriptions must be specific enough in an image to describe individual objects, like in the description ``A bag,'' but they must also be general enough to describe high-level concepts in an image, like ``An man being chased by a bear.'' Qualitatively, we note that regions that cover large portions of the image tend to be general descriptions of an image, while regions that cover only a small fraction of the image tend to be more specific. In Figure~\ref{fig:region_distributions} (a), we show the distribution of regions over the width of the region normalized by the width of the image. We see that the majority of our regions tend to be around $10\%$ to $15\%$ of the image width. We also note that there are a large number of regions covering $100\%$ of the image width. These regions usually include elements like ``sky,'' ``ocean,'' ``snow,'' ``mountains,'' etc.\ that cannot be bounded and thus span the entire image width. In Figure~\ref{fig:region_distributions} (b), we show a similar distribution over the normalized height of the region. We see a similar overall pattern, as most of our regions tend to be very specific descriptions of about $10\%$ to $15\%$ of the image height. Unlike the distribution over width, however, we do not see a increase in the number of regions that span the entire height of the image, as there are no common visual equivalents that span images vertically. Out of all the descriptions gathered, only one or two of them tend to be global scene descriptions that are similar to MS-COCO~\cite{lin2014microsoft}.
After examining the distribution of the size of the regions described, it is also valuable to look at the semantic information captured by these descriptions. In Figure~\ref{fig:region_length}, we show the distribution of the length (word count) of these region descriptions. The average word count for a description is 5 words, with a minimum of 1 word and a maximum of 12 words. In Figure~\ref{fig:region_top_phrases_words} (a), we plot the most common phrases occurring in our region descriptions, with stop words removed. Common visual elements like ``green grass,'' ``tree [in] distance,'' and ``blue sky'' occur much more often than other, more nuanced elements like ``fresh strawberry.'' We also study descriptions with finer precision in Figure~\ref{fig:region_top_phrases_words} (b), where we plot the most common words used in descriptions. Again, we eliminate stop words from our study. Colors like ``white'' and ``black'' are the most frequently used words to describe visual concepts; we conduct a similar study on other captioning datasets including MS-COCO~\cite{lin2014microsoft} and Flickr 30K~\cite{young2014image} and find a similar distribution with colors occurring most frequently. Besides colors, we also see frequent occurrences of common objects like ``man,'' ``tree,'' and ``sign'' and of universal visual elements like ``sky.''
\paragraph{Semantic diversity.}
We also study the actual semantic contents of the descriptions. We use an unsupervised approach to analyze the semantics of these descriptions. Specifically, we use word2vec~\cite{mikolov2013efficient} to convert each word in a description to a 300-dimensional vector. Next, we remove stop words and average the remaining words to get a vector representation of the whole region description. This pipeline is outlined in Figure~\ref{fig:region_clustering_pipeline}. We use hierarchical agglomerative clustering on vector representations of each region description and find 71 semantic and syntactic groupings or ``clusters.'' Figure~\ref{fig:region_clustering} (a) shows four such example clusters. One cluster contains all descriptions related to tennis, like ``A man swings the racquet'' and ``White lines on the ground of the tennis court,'' while another cluster contains descriptions related to numbers, like ``Three dogs on the street'' and ``Two people inside the tent.'' To quantitatively measure the diversity of Visual Genome's region descriptions, we calculate the number of clusters represented in a single image's region descriptions. We show the distribution of the variety of descriptions for an image in Figure~\ref{fig:region_clustering} (b). We find that on average, each image contains descriptions from 17 different clusters. The image with the least diverse descriptions contains descriptions from 4 clusters, while the image with the most diverse descriptions contains descriptions from 26 clusters.
Finally, we also compare the descriptions in Visual Genome to the captions in MS-COCO\@. First we aggregate all Visual Genome and MS-COCO descriptions and remove all stop words. After removing stop words, the descriptions from both datasets are roughly the same length. We conduct a similar study, in which we vectorize the descriptions for each image and calculate each dataset's cluster diversity per image. We find that on average, 2 clusters are represented in the captions for each image in MS-COCO, with very few images in which 5 clusters are represented. Because each image in MS-COCO only contains 5 captions, it is not a fair comparison to compare the number of clusters represented in all the region descriptions in the Visual Genome dataset. We thus randomly sample 5 Visual Genome region descriptions per image and calculate the number of clusters in an image. We find that Visual Genome descriptions come from 4 or 5 clusters. We show our comparison results in Figure~\ref{fig:region_clustering} (c). The difference between the semantic diversity between the two datasets is statistically significant ($t=-240$, $p<0.01$).
\begin{figure*}[ht]%
\centering
\subfloat[]{{\includegraphics[width=0.48\textwidth]{png_graphics/region_top_phrases.png} }}%
\qquad
\subfloat[]{{\includegraphics[width=0.43\textwidth]{png_graphics/region_top_words.png} }}%
\caption{(a) A plot of the most common visual concepts or phrases that occur in region descriptions. The most common phrases refer to universal visual concepts like ``blue sky,'' ``green grass,'' etc. (b) A plot of the most frequently used words in region descriptions. Colors occur the most frequently, followed by common objects like ``man'' and ``dog'' and universal visual concepts like ``sky.''}%
\label{fig:region_top_phrases_words}%
\end{figure*}
\begin{figure*}[t]%
\centering
\subfloat[]{{
\iftoggle{smallfigs}{
\includegraphics[width=0.96\textwidth]{pdf_graphics/cluster_infographic.pdf}
}{
\includegraphics[width=0.96\textwidth]{png_graphics/cluster_infographic.png}
}
}}%
\qquad
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/region_cluster.png} }}%
\qquad
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/cluster_coco_comparison.png} }}%
\caption{(a) Example illustration showing four clusters of region descriptions and their overall themes. Other clusters not shown due to limited space. (b) Distribution of images over number of clusters represented in each image's region descriptions. (c) We take Visual Genome with 5 random descriptions taken from each image and MS-COCO dataset with all 5 sentence descriptions per image and compare how many clusters are represented in the descriptions. We show that Visual Genome's descriptions are more varied for a given image, with an average of 4 clusters per image, while MS-COCO's images have an average of 3 clusters per image.}
\label{fig:region_clustering}
\end{figure*}
\clearpage
\newcolumntype{P}[1]{>{\centering\arraybackslash}p{#1}}
\tabcolsep=0.05cm
\begin{table*}[h]
\centering
\begin{tabular}{p{3cm}P{2cm}P{2cm}P{2cm}P{2cm}P{2cm}P{2cm}P{2cm}}
& Visual Genome & ILSVRC Det. \cite{ILSVRC15} & MS-COCO \cite{lin2014microsoft} & Caltech101 \cite{fei2007learning} & Caltech256 \cite{griffin2007caltech} & PASCAL Det. \cite{everingham2010pascal} & Abstract Scenes \cite{zitnick2013bringing} \\
\midrule
Images & 108,249 & 476,688 & 328,000 & 9,144 & 30,608 & 11,530 & 10,020 \\
Total Objects & 255,718 & 534,309 & 2,500,000 & 9,144 & 30,608 & 27,450 & 58 \\
Total Categories & 18,136 & 200 & 91 & 102 & 257 & 20 & 11 \\
Objects per Category & 14.10 & 2671.50 & 27472.50 & 90 & 119 & 1372.50 & 5.27 \\
\bottomrule
\end{tabular}
\caption{Comparison of Visual Genome objects and categories to related datasets.}
\label{fig:objects_and_categories}
\end{table*}
\begin{figure}[t!]
\centering
\subfloat[]{{\includegraphics[width=0.45\textwidth]{png_graphics/objects_region_distribution.png} }
\qquad
\subfloat[]{{\includegraphics[width=0.45\textwidth]{png_graphics/objects_image_distribution.png} }}%
\qquad
\caption{(a) Distribution of the number of objects per region. Most regions have between 0 and 2 objects. (b) Distribution of the number of objects per image. Most images contain between 15 and 20 objects.}
\label{fig:objects_region_image}
\end{figure}
\subsection{Object Statistics}
\label{sec:object_stats}
In comparison to related datasets, Visual Genome fares well in terms of object density and diversity. Visual Genome contains approximately $21$ objects per image, exceeding ImageNet~\cite{deng2009imagenet}, PASCAL~\cite{everingham2010pascal}, MS-COCO~\cite{lin2014microsoft}, and other datasets by large margins. As shown in Figure~\ref{fig:categories_and_instances}, there are more object categories represented in Visual Genome than in any other dataset. This comparison is especially pertinent with regards to Microsoft MS-COCO~\cite{lin2014microsoft}, which uses the same images as Visual Genome. The lower count of objects per category is a result of our higher number of categories. For a fairer comparison with ILSVRC 2014 Detection~\cite{ILSVRC15}, Visual Genome has about $2239$ objects per category when only the top $200$ categories are considered, which is comparable to ILSVRC's $2671.5$ objects per category. For a fairer comparison with MS-COCO, Visual Genome has about $3768$ objects per category when only the top $91$ categories are considered. This is comparable to MS-COCO's~\cite{lin2014microsoft} when we consider just the $108,249$ MS-COCO images in Visual Genome.
Objects in Visual Genome come from a variety of categories. As shown in Figure~\ref{fig:object_examples_top_objects} (b), objects related to WordNet categories such as humans, animals, sports, and scenery are most common; this is consistent with the general bias in image subject matter in our dataset. Common objects like \object{man}, \object{person}, and \object{woman} occur especially frequently with occurrences of $24$K, $17$K, and $11$K. Other objects that also occur in MS-COCO~\cite{lin2014microsoft} are also well represented with around $5000$ instances on average. Figure~\ref{fig:object_examples_top_objects} (a) shows some examples of objects in images. Objects in Visual Genome span a diverse set of Wordnet categories like food, animals, and man-made structures.
It is important to look not only at what types of objects we have but also at the distribution of objects in images and regions. Figure~\ref{fig:objects_region_image} (a) shows, as expected, that we have between 0 and 2 objects in each region on average. It is possible for regions to contain no objects if their descriptions refer to no explicit objects in the image. For example, a region described as ``it is dark outside'' has no objects to extract. Regions with only one object generally have descriptions that focus on the attributes of a single object. On the other hand, regions with two or more objects generally have descriptions that contain both attributes of specific objects and relationships between pairs of objects.
As shown in Figure~\ref{fig:objects_region_image} (b), each image contains on average around $21$ unique objects. Few images have a low number of objects, which we expect since images usually capture more than a few objects. Moreover, few images have an extremely high number of objects (e.g.\ over $40$).
\begin{figure}[t!]
\centering
\includegraphics[width=0.5\textwidth]{png_graphics/categories_instances_graph.png}
\caption{Comparison of object diversity between various datasets. Visual Genome far surpasses other datasets in terms of number of object categories.}
\label{fig:categories_and_instances}
\end{figure}
\begin{figure*}[h!]
\centering
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.45\textwidth]{pdf_graphics/object_example_classes.pdf} }}%
}{
\subfloat[]{{\includegraphics[width=0.45\textwidth]{png_graphics/object_example_classes.png} }}%
}
\subfloat[]{{\includegraphics[width=0.40\textwidth]{png_graphics/objects_top_names.png} }}%
\qquad
\caption{(a) Examples of objects in Visual Genome. Each object is localized in its image with a tightly drawn bounding box. (b) Plot of the most frequently occurring objects in images. People are the most frequently occurring objects in our dataset, followed by common objects and visual elements like \object{building}, \object{shirt}, and \object{sky}.}
\label{fig:object_examples_top_objects}
\end{figure*}
\clearpage
\subsection{Attribute Statistics}
\label{sec:attribute_stats}
Attributes allow for detailed description and disambiguation of objects in our dataset. About $45\%$ of objects in Visual Genome are annotated with at least one attribute; our dataset contains $1.6$ million total attributes with $13,041$ unique attributes. Attributes include colors (\attribute{green}), sizes (\attribute{tall}), continuous action verbs (\attribute{standing}), materials (\attribute{plastic}), etc. Each attribute in our scene graphs belongs to one object, while each object can have multiple attributes. We denote attributes as \attribute{attribute}(\object{object}).
On average, each image in Visual Genome contains $21$ attributes, as shown in Figure~\ref{fig:attribute_distribution}. Each region contains on average $1$ attribute, though about $42\%$ of regions contain no attribute at all; this is primarily because many regions are relationship-focused. Figure~\ref{fig:top_attributes} (a) shows the distribution of the most common attributes in our dataset. Colors (e.g.~\attribute{white}, \attribute{green}) are by far the most frequent attributes. Also common are sizes (e.g.~\attribute{large}) and materials (e.g.~\attribute{wooden}). Figure~\ref{fig:top_attributes} (b) shows the distribution of attributes describing people (e.g.~\object{man}, \object{girls}, and \object{person}). The most common attributes describing people are intransitive verbs describing their states of motion (e.g.\attribute{standing} and \attribute{walking}). Certain sports (\attribute{skiing}, \attribute{surfboarding}) are overrepresented due to a bias towards these sports in our images.
\vspace{-0.2cm}
\paragraph{Attribute Graphs.} We also qualitatively analyze the attributes in our dataset by constructing co-occurrence graphs, in which nodes are unique attributes and edges connect those attributes that describe the same object. For example, if an image contained a ``large black dog'' (\attribute{large}(\object{dog}), \attribute{black}(\object{dog})) and another image contained a ``large yellow cat'' (\attribute{large}(\object{cat}), \attribute{yellow}(\object{cat})), its attributes would form an incomplete graph with edges (\attribute{large}, \attribute{black}) and (\attribute{large}, \attribute{yellow}). We create two such graphs: one for both the total set of attributes and a second where we consider only objects that refer to people. A subgraph of the 16 most frequently connected (co-occurring) person-related attributes is shown in Figure~\ref{fig:attribute_graphs} (a).
Cliques in these graphs represent groups of attributes in which at least one co-occurrence exists for each pair of attributes in that group. In the previous example, if a third image contained a ``black and yellow taxi'' (\attribute{black}(\object{taxi}), \attribute{yellow}(\object{taxi})), the resulting third edge would create a clique between the attributes \attribute{black}, \attribute{large}, and \attribute{yellow}. When calculated across the entire Visual Genome dataset, these cliques provide insight into commonly perceived traits of different types of objects. Figure~\ref{fig:attribute_graphs} (b) is a selected representation of three example cliques and their overlaps. From just a clique of attributes, we can predict what types of objects are usually referenced. In Figure~\ref{fig:attribute_graphs} (b), we see that these cliques describe an animal (left), water body (top right), and human hair (bottom right).
\begin{figure}[t!]
\centering
\subfloat[]{{\includegraphics[width=0.38\textwidth]{png_graphics/attributes_per_image.png}}}
\qquad
\subfloat[]{{\includegraphics[width=0.38\textwidth]{png_graphics/attributes_per_region.png}}}
\qquad
\subfloat[]{{\includegraphics[width=0.38\textwidth]{png_graphics/attributes_per_object.png}}}
\qquad
\caption{Distribution of the number of attributes (a) per image, (b) per region description, (c) per object.}
\setlength{\belowcaptionskip}{-0.1cm}
\label{fig:attribute_distribution}
\end{figure}
Other cliques (not shown) can also uniquely identify objects. In our set, one clique contains \attribute{athletic}, \attribute{young}, \attribute{fit}, \attribute{skateboarding}, \attribute{focused}, \attribute{teenager}, \attribute{male}, \attribute{skinny}, and \attribute{happy}, capturing some of the common traits of \object{skateboarders} in our set. Another such clique has \attribute{shiny}, \attribute{small}, \attribute{metal}, \attribute{silver}, \attribute{rusty}, \attribute{parked}, and \attribute{empty}, most likely describing a subset of cars. From these cliques, we can thus infer distinct objects and object types based solely on their attributes, potentially allowing for highly specific object identification based on selected characteristics.
\begin{figure*}[t]
\centering
\subfloat[]{{\includegraphics[width=0.55\textwidth]{png_graphics/top_attributes.png}}}
\subfloat[]{{\includegraphics[width=0.45\textwidth]{png_graphics/top_people_attributes.png}}}
\qquad
\caption{(a) Distribution showing the most common attributes in the dataset. Colors (\attribute{white}, \attribute{red}) and materials (\attribute{wooden}, \attribute{metal}) are the most common. (b) Distribution showing the number of attributes describing people. State-of-motion verbs (\attribute{standing}, \attribute{walking}) are the most common, while certain sports (\attribute{skiing}, \attribute{surfing}) are also highly represented due to an image source bias in our image set.}
\label{fig:top_attributes}
\end{figure*}
\begin{figure*}[t]
\centering
\subfloat[]{{\includegraphics[width=0.9\textwidth]{png_graphics/people_attribute_graph.png}}}
\qquad
\subfloat[]{{\includegraphics[width=0.9\textwidth]{png_graphics/cliques_selected.png}}}
\qquad
\caption{(a) Graph of the person-describing attributes with the most co-occurrences. Edge thickness represents the frequency of co-occurrence of the two nodes. (b) A subgraph showing the co-occurrences and intersections of three cliques, which appear to describe water (top right), hair (bottom right), and some type of animal (left). Edges between cliques have been removed for clarity.}
\label{fig:attribute_graphs}
\end{figure*}
\clearpage
\subsection{Relationship Statistics}
\label{sec:relationship_stats}
Relationships are the core components that link objects in our scene graphs. Relationships are directional, i.e.\ they involve two objects, one acting as the subject and one as the object of a predicate relationship. We denote all relationships in the form \relationship{subject}{relationship}{object}. For example, if a \object{man} is \predicate{swinging} a \object{bat}, we write \relationship{man}{swinging}{bat}. Relationships can be spatial (e.g.~\predicate{inside\_of}), action (e.g.~\predicate{swinging}), compositional (e.g.~\predicate{part\_of}), etc. More complex relationships such as \predicate{standing\_on}, which includes both an action and a spatial aspect, are also represented. Relationships are extracted from region descriptions by crowd workers, similarly to attributes and objects. Visual Genome contains a total of $13,894$ unique relationships, with over $1.8$ million total relationships.
Figure~\ref{fig:relationship_distribution} (a) shows the distribution of relationships per region description. On average, we have $1$ relationship per region, with a maximum of $7$. We also have some descriptions like ``an old, tall man,'' which have multiple attributes associated with the \object{man} but no relationships. Figure~\ref{fig:relationship_distribution} (b) is a distribution of relationships per image object. Finally, Figure~\ref{fig:relationship_distribution} (c) shows the distribution of relationships per image. Each image has an average of $19$ relationships, with a minimum of $1$ relationship and with ax maximum of over $60$ relationships.
\begin{figure}[t]%
\centering
\subfloat[]{{\includegraphics[width=0.35\textwidth]{png_graphics/relationships_per_region.png} }}%
\qquad
\vspace{-2mm}
\subfloat[]{{\includegraphics[width=0.35\textwidth]{png_graphics/relationships_per_object.png} }}%
\qquad
\vspace{-2mm}
\subfloat[]{{\includegraphics[width=0.35\textwidth]{png_graphics/relationships_per_image.png} }}%
\caption{Distribution of relationships (a) per image region, (b) per image object, (c) per image.}%
\label{fig:relationship_distribution}%
\setlength{\belowcaptionskip}{-0.1cm}
\end{figure}
\vspace{-0.15cm}
\paragraph{Top relationship distributions.} We display the most frequently occurring relationships in Figure~\ref{fig:top_predicates} (a). \predicate{on} is the most common relationship in our dataset. This is primarily because of the flexibility of the word \predicate{on}, which can refer to spatial configuration (\predicate{on top of}), attachment (\predicate{hanging on}), etc. Other common relationships involve actions like \predicate{holding} and \predicate{wearing} and spatial configurations like \predicate{behind}, \predicate{next to}, and \predicate{under}. Figure~\ref{fig:top_predicates} (b) shows a similar distribution but for relationships involving people. Here we notice more human-centric relationships or actions such as \predicate{kissing}, \predicate{chatting with}, and \predicate{talking to}. The two distributions follow a Zipf distribution”.
\vspace{-0.15cm}
\paragraph{Understanding affordances.} Relationships allow us to also understand the affordances of objects. We show this using a specific distribution of subjects and objects involved in the relationship \predicate{riding} in Figure~\ref{fig:riding_subjects_objects}. Figure~\ref{fig:riding_subjects_objects} (a) shows the distribution for subjects while Figure~\ref{fig:riding_subjects_objects} (b) shows a similar distribution for objects. Comparing the two distributions, we find clear patterns of people-like subject entities such as \object{person}, \object{man}, \object{policeman}, \object{boy}, and \object{skateboarder} that can ride other objects; the other distribution contains objects that afford \predicate{riding}, such as \object{horse}, \object{bike}, \object{elephant}, \object{motorcycle}, and \object{skateboard}. We can also learn specific common-sense knowledge, like that \object{skateboarders} only ride \object{skateboards} and only \object{surfers} ride \object{waves} or \object{surfboards}.
\vspace{-0.15cm}
\paragraph{Related work comparison.} It is also worth mentioning in this section some prior work on relationships. The concept of visual relationships has already been explored in Visual Phrases~\cite{sadeghi2011recognition}, who introduced a dataset of $17$ such relationships such as \relationship{person}{next\_to}{bike} and \relationship{person}{riding}{horse}. However, their dataset is limited to just these $17$ relationships. Similarly, the MS-COCO-a dataset~\cite{2015arXiv150602203R} introduced $140$ actions that humans performed in MS-COCO's dataset~\cite{lin2014microsoft}. However, their dataset is limited to just actions, while our relationships are more general and numerous, with over $13$K unique relationships. Finally, VisKE~\cite{sadeghi2015viske} introduced $6500$ relationships, but in a much smaller dataset of images than Visual Genome.
\begin{figure*}[t]%
\centering
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/top_predicates.png} }}%
\qquad
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/top_people_predicates.png} }}%
\caption{(a) A sample of the most frequent relationships in our dataset. In general, the most common relationships are spatial (\predicate{on top of}, \predicate{on side of}, etc.). (b) A sample of the most frequent relationships involving humans in our dataset. The relationships involving people tend to be more action oriented (\predicate{walk}, \predicate{speak}, \predicate{run}, etc.).}%
\label{fig:top_predicates}%
\end{figure*}
\begin{figure*}[t]%
\centering
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/riding_subjects.png} }}%
\qquad
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/riding_objects.png} }}%
\caption{(a) Distribution of subjects for the relationship \predicate{riding}. (b) Distribution of objects for the relationship \predicate{riding}. Subjects comprise of people-like entities like \object{person}, \object{man}, \object{policeman}, \object{boy}, and \object{skateboarder} that can ride other objects. On the other hand, objects like \object{horse}, \object{bike}, \object{elephant} and \object{motorcycle} are entities that can afford \predicate{riding}.}%
\label{fig:riding_subjects_objects}%
\end{figure*}
\begin{figure*}[t]
\begin{center}
\iftoggle{smallfigs}{
\includegraphics[width=1.0\textwidth]{png_graphics/visual6w_examples_small.png}
}{
\includegraphics[width=1.0\textwidth]{png_graphics/visual6w_examples.png}
}
\caption{Example QA pairs in the Visual Genome dataset. Our QA pairs cover a spectrum of visual tasks from recognition to high-level reasoning.}
\label{fig:visual6w_examples}
\vspace{-1em}
\end{center}
\end{figure*}
\begin{table}%
\centering
\begin{tabular}{lccc}
& Objects & Attributes & Relationships \\
\midrule
Region Graph & 0.43 & 0.41 & 0.45 \\
Scene Graph & 21.26 & 16.21 & 18.67 \\
\bottomrule
\end{tabular}
\caption{The average number of objects, attributes, and relationships per region graph and per scene graph.}
\label{tab:graph_statistics}
\end{table}
\subsection{Region and Scene Graph Statistics}\label{sec:graph_stats}
We introduce in this paper the largest dataset of scene graphs to date. We use these graph representations of images as a deeper understanding of the visual world. In this section, we analyze the properties of these representations, both at the region level through region graphs and at the image level through scene graphs. We also briefly explore other datasets with scene graphs and provide aggregate statistics on our entire dataset.
Scene graphs by asking humans to write triples about an image~\cite{Johnson2015CVPR}. However, unlike them, we collect graphs at a much more fine-grained level, the region graph. We obtained our graphs by asking workers to create them from the descriptions we collected from our regions. Therefore, we end up with multiple graphs for an image, one for every region description. Together, we can combine all the individual region graphs to aggregate a scene graph for an image. This scene graph is made up of all the individual region graphs. In our scene graph representation, we merge all the objects that referenced by multiple region graphs into one node in the scene graph.
Each of our images has a distribution between $40$ to $50$ region graphs per image, with an average of $42$. Each image has exactly one scene graph. Note that the number of region descriptions and the number of region graphs for an image are not the same. For example, consider the description ``it is a sunny day''. Such a description contains no objects, which are the building blocks of a region graph. Therefore, such descriptions have no region graphs associated with them.
Objects, attributes, and relationships occur as a normal distribution in our data. Table~\ref{tab:graph_statistics} shows that in a region graph, there are an average of $0.43$ objects, $0.41$ attributes, and $0.45$ relationships. Each scene graph and consequently each image has average of $21.26$ objects, $16.21$ attributes, and $18.67$ relationships.
\subsection{Question Answering Statistics}
\label{sec:qa_stats}
We collected $1,773,258$ question answering (QA) pairs on the Visual Genome images. Each pair consists of a question and its correct answer regarding the content of an image. On average, every image has $17$ QA pairs. Rather than collecting unconstrained QA pairs as previous work has done~\cite{antol2015vqa,gao2015you,malinowski2014multi}, each question in Visual Genome starts with one of the six Ws -- what, where, when, who, why, and how. There are two major benefits to focusing on six types of questions. First, they offer a considerable coverage of question types, ranging from basic perceptual tasks (e.g.\ recognizing objects and scenes) to complex common sense reasoning (e.g.\ inferring motivations of people and causality of events). Second, these categories present a natural and consistent stratification of task difficulty, indicated by the baseline performance in Section~\ref{sec:answer_generation}. For instance, \emph{why} questions that involve complex reasoning lead to the poorest performance ($3.4\%$ top-100 accuracy) of the six categories. This enables us to obtain a better understanding of the strengths and weaknesses of today's computer vision models, which sheds light on future directions in which to proceed.
We now analyze the diversity and quality of our questions and answers. Our goal is to construct a large-scale visual question answering dataset that covers a diverse range of question types, from basic cognition tasks to complex reasoning tasks. We demonstrate the richness and diversity of our QA pairs by examining the distributions of questions and answers in Figure~\ref{fig:visual6w_examples}.
\paragraph{Question type distributions.} The questions naturally fall into the 6W categories via their interrogative words. Inside each of the categories, the second and following words categorize the questions with increasing granularity. Inspired by VQA~\cite{antol2015vqa}, we show the distributions of the questions by their first three words in Figure~\ref{fig:visual6w-first3-sunburst}. We can see that ``what'' is the most common of the six categories. A notable difference between our question distribution and VQA's is that we focus on ensuring that all 7 question categories are adequately represented, while in VQA, $32.37\%$ of the questions are yes/no binary questions. As a result, a trivial model can achieve a reasonable performance by just predicting ``yes'' or ``no'' as answers. We encourage more difficult QA pairs by ruling out binary questions.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=1.0\linewidth]{pdf_graphics/v6w_total_sunburst.pdf}
\caption{Distribution of question types by starting words. This figure shows the distribution of the questions by their first three words. The angles of the regions are proportional to the number of pairs from the corresponding categories. We can see that ``what'' questions are the largest category with nearly half of the QA pairs.}
\label{fig:visual6w-first3-sunburst}
\end{center}
\end{figure}
\paragraph{Question and answer length distributions.} We also analyze the question and answer lengths of each 6W category. Figure~\ref{fig:visual6w-sentence_lengths_by_question_type} shows the average question and answer lengths of each category. Overall, the average question and answer lengths are 5.7 and 1.8 words respectively. In contrast to the VQA dataset, where $.88\%$, $8.38\%$, and $3.25\%$ of the answers consist of one, two, or three words, our answers exhibit a long-tail distribution where $57.3\%$, $18.1\%$, and $15.7\%$ of the answers have one, two, or three words respectively.
We avoid verbosity by instructing the workers to write answers as concisely as possible. The coverage of long answers means that many answers contain a short description that contains more details than merely an object or an attribute. It shows the richness and complexity of our visual QA tasks beyond object-centric recognition tasks. We foresee that these long-tail questions can motivate future research in common-sense reasoning and high-level image understanding.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{pdf_graphics/sentence_lengths_by_question_type.pdf}
\caption{Question and answer lengths by question type. The bars show the average question and answer lengths of each question type. The whiskers show the standard deviations. The factual questions, such as ``what'' and ``how'' questions, usually come with short answers of a single object or a number. This is only because ``how'' questions are disproportionately counting questions that start with ``how many''. Questions from the ``where'' and ``why'' categories usually have phrases and sentences as answers.}
\label{fig:visual6w-sentence_lengths_by_question_type}
\end{figure}
\begin{figure*}[ht]
\centering
\iftoggle{smallfigs}{
\includegraphics[width=1.0\linewidth]{pdf_graphics/canonicalization_pipeline.pdf}
}{
\includegraphics[width=1.0\linewidth]{png_graphics/canonicalization_pipeline.png}
}
\caption{An example image from the Visual Genome dataset with its region descriptions, QA, objects, attributes, and relationships canonicalized. The large text boxes are WordNet synsets referenced by this image. For example, the \object{carriage} is mapped to \synset{carriage.n.02: a vehicle with wheels drawn by one or more horses.} We do not show the bounding boxes for the objects in order to allow readers to see the image clearly. We also only show a subset of the scene graph for this image to avoid cluttering the figure.}
\label{fig:canonicalization_pipeline}
\end{figure*}
\subsection{Canonicalization Statistics}
\label{sec:canonicalization_stats}
In order to reduce the ambiguity in the concepts of our dataset and connect it to other resources used by the research community, we canonicalize the semantic meanings of all objects, relationships, and attributes in Visual Genome. By ``canonicalization,'' we refer to word sense disambiguation (WSD) by mapping the components in our dataset to their respective synsets in the WordNet ontology~\cite{miller1995WordNet}. This mapping reduces the noise in the concepts contained in the dataset and also facilitates the linkage between Visual Genome and other data sources such as ImageNet~\cite{deng2009imagenet}, which is built on top of the WordNet ontology.
Figure~\ref{fig:canonicalization_pipeline} shows an example image from the Visual Genome dataset with its components canonicalized. For example, \object{horse} is canonicalized as \synset{horse.n.01: solid-hoofed herbivorous quadruped domesticated since prehistoric times}. Its attribute, \attribute{clydesdale}, is canonicalized as its breed \synset{clydesdale.n.01: heavy feathered-legged breed of draft horse originally from Scotland}. We also show an example of a QA from which we extract the nouns \object{shamrocks}, \object{symbol}, and \object{St. Patrick's day}, all of which we canonicalize to WordNet as well.
\vspace{-1em}
\paragraph{Related work.} Canonicalization, or WSD~\cite{pal2015word}, has been used in numerous applications, including machine translation, information retrieval, and information extraction~\cite{rothe2015autoextend, leacock1998using}. In English sentences, sentences like ``He scored a goal'' and ``It was his goal in life'' carry different meanings for the word ``goal.'' Understanding these differences is crucial for translating languages and for returning correct results for a query. Similarly, in Visual Genome, we ensure that all our components are canonicalized to understand how different objects are related to each other; for example, ``person'' is a hypernym of ``man'' and ``woman.'' Most past canonicalization models use precision, recall, and F1 score to evaluate on the Semeval dataset~\cite{mihalcea2004senseval}. The current state-of-the-art performance on Semeval is an F1 score of $75.8\%$~\cite{chen2014unified}. Since our canonicalization setup is different from the Semeval benchmark (we have an open vocabulary and no annotated ground truth for evaluation), our canonicalization method is not directly comparable to these existing methods. We do however, achieve a similar precision and recall score on a held-out test set described below.
\paragraph{Region descriptions and QAs.} We canonicalize all objects mentioned in all region descriptions and QA pairs. Because objects need to be extracted from the phrase text, we use Stanford NLP tools~\cite{manning-EtAl:2014:P14-5} to extract the noun phrases in each region description and QA, resulting in $99\%$ recall of noun phrases from a subset of $200$ region descriptions we manually annotated. After obtaining the noun phrases, we map each to its most frequent matching synset (according to WordNet lexeme counts).
This resulted in an overall mapping accuracy of $86\%$ and a recall of $98.5\%$. The most common synsets extracted from region descriptions, QAs, and objects are shown in Figure~\ref{fig:region_object_synset_distributions}.
\begin{table}[t]%
\centering
\begin{tabular}{l c c}
& Precision & Recall\\% & Accuracy \\
\midrule
Objects & 88.0 & 98.5\\% & 86.0 \\
Attributes & 85.7 & 95.9\\% & 83.5 \\
Relationships & 92.9 & 88.5\\% & 77.6 \\
\bottomrule
\end{tabular}
\caption{Precision, recall, and mapping accuracy percentages for object, attribute, and relationship canonicalization.}
\label{tab:canon_statistics}
\vspace{-1em}
\end{table}
\vspace{-1em}
\paragraph{Attributes.} We canonicalize attributes from the crowd-extracted attributes present in our scene graphs. The ``attribute'' designation encompasses a wide range of grammatical parts of speech. Because part-of-speech taggers rely on high-level syntax information and thus fail on the disjoint elements of our scene graphs, we normalize each attribute based on morphology alone (so-called ``stemming''). Then, as with objects, we map each attribute phrase to the most frequent matching WordNet synset. We include 15 hand-mapped rules to address common failure cases in which WordNet's frequency counts prefer abstract senses of words over the spatial senses present in visual data, e.g.\ ``short.a.01: limited in duration'' over \synset{short.a.02: lacking in length}. For verification, we randomly sample $200$ attributes, produce ground-truth mappings by hand, and compare them to the results of our algorithm. This resulted in a recall of $95.9\%$ and a mapping accuracy of $83.5\%$. The most common attribute synsets are shown in Figure~\ref{fig:attribute_relationship_synset_distributions} (a).
\paragraph{Relationships.} As with attributes, we canonicalize the relationships isolated in our scene graphs. We exclude prepositions, which are not recognized in WordNet, leaving a set primarily composed of verb relationships. Since the meanings of verbs are highly dependent upon their morphology and syntactic placement (e.g.\ passive cases, prepositional phrases), we map the structure of each relationship to the appropriate WordNet sentence frame and only consider those WordNet synsets with matching sentence frames. For each verb-synset pair, we then consider the root hypernym of that synset to reduce potential noise from WordNet's fine-grained sense distinctions. We also include 20 hand-mapped rules, again to correct for WordNet's lower representation of concrete or spatial senses; for example, the concrete \synset{hold.v.02: have or hold in one's hand or grip} is less frequent in WordNet than the abstract \synset{hold.v.01: cause to continue in a certain state}. For verification, we again randomly sample $200$ relationships and compare the results of our canonicalization against ground-truth mappings. This resulted in a recall of $88.5\%$ and a mapping accuracy of $77.6\%$. While several datasets, such as VerbNet~\cite{schuler2005verbnet} and FrameNet~\cite{baker1998framenet}, include semantic restrictions or frames to improve classification, there is no comprehensive method of mapping to those restrictions or frames. The most common relationship synsets are shown in Figure~\ref{fig:attribute_relationship_synset_distributions} (b).
\begin{figure*}[htbp]
\centering
\subfloat[]{{\includegraphics[width=0.5\linewidth]{png_graphics/region_synset_distribution.png}}}
\subfloat[]{{\includegraphics[width=0.5\linewidth]{png_graphics/object_synset_distribution.png}}}
\caption{Distribution of the 25 most common synsets mapped from (a) region descriptions and question answers and (b) objects.}
\label{fig:region_object_synset_distributions}
\end{figure*}
\begin{figure*}[htbp]
\centering
\subfloat[]{{\includegraphics[width=0.49\linewidth]{png_graphics/attribute_synset_distribution.png}}}
\subfloat[]{{\includegraphics[width=0.51\linewidth]{png_graphics/relationship_synset_distribution.png}}}
\caption{Distribution of the 25 most common synsets mapped from (a) attributes and (b) relationships.}
\label{fig:attribute_relationship_synset_distributions}
\end{figure*}
\section{Experiments}
\label{sec:experiments}
Thus far, we have presented the Visual Genome dataset and analyzed its individual components. With such rich information provided, numerous perceptual and cognitive tasks can be tackled. In this section, we aim to provide baseline experimental results using components of Visual Genome that have not been extensively studied.
Object detection is already a well-studied problem~\cite{everingham2010pascal, girshick2014rich, sermanet2013overfeat, girshick2015fast, ren2015faster}. Similarly, region graphs and scene graphs have been shown to improve semantic image retrieval~\cite{Johnson2015CVPR, schustergenerating}. We therefore focus on the remaining components, i.e. \textit{attributes}, \textit{relationships}, \textit{region descriptions}, and \textit{question answer pairs}.
In Section~\ref{sec:attribute_classification}, we present results for two experiments on attribute prediction. In the first, we treat attributes independently from objects and train a classifier for each attribute, i.e.\ a classifier for \attribute{red} or a classifier for \attribute{old}, as in~\cite{malisiewicz2008recognition, varma2005statistical, ferrari2007learning, farhadi2009describing, Johnson2015CVPR}. In the second experiment, we learn object and attribute classifiers \emph{jointly} and predict object-attribute pairs (e.g.\ predicting that an \object{apple} is \attribute{red}), as in \cite{sadeghi2011recognition}.
In Section~\ref{sec:relationship_classification}, we present two experiments on relationship prediction. In the first, we aim to predict the predicate between two objects, e.g.\ predicting the predicate \predicate{kicking} or \predicate{wearing} between two objects. This experiment is synonymous with existing work in action recognition~\cite{gupta2009observing, ramanathan2015learning}. In another experiment, we study relationships by classifying jointly the objects and the predicate (e.g.\ pre\-dicting \relationship{man}{kicking}{ball}); we show that this is a very difficult task due to the high variability in the appearance of a relationship (e.g.\ the \object{ball} might be on the ground or in mid-air above the \object{man}). These experiment are generalizations of tasks that study spatial relationships between objects and ones that jointly reason about the interaction of humans with objects~\cite{yao2010modeling, prest2012weakly}.
In Section~\ref{sec:description_generation} we present results for region captioning. This task is closely related to image captioning~\cite{chen2015microsoft}; however, results from the two are not directly comparable, as region descriptions are short, incomplete sentences. We train one of the top 16 state-of-the-art image caption generator~\cite{karpathy2014deep} on (1) our dataset to generate region descriptions and on (2) Flickr30K~\cite{young2014image} to generate sentence descriptions. To compare results between the two training approaches, we use simple templates to convert region descriptions into complete sentences. For a more robust evaluation, we validate the descriptions we generate using human judgment.
Finally, in Section~\ref{sec:answer_generation}, we experiment on visual question answering, i.e.\ given an image and a question, we attempt to provide an answer for the question. We report results on the retrieval of the correct answer from a list of existing answers.
\subsection{Attribute Prediction}
\label{sec:attribute_classification}
\begin{figure*}[t]
\centering
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{pdf_graphics/attribute_experiment_examples.pdf} }}
}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/attribute_experiment_examples.png} }}
}
\qquad
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{pdf_graphics/attribute_pair_experiment_examples.pdf}}}
}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/attribute_pair_experiment_examples.png}}}
}
\caption{(a) Example predictions from the attribute prediction experiment. Attributes in the first row are predicted correctly, those in the second row differ from the ground truth but still correctly classify an attribute in the image, and those in the third row are classified incorrectly. The model tends to associate objects with attributes (e.g. \object{elephant} with \object{grazing}). (b) Example predictions from the joint object-attribute prediction experiment.}
\label{fig:attribute_experiment}
\end{figure*}
Attributes are becoming increasingly important in the field of computer vision, as they offer higher-level semantic cues for various problems and lead to a deeper understanding of images. We can express a wide variety of properties through attributes, such as form (\attribute{sliced}), function (\attribute{decorative}), sentiment (\attribute{angry}), and even intention (\attribute{helping}). Distinguishing between similar objects~\cite{isola2015discovering} leads to finer-grained classification, while describing a previously unseen class through attributes shared with known classes can enable ``zero-shot'' learning~\cite{farhadi2009describing, lampert2009learning}. Visual Genome is the largest dataset of attributes, with $18$ attributes per image for a total of $1.8$ million attributes.
\paragraph{Setup.}
For both experiments, we focus on the $100$ most common attributes in our dataset. We only use objects that occur at least $100$ times and are associated with one of the $100$ attributes in at least one image. For both experiments, we follow a similar data pre-processing pipeline. First, we lowercase, lemmatize, and strip excess whitespace from all attributes. Since the number of examples per attribute class varies, we randomly sample $500$ attributes from each category (if fewer than $500$ are in the class, we take all of them).
We end up with around $50,000$ attribute instances and $43,000$ object-attribute pair instances in total. We use $80\%$ of the images for training and $10\%$ each for validation and testing. Because each image has about the same number of examples, this results in an approximately $80\%$-$10\%$-$10\%$ split over the attributes themselves. The input data for this experiment is the cropped bounding box of the object associated with each attribute.
We train an attribute predictor by using features learned from a convolutional neural network. Specifically, we fine-tune a 16-layer VGG network~\cite{simonyan2014very} for both of these experiments using the $50,000$ attribute and $43,000$ object-attribute pair instances respectively. We modify the network so that the learning rate of the final fully-connected layer is 10 times that of the other layers, as this improves convergence time. We use a base learning rate of 0.001, which we scale by 0.1 every $200$ iterations, and momentum and weight decays of $0.9$ and $0.0005$ respectively. We use the fine-tuned features from the network and train $100$ individual SVMs~\cite{hearst1998support} to predict each attribute. We output multiple attributes for each bounding box input. For the second experiment, we also output the object class.
\paragraph{Results.} Table~\ref{tab:attribute_experiment_results} shows results for both experiments. For the first experiment on attribute prediction, we converge after around $700$ iterations with $18.97\%$ top-one accuracy and $43.11\%$ top-five accuracy. Thus, attributes (like objects) are visually distinguishable from each other. For the second experiment where we also predict the object class, we converge after around $400$ iterations with $43.17\%$ top-one accuracy and $71.97\%$ top-five accuracy. Predicting objects jointly with attributes increases the top-one accuracy from $18.97\%$ to $43.17\%$. This implies that some attributes occur exclusively with a small number of objects. Additionally, by jointly learning attributes with objects, we increase the inter-class variance, making the classification process an easier task.
Figure~\ref{fig:attribute_experiment} (a) shows example predictions for the first attribute prediction experiment. In general, the model is good at associating objects with their most salient attributes, for example, \object{animal} with \attribute{stuffed} and \object{elephant} with \attribute{grazing}. However, there is some difference between the user-provided result and the correct ground truth, so the model incorrectly classifies some correct predictions. For example, the \attribute{white} stuffed animal is correct but evaluated as incorrect.
Figure~\ref{fig:attribute_experiment} (b) shows example predictions for the second experiment in which we also predict the object. While the results in the second row might be considered correct, to keep a consistent evaluation, we mark them as incorrect. For example, the predicted ``green grass'' might be considered subjectively correct even though it is annotated as ``brown grass''. For cases where the objects are not clearly visible but are abstract outlines, our model is unable to predict attributes or objects accurately. For example, it thinks that the ``flying bird'' is actually a ``black jacket''.
The attribute clique graphs in Section~\ref{sec:attribute_stats} clearly show that learning attributes can help us identify types of objects. This experiment strengthens that insight. We learn that studying attributes together with objects can improve attribute prediction.
\begin{table}[t]
\centering
\begin{tabular}{lccc}
& Top-1 Accuracy & Top-5 Accuracy \\
\midrule
Attribute & 18.97\% & 43.11\% \\
Object-Attribute & 43.17\% & 71.97\% \\
\bottomrule
\end{tabular}
\caption{(First row) Results for the attribute prediction task where we only predict attributes for a given image crop. (Second row) Attribute-object prediction experiment where we predict both the attributes as well as the object from a given crop of the image.}
\label{tab:attribute_experiment_results}
\end{table}
\subsection{Relationship Prediction}
\label{sec:relationship_classification}
\begin{figure*}[t]
\centering
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{pdf_graphics/relationship_experiment_examples.pdf}}}
}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/relationship_experiment_examples.png}}}
}
\qquad
\iftoggle{smallfigs}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{pdf_graphics/relationship_triple_experiment_examples.pdf}}}
}{
\subfloat[]{{\includegraphics[width=0.46\textwidth]{png_graphics/relationship_triple_experiment_examples.png}}}
}
\caption{(a) Example predictions from the relationship prediction experiment. Relationships in the first row are predicted correctly, those in the second row differ from the ground truth but still correctly classify a relationship in the image, and those in the third row are classified incorrectly. The model learns to associate animals leaning towards the ground as \predicate{eating} or \predicate{drinking} and bikes with \predicate{riding}. (b) Example predictions from the relationship-objects prediction experiment. The figure is organized in the same way as Figure (a). The model is able to predict the salient features of the image but fails to distinguish between different objects (e.g. \object{boy} and \object{woman} and \object{car} and \object{bus} in the bottom row).}
\label{fig:relationship_experiment}
\vspace{-1.0em}
\end{figure*}
While objects are the core building blocks of an image, relationships put them in context. These relationships help distinguish between images that contain the same objects but have different holistic interpretations. For example, an image of ``a man riding a bike'' and ``a man falling off a bike'' both contain \object{man} and \object{bike}, but the relationship (\predicate{riding} vs. \predicate{falling\_off}) changes how we perceive both situations. Visual Genome is the largest known dataset of relationships, with a total of $1.8$ million relationships and an average of $18$ relationships per image.
\paragraph{Setup.} The setups of both experiments are similar to those of the experiments we performed on attributes. We again focus on the top $100$ most frequent relationships. We lowercase, lemmatize, and strip excess whitespace from all relationships. We end up with around $34,000$ relationships and $27,000$ subject-relationship-object triples for training, validation, and testing. The input data to the experiment is the image region containing the union of the bounding boxes of the subject and object (essentially, the bounding box containing the two object boxes). We fine-tune a 16-layer VGG network~\cite{simonyan2014very} with the same learning rates mentioned in Section~\ref{sec:attribute_classification}.
\paragraph{Results.} Overall, we find that relationships are not visually distinct enough for our discriminative model to learn effectively. Table~\ref{tab:relationship_experiment_results} shows results for both experiments. For relationship classification, we converge after around $800$ iterations with $8.74\%$ top-one accuracy and $29.69\%$ top-five accuracy. Unlike attribute prediction, the accuracy results for relationships are much lower because of the high intra-class variability of most relationships.
For the second experiment jointly predicting the relationship and its two object classes, we converge after around $450$ iterations with $25.83\%$ top-one accuracy and $65.57\%$ top-five accuracy. We notice that object classification aids relationship prediction. Some relationships occur with some objects and never others; for example, the relationship \predicate{drive} only occurs with the object \object{person} and never with any other objects (\object{dog}, \object{chair}, etc.).
Figure~\ref{fig:relationship_experiment} (a) shows example predictions for the relationship classification experiment. In general, the model associates object categories with certain relationships (e.g.\ animals with \predicate{eating} or \predicate{drinking}, bikes with \predicate{riding}, and kids with \predicate{playing}).
Figure~\ref{fig:relationship_experiment} (b), structured as in Figure~\ref{fig:relationship_experiment} (a), shows example predictions for the joint prediction of relationships with its objects. The model is able to predict the salient features of the image (e.g. ``boat in water'') but fails to distinguish between different objects (e.g. \object{boy} vs. \object{woman} and \object{car} vs. \object{bus} in the bottom row).
\begin{table}[t]
\centering
\begin{tabular}{lccc}
& Top-1 Accuracy & Top-5 Accuracy \\
\midrule
Relationship & 8.74\% & 26.69\% \\
Sub./Rel./Obj. & 25.83\% & 65.57\% \\
\bottomrule
\end{tabular}
\caption{Results for relationship classification (first row) and joint classification (second row) experiments.}
\label{tab:relationship_experiment_results}
\end{table}
\subsection{Generating Region Descriptions}
\label{sec:description_generation}
\begin{figure}[htbp]
\centering
\iftoggle{smallfigs}{
\includegraphics[width=0.46\textwidth]{pdf_graphics/description_generation_examples.pdf}
}{
\includegraphics[width=0.46\textwidth]{png_graphics/description_generation_examples.png}
}
\caption{Example predictions from the region description generation experiment. Regions in the first column (left) accurately describe the region, and those in the second column (right) are incorrect and unrelated to the corresponding region.}
\label{fig:description_generation_examples}
\end{figure}
Generating sentence descriptions of images has gained popularity as a task in computer vision~\cite{kiros2014multimodal, mao2014explain, karpathy2014deep, vinyals2014show}; however, current state-of-the-art models fail to describe all the different events captured in an image and instead provide only a high-level summary of the image.
In this section, we test how well state-of-the-art models can caption the details of images. For both experiments, we use the NeuralTalk model~\cite{karpathy2014deep}, since it not only provides state-of-the-art results but also is shown to be robust enough for predicting short descriptions. We train NeuralTalk on the Visual Genome dataset for region descriptions and on Flickr30K~\cite{young2014image} for full sentence descriptions.
As a model trained on other datasets would generate complete sentences and would not be comparable~\cite{chen2015microsoft} to our region descriptions, we convert all region descriptions generated by our model into complete sentences using predefined templates~\cite{hou2002template}.
\paragraph{Setup.} For training, we begin by preprocessing region descriptions; we remove all non-alphanumeric characters and lowercase and strip excess whitespace from them. We have $4,158,841$ region descriptions in total. We end up with $3,150,000$ region descriptions for training -- $504,420$ each for validation and testing. Note that we ensure descriptions of regions from the same image are exclusively in the training, validation, or testing set. We feed the bounding boxes of the regions through the pretrained VGG 16-layer network~\cite{simonyan2014very} to get the 4096-dimensional feature vectors of each region. We then use the NeuralTalk~\cite{karpathy2014deep} model to train a long short-term memory (LSTM) network~\cite{hochreiter1997long} to generate descriptions of regions. We use a learning rate of $0.001$ trained with rmsprop~\cite{dauphin2015rmsprop}. The model converges after four days.
For testing, we crop the ground-truth region bounding boxes of images and extract their 4096-dimensional 16-layer VGG network~\cite{simonyan2014very} features. We then feed these vectors through the pretrained NeuralTalk model to get predictions for region descriptions.
\paragraph{Results.} Table~\ref{tab:description_generation_results} shows the results for the experiment. We calculate BLEU, CIDEr, and METEOR scores~\cite{chen2015microsoft} between the generated descriptions and their ground-truth descriptions. In all cases, the model trained on VisualGenome performs better. Moreover, we asked crowd workers to evaluate whether a generated description was correct---we got $1.6\%$ and $43.03\%$ for models trained on Flickr30K and on Visual Genome, respectively. The large increase in accuracy when the model trained on our data is due to the specificity of our dataset. Our region descriptions are shorter and cover a smaller image area. In comparison, the Flickr30K data are generic descriptions of entire images with multiple events happening in different regions of the image. The model trained on our data is able to make predictions that are more likely to concentrate on the specific part of the image it is looking at, instead of generating a summary description. The objectively low accuracy in both cases illustrates that current models are unable to reason about complex images.
Figure~\ref{fig:description_generation_examples} shows examples of regions and their predicted descriptions. Since many examples have short descriptions, the predicted descriptions are also short as expected; however, this causes the model to fail to produce more descriptive phrases for regions with multiple objects or with distinctive objects (i.e.\ objects with many attributes). While we use templates to convert region descriptions into sentences, future work can explore smarter approaches to combine region descriptions and generate a paragraph connecting all the regions into one coherent description.
\begin{table*}[t]
\centering
\begin{tabular}{lccccccc}
& BLEU-1 & BLEU-2 & BLEU-3 & BLEU-4 & CIDEr & METEOR & Human \\
\midrule
Flickr8K & 0.09 & 0.01 & 0.002 & 0.0004 & 0.05 & 0.04 & 1.6\% \\
VG & 0.17 & 0.05 & 0.02 & 0.01 & 0.30 & 0.09 & 43.03\% \\
\bottomrule
\end{tabular}
\caption{Results for the region description generation experiment. Scores in the first row are for the region descriptions generated from the NeuralTalk model trained on Flickr8K, and those in the second row are for those generated by the model trained on Visual Genome data. BLEU, CIDEr, and METEOR scores all compare the predicted description to a ground truth in different ways.}
\label{tab:description_generation_results}
\vspace{1.5em}
\end{table*}
\subsection{Question Answering}
\label{sec:answer_generation}
Visual Genome is currently the largest dataset of visual question answers with $1.7$ million question and answer pairs. Each of our $108,249$ images contains an average of $17$ question answer pairs. Answering questions requires a deeper understanding of an image than generic image captioning. Question answering can involve fine-grained recognition (e.g. ``What is the breed of the dog?''), object detection (e.g. ``Where is the kite in the image?''), activity recognition (e.g. ``What is this man doing?''), knowledge base reasoning (e.g. ``Is this glass full?''), and common-sense reasoning (e.g. ``What street will we be on if we turn right?'').
By leveraging the detailed annotations in the scene graphs in Visual Genome, we envision building smart models that can answer a myriad of visual questions. While we encourage the construction of smart models, in this paper, we provide some baseline metrics to help others compare their models.
\paragraph{Setup.} We split the QA pairs into a training set ($60\%$) and a test set ($40\%$). We ensure that all images are exclusive to either the training set or the test set. We implement a simple baseline model that relies on answer frequency. The model counts the top $k$ most frequent answers (similar to the ImageNet challenge~\cite{ILSVRC15}) in the training set as the predictions for all the test questions, where $k=100$, $500$, and $1000$. We let a model make $k$ different predictions. We say the model is correct on a QA if one of the $k$ predictions matches exactly with the ground-truth answer. We report the accuracy over all test questions. This evaluation method works well when the answers are short, especially for single-word answers. However, it causes problems when the answers are long phrases and sentences. Other evaluation methods require word ontologies~\cite{malinowski2014multi}, multiple choices~\cite{antol2015vqa,VisualMadlibs}, or human judges~\cite{gao2015you}.
\begin{table}[t!]
\centering
\begin{tabular}{l c c c}
& top-100 & top-500 & top-1000\\
\midrule
What & 0.420 & 0.602 & 0.672\\
Where & 0.096 & 0.324 & 0.418\\
When & 0.714 & 0.809 & 0.834\\
Who & 0.355 & 0.493 & 0.605\\
Why & 0.034 & 0.118 & 0.187\\
How & 0.780 & 0.827 & 0.846\\
\midrule
Overall\qquad & 0.411 & 0.573 & 0.641\\
\bottomrule
\end{tabular}
\caption{Baseline QA performances (in accuracy).}
\label{tab:qa-baseline}
\end{table}
\paragraph{Results.} Table~\ref{tab:qa-baseline} shows the performance of the open-ended visual question answering task. These baseline results imply the long-tail distribution of the answers. Long-tail distribution is common in existing QA datasets as well~\cite{antol2015vqa, malinowski2014multi}. The top 100, 500, and 1000 most frequent answers only cover $41.1\%$, $57.3\%$, and $64.1\%$ of the correct answers. In comparison, the corresponding sets of frequent answers in VQA~\cite{antol2015vqa} cover $63\%$, $75\%$, and $80\%$ of the test set answers. The ``where'' and ``why'' questions, which tend to involve spatial and common sense reasoning, tend to have more diverse answers and hence perform poorly, with performances of $0.096\%$ and $0.024\%$ top-100 respectively. The top 1000 frequent answers cover only $41.8\%$ and $18.7\%$ of the correct answers from these two question types respectively.
\section{Introduction}
\label{sec:introduction}
\begin{figure*}[t]%
\centering
\includegraphics[width=0.82\textwidth]{png_graphics/pipeline.png}
\caption{An overview of the data needed to move from perceptual awareness to cognitive understanding of images. We present a dataset of images densely annotated with numerous region descriptions, objects, attributes, and relationships. Region descriptions (e.g. ``girl feeding large elephant'' and ``a man taking a picture behind girl'') are shown (top). The objects (e.g. \object{elephant}), attributes (e.g. \attribute{large}) and relationships (e.g. \predicate{feeding}) are shown (bottom). Our dataset also contains image related question answer pairs (not shown).}%
\label{fig:pipeline}%
\end{figure*}
A holy grail of computer vision is the complete understanding of visual scenes: a model that is able to name and detect objects, describe their attributes, and recognize their relationships and interactions. Understanding scenes would enable important applications such as image search, question answering, and robotic interactions. Much pro\-gress has been made in recent years towards this goal, including image classification~\cite{deng2009imagenet, perronnin2010improving, simonyan2014very, krizhevsky2012imagenet, szegedy2014going} and object detection~\cite{everingham2010pascal, girshick2014rich, sermanet2013overfeat, girshick2015fast, ren2015faster}. An important contributing factor is the availability of a large amount of data that drives the statistical models that underpin today's advances in computational visual understanding. While the progress is exciting, we are still far from reaching the goal of comprehensive scene understanding. As Figure~\ref{fig:pipeline} shows, existing models would be able to detect discreet objects in a photo but would not be able to explain their interactions or the relationships between them. Such explanations tend to be \textit{cognitive} in nature, integrating \textit{perceptual} information into conclusions about the relationships between objects in a scene~\cite{bruner1990culture, firestone2015cognition}. A cognitive understanding of our visual world thus requires that we complement computers' ability to detect objects with abilities to describe those objects~\cite{isola2015discovering} and understand their interactions within a scene~\cite{sadeghi2011recognition}.
There is an increasing effort to put together the next generation of datasets to serve as training and benchmarking datasets for these deeper, cognitive scene understanding and reasoning tasks, the most notable being MS-COCO~\cite{lin2014microsoft} and VQA~\cite{antol2015vqa}. The MS-COCO dataset consists of $300$K real-world photos collected from Flickr. For each image, there is pixel-level segmentation of $91$ object classes (when present) and $5$ independent, user-generated sentences describing the scene. VQA adds to this a set of $614$K question-answer pairs related to the visual contents of each image (see more details in Section~\ref{sec:datasets}). With this information, MS-COCO and VQA provide a fertile training and testing ground for models aimed at tasks for accurate object detection, segmentation, and summary-level image captioning~\cite{kiros2014multimodal, mao2014explain, karpathy2014deep, vinyals2014show} as well as basic QA~\cite{ren2015image, antol2015vqa, malinowski2015ask, gao2015you, malinowski2014multi}. For example, a state-of-the-art model~\cite{karpathy2014deep} provides a description of one MS-COCO image in Figure~\ref{fig:pipeline} as ``two men are standing next to an elephant.'' But what is missing is the further understanding of where each object is, what each person is doing, what the relationship between the person and elephant is, etc. Without such relationships, these models fail to differentiate this image from other images of people next to elephants.
To understand images thoroughly, we believe three key elements need to be added to existing datasets: a \textbf{grounding of visual concepts to language}~\cite{kiros2014multimodal}, a more \textbf{complete set of descriptions and QAs} for each image based on multiple image regions~\cite{Johnson2015CVPR}, and a \textbf{formalized representation} of the components of an image~\cite{hayes1978naive}. In the spirit of mapping out this complete information of the visual world, we introduce the Visual Genome dataset. The first release of the Visual Ge\-nome dataset uses $108,249$ images from the intersection of the YFCC100M~\cite{thomee2015yfcc100m} and MS-COCO~\cite{lin2014microsoft}. Section~\ref{sec:dataset_statistics} provides a more detailed description of the dataset. We highlight below the motivation and contributions of the three key elements that set Visual Ge\-nome apart from existing datasets.
The Visual Genome dataset regards relationships and attributes as first-class citizens of the annotation space, in addition to the traditional focus on objects. Recognition of relationships and attributes is an important part of the complete understanding of the visual scene, and in many cases, these elements are key to the story of a scene (e.g., the difference between ``a dog chasing a man'' versus ``a man chasing a dog''). The Visual Genome dataset is among the first to provide a detailed labeling of object interactions and attributes, \textbf{grounding visual concepts to language}\footnotemark.
\footnotetext{The Lotus Hill Dataset~\cite{yao2007introduction} also provides a similar annotation of object relationships, see Sec~\ref{sec:datasets}.}
An image is often a rich scenery that cannot be fully described in one summarizing sentence. The scene in Figure~\ref{fig:pipeline} contains multiple ``stories'': ``a man taking a photo of elephants,'' ``a woman feeding an elephant,'' ``a river in the background of lush grounds,'' etc. Existing datasets such as Flickr 30K~\cite{young2014image} and MS-COCO~\cite{lin2014microsoft} focus on high-level descriptions of an image\footnotemark. Instead, for each image in the Visual Genome dataset, we collect more than 42 descriptions for different regions in the image, providing a much denser and \textbf{complete set of descriptions of the scene}. In addition, inspired by VQA~\cite{antol2015vqa}, we also collect an average of $17$ question-answer pairs based on the descriptions for each image. Region-based question answers can be used to jointly develop NLP and vision models that can answer questions from either the description or the image, or both of them.
\footnotetext{COCO has multiple sentences generated independently by different users, all focusing on providing an overall, one sentence description of the scene.}
With a set of dense descriptions of an image and the explicit correspondences between visual pixels (i.e.\ bounding boxes of objects) and textual descriptors (i.e.\ relationships, attributes), the Visual Ge\-nome dataset is poised to be the first image dataset that is capable of providing a structured \textbf{formalized representation} of an image, in the form that is widely used in knowledge base representations in NLP~\cite{zhou122007tree, guodong2005exploring, culotta2004dependency, socher2012semantic}. For example, in Figure~\ref{fig:pipeline}, we can formally express the relationship \predicate{holding} between the \object{woman} and \object{food} as \relationship{woman}{holding}{food)}. Putting together all the objects and relations in a scene, we can represent each image as a scene graph~\cite{Johnson2015CVPR}. The scene graph representation has been shown to improve semantic image retrieval~\cite{Johnson2015CVPR, schustergenerating} and image captioning~\cite{farhadi2009describing, chang2014semantic, gupta2008beyond}. Furthermore, all objects, attributes and relationships in each image in the Visual Genome dataset are canonicalized to its corresponding WordNet~\cite{miller1995WordNet} ID (called a synset ID). This mapping connects all images in Visual Genome and provides an effective way to consistently query the same concept (object, attribute, or relationship) in the dataset. It can also potentially help train models that can learn from contextual information from multiple images.
In this paper, we introduce the Visual Genome dataset with the aim of training and benchmarking the next generation of computer models for comprehensive scene understanding. The paper proceeds as follows: In Section~\ref{sec:data_representation}, we provide a detailed description of each component of the dataset. Section~\ref{sec:related_works} provides a literature review of related datasets as well as related recognition tasks. Section~\ref{sec:crowdsourcing_pipeline} discusses the crowdsourcing strategies we deployed in the ongoing effort of collecting this dataset. Section~\ref{sec:dataset_statistics} is a collection of data analysis statistics, showcasing the key properties of the Visual Genome dataset. Last but not least, Section~\ref{sec:experiments} provides a set of experimental results that use Visual Genome as a benchmark.
Further visualizations, API, and additional information on the Visual Genome dataset can be found online\footnote{\url{https://visualgenome.org}}.
\begin{figure*}[t]%
\centering
\iftoggle{smallfigs}{
\includegraphics[width=0.82\textwidth]{pdf_graphics/scene_graph_1.pdf}
}{
\includegraphics[width=0.82\textwidth]{png_graphics/scene_graph_1.png}
}
\caption{An example image from the Visual Genome dataset. We show 3 region descriptions and their corresponding region graphs. We also show the connected scene graph collected by combining all of the image's region graphs. The top region description is ``a man and a woman sit on a park bench along a river.'' It contains the objects: \object{man}, \object{woman}, \object{bench} and \object{river}. The relationships that connect these objects are: \relationship{man}{sits\_on}{bench}, \relationship{man}{in\_front\_of}{river}, and \relationship{woman}{sits\_on}{bench}.}%
\label{fig:scene_graph_1}%
\end{figure*}
\FloatBarrier
\begin{figure*}[t]%
\centering
\iftoggle{smallfigs}{
\includegraphics[width=0.8\textwidth]{pdf_graphics/scene_graph_2.pdf}
}{
\includegraphics[width=0.8\textwidth]{png_graphics/scene_graph_2.png}
}
\caption{An example image from our dataset along with its scene graph representation. The scene graph contains objects (\object{child}, \object{instructor}, \object{helmet}, etc.) that are localized in the image as bounding boxes (not shown). These objects also have attributes: \attribute{large}, \attribute{green}, \attribute{behind}, etc. Finally, objects are connected to each other through relationships: \relationship{child}{wears}{helmet}, \relationship{instructor}{wears}{jacket}, etc.}
\label{fig:scene_graph_2}%
\end{figure*}
\begin{figure*}[ht]
\centering
\iftoggle{smallfigs}{
\includegraphics[width=\textwidth]{pdf_graphics/data_representation.pdf}
}{
\includegraphics[width=\textwidth]{png_graphics/data_representation.png}
}
\caption{A representation of the Visual Genome dataset. Each image contains region descriptions that describe a localized portion of the image. We collect two types of question answer pairs (QAs): freeform QAs and region-based QAs. Each region is converted to a region graph representation of objects, attributes, and pairwise relationships. Finally, each of these region graphs are combined to form a scene graph with all the objects grounded to the image. \textit{Best viewed in color}}
\label{fig:data_representation}
\end{figure*}
\FloatBarrier
\section{Related Work}
\label{sec:related_works}
We discuss existing datasets that have been released and used by the vision community for classification and object detection. We also mention work that has improved object and attribute detection models. Then, we explore existing work that has utilized representations similar to our relationships between objects. In addition, we dive into literature related to cognitive tasks like image description, question answering, and knowledge representation.
\subsection{Datasets}
\label{sec:datasets}
Datasets (Table~\ref{tab:all_datasets}) have been growing in size as researchers have begun tackling increasingly complicated problems. \textit{Caltech 101}~\cite{fei2007learning} was one of the first datasets hand-curated for image classification, with 101 object categories and $15$-$30$ of examples per category. One of the biggest criticisms of Caltech 101 was the lack of variability in its examples. \textit{Caltech 256}~\cite{griffin2007caltech} increased the number of categories to 256, while also addressing some of the shortcomings of Caltech 101. However, it still had only a handful of examples per category, and most of its images contained only a single object. \textit{LabelMe}~\cite{russell2008labelme} introduced a dataset with multiple objects per category. They also provided a web interface that experts and novices could use to annotate additional images. This web interface enabled images to be labeled with polygons, helping create datasets for image segmentation. The \textit{Lotus Hill dataset}~\cite{yao2007introduction} contains a hierarchical decomposition of objects (vehicles, man-made objects, animals, etc.) along with segmentations. Only a small part of this dataset is freely available. \textit{SUN}~\cite{xiao2010sun}, just like LabelMe~\cite{russell2008labelme} and Lotus Hill~\cite{yao2007introduction}, was curated for object detection.
Pushing the size of datasets even further, \textit{$80$ Million Tiny Images}~\cite{torralba200880} created a significantly larger dataset than its predecessors. It contains tiny (i.e.\ $32\times32$ pixels) images that were collected using WordNet~\cite{miller1995WordNet} synsets as queries. However, because the data in $80$ Million Images were not human-verified, they contain numerous errors. \textit{YFCC100M}~\cite{thomee2015yfcc100m} is another large database of $100$ million images that is still largely unexplored. It contains human generated and machine generated tags.
\textit{Pascal VOC}~\cite{everingham2010pascal} pushed research from classification to object detection with a dataset containing $20$ semantic categories in $11,000$ images. \textit{Imagenet}~\cite{deng2009imagenet} took WordNet synsets and crowdsourced a large dataset of 14 million images. They started the ILSVRC~\cite{ILSVRC15} challenge for a variety of computer vision tasks. ILSVRC and PASCAL provide a test bench for object detection, image classification, object segmentation, person layout, and action classification. \textit{MS-COCO}~\cite{lin2014microsoft} recently released its dataset, with over $328,000$ images with sentence descriptions and segmentations of $91$ object categories. The current largest dataset for QA, \textit{VQA}~\cite{antol2015vqa}, contains $204,721$ images annotated with one or more question answers. They collected a dataset of $614,163$ freeform questions with $6.1$M ground truth answers and provided a baseline approach in answering questions using an image and a textual question as the input.
\begin{sidewaystable*}
\vspace{45pc}
\centering
\mbox{%
\setlength{\extrarowheight}{5pt}
\scriptsize
\begin{tabular}{lrrrrrrrrrr}
& & Descriptions & Total & \# Object & Objects & \# Attributes & Attributes & \# Relationship & Relationships & Question \\ [0ex]
& Images & per Image & Objects & Categories & per Image & Categories & per Image & Categories & per Image & Answers \\
\toprule
YFCC100M~\cite{thomee2015yfcc100m} & 100,000,000 & - & - & - & - & - & - & - & - & - \\
Tiny Images~\cite{torralba200880} & 80,000,000 & - & - & 53,464 & 1 & - & - & - & - & - \\
ImageNet~\cite{deng2009imagenet} & 14,197,122 & - & 14,197,122 & 21,841 & 1 & - & - & - & - & - \\
ILSVRC Detection (2012)~\cite{ILSVRC15} & 476,688 & - & 534,309 & 200 & 2.5 & - & - & - & - & - \\
MS-COCO~\cite{2015arXiv150602203R} & 328,000 & 5 & 27,472 & 91 & - & - & - & - & - & - \\
Flickr 30K~\cite{young2014image} & 30,000 & 5 & - & - & - & - & - & - & - & - \\
Caltech 101~\cite{fei2007learning} & 9,144 & - & 9,144 & 102 & 1 & - & - & - & - & - \\
Caltech 256~\cite{griffin2007caltech} & 30,608 & - & 30,608 & 257 & 1 & - & - & - & - & - \\
Caltech Pedestrian~\cite{dollar2012pedestrian} & 250,000 & - & 350,000 & 1 & 1.4 & - & - & - & - & - \\
Pascal Detection~\cite{everingham2010pascal} & 11,530 & - & 27,450 & 20 & 2.38 & - & - & - & - & -\\
Abstract Scenes~\cite{zitnick2013bringing} & 10,020 & - & 58 & 11 & 5 & - & - & - & - & - \\
aPascal~\cite{farhadi2009describing} & 12,000 & - & - & - & - & 64 & - & - & - & - \\
Animal Attributes~\cite{lampert2009learning} & 30,000 & - & - & - & - & 1,280 & - & - & - & -\\
SUN Attributes~\cite{patterson2014sun} & 14,000 & - & - & - & - & 700 & 700 & - & - & - \\
Caltech Birds~\cite{wah2011caltech} & 11,788 & - & - & - & - & 312 & 312 & - & - & - \\
COCO Actions~\cite{2015arXiv150602203R} & 10,000 & - & - & - & - & - & - & 140 & - & - \\
Visual Phrases~\cite{sadeghi2011recognition} & - & - & - & - & - & - & - & 17 & 1 & - \\
Viske~\cite{sadeghi2015viske} & - & - & - & - & - & - & - & 6500 & - & - \\
DAQUAR~\cite{malinowski2014multi} & 1,449 & - & - & - & - & - & - & - & - & 12,468 \\
COCO QA~\cite{ren2015image} & 123,287 & - & - & - & - & - & - & - & - & 117,684 \\
Baidu~\cite{gao2015you} & 120,360 & - & - & - & - & - & - & - & - & 250,569 \\
VQA~\cite{antol2015vqa} & 204,721 & - & - & - & - & - & - & - & - & 614,163 \\
\midrule
\textbf{Visual Genome} & 108,000 & 50 & 4,102,818 & 76,340 & 16 & 15,626 & 16 & 47 & 18 & 1,773,258 \\
\bottomrule
\end{tabular}
}
\caption{A comparison of existing datasets with Visual Genome. We show that Visual Genome has an order of magnitude more descriptions and question answers. It also has a more diverse set of object, attribute, and relationship classes. Additionally, Visual Genome contains a higher density of these annotations per image.}
\label{tab:all_datasets}
\end{sidewaystable*}
\textit{Visual Genome} aims to bridge the gap between all these datasets, collecting not just annotations for a large number of objects but also scene graphs, region descriptions, and question answer pairs for image regions. Unlike previous datasets, which were collected for a single task like image classification, the Visual Genome dataset was collected to be a general-purpose representation of the visual world, without bias toward a particular task. Our images contain an average of $21$ objects, which is almost an order of magnitude more dense than any existing vision dataset. Similarly, we contain an average of $18$ attributes and $18$ relationships per image. We also have an order of magnitude more unique objects, attributes, and relationships than any other dataset. Finally, we have 1.7 million question answer pairs, also larger than any other dataset for visual question answering.
\subsection{Image Descriptions}
One of the core contributions of Visual Genome is its descriptions for multiple regions in an image. As such, we mention other image description datasets and models in this subsection. Most work related to describing images can be divided into two categories: retrieval of human-generated captions and generation of novel captions. Methods in the first category use similarity metrics between image features from predefined models to retrieve similar sentences~\cite{ordonez2011im2text, hodosh2013framing}. Other methods map both sentences and their images to a common vector space~\cite{ordonez2011im2text} or map them to a space of triples~\cite{farhadi2010every}. Among those in the second category, a common theme has been to use recurrent neural networks to produce novel captions~\cite{kiros2014multimodal, mao2014explain, karpathy2014deep, vinyals2014show}. More recently, researchers have also used a visual attention model~\cite{xu2015show}.
One drawback of these approaches is their attention to describing only the most salient aspect of the image. This problem is amplified by datasets like Flickr 30K~\cite{young2014image} and MS-COCO~\cite{lin2014microsoft}, whose sentence desriptions tend to focus, somewhat redundantly, on these salient parts. For example, ``an elephant is seen wandering around on a sunny day,'' ``a large elephant in a tall grass field,'' and ``a very large elephant standing alone in some brush'' are 3 descriptions from the MS-COCO dataset, and all of them focus on the salient elephant in the image and ignore the other regions in the image. Many real-world scenes are complex, with multiple objects and interactions that are best described using multiple descriptions~\cite{karpathy2014deep, lebret2015phrase}. Our dataset pushes toward a complete understanding of an image by collecting a dataset in which we capture not just scene-level descriptions but also myriad of low-level descriptions, the ``grammar'' of the scene.
\subsection{Objects}
Object detection is a fundamental task in computer vision, with applications ranging from identification of faces in photo software to identification of other cars by self-driving cars on the road. It involves classifying an object into a semantic category and localizing the object in the image. Visual Genome uses objects as a core component on which each visual scene is built. Early datasets include the face detectio~ \cite{huang2008labeled} and pedestrian datasets~\cite{dollar2012pedestrian}. The PASCAL VOC and ILSVRC's detection dataset~\cite{deng2009imagenet} pushed research in object detection. But the images in these datasets are iconic and do not capture the settings in which these objects usually co-occur. To remedy this problem, MS-COCO~\cite{lin2014microsoft} annotated real-world scenes that capture object contexts. However, MS-COCO was unable to describe all the objects in its images, since they annotated only 91 object categories. In the real world, there are many more objects that the ones captured by existing datasets. Visual Genome aims at collecting annotations for all visual elements that occur in images, increasing the number of semantic categories to over 17,000.
\vspace{-0.3cm}
\subsection{Attributes}
The inclusion of attributes allows us to describe, compare, and more easily categorize objects. Even if we haven't seen an object before, attributes allow us to infer something about it; for example, ``yellow and brown spotted with long neck'' likely refers to a giraffe. Initial work in this area involved finding objects with similar features~\cite{malisiewicz2008recognition} using examplar SVMs. Next, textures were used to study objects~\cite{varma2005statistical}, while other methods learned to predict colors~\cite{ferrari2007learning}. Finally, the study of attributes was explicitly demonstrated to lead to improvements in object classification~\cite{farhadi2009describing}. Attributes were defined to be paths (``has legs''), shapes (``spherical''), or materials (``furry'') and could be used to classify new categories of objects. Attributes have also played a large role in improving fine-grained recognition~\cite{goering2014nonparametric} on fine-grained attribute datasets like CUB-2011~\cite{wah2011caltech}. In Visual Genome, we use a generalized formulation~\cite{Johnson2015CVPR}, but we extend it such that attributes are not image-specific binaries but rather object-specific for each object in a real-world scene. We also extend the types of attributes to include size (``small''), pose (``bent''), state (``transparent''), emotion (``happy''), and many more.
\subsection{Relationships}
Relationship extraction has been a traditional problem in information extraction and in natural language processing. Syntactic features~\cite{zhou122007tree, guodong2005exploring}, dependency tree methods~\cite{culotta2004dependency, bunescu2005shortest}, and deep neural networks~\cite{socher2012semantic, zeng2014relation} have been employed to extract relationships between two entities in a sentence. However, in computer vision, very little work has gone into learning or predicting relationships. Instead, relationships have been implicitly used to improve other vision tasks. Relative layouts between objects have improved scene categorization~\cite{izadinia2014incorporating}, and 3D spatial geometry between objects has helped object detection~\cite{choi2013understanding}. Comparative adjectives and prepositions between pairs of objects have been used to model visual relationships and improved object localization~\cite{gupta2008beyond}.
Relationships have already shown their utility in improving cognitive tasks. A meaning space of relationships has improved the mapping of images to sentences~\cite{farhadi2010every}. Relationships in a structured representation with objects have been defined as a graph structure called a \textit{scene graph}, where the nodes are objects with attributes and edges are relationships between objects. This representation can be used to generate indoor images from sentences and also to improve image search~\cite{chang2014semantic, Johnson2015CVPR}. We use a similar scene graph representation of an image that generalizes across all these previous works~\cite{Johnson2015CVPR}. Recently, relationships have come into focus again in the form of question answering about associations between objects~\cite{sadeghi2015viske}. These questions ask if a relationship, involving generally two objects, is true, e.g. ``do dogs eat ice cream?''. We believe that relationships will be necessary for higher-level cognitive tasks~\cite{Johnson2015CVPR, luvisualrelationship}, so we collect the largest corpus of them in an attempt to improve tasks by actually understanding relationships between objects.
\subsection{Question Answering}
Visual question answering (QA) has been recently proposed as a proxy task of evaluating a computer vision system's ability to understand an image beyond object recognition~\cite{geman2015visual,malinowski2014multi}. Several visual QA benchmarks have been proposed in the last few months. The DAQUAR~\cite{malinowski2014multi} dataset was the first toy-sized QA benchmark built upon indoor scene RGB-D images of NYU Depth v2~\cite{SilbermanECCV12}. Most new datasets~\cite{VisualMadlibs,ren2015image,antol2015vqa,gao2015you} have collected QA pairs on MS-COCO images, either generated automatically by NLP tools~\cite{ren2015image} or written by human workers~\cite{VisualMadlibs,antol2015vqa,gao2015you}.
In previous datasets, most questions concentrated on simple recognition-based questions about the salient objects, and answers were often extremely short. For instance, $90\%$ of DAQUAR answers~\cite{malinowski2014multi} and $87\%$ of VQA answers~\cite{antol2015vqa} consist of single-word object names, attributes, and quantities. This shortness limits their diversity and fails to capture the long-tail details of the images. Given the availability of new datasets, an array of visual QA models have been proposed to tackle QA tasks. The proposed models range from SVM classifiers~\cite{antol2015vqa} and probabilistic inference~\cite{malinowski2014multi} to recurrent neural networks~\cite{gao2015you,malinowski2015ask,ren2015image} and convolutional networks~\cite{ma2015cnnQA}. Visual Genome aims to capture the details of the images with diverse question types and long answers. These questions should cover a wide range of visual tasks from basic perception to complex reasoning. Our QA dataset of $1.7$ million QAs is also larger than any currently existing dataset.
\subsection{Knowledge Representation}
A knowledge representation of the visual world is capable of tackling an array of vision tasks, from action recognition to general question answering. However, it is difficult to answer ``what is the minimal viable set of knowledge needed to understand about the physical world?''~\cite{hayes1978naive}. It was later proposed that there be a certain plurality to concepts and their related axioms~\cite{hayes1985naive}. These efforts have grown to model physical processes~\cite{forbus1984qualitative} or to model a series of actions as scripts~\cite{schank2013scripts} for stories---both of which are not depicted in a single static image but which play roles in an image's story. More recently, NELL~\cite{betteridge2009toward} learns probabilistic horn clauses by extracting information from the web. DeepQA~\cite{ferrucci2010building} proposes a probabilistic question answering architecture involving over $100$ different techniques. Others have used Markov logic networks~\cite{zhu2009statsnowball,niu2012elementary} as their representation to perform statistical inference for knowledge base construction. Our work is most similar to that of those~\cite{chen2013neil,zhu2014eccv,zhu2015kb,sadeghi2015viske} who attempt to learn common-sense relationships from images. Visual Genome scene graphs can also be considered a \textit{dense} knowledge representation for images. It is similar to the format used in knowledge bases in NLP.
|
1,116,691,500,484 | arxiv | \section{Introduction}
The aim of generative models is to produce synthetic images, which are similar to samples in a dataset. Recently, remarkable approaches have been proposed and developed in this field \cite{goodfellow2014generative, kingma2018glow, kingma2016improved, oord2016pixel}. Each of these approaches has their own strengths and weaknesses. Generative Adversarial Networks (GANs) \cite{goodfellow2014generative} dominate this field in terms of generating sharp images and having a meaningful latent representation with a rich linear structure \cite{radford2015unsupervised,rosca2017variational}. GANs are composed of two networks, i.e. generator and discriminator, that compete with each other during training for learning the underlying distributions of the images. Given a latent vector $z$, which is randomly drawn from normal or uniform distribution, namely $P(Z)$, the generator produces a synthetic image. The discriminator, on the other hand, determines whether a given image is real or fake. The discriminator produces a probability/score value that shows the probability/score of the given image as being a real image. Depending on the objective function that maximizes the discrimination of real and fake images, parameters of both networks, i.e. generator and discriminator, are optimized. It’s only after the Nash equilibrium that is established between the two networks, the discriminator hardly differentiates given images as real or fake with confidence. This state is a good indicator for the generator model that the generated images are similar to the real data distribution. After the training is completed, the discriminator is discarded and the generator is used for mapping from the latent space to the image space.
Although a GAN generator provides a means for generation of different images using random latent vectors, it is difficult to model the inverse function for the generator model so that images can be manipulated in a controlled setting; especially for high resolution image generators. Existing methods apply algebraic operations in the latent vector space to encode semantically meaningful attributes to images. In DCGAN work \cite{radford2015unsupervised}, algebraic operations have been performed on latent vectors of two generated images with and without an attribute, such as a face containing sunglasses or not; the latent vector representing the attribute is added to the latent vector corresponding to a generated image to provide the image with the attribute. However, due to use of randomly generated images, such vector additions manipulate other attributes as well as the desired attribute; hence usually cause significant changes in the original face attributes. The challenge here is to find a latent vector representation and a proper direction that corresponds to the factors that changes only the desired attribute; the produced images will be the same with the original source and they are different only for the encoded attribute. This operation requires accurate mapping from the image space to the latent space and identification of the true latent direction. In this work, we propose a novel architecture for solving both problems accurately. The proposed architecture is composed of an encoder and generator networks. The training of the networks are performed in an unsupervised setting, i.e. no attribute labels are used. Semi-supervision is performed in the calculation of desired attribute directions using the encoder network. We show that the proposed solution is effective for controlled manipulation of images.
The main contributions of this work can be summarized as follows:
(1) We propose a new straightforward end-to-end architecture called Cyclic Reverse Generator (CRG) that allows learning the inverse of the generator in an unsupervised setting and allows reconstruction of both generated and real images in high quality.
(2) We obtain a latent vector direction for attribute editing using our CRG encoder using only an arbitrary pair of real images; one with and the other without an attribute.
(3) Keeping the other attributes the same, we show that it is possible to manipulate only the desired attributes at any rate. We provide an empirical analysis about determining the conservative manipulation ranges for different attributes.
(4) We give state-of-the-art results on CelebA dataset \cite{liu2015deep} to reconstruct an image at 128x128 resolution, which is the highest resolution that is obtained using an architecture with an encoder that is connected end-to-end to a generator.
The rest of the paper is organized as follows. A brief summary of the related previous works are summarized in Section \ref{sec:RelatedWorks}. The proposed method is discussed in detail in Section \ref{sec:TheMethod}. The results of our image reconstructions are provided in Section \ref{sec:experiments_and_results}. The paper is concluded with future directions.
\section{Related Works}
\label{sec:RelatedWorks}
In GANs, the studies of manipulating desired attribute(s) of images can be divided into two main groups, as semi-supervised and unsupervised methods. In the first group, training is performed using a code representing each attribute in a conditional GAN architecture \cite{mirza2014conditional, antipov2017face, perarnau2016invertible, jaiswal2018bidirectional, choi2018stargan,he2019attgan}. In these methods, after training, selected attributes of given images can be changed. However, the requirement of labeled datasets makes these methods impractical. Moreover, the attributes are limited with the trained labels. In the second group, i.e. unsupervised methods. The main purpose is to construct a model that learns the hidden structures in image generation and to inverse the generation process without using attribute labels. Our work belongs to this group. The studies in this subgroup can be examined under three categories: The first one is Gradient Based Techniques (GBTs) \cite{lipton2017precise, creswell2018inverting}. In these approaches, latent vector corresponding to an image is taken as the optimization goal; the aim is to find latent vector $z\prime$ corresponding to $\phi(z)$ image. These methods do not require an additional encoder network. A pre-trained generator network is used to perform this process. Firstly, a $z\prime$ vector is sampled randomly from a prior distribution and it is given to the generator. According to the gradient obtained from the difference between the generated image and the target image, the $z\prime$ vector is updated. After numerous iterations, the $z\prime$ vector is expected to converge to the target vector $z$ that represents the $\phi(z)$ image. In GBTs, the encoding of the latent vector representation is obtained after a great number of gradient descent steps, which makes these approaches impractical. The structure of the GBT and the optimization function is depicted in Fig. \ref{fig:GBT}.
\begin{figure}[!h]
\centering
\includegraphics[scale=.5]{Figure1.pdf}
\centering
\caption{The structure of the GBT \cite{lipton2017precise}.}
\label{fig:GBT}
\end{figure}
The second category includes architectures with an encoder network, in addition to the generator and the discriminator networks \cite{larsen2015autoencoding,donahue2016adversarial,dumoulin2016adversarially,luo2017learning}. These studies differ depending on how the encoder network is used and trained in the architecture. In the VAEGAN \cite{larsen2015autoencoding}, the generator is constructed using the decoder of a variational autoencoder (VAE), and encoder-decoder-discriminator networks are combined end-to-end as a single network. This structure enables reconstruction of more realistic and sharper images compared to VAEs. Similar to VAE-GAN, $\alpha$-GAN \cite{rosca2017variational} also combine variational and adversarial learning. This work also use additional discriminator network in the latent space and additional pixel-wise reconstruction loss term in the image space. Unlike these two models, in BiGAN \cite{donahue2016adversarial} and ALI \cite{dumoulin2016adversarially} studies, neither generator nor encoder can see each other's outputs; in both works, the discriminator is trained to distinguish tuples of samples with their latent codes. In studies belonging to the second category, generally three networks are trained simultaneously. This makes it difficult to train encoder using a pre-trained generator. Another disadvantage of studies in this category is that they are inadequate to generate high-resolution synthetic images; most of these studies cannot go beyond 64x64 resolutions \cite{heljakka2018pioneer}. Moreover, most of these studies fail to extract the latent vector representation of an image that changes only a subset of attributes while preserving the other properties as they are. Another study in this category is the AEGAN architecture \cite{luo2017learning}. The difference of this architecture from the others is that the encoder network is not trained simultaneously with the other networks. In this study, encoder-generator architecture is created by using a pre-trained generator model and an encoder network is trained from scratch. The encoder network is not trained with real images but is trained only with the generated images. The results of the reconstructions of the generated images are better than the real images; yet they are both not very successful.
The third category is Adversarial Generator-Encoder (AGE) architectures whose structure does not contain a discriminator network and an adversarial game is set up directly between the encoder and the generator \cite{ulyanov2018takes,heljakka2018pioneer}. In this category, training is performed using a combination of adversarial loss and reconstruction loss that encourages the encoder and generator to be reciprocal. There are three main advantages of these architectures: (1) Since adversarial loss is calculated between encoder and generator, there is no need for a discriminator and the number of learned parameters decreases, (2) In GANs, distributions are usually compared in high dimensional image space, in AGE, comparison is performed in a simpler latent space, (3) The encoder network is given both real and fake images as input, so the real images are better encoded. When outputs of this architecture are examined, it is seen that it produces better results than other end-to-end architectures for 64x64 resolutions, but it cannot go beyond 128x128 resolutions for reconstruction of images and the results at this resolution are not at desired level. Considering the studies in this category, it is seen that they are poor in terms of image quality and diversity compared to the models with a discriminator \cite{karras2017progressive}. Furthermore, if reconstruction loss is not used in AGE architecture, encoder and generator mapping is not reciprocal. In order to make networks reciprocal, reconstruction loss is controlled by a hyper-parameter in the objective function. However, the determination of the value of this parameter brings extra burden. To overcome the problems mentioned in this category, in this work we propose an architecture with two separate parts that have cyclic connections with each other and perform training in two steps. In the first step, a generator is trained using the original adversarial loss between the generator and discriminator \cite{karras2017progressive}. In the second step, the encoder network is trained from scratch by using the cyclic connection between the encoder and the generator. This architecture allows generator network to receive a latent vector, which is either randomly generated or encoded, as an input, and the encoder network to get either real or fake images as input during training. This approach helps convergence problems of encoders during training. Moreover, it has an additional benefit that it enables learning the inverse mapping of any pre-trained generator that are readily available for generation of images in different domains.
The success of the encoder network is directly related to the success of the generator in mapping latent vectors to images. Two main problems arise in generator training, especially when high-resolution images are used. The first problem is the stability problem; competition between the two networks, i.e. generator and discriminator, creates instability, since one dominates the other. The second problem is the mode collapse problem; the generator network produces samples in limited varieties. Recently, new objective functions \cite{arjovsky2017wasserstein,gulrajani2017improved,kodali2017convergence,mao2017least,berthelot2017began}, regularization techniques \cite{gulrajani2017improved,miyato2018spectral} and network architectures \cite{radford2015unsupervised,berthelot2017began,karras2017progressive} have been proposed to overcome the stability problem. In order to prevent mode collapse problem, generating similar images are prevented using similarities of the samples in the discriminator networks \cite{salimans2016improved,karras2017progressive}. Although there is various GAN research that generate high quality images for popular datasets, it is not clear which algorithm is superior to the others. In a recent study, the state-of-the-art models have been compared objectively using some well-known evaluation metrics \cite{lucic2018gans}. In this study, it is reported that the performance of the models largely depends on datasets and hyper-parameter optimization; there is no ideal model that consistently generate high quality images in all datasets. Therefore, it is a challenge to determine the appropriate architecture, objective function and regularization techniques for different datasets.
In addition to the vast literature about GAN training, there are recent works that focus on generating images with different attributes by encoding those attributes as part of the latent code \cite{mirza2014conditional, antipov2017face, perarnau2016invertible, jaiswal2018bidirectional, choi2018stargan,he2019attgan}. Our work is in a different line from these works; we do not explicitly represent attribute codes during training. These works simply utilize the attribute codes in the latent vectors to edit a particular attribute of an image. Yet, the attribute editing is still a challenging problem without using explicit attribute codes, since the encoding of the desired attributes in the latent space is entangled. Therefore, it is difficult to change only a subset of attributes without changing other properties of the image. Recently, different from the existing approaches, StyleGAN \cite{karras2018style} work presents an unsupervised approach to learn relevant attribute directions. In the StyleGAN, research has been carried out on making the latent vector disentangled by using the generator structure used in the style transfer literature \cite{huang2017arbitrary}. The aim is to find a latent vector that is composed of linear subspaces, such that each subspace controls an attribute. With this architecture, low, medium and high level attributes in the image can be modified by changing the determined subspaces of the latent space corresponding to the image. Still yet, the changes in this subspace changes other attributes of the image as well as the desired attributes.
\section{The Method}
\label{sec:TheMethod}
In the proposed method, we first trained a generator model in a GAN setting and used that pre-trained generator network to train an encoder that learns the inverse mapping of the generator. We propose a new cyclic parameter optimization for learning this inverse mapping. The mapping from the image space to the latent vector space is used actively to identify relevant attribute directions for image attribute editing. We show the efficacy of our proposal using the CelebA dataset.
This section is organized as follows. We first introduce the proposed Cyclic Reverse Generator (CRG) Model, which enables reconstruction of images via a cyclic error minimization function. Following that, we provide the training details of our generator and encoder networks using CelebA dataset. Then, we explain how we compute the latent attribute representations with a pair of reference images using the CRG model. Finally, we provide empirical analysis of the computed attribute directions to show that the computed attribute directions are relevant to the target attributes.
\subsection{The Cyclic Reverse Generator (CRG) Model}
\label{sec:CRG_model}
\begin{figure}[!h]
\centering
\includegraphics[scale=.8]{Figure2.pdf}
\centering
\caption{The CRG model.}
\label{fig:CRG}
\end{figure}
This section introduces the proposed model for the reconstruction of an image. Our model is composed of two parametric networks: the generator network $g_{\omega}(z)$ that maps the latent space $z$ to the data space $x$, and the encoder network $e_{\epsilon}(z)$ that maps the image $x$ from the data space to the latent space $z$. The goal is to ensure that the encoder and the generator networks have bidirectional connections for generating and encoding of either real or fake images. The design of the proposed model is shown in Fig. \ref{fig:CRG}. The reconstruction loss is calculated using the error in the image space (using $L_{x}$, in Fig. \ref{fig:CRG}) and the error in the latent space (using $L_{z}$, in Fig. \ref{fig:CRG}) together, in an order, during encoder training.
We first train the GAN architecture, i.e. generator and discriminator networks, until we obtain state-of-the-art scores for the selected datasets. We observed that, it is only after an optimum generator is trained that a successful mapping can be done from the image space, $x$, to the latent space, $z$. Our first expectation from the model is the ability to reconstruct the latent representation of randomly generated samples. For a given random $z$, the generator generates an image, $x^\prime$; giving that image to the encoder network an estimated $z$ is obtained, i.e. $z^\prime$. We minimize the objective in (\ref{egu:z_optimize}) to minimize the error between the estimated $z^\prime$, i.e. $e_{\epsilon}(g_{\omega}(z))$, and the initial $z$.
\begin{equation}
\label{egu:z_optimize}
\hat{\epsilon} =\operatorname*{argmin}_{\epsilon}(\mathbb{E}_{z \sim P(Z)}\left\|z-e_{\epsilon}(g_{\omega}(z)) \right\| _{2}^2)
\end{equation}
Our second expectation from the model is the ability to reconstruct real images. This is also important because we want to be able to compute the relevant directions related to an attribute in the latent $z$ space using real reference images. For a given image $x$, the encoded $z$ value, $z^\prime$, is used to reconstruct the image $x^\prime$, i.e. $g_{\omega}(e_{\epsilon}(x))$, we use the objective in (\ref{egu:x_optimize}) to minimize the error between the reconstructed image $x^\prime$ and the original image $x$.
\begin{equation}
\label{egu:x_optimize}
\hat{\epsilon} =\operatorname*{argmin}_{\epsilon}(\mathbb{E}_{x \sim P_{data}}\left\|x-g_{\omega}(e_{\epsilon}(x)) \right\| _{1})
\end{equation}
In each iteration, the parameters of the encoder are updated twice, using the objective in (\ref{egu:z_optimize}) and (\ref{egu:x_optimize}), respectively.
We call our proposed architecture as Cyclic Reverse Generator (CRG) (Fig. \ref{fig:CRG}), since the encoder learns the inverse function of the generator using the proposed cyclic error minimization. In the model, the encoder is given either real or generated images, and the generator is given either a randomly generated or an encoded latent vector as the input. This enables both generated and real images to be reconstructed. We use a pre-trained generator in the CRG architecture. In this setting, only the encoder parameters are updated, the generator is fixed. We also included a discussion with a non-fixed generator in Section \ref{sec:image_reconstruction_experiments}. Generally, end-to-end approaches implement simultaneous training of three networks, i.e. generator, discriminator and encoder networks, since the dominance of one of the networks in the architecture usually leads to instability problems. Therefore, a pre-trained generator is not used in these studies. However, in CRG, the encoder is trained utilizing a pre-trained generator without facing stability problems. This is advantageous since we can train an encoder for a GAN model that is already trained for different domains.
\subsection{Training The Models}
\label{sec:Training_the_models}
In this section, we provide the training details for the GAN and the encoder models using the CelebA dataset.
Recently, a lot of researchers are working actively in the GAN domain and it is a challenge to determine which architecture(s) and objective function(s) are better suited for training a GAN for a particular dataset. Authors of \cite{lucic2018gans} argue that the performance of recently proposed state of the art models heavily depends on datasets, and that no particular model is strictly dominating the others. In this work, we used a slightly modified version of the Progressive GAN (PGGAN) architecture for GAN training, since it generates high quality images\cite{karras2017progressive}.
\subsubsection{GAN Model for CelebA Dataset}
\label{sec:CelebAModel}
The CelebA dataset contains $202599$ celebrity images with large pose variations, taken from different backgrounds. In this study, we use $30000$ images in 128x128 pixel resolution. Most of the models that infer the latent $z$ vectors work with 32x32 resolution images. There are very few models that go beyond 64x64 pixel resolution \cite{heljakka2018pioneer}. Current state-of-the art models in the $z$ inference produce a maximum of 128x128 images. However, the quality of the images is not sufficiently good for the reconstruction of the real images that are necessary to find the latent vector corresponding to an attribute. The reason for this is that as image resolution increases, stability and diversity problems become more apparent in GAN training. Hence, in order to achieve the purpose of this research, we need to train a generator that enable us to generate high-resolution images without encountering stability and diversity problems.
\begin{figure}[!h]
\centering
\includegraphics[width=0.97\textwidth]{Figure3.pdf}
\centering
\caption{The GAN architecture for CelebA model.}
\label{fig:GAN_arch}
\end{figure}
\paragraph{The Generator Network:}
There are some design choices to be made when training a generator network to generate high-resolution images; selection of the objective function, used regularization technique, selected GAN architecture and selection of the hyper-parameter values. For the CelebA dataset, we use the same model that we proposed in our preliminary work \cite{doganyahya}. Our model is similar to the PGGAN model, with some additional normalization layers; the architectural details are provided in Fig. \ref{fig:GAN_arch}.
In GAN studies, as the image resolution increases, the discriminator distinguishes between real and generated images easily. This causes imbalance problem between the two networks during training. The key idea in PGGAN is to grow both the generator and the discriminator progressively to capture fine details in images. Initially, the training of both networks is started with a low spatial resolution, i.e. 4x4 pixel images, and as the training progresses, a new layer is added to both networks to increase the spatial resolution of the generated images. Differently from the PGGAN, to ensure the stability between the two networks, we apply spectral normalization \cite{miyato2018spectral} to both the generator and the discriminator as in SAGAN \cite{zhang2018self}, we keep the learning rates of the discriminator more than the generator as in TTUR \cite{heusel2017gans}, i.e. we use $1e-4$ and $2e-4$ learning rate for the generator and the discriminator, respectively. We optimize the generator and the discriminator by 1:2 rate.
\begin{table}[!h]
\caption{Performance comparison of the generator networks for 128x128 pixel resolution.}
\centering
\label{tab:table2}
\begin{tabular}{ccc}
\hline
& FID \cite{heusel2017gans} & \begin{tabular}[c]{@{}c@{}}Latent vector\\ dimension\end{tabular} \\ \hline
AGE \cite{ulyanov2018takes} & 154.79 & 512 \\ \hline
Pioneer \cite{heljakka2018pioneer} & 23.15 & 512 \\ \hline
\multirow{2}{*}{\textbf{Our model}} & \textbf{8.96} & \textbf{512} \\ \cline{2-3}
& \textbf{9.4} & \textbf{128} \\ \hline
\end{tabular}
\end{table}
In Table \ref{tab:table2}, we compare our generator with the AGE and Pioneer \cite{heljakka2018pioneer} generators for 128x128 pixel resolution, since these models contain a model that infer the latent $z$ vectors similar to ours. Pioneer is the progressive model of the AGE architecture. We use the Fréchet Inception Distance (FID) metric, which is a commonly used metric to compare generator networks in terms of image quality and diversity. A low FID score indicates a better generator. Note that our generator produces lower FID scores compared to the other methods, for both $128$ and $512$ latent vector dimensions. In the training of our CRG model, we preferred the generator network with lower dimension, i.e. $128$, since there is no significant margin in the FID scores between $128$ and $512$. The results also depict that using a model with a discriminator produces better scores. In Fig. \ref{fig:interpolation}, we show the interpolation capability between randomly generated images using our generator. The quality of the generated images is apparent in these randomly generated samples.
\begin{figure}[!h]
\centering
\includegraphics[width=.9\textwidth]{Figure4.pdf}
\caption{Sample interpolations of our generator between two generated images.}
\label{fig:interpolation}
\end{figure}
\subsubsection{The Encoder Model}
We designed our encoder model as a multi-layer convolutional neural network. The generator model described in the previous section contains approximately $22$ million trainable parameters. In general, we observe that keeping the capacity of the encoder higher than the generator provides better encoding. The encoder consists of $6$ blocks with a total of approximately $26$ million trainable parameters. Training deep CNNs from scratch can cause convergence problems during the network training and training time increases substantially \cite{zeiler2014visualizing}. Given the fact that CNNs encode low-level features in the first few layers, we use the first $4$ blocks of the pre-trained VGG-16 Net \cite{simonyan2014very} in the encoder architecture with a list of modifications: (1) we remove the fourth max-pooling layer, so as not to reduce image resolution too much, (2) we include a batch normalization layer at the end of each convolution block to overcome the vanishing gradient problem that occurs when we include additional blocks at the end, (3) to avoid overfitting, we use spatial dropout layer after the batch normalization layers. In order to increase the capacity of the network, we include two new blocks to extract features that are specific to CelebA dataset. Finally, we use global max pooling on the last layer of the encoder. There are $64$ filters in the first block and the number of filters in the consecutive blocks are doubled except for the last block; since the latent vector dimension is $128$, we use $128$ filters in the last block. We use a 3x3 filter size in the convolution layers, except for the 1x1 transposed convolution layer in the last convolution layer. The structure of the encoder is depicted in Fig. \ref{fig:encoder}.
\begin{figure*}
\centering
\includegraphics[width=.9\textwidth]{Figure5.pdf}
\caption{The Encoder network architecture.}
\label{fig:encoder}
\end{figure*}
\paragraph{Training details:}
We trained the encoder using the CRG architecture as we explained in Section \ref{sec:CRG_model}. As previously mentioned in the CRG, MAE loss is used in the image space and MSE loss is used in the latent space, and only the parameters of the encoder are optimized at each iteration. During the training, all of the weights of the convolutional layers of the encoder, including the pretrained weights of the VGG Network, are trained using the same learning rate. We train the encoder using RMSProp optimizer \cite{tieleman2012lecture} with a batch size of $128$, setting rho to $0.9$ and epsilon to $1e-08$. In order to adapt the existing VGG layer weight values properly, we use a small learning rate, i.e. $1e-4$, for optimizing all the parameters in the encoder architecture. We reduce the learning rate by a factor of $2$ when minimum validation loss stops improving for $10$ epochs. To avoid overfitting, we apply different regularizations in the training; we use 50\% spatial dropout, and data augmentation where we rotate the images in the range of $30$ degrees and apply horizontal and vertical flips. Model checkpoint is used to save the best model and using early stopping, we end the training process if no improvement is achieved in the validation loss for $20$ epochs. After the training process, we expect that the generated images and their reconstructions look almost the same, and that the reconstructions of the real images are at an acceptable level; sufficient to find accurate latent vector directions for the attribute embedding.
\subsection{Computing Latent Attribute Representations}
\label{sec:computing_latent_attribute}
In the CRG architecture, the encoder is trained completely without any supervision; we do not use any labels or encode conditions during GAN training (please refer to Section \ref{sec:Training_the_models} for details). In order to extract the latent vector corresponding to an attribute, we use an arbitrary real image as a reference and a second image that contains a particular attribute that we want to control. In Fig. \ref{fig:smile_attribute}, we demonstrate the extraction of the latent vector direction for smile attribute embedding. We first use our encoder to get the latent vectors of the reference images. Then compute the normalized direction in the latent space to determine the relevant attribute direction.
\begin{figure}[!h]
\centering
\includegraphics[width=.5\textwidth]{Figure6.pdf}
\caption{Extraction of the latent vector corresponding to smile attribute.}
\label{fig:smile_attribute}
\end{figure}
When the accuracy of the generator and its inverse function, i.e. the encoder, is high, this direction accurately changes only the relevant attribute by preserving the other attributes in the image. Equation (\ref{equ:feature_normalized}) is used to find the latent vector direction representing an attribute $f$.
\begin{equation}
\label{equ:feature_normalized}
z_{f}=\frac{z_{2}-z_{1}}{\left\|z_{2}-z_{1} \right\|_{2}^{2}}
\end{equation}
Where $z_{1}$ is the latent code that is generated by the encoder using a reference image and $z_{2}$ is the latent code obtained using an image that has the desired attribute. Note that, depending on the context, attributes can be encoded in both positive or negative directions, i.e. from blond hair to brown hair ($z_{1}-z_{2}$) or brown hair to blond hair ($z_{2}-z_{1}$); then $z_{1}$ and $z_{2}$ in Eq. (\ref{equ:feature_normalized}) needs to be configured accordingly. After $z_{f}$ is obtained, equation (\ref{equ:z_new}) is used to generate an image with the desired attribute.
\begin{equation}
\label{equ:z_new}
z_{a}=z_{p}+k*z_{f}
\end{equation}
Where $z_{p}$ is the latent vector estimated by the encoder using any arbitrary image, k is the amount of attribute that we want to impose into the image, $z_{a}$ is the latent code of the attribute encoded image. The attribute encoding can be performed in positive or negative directions, as desired, by changing the sign of k accordingly in (4). To the best of our knowledge, the existing reconstruction models using GANs are not yet capable of accurately controlling only selected attributes in images. Modifying an image this way has three main advantages: (1) there is no need to have a labeled dataset to encode attributes, (2) attribute set is not constant; it can be constructed anyhow as long as a pair of arbitrary reference images are provided, (3) the amount of attribute injection to the target image can be controlled easily by adjusting one parameter, i.e. $k$.
\subsection{Analysis of Attribute Directions ($z_{f}$):}
\label{sec:Analysis_of_attrib_dir}
In this section, we provide an empirical evaluation of a set of attributes using randomly selected real images that are unseen by our GAN training with and without the selected attributes from the CelebA dataset. Note that, these images have attribute labels in the CelebA dataset; we use these labels to easily analyze the relative positions of the images with and without attributes on the projection axis. For each attribute\footnote{For the eyeglasses attribute, due to the lack of enough samples, we use around $13000$ images.}, we selected around $20000$ neutral images, i.e. images without the selected attribute, and $20000$ images with the attribute. We want to observe the relative positions of the neutral and attributed images in the latent space by projecting them to the computed attribute vector directions for four attributes: smile, hair color, eyeglasses and beard. According to our proposal, an image without an attribute changes gradually to contain the attribute when we iterate the latent vector of the image in the computed latent attribute direction; hence we expect to see two different modalities on the projection line when we project the neutral and the attributed sample images on this line.
We computed attribute directions as we defined in Section \ref{sec:computing_latent_attribute}, using $50$ arbitrary reference images for each attribute. Then we computed an average attribute direction using these directions separately for each attribute. The projections on these directions for each attribute revealed two Normal densities, one for the neutral samples (displayed with green color) and one for the attributed samples (displayed with blue color), consistently for all the attributes (Fig. \ref{fig:histogram}).
\begin{figure}[!h]
\centering
\includegraphics[width=.9\textwidth]{Figure7.pdf}
\caption{Projections on the latent attribute vector directions: top-left: smiling attribute, top-right: hair color, bottom-left: eyeglasses, bottom-right: beard attribute.}
\label{fig:histogram}
\end{figure}
We observed a set of randomly selected samples that are projected on different parts of the attribute distributions, i.e. the ones labeled as smile, blond hair, eyeglasses and beard distributions, in Fig. \ref{fig:gaussian_tails}. We intentionally selected samples that project on the left and right tails of the Normal distributions, i.e. beyond $\mu\pm3\sigma$, and around the mean. Note that, each image used in these blue densities are labeled with the related attribute in the CelebA dataset. We wondered the appearances of images on the left tail of these distributions. In Fig. \ref{fig:gaussian_tails}, for each attribute, the images on the top row are sampled from the right tail of the distributions, i.e. above $\mu+3\sigma$. We observe that the related attribute is explicitly visible in these images, which is as we expected since they are far away from the neutral distribution. The images on the middle row are sampled around the mean of the attribute distributions; they also have the related attributes, but not as strong as the ones on the right tail. The images on the third row are sampled from the left tail of the distribution, i.e. below $\mu-3\sigma$; these samples are close to the neutral image means, and as can be seen from the images the attributes are barely visible in these samples. Although these images are labeled with the corresponding attributes in the CelebA dataset, most of them do not visually contain the related attributes. For instance, the images labeled as smiling, do not smile at all (see the third row of the smiling attribute samples). It is similar for the other attributes as well. This is due to the labeling errors in CelebA dataset, which may affect the conditional GAN based models. However, our model is not affected, since we do not use any labels during training. These random selections from left, middle and right parts of the distributions show that the generator gradually encodes the related attributes through the computed attribute directions in a visually consistent manner. In our proposal, we utilize this structure for attribute editing (see Eq. \ref{equ:z_new}).
\begin{figure}[!h]
\centering
\includegraphics[width=\textwidth]{Figure8.pdf}
\caption{Sample images from the tails of Gaussian distributions for the four attributes.}
\label{fig:gaussian_tails}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=\textwidth]{Figure9.pdf}
\caption{Sample distributions obtained using five random referance image directions with and without hair color attribute.}
\label{fig:haircolor_attrib}
\end{figure}
Instead of the average direction, when we use any random reference image direction from the set of $50$ arbitrary reference images, we still observe two separate modalities after the projections, consistent with the distribution on the average attribute direction in Fig. \ref{fig:histogram}, with only slight changes in the means and standard deviations (Fig. \ref{fig:haircolor_attrib}). These observations show that a particular attribute can be coded by a range of linear directions that lie in close proximity; hence, the related attribute direction is not actually unique, there is a set of possible directions for each attribute. We provide visual examples to display the effect of arbitrary reference image selection in attribute editing further in Section \ref{sec:experiments_and_results}.
\section{Results and Discussion}
\label{sec:experiments_and_results}
This section provides comparisons of our CRG architecture with the related models designed for the concerned datasets and the results we obtained for image attribute editing at 128x128 pixel resolution using the CelebA dataset.
\subsection{Image Reconstruction Experiments}
\label{sec:image_reconstruction_experiments}
In this section, we compare the models that are designed for reconstruction of images using generative networks, i.e. Pioneer\footnote{\url{https://github.com/AaltoVision/pioneer}}, VAE-GAN\footnote{\url{https://github.com/PrateekMunjal/Autoencoding-beyond-pixels-using-a-learned-similarity-metric}}, $\alpha$-GAN\footnote{\url{https://github.com/PrateekMunjal/Variational-Approaches-for-Auto-Encoding-Generative-Adversarial-Networks}}\cite{rosca2017variational}, AGE\footnote{\url{https://github.com/DmitryUlyanov/AGE}}, ALI\footnote{\url{https://github.com/IshmaelBelghazi/ALI}}, LR models. We generated the models using their official pre-trained models; when we can not find official models, we either retrained the models from scratch or used a pre-trained model that is configured as in the original articles. Our CRG, Pioneer and LR models were trained using 30k images while the others were trained around 200k images. The comparison of the reconstructed images for real inputs are depicted in Fig. \ref{fig:Comparison_of_all_models}. As can be seen from the images, our CRG model reconstructs high-quality images that look more similar to the given inputs.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{Figure10.pdf}
\caption{Sample reconstructions of the state of the art generative models. The images on the top row are used in the reconstructions. Pioneer, LR and our CRG model generate images in 128x128 pixel resolution, the other models generate images in 64x64 pixels.}
\label{fig:Comparison_of_all_models}
\end{figure}
Moreover, we also retrained our model to see the effect of generator parameter updates while encoder training, i.e. we update the parameters of the generator as well during encoder training. We will refer to this model as CRG(TG) from this on. Fig. \ref{fig:comparison_our_nonfixedGen}, we provide sample reconstructions for our CRG and CRG(TG) methods. Since there is no discriminator in the CRG architecture, the generated images gradually lose their sharp features during minimization of the MAE and MSE loss functions in the generator parameter updates. Hence, we believe that it is better to first train a GAN in a domain which generates realistic images; then, without changing the generator parameters, train an encoder that learns the inverse of the generator using CRG.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{Figure11.pdf}
\caption{Comparison of real image reconstructions: (first row) real images that are used in the reconstruction, (second row) our CRG model (without generator training) and (third row) CRG model with generator training and encoder training simultaneously.}
\label{fig:comparison_our_nonfixedGen}
\end{figure}
We also computed the difference hash (dhash) \cite{niu2008overview}, perceptual hash (phash) \cite{niu2008overview} and wavelet hash (whash) \cite{whash} metrics to measure the similarity between the real and the reconstructed images. In these methods, images are initially converted into a fixed hash code. Then, the similarity between the original image hash code and the regenerated image hash code determines how the two images are similar. Table \ref{tab:CELEBA_dhash_phash_whash} provides the scores of these perceptual evaluation metrics, MAE and MSE for the selected models. The dhash, phash and whash metrics generate more perceptually relevant scores; when the reconstructed image is sharp and similar to the input image, they generate high scores. On the other hand, MAE and MSE provide averages of pixelwise differences. As can be seen from the Table, the perceptual scores of the CRG is better than all the other methods; yet MAE and MSE scores of the CRG are lower than the Pioneer. The blurred reconstructions of the Pioneer model results in less error, pixelwise. When we compute the MAE and MSE scores using our CRG(TG) reconstructions, we obtain $0.215$, $0.034$, respectively. Although this scores seem to be better than all the methods, the reconstructions of CRG(TG) are obviously very blurred compared to CRG. Hence, MAE and MSE metrics are not very useful for comparing the quality of different reconstructions. The perceptual metrics validate our visual judgments in Fig. \ref{fig:Comparison_of_all_models} that the CRG model reconstructions are better than the other model reconstructions for real images.
\begin{table}[!h]
\centering
\caption{The similarities of real images and reconstructed images.}
\label{tab:CELEBA_dhash_phash_whash}
\begin{tabular}{llllll}
\hline
Model & dhash & phash & whash & MAE & MSE \\ \hline
\textbf{CRG (TG)} & 0.846 & 0.792 & 0.855 & \textbf{0.215} & \textbf{0.034} \\ \hline
\textbf{CRG} & \textbf{0.871} & \textbf{0.848} & \textbf{0.892} & 0.306 & 0.067 \\ \hline
Pioneer \cite{heljakka2018pioneer} & 0.852 & 0.804 & 0.864 & 0.261 & 0.051 \\ \hline
VAE-GAN \cite{larsen2015autoencoding} & 0.842 & 0.779 & 0.861 & 0.355 & 0.095 \\ \hline
$\alpha-$GAN \cite{rosca2017variational} & 0.837 & 0.756 & 0.855 & 0.443 & 0.122 \\ \hline
AGE \cite{ulyanov2018takes} & 0.832 & 0.754 & 0.863 & 0.449 & 0.136 \\ \hline
ALI \cite{dumoulin2016adversarially} & 0.759 & 0.668 & 0.745 & 0.487 & 0.159 \\ \hline
LR \cite{donahue2016adversarial} & 0.697 & 0.585 & 0.674 & 0.613 & 0.263 \\ \hline
\end{tabular}
\end{table}
In order to change any attribute of a generated image, it is necessary to correctly reconstruct a generated image. Fig. \ref{fig:our_model_regenerate_images_fake} shows the reconstruction of some generated images using our model. As can be seen from the samples, the encoder successfully learns the inverse mapping of the generator.
\begin{figure}[!h]
\centering
\includegraphics[width=.9\textwidth]{Figure12.pdf}
\caption{Top row: Randomly generated images, bottom row: reconstructed images using our model.}
\label{fig:our_model_regenerate_images_fake}
\end{figure}
\subsection{Image Attribute Editing Experiments}
In this section, we modify two attributes, the smile and the pose adjustment attributes, to demonstrate the attribute encoding of a source image using two arbitrary reference images. We show the results of manipulating the smile attribute in two directions and the pose adjustment of the source images with neutral and smiling gestures in Fig. \ref{fig:smile_and_pos_attribute_separate_generated}. As it is seen in the figures, with the exception of the interested attribute, other attributes of the source images are preserved substantially. These results are very promising for generating various synthetic images with desired attributes, which can be utilized in data augmentation for various problem domains. Using the proposed CRG training procedure, desired attributes can be injected to a source image without using a labeled dataset or conditional GANs architecture.
\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{Figure13.pdf}
\caption{Smile and pose adjustments using arbitrary reference images. On the left side, reference image pairs are displayed. Multiple samplings of some generated images in the reference attribute direction are depicted on the right side.}
\label{fig:smile_and_pos_attribute_separate_generated}
\end{figure}
We also made experiments on changing more than one attribute simultaneously, on purpose, while preserving the source images. In Fig. \ref{fig:smile_and_pos_attribute_together_generated}, we show manipulation of the pose adjustment and the smile attributes simultaneously. The results exemplify multiple attribute editing using again only one pair of reference images.
\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{Figure14.pdf}
\caption{Simultaneously changing smile and pose adjustments using an arbitrary reference image pair. Top row: reference image pairs, Second and third row: multiple samplings in the reference attribute direction.}
\label{fig:smile_and_pos_attribute_together_generated}
\end{figure}
Although our aim is to modify only a subset of attributes of the generated images, we also tested the model's performance on real images. As shown in Fig. \ref{fig:smile_and_pose_real_images}, we found that the results in the real images are also promising. We believe that the difference between the original source and the reconstruction is mainly due to using insufficient number of samples during our GAN training, i.e. $30000$ real images. Generator may not know producing some novel attributes that it did not see before. For example, the earrings of the second sample in Fig. \ref{fig:smile_and_pose_real_images} is not produced since the generator has not seen enough samples with large earrings during training. However, the hair styles, gesture characteristics and the pose of the input images are successfully captured in the reconstructed images. The accuracy of real image reconstruction can be increased by using more samples with many attribute variations during GAN training.
\begin{figure}[!h]
\centering
\includegraphics[width=.7\textwidth]{Figure15.pdf}
\caption{Smile attribute and pose adjustment on real images. The same smile and pose adjustment reference image pairs, depicted in Fig. \ref{fig:smile_and_pos_attribute_separate_generated}, are also used in these samples.}
\label{fig:smile_and_pose_real_images}
\end{figure}
In the Supplementary Materials Section (Appendix \ref{supp_mats}), we provided additional image attribute editing results, in total 15 attributes, that have strong visual impact. These attributes are bald, bangs, hair length, brown hair, blond hair, black hair, gender, age, straight hair, wavy hair, sunglasses, smiling, eyeglasses, beard and skin color.
\subsection{Effect of arbitrary reference images in attribute coding}
\label{sec:Consistency_of_attribute_vectors}
In this section, we elaborate the consistency of the attribute editings for an attribute using different reference image pairs. We use the same four attributes in our attribute direction analysis that we used in Section \ref{sec:Analysis_of_attrib_dir}, i.e. smile, hair color, eyeglasses and beard. The left side of each attribute frame in Fig. \ref{fig:multiple_references} shows three arbitrary reference image pairs that we prepared using real images. On the right side of each attribute window, we show the encoding of the attributes to the generated images with our method, using the attribute vector directions corresponding the reference image pairs on the left. While using Eq. \ref{equ:z_new}, we set $k$ to $2$ for all the attribute editings in these samples. Although the reference images in attribute direction computations are quite different, since the reference images are consistent in the neutral and attributed images in all but the selected attributes, the computed directions are consistent and similar. When the generated images are carefully examined, we realize some minor differences in the generated images due to the differences in the reference images, such as hair boundaries, backgrounds etc. Such minor differences are expected as a result of small variations in the computed directions. These results show that arbitrary reference image pairs can be used to encode an attribute to an input image without changing the primary visual features on the input image.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{Figure16.pdf}
\caption{Attribute editing using different reference image pairs for the four attributes. Top left frame: smile, top right frame: hair color, bottom left frame: eyeglasses, bottom right frame: beard.}
\label{fig:multiple_references}
\end{figure}
\subsection{Effective range for $k$}
\label{sec:Range_for_k}
In order to understand how the images in an attribute direction change, we performed a set of observations by changing $k$, in Eq. \ref{equ:z_new}, in positive and negative directions for a large range of values. The results for two of the selected attributes, namely the skin color attribute and beard, are depicted in Fig. \ref{fig:boundary_of_k}. As can easily be seen in skin color attribute samples, the related attribute gradually increase/decrease in positive/negative directions, as we expected. In the local neighborhoods of a generated image, the attributes other than the edited, i.e. skin color and beard in these examples, are preserved. When we continue incrementing/decrementing $k$ in the attribute vector directions, images gradually change to different faces. Hence, the editing without changing the primary face features needs to be done in the local neighborhood of a given image. When the change on the axis is implemented with high margins from the original image, by means of using large magnitudes of $k$ values, e.g. $k>20$ or $k<-20$, independent of the starting image, all images converge to a particular pair of face image on the left and right ends. Indeed, there is no end on this axis, yet after some point the interpolation on the axis does not produce new images. This is not surprising, since at the very far ends of these axes in the latent space, there are almost no observed samples by the generator due to the z sampling we use during GAN training; each component of z is sampled from a Normal distribution with $\mu_{z_i}=0$ and $\sigma_{z_i}=1$. When we go very far, i.e. a lot more than $\mu_{z_i}\pm3\sigma_{z_i}$ in the components of $z$, i.e. $z_i$, that are effected by this attribute direction, the probability of change in the generated image get very close to zero. We observe the same issue with both attributes, skin color and beard, in a similar fashion (Fig. \ref{fig:boundary_of_k}).
In general, we observed that when we go through positive or negative direction in an attribute axis with small steps, the other input features are kept very similar to the original input when $k<2$. The sequence of images in the middle sections of the Fig. \ref{fig:boundary_of_k} are sampled with $k=0.5$ to better display this observation.
\begin{figure}[!h]
\centering
\includegraphics[width=\textwidth]{Figure17.pdf}
\caption{Examination of the effect of the k parameter for a range of values. Top: change of skin color attribute for a range of k, bottom: change of beard attribute. The reference image pair are given above each sample. For both attributes, k = 0 represents the start image.}
\label{fig:boundary_of_k}
\end{figure}
In order to make a more conservative analysis to determine the ranges of $k$, we can utilize the sample distributions that we presented in Section \ref{sec:Analysis_of_attrib_dir}. Utilizing such a distribution for an attribute may be helpful to define a more systematic way for determining the boundaries of $k$. What is meant with conservative analysis is that, we want to find the effective boundaries of $k$ for an attribute so that the essential face attributes of a given image is preserved while editing only the selected attribute. As we presented in Section \ref{sec:Analysis_of_attrib_dir}, we obtain two Normal densities in each attribute direction when we project real images in the computed directions; one for the neutral images, i.e. neutral with respect to the selected attribute, and one for the attributed images. The mean and standard deviations of each distribution, for different attributes, can guide us to analyze and define the safe rages for $k$. As we have already shown in Fig. \ref{fig:gaussian_tails}, the samples beyond $3\sigma$ on the left tails of the attribute distributions contain samples with barely visible attributes, while the right tails contain images with very exaggerated attributes. Hence, we believe that going beyond $\mu\pm3\sigma$ on an attribute axis may create some noticeable changes in the given input facial attributes, which we want to preserve during editing. In this view, for any z in the latent space, we can formulate the range of $k$ by computing its projected distances to $\mu\pm3\sigma$ on the attribute axis. We then use the values in this range for safely editing only the related attributes of an image.
In order to show the idea with an example, we selected two sample images and edit them using the set of attributes that we analyzed in Section \ref{sec:Analysis_of_attrib_dir} (in Fig. \ref{fig:sigma_analysis_for_k}). We first compute the z values of these images using our Encoder. Then, we compute the projections of their z vectors on the related attribute directions, separately for each attribute, i.e. $a$. Then we compute the projected distances of the images to the $\mu_a\pm3\sigma_a$ of the attribute distributions, i.e. subscript $a$ represents the relevant attribute. We set these distances as the bounds for $k$ values in our computations in these examples. In order to observe the generated images for the computed range, we set $k$ in such a way that it coincides with $\mu_a$, $\mu_a\pm2\sigma_a$ and $\mu_a\pm3\sigma_a$ points on the attribute axes. Using the original z values of both images and our encoder, we edit the images using our simple formula, Eq. \ref{equ:z_new}, with the determined $k$ values for the related four attributes. The resultant images are depicted in Fig. \ref{fig:sigma_analysis_for_k}. As can be seen from the images, when $k$ is set to the values between $\mu\pm3\sigma$ of the attribute distributions, the attribute editing successfully preserves other facial attributes while editing only the desired attributes. When we look at the distribution parameters, the average distance of a neutral image to the right tail (i.e. $\mu+3\sigma$) of the attributed distribution changes between $1.26$ to $1.84$ for the selected four attributes. However, this range becomes smaller if the input image already has the attribute, such as the first image, which has blond hair and smile attributes already. In such cases, the projection of the original input is already very close to the attribute mean on the attribute axis, which results in very small changes in the target attribute. These values are consistent with our observations; in our experiments, for editing some attributes in neutral images, we observed that $k=2$ usually works without changing the other facial attributes. Since the range of the selected attribute distributions are less than $2$ in these experiments, the resultant images in Fig. \ref{fig:sigma_analysis_for_k} preserves other facial attributes of the image while editing.
\begin{figure}[!h]
\centering
\includegraphics[width=0.7\textwidth]{Figure18.pdf}
\caption{Left column: sample input images, right column: edited images using different values for $k$ for four attributes.}
\label{fig:sigma_analysis_for_k}
\end{figure}
\section{Conclusion and Future Works}
\label{sec:conclusion_and_future_works}
We introduced a novel CRG architecture that is effective learning the inverse of a given generator using cyclic error minimization without supervision. The proposed architecture can be used to compute the accurate latent representations of generated images for attribute editing. The attribute editing can be performed by providing a pair of real images that contains a source image with and without desired attributes. The rate of manipulation of the input images can be controlled dynamically using only one parameter. The set of attributes for editing is also dynamic and determined via reference images. The results of the experiments show that the quality of the image reconstructions is more successful than the state-of-the-art models. Although the reconstruction of the generated images is very successful, the reconstruction performance of the model with the real images may be improved by using more training data with increased attribute variation. As a future work, in addition to increasing training data variation, we are also planning to train the encoder progressively, in parallel with generator training, to increase the reconstruction performance of the real images. Moreover, we are planning to make more comprehensive analysis on the latent spaces of various GAN generators using our CRG method.
\section*{Acknowledgements}
This research is funded by Ankara University (Scientific Research Projects Grant, grant id: 18L0443010). The numerical calculations reported in this paper were partially performed at TUBITAK ULAKBIM, High Performance and Grid Computing Center (TRUBA resources). We would like to thank the anonymous reviewers for their valuable suggestions and comments. We also thank Dr. Kai Dierkes for the helpful discussions and comments during the revision process.
\bibliographystyle{ieeetr}
\footnotesize |
1,116,691,500,485 | arxiv | \section{Introduction and Preliminaries}
A loop is called an Osborn loop if it obeys any of the two
identities below.
\begin{equation}\label{eq:4.1}
\textrm{OS$_3$}~:~(x\cdot yz)x=xy\cdot [(x^\lambda \cdot xz)\cdot x]
\end{equation}
\begin{equation}\label{eq:6.1}
\textrm{OS$_5$}~:~(x\cdot yz)x=xy\cdot [(x\cdot x^{\rho}z)\cdot x]
\end{equation}
For a comprehensive introduction to Osborn loops and its
universality, and a detailed literature review on it, readers should
check Jaiy\'e\d ol\'a , Ad\'en\'iran and S\`{o}l\'{a}r\`{i}n \cite{phd195} and Jaiy\'e\d ol\'a \cite{phd195b}.
In this present paper, we shall follow the style and notations used
in Jaiy\'e\d ol\'a , Ad\'en\'iran and S\`{o}l\'{a}r\`{i}n \cite{phd195} and Jaiy\'e\d ol\'a \cite{phd195b}. The only concepts
and notions which will be introduced here are those that were not
defined in Jaiy\'e\d ol\'a , Ad\'en\'iran and S\`{o}l\'{a}r\`{i}n \cite{phd195} and Jaiy\'e\d ol\'a \cite{phd195b}.
\begin{mydef}\label{definition:bijection}
Let $(L, \cdot )$ be a loop and $U, V, W\in SYM(L, \cdot )$.
\begin{enumerate}
\item If $(U, V, W)\in AUT(L, \cdot )$ for some $V, W$, then $U$ is called autotopic.
\item If $(U, V, W)\in AUT(L, \cdot )$ such that $W=U, V=I$, then $U$ is called $\lambda $-regular.
\item If $(U, V, W)\in AUT(L, \cdot )$ such that $U=I, W=V$, then $V$ is called $\rho $-regular.
\end{enumerate}
\end{mydef}
\paragraph{}
Drisko \cite{dris} while considering the action of isotopisms and autotopisms of loops, found it convenient to think of a loop $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ in terms of the set $T_Q$ of all ordered triples $(x,y,z)$ of elements of $Q$ such that $x\cdot y=z$. An isotopism $(\alpha ,\beta ,\gamma )$ from $G$ to $H$ takes $(x,y,z)\in T_G$ to $(x\alpha,y\beta,z\gamma)\in T_H$. We shall adopt his conventions at some points in time. We shall denote by $[\alpha,\beta]$, the commutator of any $\alpha,\beta\in SYM(G,\cdot )$.
Let $(Q, \cdot ,\backslash ,/)$ be a loop, then we shall be making use of the following notations for principal isotopes of $(Q, \cdot)$.
\begin{itemize}
\item $(Q,\ast_0 )$ represents $Q_{x,v}$;
\item $(Q,\circ_0 )$ represents $Q_{_{u,\phi_0(x,u,v)}}$, $\phi_0(x,u,v)=(u\backslash
([(uv)/(u\backslash (xv))]v))$;
\item $(Q,\circ_1 )$ represents $Q_{_{u,[u\backslash (xv)]}}$;
\item $(Q,\ast_1 )$ represents $Q_{_{\phi_1(x,u,v),v}}$, $\phi_1(x,u,v)=(u\backslash
([(uv)/(u\backslash (xv))]v))$ for all
$x,u,v\in Q$;
\item $(Q,\circ_2)$ represents $Q_{_{x,\phi_2(x,u,v)}}$, $\phi_2(x,u,v)=(u\backslash[(u/v)(u\backslash
(xv))])$;
\item $(Q,\circ_3 )$ represents $Q_{_{[x\cdot
u\backslash v]/v,[u\backslash (xv)]}}$;
\item $(Q,\ast_2)$ represents $Q_{u,e}$;
\item $(Q,\ast_3)$ represents $Q_{e,v}$.
\end{itemize}
Let $(G,\cdot )$ be a loop and let
$$BS_2(G,\cdot )=\{\theta\in SYM(G)~:~G(a,b)\overset{\theta}{\cong}G(c,d)~\textrm{for some}~a,b,c,d\in
G\}.$$ As shown in Bryant and Schneider \cite{phd92},
$BS_2(G,\cdot)$ forms a group for a loop $(G,\cdot )$ and it shall
be called the second Bryant-Schneider group (2$^{\textrm{nd}}$ BSG)
of the loop.
Consider the following two notions in algebraic topology.
\begin{mydef}
Let $V_Q$ be a set of isotopes of a loop $(Q,\cdot )$ and let $S_Q\subseteq ${\large
2}$^{{}^{V_Q}}$ such that $\phi\in S_Q$. If $S_Q$ is a topology on $V_Q$, then it is called the topology of isotopes of the loop $Q$ and the pair $(V_Q,S_Q)$ is called a topological space of isotopes of $Q$ if $(V_Q,S_Q)$ is a topological space.
\end{mydef}
Based on the above notion of topological space of isotopes of a loop, the following facts are direct consequences.
\begin{mylem}
Let $(Q,\cdot )$ be a loop and let $V_Q$ be the set of isotopes of $Q$.
Then, $\bigg(V_Q,${\large
2}$^{{}^{V_Q}}\bigg)$ is a topological space of isotopes of $Q$.
\end{mylem}
\begin{mylem}
Let $(Q,\cdot )$ be a G-loop and let $V_Q$ be the set of isotopes of $Q$. Let $S_Q=\{X_i\}_{i\in\Omega}\subseteq ${\large
2}$^{{}^{V_Q}}$ such that $\phi\in S_Q$ and $x_{i_j}\cong x_{i_k}$ for all $x_{i_j},x_{i_k}\in X_i$.
Then, $(V_Q,S_Q)$ is a topological space of isotopes of $Q$.
\end{mylem}
\begin{mycor}
Let $(Q,\cdot )$ be a CC-loop or VD-loop or K-loop or Buchsteiner loop or extra loop or group. Let $S_Q=\{X_i\}_{i\in\Omega}\subseteq ${\large
2}$^{{}^{V_Q}}$ such that $\phi\in S_Q$ and $x_{i_j}\cong x_{i_k}$ for all $x_{i_j},x_{i_k}\in X_i$.
Then, $(V_Q,S_Q)$ is a topological space of isotopes of $Q$.
\end{mycor}
\begin{mydef}
A simplicial complex is a pair $(V,S)$ where $V$ is a set of points called vertices and $S$ is a given family of finite subsets, called simplexes, so that the following conditions are satisfied:
\begin{enumerate}
\item all points of $V$ are simplexes;
\item any non-empty subset of a simplex is a simplex.
\end{enumerate}
A simplex consisting of $(n+1)$ points is called $n$-dimensional simplex.
\end{mydef}
\begin{mydef}
Let $V_Q$ be a set of isotopes of a loop $(Q,\cdot )$ and let $S_Q\subseteq ${\large
2}$^{{}^{V_Q}}$. If $K_Q=(V_Q,S_Q)$ is a simplicial complex, then $K_Q$ is called a trivial simplicial complex of isotopes of the loop $Q$.
\end{mydef}
\begin{mydef}
Let $V_Q$ be a set of isotopes of a loop $(Q,\cdot )$ and let $S_Q=\{X_i\}_{i\in\Omega}\subseteq ${\large
2}$^{{}^{V_Q}}$ such that $x_{i_j}\cong x_{i_k}$ for all $x_{i_j},x_{i_k}\in X_i$. If $K_Q=(V_Q,S_Q)$ is a simplicial complex, then $K_Q$ is called a non-trivial simplicial complex of isotopes or simplicial complex of isotopes of the loop $Q$.
\end{mydef}
The facts below follow suite.
\begin{mylem}
Let $(Q,\cdot )$ be a loop and let $V_Q$ be the set of isotopes of $Q$.
Then, $\bigg(V_Q,${\large
2}$^{{}^{V_Q}}\bigg)$ is a trivial simplicial complex of isotopes of $Q$.
\end{mylem}
\begin{mylem}
Let $(Q,\cdot )$ be a G-loop and let $V_Q$ be the set of isotopes of $Q$. Let $S_Q=\{X_i\}_{i\in\Omega}\subseteq ${\large
2}$^{{}^{V_Q}}$ such that $x_{i_j}\cong x_{i_k}$ for all $x_{i_j},x_{i_k}\in X_i$.
Then, $(V_Q,S_Q)$ is a simplicial complex of isotopes of $Q$.
\end{mylem}
\begin{mycor}
Let $(Q,\cdot )$ be a CC-loop or VD-loop or K-loop or Buchsteiner loop or extra loop or group. Let $S_Q=\{X_i\}_{i\in\Omega}\subseteq ${\large
2}$^{{}^{V_Q}}$ such that $x_{i_j}\cong x_{i_k}$ for all $x_{i_j},x_{i_k}\in X_i$.
Then, $(V_Q,S_Q)$ is a simplicial complex of isotopes of $Q$.
\end{mycor}
\begin{mydef}
Let $K=(V,S)$ and $K'=(V',S')$ be two simplicial complexes. A simplicial map $f~:~K\to K'$ is a set map $f~:~V\to V'$ satisfying the property: for every simplex $x\in S$, the image $f(x)\in S'$.
\end{mydef}
In this work, the notion of simplicial complex is used to characterize universal Osborn loops. The following results are important for the set objective.
\begin{myth}\label{1:4}(Jaiy\'e\d ol\'a , Ad\'en\'iran and S\`{o}l\'{a}r\`{i}n \cite{phd195})
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop and $\gamma_0
(x,u,v)=\mathbb{R}_vR_{[u\backslash (xv)]}\mathbb{L}_uL_x$ for all
$x,u,v\in Q$, then $\mathcal{Q}$ is a universal Osborn
loop if and only if the commutative diagram
\begin{equation}\label{eq:7}
\begin{diagram}
& &(Q,\circ_0 )\\
&\ruTo^{(R_{\phi_0 (x,u,v)},L_u,I)}_{}&\dTo^{(\gamma_0,\gamma_0,\gamma_0)}_{\textrm{isomorphism}}\\
(Q,\cdot ) &\rTo^{(R_v,L_x,I)}_{\textrm{principal isotopism}} &(Q,\ast_0)
\end{diagram}
\end{equation}
holds.
\end{myth}
\begin{myth}\label{post1:4}(Ja\'iy\'e\d ol\'a \cite{phd195b})
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop and $\gamma_1
(x,u,v)=\mathbb{R}_vR_{[u\backslash (xv)]}\mathbb{L}_uL_x$ for all
$x,u,v\in Q$, then $\mathcal{Q}$ is a universal Osborn
loop if and only if the commutative diagram
\begin{equation}\label{eq:8}
\begin{diagram}
& &(Q,\ast_1 )\\
&\ruTo^{(R_v,L_{\phi_1 (x,u,v)},I)}_{}&\dTo^{(\gamma_1,\gamma_1,\gamma_1)}_{\textrm{isomorphism}}\\
(Q,\cdot ) &\rTo^{(R_{[u\backslash (xv)]},L_u,I)}_{\textrm{principal isotopism}} &(Q,\circ_1)
\end{diagram}
\end{equation}
holds.
\end{myth}
\begin{myth}\label{1:12}(Jaiy\'e\d ol\'a , Ad\'en\'iran and S\`{o}l\'{a}r\`{i}n \cite{phd195})
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop and $\gamma_0
(x,u,v)=\mathbb{R}_vR_{[u\backslash (xv)]}\mathbb{L}_uL_x$ for all
$x,u,v\in Q$, then $\mathcal{Q}$ is a universal Osborn
loop implies the commutative diagram
\begin{equation}\label{eq:7.m}
\begin{diagram}
& &(Q,\circ_2 )\\
&\ruTo^{(R_{\phi_2 (x,u,v)},L_x,I)}_{}&\uTo^{(\gamma_0,\gamma_0,\gamma_0)}_{\textrm{isomorphism}}\\
(Q,\cdot ) &\rTo^{(I,L_u,I)}_{\textrm{principal isotopism}} &(Q,\ast_2)
\end{diagram}
\end{equation}
holds.
\end{myth}
\begin{myth}\label{post1:12}(Ja\'iy\'e\d ol\'a \cite{phd195b})
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop and $\gamma_1
(x,u,v)=\mathbb{R}_vR_{[u\backslash (xv)]}\mathbb{L}_uL_x$ for all
$x,u,v\in Q$, then $\mathcal{Q}$ is a universal Osborn
loop implies the commutative diagram
\begin{equation}\label{eq:8.m}
\begin{diagram}
& &(Q,\circ_3 )\\
&\ruTo^{(R_{[u\backslash (xv)]},L_{[x\cdot
u\backslash v]/v},I)}_{}&\uTo^{(\gamma_1,\gamma_1,\gamma_1)}_{\textrm{isomorphism}}\\
(Q,\cdot ) &\rTo^{(R_v,I,I)}_{\textrm{principal isotopism}} &(Q,\ast_3)
\end{diagram}
\end{equation}
holds.
\end{myth}
\begin{mylem}\label{drispost}(Drisko \cite{dris})
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop. Then $Q_{f,g}\cong Q_{c,d}$ if and only if there exists $(\alpha , \beta ,\gamma )\in AUT(\mathcal{Q})$ such that $(f,g,fg)(\alpha ,\beta ,\gamma )=(c,d,cd)$.
\end{mylem}
\begin{myth}\label{0:3}(Bryant and Schneider \cite{phd92})
Let $(Q, \cdot ,\backslash ,/)$ be a quasigroup. If $Q_{a,b}\overset{I}{\cong}Q_{c,d}$ if and only if $c\cdot b,a\cdot d\in N_\mu (Q_{a,b})$ and $a\cdot b=c\cdot d$.
\end{myth}
\section{Main Results}
\begin{myth}\label{2post1:10}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then, the following are necessary and sufficient for each other.
\begin{enumerate}
\item $(Q,\circ_0 )\overset{I}{\cong}(Q,\circ_1 )$.
\item $(Q,\ast_0 )\overset{I}{\cong}(Q,\ast_1 )$.
\item $\mathcal{Q}$ is a boolean group.
\end{enumerate}
\end{myth}
{\bf Proof}\\
By combining the commutative diagrams in Equation~\ref{eq:7} and Equation~\ref{eq:8}, we have the commutative diagram below.
\begin{equation}\label{eq:9}
\begin{diagram}
(Q,\circ_1) & & & (Q,\circ_1)&&&&&&&&(Q,\circ_1)\\
&&&&\\
&&&&\\
& & & \uTo(0,2)^{\gamma_{01}^\circ}&\\
& & & (Q,\circ_0)\\
\uTo^{(R_{[u\backslash (xv)]},L_u,I)}&\ruTo^{(R_{\phi_0},L_u,I)} & & &\rdTo(2,2)^{\gamma_0}&\\
(Q,\cdot )&\rTo^{(R_v,L_x,I)}&&&&(Q,\ast_0)& &\\
\dTo^{(R_v,L_{\phi_1},I)} &&&&&&\rdTo(6,3)^{\gamma_{01}^*}&\\
& & & &&&&&&&&\uTo(7,0)_{\gamma_1}\\
(Q,\ast_1) & & & & &&&&&&&(Q,\ast_1)&
\end{diagram}
\end{equation}
Let
\begin{displaymath}
(Q,\circ_0 )\xrightarrow[\textrm{isotopism}]{(\delta_{01}^\circ ,\varepsilon_{01}^\circ ,\pi_{01}^\circ)}(Q,\circ_1 ).
\end{displaymath}
So, from Equation~\ref{eq:9},
\begin{gather*}
(R_{\phi_0 (x,u,v)},L_u,I)(\delta_{01}^\circ ,\varepsilon_{01}^\circ ,\pi_{01}^\circ)=(R_{[u\backslash (xv)]},L_u,I)\Rightarrow\\
(R_{\phi_0 (x,u,v)}\delta_{01}^\circ ,L_u\varepsilon_{01}^\circ,\pi_{01}^\circ)=(R_{[u\backslash (xv)]},L_u,I)\Leftrightarrow\\
R_{\phi_0 (x,u,v)}\delta_{01}^\circ =R_{[u\backslash (xv)]},~L_u\varepsilon_{01}^\circ =L_u~\textrm{and}~\pi_{01}^\circ =I\Leftrightarrow\\
\delta_{01}^\circ =R_{\phi_0 (x,u,v)}^{-1}R_{[u\backslash (xv)]},~\varepsilon_{01}^\circ =L_u^{-1}L_u=I~\textrm{and}~\pi_{01}^\circ =I.
\end{gather*}
Thus, $(Q,\circ_0 )\cong (Q,\circ_1 )$ iff $\delta_{01}^\circ =\varepsilon_{01}^\circ =I$ iff
\begin{gather*}
R_{\phi_0 (x,u,v)}^{-1}R_{[u\backslash (xv)]}=I\Leftrightarrow \phi_0 (x,u,v)=[u\backslash (xv)]\\
(u\backslash
([(uv)/(u\backslash (xv))]v))=[u\backslash (xv)]\Leftrightarrow x\backslash (uv)=u\backslash (xv).
\end{gather*}
Similarly, by using the procedure above, it can be shown that $(Q,\ast_0 )\cong (Q,\ast_1 )$ iff $x\backslash (uv)=u\backslash (xv)$.
Keeping in mind that every Osborn loop of exponent 2 is an abelian group, hence, a Boolean group. This completes the proof.
\begin{myrem}
It can be observed that in a universal Osborn loop $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ and for $\gamma_0
(x,u,v)$ and $\gamma_1(x,u,v)$ of Theorem~\ref{1:4} and Theorem~\ref{post1:4},
$\gamma_0
(x,u,v)=\gamma_1(x,u,v)$ if and only if $\Big[\mathbb{L}_uL_x,\mathbb{R}_vR_{[u\backslash (xv)]}\Big]=I$ for all $x,u,v\in Q$.
The proof of Theorem~\ref{2post1:10} can also be achieved by making use of Theorem~\ref{0:3}. Take $a=u$, $b=\phi_0 (x,u,v)$, $c=u$ and $d=u\backslash (xv)$. Then, $(Q,\circ_0 )\overset{I}{\cong}(Q,\circ_1 )$ iff
\begin{gather*}
\textrm{(i)}~u\phi_0 (x,u,v)\in N_\mu \big((Q,\circ_0 )\big),~\textrm{(ii)}~u[u\backslash (xv)]\in N_\mu \big((Q,\circ_0 )\big),~\textrm{(iii)}~u\phi_0 (x,u,v)=u[u\backslash (xv)]\Leftrightarrow
\end{gather*}
$\mathcal{Q}$ is a Boolean group.
\end{myrem}
\begin{myth}\label{2post1:11}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\circ_0 )\cong(Q,\circ_1 )$ if and only if there exists $(I,\beta ,\gamma )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:11}
uv=xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v\cdot xR_v\mathbb{L}_u=xR_v\gamma^{-1}\mathbb{R}_v\cdot xR_v\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$.
\end{myth}
{\bf Proof}\\
Following Lemma~\ref{drispost},
$(Q,\circ_0 )\cong(Q,\circ_1 )$ if and only if there exists $(\alpha,\beta ,\gamma )\in AUT(\mathcal{Q})$ such that
\begin{gather*}
(u,\phi_0 (x,u,v),u\phi_0 (x,u,v))(\alpha,\beta ,\gamma )=(u,[u\backslash (xv)],xv)\Leftrightarrow\\
(u\alpha,\phi_0 (x,u,v)\beta,(u\phi_0 (x,u,v))\gamma )=(u,[u\backslash (xv)],xv)\Leftrightarrow\\
u\alpha =u,~\phi_0 (x,u,v)\beta =[u\backslash (xv)]~\textrm{and}~(u\phi_0 (x,u,v))\gamma=xv\Leftrightarrow\\
\alpha =I,~\{u\backslash
([(uv)/(u\backslash (xv))]v)\}\beta =u\backslash (xv)~\textrm{and}~\{
[(uv)/(u\backslash (xv))]v\}\gamma=xv\Leftrightarrow\\
\alpha =I,~[(uv)/(u\backslash (xv))]R_v\mathbb{L}_u\beta =xR_v\mathbb{L}_u~\textrm{and}~[(uv)/(u\backslash (xv))]R_v\gamma=xR_v\Leftrightarrow\\
\alpha =I,~(uv)/(u\backslash (xv))=xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v
~\textrm{and}~[(uv)/(u\backslash (xv))]=xR_v\gamma^{-1}\mathbb{R}_v\Leftrightarrow\\
\alpha =I,~uv=xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v\cdot xR_v\mathbb{L}_u
~\textrm{and}~uv=xR_v\gamma^{-1}\mathbb{R}_v\cdot xR_v\mathbb{L}_u\Leftrightarrow
\end{gather*}
there exists $(I,\beta ,\gamma )\in AUT(\mathcal{Q})$ such that
\begin{displaymath}
uv=xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v\cdot xR_v\mathbb{L}_u=xR_v\gamma^{-1}\mathbb{R}_v\cdot xR_v\mathbb{L}_u.
\end{displaymath}
\begin{myrem}
If the autotopism $(\alpha,\beta ,\gamma )$ in Theorem~\ref{2post1:11} is the identity autotopism, then we shall have the equivalence of 1. and 3. of Theorem~\ref{2post1:10}.
\end{myrem}
\begin{mycor}\label{2post1:12}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\circ_0 )\cong(Q,\circ_1 )$ implies that there exists $(I,\beta ,\gamma )\in AUT(\mathcal{Q})$ such that $\gamma =\mathbb{L}_u\beta L_u$ for all $u\in Q$. Hence,
\begin{enumerate}
\item $\gamma =\beta$ iff $[\beta ,L_u]=I$ or $[\gamma ,L_u]=I$. Thence, $\beta$ is a $\rho$-regular permutation.
\item $\gamma =L_u$ iff $\beta =L_u$. Thence, $\mathcal{Q}$ is an abelian group.
\end{enumerate}
\end{mycor}
{\bf Proof}\\
The proof of these follows from the fact in Theorem~\ref{2post1:11} that
\begin{displaymath}
xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v\cdot xR_v\mathbb{L}_u=xR_v\gamma^{-1}\mathbb{R}_v\cdot xR_v\mathbb{L}_u\Rightarrow
\end{displaymath}
$\mathbb{L}_u\beta L_u=\gamma$ for all $u\in Q$.
\begin{myth}\label{2post1:11b}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\ast_0 )\cong(Q,\ast_1 )$ if and only if there exists $(\delta,I ,\pi )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:11b}
uv=x\cdot x\delta R_v\mathbb{L}_u=x\cdot xR_v\pi\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$.
\end{myth}
{\bf Proof}\\
Following Lemma~\ref{drispost},
$(Q,\ast_0 )\cong(Q,\ast_1 )$ if and only if there exists $(\delta,\varepsilon ,\pi )\in AUT(\mathcal{Q})$ such that $(x,v,xv)(\delta,\varepsilon ,\pi )=(\phi_1(x,u,v),v,\phi_1(x,u,v)v)$. The procedure of the proof of the remaining part is similar to that of Theorem~\ref{2post1:11}.
\begin{myrem}
If the autotopism $(\delta,\varepsilon ,\pi )$ in Theorem~\ref{2post1:11b} is the identity autotopism, then we shall have the equivalence of 2. and 3. of Theorem~\ref{2post1:10}.
\end{myrem}
\begin{mycor}\label{2post1:13}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\ast_0 )\cong(Q,\ast_1 )$ implies that there exists $(\delta,I ,\pi )\in AUT(\mathcal{Q})$ such that $\pi =\mathbb{R}_v\delta R_v$ for all $v\in Q$. Hence,
\begin{enumerate}
\item $\pi =\delta$ iff $[\delta ,R_v]=I$ or $[\pi ,R_v]=I$. Thence, $\delta$ is a $\lambda$-regular permutation.
\item $\delta =R_v$ iff $\pi =R_v$. Thence, $\mathcal{Q}$ is an abelian group.
\end{enumerate}
\end{mycor}
{\bf Proof}\\
The proof of these follows from the fact in Theorem~\ref{2post1:11b} that
\begin{displaymath}
x\cdot x\delta R_v\mathbb{L}_u=x\cdot xR_v\pi\mathbb{L}_u\Rightarrow
\end{displaymath}
$\pi =\mathbb{R}_v\delta R_v$ for all $v\in Q$.
\begin{myth}\label{2post1:14}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\circ_0 )\cong(Q,\circ_1 )$ and $(Q,\ast_0 )\cong(Q,\ast_1 )$ if and only if there exists $(I,\beta ,\gamma ),(\delta,I ,\pi )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:10}
uv=xR_v\mathbb{L}_u\beta^{-1}L_u\mathbb{R}_v\cdot xR_v\mathbb{L}_u=xR_v\gamma^{-1}\mathbb{R}_v\cdot xR_v\mathbb{L}_u=x\cdot x\delta R_v\mathbb{L}_u=x\cdot xR_v\pi\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$
\end{myth}
{\bf Proof}\\
This is achieved by simply combining Theorem~\ref{2post1:11} and Theorem~\ref{2post1:11b}.
\begin{myth}\label{2post1:15}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. If
$(Q,\circ_0 )\overset{\gamma_{01}^\circ}{\cong}(Q,\circ_1 )$ and $(Q,\ast_0 )\overset{\gamma_{01}^\ast}{\cong}(Q,\ast_1 )$, then $\gamma_0\gamma_{01}^\ast\gamma_1=\gamma_{01}^\circ$.
\end{myth}
{\bf Proof}\\
The commutative diagram in Equation~\ref{eq:9} proves this.
\begin{mycor}\label{2post1:16}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. If $(Q,\circ_0 )\cong(Q,\circ_1 )$ and $(Q,\ast_0 )\cong(Q,\ast_1 )$, then the following are necessary and sufficient for each other.
\begin{multicols}{3}
\begin{enumerate}
\item $\beta =I$.
\item $\gamma =I$.
\item $\delta =I$.
\item $\pi =I$.
\item $(Q,\circ_0 )\overset{I}{\cong}(Q,\circ_1 )$.
\item $(Q,\ast_0 )\overset{I}{\cong}(Q,\ast_1 )$.
\item $\mathcal{Q}$ is a boolean group.
\end{enumerate}
\end{multicols}
\end{mycor}
{\bf Proof}\\
To prove the equivalence of 1. to 4. and 7., use Equation~\ref{eq:10} of
Theorem~\ref{2post1:14}. The proof of the equivalence of 5. to 7. follows from Theorem~\ref{2post1:10}.
\begin{myrem}
Corollary~\ref{2post1:16} is a very important result in this study. It gives us the main distinctions between Theorem~\ref{2post1:10} and Theorem~\ref{2post1:14}. That is, the
necessary and sufficient condition(s) under which the isomorphisms $(Q,\circ_0 )\cong(Q,\circ_1 )$ and $(Q,\ast_0 )\cong(Q,\ast_1 )$ will be trivial. And the condition(s) is when any of the autotopic permutations of $\beta$, $\gamma$, $\delta$ and
$\pi$ of Theorem~\ref{2post1:11} and Theorem~\ref{2post1:11b} is equal to the identity mapping.
\end{myrem}
\paragraph{}
Next, it is important to deduce the actual definitions of the autotopic mappings $\beta$, $\gamma$, $\delta$, $\pi$ and the isomorphisms $\gamma_{01}^\ast$ and $\gamma_{01}^\circ$. Recall that by the necessary part of Lemma~\ref{drispost}, if $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ is a loop and $Q_{f,g}\overset{\theta}{\cong}Q_{c,d}$, then there exists $(A,B,C)\in AUT(\mathcal{Q})$ such that $(f,g,fg)(A,B,C)=(c,d,cd)$. According to the proof of this,
\begin{equation}\label{eq:10.m}
(A,B,C)=(R_g\theta R_d^{-1},L_f\theta L_c^{-1},\theta)\Leftrightarrow A=R_g\theta R_d^{-1},~B=L_f\theta L_c^{-1}~\textrm{and}~C=\theta.
\end{equation}
Thus,
\begin{gather*}
I=\alpha =R_{\phi_o(x,u,v)}\gamma_{01}^\circ R_{[u\backslash (xv)]}^{-1},~\beta=L_u\gamma_{01}^\circ L_u^{-1}~\textrm{and}~\gamma=\gamma_{01}^\circ\\
\gamma_{01}^\circ =\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]},~\beta=L_u\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}\mathbb{L}_u^{-1}~\textrm{and}~\gamma=\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}
\end{gather*}
and
\begin{gather*}
\delta =R_v\gamma_{01}^\ast R_v^{-1},~I=\varepsilon=L_x\gamma_{01}^\ast L_{\phi_1(x,u,v)}^{-1}~\textrm{and}~\pi=\gamma_{01}^\ast\\
\delta =R_v\gamma_{01}^\ast \mathbb{R}_v^{-1},~\gamma_{01}^\ast =\mathbb{L}_xL_{\phi_1(x,u,v)}~\textrm{and}~\pi=\gamma_{01}^\ast\\
\delta =R_v\mathbb{L}_xL_{\phi_1(x,u,v)}\mathbb{R}_v^{-1},~\gamma_{01}^\ast =\mathbb{L}_xL_{\phi_1(x,u,v)}~\textrm{and}~\pi=\mathbb{L}_xL_{\phi_1(x,u,v)}.
\end{gather*}
Therefore, Theorem~\ref{2post1:11} and Theorem~\ref{2post1:11b} can now be restated as follows.
\begin{myth}\label{2post1:17}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\circ_0 )\overset{\gamma_{01}^\circ}{\cong}(Q,\circ_1 )$ if and only if
\begin{equation}\label{eq:12}
y\cdot u\backslash [(uz)\psi_0]=(yz)\psi_0~\textrm{and}~uv=xR_v(R_v\psi_0)^{-1}\cdot xR_v\mathbb{L}_u
\end{equation}
where $\psi_0=\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}$ for all $x,y,z,u,v\in Q$
\end{myth}
{\bf Proof}\\
Simply substitute
\begin{displaymath}
\beta=L_u\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}\mathbb{L}_u^{-1}~\textrm{and}~\gamma=\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}
\end{displaymath}
into Equation~\ref{eq:11} of Theorem~\ref{2post1:11}.
\begin{myth}\label{2post1:17b}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\ast_0 )\overset{\gamma_{01}^\ast}{\cong}(Q,\ast_1 )$ if and only if
\begin{equation}\label{eq:12b}
[(yv)\psi_1]/v\cdot z=(yz)\psi_1~\textrm{and}~uv=x\cdot u\backslash [(xv)\psi_1]
\end{equation}
where $\psi_1=\mathbb{L}_xL_{\phi_1(x,u,v)}$ for all $x,y,z,u,v\in Q$
\end{myth}
{\bf Proof}\\
Simply substitute
\begin{displaymath}
\delta =R_v\mathbb{L}_xL_{\phi_1(x,u,v)}\mathbb{R}_v^{-1}~\textrm{and}~\pi=\mathbb{L}_xL_{\phi_1(x,u,v)}
\end{displaymath}
into Equation~\ref{eq:11b} of Theorem~\ref{2post1:11b}.
\begin{mylem}\label{2post1:17c}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop.
\begin{enumerate}
\item $\mathcal{Q}$ is a universal Osborn loop and obeys Equation~\ref{eq:12} if and only if
$\gamma_0,\gamma_{01}^\circ\in BS_2(\mathcal{Q})$.
\item $\mathcal{Q}$ is a universal Osborn loop and obeys Equation~\ref{eq:12b} if and only if
$\gamma_1,\gamma_{01}^\ast\in BS_2(\mathcal{Q})$.
\end{enumerate}
\end{mylem}
{\bf Proof}\\
This follows by combining Theorem~\ref{1:4}, Theorem~\ref{post1:4}, Theorem~\ref{2post1:11} and Theorem~\ref{2post1:11b}
\begin{myrem}
It is a self exercise to confirm if $(Q,\circ_0 )\overset{\gamma_{01}^\circ}{\cong}(Q,\circ_1 )$ and $(Q,\ast_0 )\overset{\gamma_{01}^\ast}{\cong}(Q,\ast_1 )$ in some universal Osborn loops like Moufang loops and extra loops by simply verifying Equation~\ref{eq:12} and Equation~\ref{eq:12b}. Furthermore, the relation $\gamma_0\gamma_{01}^\ast\gamma_1=\gamma_{01}^\circ$ of Theorem~\ref{2post1:15} is justifiable as well. It must be noted also, that in any universal Osborn loop $\mathcal{Q}$, Equation~\ref{eq:12} and Equation~\ref{eq:12b} are necessary and sufficient conditions for $\gamma_{01}^\ast,\gamma_{01}^\circ\in BS_2(\mathcal{Q})$.
\end{myrem}
\paragraph{}
By combining the commutative diagrams in Equation~\ref{eq:7.m} and Equation~\ref{eq:8.m}, we have the commutative diagram below.
\begin{equation}\label{eq:17}
\begin{diagram}
(Q,\circ_3) & & & (Q,\circ_3)&&&&&&&&(Q,\circ_3)\\
&&&&\\
&&&&\\
& & & \uTo(0,2)^{\gamma_{23}^\circ}&\\
& & & (Q,\circ_2)\\
\uTo^{(R_{[u\backslash (xv)]},L_{\{[x\cdot u\backslash v]/v\}},I)}&\ruTo^{(R_{\phi_2},L_x,I)} & & &\rdTo(2,2)^{\gamma_0}&\\
(Q,\cdot )&\rTo^{(I,L_u,I)}&&&&(Q,\ast_2)& &\\
\dTo^{(R_v,I,I)} &&&&&&\rdTo(6,3)^{\gamma_{23}^*}&\\
& & & &&&&&&&&\uTo(7,0)_{\gamma_1}\\
(Q,\ast_3) & & & & &&&&&&&(Q,\ast_3)&
\end{diagram}
\end{equation}
\begin{myth}\label{2post1:18}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\circ_2 )\cong(Q,\circ_3 )$ if and only if there exists $(\lambda,\mu,\nu )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:13}
\lambda =R_{u\backslash v}\mathbb{R}_v,~\mu=L_u\mathbb{L}_{u\backslash v}~\textrm{and}~
[x\cdot xR_v\mathbb{L}_u\mu^{-1}]\nu=x\lambda\cdot xR_v\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$.
\end{myth}
{\bf Proof}\\
Following Lemma~\ref{drispost},
$(Q,\circ_2 )\cong(Q,\circ_3 )$ if and only if there exists $(\lambda,\mu,\nu )\in AUT(\mathcal{Q})$ such that
$(x,\phi_2(x,u,v),x\phi_2(x,u,v))(\lambda,\mu,\nu )=([x\cdot
u\backslash v]/v,[u\backslash (xv)],\{[x\cdot
u\backslash v]/v\}[u\backslash (xv)])$. The procedure of the proof of the remaining part is similar to that of Theorem~\ref{2post1:11}.
\begin{mylem}\label{2post1:19}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. Then
$(Q,\circ_2 )\overset{\gamma_{23}^\circ}{\cong}(Q,\circ_3 )$ if and only if there exists $(\lambda,\mu,\gamma_{23}^\circ )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:14}
\gamma_{23}^\circ=\mathbb{R}_{\phi_2(x,u,v)}R_{u\backslash v}\mathbb{R}_vR_{[u\backslash (xv)]}=\mathbb{L}_xL_u\mathbb{L}_{u\backslash v}L_{\{[x\cdot
u\backslash v]/v\}}~\textrm{and}~
[x\cdot xR_v\mathbb{L}_u\mu^{-1}]\gamma_{23}^\circ=x\lambda\cdot xR_v\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$.
\end{mylem}
{\bf Proof}\\
Considering the commutative diagram in Equation~\ref{eq:17} and using Equation~\ref{eq:10.m},
\begin{displaymath}
\lambda=R_{\phi_2(x,u,v)}\gamma_{23}^\circ R_{[u\backslash (xv)]}^{-1},~\mu=L_x\gamma_{23}^\circ L_{\{[x\cdot
u\backslash v]/v\}}^{-1}~\textrm{and}~\nu=\gamma_{23}^\circ.
\end{displaymath}
The final conclusion follows from Theorem~\ref{2post1:18}.
\begin{mycor}\label{2post1:20}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a universal Osborn loop. $\gamma_{23}^\circ\in BS_2(\mathcal{Q})$ if and only if there exists $(\lambda,\mu,\gamma_{23}^\circ )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:15}
\gamma_{23}^\circ=\mathbb{R}_{\phi_2(x,u,v)}R_{u\backslash v}\mathbb{R}_vR_{[u\backslash (xv)]}=\mathbb{L}_xL_u\mathbb{L}_{u\backslash v}L_{\{[x\cdot
u\backslash v]/v\}}~\textrm{and}~
[x\cdot xR_v\mathbb{L}_u\mu^{-1}]\gamma_{23}^\circ=x\lambda\cdot xR_v\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$.
\end{mycor}
{\bf Proof}\\
This follows from Lemma\ref{2post1:19}.
\begin{mycor}\label{2post1:21}
Let $\mathcal{Q}=(Q, \cdot ,\backslash ,/)$ be a loop. $\mathcal{Q}$ is a universal Osborn loop and $\gamma_{23}^\circ\in BS_2(\mathcal{Q})$ implies $\gamma_0\in BS_2(\mathcal{Q})$ and there exists $(\lambda,\mu,\gamma_{23}^\circ )\in AUT(\mathcal{Q})$ such that
\begin{equation}\label{eq:16}
\gamma_{23}^\circ=\mathbb{R}_{\phi_2(x,u,v)}R_{u\backslash v}\mathbb{R}_vR_{[u\backslash (xv)]}=\mathbb{L}_xL_u\mathbb{L}_{u\backslash v}L_{\{[x\cdot
u\backslash v]/v\}}~\textrm{and}~
[x\cdot xR_v\mathbb{L}_u\mu^{-1}]\gamma_{23}^\circ=x\lambda\cdot xR_v\mathbb{L}_u
\end{equation}
for all $x,u,v\in Q$.
\end{mycor}
{\bf Proof}\\
This follows from Theorem~\ref{1:12} and Lemma\ref{2post1:19}.
\subsection*{Simplicial Complex of Isotopes of a Universal Osborn Loop}
\begin{myth}\label{0}
Let $(Q,\cdot )$ be a loop. Let $V_0(Q)=\big\{(Q,\cdot ),(Q,\circ_0),(Q,\ast_0)\big\}$ and $S_0(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_0)\},\{(Q,\ast_0)\},\big\{(Q,\circ_0),(Q,\ast_0)\big\}\Big\}$. Then, $(Q,\cdot )$ is a universal Osborn loop if and only if $K_0(Q)=\Big(V_0(Q),S_0(Q)\Big)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{myth}
{\bf Proof}\\
This is proved with the help of Theorem~\ref{1:4}.
\begin{myth}\label{1}
Let $(Q,\cdot )$ be a loop. Let $V_1(Q)=\big\{(Q,\cdot ),(Q,\circ_1),(Q,\ast_1)\big\}$ and $S_1(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_1)\},\{(Q,\ast_1)\},\big\{(Q,\circ_1),(Q,\ast_1)\big\}\Big\}$. Then, $(Q,\cdot )$ is a universal Osborn loop if and only if $K_1(Q)=\Big(V_1(Q),S_1(Q)\Big)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{myth}
{\bf Proof}\\
This is proved with the help of Theorem~\ref{post1:4}.
\begin{myth}\label{2}
Let $(Q,\cdot )$ be a loop. Let $V_2(Q)=\big\{(Q,\cdot ),(Q,\circ_2),(Q,\ast_2)\big\}$ and $S_2(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_2)\},\{(Q,\ast_2)\},\big\{(Q,\circ_2),(Q,\ast_2)\big\}\Big\}$. If $(Q,\cdot )$ is a universal Osborn loop, then $K_2(Q)=\Big(V_2(Q),S_2(Q)\Big)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{myth}
{\bf Proof}\\
This is proved with Theorem~\ref{1:12}.
\begin{myth}\label{3}
Let $(Q,\cdot )$ be a loop. Let $V_3(Q)=\big\{(Q,\cdot ),(Q,\circ_3),(Q,\ast_3)\big\}$ and $S_3(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_3)\},\{(Q,\ast_3)\},\big\{(Q,\circ_3),(Q,\ast_3)\big\}\Big\}$. If $(Q,\cdot )$ is a universal Osborn loop, then $K_3(Q)=\Big(V_3(Q),S_3(Q)\Big)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{myth}
{\bf Proof}\\
This is proved with the aid of Theorem~\ref{post1:12}.
\begin{mycor}\label{01}
Let $(Q,\cdot )$ be a loop. Let $V_i(Q)=\big\{(Q,\cdot ),(Q,\circ_i),(Q,\ast_i)\big\}$ and $S_i(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_i)\},\{(Q,\ast_i)\},\big\{(Q,\circ_i),(Q,\ast_i)\big\}\Big\}$ for $i=0,1$. Then, $(Q,\cdot )$ is a universal Osborn loop if and only if $K_{01}(Q)=K_0(Q)\cup K_1(Q)=\bigg(V_0(Q)\cup V_1(Q),S_0(Q)\cup S_1(Q)\bigg)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{mycor}
{\bf Proof}\\
This follows from Theorem~\ref{0} and Theorem~\ref{1}.
\begin{mycor}\label{23}
Let $(Q,\cdot )$ be a loop. Let $V_i(Q)=\big\{(Q,\cdot ),(Q,\circ_i),(Q,\ast_i)\big\}$ and $S_i(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_i)\},\{(Q,\ast_i)\},\big\{(Q,\circ_i),(Q,\ast_i)\big\}\Big\}$ for $i=2,3$. If $(Q,\cdot )$ is a universal Osborn loop, then $K_{23}(Q)=K_2(Q)\cup K_3(Q)=\bigg(V_2(Q)\cup V_3(Q),S_2(Q)\cup S_3(Q)\bigg)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{mycor}
{\bf Proof}\\
This follows from Theorem~\ref{2} and Theorem~\ref{3}.
\begin{mycor}
Let $(Q,\cdot )$ be a loop. Let $V_i(Q)=\big\{(Q,\cdot ),(Q,\circ_i),(Q,\ast_i)\big\}$ and $S_i(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_i)\},\{(Q,\ast_i)\},\big\{(Q,\circ_i),(Q,\ast_i)\big\}\Big\}$ for $i=0,1,2,3$. If $(Q,\cdot )$ is a universal Osborn loop, then $\displaystyle K_{0123}(Q)=\bigcup^3_{i=0}K_i(Q)=\bigg(\bigcup^3_{i=0}V_i(Q),\bigcup^3_{i=0}S_i(Q)\bigg)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{mycor}
{\bf Proof}\\
This is proved by combining Corollary~\ref{01} and Corollary~\ref{23}.
\begin{myth}
Let $(Q,\cdot )$ be a loop. Let $V_{01}(Q)=\big\{(Q,\cdot ),(Q,\circ_0),(Q,\ast_0),(Q,\circ_1),(Q,\ast_1)\big\}$ and $S_{10}(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_0)\},\{(Q,\ast_0)\},\{(Q,\circ_1)\},\{(Q,\ast_1)\},\big\{(Q,\circ_0),(Q,\ast_0)\big\},\\
\big\{(Q,\circ_1),(Q,\ast_1)\big\},\big\{(Q,\circ_0),(Q,\circ_1)\big\},\big\{(Q,\ast_0),(Q,\ast_1)\big\},
\big\{(Q,\circ_0),(Q,\ast_1)\big\},\big\{(Q,\circ_1),(Q,\ast_0)\big\},\\
\big\{(Q,\circ_0),(Q,\circ_1),(Q,\ast_0)\big\},
\big\{(Q,\circ_0),(Q,\circ_1),(Q,\ast_1)\big\},\big\{(Q,\ast_0),(Q,\ast_1),(Q,\circ_0)\big\},\\
\big\{(Q,\ast_0),(Q,\ast_1),(Q,\circ_1)\big\},
\big\{(Q,\circ_0),(Q,\circ_1),(Q,\ast_0),(Q,\ast_1)\big\}\Big\}$. Then, $(Q,\cdot )$ is a universal Osborn loop and obey Equation~\ref{eq:12} and Equation~\ref{eq:12b} if and only if $K_{10}(Q)=\Big(V_{01}(Q),S_{10}(Q)\Big)$ is a simplicial complex of isotopes of $(Q,\cdot )$.
\end{myth}
{\bf Proof}\\
This is proved with the aid of Theorem~\ref{0}, Theorem~\ref{1}, Theorem~\ref{2post1:17} and Theorem~\ref{2post1:17b}.
\begin{myth}\label{3.1}
Let $(Q,\cdot )$ be a universal Osborn loop. Let $V_i(Q)=\big\{(Q,\cdot ),(Q,\circ_i),(Q,\ast_i)\big\}$, $S_i(Q)=\Big\{\{(Q,\cdot )\},\{(Q,\circ_i)\},\{(Q,\ast_i)\},\big\{(Q,\circ_i),(Q,\ast_i)\big\}\Big\}$ and $K_i=(V_i(Q),S_i(Q))$ for $i=0,1,2,3$.
Define $f_{ij}~:~K_i\to K_j$ as \begin{equation*}
f_{ij}~:~\begin{cases}
(Q,\cdot ) & \longmapsto (Q,\cdot )\\
(Q,\circ_i) & \longmapsto (Q,\circ_j)\\
(Q,\ast_i) & \longmapsto (Q,\ast_j)
\end{cases}
~~i,j=0,1,2,3~\textrm{such that}~i\ne j.
\end{equation*}
Then, $f_{ij}$ is a simplicial map.
\end{myth}
{\bf Proof}\\
This is proved by Theorem~\ref{0}, Theorem~\ref{1}, Theorem~\ref{2} and Theorem~\ref{3}.
\begin{myth}
Let $(G,\cdot )$ and $(H,\star )$ be two loop isotopes under the triple $(A,B,C)$. For $D\in\{A,B,C\}$, if $D=E_1E_2\cdots E_i\cdots E_n$,~$E_i~:~G\to H,~i=1,\cdots n$ been bijections such that there does not exist $r\ge n$ for which $D=E_1E_2\cdots E_i\cdots E_r$, then the length of $D$, $|D|=n~\textrm{units}$. If $D=I$, the identity mapping, then $|D|=0$. The length of the isotopism $(G,\cdot
)\xrightarrow[\textrm{Isotopism}]{(A,B,C)}(H,\star )$ is giving by $|(A,B,C)|=|A|+|B|+|C|~\textrm{units}$.
For an isotopism $(G,\cdot
)\xrightarrow[\textrm{Isotopism}]{(A,B,C)}(H,\star )$, let the two loops $(G,\cdot )$ and $(H,\star )$ represent points in a $3$-dimensional space and let an isotopism from $(G,\cdot )$ to $(H,\star )$ be a line with $(G,\cdot )$ and $(H,\star )$ as end-points. The set of loops $V_{01}(Q)=\big\{(Q,\cdot ),(Q,\circ_0),(Q,\ast_0),(Q,\circ_1),(Q,\ast_1)\big\}$ where $(Q,\cdot )$ is a universal Osborn loop, form a rectangular pyramid with apex $(Q,\cdot )$.
\end{myth}
{\bf Proof}\\
We shall make use of the combined commutative diagram (\ref{eq:9}) as shown in the proof of Theorem~\ref{2post1:10}. There are four isotopes of $(Q,\cdot )$ as shown in the combined commutative diagram (\ref{eq:9}), namely $(Q,\circ_i),(Q,\ast_i)$ for $i=0,1$. The length of each of the isotopisms $(R_{[u\backslash (xv)]},L_u,I),(R_{\phi_0},L_u,I),(R_v,L_{\phi_1},I),(R_v,L_x,I)$ is $2~\textrm{units}$. The length of each of the isomorphisms $\gamma_0
(x,u,v)=\mathbb{R}_vR_{[u\backslash (xv)]}\mathbb{L}_uL_x$ and $\gamma_1
(x,u,v)=\mathbb{R}_vR_{[u\backslash (xv)]}\mathbb{L}_uL_x$ is $12~\textrm{units}$. The length of each of the isomorphisms $\gamma_{01}^\circ =\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}$ and $\gamma_{01}^\ast =\mathbb{L}_xL_{\phi_1(x,u,v)}$ is $6~\textrm{units}$. Hence, the four loop isotopes $(Q,\circ_i),(Q,\ast_i)$ for $i=0,1$ of $(Q,\cdot )$ form a rectangle. Thus, taking $(Q,\cdot )$ as an apex and the four isotopism as lines drawn from the apex to the four vertices of the rectangle, we have a rectangular pyramid.
|
1,116,691,500,486 | arxiv | \section{Introduction}
The standard two-sided and one-sided matching problems, and the closely related school choice problem, have been widely studied from an axiomatic viewpoint. A small number of algorithms dominate the literature. For two-sided matching, the Gale-Shapley algorithm; for one-sided matching, (random) Serial Dictatorship and Probabilistic Serial rule; for school choice, Gale-Shapley and the Boston mechanisms.
The main reason for the dominance of these algorithms is their good axiomatic behaviour with respect to notions of efficiency and strategyproofness. However if we shift the focus to fairness, social welfare, or tradeoffs between incompatible axioms, it is far from clear that these algorithms are optimal.
\subsection{Our contribution}
\label{s:contrib}
In Section~\ref{s:algo} we introduce several (in our opinion) natural algorithms for one-sided matching, several of which have not appeared (to our knowledge) in the literature before. We give a consistent derivation using specializations of the Gale-Shapley algorithm \cite{} for two-sided matching, which includes the well-known algorithms Serial Dictatorship \cite{} and Naive Boston \cite{} in a unified framework. In Section~\ref{s:props}, in addition to axiomatic properties such as efficiency and strategyproofness, we investigate welfare loss using a computational approach. We find that under truthful preferences, some of the new algorithms clearly outperform the classic ones. In particular, we recommend some new algorithms for some applications.
\section{Definitions and terminology}
Let $\mathcal{A} = \{a_1, \dots , a_n\}$ be a finite set of \emph{agents} and $\mathcal{O} = \{o_1, \dots, o_m\}$
a finite set of \emph{items}.
In general, the number of items and agents may not be equal. We focus on the case $m=n$ in the present article. The more general case involves substantial complications: different ways of assigning preferences over subsets of $\mathcal{O}$ or $\mathcal{A}$ lead to different notions of strategyproofness, for example. However all our algorithms can be modified trivially in order to work in the general case.
In the standard two-sided matching problem, each element of $\mathcal{O}$ has a complete strict preference order for elements of $\mathcal{A}$, and vice versa, while for $1$-sided matching only the latter information is required. For school choice the order of preference of items over agents depends on the preferences of agents over items --- the two sides are definitely not independent (schools typically must admit students who are qualified as long as there is capacity, and use their own preferences only when a tie must be broken). We aim to unify these three cases, and restrict to the case of strict linear orders.
Let $L(\mathcal{O})$ (respectively $L(\mathcal{A})$) denote the set of all strict linear orders on $\mathcal{O}$ (resp. agents). A preference \emph{profile} is a pair of functions $(\pi_A, \pi_I)$ where $\pi_A: A\to L(\mathcal{O})$ and $\pi_I: \mathcal{O} \to L(\mathcal{A})$. A \emph{matching} or \emph{discrete assignment} is a function $f:\mathcal{O} \leftrightarrow \mathcal{A}$. Let $S$ be the set of all doubly stochastic $n\times n$ matrices with rows indexed by agents and columns by items. A \emph{random assignment} is an element of $S$. The \emph{matching problem} is simply to output a matching given an input profile. The \emph{proportional} assignment is the random assignment in which each matrix entry equals $1/n$.
Of course, there are $n!$ discrete assignments and finding one is trivial. The point is to find one with desirable properties. A discrete assignment is \emph{efficient} if there is no other assignment which improves the outcome for some agent and does not worsen it for any agent. An algorithm for randomized assignment is \emph{ex-post efficient} if every matching occurring with positive probability is an efficient assignment.
We now review some standard algorithms from the literature.
Two algorithms for the same matching problem are \emph{equivalent} if they each produce the same output for every input. All the algorithms under study are \emph{anonymous}, meaning that a permutation of the players leads to the same permutation of the assignment. In other words, only the preferences matter, not the agents' identities. When discrete assignments are used, this means that whenever we have two agents with identical preferences, one must envy the other's assignment. A stronger condition is \emph{symmetry} (also called ``equal treatment of equals'') which says that agents with the same preferences receive the same assignment. Clearly, this can only be satisfied in the framework of random assignments.
Every algorithm that uses a fixed initial order of agents and produces a matching can yield an algorithm that produces a random matching, simply by randomizing over the initial order. This is usually done according to the uniform distribution, in order to preserve symmetry between agents. Thus every algorithm discussed in Section~\ref{s:algo} has a randomized version, which we denote by prefixing ``R" to its name.
\subsection{One-Sided Algorithms}
\label{ss:1-sided}
One-sided matching refers to the situation where the items' preferences over agents are ignored.
The most commonly discussed algorithm is Serial Dictatorship \cite{}.
\begin{eg} (Serial Dictatorship)
\label{eg:SD}
Fix an arbitrary linear ordering on $A$. The \emph{Serial Dictatorship} algorithm (SD) with respect to this ordering assigns items to agents as follows: at step $i$ , allocate to agent $i$th its most preferred item that has not already been allocated to a previous agent.
\end{eg}
The randomized version is denoted RSD, as mentioned above.
Serial Dictatorship satisfies important axiomatic properties such as \emph{ex-post efficiency} and \emph{strategyproofness} (all axiomatic properties are defined and discussed in Section~\ref{s:props}).
The next algorithm was proposed by Bogolmanaia and Moulin \cite{BoMo2001}. It is \emph{ordinally efficient} in addition to being ex-post efficient, and has a weak strategyproofness property. It can be described using a cake-eating analogy.
\begin{eg} (Probabilistic Serial)
\label{eg:PS}
The \emph{Probabilistic Serial} rule is inherently randomized, and generates a random assignment as follows. We interpret an assignment of fraction $x$ of item $j$ to agent $i$ to mean that $i$ receives fraction $x$ of $j$ (in other words we pretend that the items are infinitely divisible). All agents simultaneously begin ``eating" at unit speed, each agent at each instant eating from its most preferred item among those that have not been completely consumed. On termination we have a random assignment.
\end{eg}
A closely related problem to one-sided matching is the \emph{housing market} problem \cite{ShSc1974}. The difference is that every agent is assumed to have an initial allocated item, and we seek a method for finding an allocation that is optimal in some way.
\begin{eg} (Top Trading Cycle)
\label{eg:GTTC}
When each agent is considered to be initially assigned an entire item, the agents may trade amongst themselves as follows. Each agent $i$ points to the agent currently owning the item on the top of $i$'s preference list. By finiteness and since every node has outdegree 1, this directed graph must contain a cycle. Reallocate items according to the arcs in the cycle, and remove these agents and items from further consideration. Repeat (using pointers to the next level preference if necessary) until no items/agents remain. This \emph{TTC algorithm} \cite{ShSc1974}, attributed by Shapley and Scarf to David Gale, always yields a discrete assignment that is efficient, and the mechanism is strategyproof and individually rational. Furthermore the algorithm runs in polynomial time.
\end{eg}
\begin{eg} (sample execution of TTC)
Consider the profile where agents $1$ and $2$ have preferences $a > b > c$ and agent $3$ has preference $b > a > c$. The assignment $1:c, 2:b, 3:a$ is not ex-post efficient, because $2$ and $3$ can trade to their mutual benefit. In the first round of TTC, agents $1$ and $2$ points to agent $3$, while agent $3$ points to agent $2$. There is a cycle between agents $2$ and $3$. The agents swap along the cycle and are removed from consideration. Agent $1$ then points to itself in the next round, swaps along the cycle, remains with item $c$, and is removed from consideration. With no agents left, the TTC algorithm halts. The output is the ex-post efficient assignment $1:c, 2:a, 3:b$.
\end{eg}
When using TTC we have freedom in the choice of initial assignment. For example, choosing this uniformly at random and running TTC yields an algorithm equivalent to RSD \cite{AbSo1998}, and an adaptation of TTC to trade unit shares yields an algorithm equivalent to PS when run on the proportional endowment \cite{Kest2009}.
We find it useful to run TTC on the output of some of our algorithms in Section~\ref{s:algo} (algorithms which satisfy ex-post efficiency gain no benefit from running TTC, which terminates immediately because there is no cycle). The resulting combined algorithms, denoted XG where X is the name of the basic algorithm, are ex-post efficient and appear to have considerably better overall performance than the original algorithms.
\subsection{Two-sided algorithms}
\label{ss:2-sided}
In this case agents have complete strict preferences over items, and vice versa. The most well-known algorithm belongs to the class of \emph{deferred acceptance} algorithms. Items and agents are tentatively matched, but these ``engagements'' may be broken. In fact each item may attach to up to $n$ agents in the course of the algorithm.
In the \emph{Gale-Shapley} algorithm \cite{GaSh1962}, agents in turn approach previously unapproached items that they prefer to their currently assigned item (every agent prefers each item to not having an item, and every item prefers every agent to not being held by an agent). If the currently proposing agent is preferable to the agent currently matched with the item, the item will reject its current partner for the proposing agent. No agent may approach an item that has already rejected it (such an approach would lead to another rejection by the above rules). This ensures termination after at most $n^2$ proposals.
\begin{eg}
\label{eg:GS}
This is adapted from Example 2 in \cite{GaSh1962}.
Suppose that the proposers' preferences are as follows:
\begin{align*}
1: \quad a>b>c>d \\
2: \quad a>d>c>b \\
3: \quad b>a>c>d \\
4: \quad d>b>c>a \\
\end{align*}
and the proposees' preferences are given by
\begin{align*}
a: \quad 4>3>1>2 \\
b: \quad 2>4>1>3 \\
c: \quad 4>1>2>3 \\
d: \quad 3>2>1>4 \\
\end{align*}
\if01
The sequences of proposals are as follows:
\begin{tabular}[!ht]{|c|c|c|}
\hline
Proposal & Outcome & Current Partial Matching\\
\hline
$a_1 =>$ a & tentatively matched & $a_1$:a\\
$a_2 =>$ a & $a_2$ rejected & $a_1$:a\\
$a_3 =>$ b & tentatively matched & $a_1$:a, $a_3$:b\\
$a_4 =>$ d & tentatively matched & $a_1$:a, $a_3$:b, $a_4$:d\\
$a_2 =>$ d & $a_4$ rejected & $a_1$:a, $a_2$:d, $a_3$:b\\
$a_4 =>$ b & $a_3$ rejected & $a_1$:a, $a_2$:d, $a_4$:b\\
$a_3 =>$ a & $a_1$ rejected & $a_2$:d, $a_3$:a, $a_4$:b\\
$a_1 =>$ b & $a_1$ rejected & $a_2$:d, $a_3$:a, $a_4$:b\\
$a_1 =>$ c & tentatively matched & $a_1$:c, $a_2$:d, $a_3$:a, $a_4$:b\\
\hline
\end{tabular}
\fi
The final matching is $1:c, 2:d, 3:a, 4:b$. There are 9 proposals made during the execution of the algorithm.
\end{eg}
The Gale-Shapley algorithm has the well-known property that the output matching is \emph{stable}, meaning that there is no unmatched (agent, item) pair who each prefer each other to their current partner. Also, the output matching is optimal for proposers, meaning that each proposer receives the best possible item it can receive in a stable matching. It follows that the output of the algorithm does not depend on the order of proposals made by agents. Note that, by contrast, the output of the algorithms in Section~\ref{s:algo} will depend strongly on the order of proposals.
We can produce one-sided matching algorithms from two-sided ones by forcing the items to have specific (fictitious) preferences. We use this idea systematically in Section~\ref{s:algo}.
\subsection{School choice algorithms}
\label{ss:school}
A case intermediate between 1-sided and 2-sided matching, which is important for later, is that of \emph{school choice}. The Gale-Shapley algorithm is applicable to the case where the number of agents exceeds the number of items, provided items (schools) have capacity for some number of agents (students). A school accepts a student's proposal provisionally provided there is capacity remaining, or the student is preferable to an already tentatively accepted student. The case where each school has capacity $1$ and the numbers of schools and students are equal is the case described in Section~\ref{ss:2-sided}.
There are other algorithms for school choice that use \emph{immediate} acceptance. In this case, each student first applies to her first choice school. Each school ranks applicants and chooses as many as it can, subject to capacity. Students not accepted already then apply to their second choice school, etc. This description implicitly uses simultaneous proposing by all unmatched agents and is called the \emph{Boston mechanism} \cite{AbSo2003, MeSe2014}. There is also a sequential version in which proposals are made one agent at a time. In that case, the order of proposals clearly changes the final allocation, since no engagement can ever be broken.
The special case of the Boston mechanism in which each school has capacity $1$ and there are equal numbers of agents and items is a 2-sided matching algorithm as defined above. For the same input as in Example~\ref{eg:GS}, the final allocation using sequential offers by $1,2,3,4$ in that order is $1:a, 2:d, 3:b, 4:c$. The final allocation using simultaneous offers is $1:a, 2:c, 3:b, 4:d$.
\section{New algorithms}
\label{s:algo}
For the rest of the analysis, we construct one-sided matching algorithms by relaxing 2-sided algorithms. The Gale-Shapley algorithm generates a stable matching using a series of proposals, based on fixed preferences of the agents and items. Given the agent preferences over items as input to a 1-sided matching problem, we construct fictitious preferences for the items over the agents. For example, we can assume that all items have a fixed common preference.
\begin{prop} SD is equivalent to a special case of GS.
\label{prop:GS yields SD}
\end{prop}
\begin{proof}
Suppose that all items have the same preference order over agents, which without loss of generality we write $1>2>\dots>n$. We claim that Gale-Shapley will output an assignment that is the same as the output of Serial Dictatorship with the agent order $1, 2, \dots, n$.
The proof is inductive. The base case is that agent $1$ will get his first choice with GS. It is trivially true as agent $1$ will propose to its most preferred item, and since every item prefers agent $1$ to any other agent, they cannot be rejected later. Therefore agent $1$ will be allocated the same item under GS or SD.
Stability of GS implies that for every item that agent $i$ wants more than the item they are allocated, the item must be held by an agent ranked higher by the item. If every agent before $i$ gets its choice as per SD, agent $i$ will eventually propose to its choice under SD. As every agent after $i$ is ranked below $i$ by all items, they cannot cause that item to reject $i$. Therefore agent $i$ will have the same item under GS or SD.
\end{proof}
We can do the same thing with the (simultaneous) Naive Boston algorithm. Given an instance of 1-sided matching, we create fictitious preferences in which each item has the same preference, say $1>2>\dots >n$. The resulting algorithm we call the $1$-sided Naive Boston algorithm. Note that we can also interpret this algorithm sequentially if we ensure that agent order is $1,2,\dots, n$, but otherwise the sequential and simultaneous forms will differ in general.
Below, we generalize this fictitious preference approach by allowing each item to build its fictitious preferences dynamically, using some fixed rule, based only on the sequence of proposals that it receives from agents. Recall that any order of proposals gives the same result for the Gale-Shapley algorithm with fixed preferences, but as we see below, this is not the case in our relaxed setup.
\begin{defn}Throughout the rest of this article, for the purposes of illustration and comparison between algorithms we use what we call the \emph{standard profile} in which agents $1, 2$ and $3$ have preferences $a > b > c > d$, and agent $4$ has preferences $b > a > c > d$.
\end{defn}
\subsection{The ``permanent memory" case}
\label{ss:memory}
As the order of the agents' proposals affects the items' preferences, the order of proposals affects the final allocation. This has two consequences. The first is that the treatment of rejected agents matters. After an agent is rejected, either because the item prefers its current agent or breaks its tentative engagement, it may not necessarily be the next agent to propose. We consider two possibilities: a stack (rejected agents go to the top of the stack) or a queue (rejected agents go to the back of the queue). The other consequence is, as noted above, that the initial order of agents has an impact on the final allocation. For the purpose of the analysis, the algorithms will fix an arbitrary initial order.
We consider two rules for building preferences dynamically. These are \emph{early-proposal preference} (or \emph{Accept-First}) and \emph{late-proposal preference} (or \emph{Accept-Last}). Accept-First means that the first agent to approach an item is accepted, and subsequent proposals are rejected. Using Accept-Last, an item always breaks an engagement in favour of a new proposer, if it has not yet been held by the new proposer. In terms of the marriage interpretation often used to describe the Gale-Shapley algorithm, for Accept-First algorithms the proposees stick faithfully to their first suitor, whereas for Accept-Last algorithms the proposees are always more satisfied with a new suitor than their current fianc\'{e}.
These two dichotomies (stack/queue, Accept-First/Accept-Last) when combined with the distinction between permanent and temporary memory (explained in Section~\ref{ss:no memory}) yield 8 algorithms. We denote them by three-letter abbreviations. For example, PFS refers to permanent memory, Accept-First, stack.
Sample executions on the standard profile of all 8 algorithms introduced below can be found in Appendix~\ref{apps:examples}. The results are summarized in Table~\ref{t:stdprof}.
We consider the Accept-First algorithms. The algorithm PFS is equivalent to Serial Dictatorship. Interestingly, switching the stack to a queue results in an algorithm that is equivalent to the Naive Boston algorithm, so we obtain no new inequivalent algorithms in this case, just a unified presentation of old ones. We give the details below.
\begin{prop} PFS with order of agents $1, 2, \dots, n$ is equivalent to Serial Dictatorship with the same order of agents.
\end{prop}
\begin{proof}
By definition PFS is a special case of GS in which there is a common preference order $1>2>\dots > n$ for items over agents, because engagements are never broken (by Accept-First and the permanent memory). By Proposition~\ref{prop:GS yields SD} the latter is equivalent to SD with the agent order $1,2, \dots, n$.
\end{proof}
\begin{prop} PFQ with order of agents $1,2,\dots ,n$ is equivalent to the $1$-sided Naive Boston algorithm where the common preference order of items is $1>2>\dots >n$.
\end{prop}
\begin{proof}
It is useful to consider the above algorithm as occurring in rounds. Round $i$ ends precisely when all the remaining agents have proposed to their $i$th choice. We show by induction that during round $i$:
\begin{itemize}
\item the agents proposing are precisely those who have not been matched previously;
\item each such agent makes exactly one proposal, to its $i$th choice.
\end{itemize}
Since this is exactly the behaviour of the (simultaneous) 1-sided Boston algorithm and since assignments are never changed in either algorithm, the proposition follows.
When $i=1$, all agents have proposed to their first choice. Thus each agent has made exactly one proposal, because any agent that is accepted never makes another proposal, and any agent that is rejected must go to the back of the queue and wait until all other agents have made their first proposal.
Assuming the result holds for all rounds before $i$, then all remaining agents have proposed to and been rejected by all items down to rank $i-1$. In round $i$ they must then propose to their $i$th choice. This happens exactly once because of the queue discipline.
\end{proof}
\begin{eg} (Accept-Last permanent memory algorithms versus accept-first)
For the standard profile, the final assignment using PFS is $1:a, 2:b, 3:c, 4:d$, while using PLS it is $1:d, 2:c, 3:a, 4:b$. If we change the stack to a queue, the final assignment is $1:a, 2:c, 3:d, 4:b$ for PFQ and $1:d, 2:c, 3:b, 4:a$ for PLQ.
Note that with these preferences, some agent must receive its 4th choice and no more than two agents can receive their 1st choice. Only PFQ achieves the latter condition.
\end{eg}
Although the Accept-Last algorithms are not ex-post efficient, empirical results show that (among other things) their output, when used as an initial endowment for TTC, leads to better welfare performance than the ex-post efficient Accept-First algorithms. We discuss this in details in Section~\ref{s:props}.
\subsection{The ``temporary memory" case}
\label{ss:no memory}
The previous algorithms assume that each item retains its preferences throughout the execution of the algorithms. Other interesting algorithms can be generated by relaxing that requirement. Whenever the item resets its memory, it make sense for the agents to propose to items that rejected them before. Any rules that uses temporary memory for items must ensure that the algorithm will halt with a matching. To ensure the algorithm halts, we only allow the item to reset its memory when the number of tentative matchings has increased. As items do not go from matched to unmatched, this happens precisely when a new item is matched.
Whenever an agent proposes to an unmatched item, all items, including the new item, lose their memory of preferences. When an agent proposes to a matched item with no preferences, the item prefers the proposing agent instead of the matched agent. As the number of tentative matchings increases throughout the execution of the algorithm, there can only be $n$ resets of the preferences, and thus the number of proposals is bounded by $n^3$.
Using the above rule, four new algorithms analogous to those in Section~\ref{ss:memory} can be constructed.
We first present the Accept-First algorithms. The temporary memory analogue of Serial Dictatorship, namely TFS, is interesting. Each round of this algorithm operates like a Serial Dictatorship in which a subset of the agents repeatedly ``steal" items in chains until some agent chooses an unmatched item, whereupon a new round begins with a new agent beginning the stealing. The order of agent choices in a round is not fixed, but determined by the stealing process. We suggest the alternative name ``Iterative Dictatorship'' for this algorithm. The algorithm TFQ is the temporary memory analogue of the $1$-sided Boston algorithm. Like all queue-based algorithms it is harder to interpret than a stack-based algorithm.
The Accept-Last algorithms are harder to understand (but see the party interpretation below, which was the inspiration for our entire research program).
\begin{eg} (Temporary memory, Accept-First versus Accept-Last)
For the standard profile, the final assignment under TFS is $1:d, 2:a, 3:c, 4:b$ and the final assignment using TFQ is $1:a, 2:b, 3:d, 4:c$. By contrast, the final assignment using TLS is $1:b, 2:a, 3:d, 4:c$ while the final assignment under TLQ is $1:a, 2:b, 3:d, 4:c$.
\end{eg}
\subsection{Further comments}
\label{ss:further}
We have presented 8 algorithms in a unified framework, corresponding to the dichotomies memory/no memory, stack/queue, Accept First/Accept Last. Basic description of their behaviour on the standard profile is shown in Table~\ref{t:stdprof}. Note that all give different outputs on this input, except TFQ and TLQ, which are of course different in general. A stronger statement, namely that all randomized versions are inequivalent algorithms, is shown by example in Appendix~\ref{app:alldiff}. Also note that on this input, only PLQ fails to give an efficient allocation. Running TTC on the output of PLQ yields the allocation $1:a, 2:c, 3:b, 4:d$.
\begin{table}
\caption{Behaviour of algorithms on standard profile where $n=4$}
\label{t:stdprof}
\begin{tabular}{ccc}
\hline
Algorithm & Output matching & Number of proposals \\
\hline
PFS & 1:a, 2:b, 3:c, 4:d & 10\\
PFQ & 1:a, 2:c, 3:d, 4:b & 9\\
PLS & 1:d, 2:c, 3:a, 4:b & 9\\
PLQ & 1:d, 2:c, 3:b, 4:a & 10\\
TFS & 1:d, 2:a, 3:c, 4:b & 18\\
TFQ & 1:a, 2:b, 3:d, 4:c & 33\\
TLS & 1:b, 2:a, 3:d, 4:c & 18\\
TLQ & 1:a, 2:b, 3:d, 4:c & 21\\
\end{tabular}
\end{table}
All algorithms can be interpreted in terms of a party game, with the host providing a stash of presents. As each person arrives at the party (say though a narrow door), they take a present from the stash or (in some cases) from another person. Permanent memory algorithms have a single round, and temporary memory algorithms begin a new round every time a new present is taken. For Accept-First algorithms, in each round each present can be taken at most once.
For Accept-Last algorithms, in each round each (person, present) pair can occur at most once. The queue or stack discipline determines what happens to a partygoer when it loses its present: choose a replacement present immediately, or go to the back of the queue. The Accept-First queue-based algorithms would be uninteresting, as would PFS, but the others seem to us worth trying.
The TLS algorithm is closely related in this interpretation to the party game \emph{Yankee Swap} or White Elephant \cite{wiki:yankee}, which was the inspiration for our research program. In the real game, presents are contributed by partygoers and are wrapped, so no person has full information on their own preference. We are not aware of any other real-life party games based on the other algorithms.
\section{Properties of the algorithms}
\label{s:props}
Most of our algorithms fail to satisfy any of the common axiomatic properties. However, some have good average-case behaviour. Interestingly, they behave quite differently from each other.
\subsection{Ex-post Efficiency}
\label{ss:expost}
\begin{prop} All Accept-First algorithms are ex-post efficient.
\end{prop}
\begin{proof}
We show by induction on the round number that the partial allocation constructed so far is ex-post efficient (in each case a round ends every time a previously unmatched item is chosen --- note that this is a different usage of ``round" to that in Section~\ref{ss:further}, where permanent memory algorithms have a single round). The first round always terminates with the first agent taking its top choice, and this is obviously an ex-post efficient outcome. Suppose that the result holds for all rounds before $i$ and consider round $i$. The entering agent chooses an item and retains it throughout the round (by Accept-First policy). Each agent taking an unmatched item or stealing an item during this round (an ``active agent") chooses the best item available. Such an agent $j$ would only wish to trade with an agent $k$ who has chosen since the last memory reset (in the permanent memory case, since the beginning of the algorithm; in the temporary memory case, since the beginning of the round). However in this case $k$ will not wish to trade with $j$. Thus there can be no mutually beneficially trading cycle within the group of active agents. By inductive hypothesis there is also no such cycle within the group of inactive agents. Each agent in the active group prefers its current item to everything held by the inactive group because such items were available to steal. The result follows because TTC will terminate with no trades.
\end{proof}
\begin{eg} All Accept-Last algorithms fail ex-post efficiency. To see this, consider the profile where agents 1,2,3 have respective preference orders $a>b>c, a>b>c, b>a>c$. Direct computation shows that for each algorithm X, there is some initial agent order such that algorithm XG formed by running TTC on the output of X gives a different result. Thus X cannot be ex-post efficient.
\if01
(PM AL S) - agent 1 \& 2 a>b>c, agent 3 b>a>c, agent 1:a, agent 2:a, agent 1:b agent 3:a agent 2:b agent 1:c, 1:c 2:b 3:a, agent 2 and 3 can swap
(PM AL Q) - \jl{same preferences as MLS
Memory AcceptLast Queue (in Pref Order):
0.000000 0.500000 0.500000
0.000000 0.500000 0.500000
0.000000 1.000000 0.000000
Memory AcceptLast Queue +GTTC(in Pref Order):
0.500000 0.000000 0.500000
0.500000 0.000000 0.500000
1.000000 0.000000 0.000000 }
(TM AF S) - with 1 agent and m item, the allocation is ExEff (trivial). Assuming that i-1 agents and m item allocation is ExEff, then i agent and n item also is. Any agents that get changes his allocated item when the $i^th$ agent comes in has the same item as if the algorithm is a SD, thus those agents cannot improve on his allocation. Agents that does not change his item cannot trade with each other because the allocation with i-1 agents are ExEff. Thus at n agents and m item, the allocation is ExEff.
\fi
\if01
details
Memory AcceptLast Stack (in Pref Order):
0.166667 0.333333 0.500000
0.166667 0.333333 0.500000
0.333333 0.666667 0.000000
Memory AcceptLast Stack +GTTC(in Pref Order):
0.500000 0.000000 0.500000
0.500000 0.000000 0.500000
1.000000 0.000000 0.000000
NoMemory AcceptLast Stack (in Pref Order):
0.000000 0.500000 0.500000
0.000000 0.500000 0.500000
0.000000 1.000000 0.000000
NoMemory AcceptLast Stack +GTTC(in Pref Order):
0.500000 0.000000 0.500000
0.500000 0.000000 0.500000
1.000000 0.000000 0.000000
Memory AcceptLast Queue (in Pref Order):
0.000000 0.500000 0.500000
0.000000 0.500000 0.500000
0.000000 1.000000 0.000000
Memory AcceptLast Queue +GTTC(in Pref Order):
0.500000 0.000000 0.500000
0.500000 0.000000 0.500000
1.000000 0.000000 0.000000
NoMemory AcceptLast Queue (in Pref Order):
0.000000 0.500000 0.500000
0.000000 0.500000 0.500000
0.000000 1.000000 0.000000
NoMemory AcceptLast Queue +GTTC(in Pref Order):
0.500000 0.000000 0.500000
0.500000 0.000000 0.500000
1.000000 0.000000 0.000000
\fi
\end{eg}
\subsection{Ordinal Efficiency}
\label{ss:ordinal}
\if01 details
\jl{The NM L S/Q algorithms are almost o-eff, with 98\% and 96\% IANC preferences n=4 are o-eff. counter examples:
Preference
a>b>c>d
a>b>c>d
b>a>d>c
b>c>a>d
NoMemory AcceptLast Stack +GTTC(in Pref Order):
0.208333 0.083333 0.416667 0.291667
0.208333 0.083333 0.416667 0.291667
0.000000 0.583333 0.416667 0.000000
0.833333 0.166667 0.000000 0.000000
NoMemory AcceptLast Queue +GTTC(in Pref Order):
0.250000 0.250000 0.250000 0.250000
0.250000 0.250000 0.250000 0.250000
0.000000 0.500000 0.500000 0.000000
0.500000 0.500000 0.000000 0.000000
}
\fi
A random allocation $S$ is ordinally efficient if there is not another random allocation, $S'$ that each agent $i$ SD-prefers $S'_i$ to $S_i$, with at least 1 agent strictly SD-preferring their allocation under $S'$. Ordinal efficiency implies ex-post efficiency, but not vice versa \cite{BoMo2001}.
\begin{eg} All Accept-First algorithms fail ordinal efficiency. A counterexample (details omitted): suppose that agents 1, 2 prefer $a>b>c>d$, while agents 3, 4 have preference $a>b>d>c$. This same counterexample also works for the ex-post efficient algorithms PLSG and PLQG. A different counterexample with $n=4$ works for TLSG and TLQG, and ordinal efficiency seems to be violated less often for these two algorithms.
\if01 details
\jl{
The two memory accept last algorithm also fails o-eff on this preference.
Memory AcceptFirst Stack (in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
NoMemory AcceptFirst Stack (in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
Memory AcceptLast Stack (in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
Memory AcceptLast Stack +GTTC(in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
Memory AcceptFirst Queue (in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
NoMemory AcceptFirst Queue (in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
Memory AcceptLast Queue (in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
Memory AcceptLast Queue +GTTC(in Pref Order):
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
0.250000 0.250000 0.416667 0.083333
}
\fi
\end{eg}
\subsection{Strategyproofness}
\label{ss:strategyproof}
An assignment algorithm is \emph{strategyproof} if no agent has incentive to misreport its preferences, irrespective of what other agents do. In other words, truthful reporting is a strictly dominant strategy for each agent. This condition is of course rather strong, but is known to be satisfied by Serial Dictatorship and hence by PFS. All other algorithms discussed here fail to satisfy it as we show below. For random allocation algorithms, the incentive is understood to be expressed in terms of expected utility in the usual way.
The algorithm Probabilistic Serial satisfies \emph{weak strategyproofness} \cite{BoMo2001}, which says that no agent has incentive to deviate, where the incentive is expressed in terms of first-order stochastic dominance. In other words, for each profile and each agent, there is \emph{some} consistent choice of utility function for that agent for which deviation from truthfulness is unprofitable. Note that this is weaker than strategyproofness, which says that for each profile and agent, and for \emph{every} consistent choice of utility function for that agent, deviation from truthfulness is unprofitable.
Clearly, if algorithm X is strategyproof for every initial order of agents, so is RX.
We show by examples that none of the randomized versions of our algorithms are weakly strategyproof. Consider the case where agents $1,2,3,4$ all have preference $a>b>c>d$.
If some agent submits instead $a>c>b>d$ while the others remain truthful,
algorithms TLS, TLG, PLS, PLQ, TLQ, TLQG obtain a preferable outcome for that agent.
Similar examples show that PLQG, PLSG, TFS and TFQ, and PFQ all fail weak strategyproofness.
\if01
details
NMALS: yields 3/4 1/4 0 0
NMALS+G: 3/4 0 1/4 0
MALS: 1/4 1/2 1/4 0
MALQ: 1/4 3/4 0 0
NMALQ: 3/4 1/4 0 0
NMALQ+G: 3/4 0 1/4 0
With MALQ+G:
True Preferences:
a,b,c,d
a,b,c,d
a,d,b,c
a,c,d,b
Agent 4 gets 1/4 0 7/12 1/6
By submitting a,c,b,d Agent 4 gets 1/4 0 3/4 0
NMAFS \& NMAFQ:
True Preferences:
a,b,c,d
a,b,c,d
a,b,c,d
a,c,b,d
NMAFS: agent 4 gets 1/4 0 1/4 1/2
NMAFQ: agent 4 gets 0 1/4 0 3/4
It is trivial to see that if agent 4 submits a,b,c,d, they will get the same proportional allocation, which SD dominates the allocation with true preferences.
NB:
True preferences:
a,b,c,d
a,b,c,d
c,a,b,d
a,c,b,d
Agent 4 gets 1/3 0 0 2/3
By submitting a,b,c,d, they will get 1/3 1/3 0 1/3 which SD dominates the allocation from true preference
\fi
\if01
\jl{For MALS+G, agent 1-3 pref = a>b>c>d, agent 4 pref = b>c>d>a. allocation is
Memory AcceptLast Stack +GTTC(in Pref Order):
0.333333 0.250000 0.250000 0.166667
0.333333 0.250000 0.250000 0.166667
0.333333 0.250000 0.250000 0.166667
0.250000 0.250000 0.500000 0.000000
if agent 4 submits b>c>a>d, allocation is
Memory AcceptLast Stack +GTTC(in Pref Order):
0.333333 0.083333 0.250000 0.333333
0.333333 0.083333 0.250000 0.333333
0.333333 0.083333 0.250000 0.333333
0.750000 0.250000 0.000000 0.000000 }
\fi
\subsection{Utilitarian welfare}
\label{ss:welfare}
We give a basic analysis here, and refer the reader to our more extensive analysis \cite{WiLo2017b}. We analyse here only the average-case performance of the randomized versions of the algorithms under truthful behaviour. We impute utility values of agents by requiring them to all have the same (Borda) utility function, whereby the $i$th choice corresponds to utility $n-1+i$. We consider the \emph{utilitarian social welfare}, which is the sum over all agents of the utility of their allocation. The optimal value of the utilitarian social welfare is efficiently computable, for example using the \emph{Hungarian algorithm} \cite{}. This allows us to quantify the fraction of the maximum possible social welfare that is lost, on average, by each algorithm. We use a Java implementation by K.~Stern \cite{Ster2012}.
Results are shown in Figure~\ref{fig:util_8}. They show that RSD is outperformed substantially by PFQ, TFQ and TFS, but is better than the Accept-Last algorithms. In Figure~\ref{fig:util_TTC} we show how the welfare improves substantially when TTC is run on the output of our Accept-Last algorithms. In Figure~\ref{fig:util_5} we show how our best-performing algorithms, namely TLQG and TLQS, compare with the standard algorithms RSD, PS and Naive Boston. In fact all the Accept-Last algorithms with TTC outperform those standard algorithms.
\begin{figure}
\caption{Utilitarian welfare loss of our 8 basic algorithms}
\label{fig:util_8}
\includegraphics[width=14cm]{util_8.jpeg}
\end{figure}
\begin{figure}
\caption{Utilitarian welfare improvement for Accept-Last with TTC}
\label{fig:util_TTC}
\includegraphics[width=14cm]{util_TTC.jpeg}
\end{figure}
\begin{figure}
\caption{Utilitarian welfare loss comparison}
\label{fig:util_5}
\includegraphics[width=14cm]{util_5.jpeg}
\end{figure}
\subsection{Egalitarian welfare}
\label{ss:egal}
We also consider the egalitarian welfare, namely the welfare of the worst-off agent. Since exact computation of the optimum is difficult \cite{}, we scale by $n$ instead of the exact optimum.
Results are similar to the utilitarian case and show the non-competitiveness of RSD and Naive Boston, the positive effect of GTTC, and the overall superiority of TLSG and TLQG. In Figure~\ref{fig:egal_4} the lines for TLSG and TLQG are indistinguishable. We do not compare with PS because it is inherently random and so comparison would be unfair to the other algorithms --- the expectation of the minimum welfare is less than the same as the minimum of the expectations. Note that welfare increases with $n$ for the new algorithms, but decreases for the old ones.
\if01
\begin{figure}
\caption{Egalitarian welfare of our 8 basic algorithms}
\label{fig:egal_8}
\includegraphics[width=14cm]{egal_8.jpeg}
\end{figure}
\begin{figure}
\caption{Egalitarian welfare improvement for Accept-Last with TTC}
\label{fig:egal_TTC}
\includegraphics[width=14cm]{egal_TTC.jpeg}
\end{figure}
\fi
\begin{figure}
\caption{Normalized egalitarian welfare comparison}
\label{fig:egal_4}
\includegraphics[width=14cm]{egal_4.jpeg}
\end{figure}
\subsubsection{Egalitarian welfare bounds}
\label{sss:prop k}
Say that an algorithm satisfies a \emph{conditional egalitarian welfare bound} of $k$ if every agent receives one of its top $k$ choices, \emph{whenever that is possible under some allocation}. If $k$ happens to be the minimal possible value, this says that the algorithm yields optimal egalitarian Borda welfare on that input. All reasonable algorithms (including all those in this paper) satisfy a bound of $1$, because if all agents have different top choices, each receives its top choice. We investigated the case $k=2$ and found that none of our algorithms satisfy it in general (details omitted). However, TLS satisfies the bound with $k=2$ when $n=3$, as does its queue-based counterpart TLQ, while none of the other algorithms does. Thus for $n=3$ these algorithms are egalitarian-optimal.
\if01
\jl{each row is the probability of agent i getting their $j$th choice instead of getting item j}
I have got examples of each algorithm failing k=2.
a,b,c
a,b,c
a,c,b
PSR, AB, RSD, NMAFS, MALS, NB, NMAFQ, MALQ all give allocation:
1/3 1/2 1/6
1/3 1/2 1/6
1/3 2/3 0
when 1:a, 2:b, 3:c will have each agent getting top 2 choices.
NMALQ fails k=2 on :
a,b,c,d
a,b,c,d
a,c,d,b
b,d,a,c
allocation:
1/2 1/3 1/6 0
1/2 1/3 1/6 0
0 2/3 1/3/ 0
1/3 2/3 0 0
NMALS fails k=2 on
a,b,c,d
a,b,c,d
a,c,d,b
c,d,a,b
allocation:
1/2 1/2 0 0
1/2 1/2 0 0
0 3/4 1/4 0
1/4 3/4 0 0
The two NMAL algorithms passes k=2 for n=m=3.
\fi
\subsection{Order bias}
\label{ss:order}
In the case where all agents have the same preferences over items, some algorithms (such as serial dictatorship) are clearly biased toward the first agent while others (accept-last algorithms) are clearly biased toward the last agent. We define the \emph{order bias} of an algorithm to be the maximum over all pairs of agents of the difference of the expected (under the uniform distribution on preferences) Borda welfare gained, and normalize by $n$.
Results show that the order bias of queue-based algorithms is markedly smaller than that for stack-based algorithms. Our best welfare algorithms, namely TLSG and TLQG, have almost zero order bias (PS has zero order bias by definition), but the randomized Serial Dictatorship and Naive Boston algorithms have substantial order bias, with the former being clearly more biased than all other algorithms. Adding TTC to the Accept-Last algorithms substantially reduces order bias (not shown).
\begin{figure}
\caption{Normalized order bias of our 8 basic algorithms}
\label{fig:bias_8}
\includegraphics[width=14cm]{bias_8.jpeg}
\end{figure}
\if01
\begin{figure}
\caption{Normalized order bias improvement for Accept-Last with TTC}
\label{fig:bias_TTC}
\includegraphics[width=14cm]{bias_TTC.jpeg}
\end{figure}
\fi
\begin{figure}
\caption{Normalized order bias comparison}
\label{fig:bias_4}
\includegraphics[width=14cm]{bias_4.jpeg}
\end{figure}
\section{Conclusion}
\label{s:conc}
\subsection{Summary of results}
\label{ss:summary}
We have introduced 10 new algorithms for 1-sided matching, none of them equivalent to each other or to any algorithms in the literature, to our knowledge. Their derivation using the Gale-Shapley framework gives a unified description. Each algorithm runs in worst-case time of order $n^2$ (permanent memory) or $n^3$ (temporary memory). Although they lack strong axiomatic foundations, several of the algorithms perform well on criteria such as egalitarian welfare, utilitarian welfare, and order bias, in sharp contrast to the standard Serial Dictatorship or Boston algorithms. The algorithms TLSG and TLQG perform better overall than the standard algorithms Serial Dictatorship, Naive Boston, and Probabilistic Serial on such measures.
In this article we have introduced a new (to our knowledge) performance criterion related to symmetry, namely order bias. The queue-based algorithms perform better overall than the stack-based ones by this measure. The algorithm TLQ (and TLQG) has remarkably small order bias. Our intuition is that the queue simulates randomization of the order of choosing by agents.
The approach we use of considering algorithms that are not ex-post efficient, and then running TTC on their output, seems new to us. Ex-post efficiency is, roughly speaking, a local optimum criterion. By avoiding prematurely locking in efficiency, our new Accept-Last + TTC algorithms seem to be able to achieve efficient outcomes with higher global welfare.
The benefits of using temporary versus permanent memory are relatively small compared to the gains made by using TTC, for example. However they are real and for situations where solution quality is substantially more important than runtime, we recommend its use. The best overall algorithms in terms of solution quality are arguably TLSG and TLQG. The latter has lower order bias and the former higher welfare, although the differences between the algorithms are small.
Some of our new algorithms may be useful for specialized situations (in addition to party games). For example, TFS seems to treat all agents equally in welfare except the last, who has a definite advantage --- this may be useful, for example, when one agent is a small child. Having (almost) zero order bias is a strong fairness condition that may be very important in some applications. Our $1$-sided Naive Boston algorithm PFQ maximizes the number of agents receiving their first choice.
Enlarging the stock of basic algorithms has more benefits than simply allowing us 10 more algorithms to choose from. The concept of \emph{hybridization} has been used by Mennle \& Seuken \cite{MeSe2013}. This simply forms a new algorithm $(1-p)A + pB$ from random allocation algorithms A and B and a fixed $p\in [0,1]$ by taking the convex combination $(1-p)M_A + pM_B$ of the stochastic matrices output A and B. This allows us to trade off desirable properties such as strategyproofness and efficiency in a controlled way. Mennle \& Seuken considered only RSD, PS, Naive Boston and the algorithm maximizing utilitarian welfare as their basic algorithms. We believe that our new algorithms will prove useful as building blocks for hybrid algorithms with good overall behaviour.
\subsection{Future work}
\label{ss:future}
The Adaptive Boston school choice algorithm improves over Naive Boston by allowing agents to skip proposals that will obviously be rejected because a school has reached capacity. It satisfies a property intermediate between weak strategyproofness and strategyproofness, called \emph{partial strategyproofness} by Mennle \& Seuken \cite{MeSe2014}. Although we found a sequential interpretation of Naive Boston that avoided discussion of simultaneous proposals, we have not yet done this for Adaptive Boston.
Each of our algorithms extends to the school choice situation. The ``memory" component and the ``data structure" component translate directly with no changes required. The aceptance policy if more complicated. In school choice, schools accept applicants provisionally until capacity is reached, and then each new temporary enrolment requires an existing enrolment to be cancelled. Our Accept-Last and Accept-First policies in the capacity 1 case described above could also be termed ``Reject-Current'' or ``Reject-New". In the general school choice situation we would need to create fictitious prefernces for schools to enable them to decide which current student to reject. Some obvious ways to do that include FIFO or LIFO.
The case where the number of items exceeds the number of agents, and agents receive bundles of items chosen one at a time, is complicated. The order in which agents should choose in each round (here a round ends when all agents have incremented their previous total of items by 1) must be specified. For example, when we have 2 agents with preferences $a>b>c>d$ over 4 items, and the picking sequence $1221$, Serial Dictatorship awards $a, d$ to agent 1 and $b,c$ to agent 2. However under PFQ agent 2 attempts to get item $a$ and fails, going to the back of the queue and hence missing that turn. In the next turn it is allocated $b$, then agent 1 tries for $b$ and fails. Thus the picking sequence must be extended. Accept-Last algorithms work better in this situation.
We leave further exploration of these interesting cases with $m\neq n$ to future work.
\bibliographystyle{alpha}
|
1,116,691,500,487 | arxiv | \section{Introduction} \label{sec:introduction}
In typical microfluidic applications, the Reynolds number is very small and the flow is laminar. If chaotic mixing is not induced by the device geometry via fully three-dimensional flow fields, mixing is due to molecular diffusion only, resulting in long diffusion times, which limits the efficiency of micro-scale devices. In this context, we study here the feasibility to use a soft elastic wall to enhance mixing by inducing self-sustained chaotic velocity fluctuations also at very low Reynolds numbers.
Several strategies have been proposed in the past to increase mixing in micro-devices, which can be classified into passive and active: in the former, the mixing is enhanced through curved streamlines \citep{knight_vishwanath_brody_austin_1998a, bessoth_manz_others_1999a, stroock_dertinger_ajdari_mezic_stone_whitesides_2002a, lee_chang_wang_fu_2011a, lee_wang_liu_fu_2016a}, while in the latter the flow is made unsteady by an external actuation \citep{glasgow_aubry_2003a, bazant_squires_2004a, mensing_pearce_graham_beebe_2004a, kazemi_nourian_nobari_movahed_2017a, keshavarzian_shamshiri_charmiyan_moaveni_2018a}. Here, we focus on the possibility to enhance mixing in micro-channels by using elastic walls: the interaction between the soft wall and the flow results in a dynamical instability, which induces transition at very low Reynolds numbers \citep{verma_kumaran_2013a}. In particular, previous linear stability studies \citep{kumaran_1996a, shankar_kumaran_1999a, kumaran_muralikrishnan_2000a} have shown that the flow over elastic walls is unstable to infinitesimal disturbances when the Reynolds number exceeds a critical value which can be tuned by decreasing the shear modulus of the soft wall, thus suggesting that there is an instability even at zero Reynolds number. The existence of this instability has been proved experimentally by \citet{verma_kumaran_2012a}, reaching a transitional Reynolds number of $200$ for the softest wall used in the experiments.
Flow instabilities at low Reynolds numbers have been previously observed in the presence of elasticity; in particular, so-called purely elastic instabilities have been reported for viscoelastic fluids in a wide variety of flow configurations and they can be generally found when inertial forces are negligible compared to elasticity \citep{gardner_pike_miles_keller_tanaka_1982a, larson_1992a, shaqfeh_1996a, mckinley_pakdel_oztekin_1996a, haward_mckinley_shen_2016a}. Such instabilities are due to the non-linear coupling between the flow and the constitutive equation of the non-Newtonian fluid and lead to the so-called elastic turbulence \citep{groisman_steinberg_2000a, berti_boffetta_2010a, lim_ober_edd_desai_neal_bong_doyle_mckinley_toner_2014a, kawale_marques_zitha_kreutzer_rossen_boukany_2017a, steinberg_2019a}. Here, we will extend these works by considering a simple Newtonian fluid non-linearly coupled to a viscoelastic wall, and show that a self-sustained chaotic flow can be observed.
In this work, we present new Direct Numerical Simulations (DNS) of the flow over an incompressible hyper-elastic wall at Reynolds number where turbulence cannot be sustained in channels with rigid walls and show that fluid velocity fluctuations can be sustained by tuning the wall elasticity. In the fluid part of the channel, the full incompressible Navier--Stokes equations are solved, while momentum conservation and incompressibility constraint are enforced inside the solid material. In \secref{sec:formulation}, we first discuss the flow configuration and governing equations, and then present the numerical methodology used. The effects of an hyper-elastic wall on the channel flow are presented in \secref{sec:result}. Finally, a summary of the main findings and some conclusions are drawn in \secref{sec:conclusion}.
\begin{figure}
\centering
\includegraphics[width=0.35\textwidth]{fig1}
\caption{Sketch of the channel considered in the present work: two solid walls are located at $y=0$ and $2h+h_e$, while $y=2h$ indicates the interface between the fluid region and the elastic layer.}
\label{fig:sketch}
\end{figure}
\section{Formulation} \label{sec:formulation}
We consider the flow of an incompressible viscous fluid through a channel with an incompressible hyper-elastic wall. A sketch of the geometry and the Cartesian coordinate system are reported in \figrefS{fig:sketch}: $x$, $y$ and $z$ denote the streamwise, wall-normal and spanwise coordinates, and $u$, $v$ and $w$ the corresponding velocity components. The channel is bounded by two rigid walls located at $y=0$ and $2h+h_e$, while the elastic layer extends from $y=2h$ to $2h+h_e$, where $h_e$ represents the height of the layer, fixed here to $h_e=0.5h$. In this work, we assume the interface of the elastic layer to be initially flat. Periodic boundary conditions are imposed in the streamwise and spanwise directions.
The fluid and solid phase motion is governed by the conservation of momentum and the incompressibility constraint:
\begin{align}
\label{eq:NS}
\frac{\partial u_i^p}{\partial t} + \frac{\partial u_i^p u_j^p}{\partial x_j} = \frac{1}{\rho} \frac{\partial \sigma_{ij}^p}{\partial x_j} \;\;\; \textrm{and} \;\;\; \frac{\partial u_i^p}{\partial x_i} = 0,
\end{align}
where the suffix $^p$ is used to distinguish the fluid $^f$ and solid $^s$ phases. In the previous set of equations, $\rho$ is the density (assumed to be the same for the solid and fluid), and $\sigma_{ij}$ the Cauchy stress tensor. The two phases are coupled at the interface by the continuity of the velocity and traction force, i.e.,\ $u_i^f = u_i^s$ and $\sigma_{ij}^f n_j = \sigma_{ij}^s n_j$, where $n_i$ denotes the normal to the interface.
To numerically solve the fluid-structure interaction problem at hand, we introduce a monolithic velocity vector field $u_i$ valid everywhere, found by a volume averaging procedure. In particular, we introduce an additional variable $\phi^s$ which is the solid volume fraction; this is zero in the fluid and one in the solid, with $0\le\phi^s\le1$ around the interface. By doing so, we can now write the stress in a mixture form as
\begin{equation}
\label{eq:phi-stress}
\sigma_{ij} = \left( 1 - \phi^s \right) \sigma_{ij}^f + \phi^s \sigma_{ij}^s.
\end{equation}
This is the so-called one-continuum formulation \cite{tryggvason_sussman_hussaini_2007a}. The fluid is Newtonian and the solid is an incompressible viscous hyper-elastic material with constitutive equations
\begin{equation}
\label{eq:stress}
\sigma_{ij}^f = -p \delta_{ij} + 2 \mu \mathcal{D}_{ij} \;\;\;\; \textrm{and} \;\;\;\; \sigma_{ij}^s = -p \delta_{ij} + 2 \mu \mathcal{D}_{ij} + G \mathcal{B}_{ij},
\end{equation}
where $p$ is the pressure, $\mu$ the dynamic viscosity (assumed to be the same in the two phases), $\mathcal{D}_{ij}$ the strain rate tensor defined as $\mathcal{D}_{ij}=\left( \partial u_i/\partial x_j + \partial u_j/\partial x_i \right)/2$ and $\delta_{ij}$ is the Kronecker delta. The last term in the solid Cauchy stress tensor $\sigma_{ij}^s$ is the hyper-elastic contribution modelled as a neo-Hookean material, thus satisfying the incompressible Mooney-Rivlin law, where $\mathcal{B}_{ij}$ is the left Cauchy-Green deformation tensor and $G$ the modulus of transverse elasticity. The full set of equations can be closed in a purely Eulerian manner by updating $\mathcal{B}_{ij}$ and $\phi^s$ with the following transport equations:
\begin{equation}
\label{eq:adv}
\frac{\partial \mathcal{B}_{ij}}{\partial t} + \frac{\partial u_k \mathcal{B}_{ij}}{\partial x_k} = \mathcal{B}_{kj}\frac{\partial u_i}{\partial x_k} + \mathcal{B}_{ik}\frac{\partial u_j}{\partial x_k} \;\;\;\; \textrm{and} \;\;\;\; \frac{\partial \phi^s}{\partial t} + \frac{\partial u_k \phi^s}{\partial x_k} = 0.
\end{equation}
\subsection{Numerical implementation}
The previous set of equations are solved numerically: the time integration is based on an explicit fractional-step method \citep{kim_moin_1985a}, where all the terms are advanced with the third order Runge-Kutta scheme, except the solid hyper-elastic contribution which is advanced with the Crank-Nicolson scheme \citep{min_yoo_choi_2001a}. The governing differential equations are solved on a staggered grid using a second order central finite-difference scheme, except for the advection terms in \equref{eq:adv} where the fifth-order WENO scheme is applied. The code has been extensively validated, and more details on the numerical scheme and validation campaign are reported in Refs.\ \onlinecite{rosti_brandt_2017a, rosti_brandt_2018a, rosti_brandt_mitra_2018a, izbassarov_rosti_niazi-ardekani_sarabian_hormozi_brandt_tammisola_2018a, alghalibi_rosti_brandt_2019a, rosti_pramanik_brandt_mitra_2020a}; more details on the numerical method can be found in \citet{sugiyama_ii_takeuchi_takagi_matsumoto_2011a}.
For all the flows considered hereafter, the equations of motion are discretised on a fixed, Cartesian and uniform mesh with $1296 \times 540 \times 648$ grid points on a computational domain of size $6hk \times 2.5h \times 3hk$ in the streamwise, wall-normal and spanwise directions. $k$ is a factor used to increase the size of the domain in the homogeneous direction as the Reynolds number decreases \citep{tsukahara_seki_kawamura_tochio_2005a}; in particular, $k=1$, $4.30$, $18.5$, $79.5$ and $237$ for $Re_b=2800$, $651$, $151$, $35$ and $11$, respectively. The spatial resolution has been chosen in order to properly resolve the wall deformation for all the Reynolds numbers considered in the present study \citep{rosti_brandt_2017a}.
\section{Results} \label{sec:result}
We study laminar and turbulent channel flows over viscous hyper-elastic walls, together with the baseline cases over stationary impermeable walls. All the simulations are performed at constant flow rate, and thus the pressure gradient needed to drive the flow is determined at every time step to ensure this condition; it oscillates around a constant value at statistical state. The flow Reynolds number is defined based on the bulk velocity, i.e.,\ $Re_b=\rho U_b h/\mu$, where $U_b$ is the average value of the mean velocity computed across the whole domain occupied by the fluid phase; the choice of using $U_b$ and $h$ as reference velocity and length facilitates the comparison between the flow in a channel with elastic walls and the flow in a channel bounded by rigid walls. In the present work, we vary the bulk Reynolds number $Re_b$ and the modulus of transverse elasticity $G$. The full set of simulations is reported in \tabref{tab:cases}. All the simulations start with a fully developed turbulent flow over rigid walls, and then after an initial transient, a new statistically steady state solution is reached, either laminar or turbulent.
\begin{table}
\centering
\setlength{\tabcolsep}{5pt}
\begin{tabular}{ccccc|ccccc}
$Re_b$ & $G/\left( \rho U_b^2 \right)$ & $\overline{Re}_\tau$ & $\overline{u}_M/U_b$ & $\widehat{y}_M/h$ & $Re_b$ & $G/\left( \rho U_b^2 \right)$ & $\overline{Re}_\tau$ & $\overline{u}_M/U_b$ & $\widehat{y}_M/h$ \\
\hline
$2800$ & $\infty$ & $180.0$ & $1.16$ & $~~0.000$ & $151$ & $\infty$ & $21.3$ & $1.50$ & $~~0.000$ \\
$2800$ & $4.0$ & $180.8$ & $1.17$ & $-0.030$ & $151$ & $1.0$ & $21.8$ & $1.47$ & $-0.069$ \\
$2800$ & $2.0$ & $203.3$ & $1.19$ & $-0.076$ & $151$ & $0.5$ & $51.0$ & $1.41$ & $-0.222$ \\
$2800$ & $1.0$ & $240.5$ & $1.23$ & $-0.206$ & $35$ & $\infty$ & $10.2$ & $1.50$ & $~~0.000$ \\
$2800$ & $0.5$ & $337.0$ & $1.32$ & $-0.386$ & $35$ & $1.0$ & $11.2$ & $1.49$ & $-0.007$ \\
$651$ & $\infty$ & $49.8$ & $1.29$ & $~~0.000$ & $35$ & $0.5$ & $15.3$ & $1.48$ & $-0.014$ \\
$651$ & $2.0$ & $49.8$ & $1.30$ & $-0.010$ & $11$ & $\infty$ & $5.7$ & $1.50$ & $~~0.000$ \\
$651$ & $1.0$ & $68.3$ & $1.33$ & $-0.031$ & $11$ & $0.5$ & $5.7$ & $1.50$ & $~~0.000$ \\
$651$ & $0.5$ & $151.1$ & $1.37$ & $-0.463$ & ~ & ~ & ~ & ~ & ~
\end{tabular}
\caption{Summary of the DNSs performed, all with fixed thickness of the elastic layer $h_e=0.5h$. The table reports the bulk Reynolds number $Re_b$, the shear elastic modulus $G$, the mean friction Reynolds number $\overline{Re}_\tau$, the maximum velocity $\overline{u}_M$, and its distance from the channel centerline $\widehat{y}_M = y_M - h$.}
\label{tab:cases}
\end{table}
The friction velocity $u_\tau$ will be often employed in the following and is defined here as
\begin{equation} \label{eq:friction_velocity_total}
\overline{u}_\tau = \sqrt{\frac{\mu}{\rho} \dfrac{d \overline{u}}{d y} - \overline{u'v'} + \frac{G}{\rho} \overline{B}_{12} },
\end{equation}
where the quantities are evaluated at the mean interface location, $y=2h$. In the previous relation and in the rest of the work, the overline and the prime represent the mean and fluctuation obtained by averaging over the homogeneous directions and in time. The previous definition is used because, when the channel has moving walls, the friction velocity needs to account for the Reynolds and the elastic shear stress, that are in general non-zero at the solid-fluid interface. Note that, the actual value of the friction velocity of the elastic wall is computed from its friction coefficient, found by combining the information of the total $C_f$, obtained from the driving streamwise pressure gradient, and the one of the lower rigid wall \citep[see also][]{rosti_brandt_2017a}.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{dpdxGz}
\includegraphics[width=0.49\textwidth]{dpdx2800}
\caption{Time history of the friction Reynolds number $Re_\tau$ space averaged over the wall. The blue, magenta, red, green and cyan lines are used to distinguish the different bulk Reynolds numbers, $Re_b=2800$, $651$, $151$, $35$ and $11$, while the solid, dash-dotted, dashed and dotted lines to distinguish the different shear elastic moduli, $G/\rho U_b^2=0.5$, $1$, $2$ and $4$. In panel $(a)$, the amplitude of the fluctuations is amplified by a factor $5$ (magenta line), by $10$ (red line) and by $20$ (green line).}
\label{fig:reynoldsTime}
\end{figure}
We start our analysis by studying in \figrefS{fig:reynoldsTime} the time evolution of the friction Reynolds number $Re_\tau$, i.e.,\ $u_\tau h/\nu$. In particular, panel a) shows the friction Reynolds number for the cases with the minimum elastic modulus $G=0.5 \rho U_b^2$, thus corresponding to the most deformable wall, and for different bulk Reynolds number $Re_b$. We observe that, as expected, the friction Reynolds number decreases with the bulk Reynolds number and also the amplitude of its fluctuations. However, differently from the flow over rigid walls, the flow remains unstable even for very low Reynolds numbers, $Re_b=35$ in this case, while a further reduction of the Reynolds number leads to the flow laminarisation. If we fix the bulk Reynolds number $Re_b$ and vary only the elastic modulus $G$, three different behaviors can be observed, as shown by the space and time averaged friction Reynolds number $\overline{Re}_\tau$ pertaining all cases studied in the present work collected in \figref[$a$]{fig:reynolds}: \textit{(i)} for high $Re_b$, as $G$ increases (the wall becomes more rigid) $\overline{Re}_\tau$ decreases eventually saturating at the value obtained for a turbulent flow over rigid walls, see also the time histories in \figref[$b$]{fig:reynoldsTime}; \textit{(ii)} for intermediate $Re_b$, as $G$ increases $\overline{Re}_\tau$ decreases eventually leading to a fully laminar flow, the friction assuming the same value obtained for a laminar flow over rigid walls; \textit{(iii)} for low Reynolds numbers, the flow always becomes laminar for any initial condition and the friction Reynolds number is the same obtained for a laminar flow over rigid walls. Indeed, the thin lines in \figref[$a$]{fig:reynolds} display the characteristic values for laminar and turbulent channel flows. For every $Re_b$, reducing the wall elasticity implies a reduction of the resulting $\overline{Re}_\tau$; all cases converge to the rigid wall solution as $G$ increases, in particular, $\overline{Re}_\tau$ converges to the turbulent experimental correlation $0.09 \left( 2Re_b \right)^{0.88}$ (see e.g.,\ Ref.~\onlinecite{pope_2001a}) for $Re_b \gtrsim 482$ and to the laminar analytical solution $\sqrt{3Re_b}$ for $Re_b \lesssim 482$ as $G \rightarrow \infty$.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{reynolds}
\includegraphics[width=0.49\textwidth]{turbIntIn}
\caption{$(a)$ Mean friction Reynolds number $\overline{Re}_\tau$ and $(b)$ root mean square of the friction velocity normalised by its mean, i.e.,\ $\mathcal{I}=\sqrt{\overline{u_\tau' u_\tau'}}/\overline{u}_\tau$, as a function of the bulk Reynolds number. The blue, magenta, red, green and cyan colors are used to distinguish different bulk Reynolds numbers $Re_b=2800$, $651$, $151$, $35$ and $11$, while the upper-triangle $\blacktriangle$, circle {\Large $\bullet$}, rombus $\blacklozenge$ and lower-triangle $\blacktriangledown$ to distinguish different shear elastic moduli $G/\rho U_b^2=0.5$, $1$, $2$ and $4$. The grey and black lines in panel a) are the analytical solutions for laminar flows and the experimental correlation for turbulent flows, respectively. The inset in panel b) reports the contour of $\mathcal{I}\%$ as a function of the Reynolds number $Re_b$ and elastic shear modulus $G$ obtained by interpolation and extrapolation of our data. The black lines are separated by $0.5$.}
\label{fig:reynolds}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{turbIntElas}
\caption{Root mean square of the friction velocity normalised by its mean, i.e.,\ $\mathcal{I}=\sqrt{\overline{u_\tau' u_\tau'}}/\overline{u}_\tau$, as a function of the ratio of the normlaised wall elasticity $G$ and bulk Reynolds number, i.e.,\ $G/\rho U_b^2 Re_b$. The blue, magenta, red and green colors are used to distinguish different bulk Reynolds numbers $Re_b=2800$, $651$, $151$ and $35$, while the upper-triangle $\blacktriangle$, circle {\Large $\bullet$}, rombus $\blacklozenge$ and lower-triangle $\blacktriangledown$ to distinguish different shear elastic moduli $G/\rho U_b^2=0.5$, $1$, $2$ and $4$.}
\label{fig:elasticity}
\end{figure}
To quantify the unsteady nature of the flow, we compute the root mean square of the friction velocity $\sqrt{\overline{u_\tau' u_\tau'}}$, used here as a measure of the flow fluctuations. This is divided by its mean value and reported in \figref[$b$]{fig:reynolds} as a function of the bulk Reynolds number for all the cases considered here. Consistently with the previous discussion, we observe that reducing the wall elasticity induces a reduction of the fluctuations. This reduction is strongly non-linear, with large reduction for increment in small values of $G$ and small reduction for increment in large values of $G$. Also, we can observe again that the high Reynolds number cases converge, as $G$ increases, to a non-zero level of fluctuations, i.e.,\ the turbulent rigid wall solution, while the low Reynolds number cases tend to the laminar solution with zero fluctuations. Reducing the Reynolds number, we observe a further reduction of the fluctuation intensity; also in this case the variation is strongly non-linear with large reductions of the fluctuation intensity for large Reynolds numbers, while smaller variation are observed at small $Re_b$, when the flow tends to become laminar. The inset of \figref[$b$]{fig:reynolds} shows the same quantity, $\mathcal{I}$, as a function of both $Re_b$ and $G$ as a contour plot obtained by interpolating and extrapolating our data. We observe that, although in general $\mathcal{I}$ is a function of both $Re_b$ and $G$, i.e.,\ $\mathcal{I} = \mathcal{F} \left( Re_b, G \right)$, there is a critical value $G^* \left( Re_b\right)$ above which the solution does not significantly change anymore with the wall elasticity, and thus $\mathcal{I} = \mathcal{F}_r \left( Re_b \right)$ for $G>G^*$, where $\mathcal{F}_r$ is the solution for the flows over rigid walls. On the other hand, for $G<G^*$ the solution strongly depends on the wall elasticity: this suggests that it is possible to maintain an unsteady chaotic turbulent-like flow in principle for any Reynolds number down to $0$, as long as the wall shear elastic modulus $G$ is reduced accordingly. If we now replot the data in \figref[$b$]{fig:reynolds} as a function of a new quantity, obtained as the ratio of the wall elasticity $G/\rho U_b^2$ and the bulk Reynolds number $Re_b$, we obtain \figrefS{fig:elasticity}. By doing so, all the non-laminar cases successfully collapse onto a single master curve, decaying with power $-0.75$, i.e.,\ $\mathcal{I} \sim \left( G/\rho U_b^2 Re_b \right)^{-0.75}$. This behaviour further corroborates the idea that the level of fluctuations in the channel can be amplified either by increasing the Reynolds number (at fixed elasticity) or by increasing the wall flexibility, i.e.,\ reducing $G$ (at fixed Reynolds number).
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{mean2800}
\includegraphics[width=0.49\textwidth]{meanG}
\caption{Mean velocity profile $\overline{u}$ as a function of the wall-normal distance $y$ for $(a)$ different wall elastic moduli $G$ at $Re_b=2800$ and for $(b)$ different Reynolds numbers $Re_b$ with $G=0.5 \rho U_b^2$. The line colors and styles are the same as in \figrefS{fig:reynoldsTime}. The symbols represent the profiles from the DNS by Kim, Moin and Moser \cite{kim_moin_moser_1987a} of turbulent flow between two solid rigid walls plotted as a reference.}
\label{fig:U}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{reyK}
\includegraphics[width=0.49\textwidth]{reyKp}
\caption{Mean turbulent kinetic energy $\mathcal{K}$ as a function of the wall-normal distance $y$ for $(a)$ different wall elastic moduli $G$ at $Re_b=2800$ and for $(b)$ different Reynolds numbers $Re_b$ with $G=0.5 \rho U_b^2$. The line colors and styles are the same as in \figrefS{fig:reynoldsTime}. The symbols represent the profiles from the DNS by Kim, Moin and Moser \cite{kim_moin_moser_1987a} of turbulent flow between two solid rigid walls plotted as a reference.}
\label{fig:K}
\end{figure}
Next, we characterize the unsteady flows in terms of mean and fluctuation velocities. We start by considering the wall-normal profiles of the mean velocity $\overline{u}$ and turbulent kinetic energy $\mathcal{K} = \rho \overline{u'_i u'_i} /2$, reported in \figrefS{fig:U} and \figrefS{fig:K}. In particular, the left panels of the two figures show $\overline{u}$ and $\mathcal{K}$ at a fixed Reynolds number ($Re_b=2800$) and for all the wall elasticities $G$ studied in this work, while the right ones report $\overline{u}$ and $\mathcal{K}$ for a fixed wall elasticity ($G=0.5\rho U_b^2$) and for all the Reynolds numbers $Re_b$. From \figrefS{fig:U}, we observe that the mean velocity of the elastic wall is equal to zero \citep{rosti_brandt_2017a}; indeed, the elastic layer can only oscillates around its equilibrium position being attached to the top stationary rigid wall. Although the mean velocity is zero inside this layer, the elastic layer induces profound modification of the fluid flow in the channel. In particular, the mean velocity profile becomes more skewed, with its maximum $\overline{u}_M$ increasing and located closer to the rigid wall as the elasticity increases ($G$ decreases) as shown in \figref[$a$]{fig:U}. Note that, an inflection point in the mean profile appears within the fluid region ($0<y<2h$), usually associated to the occurrence of a Kelvin-Helmholtz instability and the formation of large scale spanwise-correlated rollers \citep{jimenez_uhlmann_pinelli_kawahara_2001a, rosti_cortelezzi_quadrio_2015a, kuwata_suga_2016a, rosti_brandt_2017a, rosti_brandt_pinelli_2018a, kuwata_suga_2019a, monti_omidyeganeh_pinelli_2019a, monti_omidyeganeh_eckhardt_pinelli_2020a}. When the Reynolds number is decreased (right panel), the asymmetry in the flow reduces with the maximum velocity increasing and its location moving back towards the channel center. Eventually, the laminar analytical profile is recovered for the smallest $Re_b$ considered.
When focusing on the velocity fluctuations in \figref[$a$]{fig:K}, we observe that the turbulent kinetic energy $\mathcal{K}$ is higher close to the elastic wall than close to the rigid wall, with the maximum value becoming almost the double of the peak close to the bottom wall for the most deformable case (left panel). This is due to the movement of the deformable wall which strongly increases the velocity fluctuations, especially the ones in the wall-normal directions, i.e.,\ $v'$ (see e.g.,\ Ref.~\onlinecite{rosti_brandt_2017a}). Furthermore, the near-wall peaks of the turbulent kinetic energy move farther from the elastic walls as the elasticity is increased. The turbulent fluctuations have non-zero values at $y = 2h$ for the elastic cases, since the no-slip condition is now enforced on a wall which is moving, i.e.,\ $u_i^f=u_i^s$. In particular, $\mathcal{K}$ does not clearly vanish until reaching the rigid top wall ($y=2.5h$), thus indicating that the fluctuations propagate deeply inside the solid layer. The asymmetry in the flow originates from the asymmetry of the geometry; this induces the shift of the minimum of $\mathcal{K}$ towards the rigid walls, as well as the shift in the same direction of the maximum velocity, as reported in \tabref{tab:cases}.
\figrefC[$b$]{fig:K} shows how the turbulent kinetic energy $\mathcal{K}$ scales with the Reynolds number, for a fixed wall elasticity; in particular, the softest wall is considered here. We observe that, as $Re_b$ decreases the peak of turbulent kinetic energy close to the rigid wall rapidly vanishes, as expected for flows over rigid walls, where the lowest Reynolds number able to sustain a turbulent flow is around $600$, as reported by \citet{tsukahara_seki_kawamura_tochio_2005a}. A similar trend is evident for the peak close to the moving wall, but the decrease is much lower than for a rigid wall. Indeed, for $Re_b<600$ the near-wall peak close to the rigid wall completely disappear, and the profiles exhibit a single peak close to elastic wall. Also, the peak moves away from the deformable wall towards the bulk of the channel as the Reynolds number reduces, indicating that all the turbulent fluctuations in the channel at low $Re_b$ are produced by the moving wall, then propagating across the channel.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{stressBALg}
\includegraphics[width=0.49\textwidth]{stressBALre}
\caption{Reynolds shear stress $-\rho \overline{u'v'}$ (solid line), viscous stress $\mu d\overline{u}/dy$ (dash-dotted line) and shear elastic stress $G\overline{B}_{12}$(dotted line) in wall units as a function of the wall-normal distance from the elastic wall $\widetilde{y}=2h-y$ normalised by $\widetilde{y}_M=2h-y_M$ for $(a)$ different wall elastic moduli $G$ at $Re_b=2800$ and for $(b)$ different Reynolds numbers $Re_b$ with $G=0.5 \rho U_b^2$. The values of $y_M$ used to normalize the different abscissa can be obtained from \tabref{tab:cases}. In particular, the blue, brown and red colors in the two panels are used for the cases $Re_b=2800$ and $G=0.5 \rho U_b^2$, $Re_b=2800$ and $G=4 \rho U_b^2$, and $Re_b=151$ and $G=0.5 \rho U_b^2$.}
\label{fig:stressBal}
\end{figure}
Apart from the diagonal components of the Reynolds stress tensor discussed above in terms of the turbulent kinetic energy, another important observable is the off-diagonal shear component of the Reynolds stress tensor $- \rho \overline{u'v'}$, which together with the mean viscous $\mu d\overline{u}/dy$ and elastic stress $G\overline{B}_{12}$ shear components form the total shear stress, i.e.,\
\begin{equation} \label{eq:shearBalance}
\tau = \mu d\overline{u}/dy - \rho \overline{u'v'} + G\overline{B}_{12}.
\end{equation}
All of these are reported in \figref[$a$]{fig:stressBal} for the cases at $Re_b=2800$ (solid lines). The cross Reynolds stress component is strongly affected by the presence of the moving wall: the maximum value increases and moves away from the wall as the elasticity increases at a fixed Reynolds number. The stress profiles vary linearly in the bulk of the channel away from the wall, although with different slopes depending on $Re_b$ and $G$. Most of these effects are well compensated in the figure by dividing $\widetilde{y}$ with $\widetilde{y}_M=2h-y_M$, i.e.,\ the distance of the location of the maximum mean velocity from the elastic wall. At the interface the value of the stress is not null as in the rigid case, however, inside the elastic layer the Reynolds shear stress vanishes quickly. The mean viscous stress is almost null in the solid and in the bulk of the channel and exhibits a small peak close to the interface which increase as $G$ increases, i.e.,\ the wall is more rigid, eventually having the maximum at the interface for the completely rigid case; the elastic stress, on the contrary, is null in the fluid region and almost the total stress in the solid layer. Thus, we can conclude that the total shear stress is dominated by the elastic stress in the solid layer, by the Reynolds stress in the bulk of the channel and by the balance of all the three components at the interface, with the relative contributions at the interface strongly changing with $G$: for rigid walls the dominant and only contribution not null at the interface is the viscous stress, while for flexible walls the Reynolds and elastic stresses grow with the wall elasticity. When the Reynolds number is varied, the balance between the three terms is significantly altered, as shown in \figref[$b$]{fig:stressBal}. Indeed, as the Reynolds numbers decreases, the Reynolds shear stress peak shifts away from the wall, thus reducing its total contribution. On the other hand, the viscous contribution increases and compensates for the loss of Reynolds shear stress. For the lowest Reynolds number (not shown in the figure), the flow is fully laminar, and the total stress is equal to the elastic stress in the solid layer and to the viscous stress in the fluid region, with the Reynolds shear stress being null. From the figure we can conclude that, differently from the flow over rigid walls, the turbulent fluctuations do not rapidly vanish when reducing the Reynolds number because of their persistence in the bulk of the channel.
To confirm these observations, we consider the turbulent kinetic energy balance. To do so, we decompose the velocity field $u_i \left( x, y, z, t \right)$ into its mean $\overline{u}_i \left( y \right)$ and fluctuation $u'_i \left( x, y, z, t \right)$ as $u_i = \overline{u}_i + u'_i$. By substituting this into the governing equation, we obtain
\begin{equation}
\rho \left( \frac{\partial u'_i}{\partial t} + \frac{\partial u'_i u'_j}{\partial x_j} + \frac{\partial \overline{u}_i u'_j}{\partial x_j} + \frac{\partial u'_i \overline{u}_j}{\partial x_j} + \frac{\partial \overline{u}_i \overline{u}_j}{\partial x_j} \right) = - \frac{\partial p}{\partial x_i} + 2\mu \frac{\partial \mathcal{D}_{ij}}{\partial x_j} + G \frac{\partial \phi^s \mathcal{B}_{ij}}{\partial x_j},
\end{equation}
which can be rewritten for later convenience as
\begin{equation}
\rho \left( \frac{\partial u'_i}{\partial t} + \frac{\partial u'_i u'_j}{\partial x_j} + u'_j \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial u'_i \overline{u}_j}{\partial x_j} + \frac{\partial \overline{u}_i \overline{u}_j}{\partial x_j} \right) = - \frac{\partial p}{\partial x_i} + 2\mu \frac{\partial \mathcal{D}_{ij}}{\partial x_j} + G \frac{\partial \phi^s \mathcal{B}_{ij}}{\partial x_j}.
\end{equation}
We now multiply the equation by $u'_i$ and obtain
\begin{equation}
\begin{split}
\rho \left( \frac{\partial u'_i u'_i/2}{\partial t} + \frac{\partial u'_i u'_i u'_j/2}{\partial x_j} + u'_i u'_j \frac{\partial \overline{u}_i}{\partial x_j} + \frac{\partial u'_i u'_i \overline{u}_j/2}{\partial x_j} + u_i \frac{\partial \overline{u}_i \overline{u}_j}{\partial x_j} \right) = \\
- \frac{\partial u'_i p}{\partial x_i} + 2\mu \frac{\partial u'_i \mathcal{D}_{ij}}{\partial x_j} - 2 \mu \mathcal{D}_{ij} \mathcal{D}_{ij} + G \frac{\partial u'_i \phi^s \mathcal{B}_{ij}}{\partial x_j} - G \phi^s \mathcal{B}_{ij} \mathcal{D}_{ij},
\end{split}
\end{equation}
where we made use of
\begin{equation}
u'_i \frac{\partial \mathcal{D}_{ij}}{\partial x_j} = \frac{\partial u'_i \mathcal{D}_{ij}}{ \partial x_j} - \mathcal{D}_{ij} \frac{\partial u'_i }{ \partial x_j} = \frac{\partial u'_i \mathcal{D}_{ij}}{ \partial x_j} - \mathcal{D}_{ij} \mathcal{D}'_{ij},
\end{equation}
and similarly of
\begin{equation}
u'_i \frac{\partial \phi^s \mathcal{B}_{ij}}{\partial x_j} = \frac{\partial u'_i \phi^s \mathcal{B}_{ij}}{ \partial x_j} - \phi^s \mathcal{B}_{ij} \frac{\partial u'_i }{ \partial x_j} = \frac{\partial u'_i \phi^s \mathcal{B}_{ij}}{ \partial x_j} - \phi^s \mathcal{B}_{ij} \mathcal{D}'_{ij},
\end{equation}
where the last substitution is possible being $\mathcal{D}_{ij}$ and $\mathcal{B}_{ij}$ symmetric tensors. The equation above can then be volume averaged with the operator
\begin{equation}
\langle \cdot \rangle = \frac{1}{\mathcal{V}} \int_\mathcal{V} \cdot \ \mathrm{d} \mathcal{V},
\end{equation}
leading to the equation
\begin{equation}
\rho \left( \frac{\partial \langle u'_i u'_i \rangle/2}{\partial t} + \langle u'_i u'_j \rangle \frac{\partial \overline{u}_i}{\partial x_j} \right) = - 2 \mu \langle \mathcal{D}'_{ij} \mathcal{D}'_{ij} \rangle - G \langle \phi^s \mathcal{B}_{ij} \mathcal{D}'_{ij} \rangle.
\end{equation}
Here, all the transport terms $\langle \partial u'_i \mathcal{F}_{ij}/\partial x_j \rangle$ vanish due to the homogeneity of the domain and to the no-slip and no-penetration boundary conditions at the rigid walls, and the terms $\langle u'_i \partial \overline{u}_i \overline{u}_j / \partial x_j \rangle$ and $\langle \overline{\mathcal{D}}_{ij} \mathcal{D}'_{ij} \rangle$ because $\langle u'_i \rangle = 0$ and $\langle \mathcal{D}'_{ij} \rangle = 0$ due to ergodicity. Finally, we obtain the turbulent kinetic energy equation
\begin{equation}
\label{eq:tke}
\frac{d \mathcal{K}}{d t} = \mathcal{P} - \varepsilon - \psi_G,
\end{equation}
where the different terms indicate the rate of change of turbulent kinetic energy $\mathcal{K}$, the turbulent production rate $\mathcal{P}$, the dissipation rate $\varepsilon$ and the power of the elastic wall $\psi_G$, defined as
\begin{equation}
\mathcal{K} = \rho \langle u'_i u'_i \rangle/2, \ \
\mathcal{P} = -\rho \langle u'_1 u'_2 \frac{\partial \overline{u}_1}{\partial x_2} \rangle, \ \
\varepsilon = 2 \mu \langle \mathcal{D}'_{ij} \mathcal{D}'_{ij} \rangle, \ \
\psi_G = G \langle \phi^s \mathcal{B}_{ij} \mathcal{D}'_{ij} \rangle.
\end{equation}
$\psi_G$ is the rate of work performed by the fluid on the elastic wall and can be either positive or negative and thus a sink or source of turbulent kinetic energy. At statistically steady state, the time derivative is obviously null, and thus \equref{eq:tke} reduces to a balance between $\mathcal{P}$, $\varepsilon$ and $\psi_G$.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{kinBalGin}
\includegraphics[width=0.49\textwidth]{kinBalIn}
\caption{Volume averaged turbulent production $\mathcal{P}$ (brown), turbulent dissipation $\varepsilon$ (orange) and power of the elastic wall $\psi_G$ (grey) as a function of the wall elasticity $G$ for a fixed Reynolds number $Re_b=2800$ $(a)$ and as a function of the Reynolds number $Re_b$ for a fixed wall elasticity $G=0.5 \rho U_b^2$ $(b)$. The symbol style is the same as in \figrefS{fig:reynolds} with the addition of the black triangles, additional simulations included for the sake of clarity. The two inset figures show the production $\mathcal{P}$ and dissipation $\varepsilon$ rates divided by the power of the elastic wall $\psi_G$.}
\label{fig:kinBal}
\end{figure}
These three terms are displayed in \figrefSC{fig:kinBal} as a function of the shear elastic modulus $G$ (left panel) and of the Reynolds number $Re_b$ (right panel). In the left panel we see that the elastic power contribution is positive, and indeed the presence of the elastic wall acts as an additional dissipation term at high Reynolds number. This term reduces as $G$ increases, eventually vanishing for perfectly rigid walls when $G \rightarrow \infty$. On the other hand, the behavior at fixed $G$ is non-monotonic with $Re_b$: as $Re_b$ decreases all the terms first increase, reach a maximum and then decreases. In particular, all the terms grow by a factor of around $10$ when decreasing the Reynolds number from $2800$ to $151$. Interestingly, while the turbulent production rapidly vanishes as the flow is approaching the laminar flow (for $Re_b\lesssim 151$), the power of the elastic walls change sign and becomes a production term for the turbulent kinetic energy. Because of this, the flow can remain turbulent at much lower Reynolds numbers than what usually found for flows over rigid walls and, by choosing properly the value of $G$, fluctuations can be sustained at any small Reynolds number. In conclusion, while at high Reynolds number the standard wall cycle \citep{jimenez_pinelli_1999a} takes place (although slightly modified by the elastic walls \citep{rosti_brandt_2017a}), at low Reynolds number a different mechanism arises to sustain the chaotic flow: this new mechanism originates from the non-linear interaction between the elastic solid and the fluid and resembles what found at low Reynolds and high Weissenberg numbers (i.e.,\ high elasticity numbers) for non-Newtonian fluids \citep{groisman_steinberg_2000a, haward_mckinley_shen_2016a, hopkins_haward_shen_2020a}.
\section{Conclusions} \label{sec:conclusion}
We have carried out a number of direct numerical simulations of laminar and turbulent channel flows over a viscous hyper-elastic wall. The flow inside the fluid region is described by the Navier--Stokes equations, while momentum conservation and incompressibility are imposed inside the solid layer. The two sets of equations are coupled using a one-continuum formulation allowing a fully Eulerian description of the multiphase flow problem. Here, we systematically reduce the Reynolds number and vary the wall elasticity to identify in which condition a chaotic unsteady flow can be sustained.
In general, the friction Reynolds number $Re_\tau$ is a function of both the bulk Reynolds number $Re_b$ and the wall shear elastic modulus $G$: we show that, reducing the the wall elasticity leads to a reduction of the resulting friction Reynolds number, with the value converging to the value of the turbulent flow over rigid walls for $Re_b\gtrsim482$ and to the laminar analytical solution for $Re_b\lesssim482$. There is therefore a critical value $G^*$ above which the solution does not change anymore with the wall elasticity and the flow behaves as in the presence of rigid walls. More interestingly, for $G<G^*$ the solution depends on the wall elasticity: the mean friction and the velocity fluctuations increase with the wall deformability and it is possible to maintain an unsteady chaotic turbulent-like flow in principle for any Reynolds number, i.e.,\ in conditions where a standard flow over rigid walls would be laminar, as long as the wall shear elastic modulus $G$ is properly reduced.
We show that, at low Reynolds number, the velocity fluctuations are mainly generated by the elastic wall, while the fluctuations close to the rigid wall rapidly vanish. As we reduce $Re_b$ to values of order $100$, we observe an increase of the velocity fluctuations due to strong wall oscillations, associated to an increase of the turbulent production $\mathcal{P}$. The power of the elastic wall is a dissipation term, approximately of the same order of the viscous dissipation, thus promoting the fragmentation of typical coherent structures and the consequent formation of small scale structures. Further reducing the bulk Reynolds number, $\mathcal{P}$ decreases as the Reynolds stresses decrease in the shear layer close to the elastic wall and remain strong only in the bulk of the channel where the mean shear is negligible. On the other hand, the power of the elastic wall changes sign and becomes a source of turbulence kinetic energy, mostly balanced by the viscous dissipation. At fixed shear elastic modulus, the flow eventually laminarises, which can be compensated by a reduction of $G$, which monotonically increases the fluctuations in the flow. Indeed, we found that the level of fluctuations scale approximately as $\sim \left( G/Re_b \right)^{-0.75}$. Thus, we can conclude that the chaotic flow at very low Reynolds numbers is mainly sustained by the elastic wall oscillations, which produce turbulent kinetic energy at the interface, then transferred to the fluid through viscous stresses; this process sustains non zero Reynolds stresses in the bulk of the channel.
The present results can have profound influence on the development of strategies to increase mixing in microfluidic devices by exploiting a dynamical instability associated to the coupling between the flow and an elastic wall.
\section*{Acknowledgments}
The authors acknowledge computer time provided by the Swedish National Infrastructure for Computing (SNIC ) and by the Scientific Computing section of Research Support Division at OIST. L.B. acknowledges financial support by the Swedish Research Council, VR 2016-06119, Hybrid multiscale modelling of transport phenomena for energy efficient processes.
\section*{Data Availability Statement}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
|
1,116,691,500,488 | arxiv | \section{Introduction}\label{sec:intro}
Wireless broadband systems are witnessing rapid growth in both the number of subscribers and the traffic volume per subscriber. More smartphone users are using web-based services such as video streaming, e.g. YouTube, and social networking, e.g. Facebook. These applications require broadband access due to multimedia-rich content. Given the limited available cellular spectrum and the aforementioned increasing demand, the need for an efficient resource allocation algorithm is of paramount importance for improving the quality of service (QoS).
The user applications running on smartphones can be divided into real-time applications, e.g. voice-over-IP (VoIP), video streaming, etc., and delay tolerant applications, e.g. application updates, emails, etc. The real-time applications are given allocation priority over the delay-tolerant applications due to latency constraints. Meanwhile, users running delay-tolerant application shouldn't be deprived from resources (i.e. no user in the network is dropped). Hence, we aim for an efficient content-aware resource allocation under the limited cellular spectrum that ensures priority to real-time applications without dropping users with delay-tolerant applications.
In addition, the cellular network traffic during peak-traffic hours can be ten times more that off-peak traffic hours \cite{pricing_survey}. This causes congestion problems that result in (a) lower QoS for subscribers and therefore wireless service providers (WSP)s have to (b) deploy more equipments and base stations to meet the increasing demand and therefore the cost of network operation increases leading to (c) higher pricing by WSP to users to cover the cost of operation. Therefore, there is a real need for a time-aware pricing model that overcome the congestion during peak traffic hours to solve the aforementioned issues.
This article aims to illustrate that context-aware resource allocation architecture with frequency reuse and its benefits to future cellular networks. The architecture considers users running real-time applications and users running delay-tolerant applications. The users running different content experience different resource allocation, due to the proposed content-aware resource allocation policy. The resource allocation optimization problem is formulated to ensure fair utility percentage allocated for active users with the available evolved-NodeB (eNodeB) spectrum resources. Therefore, the context-aware resource allocation algorithm gives priority to real-time application users over delay-tolerant application users. In addition, the optimization problem formulation guarantees that all users are assigned a fraction of the available spectrum, as the eNodeB should provide a minimum QoS for all the users subscribing for the mobile service. This allocation policy intrinsically provides time-aware pricing that
charges mobile users based on their usage time of day (i.e. peak and off-peak traffic hours).
\subsection{Related Work}\label{sec:related}
The area of resource allocation optimization has received significant interest since the seminal network utility maximization problem presented in \cite{kelly98ratecontrol}. The network utility maximization problem allocates the resources among users optimally based on bandwidth proportional fairness by using Lagrange multiplier methods of optimization theory. An iterative algorithm based on the dual problem has been proposed to solve the resource allocation optimization problem in \cite{Low99optimizationflow}. The applications considered in early research work, as in \cite{kelly98ratecontrol} and \cite{Low99optimizationflow}, are only delay-tolerant Internet traffic for wired communication networks. However, for current cellular networks both real-time and delay-tolerant applications are considered. The content-aware resource allocation architecture optimally allocates resources for these heterogeneous applications.
For earlier studies on the single carrier content-aware resource allocation, we refer to \cite{Ahmed_Utility1}. Time-aware resource pricing was introduced in \cite{Ahmed_Utility2} for single cell model. The location-aware resource allocation with carrier aggregation is studied in \cite{Ahmed_Utility4, Haya_Utility1}. The context-aware resource allocation with prioritization of mobile users based on their subscription is investigated in \cite{Ahmed_Utility3}. The context-aware resource allocation with guaranteed bit rate (GBR) to mobile users running specific services is discussed in \cite{Haya_Utility2}.
The article is organized as follows. In the next section, the context-aware resource allocation network architecture is presented along with the allocation policy. The following section provides the distribution of the context-aware resource allocation problem into cellular network entities subproblems. After that, the distributed optimal context-aware resource allocation algorithm running on various cellular network entities is presented. Then, the algorithm is investigated using simulations and the possible future extensions to the current architecture are discussed. The final section concludes the article.
\section{Context-Aware Resource Allocation Network Architecture}\label{sec:Problem_formulation}
In this section, we define the problem under investigation. We break it into utility functions, system model, resource allocation policy, and resource allocation optimization problem.
\textbf{Utility Functions:} The application utility function $U_i(r_i)$ represents the $i^{th}$ user satisfaction percentage with the allocated resources $r_i$. In our model, we assume the utility function $U_i(r_i)$ to be a strictly concave or a sigmoidal-like function. The strictly concave utility function corresponds to the delay-tolerant applications (e.g. FTP, Internet browsing, emails) and sigmoidal-like utility function corresponds to real-time applications (e.g. voice over IP and video streaming). These utility functions have the following properties: (a) $U_i(0) = 0$ and $U_i(r_i)$ is an increasing function of $r_{i}$, and (b) $U_i(r_{i})$ is twice continuously differentiable in $r_{i}$. In our model, we use the normalized sigmoidal-like utility function with $a$ as the rate of increase and $b$ as the inflection point of the function, similar to \cite{DL_PowerAllocation}, and the normalized logarithmic utility function with $k$ as the rate of increase, similar to \cite{UtilityFairness}. Some
examples of utility functions for different values of $a$, $b$ and $k$ are plotted in Figure \ref{fig:sim:Utilities}.
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{ \small
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\caption{An example of the utility functions $U_i(r_{i})$ (three sigmoidal-like functions and three logarithmic functions).}
\label{fig:sim:Utilities}
\end{figure}
\textbf{System Model:} We consider a cellular network that includes multiple cells where each cell is divided into sectors. The same frequency band is reused for the sector in same direction of different cells. In Figure \ref{fig:System_Model}, we show a diagram of the cellular network architecture consisting of $K$ eNodeBs in $K$ cells, where each cell is divided into $L$ sector (e.g. 3 sectors), and $M$ user equipments (UE)s distributed in these cells. The $i^{th}$ user rate $r_{i}^{l}$ is allocated by the $l^{th}$ sector of eNodeB, where $i = \{1,2, ...,M\}$, $l =\{1,2, ..., L\}$. Each UE has its own utility function $U_i(r_{i})$ that corresponds to the type of application running on it. Our objective is to solve for the optimal rates that eNodeB sectors should allocate to the UEs. The utility functions are given by $U_i(r_{i}^{1}+r_{i}^{2}+ ...+r_{i}^{L})$ where $\sum_{l=1}^{L}r_{i}^{l} = r_i$ and the rate vector is given by $\textbf{r} =\{{r}_1, {r}_2,..., {r}_M\}$.
\begin{figure}[]
\centering
\includegraphics[width=1.0\linewidth]{ieee_mag_model.eps}
\caption{Cellular network architecture with 3 cells and 3 sectors per cell. The cells' eNodeBs are connected to Mobility Management Entity (MME).}
\label{fig:System_Model}
\end{figure}
\textbf{Resource Allocation Policy:} The context-aware resource allocation policy is achieved by using utility proportional fairness policy, where the objective function is given by the product of utilities $\prod_{i=1}^{M}U_i$. This policy of resource allocation guarantees that the optimal users rates are allocated such that, (a) priority to real-time applications users (i.e. with sigmoidal-like utility functions) over delay-tolerant applications users (i.e. with logarithmic utility functions), (b) no user is dropped (i.e. minimum rate allocation is guaranteed and therefore minimum QoS).
\textbf{Resource Allocation Optimization Problem:} The basic formulation of the utility proportional fairness resource allocation problem is given by the following optimization problem:
\begin{equation}\label{eqn:opt_prob_fairness}
\begin{aligned}
& \underset{\textbf{r}}{\text{max}} & & \prod_{i=1}^{M}U_i(r_{i}^{1}+r_{i}^{2}+ ...+r_{i}^{L}) \\
& \text{subject to} & & \sum_{l=1}^{L}r_{i}^{l} = r_i , \;\; \sum_{i=1}^{M_k}r_{i}^{l} \leq R^{l}, \sum_{l=1}^{L} R^{l} = R, \\
& & & r_{i}^l \geq 0, \;\;\;\;\;l = 1,2, ...,L, i = 1,2, ...,M.
\end{aligned}
\end{equation}
where $R^{l}$ is the allocated rate by Mobility Management Entity (MME) to the $l^{th}$ sector of all cell, and $M_k$ is the number of users in $k^{th}$ cell. We assume that the cellular network reuses the same frequency band in similar sectors of different cells to avoid interference between cells (i.e. avoid co-channel interference). So we have the assumption that $R^{l}$ is the same for all cells and and $R$ is the sum of the allocated rates to all sectors in a cell. We assume that a UE can't co-exist in two sectors simultaneously.
\section{Context-Aware Global Optimal Solution}\label{sec:Proof}
In this section, we investigate the optimization problem and its dual. Then, we divide the optimization problem into simplified subproblems to be solved in different cellular network entities.
\textbf{Convex Optimization:} In optimization problem (\ref{eqn:opt_prob_fairness}), the solution for the objective function $\prod_{i=1}^{M}U_i$ is equivalent to the solution for the objective function $\sum_{i=1}^{M}\log U_i$. It is shown in \cite{Ahmed_Utility1, Ahmed_Utility4} that utility functions $U_i(.)$ that are logarithmic or sigmoidal-like functions have logarithms $\log U_i(.)$ that are concave functions. Therefore, the optimization problem in (\ref{eqn:opt_prob_fairness}) is convex. It follows that there exists a tractable (i.e can be solved in polynomial time) global optimal solution to the optimization problem in (\ref{eqn:opt_prob_fairness}).
\textbf{Dual Problem:} The key to a distributed and decentralized optimal solution of the primal problem in (\ref{eqn:opt_prob_fairness}) is to convert it to the dual problem using Lagrangian multiplier methods (more details on how to convert the primal problem to the dual problem can be found in \cite{Ahmed_Utility1, Ahmed_Utility4}). The dual problem of (\ref{eqn:opt_prob_fairness}) can be divided into three simpler subproblems that are solved in UEs, eNodeBs and MME in a bidding process.
\textbf{User Equipment Subproblem:} For the users in the $l^{th}$ sector of the eNodeB, the $i^{th}$ UE optimization problem is given by:
\begin{equation}\label{eqn:opt_prob_fairness_UE}
r_{i}^{l} = \arg \underset{r_{i}^{l}}\max \Big(\log(U_i(r_{i}^{1}+r_{i}^{2}+ ...+r_{i}^{L}))-\sum_{l=1}^{L}p_lr_{i}^l\Big)
\end{equation}
where $p_l$ is the shadow price (i.e. the price per resource) received from eNodeB $l^{th}$ sector. The shadow price received from eNodeB is computed by solving the eNodeB sector subproblem. In the bidding process, the $i^{th}$ UE maximizes the rate $r_i^{l}$ allocated to it by $l^{th}$ sector of eNodeB by solving the $i^{th}$ UE optimization problem (\ref{eqn:opt_prob_fairness_UE}). The UE bid $w_{i}^l$ corresponding to rate $r_{i}^{l}$ equals $p_l r_{i}^l$.
\textbf{eNodeB Sector Subproblem:} The second subproblem is the eNodeB $l^{th}$ sector optimization problem that minimizes the shadow price $p_l$. The minimization of $p_l$ is achieved by differentiating the Lagrangian of (\ref{eqn:opt_prob_fairness}) with respect to $p_l$ and setting the inequality constraints in optimization problem (\ref{eqn:opt_prob_fairness}) to equality constraints (i.e setting the slack variable to zero). Finally, we have the $l^{th}$ sector shadow price $p_l = \sfrac{\sum_{i=1}^{M}w_{i}^l}{R^l}$.
\textbf{MME Subproblem:} For achieving utility proportional fairness policy in context-aware resource allocation optimization problem (\ref{eqn:opt_prob_fairness}), MME performs fair sector total rate $R^{l}$ allocation by setting equal shadow price for all sectors (i.e. $p_l=p$ for all $l$). The aggregated bids from all users in the $l^{th}$ sectors of all eNodeBs is $W^l$ and is equal to $p R^l$. MME computes the sector total rate $R^{l}$ which is proportional to that sector aggregated bid $W^{l}$ to the sum of all sectors aggregated bids and is given by $R^l=\sfrac{W^lR}{\sum_{l=1}^{L}W^l}$ to guarantee utility proportional fairness policy.
\section{Context-Aware Resource Allocation Algorithm}\label{sec:Algorithm}
We translate the cellular entities subproblems into algorithms that run on the corresponding entities. A simplified version of UE, eNodeB and MME algorithms are shown in Algorithm \ref{alg:UE_FK}, \ref{alg:eNodeB_FK}, and \ref{alg:MME_FK}, respectively. The sequence diagram for the context-aware resource allocation algorithm is shown in Figure \ref{fig:sim:signal_flow}. It shows the interactions between UEs, eNodeBs and MME entities.
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\linewidth]{transmission_sequence.eps}
\caption{The sequence diagram of context-aware resource allocation algorithm that shows the interactions between the different entities of the cellular network.}
\label{fig:sim:signal_flow}
\end{figure}
\begin{algorithm}
\caption{The $i^{th}$ UE in $l^{th}$ sector Algorithm}\label{alg:UE_FK}
\begin{algorithmic}
\STATE {Send initial bid $w_i^{l}(1)$ to $l^{th}$ sector of eNodeB}
\LOOP
\STATE {Receive shadow price $p_{l}(n)$ from eNodeB}
\IF {STOP from eNodeB}
\STATE {STOP}
\ELSE
\STATE {Compute rate $r_{i}^{l}(n)$ using (\ref{eqn:opt_prob_fairness_UE}) and bid $w_i^{l}(n)$}
\STATE {Send new bid $w_i^{l}(n)$ to $l^{th}$ sector of eNodeB}
\ENDIF
\ENDLOOP
\end{algorithmic}
\end{algorithm}
In context-aware resource allocation algorithm, the $i^{th}$ UE starts with an initial bid $w_{i}^l(1)$ which is transmitted to the $l^{th}$ sector of $k^{th}$ eNodeB. The $l^{th}$ sector of $k^{th}$ eNodeB calculates the aggregated sector bid $W_{k}^l(n)$, where $n$ is the time index. The $k^{th}$ eNodeB sends the aggregated bids of all the sectors, $W_{k}^1(n), W_{k}^2(n), ..., W_{k}^L(n)$, to MME. MME adds the aggregated bids of similar sectors of different eNodeBs to evaluate the total aggregated sector bids $W^l(n)$. MME calculates the difference between the total sector aggregated bid $W^l(n)$ and the previously evaluated total sector aggregated bid $W^l(n-1)$ for all sectors and exits if it is less than a pre-specified threshold $\delta$ for all the sectors (i.e. exit criterion). MME evaluates the sector rates $R^l(n)$ for every sector and sends to all eNodeBs. The $l^{th}$ sector of $k^{th}$ eNodeB calculates the shadow price $p_l(n) = \sfrac{\sum_{i=1}^{M}w_{i}^l(n)}{R^l}$ and sends that value to
all
the UEs in its coverage area. The $i^{th}$ UE receives the shadow prices $p_{l}$ from corresponding sector to solve the optimization problem in (\ref{eqn:opt_prob_fairness_UE}) and send the new bid $w_{i}^l(n)$. The bidding process is repeated until $|W^l(n) -W^l(n-1)|$ is less than the threshold $\delta$.
\begin{algorithm}
\caption{The $l^{th}$ sector of eNodeB Algorithm}\label{alg:eNodeB_FK}
\begin{algorithmic}
\LOOP
\STATE {Receive bids $w_{i}^l(n)$ from UEs}
\STATE {Calculate aggregated bids $W_k^l(n)$ and send to MME}
\STATE {Receive sector rate $R^l(n)$ from MME}
\IF {STOP received from MME}
\STATE {STOP and send STOP to all UEs}
\ELSE
\STATE {Calculate $p_l(n)$ and send to all UEs}
\ENDIF
\ENDLOOP
\end{algorithmic}
\end{algorithm}
The context-aware resource allocation algorithm is set to avoid the situation of allocating zero rate to any user (i.e. no user is dropped). This is inherited from the utility proportional fairness policy in the optimization problem.
\begin{algorithm}
\caption{MME Algorithm}\label{alg:MME_FK}
\begin{algorithmic}
\STATE {Send sector rate $R^l(0)$ to $l^{th}$ sector}
\COMMENT{Let $R^l(0) = \frac{R}{L}$}
\LOOP
\STATE {Receive aggregated bids $W_k^l(n)$ from $l^{th}$ sector}
\STATE {Calculate total aggregated bids $W^l(n)$}
\COMMENT{Let $W^l(0) = 0\:\:\forall \:l$}
\IF {$|W^l(n) - W^l(n-1)|<\delta \:\:\forall l$}
\STATE {STOP and send STOP to all sectors}
\ELSE
\STATE {Calculate $R^l(n)$ and send to $l^{th}$ sector}
\ENDIF
\ENDLOOP
\end{algorithmic}
\end{algorithm}
\begin{figure}[tb]
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\caption{(a) Balanced users traffic in all sectors \{solid lines\}, (b) Unbalanced users traffic in sectors $R^l$ (when users A4, A5, A6, B4, B5, B6, C4, C5 and C6, in Figure \ref{fig:System_Model}, exit the cellular network) \{dashed lines\}, (c) Pricing scaled by 100 for the balanced traffic case \{black line\}.}
\label{fig:sim:shadow_price}
\end{figure}
\section{Experimental Results}\label{sec:sim}
In this section, we provide the performance of the context-aware resource allocation algorithm. Algorithm (\ref{alg:UE_FK}), (\ref{alg:eNodeB_FK}) and (\ref{alg:MME_FK}) were applied to various logarithmic and sigmoidal-like utility functions with different parameters in MATLAB. The simulation results showed convergence to the global optimal rates with the desired policy (a) priority to real-time users, and (b) no user is dropped. Figure \ref{fig:System_Model} illustrates the users deployment and indexes in the cellular network sectors used in the simulations. Table \ref{tab:utility} presents the users utilities used in the simulation. In the following simulations, we set $\delta =10^{-3}$ and the eNodeB rate $R$ takes values between 50 and 1150 with step of 5.
\begin {table}[t]
\caption {Users Utilities}
\label{tab:utility}
\begin{center}
\begin{tabular}{| l | l || l | l |}
\hline
\multicolumn{4}{|c|}{Sector 1 eNodeB A} \\ \hline
A1 & Sig $a=3,\:\: b=10.0$ & A4 & Log $k=1.1$ \\ \hline
A2 & Sig $a=3,\:\: b=10.3$ & A5 & Log $k=1.2$ \\ \hline
A3 & Sig $a=1,\:\: b=10.6$ & A6 & Log $k=1.3$ \\ \hline
\multicolumn{4}{|c|}{Sector 2 eNodeB A} \\ \hline
A7 & Sig $a=3,\:\: b=10$ & A10 & Log $k=1$ \\ \hline
A8 & Sig $a=3,\:\: b=11$ & A11 & Log $k=2$ \\ \hline
A9 & Sig $a=1,\:\: b=12$ & A12 & Log $k=3$ \\ \hline
\multicolumn{4}{|c|}{Sector 3 eNodeB A} \\ \hline
A13 & Sig $a=3,\:\: b=15.1$ & A16 & Log $k=10$ \\ \hline
A14 & Sig $a=3,\:\: b=15.3$ & A17 & Log $k=11$ \\ \hline
A15 & Sig $a=3,\:\: b=15.5$ & A18 & Log $k=12$ \\ \hline
\multicolumn{4}{|c|}{Sector 1 eNodeB B} \\ \hline
B1 & Sig $a=3,\:\: b=10.9$ & B4 & Log $k=1.4$ \\ \hline
B2 & Sig $a=3,\:\: b=11.2$ & B5 & Log $k=1.5$ \\ \hline
B3 & Sig $a=1,\:\: b=11.5$ & B6 & Log $k=1.6$ \\ \hline
\multicolumn{4}{|c|}{Sector 2 eNodeB B} \\ \hline
B7 & Sig $a=3,\:\: b=13$ & B10 & Log $k=4$ \\ \hline
B8 & Sig $a=3,\:\: b=14$ & B11 & Log $k=5$ \\ \hline
B9 & Sig $a=1,\:\: b=15$ & B12 & Log $k=6$ \\ \hline
\multicolumn{4}{|c|}{Sector 3 eNodeB B} \\ \hline
B13 & Sig $a=3,\:\: b=15.7$ & B16 & Log $k=13$ \\ \hline
B14 & Sig $a=3,\:\: b=15.9$ & B17 & Log $k=14$ \\ \hline
B15 & Sig $a=3,\:\: b=17.3$ & B18 & Log $k=15$ \\ \hline
\multicolumn{4}{|c|}{Sector 1 eNodeB C} \\ \hline
C1 & Sig $a=3,\:\: b=11.8$ & C4 & Log $k=1.7$ \\ \hline
C2 & Sig $a=3,\:\: b=12.1$ & C5 & Log $k=1.8$ \\ \hline
C3 & Sig $a=1,\:\: b=12.4$ & C6 & Log $k=1.9$ \\ \hline
\multicolumn{4}{|c|}{Sector 2 eNodeB C} \\ \hline
C7 & Sig $a=3,\:\: b=16$ & C10 & Log $k=7$ \\ \hline
C8 & Sig $a=3,\:\: b=17$ & C11 & Log $k=8$ \\ \hline
C9 & Sig $a=1,\:\: b=18$ & C12 & Log $k=9$ \\ \hline
\multicolumn{4}{|c|}{Sector 3 eNodeB C} \\ \hline
C13 & Sig $a=3,\:\: b=17.5$ & C16 & Log $k=16$ \\ \hline
C14 & Sig $a=3,\:\: b=17.7$ & C17 & Log $k=17$ \\ \hline
C15 & Sig $a=3,\:\: b=17.9$ & C18 & Log $k=18$ \\ \hline
\end{tabular}
\end{center}
\end {table}
\textbf{Allocated Rates:} In Figure \ref{fig:sim:rates}, we show the optimal rates of users in the $2^{nd}$ sector versus eNodeB rate $R$. The optimal resource allocation is content-aware. The users with real-time application (i.e. sigmoidal-like utilities) are allocated resources first. In real-time applications allocation, the user with the steepest utility function (largest $a$) is allocated first as shown in Figure \ref{fig:sim:rates}.
\textbf{Pricing:} In Figure \ref{fig:sim:shadow_price}, the shadow price $p$ (scaled by 100 for visibility) is plotted versus eNodeB rate $R$. The price per resource increases as the available resources for allocation in sectors are more scarce, small values of $R$, as the number of users is fixed. Similarly, if the available resources are fixed and the number of users or the traffic increases, the price per resource increases. Therefore, we have a traffic-dependent pricing. As the cellular traffic is dependent on the time of using the service (e.g. peak or off-peak traffic hours) then it is time-aware pricing which is intrinsic in the optimization problem solution. Taking advantage of the time-aware pricing, WSPs can flatten the traffic specially during peak hours by setting time-aware resource price. This will incentivize users to use the cellular network during off-peak traffic hours.
\textbf{Balanced Traffic in Sectors:} In this case, users applications rate requirements are very close or the utility parameters are approximately equal. As a result, the resources allocated by MME to different sectors in the cellular network are approximately equal. In Figure \ref{fig:sim:shadow_price}, we plot the allocated sector rates $R^{l}$ by MME versus the total eNodeB rate $R$.
\textbf{Unbalanced Traffic in Sectors:} In this case, the users in different sectors have different rate requirements. We assume in this scenario that users A4, A5, A6, B4, B5, B6, C4, C5 and C6, shown in Figure \ref{fig:System_Model} with utility parameters shown in Table \ref{tab:utility}, exit the cellular network. Therefore, the resources allocated by MME to different sectors in the cellular network are not equal. In Figure \ref{fig:sim:shadow_price}, the allocated sector rates $R^{l}$ by MME versus the total eNodeB rate $R$ are shown.
\section{Future Research Directions}\label{sec:sim}
The research directions for further improvement of the context-aware optimal resource allocation architecture are:
\textbf{Time-Aware Pricing:} The pricing results, presented in simulation section, show that solving for optimal rates provides the corresponding users the optimal price of resource (Figure \ref{fig:sim:shadow_price}) which is a time-aware price. An addition to our architecture that take advantage of the time-aware pricing is a pricing model that is available for users ahead of time \cite{pricing_survey}, e.g. day-ahead pricing. The day-ahead pricing is dependent on the history of data-usage in previous days and the expected data-usage in the next day. The user will have the option to restrict some of his applications, especially delay-tolerant ones, from running in certain time periods, especially during congested traffic hours. The advantages on the mobile user side are (a) lower resource price and (b) better QoS, and on the WSP side are (c) lower equipment deployment costs, (d) lower network operation expenses, and (d) providing a more competitive service in the wireless market.
\textbf{Location-Aware Carrier Aggregation:} In the presence of two cellular networks (e.g. two different LTE WSPs) or two heterogeneous wireless networks (e.g. WiFi and LTE) in the same cell or sector, then carrier aggregation can be performed. In this case, the UE chooses the WSP that provides the lowest price for resources in the bidding process (more details are in \cite{Ahmed_Utility4}). As the user approaches a WiFi AP, he switches from cellular network to minimize the price of resources (with the assumption that WiFi resource price is lower in this scenario) and at the same time the cellular network benefits from decreasing the traffic load especially during congested traffic hours. Hence, WSP can provide better QoS for its subscribers. So, it is a win-win situation.
\textbf{Content-Aware Discrete Resource Allocation:} An extension of this work is to include discrete resource allocation rather than the continuous one. In this case, the optimization problem is reformulated into a discrete optimization problem. That can be solved optimally using Lagrangian relaxation and branch and bound methods. This architecture modification provides better mapping to the current cellular network standards.
\section{Conclusion}\label{sec:conclude}
Context-aware resource allocation is a simple, optimal and efficient method for allocating resources in broadband wireless cellular networks. It provides the foundation for constructing a complete cellular network with all the features in the current standards e.g. carrier aggregation, frequency reuse, GBR, etc. The mathematical formulation of context-aware resource allocation optimization problem with frequency reuse is presented. The new architecture includes content-aware resource allocation and time-aware pricing and can be easily extended to include location-aware carrier aggregation. The description of the algorithms implemented in different cellular network entities and their sequence diagram are presented. The simulation results demonstrate the content-aware resource resource allocation and time-aware pricing. Hence, the cellular spectrum utilization is more efficient and the QoS to mobile users is improved.
\bibliographystyle{ieeetr}
|
1,116,691,500,489 | arxiv | \section{Introduction}
\label{sec:intro}
Machine learning with distributed databases has been a hot research area~\cite{li2014communication}.
The amount of data at each client can be large, and hence the data uploading to a central server may be constrained by energy and bandwidth limitations.
Besides, local data may contain highly sensitive information, e.g., travel records, health information and web browsing history, and thus a client may be unwilling to share it. Therefore, it is impossible or undesirable to upload distributed databases to a central server.
Recent years have witnessed the growing interest in federated learning, where data is maintained locally during the collaborative training of the server and clients~\cite{yang2019federated}. Data privacy and communication efficiency are the two main advantages of federated learning, as only model parameters or gradients are exchanged in the training process.
Most existing works for federated learning focus on solving unconstrained optimization problems using mini-batch stochastic gradient descent (SGD)~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel,yang2019federated,hardy2017private}. Depending on whether data is distributed over the sample space or feature space, federated learning can be typically classified into sample-based (horizontal) federated learning and feature-based (vertical) federated learning~\cite{yang2019federated}.
In sample-based federated learning, the datasets of different clients have the same feature space but little intersection on the sample space. Most studies on federated learning focus on this category~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel}. In the existing sample-based federated learning algorithms, the global model is iteratively updated at the server by aggregating and averaging the locally computed models at clients. Data privacy is naturally preserved as the model averaging steps avoid exposing raw data.
Specifically, at each communication round, the selected clients download the current model parameters and conduct one or multiple SGD updates to refine the local model. Multiple local SGD updates can reduce the required number of model averaging steps and hence save communication costs. However, they may yield the divergence of sample-based federated learning when local datasets across clients are heterogeneous. The most commonly used sample-based federated learning algorithm is the Federated Averaging algorithm~\cite{mcmahan2017communication}.
On the contrary, in feature-based federated learning, the datasets of different clients share the same sample space but differ in the feature space. Feature-based federated learning is more challenging, as a client cannot obtain the gradient of a loss function relying purely on its local data. In the existing feature-based federated learning algorithms~\cite{yang2019federated,hardy2017private}, intermediate parameters are exchanged for calculating the gradient before model aggregation steps.
SGD has long been used for solving unconstrained stochastic optimization problems or stochastic optimization problems with deterministic convex constraints. Recently, stochastic successive convex approximation (SSCA) is proposed to obtain Karush-Kuhn-Tucker (KKT) points of stochastic optimization problems with deterministic convex constraints~\cite{Yang} and with general stochastic nonconvex constraints~\cite{Liu,Ye}. Apparently, SSCA has a wider range of applications than SGD. It has also been shown in\cite{Yang} that SSCA empirically achieves a higher convergence speed than SGD, as SGD utilizes only first-order information of the objective function.
Some recent works have applied SSCA to solve machine learning problems~\cite{scardapane2018stochastic}. Nevertheless, SSCA has not been applied for solving federated optimization so far.
In this paper, we focus on designing sample-based federated learning algorithms using SSCA for unconstrained problems and constrained problems, respectively.
First, we propose a privacy preserving algorithm to obtain a KKT point of unconstrained sample-based federated optimization using mini-batch SSCA, and analyze its computational complexity and convergence. Such algorithm empirically converges faster (i.e., achieves a lower communication cost) than the SGD-based ones in~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel} and can achieve the same order of computational complexity as the SGD-based ones in~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel}.
Then, we propose a privacy preserving algorithm to obtain a KKT point of constrained sample-based federated optimization by combining the exact penalty method for SSCA in~\cite{Ye} and mini-batch techniques, and analyze its convergence.
Notice that federated optimization with nonconvex constraints, which can explicitly limit the cost function of a model, has not been investigated so far.
Next, we customize the two SSCA-based algorithms to two application examples, and show that all updates at each iteration have closed-form expressions.
Finally, numerical experiments demonstrate that the proposed algorithm for unconstrained sample-based federated optimization converges faster (i.e., yield lower communication costs) than the existing SGD-based ones~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel}, and the proposed algorithms for constrained federated optimization can more flexibly specify a training model.
\section{System Setting}
Consider $N$ data samples, each of which has $K$ features. For all $n\in\mathcal N\triangleq\{1,\dots,N\}$, the $K$ features of the $n$-th sample are represented by a $K$-dimensional vector $\mathbf{x}_n\in\mathbb{R}^{K}$.
Consider a central server connected with $I$ local clients, each of which maintains a local dataset.
Specifically, partition $\mathcal{N}$ into $K$ disjoint subsets, denoted by $\mathcal{N}_i$, $i\in\mathcal{I}\triangleq\{1,\dots,I\}$, where $N_i\triangleq|\mathcal{N}_i|$ denotes the cardinality of the $i$-th subset and $\sum_{i\in\mathcal{I}}N_i=N$.
For all $i\in\mathcal{I}$, the $i$-th client maintains a local dataset containing $N_i$ samples, i.e., $\mathbf{x}_n$, $n\in\mathcal{N}_i$.
For example, two companies with similar business in different cities may have different user groups (from their respective regions) but the same type of data, e.g., users' occupations, ages, incomes, deposits, etc.
The server and $I$ clients collaboratively train a model from the local datasets stored on the $I$ clients under the condition that each client cannot expose its local raw data to the others.
This training process is referred to as sample-based (horizontal) federated learning~\cite{yang2019federated}.
The underlying optimization, termed sample-based federated optimization~\cite{yang2019federated}, is to minimize the following function:
\begin{align}
&F_{0}(\boldsymbol\omega)\triangleq
\frac{1}{N}\sum_{n\in\mathcal{N}}
f_{0}(\boldsymbol\omega,\mathbf{x}_n)\label{eqn:Fs0}
\end{align}
with respect to (w.r.t.) model parameters $\boldsymbol\omega\in\mathbb{R}^d$.
To be general, we do not assume $F_{0}(\boldsymbol\omega)$ to be convex in $\boldsymbol\omega$.
In Section~\ref{sec:uncon} and Section~\ref{sec:con}, we investigate sample-based federated Learning for unconstrained optimization and constrained optimizaiton, respectively.
To guarantee the convergence of the proposed SSCA-based federated learning algorithms, we assume that $f_{0}\left(\boldsymbol\omega,\mathbf{x}_n\right)$ satisfies the following assumption in the rest of the paper.
\begin{Asump}[Assumption on $f(\boldsymbol\omega,\mathbf{x})$]\label{asump:f}\cite{Yang,Liu}
For any given $\mathbf{x}$, each $f(\boldsymbol\omega,\mathbf{x})$ is continuously differentiable, and its gradient is Lipschitz continuous.
\end{Asump}
\begin{Rem}[Discussion on Assumption~\ref{asump:f}]
Assumption~\ref{asump:f} is also necessary for the convergence of SSCA~\cite{Yang,Liu,Ye} and SGD~\cite{yu2019parallel}.
\end{Rem}
\section{Sample-based Federated Learning for Unconstrained Optimization}\label{sec:uncon}
In this section, we consider the following unconstrained sample-based federated optimization problem:
\begin{Prob}[Unconstrained Sample-based Federated Optimization]\label{Prob:uncon-sample}
\begin{align}
&\min_{\boldsymbol\omega} F_{0}(\boldsymbol\omega)\nonumber
\end{align}
where $F_{0}(\boldsymbol\omega)$ is given by~\eqref{eqn:Fs0}.
\end{Prob}
Problem~\ref{Prob:uncon-sample} (whose objective function has a large number of terms) is usually transformed to an equivalent stochastic optimization problem, and solved using stochastic optimization algorithms.
The SGD-based algorithms in~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel}, proposed to obtain a KKT point of Problem~\ref{Prob:uncon-sample}, may have unsatisfactory convergence speeds and high communication costs, as SGD only utilizes the first-order information of an objective function.
In the following, we propose a privacy-preserving sample-based federated learning algorithm, i.e., Algorithm~\ref{alg:uncon-sample}, to obtain a KKT point of Problem~\ref{Prob:uncon-sample} using mini-batch SSCA.
It has been shown in~\cite{Yang} that SSCA empirically achieves a higher convergence speed than SGD. Later in Section~\ref{sec:simu}, we shall numerically show that the proposed SSCA-based algorithm converges faster than the SGD-based algorithms in~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel}.
\begin{algorithm}[t]
\caption{Mini-batch SSCA for Problem~\ref{Prob:uncon-sample}}
\begin{algorithmic}[1]
\STATE \textbf{initialize}: choose any ${\boldsymbol\omega}^{1}$ at the server.\\
\FOR{$t=1,2,\dots,T-1$}
\STATE the server sends ${\boldsymbol\omega}^{(t)}$ to all clients.
\STATE for all $i\in\mathcal{I}$, client $i$ randomly selects a mini-batch $\mathcal{N}^{(t)}_i\subseteq\mathcal{N}_i$, computes $\mathbf q_{0}\left({\boldsymbol\omega}^{(t)},(\mathbf{x}_n)_{n\in\mathcal{N}^{(t)}_i}\right)$ and sends it to the server.
\STATE the server obtains $\bar{\boldsymbol\omega}^{(t)}$ by solving Problem~\ref{Prob:uncon-sample-ap}, and updates ${\boldsymbol\omega}^{(t+1)}$ according to \eqref{eqn:updatew}.
\ENDFOR
\STATE \textbf{Output}: ${\boldsymbol\omega}^T$
\end{algorithmic}\label{alg:uncon-sample}
\end{algorithm}
\subsection{Algorithm Description}
The main idea of Algorithm~\ref{alg:uncon-sample} is to solve a sequence of successively refined convex problems, each of which is obtained by approximating $F_{0}(\boldsymbol\omega)$ with a convex function based on its structure and samples in a randomly selected mini-batch by the server.
Specifically, at iteration $t$, we choose:
\begin{align}
\bar F^{(t)}_{0}(\boldsymbol\omega)=&(1-\rho^{(t)})\bar{F}^{(t-1)}_{0}(\boldsymbol\omega)\nonumber\\
&+\rho^{(t)}\sum_{i\in\mathcal{I}}\frac{N_i}{BN}\sum_{n\in\mathcal N_i^{(t)}}\bar{f}_{0}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n)\label{eqn:Fs0bar}
\end{align}
with $\bar F_{0}^{(0)}(\boldsymbol\omega)=0$ as an approximation function of $F_{0}(\boldsymbol\omega)$,
where $\rho^{(t)}$ is a stepsize satisfying:
\begin{align}
&\rho^{(t)}>0,\quad \lim_{t\to\infty}\rho^{(t)}=0,\quad \sum_{t=1}^\infty\rho^{(t)}=\infty,\label{eqn:rho}
\end{align}
$\mathcal N_i^{(t)}\subseteq\mathcal N_i$ is a randomly selected mini-batch by client $i$ at iteration $t$, $B\leq N_i$ is the batch size, and $\bar{f}_{0}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n)$ is a convex approximation of $f_{0}(\boldsymbol\omega,\mathbf{x}_n)$ around ${\boldsymbol\omega}^{(t)}$ satisfying the following assumptions. A common example of $\bar{f}_{0}$ will be given later.
\begin{Asump}[Assumptions on $\bar{f}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x})$ for Approximating $f(\boldsymbol\omega,\mathbf{x})$ Around $\boldsymbol\omega'$]\label{asump:fbar}~\cite{Liu}
1) $\nabla \bar{f}(\boldsymbol\omega,\boldsymbol\omega,\mathbf{x})=\nabla f(\boldsymbol\omega,\mathbf{x})$;
2) $\bar{f}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x})$ is strongly convex in $\boldsymbol\omega$;
3) $\bar{f}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x})$ is Lipschitz continuous in both $\boldsymbol\omega$ and $\boldsymbol\omega'$;
4) $\bar{f}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x})$, its derivative, and its second order derivative w.r.t. $\boldsymbol\omega$ are uniformly bounded.
\end{Asump}
Assumption~\ref{asump:fbar} is necessary for the convergence of SSCA~\cite{Yang,Liu}.
Note that for all $i\in\mathcal{I}$ and any mini-batch $\mathcal{N}'_i\subseteq\mathcal{N}_i$ with batch size $B\leq N_i$, $\sum_{n\in\mathcal{N}'_i}\bar{f}_{0}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x}_n)$ can be written as $\sum_{n\in\mathcal{N}'_i}\bar{f}_{0}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x}_n)
=p_{0}\left(\mathbf q_{0}\left(\boldsymbol\omega',(\mathbf{x}_n)_{n\in\mathcal{N}'_i}\right),\boldsymbol\omega\right)$ with $p_{0}:\mathbb{R}^{D_0+d}\to\mathbb{R}$ and $\mathbf q_{0}:\mathbb{R}^{BK+d}\to\mathbb{R}^{D_0}$.
Assume that the expressions of $\bar{f}_{0}$, $p_{0}$ and $\mathbf q_{0}$ are known to the server and $N$ clients. Each client $i\in\mathcal{I}$ computes $\mathbf q_{0}\left({\boldsymbol\omega}^{(t)},(\mathbf{x}_n)_{n\in\mathcal{N}^{(t)}_i}\right)$ and sends it to the server. Then, the server solves the following convex approximate problem to obtain $\bar{\boldsymbol\omega}^{(t)}$.
\begin{Prob}[Convex Approximate Problem of Problem~\ref{Prob:uncon-sample}]\label{Prob:uncon-sample-ap}
\begin{align}
&\bar{\boldsymbol\omega}^{(t)}\triangleq\mathop{\arg\min}_{\boldsymbol\omega} \bar F_{0}^{(t)}(\boldsymbol\omega)\nonumber
\end{align}
\end{Prob}
Problem~\ref{Prob:uncon-sample-ap} is convex and can be solved with conventional convex optimization techniques.
Given $\bar{\boldsymbol\omega}^{(t)}$, the server updates ${\boldsymbol\omega}^{(t)}$ according to:
\begin{align}
&{\boldsymbol\omega}^{(t+1)}=(1-\gamma^{(t)}){\boldsymbol\omega}^{(t)}+\gamma^{(t)}\bar{\boldsymbol\omega}^{(t)},\ t=1,2,\dots \label{eqn:updatew}
\end{align}
where $\gamma^{(t)}$ is a stepsize satisfying:
\begin{align}
&\gamma^{(t)}=0,\ \lim_{t\to\infty}\gamma^{(t)}=0,\ \sum_{t=1}^\infty\gamma^{(t)}=\infty,\nonumber\\
&\sum_{t=1}^\infty\left(\gamma^{(t)}\right)^2<\infty,\quad \lim_{t\to\infty}\frac{\gamma^{(t)}}{\rho^{(t)}}=0.\label{eqn:gamma}
\end{align}
The detailed procedure is summarized in Algorithm~\ref{alg:uncon-sample}, and the convergence of Algorithm~\ref{alg:uncon-sample} is summarized below.
\begin{Thm}[Convergence of Algorithm~\ref{alg:uncon-sample}]\label{thm:uncon-sample}
Suppose that $f_{0}$ satisfies Assumption~\ref{asump:f}, $\bar{f}_{0}$ satisfies Assumption~\ref{asump:fbar}, and the sequence $\{{\boldsymbol\omega}^{(t)}\}$ generated by Algorithm~\ref{alg:uncon-sample} is bounded. Then, every limit point of $\{{\boldsymbol\omega}^{(t)}\}$ is a KKT point of Problem~\ref{Prob:uncon-sample} almost surely.
\end{Thm}
\begin{IEEEproof}[Proof (Sketch)]
It follows from~\cite[Lemma1]{Lemma} that $\lim_{t\to\infty}\Vert\nabla\bar{F}_0^t(\boldsymbol\omega^t)$
$-\nabla F_0(\boldsymbol\omega^t)\Vert=0$. Then, the convergence of Algorithm~\ref{alg:uncon-sample} can be obtained by generalizing the analysis in \cite[Theorem~1]{Yang}.
\end{IEEEproof}
\subsection{Security Analysis}
We establish the security of Algorithm~\ref{alg:uncon-sample}.
If for all $i\in\mathcal{I}$ and any mini-batch $\mathcal{N}'_i\subseteq\mathcal{N}_i$ with batch size $B\leq N_i$, the system of equations w.r.t. $\mathbf{z}\in\mathbb{R}^{BK}$, i.e., $\mathbf q_{0}\left(\boldsymbol\omega',\mathbf{z}\right)=\mathbf q_{0}\left(\boldsymbol\omega',(\mathbf{x}_n)_{n\in\mathcal{N}'_i}\right)$, has an infinite (or a sufficiently large) number of solutions, then raw data $\mathbf{x}_n$, $n\in\mathcal{N}^{(t)}_i$ cannot be extracted from $\mathbf q_{0}\left({\boldsymbol\omega}^{(t)},(\mathbf{x}_n)_{n\in\mathcal{N}^{(t)}_i}\right)$ in Step 4 of Algorithm~\ref{alg:uncon-sample}, and hence Algorithm~\ref{alg:uncon-sample} can preserve data privacy.
Otherwise, extra privacy mechanisms, such as homomorphic encryption and secret sharing, can be applied to preserve data privacy.
\subsection{Algorithm Example}
Finally, we provide an example of $\bar{f}_{0}$ which satisfies Assumption~\ref{asump:fbar} and yields an analytical solution of Problem~\ref{Prob:uncon-sample}:
\begin{align}
\bar{f}_{0}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)}\!,\mathbf{x}_n)\!=\!&\left(\nabla f_{0}({\boldsymbol\omega}^{(t)}\!,\mathbf{x}_n)\right)^T\!\!\left(\boldsymbol\omega\!-\!{\boldsymbol\omega}^{(t)}\!\right)\!+\!\tau\Vert{\boldsymbol\omega\!-\!{\boldsymbol\omega}^{(t)}}\!\Vert_2^2, \label{eqn:fs0bar}
\end{align}
where $\tau>0$ can be any constant, and the term $\tau\Vert{\boldsymbol\omega-{\boldsymbol\omega}^{(t)}}\Vert_2^2$ is used to ensure strong convexity.
Obviously, $\bar{f}_{0}$ given by~\eqref{eqn:fs0bar} satisfies Assumption~\ref{asump:fbar}.
Notice that Problem~\ref{Prob:uncon-sample-ap} with $\bar{f}_{0}$ given by~\eqref{eqn:fs0bar} is an unconstrained convex quadratic programming w.r.t. $\boldsymbol\omega$ and hence has an analytical solution with the same order of computational complexity as the SGD-based ones in~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel} (which is $\mathcal O(d)$). The details of the analytical solution will be given in Section~\ref{sec:application}.
\section{Sample-based Federated Learning for Constrained Optimization}\label{sec:con}
In this section, we consider the following constrained sample-based federated optimization problem:
\begin{Prob}[Constrained Sample-based Federated Optimization]\label{Prob:con-sample}
\begin{align}
\min_{\boldsymbol\omega}\ &F_{0}(\boldsymbol\omega)\nonumber\\
\text{s.t.}\ &F_{m}(\boldsymbol\omega)\leq 0,\quad m=1,2,\dots,M,\nonumber
\end{align}
where $F_{0}(\boldsymbol\omega)$ is given by~\eqref{eqn:Fs0},
and
\begin{align}
F_{m}(\boldsymbol\omega)\triangleq\frac{1}{N}\sum_{n\in\mathcal{N}} f_{m}(\boldsymbol\omega,\mathbf{x}_n),\quad m=1,2,\dots,M.\nonumber
\end{align}
\end{Prob}
To be general, $F_{m}(\boldsymbol\omega)$, $m=0,\dots,M$ are not assumed to be convex in $\boldsymbol\omega$. Notice that federated optimization with nonconvex constraints has not been investigated so far. It is quite challenging, as the stochastic nature of a constraint function may cause infeasibility at each iteration of an ordinary stochastic iterative method~\cite{Ye}.
In the following, we propose a privacy-preserving sample-based federated learning algorithm, i.e., Algorithm~\ref{alg:con-sample}, to obtain a KKT point of Problem~\ref{Prob:con-sample}, by combining the exact penalty method~\cite{bertsekas1998nonlinear} for SSCA in~\cite{Ye} and mini-batch techniques.
\subsection{Algorithm Description}
\begin{algorithm}[t]
\caption{Mini-batch SSCA for Problem~\ref{Prob:con-sample}}
\begin{algorithmic}[1]
\STATE \textbf{initialize}: choose any ${\boldsymbol\omega}^{1}$ and $c>0$ at the server.\\
\FOR{$t=1,2,\dots,T-1$}
\STATE the server sends ${\boldsymbol\omega}^{(t)}$ to all clients.
\STATE for all $i\in\mathcal{I}$, client $i$ randomly selects a mini-batch $\mathcal{N}^{(t)}_i\subseteq\mathcal{N}_i$, computes $\mathbf q_{m}\left({\boldsymbol\omega}^{(t)},(\mathbf{x}_n)_{n\in\mathcal{N}^{(t)}_i}\right)$, $m=0,1,\dots,M$ and sends them to the server.
\STATE the server obtains $(\bar{\boldsymbol\omega}^{(t)}, \mathbf s^{(t)})$ by solving Problem~\ref{Prob:con-sample-ap}, and updates ${\boldsymbol\omega}^{(t+1)}$ according to \eqref{eqn:updatew}.
\ENDFOR
\STATE \textbf{Output}: ${\boldsymbol\omega}^T$
\end{algorithmic}\label{alg:con-sample}
\end{algorithm}
First, we transform Problem~\ref{Prob:con-sample} to the following stochastic optimization problem whose objective function is the weighted sum of the original objective and the penalty for violating the original constraints.
\begin{Prob}[Transformed Problem of Problem~\ref{Prob:con-sample}]\label{Prob:con-sample-ep}
\begin{align}
\min_{\boldsymbol\omega,\mathbf s}\quad &F_{0}(\boldsymbol\omega)+c\sum_{m=1}^M s_m\nonumber\\
\text{s.t.}\quad &F_{m}(\boldsymbol\omega)\leq s_m,\quad m=1,2,\dots,M,\nonumber\\
&s_m\geq 0,\quad m=1,2,\dots,M,\nonumber
\end{align}
where $\mathbf{s}\triangleq(s_m)_{m=1,\dots,M}$ are slack variables and $c>0$ is a penalty parameter that trades off the original objective function and the slack penalty term.
\end{Prob}
At iteration $t$, we choose $\bar F^{(t)}_{0}(\boldsymbol\omega)$ given in~\eqref{eqn:Fs0bar} as an approximation function of $F_{0}(\boldsymbol\omega)$, and choose:
\begin{align}
\bar F^{(t)}_{m}(\boldsymbol\omega)=&(1-\rho^{(t)})\bar{F}^{(t-1)}_{m}(\boldsymbol\omega)+\rho^{(t)}
\sum_{i\in\mathcal{I}}\frac{N_i}{BN}\nonumber\\
&\times\sum_{n\in\mathcal N_i^{(t)}}\bar{f}_{m}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n),\quad m=1,\dots,M\label{eqn:Fsmbar}
\end{align}
with $\bar F_{m}^{(0)}(\boldsymbol\omega)=0$ as an approximation function of $F_{m}(\boldsymbol\omega)$, for all $m=1,\dots,M$, where $\rho^{(t)}$ is a stepsize satisfying~\eqref{eqn:rho},
$\mathcal N^{(t)}_i$ is the randomly selected mini-batch by client $i$ at iteration $t$,
and $\bar{f}_{m}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n)$ is a convex approximation of $f_{m}(\boldsymbol\omega,\mathbf{x}_n)$ around ${\boldsymbol\omega}^{(t)}$ satisfying $\bar{f}_{m}(\boldsymbol\omega,\boldsymbol\omega,\mathbf{x})=f_{m}(\boldsymbol\omega,\mathbf{x})$ and Assumption~\ref{asump:fbar} for all $m=1,\dots,M$.
A common example of $\bar{f}_{m}$, $m=0,\dots,M$ will be given later.
Note that for all $i\in\mathcal{I}$ and any mini-batch $\mathcal{N}'_i\subseteq\mathcal{N}_i$ with batch size $B\leq N_i$, $\sum_{n\in\mathcal{N}'_i}\bar{f}_{m}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x}_n)$, $m=0,\dots,M$ can be written as $\sum_{n\in\mathcal{N}'_i}\bar{f}_{m}(\boldsymbol\omega,\boldsymbol\omega',\mathbf{x}_n)
=p_{m}\left(\mathbf q_{m}\left(\boldsymbol\omega',(\mathbf{x}_n)_{n\in\mathcal{N}'_i}\right),\boldsymbol\omega\right)$, $m=0,\dots,M$, with $p_{m}:\mathbb{R}^{D_m+d}\to\mathbb{R}$ and $\mathbf q_{m}:\mathbb{R}^{BK+d}\to\mathbb{R}^{D_m}$.
Assume that the expressions of $\bar{f}_{m}$, $p_{m}$ and $\mathbf q_{m}$, $m=0,\dots,M$ are known to the server and $N$ clients. Each client $i\in\mathcal{I}$ computes $\mathbf q_{m}\left({\boldsymbol\omega}^{(t)},(\mathbf{x}_n)_{n\in\mathcal{N}^{(t)}_i}\right)$, $m=0,\dots,M$ and send them to the server. Then, the server solves the following approximate problem to obtain $\bar{\boldsymbol\omega}^{(t)}$.
\begin{Prob}[Convex Approximate Problem of Problem~\ref{Prob:con-sample-ep}]\label{Prob:con-sample-ap}
\begin{align}
(\bar{\boldsymbol\omega}^{(t)},\mathbf{s}^{(t)})\triangleq&\mathop{\arg\min}_{\boldsymbol\omega,\mathbf{s}} \bar F^{(t)}_{0}(\boldsymbol\omega)+c\sum_{m=1}^M s_m\nonumber\\
\text{s.t.}\quad &\bar F^{(t)}_{m}(\boldsymbol\omega)\leq s_m,\quad m=1,2,\dots,M,\nonumber\\
&s_m\geq 0,\quad m=1,2,\dots,M.\nonumber
\end{align}
\end{Prob}
Problem~\ref{Prob:con-sample-ap} is convex and can be readily solved.
Given $\bar{\boldsymbol\omega}^{(t)}$, the server updates ${\boldsymbol\omega}^{(t)}$ according to \eqref{eqn:updatew}.
The detailed procedure is summarized in Algorithm~\ref{alg:con-sample}, and the convergence of Algorithm~\ref{alg:con-sample} is summarized below. Consider a sequence $\{c_j\}$. For all $j$, let $({\boldsymbol\omega}_{j}^\star,\mathbf s_{j}^\star)$ denote a limit point of $\{({\boldsymbol\omega}^{(t)},\mathbf{s}^{(t)})\}$ generated by Algorithm~\ref{alg:con-sample} with $c=c_j$.
\begin{Thm}[Convergence of Algorithm~\ref{alg:con-sample}]\label{thm:con-sample}
Suppose that $f_{m}$, $m=0,\dots,M$ satisfy Assumption~\ref{asump:f}, $\bar{f}_{0}$ satisfies Assumption~\ref{asump:fbar}, $\bar{f}_{m}$ satisfies $\bar{f}_{m}(\boldsymbol\omega,\boldsymbol\omega,\mathbf{x})=f_{m}(\boldsymbol\omega,\mathbf{x})$ and Assumption~\ref{asump:fbar} for all $m=1,\dots,M$, the constraint set of Problem~\ref{Prob:con-sample} is compact, and the sequence $\{c_j\}$ satisfies $0<c_j<c_{j+1}$ and $\lim_{j\to\infty}c_j=\infty$. Then, the following statements hold.
i) For all $j$, if $\mathbf s_{j}^\star=\mathbf0$, then ${\boldsymbol\omega}_{j}^\star$ is a KKT point of Problem~\ref{Prob:con-sample} almost surely;
ii) A limit point of $\{({\boldsymbol\omega}_{j}^\star,\mathbf s_{j}^\star)\}$, denoted by $\{({\boldsymbol\omega}_{\infty}^\star,\mathbf s_{\infty}^\star)\}$, satisfies that $\mathbf s_{\infty}^\star=\mathbf{0}$, and ${\boldsymbol\omega}_{\infty}^\star$ is a KKT point of Problem~\ref{Prob:con-sample} almost surely.
\end{Thm}
\begin{IEEEproof}
It follows from~\cite[Lemma1]{Lemma} that $\lim_{t\to\infty}\vert\bar{F}_{m}^t(\boldsymbol\omega^t)-
F_{m}(\boldsymbol\omega^t)\Vert=0$ and $\lim_{t\to\infty} \Vert\nabla\bar{F}_{m}^t(\boldsymbol\omega^t)$ $-\nabla F_{m}(\boldsymbol\omega^t)\Vert=0$. Then, we can show the first statement by generalizing the analysis in \cite[Theorem~1]{Liu} and \cite[Theorem~2]{Ye}. Moreover, we can show the second statement by generalizing the proof of~\cite[Proposition~4.4.1]{bertsekas1998nonlinear}.
\end{IEEEproof}
In practice, we can choose a sequence $\{c_j\}$ which satisfies that $0<c_j<c_{j+1}$, $\lim_{j\to\infty}c_j=\infty$ and $c_1$ is large, and repeat Algorithm~\ref{alg:con-sample} with $c=c_j$ until $\Vert\mathbf s_{j}^\star\Vert$ is sufficiently small.
\subsection{Security Analysis}
We establish the security of Algorithm~\ref{alg:con-sample}. If for all $i\in\mathcal{I}$ and any mini-batch $\mathcal{N}'_i\subseteq\mathcal{N}_i$ with batch size $B\leq N_i$, the system of equations w.r.t. $\mathbf{z}\in\mathbb{R}^{BK}$, i.e., $\mathbf q_{m}\left(\boldsymbol\omega',\mathbf{z}\right)=\mathbf q_{m}\left(\boldsymbol\omega',(\mathbf{x}_n)_{n\in\mathcal{N}'_i}\right)$, $m=0,\dots,M$ has an infinite (or a sufficiently large) number of solutions, then raw data $\mathbf{x}_n$, $n\in\mathcal{N}^{(t)}_i$ cannot be extracted from $\mathbf q_{m}\left({\boldsymbol\omega}^{(t)},(\mathbf{x}_n)_{n\in\mathcal{N}^{(t)}_i}\right)$, $m=0,\dots,M$ in Step 4 of Algorithm~\ref{alg:con-sample}, and hence Algorithm~\ref{alg:con-sample} can preserve data privacy.
Otherwise, extra privacy mechanisms need to be explored. Note that federated learning with constrained optimization has not been studied so far, let alone the privacy mechanisms.
\subsection{Algorithm Example}
We provide an example of $\bar{f}_{m}$, $m=0,\dots,M$ with $\bar{f}_{0}$ satisfying Assumption~\ref{asump:fbar} and $\bar{f}_{m}$ satisfying $\bar{f}_{m}(\boldsymbol\omega,\boldsymbol\omega,\mathbf{x})=f_{m}(\boldsymbol\omega,\mathbf{x})$ and Assumption~\ref{asump:fbar} for all $m=1,\dots,M$.
Specifically, we can choose $\bar{f}_{0}$ given by \eqref{eqn:fs0bar} and choose $\bar{f}_{m}$, $m=1,\dots,M$ as follows:
\begin{align}
\bar{f}_{m}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)}\!,\mathbf{x}_n)\!=&f_{m}({\boldsymbol\omega}^{(t)}\!,\mathbf{x}_n)\!+\!\left(\nabla f_{m}({\boldsymbol\omega}^{(t)}\!,\mathbf{x}_n)\!\right)^T\!\!\left(\!\boldsymbol\omega\!-\!{\boldsymbol\omega}^{(t)}\!\right)\nonumber\\
&+\tau\Vert{\boldsymbol\omega-{\boldsymbol\omega}^{(t)}}\Vert_2^2,\quad m=1,\dots,M, \label{eqn:fsmbar}
\end{align}
where $\tau>0$ can be any constant.
Obviously, $\bar{f}_{0}$ given by \eqref{eqn:fs0bar} satisfies Assumption~\ref{asump:fbar}, and $\bar{f}_{m}$ given by~\eqref{eqn:fsmbar} satisfies $\bar{f}_{m}(\boldsymbol\omega,\boldsymbol\omega,\mathbf{x})=f_{m}(\boldsymbol\omega,\mathbf{x})$ and Assumption~\ref{asump:fbar} for all $m=1,\dots,M$.
Note that Problem~\ref{Prob:con-sample-ap} with $\bar{f}_{0}$ given by~\eqref{eqn:fs0bar} and $\bar{f}_{m}$, $m=1,\dots,M$ given by~\eqref{eqn:fsmbar} is a convex quadratically constrained quadratic programming, and can be solved using an interior point method.
\section{Application Examples}\label{sec:application}
In this section, we customize the proposed algorithmic frameworks to some applications and provide detailed solutions for the specific problems.
Define $\mathcal{K}\triangleq\{1,\dots,K\}$, $\mathcal{J}\triangleq\{1,\dots,J\}$ and $\mathcal{L}\triangleq\{1,\dots,L\}$.
Consider an $L$-class classification problem with a dataset of $N$ samples $(\mathbf{x}_n, \mathbf y_n)_{n\in\mathcal N}$, where $\mathbf{x}_n\triangleq(x_{n,k})_{k\in\mathcal{K}}$ and $\mathbf y_n\triangleq(y_{n,l})_{l\in\mathcal{L}}$ with $x_{n,k}\in\mathbb{R}$, and $y_{n,l}\in\{0,1\}$. Consider a three-layer neural network, including an input layer composed of $K$ cells, a hidden layer composed of $J$ cells, and an output layer composed of $L$ cells.
We use the swish activation function $S(z)={z}/{(1+\exp(-z))}$~\cite{swish} for the hidden layer and the softmax activation function for the output layer.
We consider the cross entropy loss function. Thus, the resulting cost functions for sample-based and feature-based federated learning are given by:
\begin{align}
F(\boldsymbol\omega)\triangleq
-\frac{1}{N}\sum\limits_{n\in\mathcal{N}}\sum\limits_{l\in\mathcal{L}} y_{n,l}\log\left(Q_l(\boldsymbol\omega,\mathbf{x}_n)\right),
\label{eqn:Fcost}
\end{align}
with $\boldsymbol\omega\triangleq({\omega}_{1,j,k},{\omega}_{2,l,j})_{k\in\mathcal{K},j\in\mathcal{J},l\in\mathcal{L}}$ and
\begin{align}
&Q_l(\boldsymbol\omega,\mathbf{x}_n)\triangleq\frac{\exp(\sum_{j\in\mathcal{J}}{\omega}_{2,l,j} S(\sum_{k\in\mathcal{K}}{\omega}_{1,j,k}x_{n,k}))}{\sum_{h=1}^L \exp(\sum_{j\in\mathcal{J}}{\omega}_{2,h,j} S(\sum_{k\in\mathcal{K}}{\omega}_{1,j,k}x_{n,k}))},\nonumber\\
&\hspace{6cm} l\in\mathcal{L}.\label{eqn:Ql}
\end{align}
\subsection{Unconstrained Federated Optimization}\label{sec:classification}
One unconstrained federated optimization formulation for the $L$-class classification problem is to minimize the weighted sum of the cost function $F(\boldsymbol\omega)$ in~\eqref{eqn:Fcost} together with the $\ell_2$-norm regularization term $\Vert\boldsymbol\omega\Vert^2_2$:
\begin{align}
\min_{\boldsymbol\omega}\quad&F_{0}(\boldsymbol\omega)\triangleq F(\boldsymbol\omega)+\lambda\Vert\boldsymbol\omega\Vert^2_2\label{prob:class-uncon}
\end{align}
where $\lambda>0$ is the regularization parameter that trades off the cost and model sparsity.
We can apply Algorithm~\ref{alg:uncon-sample} with $\bar{f}_{0}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n)$ given by \eqref{eqn:fs0bar} to solve the problem in~\eqref{prob:class-uncon}. Theorem~\ref{thm:uncon-sample} guarantees the convergence of Algorithm~\ref{alg:uncon-sample}, as Assumption~\ref{asump:f} and Assumption~\ref{asump:fbar} are satisfied.
Specifically, the server solves the following convex approximate problem:
\begin{align}
\min_{\boldsymbol\omega}\quad&\bar F_{0}^{(t)}(\boldsymbol\omega)=\bar F^{(t)}(\boldsymbol\omega)+2\lambda(\boldsymbol\beta^{(t)})^T\boldsymbol\omega\label{prob:class-uncon-ap}
\end{align}
where
$\bar F^{(t)}(\boldsymbol\omega)$ is given by
\begin{align}
\bar F^{(t)}(\boldsymbol\omega)=\sum_{j\in\mathcal{J}}\sum_{k\in\mathcal{K}}{B_{j,k}^{(t)}}{\omega}_{1,j,k}
+\sum_{l\in\mathcal{L}}\sum_{j\in\mathcal{J}} {C_{l,j}^{(t)}}{\omega}_{2,l,j}
+\tau\Vert{\boldsymbol\omega}\Vert_2^2,\label{eqn:fbar-app}
\end{align}
and $\boldsymbol\beta^{(t)}\in\mathbb{R}^d$, ${B_{j,k}^{(t)}}$ and ${C_{l,j}^{(t)}}$ are updated according to:
\begin{align}
&\boldsymbol\beta^{(t)}=(1-\rho^{(t)})\boldsymbol\beta^{(t-1)}+\rho^{(t)}{\boldsymbol\omega}^{(t)},\nonumber\\
&{B_{j,k}^{(t)}}\!=\!(1-\rho^{(t)}){B_{j,k}^{(t-1)}}\!+\rho^{(t)}\!\left(\bar{B}_{j,k}^{(t)}\!-2\tau\omega^{(t)}_{1,j,k}\right),\label{eqn:B}\\
&{C_{l,j}^{(t)}}=(1-\rho^{(t)}){C_{l,j}^{(t-1)}}+\rho^{(t)}\left(\bar{C}_{l,j}^{(t)}-2\tau\omega^{(t)}_{2,l,j}\right),\label{eqn:C}
\end{align}
respectively, with ${\boldsymbol{\beta}}^{(0)}=\mathbf 0$ and ${B_{j,k}^{(0)}}={C_{l,j}^{(0)}}=0$.
Here, $\bar{B}_{j,k}^{(t)}$ and $\bar{C}_{l,j}^{(t)}$ are given by:
\begin{align}
\bar{B}_{j,k}^{(t)}\!=&
\sum_{i\in\mathcal{I}}\frac{N_i}{BN}\sum_{n\in\mathcal N_i^{(t)}}\sum_{l\in\mathcal{L}}\left(Q_l({\boldsymbol\omega}^{(t)},\mathbf{x}_n)-y_{n,l}\right)\nonumber\\
&\times S'\left(\sum_{k'=1}^K\omega^{(t)}_{1,j,k'}x_{n,k'}\right)\omega^{(t)}_{2,l,j}x_{n,k},\nonumber\\
\bar{C}_{l,j}^{(t)}\!=&
\sum_{i\in\mathcal{I}}\!\frac{N_i}{BN}\!\!\!\!\sum_{n\in\mathcal N_i^{(t)}}\!\!\!\left(\!Q_l({\boldsymbol\omega}^{(t)}\!,\mathbf{x}_n)\!-\!y_{n,l}\right)\!S\!\left(\sum_{k'=1}^K\!\omega^{(t)}_{1,j,k'}x_{n,k'}\!\!\right)\!,\nonumber
\end{align}
By the first-order optimality condition, the closed-form solution of the problem in~\eqref{prob:class-uncon-ap} is given by:
\begin{align}
&\bar{\omega}_{1,j,k}^{(t)}=-\frac{1}{2\tau}\left({B_{j,k}^{(t)}}+2\lambda{\beta}_{1,j,k}^{(t)}\right),\ j\in\mathcal{J},\ k\in\mathcal{K},\label{eqn:omega1-uncon}\\
&\bar{\omega}_{2,l,j}^{(t)}=-\frac{1}{2\tau}\left({C_{l,j}^{(t)}}+2\lambda{\beta}_{2,l,j}^{(t)}\right),\ l\in\mathcal{L},\ j\in\mathcal{J}.\label{eqn:omega2-uncon}
\end{align}
Thus, in Step 5 in Algorithm~\ref{alg:uncon-sample}, the server only needs to compute $\boldsymbol\omega$ according to~\eqref{eqn:omega1-uncon} and~\eqref{eqn:omega2-uncon}, respectively.
\subsection{Constrained Federated Optimization}
One constrained federated optimization formulation for the $L$-class classification problem is to minimize the $\ell_2$-norm of the network parameters $\Vert\boldsymbol\omega\Vert^2_2$ under a constraint on the cost function $F(\boldsymbol\omega)$ in~\eqref{eqn:Fcost}:
\begin{align}
\min_{\boldsymbol\omega}\quad& F_{0}(\boldsymbol\omega)\triangleq\Vert\boldsymbol\omega\Vert^2_2\label{prob:class-con}\\
\text{s.t.}\quad &F_{1}(\boldsymbol\omega)\triangleq F(\boldsymbol\omega)-U\leq0,\nonumber
\end{align}
where $U$ represents the limit on the cost.
We can apply Algorithm~\ref{alg:con-sample} with $\bar{f}_{0}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n)$ given by \eqref{eqn:fs0bar} and $\bar{f}_{m}(\boldsymbol\omega,{\boldsymbol\omega}^{(t)},\mathbf{x}_n)$ given by \eqref{eqn:fsmbar} to solve the problem in~\eqref{prob:class-con}. The convergence of Algorithm~\ref{alg:con-sample} is guaranteed by Theorem~\ref{thm:con-sample}, as Assumption~\ref{asump:f} and Assumption~\ref{asump:fbar} are satisfied.
Specifically, the server solves the following convex approximate problem:
\begin{align}
\min_{\boldsymbol\omega,s}\quad&\Vert\boldsymbol\omega\Vert^2_2+c s\label{prob:class-con-ap}\\
\text{s.t.}\quad &\bar F^{(t)}(\boldsymbol\omega)+A^{(t)}-U\leq s,\nonumber\\
&s\geq0, \nonumber
\end{align}
where $\bar F^{(t)}(\boldsymbol\omega)$ is given by~\eqref{eqn:fbar-app} with ${B_{j,k}^{(t)}}$, ${C_{l,j}^{(t)}}$ and $A^{(t)}$ updated according to~\eqref{eqn:B}, \eqref{eqn:C} and
\begin{align}
&A^{(t)}=(1-\rho^{(t)})A^{(t-1)}+\nonumber\\
&\rho^{(t)}\bigg(\bar{A}^{(t)}-\sum_{j\in\mathcal{J}}\sum_{k\in\mathcal{K}} \bar{B}_{j,k}^{(t)}\omega^{(t)}_{1,j,k}
-\sum_{l\in\mathcal{L}}\sum_{j\in\mathcal{J}} \bar{C}_{l,j}^{(t)}\omega^{(t)}_{2,l,j}\bigg), \label{eqn:A}
\end{align}
respectively, with $A^{(0)}=0$ and $\bar{A}^{(t)}$ given by:
\begin{align}
&\bar{A}^{(t)}\!=\!
\sum_{i\in\mathcal{I}}\frac{N_i}{BN}\!\!\sum_{n\in\mathcal N_i^{(t)}}\sum_{l\in\mathcal{L}} y_{n,l}\log\left(Q_l({\boldsymbol\omega}^{(t)},\mathbf{x}_n)\right)\!+\!\tau\Vert{{\boldsymbol\omega}^{(t)}}\Vert_2^2,\nonumber
\end{align}
By the KKT conditions, the closed-form solutions of the problem in~\eqref{prob:class-con-ap} is given as follows.
\begin{Lem}[Optimal Solution of Problem in~\eqref{prob:class-con-ap}]\label{lem:closedform}
\begin{align}
&\bar{\omega}_{1,j,k}^{(t)}=\frac{-\nu B_{j,k}^{(t)}}{2(1+\nu\tau)},\quad j\in\mathcal{J},\ k\in\mathcal{K},\label{eqn:omega1-con}\\
&\bar{\omega}_{2,l,j}^{(t)}=\frac{-\nu C_{l,j}^{(t)}}{2(1+\nu\tau)},\quad l\in\mathcal{L},\ j\in\mathcal{J},\label{eqn:omega2-con}
\end{align}
where
\begin{align}
&\nu=
\begin{cases}
\left[\frac{1}{\tau}\left(\sqrt{\frac{b}{b+4\tau(U-A^{(t)})}}\!-\!1\right)\right]_0^c,&b+4\tau(U-A^{(t)})>0\\
c,&b+4\tau(U-A^{(t)})\leq0,
\end{cases}\nonumber\\
&b=\sum_{j\in\mathcal{J}}\sum_{k\in\mathcal{K}} (B_{j,k}^{(t)})^2+\sum_{l\in\mathcal{L}}\sum_{j\in\mathcal{J}} (C_{l,j}^{(t)})^2.\label{eqn:b}
\end{align}
Here, $[x]^c_0\triangleq\min\left\{\max\{x,0\},c\right\}$.
\end{Lem}
Thus, in Step 5 of Algorithm~\ref{alg:con-sample}, the server only needs to compute $\boldsymbol\omega$ according to~\eqref{eqn:omega1-con} and~\eqref{eqn:omega2-con}.
\section{Numerical Results}\label{sec:simu}
In this section, we show the performance of Algorithm~\ref{alg:uncon-sample}, Algorithm~\ref{alg:con-sample} and the SGD-based algorithms~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel} in the application examples in Sections~\ref{sec:application} using numerical experiments.
We carry our experiments on Mnist data set.
For the training model, we choose, $N=60000$, $I=10$, $K=784$, $J=128$, $L=10$.
For Algorithm~\ref{alg:uncon-sample} and Algorithm~\ref{alg:con-sample}, we choose $T=100$, $\tau=0.1$, $c=10^5$,
$\rho^t=a_1/t^{\alpha}$ and $\gamma^t=a_2/t^{\alpha+0.05}$ with $a_1=0.4,0.6,0.9$, $a_2=0.4,0.9,0.9$, $\alpha=0.4,0.3,0.3$ for batch sizes $B=1,10,100$, respectively.
For the SGD-based algorithms~\cite{mcmahan2017communication,yang2019scheduling,yu2019parallel}, let $E$ denote the number of local SGD updates, and the learning rate is set as $r=\bar a/t^{\bar\alpha}$, where $\bar a$ and $\bar\alpha$ are selected using grid search method.
Note that all the results are given by the average over 100 runs.
\begin{figure}[h]
\begin{center}
\subfigure[\scriptsize{
Training cost $F(\boldsymbol\omega^t)$ vs. iteration $t$ by Algorithm~\ref{alg:uncon-sample} with $\lambda=10^{-5}$.}\label{fig:CostSam}]
{\resizebox{4.2cm}{!}{\includegraphics{eps/CostSam.eps}}}\quad
\subfigure[\scriptsize{
Training cost $F(\boldsymbol\omega^t)$ vs. iteration $t$ by Algorithm~\ref{alg:con-sample} with $U=0.13$.}\label{fig:CostSamCon}]
{\resizebox{4.2cm}{!}{\includegraphics{eps/CostSamCon.eps}}}\\
\end{center}
\vspace{-4mm}
\caption{\small{Training cost versus iteration index.}}
\label{fig:Cost}
\end{figure}
\vspace{-4mm}
\begin{figure}[h]
\begin{center}
\subfigure[\scriptsize{
Test accuracy at $\boldsymbol\omega^t$ vs. iteration $t$ by Algorithm~\ref{alg:uncon-sample} with $\lambda=10^{-5}$.}\label{fig:AccuSam}]
{\resizebox{4.2cm}{!}{\includegraphics{eps/AccuSam.eps}}}\quad
\subfigure[\scriptsize{
Test accuracy at $\boldsymbol\omega^t$ vs. iteration $t$ by Algorithm~\ref{alg:con-sample} with $U=0.13$.}\label{fig:AccuSamcon}]
{\resizebox{4.2cm}{!}{\includegraphics{eps/AccuSamcon.eps}}}\\
\end{center}
\vspace{-4mm}
\caption{\small{Test accuracy versus iteration index.}}
\label{fig:Accu}
\end{figure}
\vspace{-4mm}
\begin{figure}[h]
\begin{center}
\subfigure[\scriptsize{
$\ell_2$-norm $\Vert\boldsymbol\omega\Vert^2_2$ vs. training cost obtained by Algorithm~\ref{alg:uncon-sample}.}\label{fig:Tradeoff1}]
{\resizebox{4.2cm}{!}{\includegraphics{eps/Tradeoff1.eps}}}\quad
\subfigure[\scriptsize{
$\ell_2$-norm $\Vert\boldsymbol\omega\Vert^2_2$ vs. training cost obtained by Algorithm~\ref{alg:con-sample}.}\label{fig:Tradeoff2}]
{\resizebox{4.2cm}{!}{\includegraphics{eps/Tradeoff2.eps}}}
\end{center}
\vspace{-4mm}
\caption{\small{Model sparsity versus training cost.}}
\label{fig:Spar}
\end{figure}
Fig.~\ref{fig:Cost} and Fig.~\ref{fig:Accu} illustrate the training cost and test accuracy versus the iteration index. From Fig.~\ref{fig:Cost} and Fig.~\ref{fig:Accu}, we can see that the proposed algorithms with larger batch sizes converge faster. From Fig.~\ref{fig:CostSam} and Fig.~\ref{fig:AccuSam}, we can observe that for unconstrained federated optimization, Algorithm~\ref{alg:uncon-sample} converges faster than the SGD-based algorithm with $E=1$ at the same batch size.
In addition, Algorithm~\ref{alg:uncon-sample} with $B=10(100)$ converges faster than the SGD-based algorithm with $B=5(50)$ and $E=2$, i.e., Algorithm~\ref{alg:uncon-sample} converges faster that the SGD-based algorithm when the two algorithms induce the same computation load for each client.
Fig.~\ref{fig:Tradeoff1} and Fig.~\ref{fig:Tradeoff2} show the tradeoff curve between the model sparsity and training cost of each proposed algorithm. From Fig.~\ref{fig:Tradeoff2}, we see that with constrained sample-based federated optimization, one can set an explicit constraint on the training cost to effectively control the test accuracy. Furthermore, by comparing Fig.~\ref{fig:Tradeoff1} and Fig.~\ref{fig:Tradeoff2}, we can see that Algorithm~\ref{alg:con-sample} can achieve a better tradeoff between the model sparsity and training cost than Algorithm~\ref{alg:uncon-sample}. The main reason is that the underlying constrained sample-based federated optimization has a convex objective function and the chance for Algorithm~\ref{alg:con-sample} to converge to an optimal point is higher.
\section{Conclusions}
In this paper, we proposed two privacy preserving algorithms for unconstrained and constrained sample-based federated optimization problems, respectively, using SSCA techniques. We also showed that each algorithm can converge to a KKT point of the corresponding problem. It is worth noting that SSCA has not been used for solving federated optimization, and federated optimization with nonconvex constraints has not been investigated. Numerical experiments showed that the proposed SSCA-based algorithm for unconstrained sample-based federated optimization converges faster than the existing SGD-based algorithms, and the proposed SSCA-based algorithm for constrained sample-based federated optimization can obtain a sparser model that satisfies an explicit constraint on the model cost.
|
1,116,691,500,490 | arxiv | \section*{SUBMITTED TO "MECHANICS OF MATERIALS" ON 4th OCTOBER 2020}
\title{Rate-dependent adhesion of viscoelastic contacts. Part \textrm{I}:
contact area and contact line velocity within model multi-asperity contacts with rubber.}
\author{G. Violano}
\affiliation{Department of Mechanics, Mathematics and Management, Polytechnic
University of Bari, Via E. Orabona, 4, 70125, Bari, Italy}
\author{A. Chateauminois}
\affiliation{Soft Matter Science and Engineering Laboratory (SIMM), PSL Research
University, UPMC Univ Paris 06, Sorbonne Universités, ESPCI Paris, CNRS, 10 rue
Vauquelin, 75231 Paris cedex 05, France}
\author{L. Afferrante}
\email{[email protected]}
\affiliation{Department of Mechanics, Mathematics and Management, Polytechnic
University of Bari, Via E. Orabona, 4, 70125, Bari, Italy}
\begin{abstract}
In this work, we investigate dissipative effects involved during the detachment of a smooth spherical glass probe from a viscoelastic silicone substrate patterned with micro-asperities. As a baseline, the pull-off of a single asperity, millimeter-sized contact between a glass lens and a smooth poly(dimethylsiloxane) (PDMS) rubber is first investigated as a function of the imposed detachment velocity. From a measurement of the contact radius $a(t)$ and normal load during unloading, the dependence of the strain energy relase rate $G$ on the velocity of the contact line $v_c=da/dt$ is determined under the assumption that viscoelastic dissipation is localized at the edge of the contact. These data are incoproated into Muller's model (V.M. Muller \textit{J Adh Sci Tech} (1999) \textbf{13} 999-1016) in order to predict the time-dependence of the contact size. Similar pull-off experiments are carried out with the same PDMS substrate patterned with spherical micro-asperities with a prescribed height distribution. From \textit{in situ} optical measurements of the micro-contacts, scaling laws are identified for the contact radius $a$ and the contact line velocity $v_c$. On the basis of the observed similarity between macro and microscale contacts, a numerical solution is developed to predict the reduction of the contact radius during unloading.
\end{abstract}
\keywords{viscoelasticity, adhesion, surface roughness, energy
release rate.}
\maketitle
\section{Introduction}
Adhesion is of paramount importance in the contact mechanics of micro and
nano systems \cite{vakis2018} as, at the molecular scale, adhesive
interactions between atoms are `strong' compared to the usual forces acting
between bodies \cite{kendall}. However, adhesion is seldom observed at the
macroscopic scale due to surface roughness, which reduces the area of real
contact. Nevertheless, when dealing with very soft matter, strong adhesion
may be still detected even in presence of surface roughness \cite{tiwary2017}%
.
Soft matter adhesion finds applications in several fields, e.g. design of
pressure-sensitive-adhesives (PSA) \cite{PSA}, soft robots \cite{SOFTrobots}
and new technologies inspired by biotribological systems \cite{Biotribology}.
In most of adhesion tests on soft compliant spheres \cite%
{tiwary2017,Lorenz2013}, the measured detachment force is generally greatly
in excess of that predicted by Johnson, Kendall \& Roberts (JKR) theory \cite%
{JKR} and the detachment process is observed to be dependent on the rate of
separation \cite{GreenJohn1981}.
The JKR theory applies for purely elastic spheres and under quasi-static
conditions. In experimental investigations, the pull-off process unlikely
obeys the quasi-static conditions and the effective work of adhesion $\Delta
\gamma _{\mathrm{eff}}$ depends on the velocity $v_{\mathrm{c}}$ of the
contact line during pull-off. Namely, $\Delta \gamma _{\mathrm{eff}}$ may be
strongly increased with respect to the quasi-static value $\Delta \gamma
_{0} $ as a result of viscous dissipation, where $\Delta \gamma _{0}$
follows the well-known Dupr\'e's equation $\Delta \gamma _{\mathrm{0}}=\gamma
_{\mathrm{1}}+\gamma _{\mathrm{2}}-\gamma _{\mathrm{12}}$, being $\gamma _{%
\mathrm{1}}$, $\gamma _{\mathrm{2}}$ the adhesive energies of the two
contacting surfaces and $\gamma _{\mathrm{12}}$ the interaction term.
Gent \& Schultz (GS) \cite{GS1972} observed that viscous effects are
exclusively located close to the crack tip. Maugis \& Barquins (MB) \cite%
{MB1978} proposed a generalization of the JKR theory, showing that the
dependence of $\Delta \gamma _{\mathrm{eff}}$ on $v_{\mathrm{c}}$ can be
expressed in terms of a dissipation function\ $f(v_{\mathrm{c}},T)$ related
to the viscoelastic properties of the material and depending on the crack
tip velocity $v_{\mathrm{c}}$ and the temperature $T$. In particular, MB
showed that, for a given elastomer, the effective work of adhesion $\Delta
\gamma _{\mathrm{eff}}$ is a universal function of the crack tip velocity $%
v_{\mathrm{c}}$. Moreover, performing experimental tests on three different
geometries (spheres, punches and tapes (peeling)), MB found that the
dependence of $\Delta \gamma _{\mathrm{eff}}$ on $v_{\mathrm{c}}$ is not
affected by the geometry and loading system. In MB's solution, viscous
effects are assumed not involving bulk deformations as \textquotedblright
\textit{gross displacements must be elastic for }$G$\textit{\ to be valid in
kinetic phenomena}\textquotedblright , being $G$ the energy release rate,
i.e. the amount of energy required to advance a fracture plane by a unit
area. Robbe-Valloire \& Barquins \cite{valloire1998} extended
MB studies performing adherence experiments between a rigid cylinder and an
elastomeric solid. They confirmed the existence of a master curve for $f(v_{%
\mathrm{c}},T)$. Specifically, their results "\textit{prove once again that
the master curve drawn and its variation... is a characteristic of the
propagation in mode I at the interface of our rubber-like material, when
viscoelastic losses are closely limited to the crack tip, so that G can be
calculated from the theory of linear elasticity.}"
More recently, Muller \cite{Muller1999} showed that the process of
detachment of viscoelastic spheres can be described by a first-order
differential equation, whose solution is based on the assumption originally
proposed in Ref. \cite{GS1972}. Alternative approaches taking into account
bulk deformations were proposed by the group of Barthel in Refs. \cite%
{Barthel2002,Barthel2003,Barthel2009}.
In this work, we present an experimental investigation of dissipative effects involved in the adhesion between a rough contact interface between a smooth spherical glass probe and a viscoelastic silicone substrate patterned with a prescribed height distribution of micrometer sized spherical asperities. Taking advantage from the fact that the size of these micro-asperities (radius of 100~$\mu$m) allows for an optical measurement of the space distribution of micro-contact areas, such patterned surfaces obtained from micro-milling techniques were previously successfully used to probe the elastic interactions between micro-asperity contacts~\cite{YASHIMA2015} or to investigate adhesive equilibrium of rough contact interfaces~\cite{ACITO2019}. Here, we focus on the effects of viscoelastic dissipation on the rate-dependence of of micro-contact sizes during unloading at a imposed velocity using JKR-type experiments. The investigation of adhesive behaviour at the level of micro-contact spots is inspired by the Roberts' statement (Ref. \cite{roberts79}): "\textit{The contact of a smooth centimerer-sized rubber sphere may be regarded as that of a giant single asperity...The ability to predict the adhesion forces on a large asperity is a step towards building up a model of a real surface of micron-sized asperities, which approximate to an array of minute hemispheres of different height and radius}".\\
Accordingly, pull-off experiments have first been conducted on smooth PDMS\ surfaces to investigate the adhesion behavior at the macroscopic scale. Specifically, under the assumption of viscous effects located only near the detachment front, we propose a very simple methodology to calculate the time-dependent radius of the contacts during unloading by exploiting the Muller's approach \cite{Muller1999} which was already used to calculate the adhesion hysteresis occurring in loading-unloading tests performed on smooth viscoelastic spheres \cite{ViolAIAS2019}. Then, this approach is successfully extended at the micro-scale with no need to incorporate a size-dependence in the relationship ruling the dependence of the strain energy release rate on the velocity of the contact line.
\section{Detachment of viscoelastic spheres}
In order to detach a soft body from a rigid substrate, the force required to
create a new unit length of crack is $\left( G-\Delta \gamma _{\mathrm{0}%
}\right) $. If the energy release rate $G$ is larger than the adiabatic work
of adhesion $\Delta \gamma _{\mathrm{0}}$, the crack opens and the
detachment process advances.
Gent \& Schultz (GS) \cite{GS1972} found that%
\begin{equation}
G-\Delta \gamma _{\mathrm{0}}=\Delta \gamma _{\mathrm{0}}\cdot f(v_{\mathrm{c%
}},T) \label{G1}
\end{equation}%
where $\Delta \gamma _{\mathrm{0}}\cdot f(v_{\mathrm{c}},T)$ is the drag due
to viscoelastic losses at the crack tip, being $v_{\mathrm{c}}=-da/dt$ the
velocity of the contact line. The above relation usually works for $v_{%
\mathrm{c}}$ ranging form $10^{-5}$ \textrm{cm/s}\ to $1$ \textrm{cm/s} \cite%
{Andrews73,Gent69,Kendall73,roberts79} and allows to predict the kinetics of
detachment (see Maugis \& Barquins \cite{MB1978}).
The function $f(v_{\mathrm{c}},T)$, which is found to be independent of the
geometry and loading system, can be described by the phenomenological
equation%
\begin{equation}
f(v_{\mathrm{c}})=k(a_{\mathrm{T}}v_{\mathrm{c}})^{n}\text{,} \label{fvc}
\end{equation}%
where $k$ and $n$ are characteristic constants of the material and $a_{%
\mathrm{T}}$ is the William-Landel-Ferry (WLF)\ factor \cite{WLF1955}
accounting for the dependence of $f(v_{\mathrm{c}},T)$ on the temperature $T$%
. Eq. (\ref{fvc}) also accounts for the dependence of $G$ on the relaxed
elastic modulus $E$ (Ref. \cite{ramond1985}), whose frequency dependence
appears only at the crack tip \cite{charmet1998}.
Introducing eq. (\ref{fvc}) in (\ref{G1}), we obtain%
\begin{equation}
G=\Delta \gamma _{0}[1+c\cdot v_{\mathrm{c}}{}^{n}]\text{,} \label{G2}
\end{equation}%
with $c=k\cdot a_{\mathrm{T}}{}^{n}$.
For a given elastomer, the values of $c$ and $n$ can be obtained by fitting
the experimental data relating $G$ and $v_{\mathrm{c}}$. As observed in Ref. \cite{Lorenz2013},
the exponent $n$ "\textit{is not a universal number, but takes different values depending on viscoelastic modulus}".
\subsection{Muller's model}
Muller \cite{Muller1999} proposed a two-parameters differential equation to
describe the detachment of a viscoelastic sphere of radius $R$ and Young
modulus $E^{\ast }$ from a rigid substrate%
\begin{equation}
\frac{d\bar{a}}{d\bar{\delta}}=\left[ \frac{\Delta \gamma _{\mathrm{0}}}{%
RE^{\ast }}\right] ^{1/3}\cdot \frac{1}{\beta }\left[ \bar{a}^{3}\left( 1-%
\frac{\bar{\delta}}{3\bar{a}^{2}}\right) ^{2}-\frac{4}{9}\right] ^{1/n}\text{%
,} \label{Meq}
\end{equation}%
where $\bar{a}=a/\left[ 3R\left( \pi \Delta \gamma _{\mathrm{0}}/(6E^{\ast
}R)\right) ^{1/3}\right] $ and $\bar{\delta}=\delta /\left[ 3R\left( \pi
\Delta \gamma _{\mathrm{0}}/(6E^{\ast }R)\right) ^{2/3}\right] $ are the
dimensionless contact radius and penetration, respectively, and the
parameter $\beta $ is proportional to the driving velocity $V$%
\begin{equation}
\beta =\left( \frac{6}{\pi }\right) ^{1/3}\left( \frac{4}{9}c\right) ^{1/n}V%
\text{.} \label{beta}
\end{equation}
Muller's model moves from two assumptions: i) viscous effects are located
exclusively near the crack tip; ii) detachment occurs under constant $V$.
The energy release rate $G$, which represents the effective work of adhesion
$\Delta \gamma _{\mathrm{eff}}$ required to break the contact, can be
evaluated as%
\begin{equation}
G=\frac{\left( F_{\mathrm{H}}-F\right) ^{2}}{6\pi RF_{\mathrm{H}}}
\label{G3}
\end{equation}%
where $F_{\mathrm{H}}=4/3E^{\ast }a^{3}/R$ is the Hertzian load and $F$ is
the applied load. Eq. \ref{G3} is only valid under the assumption of viscous effects concentrated at the crack tip (see, for example, Ref. \cite{baekPDMS}).
\section{Experimental set-up}
JKR-like tests were carried out between an optical spherical glass lens and rubber substrates. The glass indenter, which is assumed
to be smooth, has a radius of curvature $R_{\mathrm{sphere}}$ of $103.7$ \textrm{mm}. Rubber substrates are made of commercially available PolyDiMethylSiloxane (PDMS) silicones.
Samples were manufactured by cross-linking at $70$ $\mathrm{%
{{}^\circ}%
C}$ for $48$ hours a mixture of Sylgard $184$ and Sylgard $527$
silicones (Dow Chemicals), with a $0.35$\textrm{:}$0.65$ weight ratio. As detailed by Palchesko et al. \cite{Palchesko}, mixing these two silicone elastomers in different ratios allows to tune the elastic modulus in between a few kPa and 3~MPa. As compared to raw Sylgard $184$, the Young's modulus of the selected Sylgard $184$:Sylgard $527$ mixture ($E=$~0.83 MPa, see below) was decreased by a factor of about $3.6$, with the aim of enhancing the adhesion properties. In addition, crosslinking simultaneously these two different silicone products was expected to result in an increased concentration of network defects such as dangling chains. As detailed in Ref. \cite{Palchesko}, such imperfections are known to enhance the viscoelastic dissipation of silicone networks.\\
Fig. \ref{Setup} shows a sketch of the experimental set-up. The spherical
indenter is fixed to a motorized vertical translation stage by means of a double
cantilever beam of known stiffness (290 \textrm{N m}$^{-1}$). The value of
the applied load is obtained from the deflection of the cantilever, as it is
measured using a high resolution optical sensor (Philtec D64-L). Due to the compliance of the double cantilever beam, the actual velocity of the lens can slightly different from the prescribed velocity. In order to account for this effect, a laser displacement sensor (Keyence LK-H057) is used to monitor the actual position of the lens. The difference between the prescribed and actual velocity of the indenter were found to be significant only for the macroscopic single asperity contact close to pull-off, when the greatest tensile normal forces are achieved.\\
The PDMS sample is fixed to two crossed motorized linear translation stages, which allow to change the relative position of the rubber sample with respect to the indenter. A LED light spot is installed to illuminate in transmission the contact area. Once illuminated, contact pictures are recorded through the transparent PDMS using a zoom objective (Leica APO Z16) and a high resolution CMOS camera (SVS Vistek Exo, $2048$ $\times $ $2048$ pixels$^{2}$, 8 bits).
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{setup_con_laser.eps}
\end{center}
\caption{The experimental setup of JKR adhesion tests.}
\label{Setup}
\end{figure}
\subsubsection{Experiments on smooth samples}
The contact radius vs. load data obtained by indentation experiments on smooth
PDMS\ samples were fitted according to the JKR theory \cite{JKR} in order to evaluate
the reduced elastic modulus ($E^{\ast }=0.83$ $\mathrm{MPa}$) and the
adhesion energy ($\Delta \gamma _{0}=0.037$ $\mathrm{J/m}^{2}$). During
loading, contact tests have been performed under fixed load conditions.
Specifically, the applied load is increased step by step and, once each load
step is reached, contact is maintained for a long time ($800$ \textrm{s}) to
ensure that adhesive equilibrium is reached (with viscoelastic effects
totally relaxed \cite{ACITO2019}).
Unloading tests are performed at imposed driving velocity of the
vertical stage, while continuously monitoring the lens position, the applied force and the contact radius. Experiments are performed at three different values of the
driving velocity $V=0.02,$ $0.002,$ $0.0002$ \textrm{mm/s}. Three contact
realizations have been carried out for each velocity.
\subsubsection{Experiments on rough samples}
PDMS\ samples were textured with a statistical distribution of spherical micro-asperities with the same radius of curvature. The
patterned surface was obtained by moulding PDMS\ in PolyMethylMethAcrylate
(PMMA) forms milled using ball-end mills with a radius of $100$ $\mathrm{\mu
m}$. In order to enhance adhesive effects, a smoothening of the spherical cavities of the PMMA\ molds has been
achieved by exposing them to a saturated \textrm{CHCl}$_{3}$ vapor for $30$
minutes. As detailed in Ref. \cite{ACITO2019}, such treatment leads to a slight increase in the radius of the spherical
bumps, up to a $10\%$ enhancement.
The patterned surface has been generated with a squared nominal area of $10$
$\mathrm{mm}^{2}$, where asperities are randomly distributed with a density
of $2\times 10^{7}$ $\mathrm{m}^{-2}$. The spherical caps present heights
distributed according to a Gaussian law with standard deviations $\sigma =5$
$\mathrm{\mu m}$.
Asperities are collocated with a non-overlapping constraint. For the considered surface density,
each contacting asperity behaves as an isolated spherical punch and
lateral interactions can be neglected as shown in Refs. \cite%
{ACITO2019,YASHIMA2015}. We stress that such assumption is no longer valid
when roughness on several length scales is considered like in the case of
self-affine fractal geometries \cite{ViolanoJKR,ViolanoDMT,PersAff}.
Fig. \ref{Microspot} shows images of contact micro-spots (blue disks) during
unloading. Accurate measurements of the area of contact spots is achieved by image processing after background removal. During unloading, experiments have been performed at
the same values of driving velocity $V$ used in the tests conducted on
smooth samples ($V=0.02,$ $0.002,$ $0.0002$ \textrm{mm/s}).
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{Microspot.eps}
\end{center}
\caption{Detachment of spherical micro-asperities. The contact spots (blue
disks) are detected after post-processing of the contact pictures. The length of the black rectangle is 1 mm.}
\label{Microspot}
\end{figure}
\section{Results}
\subsection{Smooth contact: macroscopic scale}
Fig. \ref{Fmacro} shows the contact radius $a$ as a function of the applied
load $F$, during unloading. Results are obtained for different values of the
driving velocity $V$. Three contact tests were performed for each $V$ and
the average values are reported in the plot.
The detachment process is clearly rate-dependent as great adhesion
enhancement is observed by increasing $V$, as indicated by the increase in pull-off force.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{Fmacro.eps}
\end{center}
\caption{The contact radius $a$ as a function of the applied load $F$.
Results are shown for different unloading velocities of the indenter $V=0.02$%
, $0.002$, $0.0002$ \textrm{mm/s}. For each velocity three tests have been
performed and the average values are reported in the plot.}
\label{Fmacro}
\end{figure}
Fig. \ref{RVmacro}A shows, in a semilogarithmic representation, the
reduction of the contact radius $a$ with the time $t$. Experimental data are
fitted according to the following relation%
\begin{equation}
a(t)=p_{1}\sqrt{\left( 1-\frac{t}{t_{\mathrm{po}}+1}\right) ^{p_{2}}}
\label{at}
\end{equation}%
where $t_{\mathrm{po}}$ is the instant at which detachment occurs and $%
p_{1}$, $p_{2}$ are fitting parameters.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{RVmacro.eps}
\end{center}
\caption{A: The contact radius $a$ as a function of the time $t$. Results
show the average of three contact tests. Blue, green and red markers are
referred to the unloading velocities of the indenter $V=0.02$, $0.002$, $%
0.0002$ \textrm{mm/s}, respectively. Solid lines denote the fit of
experimental data. B: The contact radius $a$ as a function of the crack tip
velocity $v_{\mathrm{c}}$. Legend symbols are the same of A.}
\label{RVmacro}
\end{figure}
During detachment, the crack tip velocity can be easily obtained as $v_{%
\mathrm{c}}=-da/dt$. Fig. \ref{RVmacro}B shows the curves $a$ vs. $v_{%
\mathrm{c}}$ obtained in the experiments at different velocities. The
maximum value of $v_{\mathrm{c}}$ is reached when abrupt pull-off occurs.
Notice that when increasing $V$ of one order of magnitude the same
enhancement in $v_{\mathrm{c}}$ is also observed.
The effective work of adhesion $G$ is calculated as a function of measured contact line velocity by eq. (\ref{G3}) using the experimental value of the normal load $F$. As discussed by Barquins \cite{stiffness1983}, the resulting $G(v_{\mathrm{c}})$ relationship is unaffected by the machine compliance. The actual stiffness of the system is taken into account because the load $F$ is read from the deflection of the double cantilever beam. The same procedure has been recently used in Ref. \cite{baekPDMS}, where the effective work of adhesion is measured in spherical contact between a glass lens and PDMS blocks.
Fig. \ref{Gmacro} shows the quantity $(G-\Delta \gamma _{\mathrm{0}})/\Delta \gamma _{\mathrm{0}}$ as a function of $v_{\mathrm{c}}$ in a double logarithmic
representation. Markers denote experimental data, while the dotted black
line is the fit obtained with eq. (\ref{G2}) using $c=31$ and $n=0.25$. Maugis $\&$ Barquins \cite{MB1978}
found $n=0.6$ for a viscoelastic polyurethane rubber. More recently, Lorenz et al.~\cite{Lorenz2013} found $n=0.19$ for polyurethane
and $n=0.12$ for Sylgard 184 PDMS rubber. As discussed by Barthel and Fr\'etigny~\cite{Barthel2009}, the dependence of $G$ on the crack tip velocity can be related to the viscoelastic creep function of the solids. Accordingly, the fact that we found for the used Sylgard 184:Sylgard 527 mixture an exponent $n$ greater than for raw Sylgard 184 probably reflects the enhanced viscoelastic dissipation of the PDMS mixture.
The values of $c$ and $n$ can be used in eq. (\ref{beta}) to calculate the
values of $\beta $ required in Muller's model.
According to Muller's model, eq. (\ref{Meq}) can be numerically integrated to obtain $a({\delta})$. Assuming that $V$ is constant during unloading, the instant $t_{\mathrm{po}}$ at which pull-off occurs can be estimated by $t_{\mathrm{po}}=(\delta_{0}-\delta _{\mathrm{po}})/V$, where $\delta _{0}$ is the initial penetration and ${\delta}_{\textrm{po}}$ the jump-off distance. However, in our experiments we found that the actual unloading velocity $V_{\mathrm{act}}(t)$ is not constant as the spherical indenter is held to the stage using a compliant double cantilever beam. Due to the deflection of the beam, the velocity $V_{\mathrm{act}}$ of the lens can be different from the imposed velocity $V$. This is especially true for the experiments on smooth PDMS, where high forces can be achieved.
However this effect is quantified by the laser displacement sensor, which monitors the actual position $z(t)$ of the lens. Hence, the actual velocity is derived as $V_{\mathrm{act}}(t)=\Delta z/\Delta t$.
In the original Muller's model, the unloading velocity $V=-d\delta /dt$ is assumed to be constant.
However, we can modify eq. (\ref{beta}) by introducing the actual velocity $%
V_{\textrm{act}}(\delta )$ in the parameter $\beta$
\begin{equation}
\beta =\left( \frac{6}{\pi }\right) ^{1/3}\left( \frac{4}{9}c\right) ^{1/n}V_{\textrm{act}}(%
\bar{\delta})\text{.} \label{betanew}
\end{equation}
where $V_{\textrm{act}}(\delta )$ is obtained by interpolating experimental data.
The time required to move from $\delta _{0}$ to a generic $\delta$ is then calculated as%
\begin{equation}
t=\int_{\delta_{0}}^{\delta}\frac{\delta }{V_{\mathrm{act}}(\delta )}d\delta.
\label{tempo}
\end{equation}
Results are shown in Figs. \ref{at_smooth}A-C, where the contact radius $a(t)$, normalized with respect to its initial value, is plotted against the time normalized with respect to the period required for pull-off to occur. Results are given for different unloading velocities and show a good agreement between experimental data and numerical predictions.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{Gmacro.eps}
\end{center}
\caption{The relative increase $\left( G-\Delta \protect\gamma _{0}\right)
/\Delta \protect\gamma _{0}$ as a function of the crack tip velocity $v_{%
\mathrm{c}}$. Results are shown for three contact realizations. Blue, green
and red markers are referred to the unloading velocities of the indenter $%
V=0.02$, $0.002$, $0.0002$ \textrm{mm/s}. The dotted line is the fit
obtained with eq. (\protect\ref{G2}).}
\label{Gmacro}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{smooth_a_t.eps}
\end{center}
\caption{A-C: The time dependence of the normalized contact radius $a(t)$. Results are shown for unloading velocity of the indenter $%
V=0.0002$, $0.002$, $0.02$ \textrm{mm/s} (figs. A,B,C respectively). Dashed lines denote Muller's model predictions, while markers experimental data, which are averaged on three contact realizations.}
\label{at_smooth}
\end{figure}
\subsection{Rough contact: microscopic scale}
In the experiments performed on rough samples the number of micro-asperities
detected in contact at the end of the loading phase is around $160$.
However, for the sake of clarity, Figs. \ref{V1micro}A and \ref{V1micro}B
show the variation of the contact radius $a$ of $8$ micro-asperities in
terms of the time and the contact line velocity $v_{\mathrm{c}}$,
respectively. Results are shown for $V=2\times 10^{-4}$ \textrm{mm/s}. In
general, asperities with a larger value of the initial contact radius $a$
require a longer time to complete their detachment process.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{V1Rmicro.eps}
\end{center}
\caption{A: The contact radius $a$ of micro-asperities as a function of time
$t$. Markers denote experimental data for a selection of eight different micro-contacts. Unloading tests are
performed at $V=0.0002$ \textrm{mm/s}. B: The contact radius $a$ as a
function of the crack tip velocity $v_{\mathrm{c}}$.}
\label{V1micro}
\end{figure}
Results on smooth and rough samples suggest the existence of scale
effects on both contact radius $a$ and detachment front velocity $v_{\mathrm{%
c}}$. For this reason, we rescale the above results introducing the factors $%
s_{\mathrm{a}}=a_{0\mathrm{macro}}/a_{0i}$, $s_{\mathrm{t}}=t_{\mathrm{%
po-macro}}/t_{\mathrm{po-}i}$ and $s_{\mathrm{v}}=s_{\mathrm{a}}/s_{\mathrm{t%
}}$. The quantity $a_{0\mathrm{macro}}$ is the initial value of the contact
radius measured at the macroscale on smooth PDMS samples at the end of the
loading phase (when unloading starts); $a_{0i}$ is instead the initial value
of the contact radius detected for the $i^{th}$ micro-asperity. Similarly, $%
t_{\mathrm{po-macro}}$ is the time at which pull-off occurs at the
macroscale (that is measured in the tests performed on smooth PDMS samples),
while $t_{\mathrm{po-}i}$ is the time required (and measured in the tests on
rough PDMS samples) to detach the $i^{th}$ micro-asperity.
Therefore, contact radius $a$, crack tip velocity $v_{\mathrm{c}}$ and time
are rescaled with the above factors. The new curves are given in Fig. \ref%
{RVmicro} for three different driving velocities $V$ and in a semi-log plot.
Solid lines denote the curves obtained at the macroscopic scale in the
experiments conducted on smooth substrate. Dashed lines identify the curves
measured for each micro-asperity during the detachment tests performed on
the rough PDMS samples. All curves obtained on the contact microspots almost
collapse on the curves measured at the macroscale (smooth samples). Such
result suggests that the distributions of the actual crack-tip velocities $%
v_{\mathrm{c}}$, which are achieved locally at micro-contact, scale during
contact unloading. This, in turn, suggests that the parameters of Muller's model identified at the macroscale can be applied to the microcontacts.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.cm]{RVmicro.eps}
\end{center}
\caption{A: The contact radius $a\times s_{a}$ \ as a function of time $%
t\times s_{t}$ (semilog scale). Solid lines denote the smooth macro-spot
detachment curve. Dashed lines denote the detachment curves of $160$
micro-asperities. Red, Green and Blue curves refer to $V=0.2,$ $2.0,$ $20,$ $%
\mathrm{\protect\mu m/s}$. A: B: The contact radius $a\times s_{a}$ \ as a
function of the crack tip velocities $v_{\mathrm{c}}\times s_{a}/s_{t}$ (semilog
scale). Legend symbols are the same of A.}
\label{RVmicro}
\end{figure}
Such an assumption finds also its motivation in recent results by Lorentz et
al. \cite{Lorenz2013}, who performed adhesion experiments on smooth spheres
of different radii (ranging from $R\approx 3$ \textrm{mm} to $R=46.5$
\textrm{mm}) and different materials. They deduced $\Delta \gamma _{\mathrm{%
eff}}$ as a function of the contact line velocity $v_{\mathrm{c}}$ using the
JKR theory and observed that the experimental data exhibited the same
velocity dependence as calculated by eq. (\ref{G1}) for $v_{\mathrm{c}%
}<10^{-4}$ \textrm{m/s} (which corresponds to the range of crack tip
velocities measured in our experiments on micro-spots).
The same plots given in Figs. \ref{at_smooth} are reported in Figs. \ref{at_rough} for each of the micro-contacts detected during the unloading phase. A satisfactory good agreement is found between experimental data and numerical predictions, which are obtained with the "macroscale" values of $c$ and $n$. Also in this case, in the Muller's model, we have introduced the actual value of the unloading velocity, which is however constant and slightly lower than the imposed one ($V_{\textrm{act}}=0.8V$).
In the $a-t$ curves, we can distinguish a period of time where stick adhesion is observed with an almost constant contact radius (i.e. the contact line velocity is zero). This "stick time" is negligibly influenced by the initial value $a_{0}$ of the contact radius \cite{Barthel2003}. For this reason, when increasing $a_{0}$, the $a-t$ curve does not scale homothetically and a master curve cannot be found. This explains the scatter in fig. \ref{at_rough}. However, an increasing trend of the pull-off time with $a_{0}$ can be observed in our experimental data, as shown in figs. \ref{at_trend}A-C, where results are presented for different unloading velocities. Anyway, as data are strongly scattered, a clear law of this increasing trend is not identified.
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{rough_a_t.eps}
\end{center}
\caption{A-C: The time dependence of the normalized contact radius $a(t)$ for micro-contacts. Results are shown for unloading velocity of the indenter $V=0.0002$, $0.002$, $0.02$ \textrm{mm/s} (figs. A,B,C respectively). Black dotted lines denote experimental data, while colored dashed lines the Muller's model predictions.}
\label{at_rough}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=15.0cm]{amax_tmax_soloexp.eps}
\end{center}
\caption{A-C: The pull-off time $t_{\mathrm{po}}$ as a function of the initial contact radius $a_{0}$. Results are shown for rough PDMS and unloading velocity of the indenter $V=0.0002$, $0.002$, $0.02$ \textrm{mm/s} (figs. A,B,C respectively). Markers denote experimental data, while colored lines the corresponding linear fit; the $R^{2}$ value is 0.73, 0.77 and 0.78 (figs. A,B,C respectively).}
\label{at_trend}
\end{figure}
\section*{Conclusions}
In this paper, we have investigated the pull-off behavior of a rough contact interface between a smooth glass lens and a nominally flat viscoelastic substrate patterned with a height distribution of spherical micro-asperities. In the absence of any elastic coupling between micro-contacts, this system allows to measure simultaneously the pull-off behavior of a collection of micro-asperities contacts differing in their initial (equilibrium) adhesive contact radius. From a comparison with macroscale pull-off experiments, it also offers the possibility to investigate the occurrence of scale effects in dissipative processes involved in adhesion.\\
Results show that the contact radius almost scales according to the ratio $%
s_{\mathrm{a}}=a_{0\mathrm{macro}}/a_{0\mathrm{micro}}$, being $a_{0}$ the initial radius measured at the beginning of the unloading process. Similarly, the contact line velocity $v_{\mathrm{c}}$\ is found scaling with a factor $s_{%
\mathrm{v}}$ depending on the ratio $s_{\mathrm{a}}/s_{\mathrm{t}}$, where $%
s_{\mathrm{t}}=t_{\mathrm{po-macro}}/t_{\mathrm{po-micro}}$ and $t_{\mathrm{%
po}}$ is the time required for the pull-off to take place.
Such results suggest that the nature of the dissipative processes involved in the pull-off of the adhesive contacts is almost scale independent from the millimeter size down to a few tens of micrometers. In other words, the assumption that viscoelastic losses are localized near the contact line in a region small with respect to the contact size remains valid at the micro-scale. Moving from this consideration, a simple theoretical procedure can be derived to evaluate the evolution of the contact radius $a$ of micro-asperities contacts during unloading.
\bigskip
|
1,116,691,500,491 | arxiv | \section{Introduction}
Star forming dwarf galaxies constitute one of the very common type of galaxies present in the local volume and increase in number and importance with redshift. They mostly have small intrinsic size and low absolute luminosity \citep{hodge1971}. Study of nearby dwarf galaxies are vital to throw light on the nature of star formation in low mass and metal poor environment \citep{weisz2011}. Dwarf galaxies show a wide variety of star formation history in terms of star formation rate (SFR) and the most recent peak in their star forming activity \citep{cignoni2018}. Perturbations due to stellar feedback, ram pressure stripping or tidal interaction can significantly affect the evolution of a dwarf galaxy \citep{revaz2018}. Many dwarfs are known to undergo bursts of star formation, with the duration of star bursts extending up to 1000 Myr in some cases. These local star bursts play a deciding role in the evolution of the galaxy in different ways. Ionizing radiation, stellar wind and supernova explosion resulting from massive stars formed in the burst, can disrupt the available gas and also alter the chemical composition of the host galaxy \citep{mcquinn2010a,mcquinn2010b}. There are different mechanisms which can trigger a local burst of star formation in a dwarf galaxy. Expanding H$~$I shell can act as a trigger for secondary star formation in a galaxy \citep{tagle2005}.\\
The H$~$I shells are shell like structure created in the interstellar medium (ISM) of a galaxy due to the combined effect of stellar wind and supernova explosion \citep{weaver1977,mccray1987,tenorio1988,chu2004,bagetakos2011}. There are also other theories like gamma ray burst \citep{loeb1998} or infall of high velocity clouds \citep{tenorio1988,murray2004}, which are proposed as the cause behind the creation of such shells. Recent study by \citet{weisz2009} shows that multiple star formation events can also provide the required energy to form shells in the ISM. H$~$I shells were first identified in the Milky Way by \citet{heiles1979}. The diameter of these shells can vary from 10 pc to more than 1000 pc. A shell with diameter more than 1 kpc are usually called as super giant shell (SGS) \citep{stewart2000}. SGSs are mostly found in dwarf galaxies, where they can sustain for a longer time due to their slow solid body-like rotation and the absence of spiral density wave. Due to the low ambient density and shallow potential well in dwarfs, a shell can also expand to larger radii. Understanding such shells is importantly relevant in the context of recent star formation in the host galaxy. The evolution of shells and their impact in the ISM of the host galaxy in terms of feedback and metal enrichment are equally important research topics. Expanding H$~$I shells have significant effect on to the ISM of a galaxy in local scales. The feedback due to the shell expansion can sweep up the ambient ISM and compress it to trigger further star formation. \citet{kawata2014} reported that star formation can also be triggered due to the collision of such expanding shells in dwarf galaxies. \citet{hopkins2008} showed that the accumulation of neutral and molecular gas along the wall of such shells can trigger further star formation. Studies on nearby dwarf galaxies, such as the Large Magellanic Cloud (LMC), IC 10, Holmberg I, Holmberg II, have confirmed the presence of star formation triggered due to expansion or collision of H~I shells \citep{yamaguchi2001,leroy2006,egorov2017,egorov2018}.\\
IC 2574 (also known as DDO 81 or UGC 5666) is a gas-rich dwarf irregular galaxy in the M81 group, located at a distance of 3.79 Mpc \citep{dalcanton2009}. The galaxy, being present $\sim$ 164 kpc away from the brightest group member M81, does not have any signature of interaction \citep{yun1999}. Parameters of this galaxy are listed in Table \ref{ic2574}. The SFR of the galaxy is found to have increased during the last 1 Gyr \citep{walter1999} and hence it is a good candidate to study recent enhancement in the star forming activities. The ISM of this galaxy is very interesting and intriguing. The distribution of H$~$I in this galaxy shows many shell like structures. Using Very Large Array (VLA) observations, \citet{walter1999} identified 48 H$~$I shells and holes in the galaxy and studied them in detail. In order to understand the connection between H~I holes and triggered star formation in the galaxy, they used the catalog of H$\alpha$ emitting regions from \citet{miller1994} and defined 40 H~II regions, and investigated their connection with the features observed in the H~I map. They reported that the H$\alpha$ emissions, which traces the current star formation, are predominantly found along the rims of larger holes. \citet{pasquali2008} used optical images taken with Large Binocular Telescope (LBT) to explore the star formation history of the galaxy and reported two recent bursts of star formation in the galaxy. They found an older burst about 100 Myr ago inside 4 kpc radius and another younger burst during last 10 Myr between radius 4 to 8 kpc. They further noticed that the younger burst of star formation is mainly located in the periphery of H$~$I shells. Therefore the galaxy IC 2574 offers a good opportunity for studying nature of star formation triggered due to the expanding H$~$I shells. \citet{pellerin2012} observed the north-eastern part of the galaxy using Hubble Space Telescope (HST) and detected 75 young stellar groups (age $\sim$ 10 Myr), which again points to the recent star formation in the galaxy.\\
Young star forming regions in galaxies often produce OB associations, a loosely bound group of O and B type stars \citep{melnik1995}. These young and massive OB stars produce copious amount of far-UV (FUV) radiation, which is an excellent proxy to trace regions of recent star formation activity and to explore the triggering mechanisms. One of the identified SGSs, located in the north-east of the galaxy, has been studied rigorously in the literature. Using UV observations from UIT, \citet{stewart2000} identified a remnant cluster inside the SGS and confirmed a causal relation between the remnant cluster and the triggered star formation seen along the rim of the shell. \citet{cannon2005} used Spitzer observations to conclude that the triggered star formation detected along the rim of the shell is found to alter the temperature of the dust present around it. An HST study by \citet{weisz2009} also revealed that star formation happened earlier inside the SGS has triggered further recent star formation along its rim. \citet{yukita2012} identified a luminous X-ray source, which is slightly offset from the remnant cluster, inside the SGS. They also showed GALEX FUV image of the SGS (their Figure 2) where the remnant cluster is shown as C2. Though the SGS located in the north-eastern part of the galaxy is studied in FUV, a detailed investigation of the FUV emission with respect to all the H~I holes in IC 2574 has not been performed.\\
In this study, a deep FUV image of this galaxy, obtained from the Ultra-Violet Imaging Telescope (UVIT), is used to understand the connection between the recent star formation and the H~I holes across the galaxy. \citet{walter1999} did a similar study connecting the H$\alpha$ emission and H~I holes, whereas we combine the FUV emission along with the H$\alpha$ emission to trace the star formation not only in and around H~I holes, but also in the entire galaxy. Here we trace star forming clumps in the FUV and check for their possible connection with H~I holes, whereas the optical study of \citet{pasquali2008} investigated star formation across the galaxy and its overall connection with the H~I holes. Our study thus aims to present a comprehensive view of the recent star formation in this galaxy, particularly using the FUV images. UVIT on-board AstroSat \citep{kumar2012} is capable of imaging in UV with a better spatial resolution than GALEX. \citet{mondal2018,koshy2018a,koshy2018b,koshy2018c} used UVIT observations to understand different characteristic of star formation in some selected galaxies and demonstrated the capability of UVIT in utilising better spatial resolution. We make use of the resolution of UVIT ($\sim 0.4\arcsec/$pixel) in the FUV to identify the star forming regions. We identify complexes of possible OB associations and their location with respect to the H~I holes and shells. We study their correlation with the H~I density as well as structural properties, and estimate size, FUV flux and SFR density. The paper is arranged as follows. The observations and data are presented in Section \ref{data_s}, extinction in UV in Section \ref{ext_s}, analysis in Section \ref{analysis_s} followed by discussions and summary in Section \ref{discussion_s} and \ref{sumarry_s} respectively.
\begin{table}
\centering
\caption{Properties of IC 2574}
\label{ic2574}
\resizebox{90mm}{!}{
\begin{tabular}{ccc}
\hline
Property & Value & Reference\\\hline
RA & 10 28 23.5 & \citet{skrutskie2006}\\
DEC & +68 24 43.7 & \citet{skrutskie2006}\\
Distance & 3.79 Mpc & \citet{dalcanton2009}\\
Metallicity (Z) & $0.006$ & \citet{cannon2005}\\
Inclination & $63^\circ$ & \cite{pasquali2008}\\
PA of major axis & $55^\circ$ & \citet{pasquali2008}\\
H~I mass & $14.8\times10^8$ $M_{\odot}$ & \citet{walter2008}\\\hline
\end{tabular}
}
\end{table}
\section{Observations and Data}
\label{data_s}
We performed a deep FUV imaging observation of the galaxy IC 2574 using the UVIT on-board AstroSat satellite \citep{kumar2012}. The instrument consists of two co-aligned telescopes, one for FUV (1300 - 1800 $\AA{}$) and another for both NUV (2000 - 3000 $\AA{}$) and Visible. It has the capability of observing sources simultaneously in all the three available bands. The visible channel is mainly used to track the drift pattern in the image produced due to the motion of the satellite. Each of the telescopes has a 28 $\arcmin$ circular field of view with an angular resolution of 1.4 $\arcsec$ in FUV and 1.2 $\arcsec$ in NUV. Both the FUV and NUV channels of UVIT are equipped with multiple photometric filters which provide a unique imaging capability in the ultra-violet bands. The galaxy IC 2574 was imaged in F148W, an FUV broad band filter of UVIT. The data in the NUV channel was not made available due to payload related problems. The observations were carried out with 9 orbits of AstroSat. The images of each orbit are drift corrected, aligned and combined with the help of a customized software CCDLAB \citep{postma2017} to produce a final deep image of 10375 seconds exposure time (Figure \ref{uv_disk}). The details of observations are given in Table \ref{uvit_obs}. The image is flat-fielded, corrected for distortion and fixed pattern noise of the detector with the help of calibration files \citep{girish2017,postma2011}. The image has 4096$\times$4096 pixel dimension with 1 pixel corresponding to 0.4 $\arcsec$ , which is equivalent to 7.6 pc at the distance of IC 2574. All the calibration measurements of UVIT used in this study are adopted from the calibration paper of the instrument by \citet{tandon2017}. We also used the moment 0 H$~$I map, obtained from The H$~$I Nearby Galaxy Survey (THINGS) \citep{walter2008}, of the galaxy in our study.
\begin{table*}
\centering
\caption{Details of UVIT observations}
\label{uvit_obs}
\begin{tabular}{p{2cm}p{2cm}p{3cm}p{3cm}p{2cm}p{3cm}}
\hline
Filter & Bandpass & ZP magnitude & Unit conversion & $\triangle\lambda$ & Exposure time\\
& ($\AA$) & (AB) & (erg/sec/cm$^2$/$\AA$) & ($\AA$) & (sec)\\\hline
F148W & 1250-1750 & 18.016 & 3.09$\times10^{-15}$ & 500 & 10375\\\hline
\end{tabular}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics[width=7in]{ic2574_color_image1.pdf}
\caption{The FUV image of the galaxy IC 2574 observed with UVIT F148W filter.}
\label{uv_disk}
\end{center}
\end{figure*}
\section{Extinction in UV}
\label{ext_s}
The nature of extinction in UV band shows characteristic variation for different external galaxies. \citet{gordon2003} studied the behaviour of extinction in the LMC, the SMC and the Milky way and found differences specifically in the UV regime. Since FUV radiation is sensitive to extinction, we considered both the Galactic extinction foreground to IC 2574 and interstellar extinction within the galaxy IC 2574 in our study. The E(B$-$V) values for Galactic foreground reddening and interstellar reddening of the galaxy IC 2574 are reported to be 0.036 and 0.013 mag respectively \citep{pasquali2008}. The nature of extinction law in IC 2574 was considered to be SMC type in the previous study by \citet{pasquali2008} and \citet{stewart2000}. In order to calculate extinction value in F148W filter, we considered Fitzpatrick law \citep{fitzpatrick1999} for foreground and SMC bar type extinction law \citep{gordon2003} for the interstellar reddening of the galaxy IC 2574. The extinction coefficients are calculated using \textit{extinction law calculator}\footnote{http://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/YorkExtinctionSolver/coefficients.cgi} of \citet{mccall2004} available in NASA/IPAC Extragalactic Database (NED). We considered $R_{V}$ values as 3.07 and 2.75 respectively for Fitzpatrick and SMC bar type extinction law and estimated the average value of extinction coefficient ($R_{F148W}$) in F148W band (for the range 1250 - 1750 $\AA$). The estimated values of $R_{F148W}$ are found to be 8.57 and 13.06 respectively for Fitzpatrick and SMC bar type law. We calculated the value of extinction using equation
\begin{equation}
\label{extinction}
A_{F148W} = R_{F148W}E(B-V),
\end{equation}
where $R_{F148W}$ is the extinction coefficient, and found it as 0.31 mag and 1.70 mag respectively for foreground and interstellar extinction. The observed fluxes are corrected for both these extinctions in our analysis. It is to be noted that the value of interstellar extinction of the galaxy may have spatial variation and that can change the value of some of our estimated parameters.
\section{Analysis}
\label{analysis_s}
\begin{figure*}
\begin{center}
\includegraphics[width=6.5in]{ic2574_fuv_contours_shells1.pdf}
\caption{The background image is the UVIT FUV image of the galaxy IC 2574 observed in F148W filter. The H$~$I holes, as identified by \citet{walter1999}, are shown by black circles/ellipses. The number corresponding to each hole is same as assigned by these authors. The green and blue contours signify pixels with FUV flux more than $2.14\times10^{-18}$ and $3.21\times10^{-19}$ $erg/sec/cm^2/\AA$ respectively. The red contours denote regions with H$~$I column density greater than $1.0 \times 10^{21} cm^{-2}$.}
\label{shell}
\end{center}
\end{figure*}
\subsection{Distribution of FUV emission and H~I gas}
\label{fuv_distribution_s}
We considered the FUV image of the galaxy IC 2574 to locate young star forming regions. The distribution of FUV bright regions in the galaxy are mostly found to be clumpy in nature (Figure \ref{ic2574}). In order to understand the extent and characteristics of these regions more clearly, we generated contours for two different threshold fluxes and displayed them in Figure \ref{shell}. The blue and green contours shown in the figure respectively signify threshold value of $3.21\times10^{-19}$ and $2.14\times10^{-18}$ $erg/sec/cm^2/\AA$ , which are respectively equivalent to 3 and 20 times the average background flux. The reason behind choosing these two different values for generating the contours is to trace both the brighter and the fainter features together in the FUV emission profile of IC 2574. By choosing this lower limit, we expect to get rid of the background emission completely and to trace emission region only due to the galaxy. The green contours which trace relatively massive and more active star forming complexes, are found to be more compact in nature whereas we noticed the blue contours, which trace the overall FUV emission profile of IC 2574, to be more extended around the green contoured regions. Therefore, the star forming regions of the galaxy are found to have an overall patchy distribution with several compact clumps dispersed throughout the disk. \\\\
FUV emission, which mimics the current SFR in a galaxy, is found to correlate well with $H_{2}$ gas, whereas it does not follow a universal relationship with H$~$I gas throughout the galaxy \citep{bigiel2008}. In order to check the correlation of identified FUV bright regions with the column density of H~I gas, we plotted contours for $n_{HI} > 10^{21} cm^{-2}$ which is shown in red in Figure \ref{shell}. These density contours are generated from the VLA moment 0 H$~$I map of the galaxy available in \citet{walter2008}. The identified FUV bright star forming regions (both green and blue contours) of the galaxy are mostly found to be present within the extent of plotted H$~$I density contour. This simply conveys that star formation is mainly happening in regions with H~I column density greater than $10^{21} cm^{-2}$, which is known as the threshold value for star formation in IC 2574 \citep{walter1999}. There is no major star formation found outside the extent of the red contours.
\subsection{H$~$I shells}
The gaseous component of the galaxy IC 2574 has many hole like structures distributed all over its disk. Using VLA H$~$I observations, \citet{walter1999} identified 48 H$~$I shells and holes in the galaxy with radius in the range 100 - 1000 pc. They also found $H\alpha$ emission along the rims of larger holes which in other word signifies propagating star formation triggered due to the expansion of these holes. An expanding H$~$I hole can compress their surrounding ISM to trigger further star formation in a galaxy \citep{tagle2005}. In order to explore this, we have shown all these 48 holes as identified by \citet{walter1999}, overlaid on the FUV image of the galaxy in Figure \ref{shell}. A careful look reveals that many holes have bright FUV emissions (green contours) in their shells whereas some show signature of FUV emission inside the hole as well. We found 30 holes with FUV emission in their periphery while 15 holes show emission inside them. This signifies that star formation in certain parts of the galaxy could be triggered due to the expansion of H~I holes. It is also possible that some of these emissions are actually coming from other parts of the galaxy and they seem to be related to the holes due to projection in the sky plane.\\
As FUV radiation traces star formation up to a few hundred Myr and the H~I holes of IC~2574 are of $\le $ 50 Myr \citep{walter1999}, a detection of FUV bright region is not sufficient to confirm the existence of hole driven star formation. In order to verify whether the FUV regions are actually young (age $<$10 Myr), it is important to check whether they also show H$\alpha$ emission. We used the catalog of H$\alpha$ emitting regions of \citet{miller1994} for cross-identification. About 54\% of the FUV bright regions, detected in the shells, are also found to have H$\alpha$ emission. These are potential locations where star formation could be triggered due to the expanding holes. The emission identified inside the holes (specifically with H$\alpha$ counter part) can actually be coming from the shell, present either in front or back along the line of sight. We discuss these in detail in the next section. It is further noticed that some of the green contours in different places across the galaxy are present in between multiple holes. This gives an appearance of star formation happening in regions which are surrounded by holes. The feedback from the expanding holes may have compressed the ISM in between to make those sites more favourable for star formation. Both the above scenarios signify the impact of hole expansion in triggering star formation inside the galaxy. \\\\
Again, the fact that star forming regions are mostly identified in regions with H$~$I density more than $10^{21} cm^{-2}$ signifies that the mechanism which has shaped the H$~$I distribution inside the galaxy has also triggered secondary star formation in some of those sites. Therefore, in majority of the cases, the FUV emission, H$~$I density and the shell structure are likely to be co-spatial.
\subsection{FUV bright star forming regions}
\label{fuv_clump_s}
\begin{figure*}
\centering
\includegraphics[width=7.0in]{collage_hole.eps}
\caption{The figure shows some selected holes along with the identified FUV bright regions. The background image is UVIT FUV image of IC 2574. The black solid circles/ellipses show the holes of radius R, where the shells are defined by black dashed lines of width R/3 between radii (R$-$R/6) to (R$+$R/6). A length scale of 100 pc is shown in each image. The brown contours denote the FUV bright regions identified for a threshold flux of $1.07\times10^{-18}$ $erg/sec/cm^2/\AA$. The numbers shown in blue and black respectively signify the ID for holes and identified regions as given in Table \ref{clumps_remark}.}
\label{uv_clumps}
\end{figure*}
In order to identify FUV bright star forming regions across the galaxy, we used the Python package \textit{astrodendro}\footnote{http://www.dendrograms.org/}. This package helps to identify structures formed by a minimum number of pixels each having a value more than a defined threshold flux. It identifies both parent (larger structure) and child structures (smaller sub-structures within the larger one) on the basis of selected parameter value. We considered the threshold flux as $1.07\times10^{-18} erg/sec/cm^2/\AA$, which is 10 times the average FUV background flux. This threshold corresponds to the flux of a B5 spectral type star at the distance of IC 2574. The minimum number of pixels for the identification of regions is considered as 10, which corresponds to a radius of $\sim$ 15 pc for a circular area. The value of this lower limit is fixed by balancing two facts. One is to resolve the smaller clumps well and other is to avoid identifying a lot of them. With these parameters, we identified a total of 403 parent structures in the galaxy within a galactocentric distance of 10 kpc. These FUV bright regions are likely to be the active star forming regions of the galaxy. Out of the 403 parent clumps, 96 (23.8\%) are found to have multiple smaller sub-structures within them whereas the rest 307 (76.2\%) clumps show no sub-structures within the detection threshold. Among all the identified parent structures, we found one, located in the north-eastern part of the galaxy, to be exceptionally larger than the rest. This parent structure contains many smaller sub-structures inside it. We selected 17 bright sub-structures instead of this single large structure for our study. Hence we have a total of 419 FUV bright regions, which we considered for the rest of our analysis.\\
In order to identify an FUV bright region to be formed due to an expanding H~I shell, we need to define a thickness to the shell and check for FUV bright regions located within this shell. Here, we considered the width of the shell as R/3 (where R is the radius of the hole), from radii (R$-$R/6) to (R$+$R/6) for each hole (Figure \ref{uv_clumps}). Implication of this assumption is discussed in section \ref{discussion_s}. If an FUV bright region (or some part of it) is present within the radii (R$-$R/6) to (R+R/6) of any hole, then it is considered as part of the shell of that hole. If it is present within radius (R$-$R/6), then we assume it to be inside the hole. In case the region is located at a distance more than (R$+$R/6) from the hole centre, we considered it to be not related to that hole. Following this methodology, we identified regions which are connected with holes (either in the shell or inside) and listed them in Table \ref{clumps_remark}. \citet{walter1999} estimated the ages of these 48 holes and also classified them in three different types (Type 1, 2, 3) on the basis of their appearance in the p-V diagram. A hole is defined as Type 1, when neither of the receding and approaching sides of the hole can be observed in p-V diagram. In the case, when p-V diagram of a hole shows deformation indicating that it is offset with respect to galaxy plane is defined as Type 2 hole. A hole is called Type 3 when the velocity of both the receding and approaching sides can be measured. We noted the information of hole types in Table \ref{clumps_remark} from \citet{walter1999}.\\
Out of 419 identified regions, we found 120 (28.6\%) to be present in shells and 53 (12.6\%) to be present inside holes. It is to be noted that the list is made as per their locations in the projected sky plane and hence there can be a possibility to have a slightly different scenario in the galaxy plane. We also found 252 (60.1\%) regions as not related to any of the holes. Therefore, 60.1\% of the star formation happening in the galaxy has no connection with the holes. In other words, 30 (62.5\%) out of 48 holes are found to have FUV emission in their shell whereas 15 (31.2\%) holes show emission inside. We noticed 16 (33.3\%) holes to have no related FUV emission. There are 13 holes which show FUV emission both in shell and inside them.\\
In Figure \ref{uv_clumps}, we have shown some selected holes of various types along with the identified FUV bright regions. In the case of Hole 3 (top-left of Figure \ref{uv_clumps}), we noticed 8 regions (Regions 22, 27, 33, 43, 44, 55, 57, 58) to be present in the shell and 9 regions (Regions 28, 35, 37, 38, 40, 42, 46, 47, 52) are identified inside it (Table \ref{clumps_remark}). The regions found inside can actually be present in the shell but it appears to be inside the hole only in the projected plane. The other holes (except Hole 11) are also found to have emissions in their shell. Hole 11 is an example of holes which do not have any related FUV emission. Hole 8 and 32 show strong FUV emission inside them. We also noticed FUV emission in region between Hole 21, 22 and 24. This is an example where star formation is possibly triggered due to expansion/collision of multiple holes.\\
We further checked the H$\alpha$ counter part for each FUV region from the catalog of \citet{miller1994}. In case there is a cross-match, we mentioned the ID number of the H$\alpha$ emitting region (from their catalog) for the corresponding FUV region in parenthesis in Table \ref{clumps_remark}. We noticed 65 (15.5\% of total) out of 120 regions identified in the shell to have H$\alpha$ emission also. The detection of H$\alpha$ emission signifies a recent star forming activity in these regions. Therefore, these are the most possible sites where star formation has been triggered recently due to expanding H~I holes. We found 16 out of 53 regions present inside the holes to have H$\alpha$ detection. These 16 regions therefore have recent star formation and hence it is more possible that they are actually present in shells, but appears to be located inside holes. The same exercise is also performed for the 252 non-related regions and we noticed 106 of them to have H$\alpha$ counter part. In Table \ref{table3_sum}, we have extracted the summary of Table \ref{clumps_remark} for different types of hole and the identified FUV regions. We found that the number of holes with emission identified in their shells are 12, 5 and 13 respectively for Type 1, Type 2 and Type 3 holes respectively. Among the 15 holes with emission inside, we found 9 of Type 3, 4 of Type 1 and 2 of Type 2. It is noticed that 23 out of 30 holes, which show FUV emission in their shell, also have $H\alpha$ emission. Therefore, 47.9\% of the holes show positive signature of triggered star formation in their shell. We did not find any significant correlation between holes with/without triggered star formation and their ages.\\
To understand the physical properties of these 419 identified regions, we estimated several parameters and listed in Table \ref{uvit_clumps_table}.
We estimated the radius (R) in pc and the galactocentric distance ($R_{gc}$) in kpc for each region. The radius, which is estimated by equating the area of the irregular shaped regions to that of a circle, is found to have a range between $\sim$ 15 - 285 pc. 95\% of our identified regions have radii smaller than 100 pc. The galactocentric distances are estimated by assuming the galaxy centre, inclination and position angle of IC 2574 from Table \ref{ic2574} and using the relation given in section 2 of \citet{marel2001}.\\\\
\subsubsection{Estimation of FUV flux, Luminosity and SFR}
The \textit{astrodendro} package, discussed in section \ref{fuv_clump_s}, also provides area and flux for the identified regions. Considering these two, we measured the background and extinction corrected FUV flux ($erg/sec/cm^2/\AA$) for the regions.
The background flux is estimated from the average flux measured in four different circular regions of radius 1 arcmin present in the galaxy field. We choose the radius to be 1 arcmin as it is around 145 pixels in the image and thus covers a good area for estimating the average background. The circle of 1 arcmin radius also fits well in the space between the edge of the detector and the extent of the galaxy. The corrected fluxes are used to estimate luminosity density ($erg/sec/pc^2$) and SFR density ($M_{\odot}/yr/kpc^2$) corresponding to each region. We used the relation given by \citet{kara2013} for estimating the SFR from the measured FUV magnitude. The relation is given in
\begin{equation}
log(SFR_{FUV} (M_{\odot}/yr)) = 2.78 - 0.4*mag_{FUV} + 2log(D),
\label{sfr_eq}
\end{equation}
where $mag_{FUV}$ is the apparent FUV magnitude (AB system) and D is the distance to the galaxy in Mpc. The measured value of SFR density shows a range between 0.0238 - 0.5409 $M_{\odot}/yr/kpc^2$ which highlights the diversity of star forming regions in the galaxy. \\
\subsubsection{Estimation of H~I density}
As the estimated values of SFR density show a wide range, it will be interesting to correlate it with the average H~I column density of these regions. In Figure \ref{uv_clumps_hi}, we have shown the H$~$I column density map of the galaxy from \citet{walter2008} along with 419 FUV bright regions (blue circles). These circles are plotted for the equivalent area of each identified region. The red contours signify regions with H$~$I column density greater than $10^{21} cm^{-2}$. The identified star forming regions are mostly found to be present in locations with dense H$~$I gas. We measured the average column density of neutral hydrogen for the FUV bright regions from the moment 0 H$~$I map. The estimated values show a range between 0.11 - 3.83 $\times10^{21} cm^{-2}$. In Figure \ref{cps_uv_hi}, we have shown the FUV luminosity density and H$~$I column density for all these 419 regions. The average H~I column density is found to be more than $10^{21} cm^{-2}$ for 345 ($\sim82.3\%$) regions. Out of these 345 regions, 241 have values between $10^{21} cm^{-2}$ and $2\times 10^{21} cm^{-2}$, 95 between $2\times10^{21} cm^{-2}$ and $3 \times 10^{21} cm^{-2}$ and 9 with value greater than $3\times 10^{21} cm^{-2}$. The rest 74 regions ($\sim17.7\%$) show column density less than $10^{21} cm^{-2}$. The estimated value of the H~I column density will differ slightly from the actual value due to the circular approximation of the regions. Similarly for FUV surface luminosity density, we found 347 regions ($\sim82.8\%$) to have value between $10^{35}$ and $2\times10^{35}$ $erg/sec/pc^2$ and 72 regions ($\sim17.2\%$) have value more than $2\times10^{35}$ $erg/sec/pc^2$ with 11 of them brighter than $10\times10^{35}$ $erg/sec/pc^2$.
\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{ic2574_uv_clumps_HI_contour.pdf}
\caption{The background image is H$~$I moment 0 map of IC 2574. The gray scale signifies flux in JY/B*M/S. The red contours signify regions having H$~$I column density more than $10^{21} cm^{-2}$. The identified FUV bright star forming regions are shown as blue circles.}
\label{uv_clumps_hi}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=3.7in]{flux_uv_HI.eps}
\caption{The measured value of FUV surface luminosity density and H$~$I column density of 419 identified FUV bright regions are shown. The inset shows the zoomed in part for FUV luminosity density between $1 - 5\times10^{35}$ $erg/sec/pc^2$.}
\label{cps_uv_hi}
\end{center}
\end{figure}
\begin{deluxetable*}{ccccc}
\centering
\tablewidth{17.0cm}
\caption{Connection between FUV bright star forming regions and the H$~$I holes. This table helps to capture the location of the FUV bright regions with respect to the H~I holes. In the second column we listed the regions present in the shell. The regions present inside each hole are shown in column 3. Each region number signifies its ID, listed in Table \ref{uvit_clumps_table}. The numbers given within parentheses are the ID of H$\alpha$ cross-identified region from the catalog of \citet{miller1994}. The age of each hole and their types are given in columns 4 and 5 respectively from \citet{walter1999}.}
\label{clumps_remark}
\startdata \\
\hline
Hole & Region in shell & Region inside hole & Age of hole (Myr)\tablenotemark{a} & Hole type\tablenotemark{a}\\\hline
1 & -- & -- & 19.6 & 3\\
2 & -- & -- & 31.3 & 3\\
3 & 22,27,33,43(3),44,55,57(5),58(9) & 28,35,37,38,40(6),42,46,47,52(4) & 29 & 3\\
4 & 123(24) & -- & 42.5 & 2\\
5 & -- & -- & 43.5 & 3\\
6 & -- & -- & 20.6 & 3\\
7 & 36(18),45,53,59(16),69,70,74(22),79(42) & 54,64,66 & -- & 1\\
8 & -- & 124(32,34,36,38,39,40,41,44) & 12.2 & 3\\
9 & -- & -- & 30.2 & 3\\
10 & -- & -- & -- & 1\\
11 & -- & -- & 44.1 & 3\\
12 & -- & -- & -- & 1\\
13 & 99(57,58,60,63),102,112(61),114,117(30,33), & -- & -- & 1\\
& 119,120,124(32,34,36,38,39,40,41,44), & & & \\
& 133(45),141,148(66),149(64),150 & & & \\
14 & -- & -- & -- & 1\\
15 & 93 & -- & 58.7 & 3\\
16 & 116 & -- & 20.6 & 3\\
17 & -- & -- & 18.1 & 3\\
18 & -- & 122 & 19.9 & 3\\
19 & 281 & -- & 14.7 & 3\\
20 & 199(92) & -- & -- & 1\\
21 & 143(80,83,84),154(78),161(94,96,102,104),162,169 & 157(89),158(87,90),160(89) & 17 & 3\\
22 & 151(103),159,162,164,167,168,170,172, & 163 & 28.9 & 3\\
& 174(107),175,176 & & & \\
23 & 232,235(109),241(109),330 & 269,287 & -- & 1\\
24 & 161(94,96,102,104) & 187,189,190,193 & -- & 1\\
25 & -- & -- & 30.1 & 3\\
26 & -- & -- & -- & 1\\
27 & 231,249 & -- & -- & 1\\
28 & 171(115) & 177(121),181(121) & 23.5 & 3\\
29 & 343(118),356(128),358(128) & 346,349,354 & 18 & 3\\
30 & 177(121),181(121),198(120),202(119),205, & -- & -- & 1\\
& 209(124),211(173),213 & & & \\
31 & 1(133),2(134) & -- & -- & 1\\
32 & 266(136),291(165) & 271(143,155),288(155) & 12.7 & 3\\
33 & 303(144),308,310(151),311,313(144),314, & 318(150),319,320(150),322,327,328,329, & 16.3 & 2\\
& 323,335,340(186),344,348,350,360 & 333,334,337,338,341,355(178) & & \\
34 & 319,327,328 & -- & 13.1 & 3\\
35 & 7(137),8(137),9(138),10(138),14(192),15(194), & 5(148),6(148),12(172),13 & 14.3 & 2\\
& 16(198,202),17(167),364,365 & & & \\
36 & -- & -- & 21.1 & 3\\
37 & 386(170),393,396 & 387,389 & -- & 1\\
\hline
\enddata
\end{deluxetable*}
\setcounter{table}{2}
\begin{deluxetable*}{ccccc}
\centering
\tablecaption{continued}
\startdata \\
Hole & Region in shell & Region inside hole & Age of hole (Myr) & Hole type\\\hline
38 & 226(181),233(193,197,199,203,206,208,218), & -- & -- & 1\\
& 234,238 & & & \\
39 & 263(184),304,306,321 & -- & -- & 1\\
40 & 3(224) & -- & 11.7 & 3\\
41 & 156(212) & -- & -- & 1\\
42 & 385(235),391(235),403 & -- & 10.8 & 2\\
43 & -- & -- & -- & 2\\
44 & -- & -- & 11.7 & 3\\
45 & 285(236),298,300(241),301,305 & 286,292,293 & 30.2 & 3\\
46 & -- & -- & 19.6 & 3\\
47 & 220 & -- & 9.8 & 3\\
48 & 394 & -- & 17.6 & 2\\
\hline
\enddata
\tablenotetext{a}{From the study of \citet{walter1999}}
\end{deluxetable*}
\begin{table*}
\centering
\caption{Summary of Table \ref{clumps_remark}. Column 2 shows the total number for holes (along with three different types) and identified regions. Columns 3 and 4 respectively show the number of holes with FUV emission in shell and inside it. Column 5 lists the number of holes with no related FUV emission. The bottom row of the table denotes the similar statistics with respect to the identified regions. The numbers shown in parenthesis denote the numbers after cross-match with H$\alpha$ emission.}
\label{table3_sum}
\begin{tabular}{p{3cm}p{3cm}p{4cm}p{4cm}p{3cm}}
\hline
Hole & Total number & FUV emission in shell & FUV emission inside hole & With no related FUV emission\\\hline
Total hole & 48 & 30(23) & 15(7) & 16\\
Type 1 hole & 16 & 12(11) & 4(0) & 4\\
Type 2 hole & 6 & 5(4) & 2(2) & 1\\
Type 3 hole & 26 & 13(8) & 9(5) & 11\\
\hline
& Total number & Present in shell & Present inside hole & Not related with hole\\\hline
FUV Region & 419 & 120(65) & 53(15) & 252(106)\\\hline
\end{tabular}
\end{table*}
\begin{table*}
\centering
\caption{Properties of FUV bright star forming regions as defined in Figure \ref{uv_clumps}. The full table containing all 419 regions is available in electronic format.}
\label{uvit_clumps_table}
\begin{tabular}{p{1.0cm}p{1.7cm}p{2.0cm}p{1.2cm}p{1cm}p{0.7cm}p{1.4cm}p{1.2cm}p{1.7cm}}
\hline
Region & RA (J2000) (hh:mm:ss.s) & DEC (J2000) (dd:mm:ss.s) & Radius (pc) & $R_{gc}$\tablenotemark{a} (kpc) & Flux\tablenotemark{b} & Luminosity density\tablenotemark{c} & SFR density\tablenotemark{d} & Average HI column density\tablenotemark{e}\\\hline
1 & 10:28:37.1 & +68:27:57.1 & 22.2 & 5.64 & 1.70 & 9.39 & 0.1910 & 2.75\\
2 & 10:28:37.2 & +68:28:1.9 & 27.4 & 5.79 & 2.38 & 8.66 & 0.1762 & 2.61\\
3 & 10:28:53.0 & +68:28:35.1 & 29.6 & 6.25 & 2.10 & 6.51 & 0.1324 & 2.19\\
4 & 10:28:53.4 & +68:28:49.8 & 33.9 & 6.67 & 4.70 & 11.13 & 0.2263 & 1.75\\
5 & 10:28:40.5 & +68:28:1.0 & 22.6 & 5.58 & 2.37 & 12.61 & 0.2564 & 0.78\\
6 & 10:28:40.1 & +68:28:3.6 & 25.3 & 5.69 & 2.76 & 11.73 & 0.2386 & 0.69\\
7 & 10:28:38.8 & +68:28:6.7 & 17.1 & 5.86 & 1.35 & 12.56 & 0.2554 & 1.21\\
8 & 10:28:38.3 & +68:28:6.7 & 19.1 & 5.89 & 1.80 & 13.41 & 0.2727 & 1.58\\
9 & 10:28:38.5 & +68:28:8.9 & 14.8 & 5.95 & 1.27 & 15.73 & 0.3199 & 1.42\\
10 & 10:28:38.9 & +68:28:9.4 & 16.6 & 5.94 & 1.60 & 15.92 & 0.3238 & 1.13\\
11 & 10:28:39.0 & +68:28:30.3 & 44.9 & 6.63 & 5.91 & 8.00 & 0.1627 & 1.06\\
12 & 10:28:44.5 & +68:28:10.4 & 24.9 & 5.72 & 5.46 & 23.93 & 0.4868 & 0.16\\
13 & 10:28:44.4 & +68:28:7.0 & 30.5 & 5.62 & 8.52 & 24.88 & 0.5061 & 0.22\\
14 & 10:28:48.1 & +68:28:4.2 & 66.9 & 5.43 & 15.79 & 9.60 & 0.1953 & 1.91\\
15 & 10:28:48.6 & +68:28:35.0 & 66.0 & 6.35 & 15.82 & 9.90 & 0.2014 & 1.94\\
16 & 10:28:49.2 & +68:28:25.0 & 87.6 & 6.03 & 46.57 & 16.52 & 0.3361 & 2.19\\
17 & 10:28:43.8 & +68:28:26.3 & 121.2 & 6.25 & 143.31 & 26.59 & 0.5409 & 0.72\\
18 & 10:27:21.8 & +68:18:43.5 & 32.0 & 9.79 & 1.14 & 3.02 & 0.0615 & 0.11\\
19 & 10:27:28.7 & +68:20:56.3 & 29.6 & 6.97 & 0.52 & 1.63 & 0.0331 & 1.15\\
20 & 10:27:32.8 & +68:20:48.6 & 18.6 & 6.81 & 0.22 & 1.76 & 0.0359 & 0.85\\
21 & 10:27:33.3 & +68:20:54.1 & 53.2 & 6.70 & 1.98 & 1.90 & 0.0387 & 0.90\\
22 & 10:27:28.3 & +68:21:10.1 & 41.9 & 6.84 & 1.16 & 1.80 & 0.0366 & 0.71\\
23 & 10:27:34.8 & +68:20:54.5 & 32.9 & 6.60 & 0.62 & 1.57 & 0.0320 & 1.01\\
24 & 10:27:34.1 & +68:20:55.3 & 21.0 & 6.63 & 0.24 & 1.49 & 0.0302 & 0.88\\
25 & 10:27:32.8 & +68:20:59.5 & 13.5 & 6.64 & 0.10 & 1.56 & 0.0318 & 1.00\\
26 & 10:27:33.4 & +68:20:59.5 & 26.0 & 6.60 & 0.35 & 1.41 & 0.0287 & 0.95\\
27 & 10:27:29.4 & +68:21:10.7 & 53.6 & 6.75 & 1.66 & 1.57 & 0.0320 & 0.64\\
28 & 10:27:27.5 & +68:21:16.6 & 21.0 & 6.85 & 0.27 & 1.66 & 0.0338 & 0.60\\
29 & 10:27:22.1 & +68:21:36.9 & 15.4 & 7.27 & 0.13 & 1.47 & 0.0298 & 1.08\\
30 & 10:27:17.0 & +68:21:51.4 & 21.0 & 7.87 & 0.34 & 2.14 & 0.0435 & 1.78\\
31 & 10:27:34.2 & +68:21:6.2 & 21.8 & 6.45 & 0.32 & 1.82 & 0.0370 & 0.95\\
32 & 10:27:20.4 & +68:21:46.7 & 54.4 & 7.45 & 1.97 & 1.82 & 0.0369 & 1.66\\
33 & 10:27:23.0 & +68:21:36.8 & 21.0 & 7.17 & 0.25 & 1.53 & 0.0311 & 0.98\\
34 & 10:27:53.1 & +68:20:18.4 & 22.6 & 7.24 & 0.36 & 1.89 & 0.0385 & 0.97\\
35 & 10:27:30.9 & +68:21:16.8 & 29.9 & 6.55 & 0.49 & 1.48 & 0.0301 & 0.58\\
36 & 10:27:37.5 & +68:21:1.6 & 22.2 & 6.34 & 0.28 & 1.56 & 0.0317 & 1.64\\
37 & 10:27:31.1 & +68:21:19.4 & 22.2 & 6.51 & 0.29 & 1.58 & 0.0320 & 0.56\\
38 & 10:27:30.7 & +68:21:20.3 & 14.8 & 6.53 & 0.11 & 1.39 & 0.0283 & 0.59\\
39 & 10:27:19.5 & +68:21:52.9 & 31.4 & 7.56 & 0.57 & 1.57 & 0.0319 & 1.76\\
40 & 10:27:30.8 & +68:21:29.6 & 104.5 & 6.43 & 8.63 & 2.15 & 0.0438 & 0.62\\\hline
\end{tabular}
\tablenotetext{a}{Galactocentric distance}
\tablenotetext{b}{Total FUV flux in $erg/sec/cm^2/\AA$ $\times 10^{-15}$}
\tablenotetext{c}{FUV Luminosity in $erg/sec/pc^2$ $\times 10^{35}$.}
\tablenotetext{d}{SFR is in $M_{\odot}/yr/kpc^2$.}
\tablenotetext{e}{Density is in $cm^{-2} \times 10^{21}$}
\end{table*}
\begin{figure*}
\begin{center}
\includegraphics[width=6.7in]{dendrogram_ls.eps}\\
\includegraphics[width=6.7in]{dendrogram_updated.eps}
\caption{The figure shows 9 selected parent structures, along with their dendrograms, identified for a threshold flux $1.07\times10^{-18} erg/sec/cm^2/\AA$ (green dashed line). The red dashed lines represent flux value of $2.14\times10^{-18} erg/sec/cm^2/\AA$. The largest parent structure and its dendrogram are shown in the upper panel of the figure. A length scale of 100 pc is shown for each region in solid black line. The Y axis of dendrogram shows the flux in terms of counts per second while the X axis denotes unique identification number of each structure which is not related to their ID presented in the paper.}
\label{dendro_all}
\end{center}
\end{figure*}
\subsection{Structure of star forming regions}
Since the structure of star forming regions in galaxies is known to be hierarchical in nature \citep{elmegreen2000}, we further explored the structural characteristics of identified star forming regions as a function of varying flux level. We selected 9 relatively larger parent structures from the list of our identified regions (Section \ref{fuv_clump_s}) and shown them in Figure \ref{dendro_all}. Among these, we have also included the largest parent structure (top panel of the figure) identified in the north-eastern part of the galaxy. Each of these parent structure contains multiple sub-structures of different size and flux level inside them. In order to explore the characteristics of sub-structures we have shown dendrogram for each region in the same figure. Dendrograms are structure trees used to highlight the parent-child connection between identified structures of different flux levels. Each tree present in Figure \ref{dendro_all} signifies a parent structure whereas the leaves connected to that tree represent child structures present within the parent structure. The selected regions have different significance as per their location in the galaxy. They are located either in shell, inside hole, in between multiple holes or away from any hole. The idea is to check the structural nature of star forming regions located in different environments of the galaxy. In the dendrograms, the y axis denotes the FUV flux level of each sub-structure. The dendrogram of the largest parent structure (top panel of the Figure \ref{dendro_all}), which is distinctly different from the rest, shows many leaves with different flux levels. The brightest star forming knots present in the galaxy are actually part of this large region. In case of other parent structures which are much smaller than this large region, we noticed similar nature of dendrogram with much reduced number of leaves, but with different flux levels. The brighter star forming clumps (sub-structure) identified throughout the galaxy are found to be present inside such large complexes (parent structure). Therefore, irrespective of location, the larger star forming complexes have multiple sub-structures inside them.
\begin{figure}
\begin{center}
\includegraphics[width=3.7in]{ic2574_l_pc2_i63_pa55_d3790.eps}
\caption{The radial surface luminosity density ($erg/sec/pc^2$) profile of the galaxy.}
\label{radial}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=3.3in]{ic2574_uv_clumps_distance_new.pdf}
\caption{The FUV image of the galaxy IC 2574. The regions highlighted as S1, S2, S3, S4, S5 and S6 are the active star forming regions of the galaxy.}
\label{uv_disk_dist}
\end{center}
\end{figure}
\subsection{FUV Luminosity density profile of IC 2574}
The H~I holes and the bright star forming regions both have a scattered distribution in the galaxy IC 2574. We also noticed the presence of active star formation from inner to far outer part of the galaxy. In order to understand the overall FUV emission profile of the galaxy as a function of galactocentric distance, we plotted radial surface luminosity density ($erg/sec/pc^2$) profile in Figure \ref{radial}. We estimated galactocentric distance to each of the pixel of the FUV image similarly as explained in section \ref{fuv_clump_s}. Starting from the galaxy centre, we considered annulus of width 0.5 kpc up to a radius 10 kpc and estimated total flux in each individual annuli. The measured fluxes are then corrected for background and extinction. The background is estimated from an annuli between radius 12 kpc and 13 kpc whereas extinction correction is done similarly as discussed in Section \ref{ext_s}. The corrected fluxes ($erg/sec/cm^2/\AA$) are converted to luminosity ($erg/sec$) by adopting a distance to the galaxy as 3.79 Mpc \citep{dalcanton2009} and bandwidth of F148W filter as 500 $\AA$ (Table \ref{ic2574} \& \ref{uvit_obs}). Then the estimated luminosity values are divided by the area of each individual annuli to calculate surface luminosity density ($erg/sec/pc^2$) at that particular radius and shown in Figure \ref{radial}. To understand the characteristics of luminosity density profile, we have shown the FUV image of IC 2574 and highlighted the location of six bright star forming regions of the galaxy as S1, S2, S3, S4, S5 and S6 in Figure \ref{uv_disk_dist}. The radial profile shows an exponential decrease up to a radius 2.5 kpc with a central peak which is due to the star forming region S1. Beyond 2.5 kpc, we noticed two peaks, one at 3.5 kpc and another at 6 kpc from the galaxy centre. Emission from the region S3 has major contribution for the peak seen at 3.5 kpc, whereas the peak at 6 kpc is mainly due to S2, the brightest star forming region of the galaxy. The nature of the radial luminosity density profile signifies the presence of active star formation in the outer part of the galaxy also.\\
We also estimated the background and extinction corrected total flux of the galaxy in F148W filter for a radius of 10 kpc and the value is 3.3$\times10^{-12}$ $erg/sec/cm^2/\AA$. The corresponding F148W magnitude and the total SFR (estimated using the relation of \citet{kara2013}) of the galaxy are estimated to be 10.45 mag and 0.57 $M_{\odot}/yr$ respectively. It is to be noted that if we use different relations for estimating SFR from the measured FUV flux, we get different values. For example, the relation provided by \citet{murphy2011} and \citet{hunter2010} results in a SFR of 0.12 and 0.18 $M_{\odot}/yr$ respectively for the same estimated flux of the galaxy IC 2574.\\
\begin{table}
\centering
\caption{Starburst99 model parameters}
\label{starburst99}
\resizebox{90mm}{!}{
\begin{tabular}{cc}
\hline
Parameter & Value\\\hline
Star formation & Instantaneous\\
Stellar IMF & Kroupa (1.3, 2.3)\\
Stellar mass limit & 0.1, 0.5, 120 $M_{\odot}$\\
Total cluster mass & $10^3 M_{\odot}$-$10^6 M_{\odot}$\\
Stellar evolution track & Geneva (high mass loss)\\
Metallicity & Z=0.004\\
Age & 10 Myr\\ \hline
\end{tabular}
}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=3.5in]{remnant_cluster.pdf}
\caption{The FUV image of H$~$I shell 35 (red ellipse) along with the remnant cluster (blue circle) is shown in the left. The zoomed in view of the cluster is shown in right where two resolved components are clearly noticed.}
\label{sgs35}
\end{center}
\end{figure}
\subsection{Remnant cluster of Super Giant Shell}
\label{section_rc}
Among the 48 identified shells, there are multiple SGSs present in the galaxy IC 2574 \citep{walter1999}. One of the present SGSs (shell 35 as per \citet{walter1999}, Figure \ref{shell}), is studied extensively in literature due to its prominent multi-wavelength characteristics. \\
\begin{table}
\centering
\caption{Properties of the remnant cluster}
\label{cluster_table}
\begin{tabular}{p{1.5cm}p{1cm}p{4cm}p{1.5cm}}
\hline
Component & Radius & FUV Flux & Mass \\
& (pc) & ($10^{-15}$ $erg/sec/cm^2/\AA$) &($10^4 M_{\odot}$) \\\hline
RC1 & 30.3 & 7.1 & 4.9\\
RC2 & 41.7 & 13.2 & 7.3\\\hline
\end{tabular}
\end{table}
In Figure \ref{sgs35}, we showed the UVIT FUV image of this SGS 35. The presence of remnant cluster as well as star forming regions along the rim are clearly noticed in the image. With the spatial resolution of UVIT, we identified two resolved components of the remnant cluster (RC1 and RC2 in Figure \ref{sgs35}) which were not detected earlier by both UIT and GALEX. The individual components are shown in the figure by blue circular apertures. In order to study the remnant cluster, we estimated the background and extinction corrected FUV fluxes for both the components shown in right panel of Figure \ref{sgs35}. \citet{stewart2000} estimated the age and mass of the central cluster and the reported values are 11 Myr and 14.2$\times10^4$ $M_{\odot}$ respectively. Assuming the age to be 10 Myr, we also estimated the masses for both the components with the help of starburst99 SSP \citep{leitherer1999} model. The model parameters assumed for this estimation are listed in Table \ref{starburst99}. All the measured parameters of the cluster are listed in Table \ref{cluster_table}. The added mass of both the components is $12.2 \times 10^4 M_{\odot}$ which matches closely with earlier estimate by \citet{stewart2000}.
\section{Discussions}
\label{discussion_s}
Giant H~I holes present in some dwarf galaxies represent a prominent feature in their interstellar medium (\citet{egorov2014} and references therein). The slow solid body-like rotation of the dwarf galaxies and the lack of strong spiral density waves, help in the formation of larger sized and long lived holes in these galaxies. Dwarf galaxies are known to sustain star formation over a very long period. It is thus important to understand how the presence of holes and the sustained star formation go hand in hand in dwarf galaxies. The key aim of this study is to identify young star forming regions in the galaxy IC 2574 and further explore their connection with the H$~$I holes. In order to do that we used deep FUV observations of the galaxy with UVIT F148W filter. A 28$\arcmin$ field of view of the telescope along with a pixel scale of 0.4$\arcsec$ has helped us to image the whole galaxy with finer details.\\
We noticed a good spatial correlation between the FUV emission and the H~I column density throughout the galaxy. The FUV bright regions of the galaxy are mostly found to have H~I column density more than $10^{21} cm^{-2}$, which is the reported threshold value for star formation in IC 2574. This signifies star formation, in this gas-rich dwarf galaxy, is mainly happening in regions which have higher gas density. We also noticed some active regions to have H~I column density less than $10^{21} cm^{-2}$. As the FUV emission in some of these regions is high, it is possible that the recent star formation has used up or ionized the neutral gas there. It is also possible that a few of these regions are not part of the galaxy and probably some background sources.\\
We identified 419 star forming regions in the galaxy by fixing the threshold flux as $1.07\times10^{-18}$ $erg/sec/cm^2/\AA{}$ (10 times average background flux) and minimum number of pixel to define a region as 10. A threshold flux lower than the selected value will result in identifying more number of regions and also the same regions with a little bigger size. In case of a higher threshold flux, the identified regions will be less in number and smaller in size. In order to keep a balance, we fixed an intermediate value for our analysis. The value of minimum number of pixel for identifying the regions will decide the minimum size of the region we want to detect. The value we adopted for our analysis can detect star forming clumps smallest up to radius $\sim$ 15 pc. The number of identified regions will increase with decreasing value of the minimum number of pixel and the vice versa. We fixed the value as 10 to identify small star forming clumps and also to avoid detecting very small regions which may not be part of the galaxy.\\
The nature of star formation shows broad characteristic variation for different dwarf galaxies. Both the external environment and internal feedback play dominant role in regulating star formation in dwarfs. Our study highlights that the expanding H$~$I holes have a major impact in triggering secondary star formation in some part of the galaxy. H$~$I holes are mainly created due to the combined effect of stellar winds and supernova explosions on to the ISM of a galaxy \citep{tagle1988}. \citet{tagle2005} concluded that the massive and compact super star clusters contribute a positive feedback for triggering further star formation around them. We find that star formation could be started either due to scooping of material by an expanding H~I shell and/or collision and compression of matter due to collision of multiple shells. We found that out of 48 holes, 30 show FUV emission in their shells, 15 holes have emission inside and 16 holes do not have any related FUV emission. This denotes that more than half of the holes (30/48) have active star formation in their shells, whereas 16/48 holes do not show any triggered star formation. We also noticed that holes with no related FUV emission mostly lie in the outer part of the galaxy. We found 9 out of 16 holes, which do not show triggered star formation, to be located outside 8 kpc radius of the galaxy, whereas only 1 out of 32 holes, with related FUV emission, is found outside 8 kpc. This signifies that the holes which are formed in the outer part of the galaxy could not trigger star formation. This may happen due to low density of available gas in the far outer part of the galaxy. Our study finds that 28.6\% of the identified star forming regions are located in shells while 12.6\% are present inside. We also found 60.1\% of the regions to be present away from holes and are not related. These numbers are estimated on the basis of our adopted shell width as R/3. The width of the shell can actually be different from our assumed value. This can slightly alter our estimated numbers for each individual holes.\\
In order to identify the star forming regions younger than 10 Myr we checked for their H$\alpha$ counter parts. It turns out that 65 of 120 regions present in the shell show both FUV and H$\alpha$ emission. In other words, 23 holes show both H$\alpha$ and FUV emission in their shells. Therefore, $\sim$ 48\% holes show signature of recently triggered star formation. As per the estimation of \citet{walter1999}, these holes cover an age range of $\sim$ 10 - 40 Myr. If star formation is triggered in some region due the expansion of a hole, then that has to be younger than the age of the hole. As the regions with both FUV and H$\alpha$ emission are likely to be younger than 10 Myr, the detection of these regions in the shell signify that star formation has been triggered there due to the expansion of holes. The regions present in the shells with only FUV emission may be a little older and hence do not show H$\alpha$ emission. Among 252 regions, which are not related with the holes, 106 found to have H$\alpha$ emission also. This means 60.1\% of the identified regions in the galaxy are undergoing star formation triggered due to other mechanisms with 25.3\% experiencing most recent trigger. Therefore, expansion or collision of H~I holes is not the only mechanism to cause recent enhancement of star formation in the galaxy. The cross-identification of H$\alpha$ emission is done from the available catalog of \citet{miller1994}. The H$\alpha$ observations have $\sim$ 4 times shallower exposure than that in FUV. In the case of some FUV bright regions, there can be faint H$\alpha$ emissions which are not detected in this H$\alpha$ image. Hence, we expect a few more cross-detection with much deeper H$\alpha$ image. That can slightly change the statistics of our results.\\
The radius of the identified regions cover a range between 15 - 285 pc, with 95\% of them to be smaller than 100 pc. The larger regions are the big parent complexes with smaller sub-structures of size 15 - 100 pc inside. As these clumps are bright in FUV, it is possible that some of them are OB associations. The sizes of OB associations cover a range between 10 - 100 pc for the Milky way and other nearby spiral and dwarf galaxies \citep{melnik1995,bresolin1996,ivanov1996,bresolin1998,bastian2007}. Therefore, the galaxy IC 2574 have produced OB associations which are similar in size with those of other nearby galaxies. In order to characterize the star forming regions, we measured their FUV surface luminosity density and the average H$~$I column density. We noticed a clear variation in the properties of these regions across the galaxy. Some regions show very high FUV luminosity with less H$~$I density, signifying vigorous recent star formation. Majority of the regions have moderate FUV luminosity with H$~$I density above the threshold, whereas we also do notice regions with very high H$~$I density and moderate FUV luminosity. These altogether indicate that the galaxy IC 2574 has a variable star forming environment throughout it. In this study, we assumed a fixed extinction value throughout the galaxy and used that to estimate corrected FUV flux, luminosity density and SFR density (Table \ref{uvit_clumps_table}). Any change in the assumed extinction value will affect the estimated values of these parameters accordingly.\\
As it has been suggested that the nature of star forming region is hierarchical from smaller scale to larger scale \citep{elmegreen2000,efremov1995}, we also explored the structural characteristics of star forming regions in the galaxy IC 2574. We noticed that majority of the large star forming complexes in IC 2574 have several smaller sub-structures of different flux levels. It is further observed that the brighter star forming clumps of the galaxy are mostly present inside larger complexes. By analysing the dendrogram of some selected large regions, we understand that in different flux levels, star forming regions form similar structure of different sizes. This highlights the hierarchical nature of these active regions. As the hierarchy is noticed for regions both related and not related to the holes, we speculate turbulence of the ISM as the primary reason behind this \citep{elmegreen2000,elmegreen2014,grasha2017}.\\
We produced the radial surface luminosity density profile of the galaxy and able to trace FUV emission at least up to a radius 10 kpc. The presence of two major bright star forming regions, at radius 3.5 kpc and 6 kpc, are identified in this radial profile. The brightest star forming region which is located at 6 kpc from the centre of the galaxy signifies that the star formation in IC 2574 is not concentrated only in the inner part of the galaxy. The presence of this active region clearly reveals that star formation in dwarfs can be dominant in any part of the galaxy depending upon a favourable star forming environment. We estimated the total FUV flux of the galaxy within 10 kpc radius and found it to be 3.3$\times10^{-12}$ $erg/sec/cm^2/\AA$. The measured SFR of the galaxy is 0.57 $M_{\odot}/yr$, which signifies a relatively active nature of IC 2574.\\
With the help of UVIT's spatial resolution, we identified two resolved components of the remnant cluster of shell 35. \citet{stewart2000} reported the discovery of this remnant cluster (single component) using UIT data and derived its properties. We derived the mass for each of the resolved cores by using starburst99 SSP model. The bigger component (M $\sim 7.3 \times 10^4$ $M_{\odot}$) is found to be 1.5 times massive than the smaller component (M $\sim 4.9 \times 10^4$ $M_{\odot}$). The added mass (M $\sim 12.2 \times 10^4$ $M_{\odot}$) of these components matches well with the mass of the remnant cluster measured earlier by \citet{walter2008}.
\section{Summary}
\label{sumarry_s}
The main results of the study are summarized below
\begin{enumerate}
\item We identified 419 FUV bright star forming regions in the galaxy IC~2574 with the help of UVIT FUV imaging data.
\item We estimated several parameters, such as size, FUV flux, surface luminosity density, H~I column density, SFR density, galactocentric distance for the identified regions.
\item We found 28.6\% of the identified regions to be located in H~I shells, 12.6\% inside holes and 60.1\% away from holes.
\item 30 out of the 48 holes show triggered star formation in their shells with 23 of them having more recent trigger, whereas 16 holes do not show any triggered star formation. We also found 15 holes to have FUV emission inside them, with 12 of those having emission in their shells as well.
\item Star formation in 60.1\% of the identified regions in IC~2574 has no connection with the H~I holes and hence it is possibly triggered due to other mechanisms.
\item The identified regions have radii between 15 - 285 pc, with 95\% of them smaller than 100 pc.
\item 82.3\% of the identified FUV bright regions have H~I column density more than $10^{21} cm^{-2}$.
\item We found sub-structures of different flux levels and sizes inside the larger star forming complexes across the galaxy. We speculate turbulence as one of the dominant drivers to build this hierarchy.
\item The galaxy is found to have active star formation in the outer part beyond 5 kpc also.
\item We resolved two individual components for the remnant cluster of shell 35 and estimated their masses. The added mass of both the component is $\sim 12.2 \times 10^4$ $M_{\odot}$ with the larger one to be 1.5 times more massive than the smaller one.
\item The star formation rate of the galaxy is found to be $\sim$0.57 $M_{\odot}/yr$.
\end{enumerate}
\acknowledgments
UVIT project is a result of collaboration between IIA, Bengaluru, IUCAA, Pune, TIFR, Mumbai, several centres of ISRO, and CSA. Indian Institutions and the Canadian Space Agency have contributed to the work presented in this paper. Several groups from ISAC (ISRO), Bengaluru, and IISU (ISRO), Trivandrum have contributed to the design, fabrication, and testing of the payload. The Mission Group (ISAC) and ISTRAC (ISAC) continue to provide support in making observations with, and reception and initial processing of the data. We gratefully thank all the individuals involved in the various teams for providing their support to the project from the early stages of the design to launch and observations with it in the orbit. This research made use of Matplotlib \citep{matplotlib2007}, Astropy \citep{astropy2013,astropy2018}, Astrodendro (http://www.dendrograms.org/), community-developed core Python packages for Astronomy. Finally, we thank the referee for valuable suggestions.
\software{CCDLAB \citep{postma2017}, SAOImageDS9 \citep{joye2003}, Matplotlib \citep{matplotlib2007}, Astropy \citep{astropy2013,astropy2018}, Astrodendro (http://www.dendrograms.org/)}
|
1,116,691,500,492 | arxiv | \section{Introduction}
Cosmic rays (CRs, hereafter) are high-energy particles that
generate randomly distributed, large signals on charge-coupled
devices (CCDs), which could affect the measured fluxes of
astronomical objects if not detected or removed properly.
Generally, CRs are removed by combining three or more
exposures of the same field \citep{whi94, fre95, fru97, gru14, des16}
, as they are unlikely
to hit the same pixel in more than one exposure. However,
multiple exposures are not always available. Furthermore, there
are certain situations in which CR detection in single exposures
is desired, such as in time domain studies.
Various methods have been developed for identifying and
replacing CRs in CCD data of single exposures, including
median filtering (e.g., Dickinson’s IRAF tasks
\scriptsize QZAP\normalsize{}, \scriptsize XZAP\normalsize{}, and \scriptsize XNZAP\normalsize{}),
applying a threshold on the contrast (e.g., IRAF
task \scriptsize COSMIC-RAYS\normalsize{ }),
trainable classification \citep{mur92, sal95, ber96},
convolution with adapted point-spread functions \citep[PSFs; ][]{rho00},
Laplacian edge detection \citep{van01},
analysis of the flux histogram \citep{pyc04}
and a fuzzy logic-based method \citep{sha05}.
All of the median filtering or PSF methods remove small CRs
from well-sampled data effectively, but problems arise when
CRs affect more than half the size of the filter or when the PSF
is smaller than the filter \citep{van01}.
All of the
methods listed above are designed for photometric data except
those of \citet{van01} and \citet{pyc04}, which work for long-slit spectroscopic data.
\citet{far05} made a comparison between different methods
including the IRAF script \scriptsize JCRREJ2 \normalsize{} of \citet{rho00},
the IRAF routine L.A.C\scriptsize{OSMIC}\normalsize{} of \citet{van01},
the C script of \citet{pyc04} and the IRAF task \scriptsize XZAP\normalsize{} on photometric images.
In that paper, Farage concluded that L.A.C\scriptsize{OSMIC}\normalsize{} provided the
best performance, with a detection efficiency of 86\% on the real
data sample, whereas other methods could at most detect 78\% of
the CRs. Increasing object density reduces the efficiency of
detection \citep{far05},
which is unfortunately unavoidable in
multi-fiber spectroscopic data where the signals are always
dense. Although L.A.C\scriptsize{OSMIC}\normalsize{} efficiently detects CRs, it replaces
the identified CR candidates with the median value of the
surrounding good pixels \citep{van01},
which is improper
when the CR hits are on the ridge or slope of the profile.
There have been no specific efforts to solve this problem on
multi-fiber spectroscopic data, which present distinct challenges
compared to photometric data. Multi-fiber images do not
have clear isolated point or extended sources as in the
photometric data, and the long stripe-like multi-fiber spectra
occupy large contiguous regions so that the available area for
the local “background” is much smaller than in the photometric
data. Methods with median filtering or interpolation of
neighboring pixels are less effective in this case.
We present an algorithm to detect and replace CRs for
Large Sky Area Multi-Object fiber Spectroscopic Telescope
(LAMOST) single-exposure images based on a two-dimen-
sional (2D) profile fitting of the spectral aperture. We first pick
out CR candidates with Laplacian edge detection and construct
a 2D function to fit the image profile in small segments along
the spectral trace with these candidates masked out; the final
CR list is generated by comparing the fitting residual with a
noise model depending on position, and the CR polluted pixels
are replaced with the corresponding value of the 2D function.
This method is applied to the data processing of LAMOST; in
principle, it can also be used for other multi-fiber spectral data
after minor modification.
We describe LAMOST data in \S2. The algorithm is explained in \S3.
In \S4, we give some examples and analyze the properties of the algorithm.
Finally, in \S5 we summarize our work.
\section {LAMOST Data}\label{sec2}
LAMOST \citep{cui12} is a fiber spectroscopic telescope
equipped with 4000 fibers feeding 16 spectrographs. Each
spectrograph, holding 250 fibers, is split into blue
(3700–5900 \AA) and red (5700–9000 \AA) arms by a dichroic
mirror. Groups of 250 spectra are recorded by two $4k{\times}4k$
CCDs at the blue and red end, respectively. The typical
duration for a single LAMOST exposure ranges from 600 to
1800s, depending on the target brightness and weather
conditions. A considerable number of CRs hit the images
during the exposure; for example, in a typical 1800s image, the
number of pixels polluted by CRs is about
$2\times10^4$.
The size of a LAMOST image, of which the dispersion
direction is along the vertical direction, is 4096$\times$4136 pixels.
In the spatial direction, the typical distance between two
adjacent fibers is 15$\sim$16 pixels. As shown in Figure \ref{fibersec}, the
cross section of the fiber profile in the spatial direction could
be well described by a S\'ersic function \citep{ser68, cle02}:
\begin{equation}\label{eq:1}
P(x)=\alpha{e}^{-\frac{|x-\beta|^\delta}{\delta\gamma^\delta}},
\end{equation}
where $\alpha$, $\beta$, $\gamma$, $\delta$ are parameters to be derived. The typical
full width at half maximum (FWHM) of the profile is about 7$\sim$8 pixels.
If $d=|x-\beta|$ is the distance from a given pixel
to the fiber profile center in the row (or horizontal/spatial)
direction, according to Figure ~\ref{fibersec}, the flux at $d=8$ is less than
0.01\% of those at the profile peak. To avoid fiber to fiber cross
talk, the magnitude range of objects observed in one LAMOST
observation is constrained to be less than 5 magnitudes. In the
extreme case, the contribution from the 5 magnitude brighter
neighbor to the pixel at $d=8$ could be ignored, so the fluxes in
the pixels of $d\leq 8$ could be considered as the flux from the
fiber itself. In 2D data reduction, $d=8$ is chosen as the
aperture for spectrum extraction. The spectral resolution of
LAMOST is about 1800, which corresponds to a FWHM about
5 pixels in the dispersion direction. The PSF changes gradually
with position on the CCD chip, but could be considered as
constant in a small region (e.g., 20 pixels); we will take
advantage of this characteristic to improve the cosmic ray
rejection.
\section {Cosmic Ray Detection and Rejection}
CRs are detected and replaced in three steps. First, we use
Laplacian edge detection \citep{van01} to generate a raw
CR candidate list. Second, for each fiber, pixels within $d=8$
of the fiber trace center are divided into small blocks; each
block is then fitted by a 2D profile with those raw CR
candidates masked out. The final CR list is determined by
comparing the fitting residual with a noise model considering
both the intrinsic noise and the uncertainty introduced by
profile fitting; the pixels polluted by CRs are replaced by the
corresponding fitted value. The details are as follows.
\subsection {Laplacian edge detection \label{sec:LACOS}}
Laplacian edge detection has been widely used for highlighting
boundaries in processing digital images \citep[e.g., ][]{gon92}.
\citet{van01} was the first to apply the
method to detect CRs in astronomical images. Their publicly
available program, L.A.C\scriptsize{OSMIC}\normalsize{}, successfully detects CRs in
both photometric and long-slit spectroscopic images. We use a
similar method to that in Section 3 of \citet{van01} to
pick out the raw CR candidates. Since further details can be
found in that paper, only basic steps are listed here.
The original image $I$ with the size of
${n_x}\times{n_y}$ is subsampled into ${2n_x}\times{2n_y}$:
\begin{equation}
I^{(2)}_{i,j}=I_{int[(i+1)/2],int[(j+1)/2]},
\end{equation}
where $i=1,\cdots,2n_x$ and $j=1,\cdots,2n_y$.
The subsampled image is then convolved with a Laplacian kernel:
\begin{equation}
\L=\mathcal{L}*I^{(2)},
\end{equation}
where $\mathcal{L}$ is the Laplace operator, $*$ denotes convolution.
The Laplace operator $\mathcal{L}$ in the above convolution is
\begin{equation}
\mathcal{L}=\frac{1}{4}\left( \begin{array}{ccc}
0 & -1 & 0 \\
-1& 4 & -1 \\
0 & -1 & 0
\end{array} \right).\
\end{equation}
Since CRs are positive in $\L$, all the negative values are set to zero.
$\L$ is then resampled to ${n_x}\times{n_y}$:
\begin{equation}
\L^{+}_{i,j}=\frac{1}{4}(\L_{2i-1,2j-1}+\L_{2i-1,2j}+\L_{2i,2j-1}+\L_{2i,2j}),
\end{equation}
where $i=1,\cdots,n_x$ and $j=1,\cdots,n_y$.
The original image is median filtered with a
${5}\times{5}$ box to construct the noise model
\begin{equation}
N_{m5}=\frac{1}{g}\sqrt{g(I_{m5})+\sigma_{rd}^2}, \label{nm5}
\end{equation}
where $g$ is the gain in electrons per ADU, $I_{m5}$ is
the image median filtered by a ${5}\times{5}$ box,
and $\sigma_{rd}$ is the readout noise in
electrons. The Laplacian image is then divided by the noise
model and the subsampling factor to obtain the deviations from
the expected Poisson fluctuations:
\begin{equation}
S=\frac{\L^{+}}{2N_{m5}}.
\end{equation}
All structures that are smooth on scales of $\geq{5}$ pixels are removed by
a ${5}\times{5}$ median filter:
\begin{equation}
S^{'}=S-S_{m5}.
\end{equation}
All pixels that meet $S^{'}>\sigma_{lim}$ are identified as CR candidates,
where $\sigma_{lim}$ is a given threshold.
We adopt $\sigma_{lim}=4.5$, similar to
L.A.C\scriptsize{OSMIC}\normalsize{}.
\subsection{Fiber profile fitting}
Fiber traces are closely aligned on multi-fiber spectral
images. In contrast to photometric images, useful signals are
quite fully rather than sparsely distributed. As pointed out by
Farage \cite{far05}, the increasing object density will certainly
reduce the efficiency of the methods designed for photometric
data. The ramp on either side of the ridge of the fiber trace is
quite steep, so it is hard for the edge detection method to
discriminate between real CR hits and good pixels on the ramp,
leading to a drop in the detection rate and a rise in the false
detection rate. Furthermore, replacing the CR polluted pixels
with the median of the surrounding pixels is seemingly unsafe.
Along the fiber trace, the shape of the PSF changes slowly. If
the PSF is well sampled, then CR discrimination could be
improved by its shape difference from the PSF.
For an image $I$ of ${n_x}\times{n_y}$ pixels with $n_f$ fibers, the pixels
close to the fiber trace center contribute the most to the
extracted flux. CRs hitting on these areas will introduce large
errors in the final spectra, while those in the trough between
fibers have much less impact. Consider a small spectral
segment centered on column $[c_{kj}]$ and row $j$,
where $k=1,\cdots,n_f$, $j=1,\cdots,n_y$ and $[c_{kj}]$ is the trace center of
the $k$th fiber at row $j$($[]$ denotes the round off of the quotation).
Since both the trace center and the shape of the PSF change
slowly inside the segment, the shape of the segment could be
fitted with a product of two orthogonal vectors:
\begin{eqnarray}
I_{xy} & = & \mathcal{F}_{xy}+\varepsilon_{xy} \nonumber\\ & = & S(x)P(y)+\varepsilon_{xy} \label{eqfit},
\end{eqnarray}
where $S(x)$ is the fiber profile in the spatial direction, $P(y)$ is a
polynomical to describe the flux variation in the dispersion
direction, and $\varepsilon_{xy}$ is noise. The size of $x$ is the same as the
aperture for flux extraction, which is set to $d=8$ for LAMOST, as discussed in Section \ref{sec2}.
The size of $y$ is chosen to be small enough to keep $P(y)$
smooth but larger than the size of single CR hits,
so that $P(y)$ can be fitted with a low-order
polynomial and the CR polluted pixels could be better
estimated by interpolation. For LAMOST, the segment size is
set $17\times 9$, i.e., $x=[c_{kj}]-8,\cdots,[c_{kj}]+8$ and
$y=j-4,\cdots,j+4$. We do not try to fit the pixels in the
bottom of the valley between fibers, since they contribute little
to the final spectrum.
The shape of $S(x)$ is determined by the output pupil of the
fiber and instrument distortion. Although the typical shape of
$S(x)$ could be described by a S\'ersic function (Eq. \ref{eq:1}),
the
actual shape deviates occasionally from the analytic function
when the optical distortion is large at the edge of the image or
the coupling between the fiber output pupil and the slit is
imperfect. Due to the above reason, $S(x)$ is constructed with an
empirical profile rather than an analytic function. All the
profiles at $y=j-10\sim{j+10}$ are first normalized, center
justified in sub-pixel scale and then averaged to derive $S(x)$
with the CR candidates masked out. Fixing the form of $S(x)$,
the polynomial coefficients of $P(y)$ are derived by least-square
surface fitting to the flux in the segment with the CR candidates
masked out. A fitted image $\mathcal{F}$ is generated after all segments are handled.
\subsection {Cosmic ray selection}
A new list of CRs is generated by comparing the noise
model with the residual of the image fitting without reference to
the old CR list. The noise or uncertainty of our method comes
from two parts: one is the intrinsic noise of the input signal, i.e.,
Poisson noise from the object and the readout noise from the
CCD circuit; the other part of the noise comes from the defect
of the profile fitting, which is larger when the profile changes
more dramatically. Basically, the first part is related to time and
the second part is related to position, which could be illustrated as:
\begin{eqnarray}
\Delta{\mathcal{F}} & = & |\frac{\partial{\mathcal{F}}}{\partial{t}}\cdot{\Delta{t}}|+\left(|\frac{\partial{\mathcal{F}}}{\partial{x}}\cdot{\Delta{x}}|+|\frac{\partial{\mathcal{F}}}{\partial{y}}\cdot{\Delta{y}|}\right)\nonumber \\
& = &\frac{1}{g} \sqrt{g\mathcal{F}+\sigma_{rd}^2}+ \left(|\frac{\partial{\mathcal{F}}}{\partial{x}}\cdot{\Delta{x}}|+|\frac{\partial{\mathcal{F}}}{\partial{y}}\cdot{\Delta{y}}|\right).
\label{newnoise}
\end{eqnarray}
The first term is sufficient to pick out CRs for most cases
when the fit is good, yet it is necessary to add the position-
dependent term to avoid false detections in regions where the
fit is not perfect. The following steps are implemented to
reject the CRs:
\begin{enumerate}
\item The position dependent terms in \ref{newnoise} are
calculated by the average gradients at each pixel.
Convolving the fitted image with the following four
arrays:
\begin{equation}
\begin{array}{cccccc}
A_1 & = & \frac{1}{2} \left(\begin{array}{ccc}
0 & 0 & 0 \\
-1& 0 & 1 \\
0 & 0 & 0
\end{array} \right)\ &
A_2 & = & \frac{1}{2} \left( \begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 0 \\
0 & -1 & 0
\end{array} \right)\ \\
A_3 & = & \frac{1}{2\sqrt{2}}\left( \begin{array}{ccc}
-1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{array} \right)\ &
A_4 & = & \frac{1}{2\sqrt{2}}\left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0
\end{array} \right)\
\end{array},
\end{equation}
a gradient array can be derived as
\begin{equation}
G=\frac{1}{2}(|A_1*\mathcal{F}|+|A_2*\mathcal{F}|+|A_3*\mathcal{F}|+|A_4*\mathcal{F}|).
\end{equation}
\item Noise models are constructed with and without the
second term in Equation \ref{newnoise}, respectively,
\begin{equation}
N_1=\frac{1}{g}\sqrt{g\mathcal{F}+\sigma_{rd}^2},
\end{equation}
and
\begin{equation}
N_2=\frac{1}{g}\sqrt{g\mathcal{F}+\sigma_{rd}^2}+G.
\end{equation}
The noise-weighted differences between the input image $I$
and the fitted image $\mathcal{F}$ are defined accordingly:
\begin{equation}
D_1=\frac{(I-\mathcal{F})}{N_1},
\end{equation}
and
\begin{equation}
D_2=\frac{(I-\mathcal{F})}{N_2}.
\end{equation}
All pixels with $D_1>20$ or $D_2>3$ are marked as CR
candidates in this step. A mask array $M$ is generated with
the CR polluted pixels set to 1.
\item{\label{item3}}
By setting the previous limits, the number of fake CRs is
greatly reduced, but real CRs with a low signal-to-noise
ratio (S/N), most of which are indiscernible from noise,
are blocked as well. Considering the consecutive pixels
occupied by a certain CR hit, the pixels at the edge of the
CR hit are more likely to be rejected due to lower signal,
though they should have higher probability to be real than
the single-pixel event. So a lower limit for those
neighbouring pixels will raise the detection rate. To do
this, all of the neighboring pixels are first added back to
the CR list by convolving the mask array $M$ with
\begin{equation}
B = \left( \begin{array}{ccc}
1 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 1
\end{array} \right)\ ,
\end{equation}
for the pixels in the expanded CR list, if the corresponding
$D_1>2$ or $D_2>2$, then they will be added to a new mask array $M{'}$.
\item Assuming there are sufficient CRs masked out, the
residual of the fit $(I-\mathcal{F})$ will represent the actual
difference between the original CR-free image and the
fitted image. The difference can be added back to
compensate for imperfect fitting, and a more accurate fit
will help to raise the CR detection rate as follows.
The difference is derived by median filtering the residual $(I-\mathcal{F})$
with the CR candidates $M{'}$ masked out:
\begin{equation}
D_m=[(I-\mathcal{F})(1-M{'})]_{m3},
\end{equation}
where m3 denotes a $3\times 3$ median filter. And the new noise-weighted residual arrays will be
\begin{equation}
D^{'}_1=\frac{(I-\mathcal{F}-D_m)}{N_1},
\end{equation}
and
\begin{equation}
D^{'}_2=\frac{(I-\mathcal{F}-D_m)}{N_2}.
\end{equation}
All pixels with $D^{'}_1>10$ or $D^{'}_2>3$ or $D_1>20$ or $D_2>3$ are masked as CR candidates in this step.
\item Rerun to Step \ref{item3} and confirm the final CR candidates.
\end{enumerate}
We do not try to fit the pixels that are either in bad fibers or in
the $d>8$ gaps between fibers because they contribute little to
the final extracted spectra. For those pixels, $\mathcal{F}$ is set to 0 and
the CR candidates are selected by simply requiring the noise-
weighted difference between the original image $I$ and the $5\times{5}$
median filtered image $I_{m5}$ to be larger than 3:
\begin{equation}
\frac{I-I_{m5}}{N_{m5}}>3,
\end{equation}
where $N_{m5}$ is the same as in Equation \ref{nm5}.
Combining the above CR candidates, the final CR mask is
generated and the value of each CR polluted pixel is replaced
by the corresponding value in the fitted image $\mathcal{F}$.
\subsection {Additional features}
With the profile fitting method, other bad pixels such as the
inherent damaged pixels could be replaced with a reasonable
value once an initial bad pixel map is known.
On a Dell Precision T5500 (eight 2.0 GHz CPUs), the IDL
implementation with one single-threaded processor requires
about 20 minutes for an image of ${4096}\times{4136}$ pixels. Most of
the time is spent on the 2D profile fitting and the timescales
linearly with the number of fibers and image size in the
dispersion direction. As the current version of our program is
not parallelized, the execution time on the current computer is
equivalent to that of a single-core processor and could be
greatly reduced after software parallelization.
\section {Examples and Application}
In this section, tests with both simulated data and real data
are carried out to illustrate the performance of our method. In
these tests, our primary concerns are the following factors: how
many pixels of real CRs are detected (efficiency), how many
pixels are falsely detected as CRs (false detection rate), and the
accuracy of the CR replacement. A better method should have
higher efficiency, lower false detection rate, and proper
restoration of the pixels polluted by CRs.
\subsection {Artificial Images}\label{arti1}
For multi-fiber spectral observations, to increase the
observation efficiency as well as to avoid fiber to fiber cross
talk, it is usually a good strategy to divide the targets into
different plates according to their brightness so that the S/N of
the targets in the same plate are similar in the same exposure
time. For plates with bright magnitude, the exposure time is
short, so the strength of the sky spectrum is low, but the S/N of
the objects is high. On the contrary, the exposure times for the
faint plates are long, so the sky spectrum is strong and the
object spectrum is weak. Since the efficiency strongly depends
on the brightness contrast between the CR and the target, plates
with different target brightness are simulated to test our method
under different situations.
First, we generate a pure CR image of ${4136}\times{4096}$ pixels
with 20,000 CR hits (approximately 10 times of those in a 30
minutes exposure LAMOST image). The shape of each CR is
set to be an ellipse with the major axis randomly distributed in
1$\sim$10 pixels and the minor axis ranges from 1 to 3 pixels, by
which almost all kinds of CRs in the real image can be
simulated. The direction of the major axis is randomly
distributed in 0$\sim 360^\circ$ and the intensities are uniformly
distributed between 0$\sim$20000 ADUs. All CRs with flux less
than 5, which is at the readout noise level, are set to 0. In total,
227,451 pixels are polluted by CR hits.
Second, to study the method’s performance with different
target brightness, two CR-free images are generated by
combining three consecutive LAMOST exposures of the same
targets. The first image (IMG600) is combined from three 600
second exposures in which the spectra are dominated by the
strong smooth continuum from the bright objects and the sky
emission lines are relatively weak. In the second one
(IMG1800), the sky emission lines are more prominent (due
to a longer exposure time of 1800 seconds) and the object
continuum is relatively weak (due the faint magnitude). The
final test images are generated by adding the pure CR image to
the CR-free images.
Both IMG600 and IMG1800 are tested by our method and
the IDL version of
L.A.C\scriptsize{OSMIC}\normalsize{}, respectively. The results are
summarized in Table 1. For IMG600, the efficiency of our
method is 1.9\% higher (73.8\% versus 71.9\%) and the number
of false detections is 5820, two orders of magnitude lower than
that of L.A.C\scriptsize{OSMIC}\normalsize{}.
For IMG1800, our efficiency is 4.
higher(80.9\% vies 76.4\%) and the number of false detections
(16,626) is less than half that of L.A.C\scriptsize{OSMIC}\normalsize{(38,912).}
The efficiency of both methods rises more than 4\% from
IMG600 to IMG1800. The reason is that the efficiency, for
those CRs falling coincidentally into the same pixel with the
object spectrum, decreases with the increasing photo noise,
while in this case, the CRs are the same in both simulations but
the object spectra are much brighter in IMG600 therefore the
noise is larger in IMG600 than in IMG1800. Compared with
our method, L.A.C\scriptsize{OSMIC}\normalsize{} is prone to mistake the wings of the
bright profiles as the sharp edges of CRs, especially when the
contrast between the background and the profile peak is high,
as in IMG600. In IMG1800, as the object brightness decreases,
the contrast and thus the number of false detections drops. Our
profile fitting method successfully bypasses this sharp edge trap
in IMG600, reducing the huge number of fake detections to a
reasonable level. As shown in Table \ref {table1}, for our method, the
number of false detections doubles from IMG600 to IMG1800.
There are two reasons for this problem. First, as the SNR of the
spectrum becomes lower, more faint pixels are mistaken as
CRs (as can be seen from Figure \ref{falsedetection}). Second, as the exposure
time increases, the intensity of the sky emission lines increases,
but the intensity of the underlying object spectra decreases (for
the magnitude gets much fainter). In this case, the relative
change at some of the exponential wings of the strong sky
emission lines becomes too dramatic to have a good
polynomial fit; the larger residual induced by the improper
fitting leads to an increase of false detections.
If we denote the SNR of a CR polluted pixel as
\begin{equation}
\phi=\frac{f_{CR}}{\sqrt{f_{clean}+\sigma_{rd}^2}},
\end{equation}
where $f_{CR}$ and $f_{clean}$ are the fluxes from the pure CR and the CR-free image, respectively, $\sigma_{rd}$ is the readout noise,
then Figure \ref{detecteff} shows the detection efficiency against $\phi$. Most of the
undetected CRs are those with low $\phi$. The efficiency
remains high for $\phi>10$ then drop quickly when $\phi<10$. The efficiency of our method is higher than that of L.A.C\scriptsize{OSMIC}\normalsize{ }\
in all situations except for $\phi<2$, where the CRs are too weak to be separated.
In Figure \ref{gooddetection}, the recovered fluxes of IMG600 are compared to
the corresponding fluxes of the CR-free image to see how well
our CR replacement works; as shown in the left and the right
panel, almost all the replacements properly follow the original
fluxes. Also shown in the middle panel is the replacement
performance of L.A.C\scriptsize{OSMIC}\normalsize{}; most of the replacements are
good, but the scatter is larger especially in the high flux region,
which is not unexpected, since its replacement method is not
specially designed for multi-fiber spectra. The performance of
both methods on IMG1800 is similar to IMG600.
Since the replacement of the false detections also changes
the flux, causing errors in the spectrum, it’s necessary to test the
replacement on those falsely detected CRs.
L.A.C\scriptsize{OSMIC}\normalsize{}\ produces too many falsely detected CRs to be
comparable with our method in IMG600, so only the results
of IMG1800 for both methods are shown in Figure \ref{falsedetection}.
As shown
in the picture, the results of both methods deviate from the true
value. Though our method systematically underestimates the
flux, most of our replaced fluxes concentrate within 80\% of
their true value and the true fluxes of most of the pixels are low,
so the influence on the extracted spectrum should be small.
L.A.C\scriptsize{OSMIC}\normalsize{} results show a large variation, with a large number
of pixels shifting from the true value to very low fluxes. Some
examples are demonstrated in Figure \ref{sam}.
For spectroscopic data, the extracted spectra are more
important than the flux of individual pixels on the 2D image.
Figure \ref{extr} compares the fluxes of the CR polluted part of the
extracted spectra with the CR-free spectra in different
situations: from top to bottom, it shows the falsely detected,
the properly detected, and the undetected CRs, respectively; in
all cases, the average difference between the CR corrected
spectra and the original spectra is less than 2.2\%, as shown
from the distribution in the right column of Figure \ref{extr} .
Figure \ref{spsam} shows an extracted spectrum sample; the residual of the CR
correction is within a few percent.
\subsection {Real Data}
We test our algorithm with real data from the LAMOST survey.
Figure \ref{realdata} shows a part of an 1800s LAMOST image
(left panel) and its reconstruction by our method (right panel).
Visual inspection of our reconstructed image shows that most
of the CR hits are properly removed. For a further comparison,
the extracted flux of the CR detected pixels vs the flux of the
classical multi-exposure-combination method are demonstrated
in Figure ~\ref{ledfop2com3}.
The results are comparable to the simulations in Figure
\ref{extr}, except that the scatter is a bit larger. The reasons for
the larger scatter could be the following: first, the simulated
image in Figure \ref{extr} has a higher SNR than the real data; second,
the falsely detected CRs cannot be discriminated from the true
CRs in this test, so the scatter should be larger than the
true-CR-only situation; and third, the sky flux varies between
exposures and the object flux gathered by LAMOST varies
with telescope pointing, which make the combined image
deviate from individual exposures, leading to a larger scatter.
\section {Conclusion}
We present a method for detecting and removing 2D profile
fitting to each segment. A new cosmic ray list is generated by
comparing the fitting residual with a noise model depending on
both the intrinsic shot noise and the relative position in the
profile. We finally produce a more accurate cosmic ray mask
table and more reasonable substitution values for CR polluted
pixels. The method is tested by both simulations and real data;
the results show that our method has a high detection rate, low
false detection rate, and proper replacement of the CR polluted
pixels.
Since this method fits the 2D profiles of the fiber spectro-
scopic data, which are different from the photometric PSF, it
cannot be applied to photometric data. However, it can be used
in slit spectroscopic data after minor modifications. The code
and samples are available at
\url{http://lamostss.bao.ac.cn/~bai/crr}.
Z. Bai acknowledges the support of the National Natural
Science Foundation of China (NSFC) (grant no. 11503054).
H. Zhang acknowledges the support of NSFC Key Program
(grant no. 11333004) and the National Key Basic Research
Program of China (grant 2014CB845700). The Guoshoujing
Telescope (the Large Sky Area Multi-Object Fiber Spectro-
scopic Telescope, LAMOST) is a National Major Scientific
Project built by the Chinese Academy of Sciences. Funding for
the project has been provided by the National Development and
Reform Commission. LAMOST is operated and managed by
the National Astronomical Observatories, Chinese Academy of
Sciences.
|
1,116,691,500,493 | arxiv | \section*{ }
By expressing the time-independent Schr\"odinger equation in one
dimension as a system of two first-order differential equations, the
transfer matrix for a rectangular potential barrier is obtained
making use of the matrix exponential. It is shown that the transfer
matrix allows one to find the bound states and the quasinormal
modes. A similar treatment for the one-dimensional propagation of
electromagnetic waves in a homogeneous medium is
also presented.\\[1ex]
{\it Keywords:} Scattering; transfer matrix; quasinormal modes;
layered systems \\[2ex]
Expresando la ecuaci\'on de Schr\"odinger independiente del tiempo
en una dimensi\'on como un sistema de dos ecuaciones diferenciales
de primer orden, se obtiene la matriz de transferencia para una
barrera de potencial rectangular haciendo uso de la exponencial de
matrices. Se muestra que la matriz de transferencia permite hallar
los estados ligados y los modos cuasinormales. Se presenta tambi\'en
un tratamiento similar para la propagaci\'on unidimensional de ondas
electromagn\'eticas en un medio
homog\'eneo.\\[1ex]
{\it Descriptores:} Dispersi\'on; matriz de transferencia; modos
cuasinormales; sistemas en capas \\[2ex]
PACS: 03.65.-w, 02.10.Ud
\section*{\tres 1. Introduction}
A standard problem in elementary quantum mechanics is that of
finding the reflection and transmission amplitudes for the
scattering produced by a potential barrier, or well, in one
dimension (see, {\em e.g.}, Refs.\ 1--4). The reflection and
transmission amplitudes are conveniently arranged in the transfer
matrix, which relates the wave function at both sides of the
potential barrier, in such a way that the effect of two or more
potential barriers is readily obtained by means of the product of
the corresponding transfer matrices (see, {\em e.g.}, Ref.\ 5 and
the references cited therein). A similar result applies for the
one-dimensional propagation of electromagnetic waves in layered
media (see, {\em e.g.}, Ref.\ 6). In fact, the transfer matrices can
be defined in all cases where there is an output that depends
linearly on an input; some important examples, apart from the two
already mentioned, are the electric circuits and optical systems. In
the cases considered here, the transfer matrices are $2 \times 2$
complex matrices but, depending on the equations involved (more
specifically, the number of variables and the differential order),
the size of the transfer matrices do vary.
The aim of this paper is to show that the transfer matrix for a
rectangular potential barrier (and, therefore, for a piecewise
constant potential) can be easily obtained integrating the
time-independent Schr\"odinger equation in one dimension by means of
the matrix exponential. The time-independent Schr\"odinger equation
in one dimension, being a second-order ordinary differential
equation, is equivalent to a system of two coupled first-order
differential equations and, only in the case of a (piecewise)
constant potential, this system can be easily integrated using the
matrix exponential. We also show that making use of the transfer
matrix one can find the bound states and the quasinormal modes. The
transfer matrix for the one-dimensional propagation of
electromagnetic waves in a medium with a piecewise constant
refractive index is obtained in a similar manner, without employing
the Fresnel coefficients.
In Sec.\ 2 an elementary discussion about the transfer matrices for
the one-dimensional Schr\"odinger equation is given (see also Ref.\
5 and the references cited therein). In Sec.\ 3 the transfer matrix
for a rectangular barrier is obtained making use of the matrix
exponential; the bound states and quasinormal modes are then found
starting from the transfer matrix. In Sec.\ 4 a similar derivation
for the case of the one-dimensional propagation of electromagnetic
waves in layered media is given.
\section*{\tres 2. Transfer matrices}
The solutions of the time-independent Schr\"odinger equation
\begin{equation}
- \frac{\hbar^{2}}{2m} \frac{{\rm d}^{2} \psi}{{\rm d}x^{2}} + V(x)
\psi = E \psi \label{sch}
\end{equation}
with a given short-range potential $V(x)$, which vanishes outside
the interval $a \leq x \leq b$, can be expressed in the form
\begin{equation}
\psi (x) = \left\{ \begin{array}{ll} A_{1} {\rm e}^{{\rm i} k(x-a)}
+ A_{2} {\rm e}^{- {\rm i} k(x-a)}, & {\rm for\ } x < a,
\\
B_{1} {\rm e}^{{\rm i} k(x-b)} + B_{2} {\rm e}^{- {\rm i} k(x-b)},
& {\rm for\ } x > b, \\
u(x), & {\rm for\ } a \leq x \leq b, \end{array}\right. \label{sol}
\end{equation}
where $k \equiv \sqrt{2mE}/\hbar$, $A_{1}$, $A_{2}$, $B_{1}$,
$B_{2}$ are constants and $u(x)$ is a function that depends on the
explicit form of the potential $V(x)$. By imposing the usual
conditions of continuity of $\psi(x)$ and its derivative at $x = a$
and $x = b$, a linear relation of the form
\begin{equation}
\left( \begin{array}{c} A_{1} \\ A_{2} \end{array} \right) = M
\left( \begin{array}{c} B_{1} \\ B_{2} \end{array} \right)
\label{tm}
\end{equation}
can be obtained, where $M$ is some $2 \times 2$ complex matrix (the
transfer matrix), which depends on $V(x)$ and the value of $k$.
Assuming that $V(x)$ is real, Eq.\ (\ref{sch}) implies that the
probability current density
\[
j(x) = \frac{\hbar}{2 {\rm i} m} \left( \psi^{*} \frac{{\rm d}
\psi}{{\rm d}x} - \psi \frac{{\rm d} \psi^{*}}{{\rm d}x} \right),
\]
where ${}^{*}$ denotes complex conjugation, satisfies the continuity
equation, ${\rm d}j/{\rm d}x = 0$, that is, $j(x) = {\rm const.}$;
then, making use of Eq.\ (\ref{sol}), one finds that, for $k$ real
\begin{equation}
|A_{1}|^{2} - |A_{2}|^{2} = |B_{1}|^{2} - |B_{2}|^{2}. \label{uni}
\end{equation}
Using the fact that
\[
|A_{1}|^{2} - |A_{2}|^{2} = \left( \begin{array}{c} A_{1} \\ A_{2}
\end{array} \right)^{\dag} \left( \begin{array}{rr} 1 & 0 \\ 0 & -1
\end{array} \right) \left( \begin{array}{c} A_{1} \\ A_{2}
\end{array} \right),
\]
where the ${}^{\dag}$ denotes the Hermitian adjoint, and Eq.\
(\ref{tm}) one finds that Eq.\ (\ref{uni}) is equivalent to
\begin{equation}
M^{\dag} \left( \begin{array}{rr} 1 & 0 \\ 0 & -1
\end{array} \right) M = \left( \begin{array}{rr} 1 & 0 \\ 0 & -1
\end{array} \right). \label{suni}
\end{equation}
The complex $2 \times 2$ matrices satisfying Eq.\ (\ref{suni}) form
a group with the usual matrix multiplication (see below). Equation
(\ref{suni}) implies that the modulus of $\det M$ is equal to 1.
The entries of the transfer matrix are related to the reflection and
transmission amplitudes of the potential $V(x)$, denoted by $r$ and
$t$, respectively. When there are no waves coming from the right
($B_{2} = 0$), there exist solutions of the Schr\"odinger equation
of the form
\begin{equation}
\psi (x) = \left\{ \begin{array}{ll} {\rm e}^{{\rm i} k(x-a)} + r
{\rm e}^{- {\rm i} k(x-a)}, & {\rm for\ } x < a, \\
t {\rm e}^{{\rm i} k(x-b)}, & {\rm for\ } x > b, \\
u_{1}(x), & {\rm for\ } a \leq x \leq b, \end{array}\right.
\label{sol1}
\end{equation}
that is, solutions of the form (\ref{sol}) with $A_{1} = 1$, $A_{2}
= r$, and $B_{1} = t$. Thus, from Eq.\ (\ref{tm}) it follows that
\[
M = \left( \begin{array}{cc} 1/t & M_{12} \\ r/t & M_{22}
\end{array} \right),
\]
with $M_{12}$ and $M_{22}$ not yet identified, and from Eq.\
(\ref{uni}) we obtain the well-known relation
\begin{equation}
1 - |r|^{2} = |t|^{2}. \label{cons}
\end{equation}
(In most textbooks the reflection and transmission amplitudes are
defined by means of expressions similar to Eq.\ (\ref{sol1}), with
$r$ and $t$ being the coefficients of ${\rm e}^{- {\rm i} kx}$ and
${\rm e}^{{\rm i} kx}$ and therefore, the amplitudes $r$ and $t$
defined by Eq.\ (\ref{sol1}) differ from those usually employed by
factors ${\rm e}^{{\rm i}ka}$ and ${\rm e}^{-{\rm i}kb}$,
respectively.)
Since $V(x)$ is real, for $k$ real the complex conjugate of the
solution (\ref{sol1})
\begin{equation}
\psi^{*} (x) = \left\{ \begin{array}{ll} r^{*} {\rm e}^{{\rm i}
k(x-a)} + {\rm e}^{- {\rm i} k(x-a)}, & {\rm for\ } x < a, \\
t^{*} {\rm e}^{- {\rm i} k(x-b)}, & {\rm for\ } x > b, \\
u^{*}_{1}(x), & {\rm for\ } a \leq x \leq b, \end{array}\right.
\label{sol2}
\end{equation}
is also a solution of the Schr\"odinger equation. Substituting the
coefficients appearing in Eq.\ (\ref{sol2}) into Eq.\ (\ref{tm}) we
find that
\begin{equation}
M = \left( \begin{array}{cc} 1/t & r^{*}/t^{*} \\ r/t & 1/t^{*}
\end{array} \right). \label{su}
\end{equation}
Then, according to Eq.\ (\ref{cons}), $\det M = 1$, which means that
(for $k$ real) the transfer matrix belongs to the group SU(1,1),
formed by the $2 \times 2$ complex matrices with unit determinant
that satisfy Eq.\ (\ref{suni}).
The reflection and transmission amplitudes of the potential $V(x)$
for waves incident from the right, $r'$ and $t'$, respectively, need
not coincide with $r$ and $t$. In fact, from Eqs.\ (\ref{tm}) and
(\ref{su}), setting $A_{1} = 0$ and $B_{2} = 1$, we must have
\[
\left( \begin{array}{c} 0 \\ t' \end{array} \right) = \left(
\begin{array}{cc} 1/t & r^{*}/t^{*} \\ r/t & 1/t^{*} \end{array}
\right) \left( \begin{array}{c} r' \\ 1 \end{array} \right),
\]
which, making use of Eq.\ (\ref{cons}), implies that
\begin{equation}
t' = t, \hspace{5ex} 0 = \frac{r'}{t} + \frac{r^{*}}{t^{*}}.
\label{symm}
\end{equation}
Hence, $r = r'$ if and only if $r/t$ is pure imaginary.
\section*{\tres 3. Rectangular barriers}
The reflection and transmission amplitudes for a given potential are
usually obtained by solving the time-independent Schr\"odinger
equation (\ref{sch}) (see, {\em e.g.}, Refs.\ 1--4). In the
exceptional case of a piecewise constant potential, the transfer
matrix (and, therefore, the reflection and transmission amplitudes)
can be readily obtained by means of matrix exponentiation.
The time-independent Schr\"odinger equation (\ref{sch}) can be
expressed as the first-order differential equation
\begin{equation}
\frac{{\rm d}}{{\rm d} x} \left( \begin{array}{c} \psi(x) \\
\psi'(x) \end{array} \right) = \left( \begin{array}{cc} 0 & 1 \\
v(x) - k^{2} & 0 \end{array} \right) \left( \begin{array}{c} \psi(x) \\
\psi'(x) \end{array} \right), \label{lin}
\end{equation}
where $v(x) \equiv 2m V(x)/\hbar^{2}$. Hence, if $V(x)$ is a
constant $V_{0}$ for $a \leq x \leq b$, the solution of Eq.\
(\ref{lin}) is
\[
\left( \begin{array}{c} \psi(x) \\ \psi'(x) \end{array} \right) =
\exp \left[ x \left( \begin{array}{cc} 0 & 1 \\ v_{0} - k^{2} & 0
\end{array} \right) \right] \left( \begin{array}{c} c_{1} \\ c_{2}
\end{array} \right),
\]
for $a \leq x \leq b$, where $v_{0} \equiv 2m V_{0}/\hbar^{2}$, and
$c_{1}$, $c_{2}$ are some constants. Thus,
\begin{equation}
\left( \begin{array}{c} \psi(a) \\
\psi'(a) \end{array} \right) = \exp \left[ - L \left(
\begin{array}{cc} 0 & 1 \\ v_{0} - k^{2} & 0 \end{array} \right)
\right] \left( \begin{array}{c} \psi(b) \\ \psi'(b) \end{array}
\right), \label{exp}
\end{equation}
with $L \equiv b-a$. Letting
\[
J \equiv \left( \begin{array}{cc} 0 & 1 \\ v_{0} - k^{2} & 0
\end{array} \right)
\]
one finds that $J^{2} = (v_{0} - k^{2}) I$, where $I$ is the unit $2
\times 2$ matrix, hence (see, {\em e.g.}, Ref.\ 7)
\begin{eqnarray}
\exp (- L J) \!\!\! & = & \!\!\! \left\{ \begin{array}{ll} \cosh (L
\sqrt{v_{0} - k^{2}}) \, I - \displaystyle \frac{\sinh (L
\sqrt{v_{0} - k^{2}})}{\sqrt{v_{0} - k^{2}}} \, J, & {\rm if\ }
v_{0} - k^{2} > 0, \\[2ex]
\cos (L \sqrt{k^{2} - v_{0}}) \, I - \displaystyle \frac{\sin (L
\sqrt{k^{2} - v_{0}})}{\sqrt{k^{2} - v_{0}}} \, J, & {\rm if\ }
v_{0} - k^{2} < 0, \\[2ex]
I - L J, & {\rm if\ } v_{0} - k^{2} = 0. \end{array} \right.
\label{sl}
\end{eqnarray}
On the other hand, from Eq.\ (\ref{sol}) we have
\[
\psi(a) = A_{1} + A_{2}, \hspace{3ex} \psi'(a) = {\rm i} k (A_{1} -
A_{2}), \hspace{3ex} \psi(b) = B_{1} + B_{2}, \hspace{3ex} \psi'(b)
= {\rm i} k (B_{1} - B_{2}),
\]
that is,
\begin{equation}
\left( \begin{array}{c} A_{1} \\ A_{2} \end{array} \right) =
\frac{1}{2} \left( \begin{array}{cc} 1 & - {\rm i}/k \\
1 & {\rm i}/k \end{array} \right) \left(
\begin{array}{c} \psi(a) \\ \psi'(a) \end{array} \right),
\hspace{5ex} \left(
\begin{array}{c} \psi(b) \\ \psi'(b) \end{array} \right) =
\left( \begin{array}{cc} 1 & 1 \\ {\rm i}k & -{\rm i}k
\end{array} \right) \left( \begin{array}{c} B_{1} \\ B_{2}
\end{array} \right). \label{chb}
\end{equation}
(Note that Eqs.\ (\ref{chb}) correspond to the continuity conditions
for $\psi$ and $\psi'$ at $x = a$ and $x = b$.)
Then, noting that
\[
\frac{1}{2} \left( \begin{array}{cc} 1 & - {\rm i}/k \\
1 & {\rm i}/k \end{array} \right) J \left( \begin{array}{cc} 1 & 1
\\ {\rm i}k & -{\rm i}k \end{array} \right) = \frac{1}{2 {\rm i} k}
\left( \begin{array}{cc} v_{0} - 2k^{2} & v_{0} \\ - v_{0} & - v_{0}
+ 2k^{2} \end{array} \right),
\]
from Eqs.\ (\ref{exp})--(\ref{chb}) one finds that, for a
rectangular potential barrier (or potential well)
\begin{equation}
V(x) = \left\{ \begin{array}{ll} 0, & {\rm if\ } x < a {\rm \ or\ }
x> b, \\ V_{0}, & {\rm if\ } a \leq x \leq b, \end{array} \right.
\label{pot}
\end{equation}
the transfer matrix is given by
\begin{equation}
M = \left\{ \begin{array}{ll} \cosh (L \sqrt{v_{0} - k^{2}}) \, I -
\displaystyle \frac{\sinh (L \sqrt{v_{0} - k^{2}})}{\sqrt{v_{0} -
k^{2}}} \frac{1}{2 {\rm i} k} \left( \begin{array}{cc} v_{0} -
2k^{2} & v_{0} \\ - v_{0} & - v_{0} + 2k^{2} \end{array} \right),
& {\rm if\ } v_{0} - k^{2} > 0, \\[3ex]
\cos (L \sqrt{k^{2} - v_{0}}) \, I - \displaystyle \frac{\sin (L
\sqrt{k^{2} - v_{0}})}{\sqrt{k^{2} - v_{0}}} \frac{1}{2 {\rm i} k}
\left( \begin{array}{cc} v_{0} - 2k^{2} & v_{0} \\ - v_{0} & - v_{0}
+ 2k^{2} \end{array} \right), & {\rm if\ } v_{0} - k^{2} < 0,
\\[3ex]
I - \displaystyle \frac{{\rm i} kL}{2} \left( \begin{array}{cc} 1 &
-1 \\ 1 & -1 \end{array} \right), & {\rm if\ } v_{0} - k^{2} = 0.
\end{array} \right. \label{trans}
\end{equation}
Note that owing to the definitions of the amplitudes $A_{1}$,
$A_{2}$, $B_{1}$, and $B_{2}$ in terms of the exponentials ${\rm
e}^{\pm {\rm i} k(x-a)}$ and ${\rm e}^{\pm {\rm i} k(x-b)}$, the
transfer matrices (\ref{trans}) depend on $a$ and $b$ only through
their difference $L = b-a$. The simplicity of the transfer matrices
(\ref{trans}) contrasts with the complexity of the expressions for
the reflection and transmission amplitudes obtained in the standard
manner (see, {\em e.g.}, Ref.\ 2, chap.\ 5). Note also that even
though we follow the conventions of Ref.\ 5, the transfer matrix
(\ref{trans}) does not agree with the amplitudes given in Eq.\ (12)
of Ref.\ 5.
It may also be noticed that, by allowing $\sqrt{v_{0} - k^{2}}$ to
become pure imaginary or taking the limit as $\sqrt{v_{0} - k^{2}}$
goes to zero, from the first expression in (\ref{trans}) one can
obtain the other two. Furthermore, one can verify directly that,
when $k$ is real, the transfer matrices (\ref{trans}) are of the
form $\left( \begin{array}{cc} \alpha & \beta \\ \beta^{*} &
\alpha^{*} \end{array} \right)$, with $|\alpha|^{2} - |\beta|^{2} =
1$ and therefore they indeed belong to SU(1,1). On the other hand,
Eqs.\ (\ref{lin})--(\ref{sl}) hold for $k$ real or complex and since
the trace of $J$ is equal to zero for any value of $k$ (even if
$V(x)$ was not real), the determinant of $\exp (-LJ)$ is equal to 1;
therefore, the transfer matrices (\ref{trans}) have determinant
equal to 1 also when $k$ is complex, though $M$ no longer belongs to
SU(1,1).
The example considered in this section also allows us to illustrate
the fact that making use of the transfer matrix one can find the
energies of the bound states or the quasinormal modes, by
considering pure imaginary or complex values of $k$, respectively.
In the case of the bound states of the potential well (\ref{pot})
with $V_{0} < 0$, we have $E < 0$ and writing $k = {\rm i} |k|$,
from Eq.\ (\ref{sol}) we see that in order for the wave function to
remain bounded, $B_{2} = 0$ and $A_{1} = 0$. Then Eq.\ (\ref{tm})
implies that $M_{11}$, the first entry of the diagonal of $M$, must
be equal to zero. Since the determinant of the transfer matrix is
equal to 1 (independent of the value of $k$), this last condition is
equivalent to saying that the off-diagonal entries of $M$ (which
have opposite signs) must be equal to $+1$ or $-1$. In the present
case $v_{0} - k^{2} < 0$, and from the second line of Eq.\
(\ref{trans}) we have
\[
\frac{\sin (L \sqrt{-|k|^{2} - v_{0}})}{\sqrt{-|k|^{2} - v_{0}}}
\frac{v_{0}}{2|k|} = \pm 1,
\]
which is equivalent to the conditions obtained in the textbooks
(see, {\em e.g.}, Refs.\ 1,2).
The so-called quasinormal modes correspond to complex values of $k$
for which there are no incident waves on the potential barrier but
only outgoing waves. In this case the solution of the
time-independent Schr\"{o}dinger equation are of the form
\begin{equation}
\psi(x) = \left\{ \begin{array}{ll} A_{2} {\rm e}^{- {\rm i}k(x-a)}, & x < a, \\
B_{1} {\rm e}^{{\rm i}k(x-b)}, & x > b, \\ u(x), & a \leq x \leq b, \\
\end{array} \right. \label{q}
\end{equation}
assuming that the real part of $k$ is positive ({\em cf.}\ Eq.\
(\ref{sol}), $A_{1}$ and $B_{2}$ are equal to zero so that there are
no ingoing waves on the barrier). Thus, as in the case of the bound
states, we have $M_{11} = 0$ and, making use of the first expression
in Eq.\ (\ref{trans}), we have
\[
\cosh (L \sqrt{v_{0} - k^{2}}) - \frac{\sinh (L \sqrt{v_{0} -
k^{2}})}{\sqrt{v_{0} - k^{2}}} \frac{v_{0} - 2k^{2}}{2 {\rm i} k} =
0
\]
which can also be expressed in the form
\begin{equation}
\cosh (L \sigma) - \frac{\sinh (L \sigma)}{\sigma} \frac{\sigma^{2}
- k^{2}}{2 {\rm i} k} = 0, \label{6}
\end{equation}
with the definition $\sigma \equiv \sqrt{v_{0} - k^{2}}$. Hence,
$\sigma^{2} + k^{2} = v_{0}$. Following Chandrasekhar [8], we
parameterize $k$ and $\sigma$ according to
\begin{equation}
k = Q \sin \alpha, \quad \sigma = Q \cos \alpha, \label{para}
\end{equation}
with $Q^{2} = v_{0}$ and $Q \geq 0$ (assuming $v_{0} \geq 0)$.
Substituting these expressions for $k$ and $\sigma$ into Eq.\
(\ref{6}) we have
\[
\cosh (L \sigma) - \sinh (L \sigma) \frac{Q^{2} \cos^{2} \alpha -
Q^{2} \sin^{2} \alpha}{2 {\rm i} Q^{2} \sin \alpha \cos \alpha} = 0,
\]
which is equivalent to
\begin{equation}
\cosh (L \sigma) + {\rm i} \sinh (L \sigma) \cot 2 \alpha = 0.
\label{8}
\end{equation}
Making use of the identities $\sin z = - {\rm i} \sinh ({\rm i} z)$,
$\cos z = \cosh ({\rm i} z)$, this last equation can be written as
$\sinh (L \sigma) \cosh ({\rm i} 2 \alpha) - \cosh (L \sigma) \sinh
({\rm i} 2 \alpha) = 0$, which is equivalent to
\[
\sinh (L \sigma - {\rm i} 2 \alpha) = 0
\]
and, therefore, $L \sigma - {\rm i} 2 \alpha = {\rm i} n \pi$, where
$n$ is an integer. Then, letting $\alpha = \alpha_{1} + {\rm i}
\alpha_{2}$ and $\sigma = \sigma_{1} + {\rm i} \sigma_{2}$, we have
\begin{equation}
L \sigma_{1} = -2 \alpha_{2}, \quad L \sigma_{2} = 2 \alpha_{1} - n
\pi. \label{11}
\end{equation}
From the relation $\sigma_{1} + {\rm i} \sigma_{2} = Q \cos
(\alpha_{1} + {\rm i} \alpha_{2})$ [see Eq.\ (\ref{para})] we obtain
\begin{equation} \label{12}
\sigma_{1} = Q \cos \alpha_{1} \cosh \alpha_{2}, \quad \sigma_{2} =
-Q \sin \alpha_{1} \sinh \alpha_{2}
\end{equation}
and, combining Eqs.\ (\ref{11}) and (\ref{12}), it follows that
\begin{equation}
-2 \alpha_{2} = L Q \cos \alpha_{1} \cosh \alpha_{2}, \quad 2
\alpha_{1} - n \pi = - L Q \sin \alpha_{1} \sinh \alpha_{2}.
\label{25}
\end{equation}
Hence,
\begin{equation}
\tan \alpha_{1} \tanh \alpha_{2} = \frac{2 \alpha_{1} - n \pi}{2
\alpha_{2}}. \label{26}
\end{equation}
Similarly, from Eq.\ (\ref{para}), we have $k_{1} + {\rm i} k_{2} =
Q \sin (\alpha_{1} + {\rm i} \alpha_{2})$, that is,
\[
k_{1} = Q \sin \alpha_{1} \cosh \alpha_{2}, \quad k_{2} = Q \cos
\alpha_{1} \sinh \alpha_{2}
\]
and, making use of Eqs.\ (\ref{25}),
\begin{equation}
k_{1} = - \frac{2 \alpha_{2} \tan \alpha_{1}}{L}, \quad k_{2} = -
\frac{2 \alpha_{2} \tanh \alpha_{2}}{L}. \label{16}
\end{equation}
By hypothesis, $k_{1} \geq 0$ and $Q \geq 0$, therefore from Eqs.\
(\ref{26}) and (\ref{16}) it follows that
\begin{eqnarray*}
{\rm if\ } \alpha_{2} > 0 \; \Rightarrow \; \tan \alpha_{1} \leq 0,
\quad 2 \alpha_{1} - n \pi \leq 0 & \Rightarrow & n > 0, \quad
\frac{\pi}{2} \leq \alpha_{1} \leq \pi, \\
{\rm if\ } \alpha_{2} < 0 \; \Rightarrow \; \tan \alpha_{1} \geq 0,
\quad 2 \alpha_{1} - n \pi \geq 0 & \Rightarrow & n \leq 0, \quad 0
\leq \alpha_{1} \leq \frac{\pi}{2}.
\end{eqnarray*}
Given a solution, $\alpha_{1}$, $\alpha_{2}$, of Eqs.\ (\ref{25}),
the values of $k_{1}$ and $k_{2}$ are determined by means of Eqs.\
(\ref{16}).
Since the time-independent Schr\"odinger equation (\ref{sch}) is
obtained assuming that the wave function has a time dependence of
the form $\exp (- {\rm i} Et/\hbar)$. When $k$ is complex, $E$ has a
negative imaginary part for $k_{1} > 0$ [$k_{2}$ is negative, see
Eq.\ (\ref{16})] that produces an exponential decay in time.
Denoting by $M^{(a,b)}$ the matrix appearing in Eq.\ (\ref{tm}), we
have the relation
\[
M^{(a,c)} = M^{(a,b)} M^{(b,c)},
\]
for any value of $c$. This relation together with Eq.\ (\ref{trans})
allow us to readily find the transfer matrix (or, equivalently, the
transmission and reflection amplitudes) for any piecewise constant
potential and from the condition $M_{11} = 0$, the bound states and
quasinormal modes can then be obtained, though the expressions will
be even more involved than the ones considered here.
\section*{\tres 4. Reflection and transmission of electromagnetic waves}
The behavior of a linearly polarized electromagnetic plane wave
normally incident on a slab of dielectric material can be found
following a procedure similar to that employed in the preceding
section. For plane monochromatic waves propagating along the
$x$-axis with the electric field parallel to the $y$-axis in a
homogeneous dielectric medium, the wave equation reduces to
\begin{equation}
\frac{{\rm d}^{2} E_{y}}{{\rm d}x^{2}} + k^{2} E_{y} = 0,
\label{em1}
\end{equation}
where $k = n \omega/c$, $n$ is the refractive index of the medium
and $\omega$ is the frequency of the wave. Equation (\ref{em1}) can
be expressed as the first-order equation
\begin{equation}
\frac{{\rm d}}{{\rm d} x} \left( \begin{array}{c} E_{y} \\ {\rm
d}E_{y}/ {\rm d}x \end{array} \right) = \left( \begin{array}{cc} 0 &
1 \\ - k^{2} & 0 \end{array} \right) \left( \begin{array}{c} E_{y}
\\ {\rm d}E_{y}/ {\rm d}x \end{array} \right), \label{em2}
\end{equation}
which is of the form (\ref{lin}) with $v = 0$; hence,
\[
\left( \begin{array}{c} E_{y}(a) \\ {\rm d}E_{y}/ {\rm d}x|_{x = a}
\end{array} \right) = \widetilde{M} \left( \begin{array}{c} E_{y}(b)
\\ {\rm d}E_{y}/{\rm d}x|_{x = b} \end{array} \right),
\]
where [see Eqs.\ (\ref{exp}) and (\ref{sl})]
\[
\widetilde{M} = \cos (kL)\, I - \frac{\sin (kL)}{k} \left(
\begin{array}{cc} 0 & 1 \\ - k^{2} & 0 \end{array} \right)
\]
and $L = b - a$.
If the slab is bounded by the planes $x = a$ and $x = b$ and, for
instance, surrounded by vacuum, Eq.\ (\ref{em1}) has solutions of
the form
\begin{equation}
E_{y} = \left\{ \begin{array}{ll} A_{1} {\rm e}^{{\rm i} k_{0}
(x-a)} + A_{2} {\rm e}^{-{\rm i} k_{0} (x-a)}, & {\rm for\ } x < a,
\\ B_{1} {\rm e}^{{\rm i} k_{0} (x-b)} + B_{2} {\rm e}^{-{\rm i}
k_{0} (x-b)}, & {\rm for\ } x > b, \end{array} \right. \label{em3}
\end{equation}
where $k_{0} \equiv \omega/c$. Faraday's law imply that the
$z$-component of the magnetic field is proportional to ${\rm
d}E_{y}/ {\rm d}x$ and, therefore, the continuity of the tangential
components of the fields at the boundary of the slab amounts to the
continuity of $E_{y}$ and ${\rm d}E_{y}/ {\rm d}x$ and from Eq.\
(\ref{em3}) we see that
\[
E_{y}(a) = A_{1} + A_{2}, \quad \frac{{\rm d}E_{y}}{{\rm d}x}(a) =
{\rm i} k_{0} (A_{1} - A_{2}), \quad E_{y}(b) = B_{1} + B_{2}, \quad
\frac{{\rm d}E_{y}}{{\rm d}x}(b) = {\rm i} k_{0} (B_{1} - B_{2}),
\]
thus, proceeding as in the previous section we obtain the relation
\begin{equation}
\left( \begin{array}{c} A_{1} \\ A_{2} \end{array} \right) = M
\left( \begin{array}{c} B_{1} \\ B_{2} \end{array} \right)
\label{tme}
\end{equation}
with the transfer matrix
\begin{equation}
M = \cos (kL) \, I - \frac{\sin (kL)}{k} \frac{{\rm i}}{2 k_{0}}
\left( \begin{array}{cc} k_{0}^{2} + k^{2} & - k_{0}^{2} + k^{2}
\\ k_{0}^{2} - k^{2} & - k_{0}^{2} - k^{2} \end{array} \right),
\label{em4}
\end{equation}
which is related to the transmission and reflection amplitudes as in
Eq.\ (\ref{su}). It may be noticed that, also in the present case,
the transfer matrix (\ref{em4}) belongs to SU(1,1) for $k$ real and
that the determinant of $M$ is equal to 1 even if $k$ is complex
(which would correspond to a nonzero conductivity).
\section*{\tres Acknowledgment}
One of the authors (I.R.G.) thanks the Vicerrector\'{\i}a de
Investigaci\'on y Estudios de Posgrado of the Universidad Aut\'onoma
de Puebla for financial support through the programme ``La ciencia
en tus manos.''
\section*{References}
\newcounter{ref} \begin{list}{\hspace{1.3ex}\arabic{ref}.\hfill}
{\usecounter{ref} \setlength{\leftmargin}{2em}
\setlength{\itemsep}{-.98ex}}
\item L.I. Schiff, {\it Quantum Mechanics}, 3rd ed.\ (McGraw-Hill, New York, 1968).
\item S. Gasiorowicz, {\it Quantum Physics}, (Wiley, New York, 1974).
\item I.I.\ Gol'dman and V.D.\ Krivchenkov, {\it Problems in
Quantum Mechanics}, (Pergamon, London, 1961; reprinted by Dover, New
York, 1993).
\item K.\ Gottfried and T-M.\ Yan, {\it Quantum Mechanics:
Fundamentals}, 2nd ed., (Springer, New York, 2003).
\item L.L.\ S\'anchez-Soto, J.F.\ Cari\~nena, A.G.\ Barriuso and
J.J.\ Monz\'on, {\it Eur.\ J.\ Phys.}\ {\bf 26} (2005) 469.
\item J.R.\ Reitz, F.J.\ Milford and R.W.\ Christy, {\it Foundations of
Electromagnetic Theory}, 4th ed.\ (Addison-Wesley, Reading, MA,
1993).
\item K.B.\ Wolf, {\it Rev.\ Mex.\ F\'{\i}s.}\ {\bf 49} (2003) 465.
\item S. Chandrasekhar, {\it Proc.\ R.\ Soc.\ London} A {\bf 344} (1975) 441,
reprinted in {\it Selected Papers}, Vol.\ 6, (Chicago University
Press, Chicago, 1991).
\end{list}
\end{document}
|
1,116,691,500,494 | arxiv | \section{Introduction}\label{Intro}
The research and development of quantum computers are currently an important and active field of quantum information science and technology \cite{QCFeynman,QCDeutsch,QCLloyd,DiVincenzoQC,QCQINandC,QSRMP2014,SCQRPP2017,
SCQNISQ20191,SCQARCMP2020,ZhugroupSQC2020,trappedionNISQ2019,AVSQSiontrapQC2020}.
On the one side, quantum computer devices have been engineered with state-of-the-art technologies using various kinds of elements including superconducting circuits \cite{NakamuraTsaigroupSC,SCQRMP2001,circuitqedreview1,SCQRPP2017,SCQNISQ20191,SCQARCMP2020,ZhugroupSQC2020,TsaigroupSCCQC2021} and trapped ions \cite{CiracZollergate,TionSMgate,trappedionsreview1A,trappedionsreview1B,trappedionsreview2,AVSQSiontrapQC2020,trappedionNISQ2019}.
On the other side, toward the application to, for example, material science, quantum chemistry, optimization problems, and quantum machine learning,
many new kinds of quantum algorithms have been recently developed such as Variational Quantum Eigensolver (VQE) \cite{VQEnc2014,VQE0,VQEnature2017,QCchemistryRMP2020,hybridQCalgorithmJPSJ2021},
Quantum Approximate Optimization Algorithm (QAOA) \cite{QAOA2014,crooks2018performance,wang2018quantum,shaydulin2019evaluating,zhou2020quantum, hybridQCalgorithmJPSJ2021,TIsing3}, and quantum circuit learning \cite{hybridQCalgorithmJPSJ2021,SchuldandPetruccionegroupQML,MitaraietalQML,QSTQML2019}.
These algorithms have characteristics such that they are constructed by the hybridization between quantum and classical computational procedures. Recently, in the task of sampling random quantum circuits, quantum supremacy has been demonstrated using the superconducting circuit device \cite{Qsup}. All these facts are implying important milestones for the advancement of the research and development of the quantum computers
and the broadening of quantum-computing applications to many fields of science and engineering.
While the above successful results of the research and development of quantum computers have been reported,
our near-term quantum computers based on circuit models have been built as intermediate-scale quantum devices yet and are fragile against quantum noise effects: they are called NISQ devices \cite{PreskillNISQ2018,SCQARCMP2020,trappedionNISQ2019}.
Quantum noise effects (decoherence) are major obstacles for performing quantum computation and historically many great efforts have been made on reducing such effects \cite{EkertgroupQCdissipation,resch1912benchmarking}.
One of the traditional and representative schemes for this are the quantum-error-correction (QEC) coding \cite{ShorPRAQEC1995,QCQINandC,NemotogroupQEC,lidar2013quantum,QECRoffe,SCQARCMP2020,ZhugroupSQC2020,TsaigroupSCCQC2021}.
Another important one is the dynamical decoupling which plays fundamental role in extending coherence times of qubits \cite{viola1998dynamical,viola1999dynamical,khodjasteh2005fault,lidar2013quantum,masuyama2018extending,SCQNISQ20191,trappedionNISQ2019,TsaigroupSCCQC2021} .
The QEC codes are, however, not implemented on NISQ devices and to obtain quantum computational results in good accuracy with NISQ devices we need to search for alternative approaches for mitigating quantum noise effects.
This research field is called quantum error mitigation (QEM), and these days, it is one of the important themes of the research and development of quantum computation \cite{EMPRL2017,EMNature2019,EMPRX2017,EMPRX2018,EMarxiv2018,bonet2018low,mcardle2019error,zlokapa2020deep,EMPRA2021,CandSQEMPRAp2021,QCchemistryRMP2020,hybridQCalgorithmJPSJ2021,OttenGrayQEM1,OttenGrayQEM2,QSEQEM, CliffordQEM,LearningBasedQEM,VirtualDistillationQEM,koczor2021exponential}.
The difficulty of the treatment of quantum noises (e.g., amplitude damping, phase damping (dephasing), depolarizing channel) is that we cannot directly construct their inverse processes by quantum gates
due to their non-unitarity.
On the other hand, it is possible to formulate quantum noise effects as quantum circuits by using ancilla bits and measurements on them \cite{QCQINandC,NorigroupNEQPRA2011,KaisgroupOpenQ2020,openQHubbard,openQsimnpj2020,drivendisspativePRB2020,koppenhofer2020quantum,de2021quantum}.
By utilizing the quantum circuits representing the quantum noise effects under consideration, we expect that we can establish QEM schemes for reducing such effects.
If this is established, we become able to mitigate the quantum noise effects by the gate operations and measurements, i.e., QEM conducted by all-quantum-computational operations.
In other words, we become able to programmably run quantum algorithms with mitigating the quantum noise effects solely by the quantum computational operations and realize high-accurate quantum computation.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.9 \textwidth]{QEMconcept.png}
\caption{ Schematic of our proposed QEM method. The original circuit represented by the blue rectangle (left side) describes the quantum circuit for the quantum algorithm to be ran and is composed of the unitary operations $U_k$
with $k=1,\ldots,d$ and $d$ denotes the depth of the quantum algorithm. It yields the expectation value $ \langle \hat{O} \rangle_{ \rho^\text{real}_{d\cdots1}} $.
On the other hand,
the quantum-noise-effect circuit group, which is represented by the orange rectangles (right side), is constructed from the original circuit by inserting an additional operation between $U_k$ and $U_{k+1}$ (gray box).
It yields the theoretically-estimated quantum computational error $\langle \hat{O} \rangle_{(\Delta_1^{\text{AD}} \rho_{d\ldots 1})^{\text{real}}}$.
By using these two expectation values, we obtain the equation for our QEM scheme in the green rectangle. Here we have taken $d=3.$}
\label{QEMconcept}
\end{figure*}
In this work, we propose our QEM schemes for quantum computational errors which occur owing to couplings with environments (decoherence) during gate operations:
errors of state preparation (initialization) and measurement, imperfections of quantum gates, and cross talks among qubits are not taken into account.
In particular, we make detailed analysis on quantum computational errors generated by amplitude damping (AD) of single-qubit states.
We show the schematic representation of our QEM scheme in Fig. \ref{QEMconcept} and it consists of two elements, the quantum circuit for the quantum algorithm under consideration (original circuit) represented by the blue rectangle
and the ensemble of quantum circuits which yields the theoretical value of the quantum computational error due to the quantum noise effect, namely, quantum-noise-effect circuit group and is represented by the orange rectangles.
By utilizing the quantum-noise-effect circuit group,
we formulate our QEM scheme as a perturbation theory with respect to a strength of quantum noise and perform it by subtracting the expectation values given by the quantum-noise-effect circuit groups from the those generated by the quantum circuit for the quantum algorithms under consideration as expressed by the formula in the green rectangle; see also the right-hand side of the first line in Eq. \eqref{QEMformula1}.
As a result, the AD effects are mitigated and we approximately obtain the ideal expectation values.
Then, we discuss the number of elementary quantum circuits which compose the quantum-noise-effect circuit group and show that it scales polynomial (linear) with respect to the product of the number of register bits
and the depth of quantum algorithm (circuit depth or the number of unitary gates composing the quantum algorithm).
Finally, we numerically demonstrate the validity of our QEM scheme by running noisy quantum simulators of qubits under the AD effects for four types of quantum algorithms in the linear-order perturbation regime.
The detailed explanation on how to extend our QEM scheme to other kinds of quantum noise effects including phase damping, generalized amplitude damping (thermalization), and depolarizing channel,
and extension of our QEM scheme to higher-order quantum noise effects are given in appendix.
The structure of this paper is given as follows.
It begins by Sec. \ref{QEMS} with our modeling of the quantum computation under the influence of the quantum noise effect.
After then we explain the formalisms of our QEM scheme.
In Sec. \ref{nss}, which presents our main results, we demonstrate numerically our QEM schemes for the noisy quantum simulations for four types of quantum algorithms.
Sec. \ref{concludediscussion} is devoted to the conclusion of this paper.
\section{QEM Schemes}\label{QEMS}
\subsection{Modeling and Formulation}\label{MandF}
Let us explain our modeling of quantum computation under the influence of quantum noise effects.
In the following, we focus on the amplitude damping (AD) effect: generalized-amplitude-damping (GAD) effect at zero temperature.
As discussed later, it is straightforward to generalize the argument for the AD effect to other quantum noise effects such as phase damping (PD) and stochastic Pauli noises.
We schematically represent such a circumstance as a quantum circuit and show it in Fig. \ref{QCunderAD1}.
\begin{figure}[h]
\includegraphics[width=0.4 \textwidth]{modelQEM.png}
\caption{Illustration of quantum computation under AD effects represented as a quantum circuit. Here we show it for $d=3$ and $N_q =2.$
The symbol $\mathcal{E}^{\text{AD}} _{ Q_{ \text{r}j}}$ expresses the occurrence of AD effect on the register bit $Q_{ \text{r}j} $ (the $j$-th register bit).
}
\label{QCunderAD1}
\end{figure}
There are $N_q$ register bits and the quantum algorithm to be run is represented by the unitary transformation $U^{ \text{QC}}$.
It is comprised of $d$ unitary transformations described by $U^{ \text{QC}}= \prod_{k=1}^d U_{ k} =U_{ d} \cdot U_{ d-1} \cdots U_{ 2} \cdot U_{ 1}$.
The unitary transformation $U_{ k} $ $(k=1,2,\ldots, d)$ is composed of single- and two-qubit gates.
We assume that the duration time (gate operation time) of the unitary transformation $U_ k$ is $\Delta t$ for any $k$.
During the time interval $\Delta t$, the register bits are influenced by the AD effects due to couplings with environments, e.g., electromagnetic field in the vacuum, phonons in solids, etc.
The quantum master equation describing the AD process in the interaction picture is given by \cite{QCQINandC,carmichaeltxb,Agarwalltxb,opendynamicstext}
\begin{align}
\frac { \partial \rho (t)}{\partial t } & = \gamma \mathcal{L} _{ \text{AD}} [\rho (t)] \notag\\
& = \gamma \sum_{j=0}^{N_q-1} \left[ \tilde{\sigma}^-_j \rho (t) \tilde{\sigma}^+_j
- \frac{1}{2} \big{\{} \tilde{\sigma}^+_j \tilde{\sigma}^-_j , \rho (t) \big{\}} \right] ,
\label{ADQME}
\end{align}
where $ \rho (t)$ is the density matrix of the $N_q$ register bits at a time $t$ and $\gamma$ is the decay rate.
The symbol $\mathcal{L} _{ \text{AD}} $ denotes the Lindblad superoperator of the AD process and the operators $\tilde{\sigma}^\pm_j = \frac{ X_j \mp i Y_j}{2}$ are the ladder (raising and lowering) operators acting on the register bit $Q_{ \text{r}j }$.
$X_j $ and $ Y_j$ are $X$ and $Y$ gates acting on $Q_{ \text{r}j }$, respectively.
$\big{\{} A, B \big{\}}$ is the anti-commutator between the operators $A$ and $B$.
In our model, we assume that the $N_q$ register bits experience homogeneously the AD effect of single-qubit state given by the decay rate $\gamma$.
At the initial time $t=0$, all the register bits are in $| 0 \rangle$ state (ground state), namely, $\rho (0) = | 0 \rangle \langle 0|^{\otimes N_q }$.
Let us write the total amount of quantum computational time (running time of the quantum algorithm under consideration) by $T(= d \cdot \Delta t)$ while we introduce the dimensionless time $ \tau = \gamma \Delta t$.
By assuming $ \tau \ll 1$, in the following let us evaluate the density matrix at the time $T$, $\rho (T) $, by using the quantum master equation \eqref{ADQME} and express it as a perturbation series with respect to $\tau$
given by
\begin{align}
\rho (T) & = \sum_{p=0} ^ \infty \frac{\tau^p}{p!} \cdot\Delta^{\text{AD}}_p \rho_{d\cdots1} \notag\\
& = \rho_{d\cdots1} + \tau\cdot\Delta^{\text{AD}}_1 \rho_{d\cdots1} + \mathcal{O}(\tau^2).
\label{ADDM}
\end{align}
Here $ \rho_{d\cdots1} = U^{ \text{QC}} \cdot \rho (0) \cdot \big{(} U^{ \text{QC}} \big{)}^\dagger$ describes the noise-free (ideal) quantum state of the register bits.
In other words, it is the ideal output quantum state generated by the quantum algorithm given by $U^{ \text{QC}}$.
The quantity $\Delta^{\text{AD}}_p \rho_{d\cdots1}$ $(p \geq1)$ is the theoretically-evaluated $p$-th-order AD effect.
Let us focus on the first-order AD effect $\Delta^{\text{AD}}_1 \rho_{d\cdots1}$ which has the form
\begin{align}
\Delta^{\text{AD}}_1 \rho_{d\cdots1} & = \sum_{k=1}^{d} \Delta^{\text{AD}}_{1,k} \rho_{d\cdots1}, \notag\\
\Delta^{\text{AD}}_{1,k} \rho_{d\cdots1} & = \left(\prod_{l=k+1}^d U_l \right) \cdot \tilde{\rho}^{\text{AD}}_{k\cdots1} \cdot \left(\prod_{l=k+1}^d U_l \right)^\dagger ,
\label{ADdeviationDM1}
\end{align}
where
\begin{align}
\tilde{\rho}^{\text{AD}}_{k\cdots1} & = \mathcal{L} _{ \text{AD}} \big{[} \rho_{k\cdots1} \big{]}
= \sum_{j=0}^{N_q-1} \left[ \tilde{\sigma}^-_j \rho_{k\cdots1} \tilde{\sigma}^+_j
- \frac{1}{2} \big{\{} P^1_{j} , \rho_{k\cdots1} \big{\}} \right], \notag\\
\rho_{k\cdots1} & = \left(\prod_{l=1}^k U_l \right) \cdot \rho (0) \cdot \left(\prod_{l=1}^k U_l \right)^\dagger,
\label{ADdeviationDM2}
\end{align}
with $ \prod_{l=k+1}^d U_l = U_d \cdot U_{d-1} \cdots U_{k+2} \cdot U_{k+1}$ and $ \prod_{l=1}^k U_l = U_k \cdot U_{k-1} \cdots U_2 \cdot U_1.$
The operator $ P^1_{j} = \tilde{\sigma}^+_j \tilde{\sigma}^-_j = \frac{\boldsymbol{1}_j - Z_j}{2}$ describes the projection onto the quantum state $|1 \rangle_j$ with $\boldsymbol{1}_j$ and $Z_j$
denoting the identity operator and the $Z$ gate acting on $Q_{ \text{r}j }$, respectively:
On the other hand, the projection operator of the quantum state $|0 \rangle_j$ is given by $ P^0_{j} = \tilde{\sigma}^-_j \tilde{\sigma}^+_j = \frac{ \boldsymbol{1}_j + Z_j}{2}$.
\subsection{QEM Scheme }\label{QEM}
Since we have evaluated the single-qubit-state AD effect, next we discuss our quantum error mitigation (QEM) scheme.
We denote the operator of which we are aiming to take an expectation value by $\hat{O}$.
When we implement the quantum state $\rho$ on a real device
what we actually obtain is a quantum state which is different from $\rho$ due to quantum noise effects: Note again that hereinafter we only consider the AD effect. Let us write it by $\rho^{\text{real}}$.
We represent the density matrix $\rho^{\text{real}}$ in terms of $\rho$ (ideal state) as $\rho^{\text{real}}= \rho + \delta^{\text{AD}} \rho$, where $\delta^{\text{AD}} \rho$ represents the deviation from $ \rho$ owing to the AD effect on a real device.
Note that we use the symbol $\delta^{\text{AD}}$ to describe the AD effect on a real device while we use $\Delta^{\text{AD}}$ to describe the theoretically-estimated AD effect like Eq. \eqref{ADDM}.
Namely, a quantum computational error occurs due to the deviation $\delta ^{\text{AD}} \rho$.
QEM is a prescription for mitigating the error coming from the deviation $\delta ^{\text{AD}} \rho$.
Mathematically, this is a task to make the value of Tr$(\hat{O}\delta ^{\text{AD}} \rho)$ as small as possible.
In our scheme, we mitigate the error Tr$(\hat{O} \delta ^{\text{AD}} \rho)$ by perturbatively treating the deviation $\delta ^{\text{AD}} \rho$ with respect to $\tau$ and using the theoretically-estimated AD effect $\Delta^{\text{AD}}_p \rho$.
In the following we show such a perturbative analysis up to the first order in $\tau$. The extension of QEM scheme to higher-order AD effect is discussed in Appendix. \ref{extdQEMSsub1}.
The key procedure of our QEM scheme is to construct quantum circuits for computing the quantity Tr$(\hat{O} \Delta^{\text{AD}}_1 \rho_{d\cdots1})$,
which describes the theoretically-estimated quantum computational error of the expectation value Tr$(\hat{O} \rho)$ in the first order of $\tau$.
For doing this, there are two difficulties:
(i) the generation of the anti-commutator term $ \big{\{} P^1_{j} , \rho_{k\cdots1} \big{\}} $ in Eq. \eqref{ADdeviationDM2}
and (ii) the implementation of the non-unitary operators $\tilde{\sigma}^-_j$ and $P^1_j$ .
Let us discuss from our solution to the difficulty (i).
We denote some sort of quantum-computational operation (gate operation or measurement) by $\mathcal{A}$.
When the operation $\mathcal{A}$ acts on the quantum state $ \rho_{k\cdots1} $ the output state we have is $ \rho_{k\cdots1} \ \to \ \mathcal{A} \rho_{k\cdots1} \mathcal{A}^\dagger$.
The anti-commutator term $ \big{\{} P^1_{j} , \rho_{k\cdots1} \big{\}} $, in contrast, is not represented in this way, and thus, it is not clear how to generate such a term by the quantum-computational operations.
We solve this in the following way. To make our argument simple, here let focus on the single-register-bit system $(N_q=1)$; the generalization to $N_q \geq2$ is straightforward and is discussed later.
First, we rewrite $\tilde{\rho}^{\text{AD}}_{k\cdots1}$ in Eq. \eqref{ADdeviationDM2} as
\begin{align}
\tilde{\rho}^{\text{AD}}_{k\cdots1} = -\frac{ \rho_{k\cdots1}}{4} + \frac{ Z \rho_{k\cdots1} Z }{4} + \tilde{\sigma}^- \rho_{k\cdots1} \tilde{\sigma}^+ - P^1 \rho_{k\cdots1} P^1.
\label{ADdeviationDM3}
\end{align}
In the above way, all the four terms in Eq. \eqref{ADdeviationDM3} are written in the form $ \mathcal{A} \rho_{k\cdots1} \mathcal{A}^\dagger$, and thus, we have solved the difficulty (i).
Let us analyze the mathematical structure of the right-hand side of Eq. \eqref{ADdeviationDM3}.
The quantum circuit for creating the first term is straightforward because it is obtained by the quantum circuit composed of $U^{ \text{QC}}$ (the quantum algorithm under consideration).
The implementation of the quantum circuit for the second term $ \frac{ Z \rho_{k\cdots1} Z }{4} $ is also straightforward because we just apply the $Z$ gate after the operation of $U_k.$
The unclear part is to find ways to construct the quantum circuits for generating the third and fourth terms given by the non-unitary operators $ \tilde{\sigma}^- $ and $P^1$
and this is nothing but the difficulty (ii).
We solve this by using an ancilla bit and a measurement on it \cite{QCQINandC}.
For the creating the operation $\tilde{\sigma}^-,$ we use the quantum circuit presented in Fig. \ref{ADcircs} (a) (AD-effect circuit A) while for the operation of $P^1$ we use the one in Fig. \ref{ADcircs} (b) (AD-effect circuit B).
\begin{figure}[h]
\includegraphics[width=0.4 \textwidth]{ADcircuits.png}
\caption{
(a) Schematic of AD-effect circuit A. When we set $\vartheta =\pi $ and post-select the measurement result of the quantum state of $Q_{a0}$ to be $|1 \rangle _{Q_{a0}}$,
we realize the operation of $ \tilde{\sigma}^-$ on $Q_{r0}$.
(b) Schematic of AD-effect circuit B. By setting $\vartheta =\pi $ and post-selecting the measurement result of the quantum state of $Q_{a0}$ to be $|0 \rangle _{Q_{a0}}$,
we the operation of $ P^1$ on $Q_{r0}$ is created.
}
\label{ADcircs}
\end{figure}
In both quantum circuits, we regard the ancilla bit $ Q_{a0} $ as the environment which induces the AD effect on the register bit $Q_{r0}$.
The interactions between these two qubits are represented by the controlled-rotational gate $U_{CR_y}[Q_{r0};Q_{a0}](\vartheta)$ and the controlled-not gate $U_{CX}[Q_{a0};Q_{r0}]$.
The controlled-rotational gate $U_{CR_y}[Q_{r0};Q_{a0}](\vartheta)$ describes the rotation about $y$ axis by the rotational angle $\vartheta$ and it is composed of the control bit $Q_{r0}$ and the target bit $Q_{a0}$.
On the other hand, for the gate operation $U_{CX}[Q_{a0};Q_{r0}]$ the ancilla bit $Q_{a0}$ is the control bit while the register bit $Q_{r0}$ is the target bit.
We have used the notation such that the control bit(s) comes before the semicolon while the target bit(s) comes after.
Let us explain the output states generated by the AD-effect circuits A and B.
For both quantum circuits, the initial quantum states of $ Q_{r0} $ and $ Q_{a0} $ are the same and it is
$\rho_{Q_{r0}Q_{a0}}(0) = \rho_{Q_{r0}} (0) \otimes \rho_{Q_{a0}}(0)$ with $\rho_{Q_{r0}} (0) = |0\rangle_{Q_{r0}} \langle0|$ and $ \rho_{Q_{a0}}(0)= |0\rangle_{Q_{a0}} \langle0|$.
The AD-effect circuit A is given by the unitary operation $U_{\text{ADA}} =U_{CX}[Q_{a0};Q_{r0}] \cdot U_{CR_y}[Q_{r0};Q_{a0}](\vartheta)$ while
the AD-effect circuit B is given by $U_{\text{ADB}} =U_{CX}[Q_{a0};Q_{r0}] \cdot (\boldsymbol{1}_{2\times2} \otimes X_{ Q_{a0} })\cdot U_{CR_y}[Q_{r0};Q_{a0}](\vartheta)$, where $\boldsymbol{1}_{2\times2}$ is the two by two identity operator.
Owing to these unitary operations, the quantum state generated by the AD-effect circuit A is given by $ \rho_{\text{ADA},Q_{r0}Q_{a0}} (\vartheta) =
U_{\text{ADA}}\cdot \rho_{Q_{r0}Q_{a0}}(0) \cdot(U_{\text{ADA}})^\dagger $ while the quantum state
created by the AD-effect circuit B is $\rho_{\text{ADB},Q_{r0}} (\vartheta) = U_{\text{ADB}}\cdot \rho_{Q_{r0}Q_{a0}}(0) \cdot(U_{\text{ADB}})^\dagger $.
At the end, we measure the ancilla bit $Q_{a0}.$ Then the quantum states of $Q_{r0}$ (reduced density matrices) are described by the Kraus operators \cite{QCQINandC,KrausRepresentationref}.
\begin{align}
\mathcal{K} ^{\text{ADA}} _{0} & = {}_{Q_{a0}} \langle 0| U_{\text{ADA}} |0 \rangle _{Q_{a0}}
= P^0 + \cos \left( \frac{ \vartheta }{2} \right)P^1 \notag\\
\mathcal{K} ^{\text{ADA}} _{1} & = {}_{Q_{a0}} \langle 1| U_{\text{ADA}} |0 \rangle _{Q_{a0}}
= \sin \left( \frac{ \vartheta }{2} \right)\tilde{\sigma}^-, \notag\\
\mathcal{K} ^{\text{ADB}} _{0} & = {}_{Q_{a0}} \langle 0| U_{\text{ADB}} |0 \rangle _{Q_{a0}}
= \sin \left( \frac{ \vartheta }{2} \right)P^1 , \notag\\
\mathcal{K} ^{\text{ADB}} _{1} & = {}_{Q_{a0}} \langle 1| U_{\text{ADB}} |0 \rangle _{Q_{a0}}
= \tilde{\sigma}^+ + \cos \left( \frac{ \vartheta }{2} \right)\tilde{\sigma}^- . \label{Kraus1}
\end{align}
For the AD-effect circuit A (B) the Kraus operators $ \mathcal{K} ^{\text{ADA}} _{0} $ ($\mathcal{K} ^{\text{ADB}} _{0}$) acts
on the register bit $Q_{r0}$ when the measurement outcome of the quantum state of the ancilla bit $Q_{a0}$ is
$|0 \rangle _{Q_{a0}}$ while $ \mathcal{K} ^{\text{ADA}} _{1} $ $(\mathcal{K} ^{\text{ADB}} _{1}) $ operates when the measurement outcome is $|1 \rangle _{Q_{a0}}$.
When we average these two outcomes,
the quantum state of $Q_{r0}$ created by the AD-effect circuit A is given by$ \rho_{\text{ADA},Q_{r0}} (\vartheta) = \text{Tr}_{Q_{a0}}\big{[} \rho_{\text{ADA},Q_{r0}Q_{a0}} (\vartheta) \big{]}=
\sum_{\mu=0,1} \mathcal{K} ^{\text{ADA}} _{\mu} \cdot \rho_{Q_{r0}} (0) \cdot ( \mathcal{K} ^{\text{ADA}} _{\mu})^\dagger$,
where the symbol $\text{Tr}_{Q_{a0}}$ denotes the partial trace with respect to $Q_{a0}$ degrees of freedom.
Similarly, for the AD-effect circuit B we have $ \rho_{\text{ADB},Q_{r0}} (\vartheta) =
\sum_{\mu=0,1} \mathcal{K} ^{\text{ADB}} _{\mu} \cdot \rho_{Q_{r0}} (0) \cdot ( \mathcal{K} ^{\text{ADB}} _{\mu})^\dagger$.
In particular, for the AD-effect circuit A when we take $\vartheta$ to be $\vartheta_t$ such that $ \cos^2\big{(} \frac{\vartheta_t}{2}\big{)} = e^{-\gamma t} $ \cite{QCQINandC},
the matrix representation of $\rho_{\text{ADA},Q_{r0}} (\vartheta_t)$ is given by
\begin{widetext} \begin{align}
\rho_{\text{ADA},Q_{r0}} (\vartheta_t) =
\left (
\begin{array}{cc}
\left[ \rho_{\text{ADA},Q_{r0}} (0) \right]_{00} + \left[ \rho_{\text{ADA},Q_{r0}} (0) \right]_{11} (1- e^{-\gamma t} )& \left[ \rho_{\text{ADA},Q_{r0}} (0) \right]_{01} e^{- \frac{\gamma t}{2}} \\
\left[ \rho_{\text{ADA},Q_{r0}} (0) \right]_{10} \cdot e^{- \frac{\gamma t}{2}} & \left[ \rho_{\text{ADA},Q_{r0}} (0) \right]_{11} e^{-\gamma t}
\end{array}
\right ).
\label{outRDMQr0}
\end{align}\end{widetext}
The matrix element $ \left[ \rho_{\text{ADA},Q_{r0}} (0) \right]_{nn^\prime} $ $(n,n^\prime=0,1)$ denotes the $(n,n^\prime)$-element of $ \rho_{\text{ADA},Q_{r0}} (0).$
The reduced density matrix $\rho_{\text{ADA},Q_{r0}} (\vartheta_t)$ in Eq. \eqref{outRDMQr0} is nothing but the solution of the quantum master equation \eqref{ADQME}.
Further, when we take $\vartheta_t \to \pi$, the Kraus operators in Eq. \eqref{Kraus1} becomes
$\big{\{}\mathcal{K} ^{\text{ADA}} _{0} ,\mathcal{K} ^{\text{ADA}} _{1} \big{\}} \to \big{\{} P^0 , \tilde{\sigma}^- \big{\}},
\big{\{}\mathcal{K} ^{\text{ADB}} _{0} ,\mathcal{K} ^{\text{ADB}} _{1} \big{\}} \to \big{\{} \tilde{\sigma}^+,P^1 \big{\}}. $
Therefore, for the case of the AD-effect circuit A by using the measurement result of $Q_{a0}$ such that we post-select the output state of $Q_{a0}$ to be $|1 \rangle _{Q_{a0}}$
we can realize the operation of $\tilde{\sigma}^- $ on $Q_{r0}$.
On the other hand, for the AD-effect circuit B by post-selecting the output state of $Q_{a0}$ to be $|0 \rangle _{Q_{a0}}$ we realize the operation of $P^1 $ on $Q_{r0}$.
To show the above things concretely, let us present the examples of the quantum circuits for the generation of
$ \left(\prod_{l=k+1}^d U_l \right) \cdot \left( \tilde{\sigma}^- \rho_{k\cdots1} \tilde{\sigma}^+ \right) \cdot \left(\prod_{l=k+1}^d U_l \right)^\dagger $
and $ \left(\prod_{l=k+1}^d U_l \right) \cdot \left( P^1 \rho_{k\cdots1} P^1 \right) \cdot \left(\prod_{l=k+1}^d U_l \right)^\dagger $ for $k=2, d=3$, and we show them in Figs. \ref{exspk2d3AD} (a) and \ref{exspk2d3AD} (b), respectively.
As a result, by using the AD-effect circuits A and B we can perform the actions of $\tilde{\sigma}^- $ and $P^1 $ as described by the third and four terms in Eq. \eqref{ADdeviationDM3},
and thus, we have solved the second difficulty (ii).
%
\begin{figure}[h]
\includegraphics[width=0.4 \textwidth]{exspk2d3AD.png}
\caption{
(a) Schematic of an elementary quantum circuit in the AD-effect circuit group given by the AD-effect circuit A which generates $ \left(\prod_{l=k+1}^d U_l \right) \cdot \left( \tilde{\sigma}^- \rho_{k\cdots1} \tilde{\sigma}^+ \right) \cdot \left(\prod_{l=k+1}^d U_l \right)^\dagger $.
For doing so, we set $\vartheta =\pi $ and post-select the measurement result of the quantum state of $Q_{a0}$ to be $|0 \rangle _{Q_{a0}}$.
(b) Schematic of an elementary quantum circuit in the AD-effect circuit group given by the AD-effect circuit B. When we post-select the measurement outcome of $Q_{a0}$ to be $|1 \rangle _{Q_{a0}}$, we have $ \left(\prod_{l=k+1}^d U_l \right) \cdot \left( P^1 \rho_{k\cdots1} P^1 \right) \cdot \left(\prod_{l=k+1}^d U_l \right)^\dagger $. Both of these quantum circuits in (a) and (b) are for $k=2, d=3$ with $\vartheta=\pi$. }
\label{exspk2d3AD}
\end{figure}
Since the difficulties (i) and (ii) have been solved, we are now ready to establish our QEM scheme.
To compute the quantity Tr$(\hat{O} \cdot \Delta^{\text{AD}}_1 \rho_{d\cdots1})$ we need four types of quantum circuits:
the original circuit given by $U^{\text{QC}}$, the quantum circuit where the additional $Z$ gate is performed, and the AD-effect circuits A and B.
The latter three quantum-circuit ensembles composed of the additional $Z$-gate, $\tilde{\sigma}^-, $ and $P^1$ operations form the quantum-noise-effect circuit group for the AD effect, i.e, AD-effect circuit group.
Hereinafter, let us write the trace of the product between the operator $\hat{O}$ and $\rho$ by $ \text{Tr}(\hat{O} \rho) = \langle \hat{O} \rangle_\rho$.
We perturbatively express the quantum states $ \rho ^{\text{real}} _{d\cdots1} $ with respect to $\tau$ as
$ \rho ^{\text{real}} _{d\cdots1} = \rho _{d\cdots1} + \delta^{\text{AD}} ( \rho _{d\cdots1} ) $ with
$ \delta^{\text{AD}} ( \rho _{d\cdots1} ) = \sum_{p=1}^\infty \frac{ \tau^p }{p!} \cdot \delta^{\text{AD}}_p ( \rho _{d\cdots1} )$.
Furthermore, we write the quantity which is obtained by the implementation of $\Delta^{\text{AD}}_1 \rho_{d\cdots1}$ on a real device by $ (\Delta^{\text{AD}}_1 \rho_{d\cdots1})^{\text{real}} $.
With similar to $ \rho ^{\text{real}} _{d\cdots1}$, we perturbatively express $ (\Delta^{\text{AD}}_1 \rho_{d\cdots1})^{\text{real}} $ in terms of
$ \Delta^{\text{AD}}_1 \rho_{d\cdots1} $ and $\tau$ as
$ (\Delta^{\text{AD}}_1 \rho_{d\cdots1})^{\text{real}} = \Delta^{\text{AD}}_1 \rho_{d\cdots1} + \delta^{\text{AD}}( \Delta^{\text{AD}}_1 \rho_{d\cdots1})$
with $ \delta^{\text{AD}} ( \Delta^{\text{AD}}_1 \rho_{d\cdots1} ) = \sum_{p=1}^\infty \frac{ \tau^p }{p!} \cdot \delta^{\text{AD}}_p (\Delta^{\text{AD}}_1 \rho_{d\cdots1} ). $
Then, by using Eqs. \eqref{ADDM}, \eqref{ADdeviationDM1}, \eqref{ADdeviationDM2}, and \eqref{ADdeviationDM3} we obtain
the quantum-error-mitigated expectation value of $\hat{O}$ given by
\begin{align}
\langle \hat{O} \rangle^{\text{QEM}}_{\rho _{d\cdots1}} & \equiv
\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} - \tau \langle \hat{O} \rangle_{ (\Delta^{\text{AD}}_1 \rho_{d\cdots1})^{\text{real}} } \notag\\
& = \langle \hat{O} \rangle_{\rho _{d\cdots1}}
+ \tau \Big{(} \langle \hat{O} \rangle _{ \delta^{\text{AD}}_1 ( \rho _{d\cdots1} )} - \langle \hat{O} \rangle_{ \Delta^{\text{AD}}_1 \rho_{d\cdots1} } \Big{)} \notag\\
&+ \mathcal{O}(\tau^2). \label{QEMformula1}
\end{align}
The idea of our QEM scheme is clearly represented in the second line of Eq. \eqref{QEMformula1}.
The first term is the ideal expectation value while the second term represents the conduction of our QEM scheme.
It is described as the subtraction between $\langle \hat{O} \rangle _{ \delta^{\text{AD}}_1 ( \rho _{d\cdots1} )}$ (the quantum computational error occurring on a real device) and
$ \langle \hat{O} \rangle_{ \Delta^{\text{AD}}_1 \rho_{d\cdots1} } $ (theoretically-evaluated quantum computational error), which is computed by the AD-effect circuit group.
The heart of the idea for doing this is that we have considered that the real noise effect $\delta^{\text{AD}}_1 ( \rho _{d\cdots1} )$ is (approximately) equivalent to the theoretically-estimated noise effect $\Delta^{\text{AD}}_1 ( \rho _{d\cdots1} )$.
When the second term in the second line of Eq. \eqref{QEMformula1} becomes small enough,
we consider that we have accomplished in mitigating the error of quantum computation on a real device.
Note that the quantum noise effects coming from $ \delta^{\text{AD}} ( \Delta^{\text{AD}} \rho_{d\cdots1} )$ are suppressed by multiplying $ \langle \hat{O} \rangle_{ (\Delta^{\text{AD}} \rho_{d\cdots1})^{\text{real}} } $
by $\tau$ (the second term in the right-hand side of the first line of Eq. \eqref{QEMformula1}).
This is because the lowest-order error AD effect on the implementation of $ \Delta^{\text{AD}}_1 \rho_{d\cdots1}$, which is $\delta^{\text{AD}}_1 (\Delta^{\text{AD}}_1 \rho_{d\cdots1} ),$ becomes $\mathcal{O}(\tau^2)$ due to the multiplication by $\tau$:
$ \delta^{\text{AD}} ( \Delta^{\text{AD}}_1 \rho_{d\cdots1} ) \to
\tau \cdot \delta^{\text{AD}} ( \Delta^{\text{AD}}_1 \rho_{d\cdots1} ) = \sum_{p=1}^\infty \tau^{p+1} \cdot \delta^{\text{AD}}_p (\Delta^{\text{AD}}_1 \rho_{d\cdots1} ). $
Consequently, in the first-order perturbation theory with respect to $\tau$ we have established our QEM scheme by the usage of the AD-effect circuit group and is expressed by the formula given by Eq. \eqref{QEMformula1}.
The above argument on QEM-scheme derivation can be straightforwardly generalized to register-bit systems for $N_q \geq2.$
In this case, $\tilde{\rho}^{\text{AD}}_{k\cdots1} $ in Eq. \eqref{ADdeviationDM1} is represented as
\begin{align}
\tilde{\rho}^{\text{AD}}_{k\cdots1} & = \mathcal{L} _{ \text{AD}} [ \rho_{k\cdots1}] \notag\\
& = \sum_{j=0}^{N_q-1} -\frac{ \rho_{k\cdots1}}{4} + \frac{ Z_j \rho_{k\cdots1} Z_j }{4} + \tilde{\sigma}^-_j \rho_{k\cdots1} \tilde{\sigma}^+_j \notag\\
& - P^1_j \rho_{k\cdots1} P^1_j.
\label{ADdeviationDM4}
\end{align}
We can apply our QEM scheme to the $N_q$ register-bit system in the following way.
We prepare $N_q$ register bits and one ancilla bit $\big{\{} Q_{r0}, Q_{r1}, \ldots, Q_{rN_q-1}, Q_{a} \big{\}}$.
Then we create an ensemble of quantum circuits composed of the $j$-th register bit $Q_{rj}$ $(j=0,1, \ldots,N_q-1)$ and the ancilla bit $Q_{a}$ which describes that
$Q_{rj}$ is subject to the AD effect induced by the ancilla bit $Q_{a}$.
Namely, we create the ensemble of four types of quantum circuits composed of $Q_{rj}$ and $Q_{a}$, the original quantum circuits given by $U^{ \text{QC}}$, the quantum circuits with additional $Z$-gate operations, and the AD-effect circuits A and B.
By summing up all these quantum circuits, we obtain the AD-effect circuit group which enables us to perform QEM for $N_q$-register-bit system under the AD effect,
The total number of quantum circuits which compose the AD-effect circuit group is $3dN_q+1,$ and thus, it scales polynomial in $dN_q$, which is not so high-cost computational performance.
Several comments are given in order.
Firstly, we can extend our QEM formalism into cases of other quantum noises including generalized amplitude damping (GAD), phase damping (PD), their composite channels,
and stochastic Pauli noise models such as bit flip, phase flip, bit-phase flip, and depolarizing channel.
Secondly, we can create quantum-noise-effect circuit groups which enable us to perform QEM for higher-order quantum noise effects.
We present the arguments on these two cases in Appendix \ref{appendix1}.
Thirdly, we have set the gate time $\Delta t$ to be equivalent for all the gates.
For real quantum devices, however, the gate times are not equivalent, for instance, the gate times for the single-qubit gates are shorter than those of the two-qubit gates.
Despite this situation, we can extend our QEM formalism for cases of gate times not being equivalent since the orders of the gate times are approximately the same as we discuss in subSec. \ref{discussionQEMNISQ}.
In addition, in the above analysis we have considered the situation where the quantum noise strength is homogeneous for all qubits. In real situations, however, quantum noise strengths might be inhomogeneous due to imperfect fabrication of quantum devices. Nevertheless, we can similarly and easily extend our QEM scheme to such cases since the quantum noise strengths are considered to be in the same order.
%
\begin{table}[hbtp]
\caption{Comparison of the QEM methods}
\label{table:comparison_of_qem}
\centering
\begin{tabular}{lcc}
\hline
Method & $N_a$ & $N_{\text{AQC}}$ \\
\hline
ZNE \cite{EMPRL2017,EMPRX2017} & 0 & $N_\tau$ \\
PEC \cite{EMPRL2017, EMPRX2018} & 0 & $\mathcal{O}(e^{2b\varepsilon d})$ \\
Our proposed method & $k$ & $\mathcal{O}(\{d N_q\}^{k})$ \\
\hline
\end{tabular}
\end{table}
Finally, we stress the feature of our proposed method by comparing with
two other representative QEM methods using the following quantities,
the number of register qubits $N_q$, the depth of quantum algorithm to be ran, the number of ancillary qubits $N_a$,
and the number of additional quantum circuits $N_{\text{AQC}}$,
In Table \ref{table:comparison_of_qem}, we summarize and present the comparison
between our QEM method and the others. Here $k$ is the order of the perturbation theory of our QEM method.
First, let us make a comparison with zero noise extrapolation (ZNE) \cite{EMPRL2017,EMPRX2017}.
This method does not require ancilla bits but requires additional quantum circuits for various noise strengths, from which ideal values are estimated. We write the number of noise strengths to perform ZNE by $N_\tau$ and denote them by $\gamma_0,\gamma_1,\ldots< \gamma_{N_\tau-1}:$ for ZNE $N_{\text{AQC}} = N_\tau$.
In practice, $N_\tau$ is not relatively a large number and ZNE can be conducted
without depending on $N_q$ and $d$ which means that QEM cost is relatively small
but its quality gets worse provided that $\gamma_0$ is small enough.
Our QEM method, on the other hand, requires ancilla qubits and the value of $N_{\text{AQC}}$
for doing $k$-th order perturbation calculation is
$\mathcal{O}(\{d N_q\}^{k})$, which is in general higher than $N_\tau$.
Nevertheless, in principle our QEM method can be applied for any value of noise strength $\gamma$
by doing higher-order perturbation calculation and the quality of our method becomes higher for larger $k$.
Second, let us make a comparison with probabilistic error cancellation (PEC) \cite{EMPRL2017, EMPRX2018}.
Like ZNE, PEC can recover zero-noise values without any ancillary qubit while
it is necessary to prepare additional quantum circuits and the number of them
scales exponentially with respect to the product of error rate of gates $\varepsilon$ and $d$, i.e., $N_{\text{AQC}} \sim \mathcal{O}(e^{2b\varepsilon d})$, where $b$ is real positive value \cite{hybridQCalgorithmJPSJ2021}.
As discussed previously, in our QEM method although we need ancilla bits
we can do higher-order perturbation calculation such that $ N_{\text{AQC}} \sim \mathcal{O}(\{dN_q\}^{k})$, which is polynomial in $dN_q$ and is lower than $N_{\text{AQC}}$ for PEC.
%
\section{Numerical Simulations}\label{nss}
In this section, we numerically demonstrate our QEM scheme for four types of algorithms.
For the quantum noise effect we choose AD effect. In subSec. \ref{QEMpre}, as a preliminary of our QEM demonstration,
we present the results of two algorithms: the algorithm composed of the initial $X$-gate operation and the repetition of the Hadamard gate acting on a single register bit
and that composed of the initial $X\otimes X$ operation and the controlled-Hadamard gate acting on two register bits.
In subSec. \ref{QEMQAAalg}, we show the results of QEM for a long-term quantum algorithm and here we choose quantum amplitude amplification (QAA) algorithm (Grover's search algorithm).
In subSec. \ref{QEMqaoa}. we show the results of recently developed NISQ quantum algorithms, Quantum Approximate Optimization Algorithm (QAOA).
In subSec. \ref{discussionQEMNISQ} , we discuss the validity of our QEM scheme under the conditions of NISQ-device parameters.
In the following, let us explain the formalism of our noisy quantum simulations (numerical simulations of running quantum algorithms with real quantum devices performed by classical computers).
As shown in Fig. \ref{QCunderAD1}, every time we apply an unitary (gate) operation, the $N_q$ register bits experience AD effects.
Suppose that at a time $t_0$ the quantum states of the register bits were given by the density matrix $\rho(t_0)$. According to the quantum master equation \eqref{ADQME},
when the unitary gate $U$ has been applied to the register bits within the time interval $\Delta t$ the quantum state of the register bits at $t=t_0 + \Delta t$
is expressed by the density matrix
\begin{widetext}
\begin{align}
\rho^{\text{AD}} (t_0 + \Delta t) = \mathcal{E}^{\text{AD}} [U\rho(t_0)U^\dagger ],
\label{noisyQsimformula1}
\end{align}
where $\mathcal{E}^{\text{AD}}[\cdots]$ is the superoperator which describes the AD effect on the $N_q$ register bits and it is given by
\begin{align}
& \mathcal{E}^{\text{AD}} [\rho] = \sum_{ n_{Q_{\text{r}0}}, \ldots, n_{Q_{\text{r}N_q-1}} } \mathcal{M}_{ n_{Q_{\text{r}0}}, \ldots, n_{Q_{\text{r}N_q-1}} } \cdot
\rho \cdot
\mathcal{M}^\dagger_{ n_{Q_{\text{r}0}}, \ldots, n_{Q_{\text{r}N_q-1}} }, \notag\\
& \mathcal{M}_{ n_{Q_{\text{r}0}}, \ldots, n_{Q_{\text{r}N_q-1}} } = \mathcal{M}^{\text{AD}}_{ n_{Q_{\text{r}0}} } \otimes \cdots \otimes \mathcal{M}^{\text{AD}}_{ n_{Q_{\text{r}N_q-1}} }, \notag\\
& \mathcal{M} ^{\text{AD}} _{0} = \left [
\begin{array}{cc}
0 & \sin\left( \frac{\vartheta_\tau}{2} \right) \\
& 0
\end{array}
\right ], \quad
\mathcal{M} ^{\text{AD}} _{1} = \left [
\begin{array}{cc}
1 & 0 \\
0 & \cos\left( \frac{\vartheta_\tau}{2} \right)
\end{array}
\right ],
\label{noisyQsimformula2}
\end{align}\end{widetext}
where $n_{Q_{\text{r}0}}, \ldots, n_{Q_{\text{r}N_q-1}} =0,1$, and $ \cos^2\left( \frac{\vartheta_\tau}{2} \right) = e^{-\tau}$.
The operators $ \mathcal{M} ^{\text{AD}} _{0}$ and $ \mathcal{M} ^{\text{AD}} _{1}$ are the Kraus operators acting on single-qubit states and describe the influence of the AD effect on a single register bit
during the time interval $\Delta t$.
Here we assume that the $N_q$ register bits homogeneously experience
the single-qubit-state AD effect as described in Eq. \eqref{noisyQsimformula2}.
For later convenience, let us introduce the notation which describes the operation of the unitary operator $U$ on the quantum state $\rho$ by
$\mathcal{T}[\rho,U] (=U\rho U^\dagger)$.
Let us write the quantum state generated by the unitary transformation $U^{ \text{QC}}$ under the AD effect by $\rho^{\text{AD}}_{d\cdots1}$.
By using the superoperator $\mathcal{E}^{\text{AD}}[\cdots]$, the output state $\rho^{\text{AD}}_{d\cdots1}$ is represented as
\begin{widetext}
\begin{align}
\rho^{\text{AD}}_{d\cdots1} = \mathcal{E}^{\text{AD}}\big{[} \mathcal{T}[ \cdots \mathcal{E}^{\text{AD}}\big{[} \mathcal{T}[ \mathcal{E}^{\text{AD}}\big{[} \mathcal{T}[ \rho(0), U_1] \big{]}, U_2 ] \big{]}, \cdots U_d ] \big{]}.
\label{noisyQsimformula3}
\end{align}\end{widetext}
Eqs. \eqref{noisyQsimformula2} and \eqref{noisyQsimformula3} are the basic equations of our noisy quantum simulations.
Namely, we perform our noisy quantum simulations by identifying the quantum state $\rho^{\text{AD}}_{d\cdots1} $ in the above equation with $\rho^{\text{real}}_{d\cdots1}$, which is the AD-affected quantum state generated by the unitary transformation $U^{ \text{QC}}$ on real devices.
We conduct QEM described by Eq. \eqref{QEMformula1} for various values of $\tau$ by tuning the value of $\vartheta_\tau$.
For performing our noisy quantum simulations, we have created two types of numerical codes.
The first one is our original numerical code and the second one is the numerical code created by Qiskit \cite{Qiskit}.
The difference between these two codes is that the latter is programmed by the two numbers, $N_\text{QC}$ and $N_\text{samp}$.
The number $N_\text{QC}$ describes the repetition number of quantum computation owing to the given quantum circuit whereas the number $N_\text{samp}$ describes how many times you conduct the sampling for the expectation values of physical operators obtained by the $N_\text{QC}$-repeated quantum computation.
Owing to this sampling, the repetitive number of the quantum computations effectively becomes $N_\text{QC} \times N_\text{samp}$, and our simulation results become more trustable.
In our simulations we take $N_\text{QC} = 2^{10}$ and $ N_\text{samp}=100$.
Our original numerical code, on the other hand, is the code for a noisy quantum simulation in the limit $N_{\text{QC}} \to \infty$, and basically, it performs pure linear algebraical computations such as matrix-product and trace operations.
We note that when we create the numerical codes with Qiskit, we need to be careful with how controlled-unitary operators are implemented.
On Qiskit program, the controlled-unitary operators are implemented as the decomposition of $U_{CX}$ gates and single-qubit unitary gates.
For example, the control-$R_y$ gate $U_{CR_{y}(\vartheta)} [Q_{r1};Q_{r0}]$
is decomposed as $U_{CR_{y}(\vartheta)} [Q_{r1};Q_{r0}] = \left( {\boldsymbol{1}_{2\times2}}_{Q_{r0}} \otimes R_{y}(-\vartheta/2)_{Q_{r1}} \right) \cdot U_{CX} [Q_{r1};Q_{r0}] \cdot \left( {\boldsymbol{1}_{2\times2}}_{Q_{r0}} \otimes R_{y}(\vartheta/2)_{Q_{r1}} \right) \cdot U_{CX}[Q_{r1};Q_{r0}]$.
Therefore, when we simulate QAA for three-qubit systems and QAOA with our original code we implement $U_{CR_{y}(\vartheta)} [Q_{r1};Q_{r0}]$ in the same way as Qiskit program does.
In order to quantitatively describe the validity of our QEM scheme we introduce the measure defined by
\begin{align}
\text{RT}_{\text{QEM}} = \frac{| \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |}{| \langle \hat{O} \rangle_{\rho _{d\cdots1}} - \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} |}.
\label{QEMratio}
\end{align}
The numerator of $\text{RT}_{\text{QEM}}$ in Eq. \eqref{QEMratio} describes the absolute of the difference between the expectation value owing to the noisy quantum simulation ($\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}}$) and the ideal expectation value ($\langle \hat{O} \rangle_{\rho^{\text{ideal}} _{d\cdots1}}$) .
On the other hand, the denominator represents the absolute of the difference between the expectation value obtained by our QEM scheme ($\langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$, see Eq. \eqref{QEMformula1} and the ideal expectation value. In other words, the measure $\text{RT}_{\text{QEM}}$ in Eq. \eqref{QEMratio} is the ratio between the absolute of the error without QEM and the one with QEM.
Thus, when $\text{RT}_{\text{QEM}} >1$ the expectation value $\langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$ is closer to the ideal value than $\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}}$,
which means that our QEM scheme is working.
In addition to the ratio $\text{RT}_{\text{QEM}} $ in Eq. \eqref{QEMratio}, we display the results of ideal simulations, noisy simulations, and noisy simulations with QEM, and show explicitly the validity of our QEM scheme.
Note that for our original code we take the noise-strength parameter to be $\vartheta_\tau = i_\text{o} \times 0.01$
with $i_\text{o}=0,1,\ldots,50$ while for Qiskit code we take $\vartheta_\tau = i_\text{Q} \times 0.05$
with $i_\text{Q}=0,1,\ldots,10$. For computing the expectation values we include the cases $i_\text{o}=0$ and $i_\text{Q}=0$
whereas for computing the ratio $\text{RT}_{\text{QEM}}$ we omit $i_\text{o}=0$ and $i_\text{Q}=0$.
This is because in these two cases we have
$ \langle \hat{O} \rangle_{\rho _{d\cdots1}} = \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} = \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} $ and we encounter in the indefinite $\frac{0}{0}$.
In the following, we create a subsection for each quantum algorithm and discuss the results in detail.
\begin{figure}[!h]
\includegraphics[width=0.3 \textwidth]{XHXXCHqcircs.png}
\caption{
Quantum circuits for (a) $U^{ \text{QC}}_{ \text{pre}1}=H^{ \otimes d-1} \cdot X $ and
(b) $U^{ \text{QC}}_{ \text{pre}2}=(U_{\text{C}H}[Q_{r0};Q_{r1}] )^{ \otimes d-1} ) \cdot X_{Q_{r0}} \otimes X_{Q_{r1}}$.
}
\label{XHXXCHqcircs}
\end{figure}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.9 \textwidth ]{XHnoisyQsims.png}
\caption{ Quantum simulations for the quantum algorithm $U^{ \text{QC}}_{ \text{pre}1}=H^{ \otimes d-1} \cdot X$.
Plots in (a) and (b) are the results of QEM for the expectation value of $X$ and $Z$, respectively.
The dotted lines are the expectation values without QEM while the solid lines represent the expectation values with QEM.
The black dotted lines are the results of the ideal simulations. In (c) and (d), we show the ratio $\text{RT}_{\text{QEM}} $ for the expectation values of $X$ and $Z$
respectively. The red, blue, and green plots are the results for $d = 1+2^3,d = 1+2^4,$ and $d = 1+2^5$, respectively.
}
\label{XHnoisyQsims}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.8 \textwidth]{XXCHnoisyQsims.png}
\caption{ Quantum simulations for the quantum algorithm $U^{ \text{QC}}_{ \text{pre}2}=(U_{\text{C}H}[Q_{r0};Q_{r1}] )^{ \otimes d-1} ) \cdot X_{Q_{r0}} \otimes X_{Q_{r1}}$.
Plots in (a) and (b) are the results of QEM for the expectation value of $ZX$ and $ZZ$, respectively.
The dotted lines are the expectation values without QEM while the solid lines represent the expectation values with QEM.
The black dotted lines are the ideal simulation results.
We plot the ratio $\text{RT}_{\text{QEM}}$ for expectation value of $ZX$ and $ZZ$ in (c) and (d), respectively. The red, blue, and green plots are the results for $d = 1+2^3,d = 1+2^4,$ and $d = 1+2^5$, respectively.}
\label{XXCHnoisyQsims}
\end{figure*}
\subsection{Preliminary}\label{QEMpre}
As a preliminary, let us conduct the noisy quantum simulations for two simple algorithms.
The first algorithm is given by the unitary operation $U^{ \text{QC}}_{ \text{pre}1}=H^{ \otimes d-1} \cdot X $, where $H$ denotes the Hadamard gate.
The second one is given by $U^{ \text{QC}}_{ \text{pre}2}=(U_{\text{C}H}[Q_{r0};Q_{r1}] )^{ \otimes d-1} ) \cdot X_{Q_{r0}} \otimes X_{Q_{r1}}$.
The unitary operator $U_{CH}[Q_{r0};Q_{r1}]$ is the controlled-Hadamard gate composed of the control bit $Q_{r0}$ and the target bit $Q_{r1}$.
Here we take the circuit depth $d$ to be $d = 1 + 2^{n}$ with $n$ being positive integers.
Let us show the quantum circuits for these two algorithms in Fig. \ref{XHXXCHqcircs}.
By changing the values of $d$ and $\tau$, we numerically analyze how well our QEM scheme works in terms of these two parameters.
Let us discuss from the results of noisy quantum simulations conducted by the quantum circuit in Fig. \ref{XHXXCHqcircs} (a) and show them in Fig. \ref{XHnoisyQsims}.
We have taken the physical operators $\hat{O}$ as $\hat{O} = X,Z$.
The horizontal axis represents $\vartheta_\tau$, which can be regarded as the strength of AD effect.
On the other hand, the vertical axis in Figs. \ref{XHnoisyQsims}(a,b) denote the expectation values of $\hat{O}$ while those in Figs. \ref{XHnoisyQsims}(c,d) describe the ratio $\text{RT}_{\text{QEM}} $ given in Eq. \eqref{QEMratio}:
Figs. \ref{XHnoisyQsims}(a,c) are the results for $ \hat{O}=X$ while Figs. \ref{XHnoisyQsims}(b,d) are those for $ \hat{O}=Z$.
The dotted lines in Figs. \ref{XHnoisyQsims}(a,b)
describe the expectation values without QEM being performed whereas the solid lines represent the expectation values with QEM being performed.
For both the dotted and solid lines the red, blue, and green plots in Figs. \ref{XHnoisyQsims} (a) and (b) are the computational results of $\langle X \rangle$ and $\langle Z \rangle$
for $d=2^3 +1$, $2^4 +1$, and $2^5 +1$, respectively. The black dotted lines are the results of the ideal simulations.
Let us analyze our simulation results by comparing the behaviors of the expectation values and the ratio $\text{RT}_{\text{QEM}} $ in Eq. \eqref{QEMratio}
as functions of $\vartheta_\tau$. In this way, we can clearly see whether our QEM scheme is working or not, and for this purpose in the following we rewrite
$\langle \hat{O} \rangle_{\rho _{d\cdots1}}, \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}}, \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}},$
and $\text{RT}_{\text{QEM}}$ as $\langle \hat{O} \rangle_{\rho _{d\cdots1}} (\vartheta_\tau), \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}}(\vartheta_\tau), \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}(\vartheta_\tau),$ and $\text{RT}_{\text{QEM}}(\vartheta_\tau)$, respectively, to emphasize that they are the functions of $\vartheta_\tau$.
Furthermore, we introduce the angle $\vartheta^{\text{c}}_\tau$ such that $\text{RT}_{\text{QEM}} (\vartheta_\tau =\vartheta^{\text{c}}_\tau)=1$, which indicates that the point $ \vartheta_\tau = \vartheta^{\text{c}}_\tau $ is the critical point of our QEM to become failed.
Let us look from the results shown in Figs. \ref{XHnoisyQsims}(a,c) by focusing on how the behaviors of $\langle \hat{O} \rangle_{\rho _{d\cdots1}}, \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}}, \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}},$ and $\text{RT}_{\text{QEM}}$ change by increasing $\vartheta_\tau$.
In Fig. \ref{XHnoisyQsims}(a), as the definition of $ \vartheta^{\text{c}}_\tau $ we certainly see that in the range $0< \vartheta_\tau \leq \vartheta^{\text{c}}_\tau$
the absolute $| \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |$ is bigger than
$| \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |$, which implies that $\langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} $ is numerically closer to $\langle \hat{O} \rangle_{\rho _{d\cdots1}} $ than $\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}}$, and correspondingly, in Fig. \ref{XHnoisyQsims}(c) we see $\text{RT}_{\text{QEM}} (\vartheta_\tau) \geq 1$.
As we increase the value of $\vartheta_\tau$ from $ \vartheta^{\text{c}}_\tau $, the absolute $| \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |$ becomes smaller than $| \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |$, and correspondingly, the ratio $\text{RT}_{\text{QEM}}(\vartheta_\tau)$ decreases monotonically from one.
For the region $ \vartheta_\tau \geq \vartheta^{\text{c}}_\tau $ to improve the quality of our QEM we need take into account higher-order AD effects and establish QEM schemes for mitigating them and we expect the value of $\vartheta^{\text{c}}_\tau$
to become larger. Next, let us analyze how the quality of our QEM becomes when we vary the circuit depth $d$.
We see that for every $\vartheta_\tau$ both $| \langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |$ and $| \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} - \langle \hat{O} \rangle_{\rho _{d\cdots1}} |$ become bigger and $\text{RT}_{\text{QEM}}(\vartheta_\tau)$ decreases as we increase $d$. This is reasonable because when $d$ gets larger the amount of error gets bigger.
For the results in Figs. \ref{XHnoisyQsims}(b,d),
basically we see that both the expectation values of $Z$ and $\text{RT}_{\text{QEM}}(\vartheta_\tau)$ show the similar behaviors as those for $ \hat{O}=X$:
(I) the validity of QEM ($\text{RT}_{\text{QEM}} (\vartheta_\tau) \geq 1$) in the range $0< \vartheta_\tau \leq \vartheta^{\text{c}}_\tau$ and monotonic decrease of $\text{RT}_{\text{QEM}}(\vartheta_\tau)$ for $ \vartheta_\tau > \vartheta^{\text{c}}_\tau$, and
(II) worsening of the quality of our QEM for large $d$.
In contrast to the above characteristics of $\langle X \rangle$ and $\langle Z \rangle$, we have numerically verified that the expectation value of $Y $ takes zero for any $ \vartheta_\tau$.
This is because when the density matrix $\rho$ is a real matrix the expectation value $\langle Y \rangle $ is zero.
Since both the quantum algorithm and the AD effect are described by real numbers (see also Eq. \eqref{outRDMQr0} or the Kraus operators in Eq. \eqref{noisyQsimformula2} the density matrix generated by these two things is real and we have $\langle Y \rangle =0$.
Let us discuss the results in Fig. \ref{XXCHnoisyQsims}.
They are the noisy simulation results of the quantum algorithm given by $U^{ \text{QC}}_{ \text{pre}2}$ (see the quantum circuit in Fig. \ref{XHXXCHqcircs}) and we have taken $d = 1 + 2^{n}$ as in the case of simulations for $U^{ \text{QC}}_{ \text{pre}1}$.
Here we have simulated $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ for the expectation values of the operators $\hat{O} = ZX, ZZ$.
Overall, we see the same characteristics with the cases of $ \hat{O}=X,Z$: the characteristics (I) and (II) mentioned above.
For any $\vartheta_\tau$, the ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ for the noisy simulations of $U^{ \text{QC}}_{ \text{pre}2}$ are smaller than those of $U^{ \text{QC}}_{ \text{pre}1}$.
This is because $U^{ \text{QC}}_{ \text{pre}1}$ is solely comprised of the single-qubit gates ($X$ and $H$) while $U^{ \text{QC}}_{ \text{pre}2}$ is constructed by
$n$-operation of the controlled-Hadamard gate (two-qubit gate), and thus, the bigger amount of errors are accumulated in the latter case.
The difference between the characteristics of noisy simulations for $U^{ \text{QC}}_{ \text{pre}1}$ and those for $U^{ \text{QC}}_{ \text{pre}2}$,
although it is not an essential point for the validity of our QEM, is that we see one minima and one maxima in the plots for $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ (Figs. \ref{XXCHnoisyQsims}(c,d)).
Let us denote the point where $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ takes the minimum (maximum) by
$\vartheta^{\text{min}}_\tau$ ($\vartheta^{\text{max}}_\tau$): note that these values depend on $d$.
We can understand why these points emerge by looking at Figs. \ref{XXCHnoisyQsims}(a,b).
In the range $0< \vartheta_\tau \leq \vartheta^{\text{min}}_\tau$ we have $\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} < \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} $
whereas in the range $\vartheta^{\text{min}}_\tau < \vartheta_\tau \leq \vartheta^{\text{max}}_\tau$ we have
$\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} > \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} $.
Then, in the range $\vartheta^{\text{max}}_\tau < \vartheta_\tau$ we have $\langle \hat{O} \rangle_{\rho^{\text{real}} _{d\cdots1}} < \langle \hat{O} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} $.
As a result, the minima appears at $ \vartheta_\tau = \vartheta^{\text{min}}_\tau$ whereas the maxima emerges at $ \vartheta_\tau = \vartheta^{\text{max}}_\tau$.
Likewise the case of the noisy simulations of $U^{ \text{QC}}_{ \text{pre}1}$, the density matrices are generated as real matrices (the unitary transformation $U^{ \text{QC}}_{ \text{pre}2}$ as well as the AD effects are described by real numbers),
and the expectation values of the Pauli operators, $ IY, X Y, YI, YX, YZ, ZY$ are zero for both ideal and noisy simulations. Here we have rewritten $\boldsymbol{1}_{2\times2} $ as $I$ for convenience.
Note that the expectation value of the identity operator ($= \boldsymbol{1}_{4\times4}:$ four by four identity operator) is one for any quantum state including noise-affected quantum states since the trace of density matrix is one for any quantum state.
In other words, it is unnecessary to do QEM for the expectation value of the identity operator.
%
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.7 \textwidth]{QAAQcirc.png}
\caption{ Quantum circuits for $ U^{ \text{QAA}}$ in Eq. \eqref{UnitaryQAA}. }
\label{QAAQcirc}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.8 \textwidth]{groverq3probsandratios.png}
\caption{ Quantum simulation results for the quantum algorithm $ U^{ \text{QAA}}$ in Eq. \eqref{UnitaryQAA}.
Plots in (a) and (b) are the results of the probabilities $P_{110}$ (probability of $| 110 \rangle$) and $P_{111}$ (probability of $| 111 \rangle$), respectively.
The dotted black lines are the ideal simulation results whereas the blue and orange curves are the noisy simulation results and the ones with QEM, respectively. All of them are obtained by our original code.
The blue and orange circles are the noisy simulation results and the ones with QEM, respectively, and they are obtained by our Qiskit code.
Plots in (c) and (d) are the results of the ratio $\text{RT}_{\text{QEM}}$ for $P_{110}$ and $P_{111}$, respectively. The black curves are obtained by our original code while the red circles by our Qiskit code.
For each $\vartheta_\tau$ we have plotted 100 circles in (a) - (d), i.e., $N_{\text{samp}}=100.$
}
\label{groverq3probsandratios}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1.0 \textwidth]{groverq3histogram.png}
\caption{ Histogram of the probability distribution of the computational basis states for QAA simulations given by $ U^{ \text{QAA}}$ in Eq. \eqref{UnitaryQAA}.
Here we have set $\vartheta_\tau = 0.2$.
}
\label{groverq3histogram}
\end{figure*}
\subsection{Quantum Amplitude Amplification}\label{QEMQAAalg}
By taking account of the previous analysis, let us apply our QEM scheme to Quantum Amplitude Amplification (QAA) \cite{TIsing3} for three-qubit systems: two-register bits and one oracle bit.
One of the important application of QAA is the database retrieval and the quantum algorithms for this is called the Grover's search algorithm \cite{TIsing3,GroverQA,Grover1997,montanaro2016quantum,QAZoo}.
Let us denote the (classical) oracle function by $f$ and a binary by $x$ which takes $``00",``01",``10",``11".$
We consider that we have only one solution of $f$ and write it by $x^\ast$, which satisfy $f(x^\ast)=1$, and assume $x^\ast=``11"$: for $x= ``00",``01",``10"$ we have $f(x)=0$.
The oracle operator $O_{\text{QAA}}$ can be implemented on a quantum circuit by using one oracle bit $Q_\text{o}$ such that $O_{\text{QAA}}\left[ | x \rangle_{Q_{\text{r}0}Q_{\text{r}1}}\otimes \left( \frac{ |0 \rangle_ {Q_\text{o}} - |1 \rangle_{Q_\text{o}} } {\sqrt{2} } \right) \right]
= (-1)^{f(x)} \left[ | x \rangle_{Q_{\text{r}0}Q_{\text{r}1}}\otimes \left( \frac{ |0 \rangle_ {Q_\text{o}} - |1 \rangle_{Q_\text{o}} } {\sqrt{2} } \right) \right],$
where the superposition state $\frac{ |0 \rangle_ {Q_\text{o}} - |1 \rangle_{Q_\text{o}} } {\sqrt{2} }$ is created by applying $H\cdot X$ on the oracle-bit state $|0 \rangle_ {Q_\text{o}}.$
In our case, $(-1)^{f(x)} =-1$ when $ | x \rangle_{Q_{\text{r}0}Q_{\text{r}1}} = | 11 \rangle_{Q_{\text{r}0}Q_{\text{r}1}}$, and the oracle operator $O_{\text{QAA}}$ is equivalent to the Toffoli gate comprised of the two controlled bits $Q_{\text{r}0}$ and $Q_{\text{r}1}$ and the target bit $Q_\text{o}$ \cite{TIsing3}, and write it by $U_{\text{C}X}[Q_{\text{r}0}Q_{\text{r}1}; Q_\text{o}]$.
To construct QAA we need one more unitary transformation and that is $U_\psi = \left(\boldsymbol{1}_{4\times4}-2| \psi \rangle\langle \psi|\right) \otimes \boldsymbol{1}_{2\times2}$, where $|\psi \rangle =H^{\otimes 2} | 00\rangle_{Q_{\text{r}0}Q_{\text{r}1}}$.
By introducing $U_{\text{init}}= H^{\otimes 2} \otimes (H\cdot X)_{ Q_\text{o} } $, QAA is given by the unitary operation \cite{TIsing3}
\begin{widetext}
\begin{align}
U^{ \text{QAA}} & = ( U_{\text{G}} )^k \cdot U_{\text{init}}, \notag\\
U^{ \text{G}} & = - U_\psi \cdot O_{\text{QAA}}, \notag\\
U_\psi & = \big{(} (X \cdot H)^{\otimes 2} \otimes \boldsymbol{1}_{2\times2, Q_\text{o}} \big{)}
\cdot
\big{(} \boldsymbol{1}_{2\times2, Q_{\text{r}0}} \otimes H_{ Q_{\text{r}1} } \otimes \boldsymbol{1}_{2\times2, Q_\text{o}} \big{)}
\cdot \big{(} U_{\text{C}X}[Q_{\text{r}0};Q_{\text{r}1}] \otimes \boldsymbol{1}_{2\times2, Q_\text{o}} \big{)} \notag\\
&\cdot \big{(} \boldsymbol{1}_{2\times2, Q_{\text{r}0}} \otimes H_{ Q_{\text{r}1} } \otimes \boldsymbol{1}_{2\times2, Q_\text{o}} \big{)}
\cdot \big{(} (X \cdot H)^{\otimes 2} \otimes \boldsymbol{1}_{2\times2, Q_\text{o}} \big{)}
\label{UnitaryQAA}
\end{align} \end{widetext}
We show the quantum circuit for the unitary operation $U^{ \text{QAA}} $ in Fig. \ref{QAAQcirc} \cite{TIsing3}: the quantum circuit for the oracle operator $O_{\text{QAA}}$ ($=U_{\text{C}X}[Q_{\text{r}0}Q_{\text{r}1}; Q_\text{o}]$) is shown in Appendix. \ref{appendix2}. We note that on the quantum circuit in Fig. \ref{QAAQcirc}, what is actually implemented is $-U^{ \text{G}} = U_\psi \cdot O_{\text{QAA}}$ and the
global phase factor $(-1)$ does not affect our result.
The unitary operation $U_{\text{G}} $ is called the Grover operator and $k$ is the repetitive number number of its operation
and here we have $k=1.$ After the operation of $U^{ \text{QAA}} $ in Eq. \eqref{UnitaryQAA}, ideally
both of the probability weights of $|110\rangle$ and $|111\rangle$ are $\frac{1}{2}$, and thus, the probability of obtaining the quantum state $|11\rangle$
as the output state is $\frac{1}{2}\times 2=1$, which implies the success of searching the solution $x^\ast=``11"$.
By taking account of the above theoretical framework, we examine whether our QEM scheme works or not for QAA given by $U^{ \text{QAA}} $ in Eq. \eqref{UnitaryQAA}
by computing the probability weights of $|110\rangle$ and $|111\rangle$ which we name as $P_{110}$ and $P_{111}$, respectively,
and show these results in Fig. \ref{groverq3probsandratios}.
Solid lines in Figs. \ref{groverq3probsandratios}(a) and (b) describes the probability weights obtained by our original numerical code and we have denoted $\langle P_{110(111)} \rangle_{\rho^{\text{ideal}} _{d\cdots1}}, \langle P_{110(111)} \rangle_{\rho^{\text{real}} _{d\cdots1}},$ and $\langle P_{110(111)} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$ by $ P_\text{ideal}, P_\text{real}$, and $ P_\text{QEM} $, respectively.
On the other hand, the blue and orange circles are calculated by our Qiskit code and we have denoted $\langle P_{110(111)} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ and $\langle P_{110(111)} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$ by $P_{\mathrm{noisy}}^{\mathrm{Qiskit}}$ and $P_{\mathrm{QEM}}^{\mathrm{Qiskit}}$, respectively.
Similarly, in Figs. \ref{groverq3probsandratios}(c) and (d), we have denoted the ratio $\text{RT}_{\text{QEM}}$ calculated by our Qiskit code by $\text{RT}_{\text{QEM}}^{\text{Qiskit}}$:
for the results obtained by our original code we have just used the notation $\text{RT}_{\text{QEM}}$ for describing them.
Let us look from the simulation results of $P_{110}$ and the associated ratio $\text{RT}_{\text{QEM}}$ given by Figs. \ref{groverq3probsandratios}(a) and (c), respectively.
In the range $0 \leq \vartheta_\tau \leq 0.5$, overall the simulation results with QEM are numerically close to the ideal values than the noisy simulation results, and correspondingly, we have $\text{RT}_{\text{QEM}} >1$.
The similar characteristics can be seen in Figs. \ref{groverq3probsandratios}(b) (simulation results of $P_{111}$) and (d) ($\text{RT}_{\text{QEM}} $ for $P_{111}$).
For the results obtained by our Qiskit code, the deviation between $\text{RT}_{\text{QEM}}$ and $\text{RT}_{\text{QEM}}^{\text{Qiskit}}$ becomes prominent in the small $\vartheta_\tau$ region and we consider this as follows.
When noise strength $\vartheta_\tau$ is weak enough, on the Qiskit code the difference between the noisy value and the ideal value is very tiny such that our QEM becomes invalid and $\text{RT}_{\text{QEM}}^{\text{Qiskit}}$
gets lower than one. On the other hand, we see that some red points are above $\text{RT}_{\text{QEM}}=1.$
We consider that by greatly increasing $N_\text{QC}$, we expect that $\text{RT}_{\text{QEM}}^{\text{Qiskit}}$
approaches to $\text{RT}_{\text{QEM}}$.
As a result, our QEM works for the noisy simulations of both $P_{110}$ and $P_{111}$.
Let us also show the simulation results of the rest of the six probabilities of the computational basis states for $\vartheta_\tau = 0.2$ as the histogram in Fig. \ref{groverq3histogram}, which also includes $P_{110}$ and $P_{111}$.
The ideal values of these six probabilities are all zero and we see that the simulation results with QEM are numerically closer to them compared to the noisy simulation results, which indicates that our QEM scheme also works for the other six probabilities.
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.8 \textwidth]{circuitGSA.png}
\caption{ Quantum circuit for QAA given by $V^{ \text{QAA}} $ with $k=1$.}
\label{circuitQAA}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width=0.8 \textwidth]{GrovernoisyQsims.png}
\caption{ Noisy quantum simulations for QAA given by $V^{ \text{QAA}}$ for $\hat{O}=P_{00}, P_{11}$.
All these results are obtained by our numerical code.
(a) Plots of the results of $P_{\mathrm{ideal}} = \langle P_{00} \rangle_{\rho _{d\cdots1}}, P_{\mathrm{noisy}} = \langle P_{00} \rangle_{\rho^{\text{real}} _{d\cdots1}},$ and
$P_{\mathrm{QEM}} = \langle P_{00} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$
presented by the black dashed line, the blue solid line, and the orange solid line, respectively.
(b) Plots of the results of $P_{\mathrm{ideal}} = \langle P_{11} \rangle_{\rho _{d\cdots1}}, P_{\mathrm{noisy}} = \langle P_{11} \rangle_{\rho^{\text{real}} _{d\cdots1}},$ and $P_{\mathrm{QEM}} = \langle P_{11} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$
presented by the black dashed line, the blue solid line, and the orange solid line, respectively.
(c) The ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ for $P_{00}$.
(d) The ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ for $P_{11}$.
}
\label{GrovernoisyQsims}
\end{figure*}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45 \textwidth]{noisyQAAhistogram.png}
\caption{ Histogram of the probability distribution of the computational basis states for two-qubit-system QAA simulations. We have set $\vartheta_\tau = 0.2$.}
\label{QAAhistogram}
\end{figure}
In addition to the above simulation, let us present the simplified version of QAA for the two-qubit systems \cite{2021ibm}.
In this problem setting, we consider the effective two-dimensional space spanned by the two Bell states $| \Psi^+ \rangle = \frac{|01\rangle + |10\rangle}{\sqrt{2}} $
and $| \Phi^+ \rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}} $, and write their superposition state by
$|\Sigma \rangle = c_{\Psi^+} | \Psi^+ \rangle + c_{\Phi^+} | \Phi^+ \rangle $, where the complex coefficients $ c_{\Psi^+} , c_{\Phi^+} $
satisfy $|c_{\Psi^+}|^2 + |c_{\Phi^+}|^2=1.$
Our goal is to amplify the probability amplitude $c_{\Phi^+} .$
Here we take the initialization operator to be $V_\text{init} = U_{CX}[Q_{r1};Q_{r0}] \cdot ( R_y\left(\frac{2\pi}{3}\right)_{Q_{r0}} \otimes H_{Q_{r1}} )$
while we take the Grover operator to be $V_\text{G} = (V_\text{s} V_\omega)^k$, where
$ V_\omega = ( X \cdot Z \cdot X )_{Q_{r0}} \otimes Z_{Q_{r1}}$ and
$V_\text{s}= V_\text{init} \cdot( 2|0\rangle^{\otimes 2}\langle 0| - \boldsymbol{1}_{4\times4}) \cdot V^\dagger_\text{init} \equiv -\tilde{V}_\text{s}$
with $\tilde{V}_\text{s} = V_\text{init} \cdot( X\otimes X \cdot U_{CZ}[Q_{r0};Q_{r1}] \cdot X\otimes X) \cdot V^\dagger_\text{init}$ [xxx].
$k$ is the number of $V_\text{G}$ to be applied. In total, the unitary operation for running QAA is given by $V^{ \text{QAA}} = ( V_{\text{G}} )^k \cdot V_{\text{init}}.$
We present the corresponding quantum circuit in Fig. \ref{circuitQAA}.
As similar to the above case, on the quantum circuit we implement $\tilde{V}_{\text{s}}$ instead of $V_{\text{s}}$.
In the following, let us take a look at the meanings of the three unitary operations $V_{\text{init}}, V_{\text{s}}$, and $V_{\omega}$.
First, the unitary operation $V_{\text{init}}$ generates the superposition of $| \Phi^+ \rangle $ and $| \Psi^+ \rangle $ as
$|00\rangle \to |s\rangle = V_{\text{init}}|00\rangle =
\cos \frac{\theta_V}{2}|\Psi^{+}\rangle + \sin \frac{\theta_V}{2}|\Phi^{+}\rangle$ with $\theta_V=\pi/3$.
Second, the unitary operation $V_{\omega} $ is the oracle operator and when it is applied to the initial state $| \text{s} \rangle$
we have $V_{\omega}| \text{s} \rangle = \cos\frac{\theta_V}{2}| \Psi^+\rangle - \sin\frac{\theta_V}{2}| \Phi^+\rangle$, i.e., the oracle operator $V_{\omega} $ is the operator such that it reverses the sign of the Bell state $ | \Phi^+\rangle$.
Third, $V_\text{s} = 2|\text{s}\rangle\langle \text{s} | - \boldsymbol{1}_{4\times4}$ is the reflection of the vector $V_{\omega}|\text{s} \rangle$ with respect to the vector $|\text{s}\rangle$.
When we operate $V_{\text{G}} $ on $|s\rangle$ for $k$ times we have
$(V_{\text{G}})^k |s\rangle = \cos\frac{(2k+1)\theta_V}{2}| \Psi^+\rangle + \sin\frac{(2k+1)\theta_V}{2}| \Phi^+\rangle,$ and since $\theta_V=\pi/3$ we have $k=1.$
We can understand the geometrical meaning of the operation $V_{\text{G}} $ as follows.
Let us consider the effective three-dimensional space spanned by the two Bell states $| \Phi^+ \rangle$ and $| \Psi^+ \rangle$ and the vector $|\xi \rangle$ which is
perpendicular to both $| \Phi^+ \rangle$ and $| \Psi^+ \rangle$.
Furthermore, we call the axis which is parallel to $|\xi \rangle$ (perpendicular to the two-dimensional plane spanned by $| \Phi^+ \rangle$ and $| \Psi^+ \rangle$) as $\xi$-axis.
The unitary operation $V_{\text{G}} $ is the rotation about $\xi$-axis by the angle $2\vartheta$ in this effective three-dimensional space.
We can verify whether $| \Phi^+\rangle$ has been generated as the output state or not by measuring the probabilities of $|00\rangle$ state and $|11\rangle$ state,
which are denoted by $p_{00}$ and $p_{11}$, respectively.
In other words, the probabilities $p_{00}$ and $p_{11}$ are the expectation values of the projection operators
$P_{00}$ and $P_{11}$, respectively, where $P_{00}$ ($P_{11}$) is the projection operator of $|00\rangle$ ($|11\rangle$) state.
Namely, we run noisy quantum simulation for $\hat{O}=P_{00}, P_{11}$ and perform QEM on them: Note that in the case of the ideal simulation we obtain $p_{00} = p_{11} = \frac{1}{2}$.
We plot the numerical results of $p_{11}$ and $p_{00}$ for the range $0 \leq \vartheta_\tau \leq 0.5$ in Figs. \ref{GrovernoisyQsims}(a) and (b),respectively,
and in Figs. \ref{GrovernoisyQsims}(c) and (d) we plot $\text{RT}_{\text{QEM}} (\vartheta_\tau)$
for $p_{11}$ and $p_{00}$, respectively.
All these results shown here are obtained by our original numerical code and we have denoted $\langle P_{00(11)} \rangle_{\rho^{\text{ideal}} _{d\cdots1}}, \langle P_{00(11)} \rangle_{\rho^{\text{real}} _{d\cdots1}},$ and $\langle P_{00(11)} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$
by $ P_\text{ideal}, P_\text{real}$, and $ P_\text{QEM} $, respectively.
Let us look from the results of the probability $p_{11}$.
In Fig. \ref{GrovernoisyQsims}(a) we see that for any $\vartheta_\tau$ the absolute of the deviation
$\big{|} \langle P_{11} \rangle_{\rho^{\text{real}} _{d\cdots1}} - \langle P_{11} \rangle_{\rho _{d\cdots1}} \big{|}$ is bigger than $\big{|} \langle P_{11} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} - \langle P_{11} \rangle_{\rho _{d\cdots1}} \big{|}$,
and correspondingly, as we see in Fig. \ref{GrovernoisyQsims}(d)
the ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ is greater than one. Therefore, our QEM scheme works well for the noisy simulation of $p_{11}$.
In contrast, in Fig. \ref{GrovernoisyQsims}(b) an (d) we see that the probability $p_{00}$ shows essentially a different behavior.
That is the expectation value without QEM $ \langle P_{00} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ is numerically closer to the ideal value $\langle P_{00} \rangle_{\rho _{d\cdots1}}$ ($=0$) compared to the QEM-performed expectation value $ \langle P_{00} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$, and correspondingly,
the ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ is lower than one.
Such a characteristic is understood as follows.
Firstly, we have analytically examined that
the expectation value $ \langle P_{00} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ does not include the first-order term in $\tau$, i.e., $ \langle P_{00} \rangle_{\Delta^{\text{AD}}_1 \rho _{d\cdots1}} =0$.
The lowest-order term included in the numerator of $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ is $\mathcal{O}(\tau^2)$.
Secondly, due to our QEM the lowest order of the denominator of $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ is also $\mathcal{O}(\tau^2)$.
As a result, the ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$ becomes lower than one, which indicates that it is not appropriate to adapt our QEM scheme.
We consider that this is because our QEM scheme described by Eq. \eqref{QEMformula1} is the scheme for mitigating the first-order AD effect.
We note that in the limit of $\vartheta_\tau \to 0$, the ratio $\text{RT}_{\text{QEM}} $ takes finite value, and analytically it is the ratio between
the absolute of the coefficient of $ \langle P_{00} \rangle_{\delta^{\text{AD}}_2 \rho _{d\cdots1}} $ and that of $ \langle P_{00} \rangle_{\delta_1(\Delta^{\text{AD}}_1 \rho _{d\cdots1})}.$
Moreover, when we use the Qiskit code the first-order term in $\tau$ appears for $ \langle P_{00} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ owing to the
decomposition of $U_{CZ}[Q_{r1};Q_{r0}] $ gate given by $\left({\boldsymbol{1}_{2\times2}}_{Q_{r0}} \otimes H_{Q_{r1}} \right) \cdot U_{CX}[Q_{r1};Q_{r0}] \cdot \left({\boldsymbol{1}_{2\times2}}_{Q_{r0}} \otimes H_{Q_{r1}} \right)$,
and in such a case our QEM works well.
Besides $p_{00}$ and $p_{11}$, let us briefly discuss the noisy simulation results of the probability weights of $|01\rangle$ and $|10\rangle$ and write them by $p_{01}$ and $p_{10},$ respectively.
We show them in the histogram in Fig. \ref{QAAhistogram} which describes the probability distribution of the computational basis states of the two register qubits $Q_{\text{r}0}$ and $Q_{\text{r}1}$.
Here we have taken $\vartheta_\tau=0.2$.
We see that like the probability weight of $|11\rangle$,
our QEM scheme works for the probability weights of $|01\rangle$ and $|10\rangle$.
Let us end this subsection by giving the following comment.
In the previous subsection, we have seen that our QEM scheme becomes meaningless in the cases where the expectation values of the ideal simulations are equivalent to those of noisy simulations such as the simulation for the expectation value $\langle Y \rangle$. Besides these cases, our QEM scheme represented by the formula in Eq. \eqref{QEMformula1} does not work when noisy expectation values do not include the first-order term in $\tau$ like the noisy simulation for the probability $p_{00}$ discussed above. In other words, if we construct the QEM formula which describes the mitigation for a higher-order quantum noise effect, which is discussed in Appendix \ref{extdQEMSsub1}, by using it we become able to accomplish the noisy quantum simulation obtaining $\text{RT}_{\text{QEM}} >1.$
\begin{figure}[b]
\centering
\includegraphics[width=0.35 \textwidth]{QAOAgraph.png}
\caption{ Structure of the graph $G = (V,E)$. It is the square composed of the four vertices $v_0, v_1, v_2,$ and $v_3$
and the four edges $ \langle v_0 v_{1} \rangle, \langle v_1 v_2 \rangle, \langle v_2 v_3 \rangle,$ and $ \langle v_3 v_0 \rangle$.
For each vertex $v_i$ ($i=0,1,2,3$) the binary value $z_i=\pm1$ is assigned and the set $(v_i,z_i)$ is encoded in the qubit $Q_{\text{r}i}$ in the QAOA simulation.}
\label{QAOAgraph}
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[width=0.475 \textwidth]{QAOAqcirc.png}
\caption{ Quantum circuit for QAOA.}
\label{QAOAqcirc}
\end{figure}
\subsection{ QAOA}\label{QEMqaoa}
As a final example, let us apply our QEM scheme to the noisy simulation of the variational quantum algorithm called
Quantum Approximate Optimization Algorithm (QAOA) \cite{hybridQCalgorithmJPSJ2021,QAOA2014,crooks2018performance,wang2018quantum,shaydulin2019evaluating,zhou2020quantum,TIsing3}. In the following, we analyze QAOA for a max-cut problem which is to divide vertices (nodes) of a given graph into two groups
so that the number of edges connecting two vertices belonging to the different groups is maximized and is a NP (Non-deterministic Polynomial time)-hard problem.
First, we discuss from a theoretical framework of a classical approximate optimization.
We express the given graph $G$ as $G = (V,E)$, where $V = \{ v_0, \ldots, v_i, \ldots, v_{N_\text{V}-1} \}$ is the set of vertices with $N_\text{V}$ denoting their total number and for each vertex $v_i$ the binary value $z_i = \pm 1$ is assigned.
$E = \Big{\{} \big{\{} \langle v_0 v_1 \rangle, C_{0,1} \big{\}}, \ldots, \big{\{} \langle v_i v_{i+1} \rangle, C_{i,i+1} \big{\}}, \ldots,$ $\big{\{} \langle v_{N_\text{V}-2} v_{N_\text{V}-1} \rangle, C_{N_\text{V}-2,N_\text{V}-1} \big{\}} \Big{\}}$
is the set of the edges with $\langle v_i v_{i+1} \rangle$ denoting the edge connected by the vertices $v_i$ and $v_{i+1} $.
The quantity $C_{ij}$ ($i,j=0,\ldots, N_\text{V}-1$ with $i\neq j$) is the adjacency matrix element (weight) for the edge $\langle v_i v_j \rangle$ which is semi-positive.
Let us write the $N_\text{V}$ strings of $z_i$ by $\boldsymbol{z} = (z_0,\ldots, z_{N_\text{V}-1})$. The goal of a classical approximate optimization is to minimize the cost function
\begin{align}
C(\boldsymbol{z})= \frac{1}{2}\sum_{ (v_i, v_j)} C_{ij} \big{(} z_iz_j -1 \big{)} , \label{AOAcostfunction}
\end{align}
or equivalently to maximize the ratio $r_\text{CAO}$ ($\leq1$) which satisfies
\begin{align}
\frac { C(\boldsymbol{z}) }{C_\text{min}} \geq r_\text{CAO}, \label{AOAcostfunction2}
\end{align}
where $C_\text{min}$ is the minimum value of $ C(\boldsymbol{z})$.
In our simulation, as illustrated in Fig. \ref{QAOAgraph} we adopt the square graph given by the four vertices $v_0, v_1, v_2,$ and $v_3,$
and the edges are $ \langle v_0 v_{1} \rangle, \langle v_1 v_2 \rangle, \langle v_2 v_3 \rangle,$ and $ \langle v_3 v_0 \rangle$, and take $C_{ij}=1$ for any edge $\langle v_i v_{j} \rangle$.
Next, we discuss the theoretical framework for QAOA. The four vertices $v_0, v_1, v_2,$ and $v_3$ are encoded in four qubits $Q_{ \text{r} 0}, Q_{ \text{r} 1}, Q_{ \text{r} 2},$ and $Q_{ \text{r} 3}$, respectively,
and the values $z_i z_j $ in the expectation values of the operators $Z_{Q_i} \otimes Z_{Q_j}$.
The cost function $C(\boldsymbol{z})$ in Eq. \eqref{AOAcostfunction} is given by the expectation of the Hamiltonian
\begin{align}
H_{\text{C}} = \frac{1}{2}\sum_{(i,j) } \big{(} Z_{Q_{\text{r}i}} \otimes Z_{ Q_{\text{r}j} } -1 \big{)} , \label{QAOAHamiltonian}
\end{align}
where the symbol $(i, j)$ ($i, j =0,1,2,3$) denotes the summation for the edges connected by the qubits $Q_{ \text{r} i}$ and $Q_{ \text{r} j}$ under the square-graph structure in Fig. \ref{QAOAgraph}.
In this simulation the physical operator $\hat{O}$ is the Hamiltonian $H_{\text{C}} $ in Eq. \eqref{QAOAHamiltonian}.
The unitary operation for running QAOA, which we denote by $U^{ \text{QAOA}}$, consists of three elements.
The first one is the unitary operation for creating the reference state and is given by the Hadamard-gate operation on all four qubits, $ U_\text{int} = H^{\otimes 4}$.
The other two are the unitary operations $U_{ \text{C}}(\vartheta^{ \text{QAOA}}_j)$ and $U_{ X}(\varphi^{ \text{QAOA}}_j)$ which are generated
by the Hamiltonian $H_{\text{C}} $ in Eq. \eqref{QAOAHamiltonian} with the angle $\vartheta^{ \text{QAOA}}_j$ and the term $H_X = \sum_{i} X_{Q_{\text{r}i}}$
with the angle $\varphi^{ \text{QAOA}}_j$, respectively: in QAOA $H_X$ is called the transverse-field (mixing or driving) term.
The two types of angles $\vartheta^{ \text{QAOA}}_j$ and $\varphi^{ \text{QAOA}}_j$ ($j=1,\ldots, p$) are the variational parameters
and $p$ is the repetition number of applying the unitary operation $U_{ X}(\varphi^{ \text{QAOA}}_j) \cdot U_{ \text{C}}(\vartheta^{ \text{QAOA}}_j) $ (the number of iteration), which determines the accuracy of QAOA.
In total, $U^{ \text{QAOA}} $ is given by
\begin{align}
& U^{ \text{QAOA}}
=
\left[ \prod_{j=1}^{p} \left( U_X (\varphi^{\text{QAOA}}_j) \cdot U_C (\vartheta^{\text{QAOA}}_j) \right) \right]
\cdot
\left[\bigotimes_{a=0}^3 H_{Q_{ra}} \right], \notag\\
& U_C (\vartheta^{\text{QAOA}}_j) = e^{-i \vartheta^{\text{QAOA}}_j H_C}, \quad U_X (\varphi^{\text{QAOA}}_j) = e^{-i \varphi^{\text{QAOA}}_j H_X}.
\label{UQAOA1}
\end{align}
The quantum circuit for $U^{ \text{QAOA}} $ in Eq. \eqref{UQAOA1} is presented in Fig. \ref{QAOAqcirc}.
In our simulation we set $p=2$ and the circuit depth is $d=15$.
The unitary operation $U_X (\varphi^{\text{QAOA}}_j)$ is implemented by the $R_x$ gate (rotation about $x$ axis) with the angle $2\varphi^{\text{QAOA}}_j$.
Meanwhile, the quantum circuit for $U_C (\vartheta^{\text{QAOA}}_j)$ is composed of the sets of the quantum gates $\big{[} U_{CX}[Q_{\text{r}i}; Q_{\text{r}i} ], R_z (2\vartheta^{\text{QAOA}}_j) \big{]}$,
where $R_z$ denotes the rotational gate about $z$ axis and the associated angle is $2\vartheta^{\text{QAOA}}_j$.
Corresponding to Eq. \eqref{AOAcostfunction2}, the goal of QAOA simulation is to compute and minimize the expectation value
\begin{align}
C\big{(}\boldsymbol{\vartheta}^{\text{QAOA}}, \boldsymbol{\varphi}^{\text{QAOA}}\big{)} = \langle \boldsymbol{\vartheta}^{\text{QAOA}}, \boldsymbol{\varphi}^{\text{QAOA}}| H_{\text{C}}
| \boldsymbol{\vartheta}^{\text{QAOA}}, \boldsymbol{\varphi}^{\text{QAOA}} \rangle ,
\label{QAOAcostfunction}
\end{align}
%
\begin{figure}[!htb]
\centering
\includegraphics[width=0.5 \textwidth]{QAOAnoisQsimresults1.png}
\caption{ Quantum simulations for QAOA.
In (a) we show the results of QEM for the cost function $ C\big{(}\boldsymbol{\vartheta}^{\text{QAOA}}, \boldsymbol{\varphi}^{\text{QAOA}}\big{)} $.
The blue and orange solid lines are $C_\text{noisy} = \langle H_{C} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ and $C_\text{QEM} = \langle H_{C} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$, respectively, and they obtained by our original code.
The blue and orange circles are the results of $ \langle H_{C} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ and $\langle H_{C} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$ obtained by our Qiskit code, respectively.
For this simulation, we have described $\langle H_{C} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ and $ \langle H_{C} \rangle_{\rho^{\text{QEM}} _{d\cdots1}} $
as $ C_\text{noisy}^{\text{Qiskit}}$ and $ C_\text{QEM}^{\text{Qiskit}}$, respectively.
The dotted black line is the ideal expectation value $C_\text{ideal} = \langle H_{C} \rangle_{\rho _{d\cdots1}}$.
In (b) we have plotted the results of the ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$. The black curve is the result obtained by our original code whereas the red circles are the one obtained by our Qiskit code.
For each $\vartheta_\tau$, we have plotted 100 circles. }
\label{QAOAnoisyQsims}
\end{figure}
\begin{figure*}[!htb]
\centering
\includegraphics[width=1.0 \textwidth]{QAOAQstateprobhistogram.png}
\caption{ Histogram of the probability distribution of the computational basis states for QAOA simulations. We have set $\vartheta_\tau = 0.2$.
$P_\text{ideal}, P_\text{noisy},$ and $P_\text{QEM}^\text{Qiskit} $ are obtained by our original code while $P_\text{noisy}^\text{Qiskit}$ and $P_\text{QEM}^\text{Qiskit} $ are obtained by the Qiskit code.}
\label{QAOAhistogram}
\end{figure*}
where $ | \boldsymbol{\vartheta}^\text{QAOA}, \boldsymbol{\varphi}^\text{QAOA} \rangle = U^\text{QAOA} |0\rangle^{\otimes 4}$
with $ \boldsymbol{\vartheta}^\text{QAOA} = (\vartheta^\text{QAOA}_1, \vartheta^\text{QAOA}_2)$ and
$ \boldsymbol{\varphi}^\text{QAOA} = (\varphi^\text{QAOA}_1, \varphi^\text{QAOA}_2)$.
Let us also call the expectation value $ C\big{(}\boldsymbol{\vartheta}^\text{QAOA}, \boldsymbol{\varphi}^\text{QAOA}\big{)} $ in Eq. \eqref{QAOAcostfunction} as the cost function
and its minimization is equivalent to the optimization of the variational parameters $\vartheta_j^\text{QAOA}, \varphi_j^\text{QAOA} $,
and write them by $\vartheta_j^\text{QAOA,opt}, \varphi_j^\text{QAOA,opt} $. We show their values in Appendix \ref{appendix3}.
Let us now discuss our simulation results shown in Figs. \ref{QAOAnoisyQsims}-\ref{QAOAhistogram}.
All these results are obtained by computing the probability distribution of the computational basis states since the Hamiltonian $H_{\text{C}} $ in Eq. \eqref{QAOAHamiltonian} is given by the $Z$ gate operations.
First, let us take a look at the simulation results in Fig. \ref{QAOAnoisyQsims}.
In Fig. \ref{QAOAnoisyQsims} (a), we have plotted the results of the cost function $C\big{(}\boldsymbol{\vartheta}^{\text{QAOA}}, \boldsymbol{\varphi}^{\text{QAOA}}\big{)} $
for the ideal simulation ($C_\text{ideal} = \langle H_{C} \rangle_{\rho _{d\cdots1}}$), the noisy simulation ($C_\text{noisy} = \langle H_{C} \rangle_{\rho^{\text{real}} _{d\cdots1}}$),
and the simulation with QEM ($C_\text{QEM} = \langle H_{C} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$).
The dashed black line, the blue solid line, and the orange solid line are $C_\text{ideal}, C_\text{noisy}$, and $C_\text{QEM}$, respectively, and they are all obtained by our original code.
On the other hand, the blue and orange circles are $ \langle H_{C} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ and $ \langle H_{C} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$, respectively, and they have been calculated by our Qiskit code.
We have denoted $\langle H_{C} \rangle_{\rho^{\text{real}} _{d\cdots1}}$ and $ \langle H_{C} \rangle_{\rho^{\text{QEM}} _{d\cdots1}}$ by $C_{\mathrm{noisy}}^{\mathrm{Qiskit}}$ and $C_{\mathrm{QEM}}^{\mathrm{Qiskit}}$, respectively.
We have plotted 100 circles for each $\vartheta_\tau$. We see that in the range $0.1 \leq\vartheta_\tau \leq 0.5$ our QEM works well.
In Fig. \ref{QAOAnoisyQsims} (b), we have plotted the ratio $\text{RT}_{\text{QEM}} (\vartheta_\tau)$.
The black curve is the result obtained by our original code while the red circles are those obtained by our Qiskit code and 100 circles are plotted for each $\vartheta_\tau$.
Corresponding to the result shown in Fig. \ref{QAOAnoisyQsims} (a), the ratio satisfies $\text{RT}_{\text{QEM}}(\vartheta_\tau) >1$.
Finally, let us explain the results in Fig. \ref{QAOAhistogram}.
Here we have presented the histogram of the probability distribution of the computational basis states for the ideal case $(P_\text{ideal})$, the noisy case $(P_\text{noisy}, P_\text{noisy}^\text{Qiskit})$, and the case with QEM being performed
$(P_\text{QEM}, P_\text{QEM}^\text{Qiskit})$,
with setting $\vartheta_\tau=0.2$.
We see that the probabilities of the two quantum states $|0101\rangle$ and $|1010\rangle$ are both equal to 0.5.
This implies that under the optimized variational parameters the cost function $C\big{(}\boldsymbol{\vartheta}^{\text{QAOA}}, \boldsymbol{\varphi}^{\text{QAOA}}\big{)} $
becomes minimized such that the four qubits $Q_{\text{r}i}$ are partitioned into the two groups $\big{[}Q_{\text{r}0},Q_{\text{r}2}\big{]}$ and $\big{[}Q_{\text{r}1},Q_{\text{r}3}\big{]}$, and it implies that
all the edges of the square are to be cut, i.e., the maximum number of edges to be cut is four.
Correspondingly, as shown in Fig. \ref{QAOAnoisyQsims} (a) the ideal minimum value of the cost function is $-4.0,$ and we obtain the maximum cut number four by multiplying minus one.
Consequently, our QEM scheme works for the noisy QAOA simulation.
\subsection{ Discussion for QEM under NISQ Device Parameters}\label{discussionQEMNISQ}
Let us end this section by discussing the validity of our QEM scheme under the conditions of physical parameters of NISQ devices such as gate times and $T_1$ and $T_2$ (transverse and longitudinal) times.
First, let us discuss the case of superconducting qubit systems. Typically, $T_1$ times of superconducting qubits are around 100 $\mu$sec while two-qubit gate times are about 100 nsec \cite{SCQARCMP2020,ZhugroupSQC2020}.
For simplicity, let us set $T_1 = \big{(}2\gamma (2\bar{n}+1) \big{)}^{-1} \approx (2\gamma)^{-1}$. Here we have taken the qubit frequency and temperature to be 5.0 GHz and 10 mK \cite{SCQNISQ20191,TsaigroupSCCQC2021}, respectively, and we obtain $\bar{n}\sim10^{-11}$.
Then, the noise strength $\vartheta_\tau$ is estimated to be $\vartheta_\tau \approx 0.04$.
From the results in Figs. \ref{groverq3probsandratios}, \ref{GrovernoisyQsims} and \ref{QAOAnoisyQsims}, we expect that our QEM works for noisy quantum computing on superconducting-qubit NISQ devices.
Next, let us discuss the case of ion-trap qubit systems. For instance, quantum noise effects occurring in hyperfine-state type ion-trap qubit systems are considered to be the phase damping and
$T_2$ times are about 10 sec while two-qubit gate times are about 100 $\mu$sec \cite{trappedionNISQ2019}.
By setting $T_2 = (2\gamma_\text{p})^{-1}$, we have $\tau_\text{p} = \gamma_\text{p} \Delta t \approx 5\times 10^{-6}$, which is sufficiently small and therefore, we expect that our QEM scheme also works
for ion-trap NISQ devices.
\section{ Conclusion and Outlook}\label{concludediscussion}
In this paper, we have established our QEM scheme for reducing the quantum noise (decoherence) effects on the single-qubit states which occur during the gate operations.
We have formulated it as the perturbation theory with respect to the noise strength (in the case of AD effect it is $\tau$), which are evaluated by the gate time and decay rate ($T_1$ time and/or $T_2$ time),
and is represented by the ensemble of quantum circuits, namely the quantum-noise-effect circuit groups.
The numbers of quantum circuits composing the quantum-noise-effect circuit groups are polynomial with respect to the product of the depth of the quantum algorithm under consideration
and the number of register bits, which can be considered that the conduction of our QEM scheme is not so high-cost computational performance.
To demonstrate the validity of our QEM scheme, we have performed the noisy quantum simulations of the qubits under the AD effects for four types of quantum algorithms based on the linear-order perturbation theory.
It is to be noted that before we conduct our QEM scheme, we need to be careful with if the expectation values on which we are aiming to perform QEM do not include the linear-order term in $\tau$ or
if they are equivalent to the ideal simulation results like the computation of the expectation value of the identity operator.
As long as these two are not the cases then our linear-order QEM scheme works as we have demonstrated in Sec. \ref{nss} and it is valid in a broad region of $\tau$, which implies its effectiveness and powerfulness.
Our QEM scheme can be generalized to error mitigation for other types of quantum noises including the generalized amplitude damping, the phase damping, the composition of these two, and the stochastic Pauli noises like the depolarizing channel. Furthermore, it can be extended to cases of error mitigation for higher-order quantum noise effects and once this is established we expect that we become able to perform quantum computations in high accuracy even for long-depth quantum algorithms.
Our QEM scheme is solely conducted by gate operations and measurements on ancilla bits and can be applied to any type of quantum algorithm.
Therefore, by using our QEM scheme we become able to programmably perform high-accurate quantum computing solely by the quantum-computational operations (software manipulations)
and it can be conducted without having the limitations of the number of qubits and error rates of gate operations.
Furthermore, our QEM scheme can be performed with any type of quantum hardware such as solid-state systems and atomic-molecule and optical systems and with quantum devices of any generation.
These two characteristics are the big advantages of our scheme.
We expect that we become able to obtain quantum computational results with extremely high qualities by combining our QEM scheme with other error-mitigation tasks such as those for state preparation (initialization) and measurement and imperfections of gate operations.
One of the important outlook of this work is quantum computations by large-scale quantum devices or future (next-generation) quantum devices.
In such a case, we consider that we also need to take into account quantum noises acting on many-body quantum states such as collective quantum noises \cite{carmichaeltxb,Agarwalltxb,opendynamicstext,superradiance1,GH82,DuancollectivedecohePRA1998,ViolagroupNJP2002} and correlated noises \cite{EMPRA2021,EMarxiv2018,TCADPRA2002R,TCADPRA2003,Cnoisepra2005,NoviasprlCnoise,KitaevgroupCnoiseprl2006}.
Our QEM scheme can be extended to mitigation of these many-body quantum noise effects provided that they are formulated as groups of quantum circuits.
When such formalisms are being constructed, we expect that we become able to realize QEM scheme which mitigates both the single-qubit-state and the many-body-state quantum noise effects.
We expect that this leads to conduction of quantum computing for big-size problems with high-quality results being obtained.
We believe that this paves the way to realize high-quality quantum computing for application to problems in many branches of science and engineering
including material science, quantum chemistry, combinatorial optimization problems, and machine learning using large-scale quantum computers.
\acknowledgements
We thank all the other members of Quemix Inc. for giving us the fruitful comments and reading this manuscript carefully.
This work was supported by MEXT as ”Program for Promoting Researches on the Supercomputer Fugaku” (JP-MXP1020200205) and JSPS KAKENHI as ”Grant-in- Aid for Scientific Research(A)” Grant Number 21H04553. The computation in this work has been done using Supercomputer Center at the Institute for Solid State Physics in the University of Tokyo.
\begin{widetext}
|
1,116,691,500,495 | arxiv |
\section{Introduction}
The private retrieval of information from public databases has received
significant attention already for several decades from researchers in the
computer science community (see, e.g., \cite{Chor_etal1998,
Yekhanin2010}). While this line of work, commonly known as private information
retrieval (PIR), is concerned with downloading individual messages in a
private manner from databases, a recently proposed generalization of this
problem \cite{Maddah-Aliarxiv2017, SunJafararxiv_2017} addresses the private
computation of functions of these messages. In accordance with \cite{Maddah-Aliarxiv2017} we denote this approach as private
function retrieval (PFR) in the following. In PFR a user has access
to a given number of databases and intends to compute a function of messages
stored in these databases. This function is kept private from the
databases, as they may be under the control of an eavesdropper. Both works \cite{Maddah-Aliarxiv2017, SunJafararxiv_2017}
characterize the fundamental information theoretic communication overhead
needed to reliably compute the given function and specify the corresponding
capacity and achievable rates as a function of the message size, the number
of messages, and the number of databases, respectively. Further, the authors
assume that the data is replicated on each database. Surprisingly, the obtained PFR capacity
result is equal to the PIR capacity of
\cite{Sun_Jafar2015}.
However, although repetition coding adds the largest amount of redundancy and
thus protects effectively against erasures, it is associated with a large
storage cost. A more general way to optimally trade-off the available
redundancy (or rate) versus the erasure correcting capability is given by MDS
codes. In particular, for an $(N,K)$ MDS code with $N$ code symbols and $K$
information symbols and rate $R_c=K/N$ $N-K$ erasures can be
recovered from any $K$ code symbols. Coded PIR has been
addressed in two different lines of work. Achievable schemes for MDS coded
PIR have been presented in \cite{Chanetal_ISIT2015, Tajeddineetal_ISIT2016}
and the capacity has been established in
\cite{BanawanUlukus_Globecom2017}. On the other hand, in \cite{Yaakoobi_ISIT2015} linear codes with $k$
different reconstruction sets for each code symbol have been proposed in
form of so called $k$-server PIR.
In this paper we propose coded PFR, which to the best of our knowledge has
not been addressed yet in the recent literature, with the notable exception
of the parallel work in \cite{Karpuk}, which is based on a fixed ($k$-)
server PIR scheme with the inclusion of colluding databases. Our scheme is
based on MDS codes which in contrast to \cite{Karpuk} minimize the storage
overhead and maximize the achievable download rate. In particular, we provide a
characterization of the achievable rate of MDS coded PFR if the user wishes
to compute an arbitrary linear combination of $M$ independent equal-sized
messages over some finite field $\mathbb{F}_q$, distributed over $N$
non-colluding MDS-coded databases. Surprisingly, our achievable rate matches
the capacity for MDS coded PIR in \cite{BanawanUlukus_Globecom2017}. This
demonstrates that, compared to the naive scheme, where $M$ coded messages are
downloaded and linearly combined offline at the user (requiring $M$-times
the coded PIR rate), downloading the result of the computation privately and
directly from the databases does not incur any penalty in rate compared to
the coded PIR case. Thus, our result strictly generalizes the achievable schemes in
\cite{Maddah-Aliarxiv2017, SunJafararxiv_2017} which represent special
cases of our proposed PFR scheme.
\vspace*{-0.5ex}
\section{Problem Statement} \vspace*{-0.5ex}\label{ProblemStatment}
In the following, we use {$[1:X]$} to denote the set $\{1,\dots,X\}$. Similarly, $X_{1:N}=\{X_1,\dots, X_N\}$.
\vspace{-0.5ex}
\subsection{System Model}\vspace{-0.5ex}
In coded PFR, a user wishes to privately retrieve a linear combination of the messages stored in the databases such that the coefficients of the linear combination are kept secret from each individual database. Consider a linear distributed storage system storing $M$ equal-sized messages on $N$ non-colluding databases. The message $W_m, m\in [1:M],$ is composed from $L$ symbols chosen independently and uniformly at random from the finite field $\mathbb{F}_{q}$ with
\begin{align}
H(W_1)=& \dots = H(W_M)=L \log q,\\
\hspace{-0.45ex}H(W_1,\!\dots\!,W_M)=& H(W_1)+\!\dots\! + H(W_M)=ML \log q. \vspace*{-0.5ex}
\end{align}
Each message is divided into $\tilde{L}$ segments, each of $K$ symbols, forming a $\tilde{L}\times K$ matrix, where $L=\tilde{L} K$. The messages are stored using an $(N,K)$ MDS code with the full rank generator matrix defined by
\begin{equation} \vspace*{-0.5ex}
{\bf G}=\big[ {\bf{g}}_1\quad {\bf{g}}_2\quad \dots\quad {\bf{g}}_N\big]_{K\times N},
\end{equation}
with ${\bf{g}}_n,\;n\in[1:N],$ denoting the $n$-th column vector of ${\bf G}.$
The generator matrix produces a code that can tolerate up to $N-K$ erasures by retrieving data from any set ${\mathcal{K}}\subset \{1,\dots,N\}$ databases, where $|{\mathcal K}|\geq K$. The encoding process for message $W_m$ is defined as follows:
\begin{equation}
\hspace{-4ex}\begin{aligned}
{\begin{bmatrix}{\bf{w}}_{m,t}\end{bmatrix}}_{1\times K} &{\begin{bmatrix} {\bf{g}}_1& {\bf{g}}_2& \dots& {\bf{g}}_N\end{bmatrix}}_{K\times N}\\
&\quad\qquad= {\begin{bmatrix} {\bf{g}}_1^T{\bf{w}}_{m,t}&\dots&{\bf{g}}_N^T{\bf{w}}_{m,t}\\ \end{bmatrix}}_{1\times N},
\end{aligned}
\end{equation}
where ${\bf{w}}_{m,t},$ $\forall m\in[1:M], \forall t\in[1:\tilde{L}],$ denotes the $K$-dimensional vector of symbols of the $t$-th segment from the message $W_m$. The resulting $N$ coded symbols for each segment are then distributed over the $N$ databases, and the code rate is given by $R_c=\frac{K}{N}.$
Consequently, the code symbols stored at each database $n\in [1:N]$ are given by
\begin{equation}\label{eq:codedDatabse}
{\bf W}_{DB_n}=
\begin{bmatrix}
{\bf{g}}_n^T{\bf{w}}_{1,1} & {\bf{g}}_n^T{\bf{w}}_{1,2} &\dots& {\bf{g}}_n^T{\bf{w}}_{1,\tilde{L}}\\
\vdots &\vdots& \ddots & \vdots\\
{\bf{g}}_n^T{\bf{w}}_{M,1} & {\bf{g}}_n^T{\bf{w}}_{M,2} &\dots& {\bf{g}}_n^T{\bf{w}}_{M,\tilde{L}}\\
\end{bmatrix},
\end{equation}
\noindent where we use ${\bf{W}}[t]$ to denote the $t$-th column, and $W_{m}(t)$ for the element of the $m$-th row and $t$-th column of the database, respectively.
In PFR, the linear combination $\nu$ the user intends to retrieve is represented as
\begin{align}
\widetilde{W}_\nu &= {\bf{v_\nu} }[W_1,\dots,W_M]^T \label{eq:v_msg}\\
&=v_{\nu}(1)W_1+\dots+v_{\nu}(M)W_M \label{eq:v_msg1}\\
&= \begin{bmatrix}{\bf{v_\nu} }{\bf{W}}[1]& \dots & {\bf{v_\nu} }{\bf{W}}[\tilde{L}]\end{bmatrix}\!, \label{eq:v_ms3}
\end{align}
where ${\bf v_\nu}$ is an $M$-dimensional non-zero coefficient vector of the linear combination (row vector) indexed by $\nu$, the coefficients $v_{\nu}(m), \;\forall m\in[1:M],$ are chosen from the finite field $\mathbb{F}_{q}$, and the addition ``$+$''~is done element-wise over the same field. We assume that the vector ${\bf v_\nu}$ is an element of the set ${\cal V}$ that contains all possible distinct $M$-dimensional vectors defined over ${\mathbb F_q}$ where $\nu \in [1:V],\;V= |{\cal V}|=\frac{q^M-1}{q-1}$.
In order for the user to retrieve the linear combination $\widetilde{W}_\nu$, while keeping $\nu$ secret from each database, it generates $N$ query matrices for the databases $\{Q_1^{[\nu]},\dots,Q_N^{[\nu]}\}$. Since the query matrices are generated by the user without prior knowledge of the realizations of the stored messages, the queries must be independent of the messages,
\begin{equation}\vspace*{-0.5ex}
I(Q_1^{[\nu]},\dots,Q_N^{[\nu]};W_1,\dots,W_M)=0, \quad \forall \nu\in[1:V].
\end{equation}
Upon the reception of the query $Q_n^{[\nu]}$, the $n$-th database generates an answer string $A_n^{[\nu]}$ as a deterministic function of the received query and the stored symbols from each message. Hence,
\begin{equation}\vspace*{-0.5ex}
\!H(A_n^{[\nu]}|Q_n^{[\nu]},{\bf W}_{DB_n})=0, \quad\! \forall \nu\in[1:V],\forall n\in[1:N]. \!\!
\end{equation}
To maintain user privacy, the query-answer function must be identically distributed for each possible linear combination $\nu\in[1:V]$ from the perspective of each database $n\in[1:N]$. In other words, the scheme's queries and answers strings must be independent from the desired linear combination index, therefore the following privacy constraint must be satisfied:\vspace*{-1ex}
\begin{equation}\label{privacy_const}
\begin{aligned}
I(A_n^{[\nu]},Q_n^{[\nu]},{\bf W}_{DB_n};\nu)=0, \!\quad\! \forall \nu\in[1:V].\!
\end{aligned}
\end{equation}
After the user receives all answer strings from each database, the user must be able to reliably decode the desired linear combination message $\widetilde{W}_\nu$ with a probability of error $P_e$ that goes to zero as the message size $L$ approaches infinity. Following Fano's inequality, this translates to the decodability constraint
\vspace*{-1ex}\begin{equation}\label{decoding_const}
H(\widetilde{W}_\nu |A_{1:N}^{[\nu]},Q_{1:N}^{[\nu]})= o(L),
\end{equation}
\noindent where $o(L)$ represents any function of $L$, $f(L)$, that satisfies $\lim_{L\rightarrow \infty} f(L)/L\rightarrow 0.$
The retrieval rate of the coded PFR scheme is characterized by the message length $L$, the query structure $Q,$ and the query-answer function, and is defined as the ratio between the size of the desired linear combination message and the total number of downloaded symbols in bits as
\vspace*{-1ex}\begin{equation}\label{eq:PFR_rate_def}
R=\frac{H(\widetilde{W}_\nu)}{\sum_{n=1}^{N} H(A_n^{[\nu]})} .
\end{equation}
A rate $R$ is said to be achievable if there exist a sequence of coded PFR schemes that satisfy the privacy and correctness constraints of \eqref{privacy_const}, \eqref{decoding_const} for $P_e\rightarrow 0$ as $L\rightarrow \infty$.
\vspace*{-0.5ex}
\section{Achievable Rate of MDS Coded PFR} \label{MainResults}
\begin{theorem}
For an $(N,K)$ coded distributed storage system with code rate $R_c=\frac{K}{N}$, $M$ messages and a set of $V$ linear combinations defined over the field $\mathbb{F}_{q}$, a PFR achievable rate is given as \vspace*{-2ex}
\begin{align}
\qquad \qquad R&\leq \frac{1-R_c}{1-R_c^M} \label{eq:capacity1}\\
&=\Big(1+\frac{K}{N}+\frac{K^2}{N^2}+\dots+\frac{K^{M-1}}{N^{M-1}}\Big)^{-1}. \label{eq:capacity}
\end{align}\vspace{-2ex}
\end{theorem}
\begin{remark}
This achievable rate generalizes the achievable rate of repetition coded PFR \cite{SunJafararxiv_2017} which corresponds to the special case of $K\!=\!1$.
Also, \eqref{eq:capacity1} is only a function of the distributed storage
coding rate $R_c$ and the number of stored independent messages $M,$ and is
universal in the sense that it does not depend on the number of linear
combinations $V$ defined over the finite field $\mathbb{F}_{q}$ nor on the
explicit structure of the code.
\end{remark}
\begin{remark}
If we consider each of the $V$ linear combinations of messages in \eqref{eq:v_msg} as a new \emph{virtual message} $\widetilde{W}_\nu$, and then apply the coded PIR scheme of \cite{BanawanUlukus_Globecom2017}, the scheme rate will be $\frac{1-R_c}{1-R_c^V}$ which is smaller than \eqref{eq:capacity1} since $M\leq V$.
\end{remark}
\begin{remark}
When the linear combination set ${\cal V}$ is reduced to the first $M$
linear combinations
(i.e.,~${\bf{v}}_{1:M} \in {\cal V}:
[{\bf{v}}_1\;{\bf{v}}_2\;\dots\;{\bf{v}}_M]={\bf I}_M$),
the achievable rate of \eqref{eq:capacity1} is tight. That is
because in this setting the problem of coded PFR is reduced to coded PIR
where the converse is implied from \cite{BanawanUlukus_Globecom2017}.
Also, we note that \eqref{eq:capacity1} is equivalent to the coded PIR
capacity \cite{BanawanUlukus_Globecom2017}, which has been observed in
\cite{SunJafararxiv_2017} for $K=1$. Thus,
downloading linear combinations of messages does not incur additional costs
over downloading individual messages.
\end{remark}
\begin{remark}
Eq. \eqref{eq:capacity1} is a strictly decreasing function in the number of messages $M$ for fixed $R_c$. As the number of messages increases $M\rightarrow \infty$, the achievable rate approaches $1-R_c$. Moreover, as $R_c \rightarrow 1$ in \eqref{eq:capacity}, $R \rightarrow \frac{1}{M},$ indicating that to maintain the privacy of the desired linear combination, the user must download all the messages and perform the computation off-line.
\end{remark}
\vspace*{-1ex}
\section{Proof of Theorem~1} \label{AchievableScheme}
\vspace*{-0.5ex}
\subsection{Query generation}
\vspace*{-0.5ex}
The generation of the queries is shown in Algorithm~1.
Let $B\in[1:V]$ be the block indicator and $R\in[1:K]$ be the repetition indicator, respectively. Let the $v$-sum be the combination of $v$ distinct elements out of $V$ elements. Since we have $V \choose v$ different combinations, we denote each different combination as a \emph{type} of the $v$-sum. Let the components of these combinations be symbols of the $V$ virtual messages. As mentioned above, we generate the query set for each database in blocks, where a block represents a group of all ${V\choose v}$ types of $v$-sums for all $v \in[1:V],$ resulting in $V$ blocks in total. To this end, we let the size of the \emph{dependent} virtual messages to be $L=KN^V$ (i.e.,~${\tilde{L}=N^V}$).
For a desired linear combination $\nu \in[1:V]$ we use the notation $Q^{[\nu]}(DB_B)$ to indicate the query set of the database $DB_B\in[1:N]$. This set is composed from $VK$ disjoint subsets $Q^{[\nu]}_{B,R}(DB_B)$ generated for each block $B$ and repetition $R$. We require $K^{V-B}(N-K)^{B-1}$ distinct instances of each type of $v$-sum for every set $Q^{[\nu]}_{B,R}(DB_B)$. Each block and repetition subset is further subdivided into two subsets: the first subset $Q^{[\nu]}_{B,R}(DB_B,{\cal M})$ consists of the $v$-sum types with symbols from the desired linear combination, and the second subset $Q^{[\nu]}_{B,R}(DB_B,{\cal I})$ contains only $v$-sum types with symbols from undesired linear combinations. The query sets for all databases are generated by Algorithm 1 with the following procedures.
\indent {\bf \textit{1) Index assignment}}: In the MDS-coded PIR scheme
\cite{BanawanUlukus_Globecom2017}, the user privately applies a random
permutation over the coded symbols of each message independently. The goal
is to make the coded symbols queried from each database to appear to be
chosen randomly and independently from the desired message. However, for the
PFR problem the linear function is computed element-wise, thus there is a
dependency across the symbols with the same index, which must be maintained
under a permutation. To this end, we modify the permutation to be fixed
across all messages. Let $\pi(\cdot)$ be a random permutation function over
$[1:\tilde{L}]$. We use the notation $U_\nu(t)$, where
\begin{equation}
U_\nu(t)\triangleq \sigma_t\widetilde{W}_\nu(\pi(t))=\sigma_t{\bf v}_{\nu}{\bf{W}}[\pi(t)],
\end{equation}
to indicate the permuted message symbol from the virtual message $\widetilde{W}_\nu$. The random variable $\sigma$ is used to indicate the sign assigned to each individual virtual message symbol, $\sigma_t \in \{+1,-1\}$ \cite{SunJafararxiv_2017}. Both $\sigma_t$ and $\pi$ are randomly selected privately and uniformly by the user.
\indent {\bf \textit{2) Block ${B=1}$}}: This block is described by Steps {3} to {10} of Algorithm 1, where we have $v=1$ for the $v$-sum.\\
\indent {\textit{Initialization:}}
In the initialization step, the user queries the first database {${DB_1=1}$} for $K^{V-1}$ distinct symbols from the desired linear combination $U_\nu(i)$. This is done by calling the function "$\text{new}(U_\nu)$" that will select a symbol from message $U_\nu$ with a new index $i$ each time it is called (Step 6).
\indent {\textit{Database symmetry:}}
Database symmetry is obtained via the ``For'' loop in Step {3}, resulting in a total number of $N K^{V-1}$ symbols over all databases.
\indent {\textit{Message symmetry:}}
In Step {7}, to maintain message symmetry, the user ask each database for the same number of distinct symbols of all other linear combinations $U_\theta(i),\;\theta\in\{1,\dots,V\}\!\setminus\!\{\nu\},$ resulting in a total number of $NVK^{V-1}$ symbols. As a result, the query sets for each database are symmetric with respect to all linear combination vectors in $[1:V]$. We associate the symbols of undesired messages in $K$ groups $G\in{[1:K]}$ to be exploited as distinct side information for different rounds of the scheme as shown in Step {7.}
\indent {\bf \textit{3) Side-information exploitation:}}
In Steps {11} to {20}, we generate the blocks {$B\in[2:V]$} by applying two subroutines ``Exploit-SI'' and ``M-Sym'', respectively. We first use the subroutine ``Exploit-SI'' \cite{SunJafararxiv_2017} to generate queries for new symbols of the desired linear combination $U_\nu$ by combining these symbols with different side information groups from the previous block associated with $N-K$ neighboring databases, as shown in Step {13}. This is required by our proposed MDS coded scheme to ensure privacy and is in contrast to \cite{SunJafararxiv_2017}, where the side information of previous blocks from \emph{all} databases is utilized.
Then, the subroutine ``M-Sym'' \cite{SunJafararxiv_2017} is used to generate side information to be exploited in the following blocks. This subroutine select symbols of undesired messages to generate $v$-sums that enforce symmetry in the block queries. For example in $B=2$, if we have the queries $U_\nu(i)+U_2(j)$, and $U_\nu(l)+U_3(r) \in Q^{[\nu]}_{2,R}(DB_2,{\cal M})$, this subroutine will generate $U_2(l)+U_3(i)$. As a result, we can show that the symmetry over the linear combinations and databases is maintained. By the end of this step we have in total $N{V\choose B} K^{V-B}(N-K)^{B-1}$ queries for each block from all databases.
\indent {\bf \textit{4) Generation of further query rounds:}}
We require further query rounds to obtain $K$ linear equations for each coded symbol to be able to decode. To this end, we circularly shift the order of the database at each repetition. The shift is done for the initial block, $B=1$, in Steps {22} to {25}. However, for the following blocks we only rotate the indices of desired messages $U_\nu$ and combine them with new groups of side information from the neighboring databases from the first round as seen in Steps {26} to {33}. This rotation and side information exploitation for $B\in[2:V]$ is done using the subroutine ``Reuse-SI'' (omitted in the interest of space).
\indent {\bf \textit{5) Query set assembly:}}
Finally, in Steps {35} to {37}, we assemble each query set from the queries disjoint subsets obtained in the previous blocks and rounds.
\begin{remark}
Note that the proposed scheme significantly differs from the one presented in \cite{SunJafararxiv_2017} in terms of how the side information is exploited due to coding. In particular, we distribute the side information over $K$ rounds such that every database is queried for each message and linear combination only once.
\end{remark}
\begin{table}[ht!]
\small
\begin{tabular}{p{0.965\linewidth}}
\specialrule{.1em}{.05em}{.05em}
\rule{0pt}{2.5ex}
\hspace{-0.5em}\noindent\textbf{Algorithm 1:} Query set generation algorithm\\
\specialrule{.1em}{.05em}{.05em} \specialrule{.1em}{.05em}{.05em}
\rule{0pt}{2.5ex}
\hspace{-0.35em}\textbf{Input:} $\nu, K, N, M,$ and $V.$\\
\textbf{Output:} ${Q}^{[\nu]}(1),\dots,{Q}^{[\nu]}(N)$\\
\begin{hangparas}{.15in}{1}1.~\textbf{Initialize:} All query sets are initialized as a null set $Q^{[\nu]}(1),\dots,Q^{[\nu]}(N)\leftarrow \emptyset$, the block counter $B=1,$ and repetition counter $R=1.$ Let number of neighboring databases $Nb= {N-K}$ \end{hangparas}
2.~\textbf{Let} repetition $R_{B}= K^{V-B}(N-K)^{B-1}\quad \forall B\in[1:V]$\\
3.~\textbf{For} first database block $DB_1=1:N$ \textbf{do}\\
4.\hspace{2ex}\textbf{For} side information group $G=1:K$ \textbf{do} \\
5.\hspace{4ex}\textbf{For} repetition group $RG=1:(R_{1}/K)$ \textbf{do} \vspace{-1ex}
\begin{equation*}\vspace{-0.5ex}
\begin{aligned}
\hspace{-2ex}\text{6.} \qquad\qquad Q^{[\nu]}_{1,R}(DB_1,{\cal M})\!\leftarrow& \{u_{\nu}\}, u_\nu =\text{new}(U_{\nu})\\
\hspace{-2.8ex}\text{7.} \qquad\qquad Q^{[\nu]}_{1,R}(DB_1,{\cal I}_{G})\!\leftarrow& \{\text{new}(U_{1}),\dots,\text{new}(U_{V})\}\!\setminus\!\{u_{\nu}\}
\end{aligned}
\end{equation*}
8.\hspace{4ex}\textbf{End For} (repeat within the same SI group)\\
9.\hspace{2ex}\textbf{End For} (repetition for SI groups)\\
\hspace{-1.1ex}10.~\textbf{End For}\\
\hspace{-1.1ex}11.~\textbf{For} block $B=2:V$ \textbf{do}\\
\hspace{-1.1ex}12.\hspace{2ex}\textbf{For} $DB_{B}=1:N$ \textbf{do} \vspace{-1ex}
\begin{equation*}
\hspace{3.7ex}
\begin{aligned}
\hspace{-6.4ex}\text{13.} \quad\quad Q^{[\nu]}_{B,R}(DB_{B},{\cal M})\leftarrow &\text{\bf{Exploit-SI}}\big(Q^{[\nu]}_{B-1,R}(DB_{B}\!+\!1,{\cal I}_{Nb})\\[-0.7ex]
&\hspace{1.5ex}\cup\dots\!\cup Q^{[\nu]}_{B-1,R}(DB_{B}\!+\!Nb,{\cal I}_{1})\big)
\end{aligned} \end{equation*}
\hspace{-1.1ex}14.\hspace{4ex}\textbf{For} side-information group $G=1:K$ \textbf{do} \\
\hspace{-1.1ex}15.\hspace{6ex}\textbf{For} $RG=1:(R_{B}/K)$ \textbf{do} \vspace{-1ex}
\begin{equation*} \vspace{-0.6ex}
\hspace{-6.1ex}\text{16.} \qquad\qquad Q^{[\nu]}_{B,R}(DB_{B},{\cal I}_{G})\leftarrow \text{\bf{M-Sym}} (Q^{[\nu]}_{B,R}(DB_B,{\cal M}) )
\end{equation*}
\hspace{-1.1ex}17.\hspace{6ex}\textbf{End For} (repeat within the same SI group)\\
\hspace{-1.1ex}18.\hspace{4ex}\textbf{End For} (repeat for SI groups)\\
\hspace{-1.1ex}19.\hspace{2ex}~\textbf{End For} (repeat for each database)\\
\hspace{-1.1ex}20.~\textbf{End for} (repeat for each block)\\
\hspace{-1.1ex}21.~\textbf{For} query round $R=2:K$ \textbf{do}\\
\hspace{-1.1ex}22.\hspace{2ex}\textbf{For} $DB_1=1:N$ \textbf{do}\vspace{-1ex}
\begin{equation*}\vspace{-0.5ex}\hspace{-5ex}
\begin{aligned}
\hspace{-6.2ex}\text{23.} \qquad\qquad& Q^{[\nu]}_{1,R}(DB_1,{\cal M})\leftarrow Q^{[\nu]}_{1,R-1}(DB_1-1,{\cal M})\\
\hspace{-6.2ex}\text{24.} \qquad\qquad& Q^{[\nu]}_{1,R}(DB_1,{\cal I}_G)\leftarrow Q^{[\nu]}_{1,R-1}(DB_1-1,{\cal I}_G)
\end{aligned}
\end{equation*}
\hspace{-1.1ex}25.\hspace{2ex}\textbf{End For} (initializing rounds) \\
\hspace{-1.1ex}26.\hspace{2ex}\textbf{For} block $B=2:V$ \textbf{do}\\
\hspace{-1.1ex}27.\hspace{3ex}\textbf{For} $DB_B=1:N$ \textbf{do}\\
\hspace{-1.1ex}28.\hspace{4ex}\textbf{For} side information group $G=1:K$ \textbf{do} \vspace{-1ex}
\begin{equation*} \vspace{-0.5ex}
\begin{aligned}
\hspace{-7.0ex}\text{29.} \quad\qquad Q^{[\nu]}_{B,R}(&DB_B,{\cal I}_{G})\leftarrow Q^{[\nu]}_{B,R-1}(DB_B-1,{\cal I}_{G}) \\
\hspace{-7.0ex}\text{30.} \quad\qquad Q^{[\nu]}_{B,R}(&DB_B,{\cal M})\leftarrow \text{\bf{Reuse-SI}}\big(Q^{[\nu]}_{B,R}(DB_B,{\cal I}_{G}),\\[-0.7ex]
&Q^{[\nu]}_{B-1,1}(DB_{B}+1,{\cal I}_{Nb+R-1})\cup\dots \\[-0.7ex]
&\quad\qquad\dots\cup Q^{[\nu]}_{B-1,1}(DB_{B}+Nb,{\cal I}_{R})\big)
\end{aligned}
\end{equation*}
\hspace{-1.1ex}31.\hspace{4ex}\textbf{End For} (SI groups) \\
\hspace{-1.1ex}32.\hspace{3ex}\textbf{End For} (repeating for each database) \\
\hspace{-1.1ex}33.\hspace{2ex}\textbf{End For} (repeating for each block) \\
\hspace{-1.1ex}34.~\textbf{End For} (repeating for each round) \\
\hspace{-1.1ex}35.~\textbf{For} $DB_B=1:N$ \textbf{do}\vspace{-1ex}
\begin{equation*}\hspace{1ex}\vspace{-0.5ex}
\begin{aligned}
\hspace{-2.1ex}\text{36.} \quad\quad\! &Q^{[\nu]}(DB_B)\!\leftarrow\!\!\bigcup\limits_{B=1}^{V} \!\bigcup\limits_{R=1}^{K}\!\! \big(Q^{[\nu]}_{B,R}(DB_B,{\cal I})\cup Q^{[\nu]}_{B,R}(DB_{B},{\cal M}) \big)
\end{aligned} \end{equation*}
\hspace{-1.1ex}37.~\textbf{End For} (assembling the query sets)\\
\specialrule{.1em}{.05em}{.05em}
\end{tabular}
\normalsize
\vspace*{-4ex}
\end{table}
\vspace{-1ex}
\subsection{Sign assignment and redundancy elimination}\label{sec:SignAssyment} \vspace{-0.5ex}
We carefully assign an alternating sign $\sigma_t \in [+1,-1]$ to each symbol in the query set, based on the desired linear combination index $\nu$ \cite{SunJafararxiv_2017}. The intuition behind the sign assignment is to introduce a uniquely solvable linear equation system from the different $v$-sum types. By obtaining such an equation system in each block, the user can opt from downloading these queries, compute them off-line, and thus reduce the download rate. Based on this insight we can state the following lemma.
\begin{lemma}[\hspace{-0.1ex}\cite{SunJafararxiv_2017}]
For all {$\nu\in[1:V]$}, each database $n\in[1:N]$, and based on the side information available from the neighboring databases, there are $V-M\choose v$ redundant $v$-sum types out of all possible types $V\choose v$ in each block {${v\in[1:V-M]}$} of the query sets.
\end{lemma}
Lemma~1 is also applicable when the desired linear function is performed over MDS-coded databases due to the fact that each MDS-coded symbol is itself a linear combination. That is, the MDS code can be seen as an inner code and the desired linear function as an outer ``code'' with respect to the databases. Hence, the redundancy resulting from the linear dependencies between messages is also present under MDS coding and we can extend Lemma~1 to our scheme. We now make the final modification to our PFR query sets. We first directly apply the sign assignment $\sigma_t$, then remove the redundant $v$-sum types from every block $B\in{[1:V]}$. Finally, we generate the query matrices $Q^{[\nu]}_{1:N}$ using a one-to-one mapping function $f,$ for which $Q^{[\nu]}(DB_B)$ is the preimage.
\begin{proof}
The proof of optimality for arbitrary $N,K,M$ and $V$ follows from the structure of the query and Lemma~1. The achievable rate is given as
\begin{equation*}\label{eq:Cap_proof}
\begin{split}
&R\stackrel{(a)}{\leq}\frac{KN^V}{KN\sum_{v=1}^{V}\Big( {V\choose v}-{V-M\choose v}\Big) K^{V-v}(N-K)^{v-1}}\\
&=\frac{N^V\big(\frac{N-K}{N}\big)}{\sum_{v=1}^{V}\Big( {V\choose v} K^{V-v}(N-K)^{v}-{V-M\choose v} K^{V-v}(N-K)^{v}\Big)}\\
&\stackrel{(b)}{=}\frac{N^V\big(\frac{N-K}{N}\big)}{(N^V\!-K^V)-\sum_{v=1}^{V-M} {V-M\choose v} K^{V-v}(N-K)^{v}}\\
&=\frac{N^V\big(\frac{N-K}{N}\big)}{(N^V\!-K^V)-K^M\sum_{v=1}^{V-M} {V-M\choose v} K^{V-M-v}(N-K)^{v}}\\
&=\frac{N^V\big(1-\frac{K}{N}\big)}{(N^V\!-K^V)-K^M\big( N^{V-M}-K^{V-M}\big)}\\
&=\frac{N^V\big(1-\frac{K}{N}\big)}{(N^V\!-K^V)-K^M N^{V-M}+K^{V}}\\
&=\frac{N^V\big(1-\frac{K}{N}\big)}{N^V\!-K^M N^{V-M}}=\frac{1-R_c}{1-R_c^M};
\end{split}
\end{equation*}
where (a) follows from the definition of the PFR rate
\eqref{eq:PFR_rate_def}; (b) follows from the fact that the second
term of the summation in the denominator is equal to zero for
$v>V-M$ and consequently we can change the upper bound of the summation; and the first term of the summation follows from the binomial theorem.
\end{proof}
\vspace{-1.5ex}
\subsection{Correctness (decodability)}\vspace{-0.5ex}
To prove correctness, we show that the user can obtain the desired linear combination $\widetilde{W}_\nu$ from the answers retrieved from $N$ databases. From the query answers $A^{[\nu]}_{1:N}$, we group the $K$ identical queries from different rounds and databases. Each group will result in $K$ linearly independent equations that can be uniquely solved. We decode, block by block, starting from block one, which we directly decode and obtain
$K N \big(\!{V\choose v}-{V-M\choose v}\!\big) K^{V-v}(N-K)^{v-1}$ decoded symbols. Now, using these symbols we regenerate ${V-M\choose v}$ redundant symbols according to Lemma~1 and obtain
{$K N {V\choose v} K^{V-v}(N-K)^{v-1}$} symbols in total. Out of these queries there are
$K N \big(\!{V\choose v}-{V-1\choose v}\!\big) K^{V-v}(N-K)^{v-1}$ symbols from $\widetilde{W}_\nu$.
Next, for blocks $B\in[2:V]$, we use the symbols obtained in the previous block $B-1$ to remove the side information associated with the desired linear combination symbols of the current block $B$, then the operations from the first block (decode and retrieve redundancy) are repeated. As a result, we obtain a total number of symbols equal to $K N\sum_{v=1}^{V}\big(\! {V\choose v}-{V-1\choose v}\!\big) K^{V-v}(N-K)^{v-1} =\frac{N}{N-K} \big( N^V- KN^{V-1}\big) = K N^V$ denoting precisely the number of symbols in $\widetilde{W}_\nu$.
\vspace{-0.5ex}
\subsection{Privacy}
\vspace*{-0.5ex}
Privacy is guaranteed by preserving an equal number of requests for any
linear combination $\widetilde{W}_\nu,$ where the requests are symmetric from the
perspective of the accessed virtual messages. As the MDS code can be seen as
an outer code, the arguments in \cite{Maddah-Aliarxiv2017,
SunJafararxiv_2017} apply here as well. In particular, each database is queried with precisely the same $v$-sum type
components, i.e., $U_\nu(t)$, which ensures symmetry. This can be seen from Step 16 in Algorithm 1 where the same subroutine ``M-Sym'' is used for each block and database. By selecting a permutation $\pi(t)$ and a sign assignment $\sigma_t$ uniformly at random, queries for code symbols are permuted in the same way over all databases. With other words, for any
$U_\nu(t)=\sigma_t {\bf v}_\nu {\bf W}[\pi(t)]$ there exist $\sigma_t,
\pi(t)$ such that $Q^{[\theta]}(DB_B) \leftrightarrow Q^{[\nu]}(DB_B) \quad
\forall \nu, \theta \in [1:V]$. Thus, $A_n^{[\nu]}$, and $Q_n^{[\nu]}$
are statistically independent of $\nu$ and \eqref{privacy_const} holds.
\vspace{-1ex}
\section{Example} \label{EX} \vspace{-0.5ex}
We consider $M=2$ messages stored using a $(3,2)$ MDS code. The user wishes to obtain a linear combination over the binary field (i.e.,~${\bf{v}_\nu}\in {\mathbb F}_2^M, \quad V=3$). Therefore, we have the linear combinations ${{\bf{v}}_1=[1 \;0], {\bf{v}}_2=[0 \; 1], {\bf{v}}_3=[1\; 1]},$ and each message must be of length $L=KN^V=54$ symbols. We simplify the notation by letting $a_t=U_1(t), \; b_t=U_2(t),$ and $c_t=U_3(t)$ for all $t\in[1:27]$. Let $\sigma_t=1\;\forall t$ and let the desired linear combination index be $\nu=3$.
\indent{\textit{Query set construction:}}
Algorithm~1 starts with $B=1$ by generating queries for each database and {$K^{V-1}=4$} distinct instances of $c_t$ (i.e.,~from database~1 query ${\bf{g}}_1^T(c_{1:4})\triangleq\{{\bf{g}}_1^Tc_1,\dots, {\bf{g}}_1^Tc_4\}$). By message symmetry this also applies for $a_t$ and $b_t$ to form two groups of side information sets to be used in the next block with $NK^{V-1}=12$ symbols in total from each linear combination. Next, one group of side information is queried jointly with a new instant of the desired message. For example, for database~1 and type $b+c$ we have ${\bf{g}}_1^T(b_{5:6}-c_{13:14}) \triangleq \{{\bf{g}}_1^Tb_5-{\bf{g}}_1^Tc_{13}, {\bf{g}}_1^Tb_6-{\bf{g}}_1^Tc_{14}\}$. The remaining blocks and rounds follow from Algorithm~1. After generating the query set for each database, we apply the sign assignment and remove the redundant queries.
\indent{\textit{Decoding:}}
The answer strings from each query are shown in Table~1. Note that there is
no $c_t$ in the first block as they are redundant and can be generated by
the user. To decode we start with Block 1, and we obtain ${\bf{g}}_1^Ta_1$
from database~1 and ${\bf{g}}_2^Ta_1$ from database 2. Thus, by the MDS code
properties we can decode and obtain the segment $a_1$; similarly for all
other queries in this block. Now for Block~2, we first remove the side
information from the types containing symbols of $c_t$. For example, for
${\bf{g}}_1^Tb_5-{\bf{g}}_1^Tc_{13}$ from database~1, we have $b_5$ from the
previous block. As a result, we obtain ${\bf{g}}_1^Tc_{13}$. Similarly we
obtain ${\bf{g}}_2^Tc_{13}$ from database~2, and $c_{13}$ can be recovered.
\input{Ex1_table}
\vspace{-2ex}
\section{Outer Bound for the Special Case $V=2$}
\vspace{-0.5ex}
In the special case of $V=2,$ $M$ independent messages, and any $(N,K)$ MDS
code an outer bound for the coded PFR problem is obtained by combining the
independence of answer strings from any $K$ databases \cite[Lemma
2]{BanawanUlukus_Globecom2017} with
\cite{SunJafararxiv_2017}. Thus, we can show that the retrieval rate is
upper
bounded as $R \leq
\frac{NH(\omega_1)}{KH(\omega_1,\omega_2)+ H(\omega_1)(N-K)}$, where the
joint distribution of $(\omega_1,\omega_2)$ is the joint distribution of
$(\widetilde{W}_{1,\ell}, \widetilde{W}_{2,\ell})$ for all $\ell \in [1:\tilde{L}]$ selected iid~with respect to the symbols of the messages.
\vspace{-1ex}
\bibliographystyle{IEEEtran}
|
1,116,691,500,496 | arxiv | \section{Introduction}
Theoretical modeling of structure formation in a $\Lambda$-CDM
cosmology cannot match observed galaxy luminosity functions locally
unless some form of heating or ``feedback'' is included in the
simulations \citep[e.g.,][]{GranatoEtal2004}. This strong theoretical
requirement coupled with (1) the empirical discoveries of the strong
correlations between black hole masses and the properties of their
galactic bulges \citep[e.g.,][]{FerMer2000,GebhardtEtal2000} and (2)
the observation that growing supermassive black holes reveal
themselves as luminous quasars \citep[e.g.,][]{Soltan1982,YuTre2002}
have led to quasar winds becoming promising sources of feedback in
massive ($>L^{\ast}$) galaxies \citep[e.g.,][]{SiRe1998}. Attempts to
model these outflows in cosmological simulations currently employ
simplifying and inaccurate assumptions without incorporating empirical
constraints. Quasar winds are directly observed in Broad Absorption
Line (BAL) quasars; this $\sim15$--20\% of the luminous, radio-quiet
quasar population exhibits deep troughs from high-ionization ultraviolet\
resonance transitions such as C~{\sc iv}\ and
O~{\sc vi}. Such absorption features appear blueshifted along lines of sight
passing through winds with terminal velocities reaching 0.03--0.3$c$.
Constraining the nature of quasar outflows, including the geometry,
acceleration mechanism, ionization state, and mass outflow rate, is
fundamental both for understanding the role of quasars in galaxy
evolution as well as accretion physics. Quasars are by nature
multiwavelength, and a complete accounting of the outflow requires a
panchromatic approach.
\section{Empirical Constraints on Wind Properties}
BAL quasars have been the targets of surveys at all wavelengths. As
the sensitivity of available facilities increases over time, the
empirical data on distinct outflow components have become more
constraining. Below, we briefly summarize the conclusions and
implications from studies in three regimes, the ultraviolet, the X-ray, and
the infrared; these results are compiled in Table~1. Constraints on the
launching radius ($R_{\rm launch}$), covering fraction ($f_{\rm
cov}$), column density ($N_{\rm H}$), and velocity ($v$) of the wind probed in
these regimes is of particular interest for determining the structure
and kinetic energy of the flow.
\subsection{The Ultraviolet Line-Driven Wind}
BAL quasars by definition show outflows in the ultraviolet, and there is
compelling evidence that these are radiatively driven. The momentum
from photons absorbed from the quasar continuum is sufficient to push
gas to the high observed velocities of up to $10^{4.5}$~km~s$^{-1}$\
\citep[e.g.,][]{MuChGrVo1995,deKool1997}.
In fact, the BAL absorption features represent photon momentum absorbed from
the ultraviolet\ continuum. In addition, line-locked systems
detected in some objects provide direct evidence for the importance of
line driving
\citep[e.g.,][]{BrMi89}.
Stable absorption-line locking occurs when the relative Doppler shift
of gas at different distances from the continuum source is
approximately equal to the wavelength separation of two strong
absorption lines (e.g., Ly$\alpha$ and N~{\sc v}). These systems then become
locked into approximately this velocity separation, with the system
closer to the continuum source absorbing the photons that would
otherwise continue to accelerate the more distant, higher velocity
system \citep[e.g.,][]{Scargle1973}. Line-locked systems do not occur
unless line-driving plays an important role in the dynamics of the
outflow \citep[e.g.,][]{KoVoMoWe1993,Arav1996,ChNe03}.
In fact, the line-locked transitions themselves have to play an important
role in the dynamics for line-locking to operate.
The geometry most frequently associated with BAL outflows is
equatorial \citep[e.g.,][]{deKBe1995,MuChGrVo1995}, and
spectropolarimetry of BAL and non-BAL~quasars supports this generic
picture \citep[e.g.,][]{OgCoMiTr1999}. In this scenario, the material
in the wind originates in the accretion disk. Some vertical pressure,
either thermal, magnetic, or radiative, pushes gas upwards, to be
illuminated by the continuum emission generated at smaller
radii. Radiation pressure then accelerates the gas radially; this is
most efficient when the photon energies match those of strong
resonance atomic transitions. The covering fraction of the outflow in
this picture is determined by the ratio of the (vertical) disk
pressure to the (radial) central continuum pressure. As ultraviolet\ emission
lines are often absorbed, the BAL wind must be outside of, or perhaps
co-spatial with, the broad emission line (BEL) region. Using the
$L_{\rm UV}$--$R_{\rm CIV}$ reverberation mapping relationship
measured by Kaspi et al. (2007)\nocite{KaspiEtal2007}, the C~{\sc iv}\ BEL
region radius is $\sim2\times10^{17}$~cm for a luminous ($\lambda
L_{1350 \AA}=10^{46}$ erg~s$^{-1}$) quasar. For a BEL region at the
base of the outflow, the C~{\sc iv}\ BAL radius is approximately the same
\citep{MC98}.
As mentioned above, because the ultraviolet\ continuum generates the radiative
pressure on the BAL gas, the distance between the continuum and the
BAL gas will ultimately affect the covering fraction of the wind.
This is an area where important observational constraints can perhaps
be brought to bear. Though \mbox{$\alpha$-disk} models predict that $\sim
90\%$ of the optical/ultraviolet\ continuum is generated within $7 \times
10^{15}$~cm for $M_{\rm BH}=3\times10^8$~$M_{\odot}$, recent constraints
from microlensing of quasar accretion disks suggests the continuum
emitting region is actually significantly larger, $\sim 5 \times
10^{16}$~cm \citep{PoEtAl2006,KoEtAl2006}.
\subsection{The Shielding Gas}
At first glance, a quasar appears well-suited to radiative gas
acceleration given the strong ultraviolet\ radiation field. However, unlike
\mbox{O stars}, quasars are also strong X-ray sources. This high flux
of X-ray photons ionizes the wind, thus eliminating ultraviolet\ resonance
lines. Highly ionized gas can only be driven radiatively by radiation
pressure on electrons, which is much less efficient than resonance
line pressure. To prevent overionization, some material is needed to
protect the wind from the ionizing far-ultraviolet\ and X-ray continuum
\citep[][]{ShViSh85}. In the context of continuous winds, a layer of
shielding gas was hypothesized by Murray et al. (1995; who dubbed it
``hitchhiking gas'')\nocite{MuChGrVo1995} as a thick, highly ionized
layer of gas interior to the ultraviolet\ BAL wind. Their model required
shielding gas in order to launch the wind from small radii
($\sim10^{16}$ cm). Though initially introduced in a rather ad hoc
manner, the empirical evidence for the existence of the shielding gas
has become quite compelling. In particular, measurements of the
column density of X-ray absorbing gas in radio-quiet BAL~quasars find
a range of $N_{\rm H}$=10$^{22}$--10$^{24}$~cm$^{2}$
\citep[e.g.,][]{GreenEtal2001,GaBrChGa2002}. These values are one to
two orders of magnitude larger than the best constraints from careful
modeling of the ultraviolet\ absorption lines \citep[e.g.,][]{AravEtal2001}.
This discrepancy is most dramatic in the BAL quasars whose extreme
X-ray weakness indicates they are likely to host Compton-thick
($N_{\rm H}$\,$>1.5\times10^{24}$~cm$^{-2}$) absorbers.
To date, Compton-thick absorption has only been confirmed for one BAL
quasar, Mrk~231, with the detection of its direct continuum above
10~keV by Braito et al. (2004)\nocite{braito+04}; at softer X-ray
energies only scattered and starburst emission is seen. Notably, the
putative X-ray Compton-thick BAL~quasars show broad emission lines and
often blue ultraviolet-optical continua
(e.g., Clavel et al. 2006; Gallagher et al. 2006).\nocite{clavel+06,gall+06}
As first seen by Green et al. (2001)\nocite{GreenEtal2001}, the
$\sim10\%$ of BAL~quasars with low-ionization (Mg~{\sc ii}) BALs may
typically have Compton-thick X-ray absorbers, and Gallagher
et~al. (2006) speculated that Mg~{\sc ii} BALs might require such
X-ray absorption for the low-ionization gas to exist in the outflow.
The converse is not true, however, and so Compton-thick X-ray absorbers
may be a necessary but not sufficient condition for low-ionization
BALs.
Given that an absorber with $N_{\rm H}$$>1.5\times10^{24}$~cm$^{-2}$ is
optically thick to the ultraviolet/X-ray continuum in a quasar, such X-ray
absorbers must not fully cover the ultraviolet\ continuum-emitting region.
Though a (less than Compton-thick) highly ionized shielding gas
component might be expected to allow a significant ultraviolet\ photon flux
through the wind \citep{MuChGrVo1995}, radiative transfer calculations
of highly ionized magneto-hydrodynamic (MHD) disk winds hint that some
less ionized gas would still be present to block a large fraction of
the ultraviolet\ flux \citep{Everett05}. This implies that the ultraviolet\ BAL wind
lies along a distinct path to the ultraviolet\ continuum in these systems
compared to the absorber blocking the X-ray continuum. If this is
generically true (though only evident for the most extreme examples),
this result provides an important constraint on the relative location
of the ultraviolet\ and X-ray continuum sources. For an X-ray continuum
generated on smaller spatial scales than the ultraviolet, as implied by recent
constraints from microlensing \citep{PoEtAl2006,KoEtAl2006}, a
stratified wind can account for discrepancies in the ultraviolet\ and X-ray
absorber properties. We explore this further in \S\ref{sec:model}
To date, the best evidence from X-ray spectral modeling indicates that
the absorbers are plausibly highly ionized such that the soft X-ray
opacity would be dominated by O~{\sc vii}\ and O~{\sc viii}\ absorption edges.
Strong X-ray absorption variability unmatched by changes in the ultraviolet\
BALs also points towards distinct ultraviolet\ and X-ray absorbing material
\citep[e.g.,][]{gall+04} with the X-ray absorber closer than the ultraviolet\
absorber to the central X-ray continuum. The bulk of the X-ray data
to date thus support the identification of the X-ray absorber with the
putative shielding gas.
The recent discovery of a correlation between the maximum terminal
velocity of the ultraviolet\ BAL (as measured for C~{\sc iv}), $v_{\rm max}$, and
X-ray weakness in BAL quasars indicates the importance of shielding in
the outflow (Gallagher et al. 2006). Without enough signal in this exploratory
survey for spectral fitting, X-ray weakness was taken
to indicate strong X-ray absorption. It is a generic property
of gas escaping from the vicinity of the black hole that the terminal
velocity will be of order the Keplerian velocity of the radius from
which it was launched. Gas that obtains the highest velocities then
might have originated at the smallest radii where the photon densities
are highest. However, radiative line-driving is only efficient if the
gas does not become overionized; this can be accomplished with a thick
layer of shielding gas. This scenario might explain the correlation
between extreme X-ray weakness and the highest values of $v_{\rm
max}$. Figure~\ref{shield} shows a schematic of this situation.
To date, the largest unknown in the properties of the shielding gas is
its velocity. For the bulk of BAL quasars with X-ray spectra, the
velocity cannot be measured with the current generation of
observatories; only a handful are bright enough in X-rays to search
for X-ray BALs. Two BAL quasars with putative Fe~{\sc XXV} BALs,
PG1115$+$080 and APM08279$+$5255, indicate high (tenths of $c$)
blueshifts \citep{chartas+07}, however, the frequency of such features
is unknown.
\begin{figure}
\centerline{\psfig{figure=gallagher_fig1.eps,width=15cm}}
\caption{
Two diagrams of the inner part of the accretion disk illustrating the
possible connection between the presence of X-ray shielding gas (solid
shape) and the velocity of the ultraviolet\ BAL outflow (solid curves). In
both panels, the black hole is to the right, and the continuum
emission is generated in the accretion disk. The horizontal white bar
serves as a scale marker, and the size of the arrows corresponds to
the velocity of the outflowing BAL gas. {\em Left:} In this case, a
thin shield does not allow gas to be accelerated until larger radii.
{\em Right:} A thick shield prevents overionization of the ultraviolet\ BAL
wind at smaller radii where the higher photon densities allow the material
to be launched to larger terminal velocities. A larger vertical
contribution from the accretion disk radiation can also change the
covering fraction of the outflow.
\label{shield}
}
\end{figure}
\subsection{Dusty Outflows}
Outflows may exist on larger scales, as well. K\"onigl \& Kartje
(1994)\nocite{KoKa1994} first proposed that the so-called ``dusty
torus'' -- the structure consisting of cold material on parsec scales
that reprocesses direct accretion power into thermal infrared\ emission --
is actually an outflow. In their model (applied to Seyfert galaxies),
the wind is uplifted vertically from the accretion disk along magnetic
field lines as a magneto-centrifugally launched outflow \citep[as in,
e.g.,][]{BP82,EmBlSh92,Bottorff97}. At large launching radii, dust
can survive in this outflow, and when this dusty gas attains a
sufficient vertical height, it becomes illuminated by the central ultraviolet\
continuum. At that point, radiation pressure accelerates the dusty
gas radially outward, flattening the wind. This elegant model avoids
problems with explaining the torus as a large ring of (clumpy) cold
material in the center of the galaxy; such gas is dynamically unstable
and will collapse \citep[c.f.,][]{KrBe1988}.
While the role of magnetic fields in driving quasar outflows remains
to be observationally constrained,
ultraviolet\ photons certainly will efficiently accelerate
dust grains. The inner wall of this dusty outflow is set by the
temperature at which refractory dust grains (likely graphites)
sublimate, $T_{\rm sub}\sim1500$~K. As the dust is heated by the
radiant power of the quasar continuum, the sublimation radius, $R_{\rm
sub}$, is proportional to $L_{\rm UV}^{0.5}$ \citep{barvainis87}. For
a quasar with $L_{\rm UV}\sim10^{46}$~erg~s$^{-1}$, the inner wall of
the dusty outflow is at 1--2~pc. At these radii, the supermassive
black hole will dominate the dynamics of the gas, and the
Keplerian velocity is $\sim10^3$~km~s$^{-1}$. The velocity dispersion of the
bulge of a massive host galaxy, 200--300~km~s$^{-1}$, provides a plausible
lower bound to the velocity of this material.
BAL~quasar ultraviolet\ spectra typically show evidence for reddening and
extinction in comparison with non-BAL quasars (Sprayberry \& Foltz 1992;
Reichard et al. 2003)\nocite{SpFo1992,reich+03b}, and this could be
taken to imply that some part of this dusty outflow is perhaps just
the outer regions of the
ultraviolet\ BAL wind. However,
a recent study of 9.7\micron\ silicate features in infrared\ quasar spectra
by Shi et al. (2006)\nocite{shi+06} found that BAL quasars typically
show very prominent silicate emission. This is in contrast to type~2
(narrow emission line) Seyfert galaxies which usually show strong
silicate absorption, as predicted by K\"onigl \& Kartje (1994) and
others. The detection of silicate emission in BAL quasars indicates
that the dusty outflow therefore is distinct from the ultraviolet\ BAL wind.
If the silicate grains (at $T_{\rm sil}\sim200$~K; Hao et
al. 2005)\nocite{HaoEtal2005} were instead carried in the BAL wind,
the line of sight to the infrared\ continuum source generated by the warmer
dust at smaller radii would pass through the silicate region. In this
case, silicate absorption is expected.
Within the unified quasar picture, some fraction of the sky is
obscured by the dusty outflow such that the broad emission line region
and central continuum are hidden from the direct line of sight. In
this case, the ratio of type~2 to type~1 (broad emission line) quasars
gives the value for the covering fraction of the dusty outflow
\citep[e.g.,][]{ric+06}.\\
\begin{centering}
\begin{tabular}{lccccc}
\multicolumn{6}{c}{Properties of the Stratified Quasar Wind}\\
\hline
Wind & $R_{\rm launch}$ & $f_{\rm cov}$ & $N_{\rm H}$$^{\rm a}$ &
ion. & $v$ \\
Component & (cm) & & (cm$^{-2}$) &
state$^{\rm b}$ &(km~s$^{-1}$) \\
\hline
Shielding Gas & 10$^{15-16}$ & $>f_{\rm cov,UV}$ & 10$^{22-24}$ &
O~{\sc vii}, O~{\sc viii}\ &? \\
UV BAL Wind & 10$^{17}$ & 0.2(1--$f_{\rm type2}$) & 10$^{21-22}$
& C~{\sc iv}, O~{\sc vi}\ & 10$^{3-4}$ \\
Dusty Outflow & 10$^{18.5}$ & $f_{\rm type2}$ & $\cdots$ &
neutral & 10$^{2-3}$ \\
\hline
\end{tabular}
$^{\rm a}$Line-of-sight column density. $^{\rm b}$Common ions
representing the ionization state.
\end{centering}
\section{The Role of Winds}
In the simplest disk-wind paradigm, all quasars host outflows, but
only in BAL~quasars are these driven along the line of sight.
Therefore, the fraction of type~1 quasars with BALs, $\sim15$--20\%
\citep{reich+03b,HewFol2003}, corresponds to the covering fraction of
the BAL wind. The opposite case would be that only a subset of
quasars host BAL winds, but these dusty shrouds cover a large fraction
of the sky -- the ``cocoon'' picture \citep[e.g.,][]{BeckerEtal2000}.
In this latter situation, BAL quasars would be expected to be mid-infrared\
bright relative to non-BAL~quasars with little or no wind because a
larger fraction of the accretion power is captured and reprocessed
into the thermal infrared\ by dust. The recent {\em Spitzer} survey of 38
BAL~quasars by Gallagher et al. (2007)\nocite{gall+06c} disputes this
latter view, as they found that the mid-infrared\ properties of BAL~quasars
are consistent with non-BAL~quasars of comparable luminosity. In
particular, the relative power in the optical and mid-infrared\ in the two
populations is indistinguishable. Coupled with clear evidence from
spectropolarimetry that there are lines of sight to BAL quasars that
are not covered by the ultraviolet\ outflow \citep[e.g.,][]{OgCoMiTr1999}, it
seems quite likely that most luminous quasars host BAL outflows, and
only in BAL~quasars are we actually looking through them
\citep{WeMoFoHe1991}.
Though the first ultraviolet\ spectroscopic comparisons of the emission-line
and continuum properties of BAL versus non-BAL quasars found them to
be ``remarkably similar'' \citep{WeMoFoHe1991}, spectral studies with
much larger samples revealed that BAL are more often found in quasars
with intrinsically blue ultraviolet-optical continua and broader emission lines
\citep{ric+02}. We point out that the covering
fraction of the wind in any given quasar is likely to vary, and so
those quasars with the largest covering fractions are most likely to
be identified as BAL quasars. This will skew the ``average''
continuum and emission-line properties of BAL quasars to be
representative of quasars with more substantial outflows, rather than
the typical outflow. A discussion of possible links between active
winds, continuum properties, and ultraviolet\ emission-line properties is
presented in Richards (2006).\nocite{ric06}
\section{Constructing a Consistent Geometry}
\label{sec:model}
Based on the empirical constraints outlined in \S2, we construct the
diagram of the stratified wind presented in Figure~2 with
approximately three zones: the shielding gas, the ultraviolet\ BAL wind, and
the dusty outflow. Each zone is spatially distinct and can be
characterized by distinct covering fractions, column densities,
ionization states, and probably velocities. The acceleration
mechanisms also differ. While there is compelling evidence that
radiative line pressure dominates for the ultraviolet\ BAL wind, the dynamical
state of the shielding gas remains uncertain. For gas characterized
by atomic species from
O~{\sc vii}\ up to Fe~{\sc xxv}, the gas is too ionized for line driving to
be effective, and X-ray continuum driving is also insufficient
\citep{EvBa2004}. Therefore, if the shielding gas velocities are
typically $\sim0.1c$ as seen in the two known cases of X-ray BALs, MHD
forces are likely to dominate the acceleration. However, the
shielding gas might instead be stalled or even infalling
\citep{PrStKa2000}. Meanwhile, for the dusty outflows on large
scales, ultraviolet\ continuum pressure on dust grains overrides electron
continuum pressure by approximately a factor of 850 \citep{KoKa1994}.
Efficient acceleration combined with the large launching radius for
the dusty component make it the most equatorial part of the outflow in
a luminous quasar.
Observationally, panchromatic observations are very important in
constructing this picture, as each wind component is viewed primarily
in a distinct portion of the spectral energy distribution. While the
X-ray continuum is imprinted by both the shielding gas and the ultraviolet\
BAL wind, the larger column density of the shielding gas makes its
effect more pronounced. The ultraviolet\ continuum is likely not completely
covered by the shielding gas, which in any case is highly ionized and
would be nearly invisible in the ultraviolet\ for $N_{\rm H}$$\ll 10^{24}$~cm$^{-2}$.
The BALs affecting the ultraviolet\ continuum are the clear signatures of this
component. The dusty outflow, meanwhile, is detected and probed via
its infrared\ emission in luminous BAL quasars.
We emphasize that this proposed picture is based on empirical data for
luminous, radio-quiet BAL quasars. Outflows in both lower luminosity
Seyfert galaxies and radio-loud quasars are likely to be qualitatively
distinct because of differences in both luminosity and spectral energy
distributions. Specifically, Seyfert galaxies and radio-loud quasars
emit a larger fraction of their radiant power in the X-rays than
luminous radio-quiet quasars \citep[e.g.,][]{BrYuSi1997,steffen+06}.
As discussed by Murray et al. (1995), X-ray loud active galactic
nuclei will have difficulty launching radiatively driven winds.
These differences in wind driving are observationally supported: for
instance, the ultraviolet\ absorbing outflows in the lower luminosity Seyfert
galaxy NGC 4151 are likely MHD-driven \citep{CrKr2006}. In addition,
magneto-centrifugally dominated wind models have successfully fit
emission line variations in the Seyfert 1 galaxy NGC 5548
\citep{Bottorff97}; these models also yield a stratified wind
structure \citep{BoKoSh2000}.
\begin{figure}
\psfig{figure=gallagher_fig2.eps,width=15cm}
\caption{
A diagram of the black hole and accretion disk of a luminous,
radio-quiet quasar illustrating the separate components of the outflow
as described in the text: the shielding gas (solid shapes), the ultraviolet\
BAL wind (solid curves), and the dusty outflow (dotted curves). The
straight, labeled lines indicate the observer's lines of sight through
the stratified wind probed by studies in the X-ray, ultraviolet, and infrared;
these arrows point to the approximate location of the continuum source
in each regime.
\label{model}
}
\end{figure}
\section{Conclusions}
Figure~2 is a schematic, simplified view of the stratified wind drawn
in an attempt to incorporate the growing body of multiwavelength data
as well as modeling of disk winds from the past decade or so. As
such, it requires further theoretical and empirical elaboration. For
example, the outflowing (magneto-centrifugal) wind is likely to be
clumpy, as suggested by comparisons of models with observations
\citep{NeIvEl2002,ElSh2006}. Furthermore, the shielding gas is
probably not discontinuous from the ultraviolet\ BAL wind, but is instead the
highly ionized inner region
\citep[e.g.,][]{KoKa1994,MuChGrVo1995,BoKoSh2000,PrStKa2000,Everett05}.
Outflows are most easily studied in BAL quasars where the absorbing
gas is obviously along the line of sight. However, there should be
signatures of outflows in non-BAL quasars if the disk-wind paradigm is
generally correct. For example, X-rays absorbed by the shielding gas
will be emitted along other lines of sight; this contribution to
non-BAL quasar X-ray spectra at soft energies will depend on the
covering fraction and geometry of the shield. In this case, high
signal-to-noise X-ray spectroscopy may reveal a variable scattered
light component in luminous type~1 quasars.
In the near future, high quality near and mid-infrared\ spectra will offer
new insights into the hottest dust at the inner boundary of the dusty
outflow. Furthermore, in-depth analysis of solid state features such
as 9.7 and 18~\micron\ silicate emission can provide constraints on
dust processing and perhaps grain formation within the quasar
environment.
At present, it appears that neither the dusty outflow nor the ultraviolet\ BAL
wind carries sufficient kinetic luminosity to account for the feedback
required to affect galaxy evolution. The shielding gas, with its high
column density and currently unknown velocity, is therefore the most
promising component to dominate the energetics. Constraining these
velocities will require the high spectral resolution and large
effective area of the next generation of X-ray observatories such as
{\em Constellation-X}, as well as continued modeling efforts that
incorporate all phases of the outflow.
\acknowledgements
We thank M. Elitzur, F. Hamann, D. Hines, A. K\"{o}nigl, and
G. Richards for helpful discussions that contributed to the picture
presented in this paper, and O. Blaes and S. Kaspi for pointing us
towards useful empirical results. Support for S.C.G. was provided
by NASA through the {\em Spitzer} Fellowship Program, under award
1256317; this project was made possible by {\em Chandra} X-Ray Center
grant GO4--5113X. J.E.E was supported by NSF AST--0507367 and NSF
PHY--0215581 (to the Center for Magnetic Self-Organization in
Laboratory and Astrophysical Plasmas).
|
1,116,691,500,497 | arxiv | \section{Introduction}
\label{sec:intro}
Over the past several decades, weakly interacting massive particles (WIMPs) have been the leading class of candidates for our universe's dark matter. This paradigm has been motivated primarily by
the fact that a stable particle species with a weak-scale mass and interaction strength is predicted to freeze-out of thermal equilibrium in the early universe with a relic abundance that is comparable to the measured cosmological density of dark matter. As such particles are also often found within frameworks that address the electroweak hierarchy problem (including, but not limited to, weak-scale supersymmetry), this connection has become commonly known as the ``WIMP miracle''~\cite{Bertone:2016nfn}.
The WIMP paradigm has motivated an expansive experimental program, consisting of direct detection, indirect detection, and collider searches. As these efforts have progressed, however, no conclusive detections have been made, and increasingly powerful bounds have been placed on dark matter's non gravitational interactions with the Standard Model (SM). Over the traditional range of WIMP masses ($\sim$10-1000 GeV), direct detection experiments now strongly constrain the dark matter's elastic scattering cross section with nuclei~\cite{Tan:2016zwf,Akerib:2015rjg,Akerib:2016vxi,Agnese:2015nto}, and astrophysical observations by gamma-ray telescopes~\cite{Hooper:2012sr,Ackermann:2015zua} and cosmic ray detectors~\cite{Bergstrom:2013jra,Giesen:2015ufa,Cirelli:2013hv} have also begun to constrain the WIMP parameter space. Although many WIMP models remain viable, it is perhaps surprising that no definitive detection of particle dark matter has yet been made.
In light of this experimental situation, it has become increasingly interesting to consider dark matter scenarios beyond the conventional WIMP paradigm. In this paper, we focus on dark matter candidates with negligible couplings to the SM and that reside within a sector that is thermally decoupled from the visible matter in the early universe. In doing so, we build upon our previous recent work~\cite{Berlin:2016vnh} by considering a wider range of models and discussing their phenomenology in greater detail.
Throughout this study, we assume that the visible sector, which contains the SM, is supplemented by a decoupled hidden sector, which contains the dark matter. We further assume that both sectors are
thermally populated during post-inflation reheating and maintain separate temperatures throughout cosmological evolution~\cite{Allahverdi:2010xz,Adshead:2016xxj}.
Although sequestered from the SM, the hidden sector may consist of many new additional particle species with sizable mutual interaction rates. In particular, it is possible that the lightest stable hidden species, $X$, freezes out via $XX \to YY$ annihilation, where $Y$ is a lighter hidden sector species that ultimately decays into SM particles. Being stable, we take $X$ to be our dark matter candidate.
If the $Y$ is short-lived, it will never dominate the energy density of the universe, and will have little effect on cosmological evolution.
In this regime, $X$ will freeze out with the observed dark matter abundance only if its mass and couplings are similar to those of traditional WIMPs. Although, in principle, such a scenario can be viable for a wide range of masses, constraints from perturbative unitarity typically require $m_X \lesssim \order{100}$ TeV~\cite{Griest:1989wd} (see, however, Ref.~\cite{Harigaya:2016nlg}). This bound can be circumvented, however, if the entropy of the visible sector increases appreciably after the freeze-out of $X$~\cite{Dev:2016xcp,Fornengo:2002db,Gelmini:2006pq,Kane:2015jia,Hooper:2013nia,Patwardhan:2015kga,Randall:2015xza,Reece:2015lch,Lyth:1995ka,Davoudiasl:2015vba,Cohen:2008nb,Yamanaka:2014pva}.
For instance, a heavy and long-lived species in the hidden sector could come to dominate the energy density of the universe before decaying to SM particles, thereby diluting all relic abundances, including that of $X$. As we will see in Sec.~\ref{sec:decay}, the increase in the visible sector entropy from $Y$ decay scales as $\propto \tau_Y^{1/2}$, where $\tau_Y$ is the lifetime of the unstable species. Thus, for sufficiently large $\tau_Y$, it is possible to significantly dilute the abundance of $X$, thereby achieving an acceptable density of dark matter, even for masses well above the conventional limit from perturbative unitarity, $m_X \gg 100$ TeV.
Long lifetimes are straightforwardly realized if the decaying particle is the lightest hidden sector state. In fact, if the hidden and visible sectors are highly decoupled, the lightest hidden sector state will automatically be long-lived, since its width relies on a coupling that is too small to sustain thermal equilibrium between the two sectors. We emphasize that this picture is relatively universal, and can be found within any model in which the dark matter freezes out through annihilations in a heavy and highly decoupled hidden sector that is populated after inflation. In contrast to scenarios in which an additional out-of-equilibrium decay is invoked solely to dilute the initial cosmological abundances of various species, dilutions of the type considered in this paper are an inevitable consequence of thermal decoupling.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.6\textwidth]{Figures/cartoon}
\caption{\label{fig:schematic} A schematic diagram of the processes that we will consider in this study. Here $X$, the dark matter candidate, annihilates into pairs of metastable hidden sector $Y$ particles. If the hidden sector is heavy and extremely decoupled from the visible sector (which contains the Standard Model), then $Y$ will be long-lived, and may eventually dominate the universe's energy density. Upon its decay into Standard Model particles, $Y$ reheats the visible universe and dilutes all particle abundances, including the relic density of $X$.}
\end{center}
\end{figure}
The remainder of this paper is structured as follows. In Sec.~\ref{sec:thermo}, we review the early universe thermodynamics of scenarios with a decoupled hidden sector. We then discuss in detail the processes of thermal freeze-out and out-of-equilibrium decay in Secs.~\ref{sec:fo} and~\ref{sec:decay}, respectively.
In Sec.~\ref{sec:Neff}, we discuss possible contributions to the effective number of neutrino species within this class of scenarios. In Sec.~\ref{sec:models}, we describe three concrete realizations of dark matter in a decoupled hidden sector, in which the hidden and visible sectors interact through the vector portal, Higgs portal, or lepton portal. Finally, we briefly summarize our results and conclusions in Sec.~\ref{sec:conclusion}.
\section{Hidden Sector Thermodynamics}
\label{sec:thermo}
In this section, we review the thermodynamic evolution of a generic hidden sector, whose constituents interact very feebly with the visible sector. In the decoupled limit,
these sectors influence each other's evolution only indirectly by either modifying the cosmic expansion rate, or by injecting energy through any decays of hidden sector particles into the visible sector.
We begin by considering two particle species within the hidden sector: the lightest hidden sector particle, $Y$, and the lightest stable hidden sector particle, $X$. The stable species will annihilate through processes such as $XX \to YY$ until its abundance freezes out of equilibrium, in analogy with conventional WIMP freeze-out. Since $Y$ is the lightest particle in the hidden sector, $Y$ can only decay to the SM, either directly or through a multi-step cascade, e.g., $Y \to \cdots \to \text{SM}$; this setup is depicted schematically in Fig.~\ref{fig:schematic}. For simplicity, we will assume for the moment that the interactions between these two sectors are too feeble to reach equilibrium. Such feeble interactions could arise, e.g., through mass-mixing, loop-induced effects, or suppressed tree-level interactions, and may be sufficiently small such that $Y$ will be relatively long-lived, with a lifetime as long as $\tau_{\, Y} \sim \order{1}$ second.
If kinetically decoupled, the hidden and visible sectors will each be described by distinct thermal distributions whose respective temperatures evolve differently over time. It is useful to define the ratio of the hidden and visible sector temperatures, $\xi \equiv T_h/T$. Here and throughout this paper, quantities pertaining to hidden sector dynamics are labelled with a subscript or superscript ``$h$", while those without such a label denote visible sector quantities.
For our initial conditions, we take $\xi = \xi_\text{inf}\, $, where the subscript denotes the value immediately following post-inflation reheating. At early times, significantly before the decay of $Y$, entropy is approximately conserved independently in both sectors. Hence, the evolution of $\xi$ can be tracked using the forms for the entropy densities, $s = (2 \pi^2 / 45) \, g_*(T) \, T^3$ and $s_h = (2 \pi^2 / 45) \, g_*^h(T_h) \, T_h^3\, $, where $g_*$ and $g_*^h$ correspond to the effective relativistic degrees-of-freedom in equilibrium with the visible and hidden sectors, respectively. Conservation of entropy implies that $s_h / s = s_h / s |_\text{inf}\, $, from which it follows that $\xi$ evolves as
\begin{eqnarray}
\label{eq:xi1}
\xi = \left( \frac{g_*(T)}{g_{*\, \text{inf}}} \right)^{1/3} \left( \frac{g^h_{* \, \text{inf}}}{g^h_*(T_h)} \right)^{1/3} ~ \xi_\text{inf}
~.
\end{eqnarray}
For the most part, we will be interested in $T \gg \order{100} \text{ GeV}$, for which $g_* \simeq g_{* \, \text{inf}} \approx 106.75$. For the case of $m_Y \ll T_h \ll m_X$, $g_{* \, \text{inf}}^h = c_Y g_Y + c_X g_X$ and $g_*^h = c_Y g_Y$, where $c_{X,Y} = 1 \, (7/8)$ for bosonic (fermionic) $X$, $Y$, and $g_{X,Y}$ are the number of internal degrees-of-freedom of $X$, $Y$, respectively. Under these assumptions, Eq.~(\ref{eq:xi1}) reduces to
\begin{eqnarray}
\label{eq:xi2}
\xi = \left( 1 + \frac{c_X \, g_X}{c_Y \, g_Y} \right)^{1/3} ~ \xi_\text{inf}
~.
\end{eqnarray}
This behavior is exhibited in the solid orange line of Fig.~\ref{fig:xi}, corresponding to the case of $m_Y \ll m_X$, for which $\xi/\xi_{\rm inf}$ is nearly constant when $T_h \ll m_X$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.7\textwidth]{Figures/xi_plot}
\caption{\label{fig:xi} Temperature dependence of $\xi\equiv T_h/T$, for the case that $X$ is a Dirac fermion and $Y$ is a massive neutral vector boson. RFO (NRFO) denotes that $Y$ freezes out while (non-)relativistic. Otherwise, $Y$ is assumed to be in chemical equilibrium. Cannabilism occurs indefinitely if $Y$ remains in chemical equilibrium once $T_h \lesssim m_Y$, as seen by the sharp rise in $\xi$ for the blue and dashed-red lines, corresponding to $m_Y = m_X / 10$ and $m_Y = m_X / 50$, respectively. Similarly, for the yellow-dashed line, we once again take $m_Y = m_X / 10$, but assume that once $T_h \lesssim m_Y$, $Y$ only remains in chemical equilibrium up until it freezes out at $T_h \sim m_Y / 5$, at which point $\xi \sim 1 / R$. Also, as illustrated by the dashed light-blue line, we fix $m_Y = m_X / 50$ and assume that $Y$ freezes out while still relativistic. In this case, $\xi$ is truncated by Eq.~(\ref{eq:xi2}), up until $T_h \lesssim m_Y$, at which point $\xi \sim 1 / R$. Finally, we show the limiting case of $m_Y \ll m_X$ as depicted by the solid orange line.}
\end{center}
\end{figure}
The ratio $\xi/\xi_{\rm inf}$ can evolve quite differently, however, if the conditions described above are not met; for example, if we relax the assumption that $T_h \gg m_Y$. In this case, if $Y$ is in equilibrium for temperatures $T_h \ll m_Y$, then its entropy density is given by
\begin{eqnarray}
s_h = \frac{m_Y n_Y}{T_h} = g_Y \bigg(\frac{m_Y^{5} T_h}{ 8\pi^{3}}\bigg)^{1/2} \, e^{-m_{Y}/T_h}~,
\end{eqnarray}
causing the hidden sector to enter into a state of ``\emph{cannibalism}," (see, e.g., Refs.~\cite{Carlson:1992fn,Pappadopulo:2016pkp,Kuflik:2015isi,Farina:2016llk}). In this case, conservation of hidden sector entropy, $s_h R^3$, implies that
\begin{eqnarray}
R^3 ~ T_h^{1/2} ~ e^{-m_Y / T_h} = \text{constant}~,
\end{eqnarray}
where $R$ is the scale factor. In the limit that $T_h \ll m_Y$, the variation of the exponential dominates, and the above expression can be approximated as $e^{m_Y / T_h} \propto R^3$, or $T_h \propto m_Y / \ln{R}$. As a result, $\xi$ increases rapidly as a function of the scale factor, such that
\begin{eqnarray}
\xi \propto \frac{R}{\ln{R}}
~.
\end{eqnarray}
This behavior is exhibited by the solid blue, dashed red, and dashed yellow lines in Fig.~\ref{fig:xi}, each of which depict periods of cannibalism in the hidden sector.
Alternatively, if $Y$ freezes out of chemical equilibrium while still relativistic, the value of $\xi$ will be held to that described in Eq.~(\ref{eq:xi2}) until $Y$ becomes non-relativistic, at which point $\xi \propto 1 / R$. This can be seen from the phase-space density of $Y$. Once a relativistic species has frozen out in the hidden sector, its comoving number density is conserved and, as a result, the phase space density, $f$ (or equivalently $(E-\mu)/ T_h$), is held constant,
\begin{eqnarray}
f \sim e^{-(E-\mu)/T_h} \sim dn / d^3 p \sim R^{-3} / R^{-3} = \text{constant}.
\end{eqnarray}
Imagine that $Y$ freezes out at $T^i$, $T_h^i \gg m_Y$, and $E_Y^i \approx p_Y^i \gg m_Y$, and consider later times before $Y$ becomes non-relativistic. Assuming that $E_{Y} \gg \mu_{Y}$ and using the fact that $E_{Y}/T_h$ is fixed, the temperature of $Y$ evolves as
\begin{eqnarray}
T_h = T_h^i ~ \frac{E_Y}{E_Y^i} \approx T_h^i ~ \frac{p_Y}{p_Y^i} \approx T_h^i ~ \frac{R^i}{R} \approx T_h^i ~ \frac{T}{T^i}
~,
\end{eqnarray}
and, hence, $\xi = \xi^i$. Alternatively, imagine that while $Y$ is non-relativistic, its comoving number density becomes or is already fixed. In this case, its kinetic energy scales as $E_{Y, \text{kin}} \propto 1 / R^2$, and hence so does $T_h$. From this it follows that $\xi = \xi^i ~ R^i / R = \xi^i ~ T / T^i$. Furthermore, through a similar argument as above, in the non-relativistic limit,
\begin{eqnarray}
T_h = T_h^i ~ \frac{m_Y - \mu_Y}{m_Y - \mu_Y^i}
~,
\end{eqnarray}
which gives $\mu_Y = m_Y + (\mu^i_Y - m_Y) ~ T_h / T_h^i$.
In Fig.~\ref{fig:xi}, we illustrate the behavior of $\xi$ for a number of possible scenarios. Although we have assumed in generating this figure that $X$ is a Dirac fermion and $Y$ is a neutral vector boson, the discussion in this section is more general, and applies to $X$ and $Y$ of any spin. In evaluating $\xi$, we have numerically solved the equation $s_h (\xi \, T) / s_h (\xi_\text{inf} \, T_\text{inf}) = s (T) / s (T_\text{inf})$, along with $s = (\rho + P)/T$ and $s_h = (\rho_h + P_h)/T_h$, and the general forms for energy density and pressure of a species, $i$~\cite{Kolb:1990vq}
\begin{eqnarray}
\rho_i = \frac{g_i}{2\pi^2} \int^{\infty}_{m_i} \frac{(E^2-m^2_{i})^{1/2}}{\exp[(E-\mu_{i})/T_h] \pm 1} E^2 dE ~~,~~
P_i = \frac{g_{i}}{6\pi^2} \int^{\infty}_{m_i} \frac{(E^2-m^2_{i})^{3/2}}{\exp[(E-\mu_{i})/T_h] \pm 1} dE,
\end{eqnarray}
where $\mu_{i}$ denotes the chemical potential and the $\pm1$ in the demoninators is positive in the case of fermions and negative for bosons.
\section{Hidden Sector Freeze-Out}
\label{sec:fo}
Chemical equilibrium in the hidden sector is governed by processes such as~ $\overbrace{Y ~Y \cdots}^{n} \leftrightarrow \overbrace{Y ~ Y \cdots}^{n-1}\, $, $XX \leftrightarrow YY$, and $X Y Y \leftrightarrow X Y$. When the rate of these reactions is overtaken by Hubble expansion, the corresponding comoving number densities become fixed (until the time at which $Y$ begins to decay). In this section, we review this process of chemical freeze-out for the case of a hidden sector that is thermally decoupled from the SM~\cite{Feng:2008mu,Cheung:2010gj,Sigurdson:2009uz}.
The coupled system of Boltzmann equations for the number densities of $X$ and $Y$ is given by
\begin{align}
\label{eq:boltz1}
\dot{n}_X + 3 H n_X &= - \langle \sigma v \rangle_X \, \big( n_X^2 - \frac{n_Y^2}{n_Y^{\text{eq}\, 2}} \, n_X^{\text{eq} \, 2}\big) + \cdots
\nonumber \\
\dot{n}_Y + 3 H n_Y &= + \langle \sigma v \rangle_X \, \big( n_X^2 - \frac{n_Y^2}{n_Y^{\text{eq}\, 2}} \, n_X^{\text{eq} \, 2}\big) - \Gamma_Y \, n_Y - \langle \sigma v^2 \rangle_X \, \big( n_X n_Y^2 - n_Y^\text{eq} \, n_X n_Y \big)
\nonumber \\
& ~~ - \langle \sigma v^2 \rangle_Y \, \big( n_Y^3 - n_Y^\text{eq} \, n_Y^2 \big) - \langle \sigma v^3 \rangle_Y \, \big( n_Y^4 - n_Y^{\text{eq} \, 2} \, n_Y^2 \big) + \cdots
~,
\end{align}
where $n_{X,Y}^\text{eq}$ denotes an equilibrium number density, $\langle \sigma v \rangle_X$ is the thermally averaged cross section for $XX \to YY$, and $\Gamma_Y$ is the decay rate for $Y$ into SM particles. The quantities $\langle \sigma v^2 \rangle_X$, $\langle \sigma v^2 \rangle_Y$, and $\langle \sigma v^3 \rangle_Y$ are the thermally averaged ``\emph{cross sections}" for $XYY \to XY$, $YYY \to YY$, and $YYYY \to YY$, respectively. For brevity, we have not included symmetry factors; for example, if $X$ is not self-conjugate, then $\langle \sigma v \rangle_X$ should be replaced with $\langle \sigma v \rangle_X\, /\, 2$. The ellipses denote higher order processes that are sub-dominant.
The Boltzmann equations in Eq.~(\ref{eq:boltz1}) are greatly simplified in the case of entropy conservation. This is valid at times significantly before the decay of $Y$, or in cases in which $Y$ never dominates the energy density. In particular, we will recast the above equations in terms of the yield or comoving number densities, $Y_{X,Y} \equiv n_{X,Y} / s$ (not to be confused with the species $Y$). Taking the time-derivative of $Y_{X,Y}$ gives
\begin{eqnarray}
\label{eq:sdotY1}
s \dot{Y}_{X,Y} = \dot{n}_{X,Y} - n_{X,Y} \, \frac{\dot{s}}{s}
~.
\end{eqnarray}
Conservation of visible sector entropy, $s R^3 = \text{constant}$, implies that
\begin{eqnarray}
\label{eq:sdotovers}
\frac{\dot{s}}{s} = - 3 H
~.
\end{eqnarray}
Then, substituting Eq.~(\ref{eq:sdotovers}) into Eq.~(\ref{eq:sdotY1}) gives
\begin{eqnarray}
\label{eq:sdotY2}
s \dot{Y}_{X,Y} = \dot{n}_{X,Y} + 3 H n_{X,Y}
~.
\end{eqnarray}
Invoking entropy conservation once again, $S \propto T^3 R^3 = \text{constant}$, gives $\dot{T}/T = - H$, which can be rewritten in terms of $x \equiv m_X / T$,
\begin{eqnarray}
\label{eq:xdot}
\dot{x} = H x
~.
\end{eqnarray}
Using the chain rule and Eq.~(\ref{eq:xdot}), we then have
\begin{eqnarray}
\label{eq:dotY}
\dot{Y}_{X,Y} = H \, x \, \frac{dY_{X,Y}}{dx}
~.
\end{eqnarray}
By substituting Eqs.~(\ref{eq:sdotY2}) and~(\ref{eq:dotY}) into Eq.~(\ref{eq:boltz1}), we find
\begin{align}
\label{eq:boltz3}
\frac{dY_X}{dx} &= \frac{- s \, \langle \sigma v \rangle_X}{H \, x}~ \big(Y_X^2 - \frac{Y_Y^2}{Y_Y^{\text{eq} \, 2}} ~ Y_X^{\text{eq} \, 2} \big) + \cdots
\nonumber \\
\frac{dY_Y}{dx} &= \frac{1}{H \, x}~ \bigg[ s \, \langle \sigma v \rangle_X \big(Y_X^2 - \frac{Y_Y^2}{Y_Y^{\text{eq} \, 2}} ~ Y_X^{\text{eq} \, 2} \big) - \Gamma_Y Y_Y - s^2 \langle \sigma v^2 \rangle_X \big( Y_X Y_Y^2 - Y_Y^\text{eq} Y_X Y_Y \big)
\nonumber \\
& \qquad \qquad - s^2 \langle \sigma v^2 \rangle_Y \big( Y_Y^3 - Y_Y^\text{eq} Y_Y^2 \big) - s^3 \langle \sigma v^3 \rangle_Y \big( Y_Y^4 - Y_Y^{\text{eq} \, 2} Y_Y^2 \big) + \cdots \bigg]
~.
\end{align}
The Hubble parameter, $H$, is given in terms of the visible and hidden sector energy densities
\begin{eqnarray}
H^2 = \frac{8 \pi}{3 m_\text{pl}^2} \left( \rho + \rho_h \right) = \frac{8 \pi}{3 m_\text{pl}^2} \, \frac{\pi^2}{30} \, \left( g_* \, T^4 + g_*^h \, T_h^4 \right) \equiv \frac{4 \pi^3}{45} ~ \frac{m_X^4}{m_\text{pl}^2} ~ \frac{g_*^{\text{eff}}}{x^4}
~,
\end{eqnarray}
where $m_\text{pl} = 1.22 \times 10^{19}$ GeV, and we have defined $g_*^{\text{eff}} \equiv g_* + g_*^h ~ \xi^4$.
The final abundances of $X$ and $Y$ can be found by numerically solving either Eq.~(\ref{eq:boltz1}) or~(\ref{eq:boltz3}). However, it is often the case that processes responsible for depleting the number density of $Y$ at temperatures $T_h \lesssim m_Y$ are suppressed relative to those governing the freeze-out of $X$. If there also exists the hierarchy, $m_X \gg m_Y \gg \order{100} \text{ GeV}$, it is sensible to assume that $Y$ freezes out when it is relativistic at temperatures significantly above the weak scale. Approximating $n_Y$ with the relativistic expression $n_Y \approx c_Y^\prime \zeta (3) g_Y T_h^3 / \pi^3$, where $c_Y^\prime = 1 \, (3/4)$ for bosonic (fermionic) $Y$, and $g_* = 106.75$, we have $Y_Y = Y_Y^\text{eq} \approx 0.0026 \, c_Y^\prime \, \left( g_Y + g_X \, c_X / c_Y \right) \xi_\text{inf}^3$, where we have also used Eq.~(\ref{eq:xi2}). Changing variables once again to $\Delta = Y_X - Y_X^\text{eq}$, the first line of Eq.~(\ref{eq:boltz3}) can be rewritten as
\begin{eqnarray}
\label{eq:delta}
\frac{d\Delta}{dx} = - \frac{d Y_X^\text{eq}}{dx}- f(x) \, \Delta \, \big[ \Delta + 2 \, Y_X^\text{eq} \big]
~,
\end{eqnarray}
where we have defined
\begin{eqnarray}
\label{eq:FofX}
f(x) \equiv \frac{s \, \langle \sigma v \rangle_X}{H \, x} = \sqrt{\frac{\pi}{45}} ~ \frac{g_*}{\sqrt{g_*^\text{eff}}} ~ m_X \, m_\text{pl} ~ \frac{a+6 \, \xi \, b / x}{x^2}
~,
\end{eqnarray}
and where $\sigma v_X \equiv a + b v^2$ is the cross section for $XX \to YY$ prior to thermal averaging.
It will suffice to solve Eq.~(\ref{eq:delta}) semi-analytically. To do so, first, consider its form before $X$ departs from chemical equilibrium. At this point, $Y_X$ tracks $Y_X^\text{eq}$ very closely and hence $d \Delta / d x$ is negligible, giving
\begin{eqnarray}
\label{eq:earlytimes}
\Delta = - \, \frac{dY_X^\text{eq}}{dx} ~ \frac{1}{f(x) \big[\Delta + 2 Y_X^\text{eq} \big]}
~.
\end{eqnarray}
Freeze-out occurs when $Y_X$ no longer tracks $Y_X^\text{eq}$, or in other words, when $\Delta$ is comparable to $Y_X^\text{eq}$. Specifically, freeze-out is defined by $\Delta = c \, Y_X^\text{eq}$,
where $c$ is some order one number chosen to match numerical solutions. We will take $c \approx 0.4$ for $s$-wave annihilation~\cite{Kolb:1990vq}. Assuming that $X$ freezes out when non-relativistic at $x = x_f$, Eq.~(\ref{eq:earlytimes}), along with $n_X^\text{eq} \approx g_X(m_X^2 / 2 \pi x_f)^{3/2} e^{-x_f}$ and $\Delta = c \, Y_X^\text{eq}$, then imply that
\begin{eqnarray}
x_f = \xi \, \ln{\left( \frac{c(c+2)}{4 \pi^3} \, \sqrt{\frac{45}{2}} \, \frac{g_X}{\sqrt{g_*^\text{eff}}} \, m_X \, m_\text{pl} \, \frac{\xi^{5/2} (a+6 \xi b / x_f)}{\sqrt{x_f} (1- 3 \xi / 2 x_f)} \right)}
~,
\end{eqnarray}
where $g_*^\text{eff}$ and $\xi$ are evaluated at freeze-out. In practice, the above equation may be solved numerically for $x_f$.
Now, consider the form of Eq.~(\ref{eq:delta}) after $X$ departs from chemical equilibrium. At this point, $Y_X^\text{eq}$ is negligible due to Boltzmann suppression, and hence $\frac{d \Delta}{dx} ~ \Delta^{-2} = - f(x)$. Integrating this from $x= x_f$ to $x=\infty$, and using the fact that $ \Delta (x = \infty) \ll \Delta (x = x_f)$, we find
\begin{eqnarray}
\label{eq:Ysimple}
Y_X (x=\infty)^{-1} = \int_{x_f}^\infty \, dx \, f(x) \approx \sqrt{\frac{\pi}{45}} ~ \frac{g_*}{\sqrt{g_*^\text{eff}}} ~ m_X \, m_\text{pl} ~ \frac{a+3 \, \xi \, b / x_f}{x_f}
~,
\end{eqnarray}
where $g_*$ and $g_*^\text{eff}$ are evaluated at freeze-out. Note that in Eq.~(\ref{eq:Ysimple}), we have ignored variation of $g_*^\text{eff}$ from $x=x_f$ to $x = \infty$. For $\xi_\text{inf} \gg 1$, it is possible that $g_*^\text{eff}$ varies significantly over this domain, in which case we will instead use the more general form
\begin{eqnarray}
Y_X (x=\infty)^{-1} \approx \sqrt{\frac{\pi}{45}} ~ g_* ~ m_X \, m_\text{pl} ~ \int_{x_f}^\infty \, dx ~ \frac{a+ 6 \xi b /x}{x^2 \, \sqrt{g_*^\text{eff}}}
~.
\end{eqnarray}
The relic abundance today is evaluated as $\Omega_X = m_X s_0 Y_X (x=\infty) / \rho_c$, where $s_0 = 2891.2$ cm$^{-3}$ is the visible sector entropy density today and $\rho_c = 1.05375 \times 10^{-5} \, h^2$ GeV cm$^{-3}$ is the critical energy density~\cite{Agashe:2014kda}. When Eq.~(\ref{eq:Ysimple}) applies, this leads to
\begin{eqnarray}
\label{eq:relicab}
\Omega_X h^2 = 8.5 \times 10^{-11} ~ \frac{x_f \sqrt{g_\star^\text{eff}}}{g_*} ~ \left( \frac{a+3 \xi b / x_f}{\text{GeV}^{-2}} \right)^{-1}
~.
\end{eqnarray}
This will constitute the final abundance of $X$, provided that no entropy is transferred into the visible sector. If instead the SM entropy increases by a factor $S_f / S_i$, $\Omega_X h^2$ is effectively reduced by the same factor. This is simple to see from the following argument. Imagine that the visible sector has an initial entropy of $S_i$, which is later raised to $S_f$ through some unspecified process. Before this entropy increase, $X$ has an energy density $\rho_X^i = m_X s_i Y_X$, where $s_i = S_i / R_i^3$. Expansion of the universe dilutes the energy density such that
\begin{align}
\rho_X^f &= \rho_X^i \, \frac{R_i^3}{R^3} = m_X \, Y_X \, \frac{s_i \, R_i^3}{R^3} = m_X \, Y_X \, \frac{S_i}{R^3} = m_X \, Y_X \, \frac{S_i}{R^3} \, \frac{S_f}{S_f} = \frac{m_X \, s_f \, Y_X}{S_f / S_i}
~,
\end{align}
where $s_f = S_f / R^3$. Therefore, the dark matter energy density today is $\rho_X = m_X \, s_0 \, Y_X \, / (S_f / S_i)$. Hence, $\Omega_X h^2$, as written in Eq.~(\ref{eq:relicab}), is diluted by the factor $S_f / S_i$. As we will show in the next section, the radiation coming from the late-time out-of-equilibrium decay of $Y$ naturally generates such an increase in entropy.
\section{Out-of-Equilibrium Decay}
\label{sec:decay}
In the previous section, we described the thermal freeze-out of a dark matter candidate, $X$, which resides in a sector that is highly decoupled from the SM. We now turn our attention to the lightest particle species in the hidden sector, $Y$, which is assumed to be unstable and will eventually decay into SM particles. Due to the highly decoupled nature of the hidden sector, however, we expect such decays to be highly suppressed, leading $Y$ to be long-lived. Furthermore, upon becoming non-relativistic, the energy density of $Y$ scales as $\rho_Y \propto R^{-3}$, while the visible bath instead evolves as $\rho_\text{SM} \propto R^{-4}$. As a result, $\rho_Y / \rho_\text{SM}$ scales linearly with $R$, thus making it possible for the $Y$ population to come to dominate the energy density of the early universe, and significantly reheating the SM bath upon its eventual decay. In this section, we investigate the consequences arising from this out-of-equilibrium decay, closely following the approach described in Ref.~\cite{Kolb:1990vq}.
Using the sudden-decay approximation, it is simple to work out an estimate for the reheating of the visible sector. Imagine that $Y$, which is non-relativistic, comes to dominate the energy density of the universe up until time $t=\tau_Y$, at which point it decays into SM particles which quickly thermalize with the visible bath. Using conservation of energy, the energy density of the universe immediately prior to the decay, $\rho_Y$, should equal the energy density in radiation immediately after the decay. We will denote these two snapshots in time as $t = \tau_Y - \epsilon_t$ and $t = \tau_Y + \epsilon_t$, respectively, where $\epsilon_t$ is some small positive time-scale relative to $\tau_Y$. We will also use notation such that the label ``$i$" corresponds to $t = \tau_Y-\epsilon_t$, while ``$f$" corresponds to $t = \tau_Y+\epsilon_t$. Immediately prior to decay, the Friedmann equation gives
\begin{eqnarray}
\label{eq:Hdecay}
H^2(t = \tau_Y - \epsilon_t) = \frac{4}{9 \, \tau_Y^2} = \frac{8 \pi}{3\, m_\text{pl}^2} \, \rho_Y = \frac{8 \pi}{3\, m_\text{pl}^2} \, s_i m_Y Y_Y = \frac{16 \pi^3}{135 \, m_\text{pl}^2} \, g_* T_{i}^3 m_Y Y_Y
~,
\end{eqnarray}
or equivalently,
\begin{eqnarray}
\label{eq:TRi}
T_{i}^3 = \frac{15 m_\text{pl}^2}{4 \pi^3 g_* m_Y Y_Y \tau_Y^2}
~.
\end{eqnarray}
Solving for $\rho_Y$ in terms of $\tau_Y$ in Eq.~(\ref{eq:Hdecay}) and enforcing energy conservation leads to
\begin{eqnarray}
\rho_Y = \frac{m_\text{pl}^2}{6 \pi \tau_Y^2} = \frac{\pi^2}{30} \, g_* \, T_{f}^4
~,
\end{eqnarray}
or equivalently for the reheat temperature,
\begin{eqnarray}
\label{eq:TRf}
T_{f}^3 = \left( \frac{5 m_\text{pl}^2}{g_* \pi^3 \tau_Y^2} \right)^{3/4}
~.
\end{eqnarray}
The increase in SM entropy, in the sudden-decay approximation, is then found by taking the ratio of $T_{f}^3/ T_{i}^3$,
\begin{eqnarray}
\label{eq:suddendecay}
\frac{S_f}{S_i} = \frac{T_{f}^3}{T_{i}^3} \approx 2.1 ~ g_*^{1/4} ~ \frac{m_Y Y_Y \tau_Y^{1/2}}{m_\text{pl}^{1/2}}
~.
\end{eqnarray}
We will now derive the change in entropy more systematically, no longer relying on the sudden-decay approximation. From the definition of $\tau_Y$, $N_Y \propto e^{-t / \tau_Y}$, we obtain the differential equation,
\begin{eqnarray}
\frac{d(R^3 n_Y)}{dt} = - \frac{1}{\tau_Y} \, R^3 n_Y
~,
\end{eqnarray}
which when expanded and divided by $R^3$ gives
\begin{eqnarray}
\dot{n}_Y + 3 H n_Y = - n_Y / \tau_Y
~.
\end{eqnarray}
Since $Y$ is assumed to be non-relativistic, $\rho_Y = m_Y n_Y$, and the above equation is equivalent to
\begin{eqnarray}
\label{eq:doe1}
\dot{\rho}_Y + 3 H \rho_Y = - \rho_Y / \tau_Y
~,
\end{eqnarray}
which has the general solution
\begin{eqnarray}
\label{eq:rho1}
\rho_Y (R) = \rho_Y (R_i) \left( \frac{R_i}{R} \right)^3 e^{-(t-t_i) / \tau_Y}
~.
\end{eqnarray}
Now, imagine that as $Y$ decays, the energy deposited is rapidly converted into relativistic thermalized particles. It follows from the second law of thermodynamics that
\begin{eqnarray}
dS = \frac{dQ}{T} = \frac{-d(R^3 \rho_Y)}{T} = \frac{-R^3}{T} \, dt \, (\dot{\rho}_Y + 3 H \rho_Y) = \frac{R^3 \rho_Y}{T} \, (dt / \tau_Y)
~,
\end{eqnarray}
where in the last equality we have used Eq.~(\ref{eq:doe1}). Solving $S = (2\pi^2 / 45) g_* T^3 R^3$ for $T$ and substituting into the equation above,
\begin{align}
\label{eq:doe2}
S^{\, 1/3} \, \dot{S} = S^{\, 1/3} \,\frac{R^3}{T} \, \frac{\rho_Y}{\tau_Y} = \left( \frac{2 \pi^2}{45} g_* \right)^{1/3} \, \frac{R^4 \rho_Y}{\tau_Y} = \left( \frac{2 \pi^2}{45} g_* \right)^{1/3} \, \frac{R \, R_i^3}{\tau_Y} ~ \rho_Y (R_i) ~ e^{-(t-t_i) / \tau_Y}
~,
\end{align}
where in the last equality we used Eq.~(\ref{eq:rho1}). A formal solution to Eq.~(\ref{eq:doe2}) is
\begin{align}
\label{eq:S}
S^{4/3} &= S_i^{4/3} + \frac{4}{3} \, \rho_Y (R_i) R_i^4 ~~\tau_Y^{-1} \int_{t_i}^t \, d t^\prime \, \left( \frac{2 \pi^2}{45} g_* \right)^{1/3} \, \frac{R(t^\prime)}{R_i} \, e^{-(t^\prime-t_i) / \tau_Y}
\nonumber \\
&\equiv S_i^{4/3} + \frac{4}{3} \, \rho_Y (R_i) R_i^4 ~ I
~.
\end{align}
To simplify Eq.~(\ref{eq:S}), we take note of two important relations involving the energy density of SM radiation, $\rho_R$, and the visible sector entropy, $s\, $:
\begin{eqnarray}
\label{eq:rhoR1}
s \, T = \frac{4}{3} \, \rho_R
~,
\end{eqnarray}
and
\begin{eqnarray}
\label{eq:rhoR2}
\rho_R = \frac{3}{4} \left( \frac{45}{2 \pi^2 g_*} \right)^{1/3} S^{4/3} \, R^{-4}
~.
\end{eqnarray}
We then have
\begin{align}
\label{eq:rhoR3}
\rho_Y (R_i) R_i^4 S_i^{-4/3} &= m_Y R_i Y_Y S_i^{-1/3}
\nonumber \\
&= m_Y R_i Y_Y S_i^{-1/3} ~ \times ~ \frac{4 \rho_R (R_i) / 3}{S_i T_i / R_i^3}
\nonumber \\
&= m_Y R_i Y_Y S_i^{-1/3} ~ \times ~ \frac{1}{S_i T_i / R_i^3} ~ \times ~ \left( \frac{45}{2 \pi^2 g_*(T_i)} \right)^{1/3} ( S_i^{4/3} / R_i^4 )
\nonumber \\
&= \frac{m_Y Y_Y}{T_i} \left( \frac{45}{2 \pi^2 g_*(T_i)} \right)^{1/3}
~,
\end{align}
where we used Eq.~(\ref{eq:rhoR1}) and Eq.~(\ref{eq:rhoR2}) in the second and third lines, respectively. Taking $t_i << \tau_Y$, $t_f \gg \tau_Y$ and substituting Eq.~(\ref{eq:rhoR3}) into Eq.~(\ref{eq:S}) then implies
\begin{eqnarray}
\label{eq:S2}
\frac{S_f}{S_i} = \left[ 1 + \frac{4}{3} \left( \frac{45}{2 \pi^2 g_*(T_i)} \right)^{1/3} \, \frac{m_Y Y_Y}{T_i} \, I \right]^{3/4}
~,
\end{eqnarray}
where now $I$ is defined to be
\begin{eqnarray}
\label{eq:I_int}
I \equiv \tau_Y^{-1} \int_0^\infty dt \left( \frac{2 \pi^2}{45} g_* \right)^{1/3} \frac{R(t)}{R(0)} e^{-t/\tau_Y}
~.
\end{eqnarray}
In the limit that $Y$ dominates the energy density before its decay, a numerical form of $I$ is sufficient and Eq.~(\ref{eq:S2}) can be approximated as
\begin{eqnarray}
\frac{S_f}{S_i} \approx 1.83 ~ \langle g_*^{1/3} \rangle^{3/4} \, \frac{m_Y Y_Y \tau_Y^{1/2}}{m_\text{pl}^{1/2}}
~,
\end{eqnarray}
where the brackets indicate time-averaging over the decay~\cite{Kolb:1990vq}. Note that the difference between this numerical solution and that found using the sudden-decay approximation is at most $\order{1}$. In practice, throughout this study, we will numerically solve the system of equations, consisting of Eqs.~(\ref{eq:doe2}), (\ref{eq:S2}), and~(\ref{eq:I_int}), and the Friedmann equation,
\begin{eqnarray}
H^2 = \frac{8 \pi}{3 m_\text{pl}^2} \left( \rho_X + \rho_Y + \rho_R \right)
~,
\end{eqnarray}
where $\rho_Y$ is determined from Eq.~(\ref{eq:rho1}), $\rho_R = \pi^2 g_* T^4 / 30$, $\rho_X \propto R^{-3}$, and $S = (2 \pi^2 / 45) g_* T^3 R^3$.
\section{The Effective Number of Neutrino Species}
\label{sec:Neff}
In models with a decoupled sector, there may be additional relativistic particles present during or after Big Bang Nucleosynthesis (BBN), with the potential to impact the measured expansion history of the universe. In this section, we briefly discuss this possibility within the context of the class of models under consideration here.
In generality, the effective number of neutrino species, $N_\text{eff}$, is defined in terms of the energy density of the universe, or equivalently in terms of $g_*^\text{eff}$. Allowing the neutrino temperature to be different than that of the SM plasma, we have
\begin{eqnarray}
g_*^\text{eff} = g_*^{\text{SM} - \nu} + g_*^\nu \, \xi_\nu^4 + g_*^h \, \xi_h^4 \equiv g_*^{\text{SM} - \nu} + \frac{7}{8} \times 2 \times N_\text{eff} \times ( \xi_\nu^0 )^4
~,
\end{eqnarray}
where $\text{SM} - \nu$ denotes the SM \emph{omitting} the three species of neutrinos, $\xi_\nu \equiv T_\nu / T$, $\xi_h \equiv T_h / T$~(we have restored the $h$ subscript for clarity), and $\xi_\nu^0$ is $T_\nu / T$ in the SM when neutrino reheating from electron-positron annihilations is neglected, i.e., $\xi_\nu^0 = (4 / 11)^{1/3} \approx 0.714$ for $T \lesssim m_e$ and $\xi_\nu^0 = 1$ for $T \gtrsim m_e$.
For $n_\nu$ flavors of neutrinos, we have
\begin{eqnarray}
\frac{7}{8} \times 2 \times N_\text{eff} \times ( \xi_\nu^0 )^4 = \frac{7}{8} \times 2 \times n_\nu \times \xi_\nu^4 ~ + ~ g_*^h \, \xi_h^4
~.
\end{eqnarray}
Solving for $N_\text{eff}$ yields
\begin{eqnarray}
N_\text{eff} = n_\nu \left( \frac{\xi_\nu}{\xi_\nu^0} \right)^4 + \frac{4}{7} ~ g_*^h ~ \left( \frac{\xi_h}{\xi_\nu^0} \right)^4
~.
\end{eqnarray}
In the SM, $g_*^h = 0$, $n_\nu = 3$, and when $T \lesssim m_e$, $\xi_\nu$ is slightly larger than $\xi_\nu^0$, so that $N_\text{eff} \approx 3.046$.
Consider the case of 3 neutrino flavors and standard cosmology ($\xi_\nu = \xi_\nu^0$) with an additional decoupled hidden sector. At early times, around BBN, for example, $T_\nu = T_\gamma$ and so the analogous calculation yields
\begin{eqnarray}
N_\text{eff} \approx 3 + \frac{4}{7} ~ g_*^h ~ \xi_h^4 ~~~ \text{(BBN)}
~.
\end{eqnarray}
Alternatively, after neutrino decoupling, for instance at recombination,
\begin{eqnarray}
N_\text{eff} \approx 3.046 + \frac{4}{7} ~ \left(\frac{11}{4}\right)^{4/3} g_*^h ~ \xi_h^4 ~~~ \text{(CMB)}
~,
\end{eqnarray}
in agreement with Ref.~\cite{Feng:2008mu}.
Alternatively, we can also consider contributions to $N_{\rm eff}$ that arise from the decay products of the long-lived particle species, $Y$. More specifically, consider a scenario in which $Y$ has a finite branching fraction, $B_a$, to a light and decoupled state, $a$. For as long as this population of decay products remains relativistic, they will continue to contribute to $N_{\rm eff}$ (after which they will behave like matter). This will be the case so long as $T \gg T_{f} m_a / f_a m_Y$, where $T_{f}$ is the temperature of the universe following the decays of $Y$ and $f_a$ is fraction of $m_Y$ that goes into an individual $a$ (for example, for $Y \rightarrow aa$, $f_a=0.5$).
Including the contribution from these decay products, the effective number of neutrino species is given by
\begin{equation}
N_{\rm eff} \approx 3.046 + \frac{43}{7} \bigg(\frac{B_a}{1-B_a}\bigg) \bigg(\frac{g_{\star}(T_{\nu, {\rm dec}})}{g_{\star} (T_{f})}\bigg)^{1/3},
\end{equation}
where $g_{\star}(T_{\nu, {\rm dec}}) \approx 10.75$ and $T_{\nu, {\rm dec}}$ is the temperature at neutrino decoupling. Comparing this expression to constraints on $N_{\rm eff}$ from measurements of the CMB ($N_{\rm eff} = 3.15 \pm 0.23$)~\cite{Ade:2015xua}, we conclude that $B_a \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 0.1 \, [g_{\star}(T_{f})/100]^{1/3}$. Next generation CMB experiments are anticipated to improve significantly upon this constraint~\cite{Dodelson:2013pln,Feng:2014uja,Wu:2014hta}.
\section{Models of Dark Matter in a Decoupled Sector}
\label{sec:models}
The scenario described above is generic and can be applied to several different classes of models. If the hidden sector is composed of SM gauge singlets, it is natural for it to be very weakly coupled to the visible bath. However, in order to facilitate the decay of the metastable state, $Y$, into SM particles, some portal between the two sectors must be introduced. At the renormalizable level, such decays can proceed through the following three operators: $B_{\mu \nu}$, $|H|^2$, and $HL$, known as the vector, Higgs, and lepton portals, respectively.
In this section, we will investigate models utilizing each of these portals in turn, focusing on the phenomenology outlined in Secs.~\ref{sec:thermo}-\ref{sec:decay}. Each model contains unique features and introduces complications beyond the simplest possible realization. We will proceed in order of increasing complexity. In particular, in Sec.~\ref{sec:vector} we explore the vector portal, which serves as a simple and concrete manifestation of the generic scenario described in the previous sections. In Sec.~\ref{sec:higgs}, we proceed to the Higgs portal, which necessitates a careful treatment of the freeze-out process, due to the fact that the singlet-like scalar mediator may remain in chemical equilibrium while non-relativistic. Sec.~\ref{sec:lepton} presents the lepton portal model, whose ultraviolet structure incorporates a heavy right-handed neutrino which may have potential implications for leptogenesis.
\subsection{Vector Portal}
\label{sec:vector}
In the vector portal scenario~\cite{Pospelov:2007mp,Krolikowski:2008qa}, a new spontaneously broken $U(1)_X$ gauge symmetry is introduced, along with a corresponding massive neutral gauge boson, $Z^\prime$. As our dark matter candidate, we add to this model a complex scalar, $\phi$, which has a unit charge under $U(1)_X$ and couples to the $Z^\prime$ through the gauge coupling $g_{Z^\prime}$. $\phi$ does not acquire a vacuum expectation value (VEV) and is independent of the breaking of $U(1)_X$. Alternatively, one could also consider dark matter in the form of a Dirac fermion, as we explored previously in Ref.~\cite{Berlin:2016vnh}. If there exist particles charged under $U(1)_X \times U(1)_Y$, a small degree of kinetic mixing between the $Z^\prime$ and the SM hypercharge gauge boson can be radiatively generated. The hidden sector Lagrangian then contains the following interactions
\begin{eqnarray}
\mathcal{L} \supset - \frac{\epsilon}{2} \, B^{\mu \nu} Z_{\mu \nu}^\prime + i g_{Z^\prime} ~ Z_\mu^\prime (\phi^* \partial_\mu \phi - \phi \partial_\mu \phi^*) + g_{Z^\prime}^2 ~ Z^{\prime \mu} Z_\mu^\prime ~ |\phi|^2
~.
\end{eqnarray}
There may also exist direct couplings between $\phi$ and the SM Higgs through the interaction, $|\phi|^2 |H|^2$. However, since the hidden and visible sectors are thermally decoupled, this interaction must be significantly suppressed. In this section, we take the kinetic mixing parameter, $\epsilon$, to be the only relevant coupling between the two sectors.
In the limit that $m_{Z^\prime} \gg m_Z$, mixing through $\epsilon$ generates an effective interaction between the $Z^\prime$ and SM fermions,
\begin{eqnarray}
\mathcal{L} \supset - \epsilon \, g_1 \, \sum\limits_{f} \, Y_f ~ Z_\mu^\prime ~ \bar{f} \gamma^\mu f + \order{m_Z / m_{Z^\prime}}
~,
\end{eqnarray}
where $g_1$ is the hypercharge gauge coupling and $Y_f$ is the hypercharge of the SM fermion, $f$~\cite{Hoenig:2014dsa}. This allows the $Z^\prime$ to decay to SM fermions with a width given by
\begin{eqnarray}
\label{eq:gammaZp}
\Gamma_{Z^\prime} = \frac{5}{3} \, \alpha_1 \, \epsilon^2 \, m_{Z^\prime} + \order{m_Z / m_{Z^\prime}}
~.
\end{eqnarray}
Similarly, $\phi$ couples to the SM $Z$ through the terms
\begin{eqnarray}
- \mathcal{L} \supset \frac{i g_{Z^\prime} \epsilon s_w m_Z^2}{m_{Z^\prime}^2} ~ Z_\mu (\phi^* \partial_\mu \phi - \phi \partial_\mu \phi^*) + \frac{2 g_{Z^\prime}^2 \epsilon s_w m_Z^2}{m_{Z^\prime}^2} ~ Z^{\prime \mu} Z_\mu ~ |\phi|^2 + \order{\epsilon^2}
~,
\end{eqnarray}
where $s_w$ is sine of the Weinberg angle. Through $Z$ and $Z^\prime$ exchange, these interactions allow $\phi$ to scatter off protons in underground direct detection experiments, leading to a spin-independent cross section given by
\begin{eqnarray}
\sigma_p = 4 ~ g_1^2 \, c_w^4 \, \alpha_X \, \epsilon^2 ~ \frac{\mu^2}{m_{Z^\prime}^4}
~,
\end{eqnarray}
where $\alpha_X \equiv g_{Z^\prime}^2 / 4 \pi$, $\mu$ is the reduced mass of the proton and $\phi$, and $c_w$ is cosine of the Weinberg angle.
Before any large increase in entropy occurs from $Z^\prime$ decays, $\phi$ freezes out through the process $\phi \bar{\phi} \to Z^\prime Z^\prime$, with an initial abundance given by Eq.~(\ref{eq:relicab}). In particular, $\frac{1}{2} \, \sigma v (\phi \bar{\phi} \to Z^\prime Z^\prime) = a + b \, v^2$, where
\begin{align}
a &= \frac{\pi \alpha_X^2}{2 m_\phi^2} ~ \sqrt{1-r^2} ~ \bigg( 2 + \frac{r^4}{\left(2 - r^2\right)^2} \bigg) \approx \frac{\pi \alpha_X^2}{m_\phi^2} + \mathcal{O}(r^2)~,
\nonumber \\
\nonumber \\
b &= \frac{\pi \alpha_X^2}{48 m_\phi^2} ~ \bigg(\frac{27 r^{10}-254 r^8+900 r^6-1528 r^4+1312 r^2-448}{\left(1-r^2\right)^{1/2} \left(2 - r^2\right)^4} \bigg) \approx -\frac{7 \pi \alpha_X^2}{12 m_\phi^2} + \mathcal{O}(r^2)
~,
\end{align}
where $v$ is the relative $\phi$ velocity, and $r \equiv m_{Z^\prime} / m_\phi \, $.
\begin{figure}[t]
\begin{center}
\includegraphics[width=1\textwidth]{Figures/Zp_FO_Feynman}
\caption{\label{fig:Zp_Feynman} Representative Feynman diagrams for processes that could potentially maintain the chemical equilibrium of the $Z^\prime$ population for $T_h \lesssim m_{Z^\prime}$.}
\end{center}
\end{figure}
The dilution of the $\phi$ density from late-time $Z^\prime$ decays directly follows the discussion in Sec.~\ref{sec:decay}. As seen from Eq.~(\ref{eq:suddendecay}), the required inputs are $\tau_{Z^\prime}$ and $Y_{Z^\prime}$, the former of which is given by the inverse of Eq.~(\ref{eq:gammaZp}). Various processes may keep $Z^\prime$ in chemical equilibrium (with respect to the rest of the hidden sector) as the hidden sector cools. Representative diagrams that deplete the $Z^\prime$ number density are shown in Fig.~\ref{fig:Zp_Feynman}. In the discussion preceding Eq.~(\ref{eq:delta}), we noted that solving the Boltzmann equation is immensely simplified if the $Z^\prime$ departs from chemical equilibrium while it is still relativistic. Alternatively, in order for the $Z^\prime$ to remain in chemical equilibrium while non-relativistic, the rate, $\Gamma$, for a process that depletes the $Z^\prime$ number density must overcome Hubble expansion at or before the critical temperature, $T_h = m_{Z^\prime}$. Therefore, the quantity of interest is $\Gamma / H$, as evaluated at $T_h = m_{Z^\prime}$. If $\Gamma / H \ll 1$, it is safe to assume that the $Z^\prime$ population freezes out while still relativistic.
We first consider the process $Z^\prime Z^\prime Z^\prime Z^\prime \to Z^\prime Z^\prime$ mediated by a $\phi$ loop.
Gauge invariance and dimensional analysis suggests that the rate for this process will scale as follows:
\begin{eqnarray}
\Gamma (Z^\prime Z^\prime Z^\prime Z^\prime \to Z^\prime Z^\prime) \sim n_{Z^\prime}^3 \, \frac{\alpha_X^6 m_{Z^\prime}^8}{m_\phi^{16}}
~.
\end{eqnarray}
Similarly, the rates for the tree-level processes $Z^\prime Z^\prime \phi \phi \to \phi \phi$ and $Z^\prime Z^\prime Z^\prime \phi \to Z^\prime \phi$ can be written as
\begin{eqnarray}
\Gamma (Z^\prime Z^\prime \phi \phi \to \phi \phi) \sim n_{Z^\prime} n_\phi^2 \, \frac{\alpha_X^4}{m_\phi^8} ~ ,\quad \Gamma (Z^\prime Z^\prime Z^\prime \phi \to Z^\prime \phi) \sim n_{Z^\prime}^2 n_\phi \, \frac{\alpha_X^4}{m_\phi^8}
~.
\end{eqnarray}
In Fig.~\ref{fig:Zp_FO}, we plot the quantity $\Gamma / H $, evaluated at $T_h = m_{Z^\prime}$, as a function of $\alpha_X$ for each of these three interactions. As illustrated in this figure, for $\alpha_X \lesssim 0.5$ and $m_\phi / m_{Z^\prime} \gtrsim 10$, $\Gamma / H \lesssim 10^{-1}$, and the $Z^\prime$ population is not maintained in chemical equilibrium. For the remainder of our analysis, we will therefore assume that the $Z^\prime$ freezes out while it is relativistic. Following the discussion above Eq.~(\ref{eq:delta}), this implies that the $Z^\prime$ comoving number density is $Y_{Z^\prime} \approx 0.013 ~ \xi_\text{inf}^3 \, $. Assuming that $Z^\prime$ freezes out while relativistic allows us to focus solely on the first line of Eq.~(\ref{eq:boltz3}). Despite this simplification, the term proportional to $( Y_{Z^\prime}/ Y_{Z^\prime}^\text{eq} )^2$ deviates from unity when $T_h \lesssim m_{Z^\prime}$ and the equilibrium comoving number density becomes Boltzmann suppressed. By numerically solving the Boltzmann equation, we find that the inclusion of this effect alters our results by $\order{5 \%}$ for $m_{\phi} / m_{Z^\prime} \approx 5$ and by only $\order{1 \%}$ for $m_{\phi} / m_{Z^\prime} \approx 20$, relative to that obtained using the semi-analytic approximation.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.497\textwidth]{Figures/Zp_FO_M20}
\includegraphics[width=0.497\textwidth]{Figures/Zp_FO_M10}
\caption{\label{fig:Zp_FO} $\Gamma / H$ evaluated at $T_h = m_{Z^\prime}$ as a function of the coupling $\alpha_X$, for the processes $Z^\prime Z^\prime Z^\prime Z^\prime \to Z^\prime Z^\prime$ (red), $Z^\prime Z^\prime \phi \phi \to \phi \phi$ (orange), and $Z^\prime Z^\prime Z^\prime \phi \to Z^\prime \phi$ (blue), assuming that the hidden and visible sectors are thermally decoupled. We have taken $m_\phi = 1$ PeV, and $m_{Z^\prime}= 50 \, (100)$ TeV in the left (right) panels. The width of the bands corresponds to $\xi_{\text{inf}}=0.1 - 10$. Larger values of $\xi_\text{inf}$ lead to larger rates relative to that of Hubble expansion. For $m_{Z^\prime}= 50$ TeV, corresponding to the left panel above, $\Gamma / H \ll 1$ and hence the $Z^\prime$ population departs from chemical equilibrium while still relativistic. For smaller ratios of $m_\phi / m_{Z^\prime}$, corresponding to the right panel, processes that deplete the $Z^\prime$ number density allow the $Z^\prime$ to remain in chemical equilibrium while non-relativistic for $\alpha_X \gtrsim 0.5$.}
\end{center}
\end{figure}
Throughout, we have assumed that $\epsilon$ is sufficiently small such that $\phi$ and $Z^\prime$ are thermally decoupled from the SM bath. We now revisit this assumption, and consider scattering processes that could potentially equilibrate the two sectors for sufficiently large values of $\epsilon$. The dominant interactions are $Z^\prime f \to (\gamma / g) f$ and $( \gamma / g) f \to Z^\prime f$, where $f$ is some SM fermion. At leading order in $m_f / m_{Z^\prime}$, we find
\begin{align}
\hspace{-0.1cm}\sigma v (Z^\prime f \to \gamma f) &\approx \frac{\alpha_\text{em} \, Q_f^2 \, \left(g_v^2+g_a^2\right)}{6 \left(s-m_{Z^{\prime}}^2\right)^2} \, \Bigg[ s+ 6 m_{Z^{\prime}}^2 -
\frac{7 m_{Z^{\prime}}^4 }{ s} + 2 \left( s - 2 m_{Z^{\prime}}^2 + \frac{2 m_{Z^{\prime}}^4 }{ s} \right) \log{\frac{s (1-m_{Z^{\prime}}^2/s )^2}{m_f^2}} \Bigg]~,
\end{align}
while for the reverse process
\begin{align}
\sigma v (\gamma f \to Z^\prime f) &= \frac{3(s-m_{Z^{\prime}}^2)^2}{2s^2} \sigma v (Z^\prime f \to \gamma f)~.
\end{align}
Here $\sqrt{s} \approx 4 T$ is the center of mass energy\footnote{For thermal distributions of bosons and fermions, the average energy per particle is approximately $\rho/n = 2.70~T$ and $3.15~T$ respectively, so
for fermion-boson scattering, the angle averaged $s = (p_1 + p_2)^2 \to 2 E_1 E_2 \approx (4 T)^2$.}, $Q_f$ is the electric charge of $f$, and $g_{v,a} = -\epsilon \, g_1 \, (Y_{f_R} \pm Y_{f_L}) \, $, where $Y_{f_{L/R}}$ is the hypercharge of SM fermion $f_{L/R}$. For processes involving gluons instead of photons, one simply replaces the quantity $\alpha_\text{em} \, Q_f^2$ with $4 \, \alpha_s$. If $n_f \, \sigma v \lesssim H$ at $T = m_\phi / x_f$, then the hidden and visible sectors do not equilibrate before the freeze-out of the dark matter abundance.
For our numerical results, we include contributions from all SM fermions, $f$, and all gauge interactions involving gluons and electroweak gauge bosons. We safely neglect contributions from pure gauge boson external states ({\it i.e.} $Z^\prime \gamma \to ff$) since, for $T \gg v$, these are highly subdominant to the total contribution from $Z^\prime f$ initiated rates and the corresponding reverse processes on account of $g_*(T) \simeq 100$.
\begin{figure}[h!]
\begin{center} \hspace{-0.7cm}
\includegraphics[width=3.2in]{Figures/Vector_M20_xi1}
\includegraphics[width=3.2in]{Figures/Vector_M10_xi1} \hspace{-0.7cm} \\
\hspace{-0.7cm}
\includegraphics[width=3.2in]{Figures/Vector_M20_xi0p1}
\includegraphics[width=3.2in]{Figures/Vector_M20_xi10}
\caption{\label{fig:vector} Selected regions of parameter space in the vector portal model. The black contours ($\Omega_\phi h^2=0.12$) correspond to regions in the $m_\phi - \epsilon$ plane where the final $\phi$ abundance matches the observed dark matter density for three different values of the $Z^\prime$ coupling, $\alpha_X = 0.03$, 0.1, and 0.3. For larger values of $\epsilon$, and for the same three values of $\alpha_X$, the red regions (LUX) are currently ruled out by direct detection constraints from LUX and/or PandaX~\cite{Akerib:2015rjg,Tan:2016zwf}. On the other hand, in the shaded blue region (BBN) the $Z^\prime$ decays reheat the SM plasma to a temperature below 10 MeV, in potential tension with the successful predictions of BBN. In and above the brown region ($\rho_{Z^\prime} < \rho_\text{SM}$), the $Z^\prime$ population never comes to dominate the energy density of the universe, while in and above the yellow region ($\tau_{Z^\prime} < H_\text{FO}^{-1}$) $Z^\prime$ dominates the energy density but decays before the freeze-out of $\phi$. The shaded orange region (KE) corresponds to values of $\epsilon$ for which kinetic equilibrium between the hidden and visible sectors is established. In the top-left and top-right panels, we have fixed $\xi_\text{inf}=1$ and $m_{\phi} / m_{Z^\prime}= 20$ and 10, respectively. The bottom-left and bottom-right panels illustrate the effect of varying $\xi_\text{inf}$ while fixing $m_{\phi} / m_{Z^\prime}= 20$.}
\end{center}
\end{figure}
We illustrate the phenomenology of this model in Fig.~\ref{fig:vector}, as a function of the dark matter mass, $m_\phi$, and kinetic mixing parameter, $\epsilon$, for various values of $\alpha_X$, $m_\phi / m_{Z^\prime}$, and $\xi_\text{inf}$. The abundance of dark matter in the vector portal scenario diverges from the typical WIMP estimate for sufficiently small values of $\epsilon$. In this case, $\phi$ can be as heavy as $\order{10}$ PeV before running afoul of constraints from Big Bang Nucleosynthesis (BBN). Along the black contours, the final abundance of $\phi$ matches the observed dark matter density, $\Omega_\phi h^2 \sim 0.12$. For longer lifetimes of the $Z^\prime$ (smaller values of $\epsilon$), $Z^\prime$ can come to dominate the energy density of the universe, corresponding to the parameter space below the brown shaded region. In this case, the entropy dump from the $Z^\prime$ decay significantly dilutes the $\phi$ abundance, allowing for large values of $m_\phi$ which would otherwise be inconsistent with the observed density of dark matter. For lifetimes longer than $\order{1}$ second, however, the reheating temperature after the $Z'$ decay is significantly less than 10 MeV, leading to potential tension with the successful predictions of BBN (shaded blue). For a sufficient degree of kinetic mixing, the hidden sector and SM bath are maintained in kinetic equilibrium in the early universe (shaded orange), and may potentially fall within the reach of direct detection experiments, such as LUX~\cite{Akerib:2016vxi,Akerib:2015rjg} and PandaX~\cite{Tan:2016zwf} (shaded red). We also highlight the parameter space in which the $Z^\prime$ population dominates the energy density and decays before the freeze-out of $\phi$ (shaded yellow). In this case, the hidden and visible sector entropies are no longer conserved during the freeze-out of $\phi$, invalidating the assumption that led to the derivation of Eqs.~(\ref{eq:boltz3}) and (\ref{eq:relicab}). Since, in most cases, the $Z^\prime$ abundance does not dominate the energy density of the universe when it decays before the freeze-out of $\phi$, we expect the resulting correction to be small.
For $\xi_\text{inf} \ll 1$, as considered in the bottom-left panel of Fig.~\ref{fig:vector}, the hidden sector is only modestly populated (relative to the SM) after inflation. As a result, the effects of the $Z^\prime$ decay are reduced, and only for much longer lifetimes does the $Z^\prime$ population come to dominate the energy density of the universe. Regardless, compared to the standard thermal WIMP calculation, thermal decoupling in this scenario results in the underproduction of the hidden sector and thus allows for larger dark matter masses, without exceeding the observed dark matter density.
Although we have focused on scalar dark matter in this section, fermonic dark matter is also a viable possibility within the context of vector portal scenarios~\cite{Berlin:2016vnh}. Qualitatively, very similar conclusions are reached in these two cases. In particular, Fig.~2 of Ref.~\cite{Berlin:2016vnh} can be directly compared to the results shown in Fig.~\ref{fig:vector} of this paper.
\subsection{Higgs Portal}
\label{sec:higgs}
In the Higgs portal scenario, a real scalar singlet, $\phi$, couples to the SM Higgs at tree-level~\cite{Pospelov:2007mp,Burgess:2000yq,Davoudiasl:2004be,Bird:2006jd,Kim:2006af,Finkbeiner:2007kk,D'Eramo:2007ga,Barger:2007im,SungCheon:2007nw,MarchRussell:2008yu,McDonald:2008up,Pospelov:2011yp,Piazza:2010ye,Kouvaris:2014uoa,Kainulainen:2015sva,Krnjaic:2015mbs,Batell:2012mj,Tenkanen:2016jic}. Working in the basis where $\phi$ does not acquire a VEV, the general scalar potential is given by
\begin{eqnarray}
\label{eq:FullScalarPotential}
V (\phi, H) = - \mu^2 |H|^2 + \lambda |H|^4 + \frac{\delta_1}{2} \, |H|^2 \, \phi + \frac{\delta_2}{2} \, |H|^2 \, \phi^2 - \frac{\delta_1 v^2}{4} \, \phi + \frac{\kappa_2}{2} \, \phi^2 + \frac{\kappa_3}{3!} \, \phi^3 + \frac{\kappa_4}{4!} \, \phi^4
~,
\end{eqnarray}
where $v \equiv \mu/\sqrt{\lambda} \approx 246 $ GeV and the tadpole coefficient is chosen to prevent $\phi$ from getting a VEV.
After electroweak symmetry breaking, mass mixing between the SM Higgs, $h$, and $\phi$ is controlled solely by the dimensionful parameter $\delta_1$. In the limit that $m_\phi \approx \sqrt{\kappa_2} \gg m_h$, the mixing angle, $\epsilon$, is approximated as
\begin{eqnarray}
\epsilon = -\frac{v \, \delta_1}{2 m_\phi^2} + \order{m_h / m_\phi}~,
\end{eqnarray}
so we will therefore focus on the phenomenology that arises from the following simplified scalar potential
\begin{eqnarray}
\label{eq:ScalarPotential}
V (\phi,H) = V_\text{SM} (H) + \frac{\delta_1}{2} \, |H|^2 \, \phi - \frac{\delta_1 \mu^2}{4} \, \phi + \frac{\kappa_2}{2} \, \phi^2
~,
\end{eqnarray}
where $V_\text{SM} (H)$ is the SM Higgs potential. Note that it is technically natural for $\delta_1$ to be very small. In particular, the quantum correction to $\delta_1$ via a SM Higgs loop scales as $\Delta \delta_1 \sim \delta_1 \, \lambda \, \log{(\Lambda_\text{UV}/m_{\phi})}/16\pi^2 \, $, where $\Lambda_\text{UV}$ is the high-energy cutoff of the theory. Since we will be most interested in regions of small mixing, $\epsilon \ll 1$, LHC constraints on SM Higgs couplings are negligible~\cite{Aad:2015gba,Khachatryan:2014jba}.
In this model, we assume that $\phi$ is odd under an approximate $Z_2$ symmetry, which is softly broken only by the super-renormalizable portal coupling, $\delta_1$, in the simplified potential of Eq.~(\ref{eq:ScalarPotential}). For sufficiently long $\phi$ lifetimes, corresponding to small values of $\epsilon$, we typically need $\delta_1$ to be in the neighborhood of
\begin{eqnarray}
\delta_1 \simeq {\rm GeV} \left(\frac{\epsilon}{10^{-10}}\right)\left(\frac{m_\phi}{\rm PeV}\right)^2~,~
\end{eqnarray}
in the general vicinity of the weak scale. In the full potential of Eq.~(\ref{eq:FullScalarPotential}), there is an additional $Z_2$ breaking coupling, $\kappa_3$, which renormalizes the value of $\delta_1$ at the one loop level. To ensure that this does not significantly increase the $\phi$ width, this correction must not exceed $ \sim \delta_1$, which implies
\begin{eqnarray}
\kappa_3 \lesssim \frac{16 \pi^2 }{ \log \frac{\Lambda_{\rm UV}}{m_\phi} } \frac{\delta_1}{\delta_2}~.~
\end{eqnarray}
As our dark matter candidate, we introduce a singlet Majorana fermion, $\chi \, $, which couples to $\phi$ through the interactions
\begin{eqnarray}
\mathcal{L} \supset ~ \phi ~ \bar{\chi} (\lambda_s + \lambda_p \, i \gamma^5) \chi
~.
\end{eqnarray}
After EWSB, mass mixing leads to the substitution $\phi \to \phi - \epsilon ~ h$, which generates an effective dark matter coupling to the SM Higgs, allowing direct detection experiments to constrain the quantity $\lambda_s \, $. In particular, $\chi$ scatters off nucleons through SM Higgs exchange with a spin-independent cross section of
\begin{eqnarray}
\sigma_\text{SI} \approx 2 \times 10^{-46} \text{ cm}^2 \times \left( \frac{\epsilon}{0.1} \right)^2 ~ \left( \frac{\lambda_s}{0.1} \right)^2
~.
\end{eqnarray}
Similarly, $\phi$ couples directly to the SM through
\begin{eqnarray}
\mathcal{L} \supset - \frac{\delta_1}{4} \, \phi \, h^2 - \, \frac{\epsilon}{v}\sum\limits_f m_f ~ \phi \, \bar{f} f + \frac{2 \epsilon}{v} \left( m_W^2 \, W^{+ \mu} W_\mu^- + \frac{1}{2} m_Z^2 \, Z^\mu Z_\mu \right) \left( \phi + \frac{1}{v} \, h \, \phi \right) + \order{\epsilon^2}
~.
\end{eqnarray}
At leading order in $\epsilon$, $\phi$ decays to pairs of Higgs bosons, SM fermions, and gauge bosons, with partial widths given by
\begin{align}
\Gamma (\phi \to h h) & = \frac{m_\phi^3 \epsilon^2}{32 \pi v^2} + \order{m_h / m_\phi}
\nonumber \\
\Gamma (\phi \to f \bar{f}) &= \frac{n_c m_\phi m_f^2 \epsilon^2}{8 \pi v^2} + \order{m_f / m_\phi}
\nonumber \\
\Gamma (\phi \to V V ) &=
\frac{m_\phi^3 \epsilon^2}{16 \pi (1 + \delta_{V Z}) v^2} + \order{m_V / m_\phi}
~,
\end{align}
where $n_c$ is the number of colors of the SM fermion, $f$, and $\delta_{V Z} = 1 (0)$ for $Z$ ($W^\pm$) final states.
%
As seen from the limiting forms above, $\Gamma (\phi \to h h) / \Gamma (\phi \to f \bar{f}) \sim (m_\phi / m_f)^2$ and $\Gamma (\phi \to h h) / \Gamma (\phi \to V V) \sim (m_\phi / m_V)^4$. Since we will focus here on cases in which $m_\phi \gg 100$ GeV, the dominant decay channel is to pairs of SM Higgs bosons.
Prior to the decay of $\phi$, $\chi$ freezes out through its annihilations within the hidden sector, $\chi \chi \to \phi \phi$. In particular, $\sigma v (\chi \chi \to \phi \phi) = a + b \, v^2$, where
\begin{align}
a &= \frac{2\sqrt{1-r^2} \lambda_p^2 \lambda_s^2}{m_\chi^2 \pi \left(r^2-2\right)^2} \approx \frac{\lambda_p^2 \lambda_s^2}{2 \pi m_\chi^2} + \mathcal{O}(r^2)
\nonumber \\
\nonumber \\
b &= \frac{-2 \left(r^2-1\right)^3 \lambda_p^4+3 \left(r^6-8 r^4+20 r^2-12\right) \lambda_s^2 \lambda_p^2+2 \left(-2 r^6+10 r^4-17 r^2+9\right) \lambda_s^4}{12 m_\chi^2 \pi \sqrt{1-r^2} \left(r^2-2\right)^4}
\nonumber \\
&\approx \frac{\lambda_p^4-18 \lambda_p^2 \lambda_s^2+9 \lambda_s^4}{96 \pi m_\chi^2} + \mathcal{O}(r^2)
~,
\end{align}
where $v$ is the relative $\chi$ velocity, and $r \equiv m_{\phi} / m_\chi \, $. If $\phi$ departs from chemical equilibrium while relativistic, the initial abundance of $\chi$ is given by Eq.~(\ref{eq:relicab}).
For the case of the vector portal, as discussed in Sec.~\ref{sec:vector}, $Z^\prime$ depleting processes were suppressed. As we shall see below, however, $\phi$ is able to maintain chemical equilibrium in the Higgs portal case when $T_h \lesssim m_\phi$ for sufficiently large values of $\lambda_s$ or $\lambda_p\, $. This is directly tied to the fact that these interactions involve scalars and correspond to operators of lower dimension. Similar to as in the previous subsection, we consider the process $\phi \phi \phi \to \phi \phi$ mediated by a $\chi$ loop.
Following the approach described in Appendix~\ref{sec:app1}, we find by explicit calculation the rate for this process in the non-relativistic limit:
\begin{eqnarray}
\Gamma (\phi \phi \phi \to \phi \phi) = n_\phi^2 ~ \frac{784 \sqrt{5}}{3 \pi^5} \frac{\lambda^{10}}{m_\chi^2 m_\phi^3}
~,
\end{eqnarray}
where, for simplicity, we have taken $m_\chi \gg m_\phi$ and $\lambda_s = \lambda_p = \lambda\, $. In Fig.~\ref{fig:Phi_FO}, we show $\Gamma (\phi \phi \phi \to \phi \phi) / H$ evaluated at $T_h = m_\phi$ as a function of $\lambda_s = \lambda_p\, $. It is apparent that if $\lambda_{s,p} \gtrsim \order{0.1}$, then $\Gamma / H \gtrsim 1$, indicating that $\phi$ freezes out while non-relativistic. In this case, instead of using Eq.~(\ref{eq:relicab}), we numerically solve the coupled Boltzmann system in Eq.~(\ref{eq:boltz3}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.497\textwidth]{Figures/Phi_FO}
\caption{\label{fig:Phi_FO} $\Gamma / H$ evaluated at $T_h = m_\phi$, as a function of $\lambda_s = \lambda_p$, for the process $\phi \phi \phi \to \phi \phi$, assuming that the hidden and visible sectors are thermally decoupled. We have taken $m_\chi = 1$ PeV throughout and $m_\phi= 200$ TeV (blue) and 10 TeV (red). The width of the bands corresponds to $\xi_{\text{inf}}=0.1 - 10$. Larger values of $\xi_\text{inf}$ lead to larger rates relative to Hubble expansion. $\Gamma / H \lesssim 1$ only for $\lambda_{s,p} \lesssim \order{0.1}$. For larger values of $\lambda_{s,p}$, $\phi$ departs chemical equilibrium after becoming non-relativistic, and one must numerically solve the coupled Boltzmann equations for $\chi$ and $\phi$.}
\end{center}
\end{figure}
If $\chi$ and $\phi$ are to remain thermally decoupled from the SM during dark matter freeze-out, the scattering processes, $\phi h \leftrightarrow t \bar{t}$ and $\phi t \leftrightarrow h t$, must not exceed the rate of Hubble expansion before $T = m_\chi / x_f$. At temperatures significantly above 100 GeV, the SM Higgs VEV, $v$, is suppressed, and hence, we will consider processes that do not depend explicitly on electroweak symmetry breaking, such as the $\phi-h-h$ cubic term in Eq.~(\ref{eq:ScalarPotential}) which is controlled by the dimensionful coupling, $\delta_1$. In the limit $m_\phi \gg m_t, m_h$, we find that the scattering processes are approximated as
\begin{align}
\sigma v (\phi \, h \to t \, \bar{t}) &\approx \frac{3 \delta_1^2 m_t^2}{32 \pi v^2 \, s (s - m_\phi^2)}~,
\nonumber \\
\sigma v (t \, \bar{t} \to \phi \, h) &\approx \frac{\delta_1^2 m_t^2 (s-m_\phi^2)}{128 \pi v^2 \, s^3}~,
\nonumber \\
\sigma v (\phi \, t \to h \, t) &\approx \frac{\delta_1^2 \, m_t^2}{64 \pi \, v^2 \, (s-m_\phi^2)} ~ \Bigg\{ ~ \frac{\left(4-x_h^2\right) s}{m_\phi^4 + x_h^2 \, s \, \left(s-m_\phi^2\right)} + \frac{1}{s-m_\phi^2} ~ \log{ \left[ \frac{s (s-m_\phi^2)^2}{m_t^2 \left(m_\phi^4 + x_h^2 s (s-m_\phi^2)\right)} \right] } ~ \Bigg\}~,
\nonumber \\
\sigma v (h \, t \to \phi \, t) &\approx \frac{\delta_1^2 \, m_t^2}{64 \pi \, v^2 \, s} ~ \Bigg\{ ~ \frac{\left(4-x_h^2\right) (s-m_\phi^2)}{m_\phi^4 + x_h^2 \, s \, \left(s-m_\phi^2\right)} + \frac{1}{s} ~ \log{ \left[ \frac{s (s-m_\phi^2)^2}{m_t^2 \left(m_\phi^4 + x_h^2 s (s-m_\phi^2)\right)} \right] } ~ \Bigg\}
~,
\end{align}
where $x_h \equiv m_h / m_t$, and the ``$v$" on the right-hand side denotes the SM Higgs VEV. If $n_{h,t} \, \sigma v \lesssim H$ at $T = m_\chi / x_f$, then the hidden sector and the SM do not equilibrate before the freeze-out of the dark matter abundance.
\begin{figure}[h!]
\begin{center} \hspace{-0.7cm}
\includegraphics[width=3.2in]{Figures/Higgs_M20_xi1}
\includegraphics[width=3.2in]{Figures/Higgs_M10_xi1} \\
\hspace{-0.7cm}
\includegraphics[width=3.2in]{Figures/Higgs_M20_xi0p1}
\includegraphics[width=3.2in]{Figures/Higgs_M20_xi10}
\caption{\label{fig:higgs} Selected regions of parameter space in the Higgs portal model. The black contours ($\Omega_\phi h^2=0.12$) correspond to regions in the $m_\phi - \epsilon$ plane where the final $\phi$ abundance matches the observed dark matter density for three different values of the $\chi-\phi$ couplings, $\lambda_s = \lambda_p = 0.25$, 0.5, and 1. The cross section for dark matter-nucleon scattering is beyond the reach of LUX or PandaX throughout the parameter space shown. In the shaded blue region (BBN) the $\phi$ decays reheat the SM plasma to a temperature below 10 MeV, in potential tension with the successful predictions of BBN. In and above the brown region ($\rho_\phi < \rho_\text{SM}$), the $\phi$ population never comes to dominate the energy density of the universe, while in and above the yellow region ($\tau_\phi < H_\text{FO}^{-1}$) $\phi$ dominates the energy density but decays before the freeze-out of $\chi$. The shaded orange region (KE) corresponds to values of $\epsilon$ for which kinetic equilibrium between the hidden and visible sectors is established. In the top-left and top-right panels, we have fixed $\xi_\text{inf}=1$ and $m_{\chi} / m_{\phi}= 20$ and 10, respectively. The bottom-left and bottom-right panels illustrate the effect of varying $\xi_\text{inf}$ while fixing $m_{\chi} / m_{\phi}= 20$. The jagged features depicted in some of these curves are the result of kinematic thresholds for $\phi$ decay, which predominantly proceed to heavy SM states.}
\end{center}
\end{figure}
Similar to as in the previous section, Fig.~\ref{fig:higgs} illustrates the phenomenology of this model as a function of the dark matter mass, $m_\chi$, and singlet-SM Higgs mixing parameter, $\epsilon$, for representative values of the quantities $m_\chi / m_\phi$, $\xi_\text{inf}$ and $\lambda_{s,p}$. For simplicity, we consider the case that $\lambda_s = \lambda_p$. As discussed above, for $\lambda_{s,p} \gtrsim \order{0.1}$, the abundances of $\chi$ and $\phi$ (prior to the decay of $\phi$) are calculated by numerically solving the coupled Boltzmann equations, Eq.~(\ref{eq:boltz3}), incorporating the dominant processes ($\chi \chi \to \phi \phi$ and $\phi \phi \phi \to \phi \phi$) that are responsible for the depletion of both species. Similar to as in Sec.~\ref{sec:vector}, for sufficiently suppressed values of the singlet-SM Higgs mixing parameter, $\epsilon$, $\phi$ is long-lived and comes to dominate the energy density of the universe, diluting the relic abundance of $\chi$ upon its decay. However, compared to the vector portal scenario, this effect is suppressed, largely due to the enhanced strength of the process $\phi \phi \phi \to \phi \phi$. In particular, although larger values of $\lambda_{s,p}$ deplete the initial freeze-out abundance of $\chi$ through the annihilations $\chi \chi \to \phi \phi$, such couplings also enhance the $3 \to 2$ self-annihilation for $\phi$, effectively depleting the comoving number density, $Y_\phi$, and softening the dilution from its decay, as seen from Eq.~(\ref{eq:suddendecay}). As a result, for $\lambda_{s,p} \sim \order{1}$, $\Omega_\chi h^2$ matches the observed dark matter abundance without running afoul of constraints from BBN only for $m_\chi \lesssim \order{100}$ TeV, when $\xi_\text{inf} = 1$ and $m_\chi / m_\phi = 20$.
\subsection{Lepton Portal}
\label{sec:lepton}
The gauge singlet operator, $LH$, allows for the simple construction of a model that links the hidden and visible sectors through the lepton portal~\cite{Pospelov:2007mp,Bai:2014osa}. This same operator is often invoked in seesaw models as an explanation for the smallness of the SM neutrino masses~\cite{Minkowski:1977sc,Yanagida:1979as,Mohapatra:1979ia,GellMann:1980vs,Schechter:1980gr}. For realistic models of neutrino masses and mixing angles, there must be at least two right-handed SM singlet neutrinos, $N_{1,2}$, with Yukawa couplings to the SM lepton and Higgs doublets. As a result, models of neutrino masses often involve adding several new parameters to the SM Lagrangian, most of which are irrelevant to the dark matter phenomenology. Therefore, we will choose to focus on a simplified model involving only a single sterile neutrino, $N$, which couples to a single lepton doublet, $L$, where $L$ is one of the SM leptons, $L_e, L_\mu, L_\tau$~\cite{Batell:2016zod}. Additionally, as our dark matter candidate, we will add a SM singlet Weyl fermion, $\chi$, and a real scalar, $\phi$, which will allow $\chi$ to annihilate through the process $\chi \chi \to N N$.
The relevant terms in the simplified Lagrangian take the form,
\begin{eqnarray}
- \mathcal{L} \supset y_\nu \, N \, L \, H + \frac{1}{2} M_N \, N^2 + \lambda \, \phi \, \chi \, N + \text{h.c.}
~,
\end{eqnarray}
where 2-component Weyl and $SU(2)_L$ indices are implied. For generality, and in light of the necessity of CP violation for leptogenesis, we will allow for $y_\nu$ and $M_N$ to be complex, but for simplicity take $\lambda$ to be real. In particular, the phases are parameterized as
\begin{eqnarray}
y_\nu = |y_\nu| \, e^{i \phi_\nu} ~,\quad M_N = |M_N| \, e^{i \phi_N}
~.
\end{eqnarray}
Although only one of these two phases is physical, we will allow for the presence of both explicitly in our calculation. We will also assume that $m_\phi > m_\chi + |M_N|$ so that the $\phi$ decays promptly through $\phi \to \chi N$ and hence does not repopulate the dark matter, $\chi$, out of equilibrium.
After electroweak symmetry breaking, the neutrino mass matrix is given by
\begin{eqnarray}
\label{eq:NeutrinoMassMatrix}
- \mathcal{L} \supset \frac{1}{2} \begin{pmatrix} \nu & N \end{pmatrix} \begin{pmatrix} 0 & y_\nu v / \sqrt{2} \\ y_\nu v / \sqrt{2} & M_N \end{pmatrix} \begin{pmatrix} \nu \\ N \end{pmatrix} + \text{h.c.}
~.
\end{eqnarray}
The physical masses are given by the square roots of the eigenvalues of $M^\star M$, where $M$ is the mass matrix in Eq.~(\ref{eq:NeutrinoMassMatrix}). In the limit that $|M_N| \gg |y_\nu| v$, the physical masses are
\begin{eqnarray}
m_{\nu_{_\text{SM}}} \approx \frac{ |y_\nu|^2 v^2 }{2 |M_N|} ~,\quad m_{\nu_s} \approx | M_N |
~.
\end{eqnarray}
The mass eigenstate basis is defined by,
\begin{eqnarray}
\nu \approx - e^{i (\phi_N/2 - \phi_\nu)} \left( i ~ \nu_{_\text{SM}} - \epsilon ~ \nu_s \right) ~ , \quad N \approx e^{- i \phi_N/2 } \left( \nu_s + i \, \epsilon ~ \nu_{_\text{SM}} \right)
~,
\end{eqnarray}
where
\begin{eqnarray}
\epsilon \equiv \frac{|y_\nu| \, v}{\sqrt{2} \, |M_N|} \approx \left(\frac{m_{\nu_{_\text{SM}}}}{m_{\nu_s}}\right)^{1/2}
~,
\end{eqnarray}
and $\nu_{_\text{SM}}$ and $\nu_s$ are predominantly SM-like and singlet-like, respectively.
Now, let us rewrite the relevant Lagrangian interactions (to leading order in $\epsilon$) in 4-component notation, taking into account all of the necessary field redefinitions to a basis in which $ \nu_{_\text{SM}}$, $\nu_s$, and $\chi$ are now Majorana spinors. We find
\begin{align}
\label{eq:lepton_interactions}
\mathcal{L} &\supset - \frac{|y_\nu|}{\sqrt{2}} ~ h ~ \bar{\nu}_{_\text{SM}} i \gamma^5 \nu_s - \lambda ~ \phi ~ \bar{\nu}_s \left( \cos{\frac{\phi_N}{2}} + \sin{\frac{\phi_N}{2}} ~ i \gamma^5 \right) \chi - \epsilon ~ \lambda ~ \phi ~ \bar{\nu}_{_\text{SM}} \left( \sin{\frac{\phi_N}{2}} - \cos{\frac{\phi_N}{2}} ~ i \gamma^5 \right) \chi
\nonumber \\
& + \frac{\epsilon \, g_2}{2 c_w} ~ Z_\mu ~ \bar{\nu}_{_\text{SM}} i \gamma^\mu \nu_s
+ \frac{\epsilon \, g_2}{2 \sqrt{2}} ~ \left[ e^{i (\phi_\nu - \phi_N / 2)} ~ W_\mu^+ ~ \bar{\nu}_s \gamma^\mu (1- \gamma^5) \ell + \text{h.c.} \right]
~.
\end{align}
For the moment, we will ignore aspects relevant to leptogenesis, e.g., CP violation, so that the phases $\phi_{\nu,N}$ are set to zero, and the Lagrangian takes a more simplified form
\begin{align}
\mathcal{L} &\supset - \frac{|y_\nu|}{\sqrt{2}} ~ h ~ \bar{\nu}_{_\text{SM}} i \gamma^5 \nu_s - \lambda ~ \phi ~ \bar{\nu}_s \, \chi + \epsilon ~ \lambda ~ \phi ~ \bar{\nu}_{_\text{SM}} i \gamma^5 \chi
\nonumber \\
& + \frac{\epsilon \, g_2}{2 c_w} ~ Z_\mu ~ \bar{\nu}_{_\text{SM}} i \gamma^\mu \nu_s
+ \frac{\epsilon \, g_2}{2 \sqrt{2}} ~ \left[ W_\mu^+ ~ \bar{\nu}_s \gamma^\mu (1- \gamma^5) \ell + \text{h.c.} \right]
~.
\end{align}
Before the decay of $\nu_s$, $\chi$ freezes out via $\chi \chi \to \nu_s \nu_s$ through the $t$-channel exchange of $\phi$ with an initial abundance that is dictated by $\sigma v (\chi \chi \to \nu_s \nu_s) = a + b v^2$, where
\begin{align}
a &= \frac{\lambda ^2 m_\chi^2 (r+1)^2 \sqrt{1-r^2}}{16 \pi \left(m_\chi^2 \left(r^2-1\right)-m_\phi^2\right)^2}
\nonumber \\
&\approx \frac{\lambda ^2 m_\chi^2}{16 \pi \left(m_\chi^2+m_\phi^2\right)^2} + \mathcal{O}(r^2)~,
\nonumber \\
\nonumber \\
b &= \frac{\lambda ^2 m_\chi^2 (r+1)^{3/2} \left(m_\chi^4 \left(r^2-1\right)^2 (r (23 r-8)+4)-2 m_\chi^2 m_\phi^2 (r-1) (r+1) (r (23 r-24)-8)+m_\phi^4 (r (23 r-40)+20)\right)}{384 \pi \sqrt{1-r} \left(m_\phi^2-m_\chi^2 \left(r^2-1\right)\right)^4}
\nonumber \\
&\approx \frac{\lambda ^2 m_\chi^2 \left(m_\chi^4-4 m_\chi^2 m_\phi^2+5 m_\phi^4\right)}{96 \pi \left(m_\chi^2+m_\phi^2\right)^4} + \mathcal{O}(r^2)
~,
\end{align}
$v$ is the relative $\chi$ velocity, and $r \equiv m_{\nu_s} / m_\chi$. If $\nu_s$ departs from chemical equilibrium while still relativistic, the initial abundance of $\chi$ is well approximated by the semi-analytic form in Eq.~(\ref{eq:relicab}).
The interactions in Eq.~(\ref{eq:lepton_interactions}) allow $\nu_s$ to decay to electroweak Higgs/gauge bosons and SM leptons. To leading order in $m_h / m_{\nu_s}$, the corresponding rates are given by
\begin{align}
\Gamma (\nu_s \to h ~ \nu_{_\text{SM}}) &\approx \Gamma (\nu_s \to Z ~ \nu_{_\text{SM}}) \approx \Gamma (\nu_s \to W^\pm \ell^\pm) \approx \frac{ \epsilon^2 m_{\nu_s}^3}{16 \pi v^2}
~.
\end{align}
Hence, in the case that $m_{\nu_s} \gg m_h$, the total width is approximated as
\begin{eqnarray}
\Gamma_{\nu_s} \approx \frac{ 3 \epsilon^2 m_{\nu_s}^3}{16 \pi v^2}
~.
\end{eqnarray}
The elastic scattering of $\chi$ with nuclei proceeds through loops involving $\phi$ and $\nu_s$ at leading order, resulting in rates that are well below the irreducible neutrino background.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.497\textwidth]{Figures/Nu_FO}
\caption{\label{fig:Nu_FO} $\Gamma / H$ evaluated at $T_h = m_{\nu_s}$ as a function of the coupling $\lambda$ for the processes $\nu_s \nu_s \nu_s \nu_s \to \nu_s \nu_s$ (red), and $\phi \nu_s \nu_s \to \chi \nu_s$ (blue), assuming that the hidden and visible sectors are thermally decoupled. We have taken $m_\phi = 2$ PeV, and $m_{\chi}= 1$ PeV, and $m_{\nu_s} = 100$ TeV. The width of the bands corresponds to $\xi_{\text{inf}}=0.1 - 10$. Larger values of $\xi_\text{inf}$ lead to larger rates relative to Hubble expansion. For $\lambda \lesssim \order{1}$, $\Gamma / H \ll 1$ and hence $\nu_s$ departs chemical equilibrium while it is still relativistic.}
\end{center}
\end{figure}
After the freeze-out of $\chi$, $\nu_s$ remains in chemical equilibrium until the rate for processes that deplete its number density falls below that of Hubble expansion. The process $\nu_s \nu_s \nu_s \nu_s \to \nu_s \nu_s$ is mediated by a $\chi-\phi$ loop, similar to the left-most diagram of Fig.~\ref{fig:Zp_Feynman}. Assuming that $m_\phi \gg m_\chi$, the rate for this process scales as
\begin{eqnarray}
\Gamma (\nu_s \nu_s \nu_s \nu_s \to \nu_s \nu_s) \sim n_{\nu_s}^3 ~ \frac{\lambda^{12} \, m_{\nu_s}^2}{m_\phi^{10}}
~.
\end{eqnarray}
Similarly, $\phi \nu_s \nu_s \to \chi \nu_s$ may proceed, e.g., through an $s$-channel $\phi$, analogous to the center and right-most diagrams of Fig.~\ref{fig:Zp_Feynman}. By dimensional analysis, we estimate the corresponding rate as
\begin{eqnarray}
\Gamma (\phi \nu_s \nu_s \to \chi \nu_s) = n_{\nu_s} n_\phi ~ \frac{\lambda^6}{m_\phi^5}
~.
\end{eqnarray}
In Fig.~\ref{fig:Nu_FO}, we plot the quantity $\Gamma / H$, evaluated at $T_h = m_{\nu_s}$, as a function of $\lambda$ for these two processes. As illustrated in this figure, for $\lambda \lesssim \order{1}$, $\Gamma / H \lesssim 10^{-2}$, and $\nu_s$ is not maintained in chemical equilibrium. For the remainder of our analysis, we will therefore assume that $\nu_s$ freezes out while relativistic. Following the discussion above Eq.~(\ref{eq:delta}), this implies that the $\nu_s$ comoving number density is fixed as $Y_{\nu_s} \approx 0.01 ~ \xi_\text{inf}^3 \, $ and, as in Sec.~\ref{sec:vector}, this justifies calculating the initial freeze-out abundance of $\chi$ through the use of the semi-analytic form in Eq.~(\ref{eq:relicab}).
\begin{figure}[!t]
\begin{center}
\includegraphics[width=3.2in]{Figures/Lepton_M20_xi1}
\includegraphics[width=3.2in]{Figures/Lepton_M10_xi1}
\includegraphics[width=3.2in]{Figures/Lepton_M20_xi0p1}
\includegraphics[width=3.2in]{Figures/Lepton_M20_xi10}
\caption{\label{fig:lepton}
Selected regions of parameter space in the lepton portal model. The black contours ($\Omega_\chi h^2 = 0.12$) correspond to regions in the $m_\chi - \epsilon$ plane where the final $\chi$ abundance matches the observed dark matter energy density for three different values of the $\chi-\nu_s$ coupling, $\lambda = 0.1$, 0.5, and 1. The cross section for dark matter-nucleon scattering is beyond the reach of LUX or PandaX throughout the parameter space shown. In the shaded blue region (BBN) the $\nu_s$ decays reheat the SM plasma to a temperature below 10 MeV, in potential tension with the successful predictions of BBN. In and above the brown region ($\rho_{\nu_s} < \rho_\text{SM}$), the $\nu_s$ population never comes to dominate the energy density of the universe, while in and above the yellow region ($\tau_{\nu_s} < H_\text{FO}^{-1}$) $\nu_s$ dominates the energy density but decays before the freeze-out of $\chi$. The shaded orange region (KE) corresponds to values of $\epsilon$ for which kinetic equilibrium between the hidden and visible sectors is established. The green dotted line ($T(\tau_{\nu_s}) > 100$ GeV) corresponds to the boundary of the parameter space in which the temperature of the SM plasma is above 100 GeV at the time of $\nu_s$ decay, representing a favorable condition for leptogenesis. In the top-left and top-right panels, we have fixed $\xi_\text{inf}=1$ and $m_{\chi} / m_{\nu_s}= 20$ and 10, respectively, while the bottom-left and bottom-right panels illustrate the effect of varying $\xi_\text{inf}$ while fixing $m_{\chi} / m_{\nu_s}= 20$. The mass ratio, $m_{\phi}/m_{\chi} = 1.1$, is fixed throughout all panels.
}
\end{center}
\end{figure}
The hidden sector will remain thermally decoupled from the SM during the dark matter freeze-out process if the scattering processes $\nu_s \nu_{_\text{SM}} \leftrightarrow t \bar{t}$ and $\nu_s t \leftrightarrow \nu_{_\text{SM}} t$ do not exceed Hubble expansion before $T = m_\chi / x_f$. For temperatures significantly above the electroweak scale, $\nu_s - \nu_{_\text{SM}}$ mixing is suppressed, and hence, we will refrain from considering processes that depend explicitly on such mixing. In the limit that $m_{\nu_s} \gg m_t$, we find that the scattering cross sections for $\nu_s \nu_{_\text{SM}} \leftrightarrow t \bar{t}$ and $\nu_s t \leftrightarrow \nu_{_\text{SM}} t$ are approximated as
\begin{align}
\sigma v (\nu_s \nu_{_\text{SM}} \to t \bar{t}) \approx \frac{3 |y_\nu|^2 m_t^2 s}{32 \pi v^2 (s-m_h^2)^2}~,~~
\sigma v (t \bar{t} \to \nu_s \nu_{_\text{SM}}) \approx \frac{|y_\nu|^2 m_t^2 (s-m_{\nu_s}^2)^2}{32 \pi v^2 s (s-m_h^2)^2}~, \nonumber
\end{align}
\begin{align}
\hspace{-0.8cm}\sigma v (\nu_s t \to \nu_{_\text{SM}} t) &\approx \frac{m_t^2 |y_\nu|^2}{32 \pi v^2 (s-m_{\nu_s}^2)} \Bigg\{ \frac{m_{\nu_s}^4-2 m_{\nu_s}^2 s \left(x_h^2-2\right)+s^2 x_h^2}{m_{\nu_s}^4+s x_h^2 \left(s-m_{\nu_s}^2\right)} + \frac{m_{\nu_s}^2}{s-m_{\nu_s}^2} \log \left[ \frac{s \left(s-m_{\nu_s}^2\right)^2}{m_t^2 \left(m_{\nu_s}^4+s x_h^2 (s-m_{\nu_s}^2)\right)} \right] \Bigg\}~, \!\!\!\!\!\!\!\!\!\!\!\!\! \nonumber
\end{align}
\begin{align}
\sigma v ( \nu_{_\text{SM}} t \to \nu_s t) \approx \frac{(s-m_{\nu_s}^2)^2}{s^2} \sigma v (\nu_s t \to \nu_{_\text{SM}} t)~,
\end{align}
where $x_h \equiv m_h / m_t$, and the ``$v$" on the right-hand side is the SM Higgs VEV. If $n_{ \nu_{_\text{SM}}} \, \sigma v \lesssim H$ at $T = m_\chi / x_f$, then the hidden sector and the SM do not equilibrate before the freeze-out of the dark matter abundance.
In Fig.~\ref{fig:lepton}, we plot some of the phenomenological features of this model as a function of $m_\chi$ and $\epsilon$, fixing $m_\phi = 1.1 ~ m_\chi$ and for various choices of $m_\chi / m_{\nu_S}$ and $\xi_\text{inf}$. In most respects, this resembles the results shown in the previous two subsections. In this case, however, we also show as a green dotted line the boundary of the region in which the temperature of the SM plasma is reheated to above 100 GeV through $\nu_s$ decays. Above this approximate temperature, electroweak sphalerons are in thermal equilibrium with the SM plasma, and are thus potentially able to convert a lepton-antilepton asymmetry (such as one generated through $\nu_s$ decays) into a baryon asymmetry.
\section{Summary and Conclusions}
\label{sec:conclusion}
Motivated by the increasingly stringent constraints that have been placed in recent years on dark matter in the form of WIMPs, we consider in this study dark matter candidates that are part of a larger sector with no sizable interactions with the Standard Model. Such a hidden sector could very plausibly be populated after inflation, and will undergo a thermodynamic history that is independent of the visible sector (which contains the Standard Model). As the hidden sector cools, its lightest particles will become non-relativistic and may come to dominate the energy density of the universe. When these particles ultimately decay, they reheat the universe and dilute the abundances of any previously frozen-out relics, including that of the dark matter itself. This sequence of events is a generic consequence of the hidden sector's highly decoupled nature, and phenomenology of this type can be found within a wide range of theoretical frameworks.
In this study, we have described in some detail the thermodynamics and cosmological evolution of models that feature a highly decoupled hidden sector. After presenting a more general discussion, we have considered three simple, representative models, in which the hidden and visible sectors interact through what are known as the vector, Higgs, and lepton portals. In each of these cases, we identify significant parameter space in which the decoupled cosmological history considered here is viably realized. Furthermore, due to the dilution that results from the decays of long-lived hidden sector particles, the dark matter can be as heavy as $\sim$1-100 PeV in these scenarios, without generating a dark matter abundance in excess of the measured value.
\bigskip
\bigskip
\textbf{Acknowledgments.} AB is supported by the Kavli Institute for cosmological physics at the University of Chicago through grant NSF PHY-1125897. Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE-AC02-07CH11359 with the US Department of Energy.
\begin{appendix}
\section{$3 \to 2$ Scattering Rates}
\label{sec:app1}
In this appendix, we will derive a general form for $3 \to 2$ scattering rates, $\sigma v^2 (\, X_1 \, X_2 \, X_3 \to X_1^\prime \, X_2 ^\prime \, )$. Let $| i \rangle$ and $|f \rangle$ abbreviate the initial and final states, respectively. The relevant matrix element is related to the amplitude, $i \mathcal{M}$, by
\begin{eqnarray}
\langle f | i \rangle = (2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~ i \mathcal{M}
~,
\end{eqnarray}
where $k_\text{in,out}^\mu$ is the total incoming or outgoing 4-momenta. The probability, $P$, for this process to occur is then given by
\begin{eqnarray}
P = \frac{|\langle f | i \rangle|^2}{\langle f | f \rangle \langle i | i \rangle}
~.
\end{eqnarray}
Imagining that the scattering occurs in a spacetime box of spatial volume $V$ and time $T$, the numerator above can then be written as
\begin{align}
|\langle f | i \rangle|^2 &= \left[ (2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) \right]^2 ~ |\mathcal{M}|^2
\nonumber \\
&= (2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~ (2 \pi)^4 ~ \delta^4 (0) ~ |\mathcal{M}|^2
\nonumber \\
&= (2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~ V \, T ~ |\mathcal{M}|^2
~.
\end{align}
The single-particle states are normalized as
\begin{eqnarray}
\langle k | k \rangle = (2 \pi)^3 ~ 2 E_k ~ \delta^3(0) = 2 ~ E_k ~ V
~.
\end{eqnarray}
Therefore, $\dot{P} \equiv P / T$ can be expressed as
\begin{eqnarray}
\dot{P} = \frac{ (2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~ |\mathcal{M}|^2}{8 E_1 E_2 E_3 \times 4 E_1^\prime E_2^\prime \times V^4 }
~.
\end{eqnarray}
Summing over the outgoing momenta results in a factor of $V \times d^3 k_i^\prime / (2 \pi)^3$ for each outgoing particle. This gives
\begin{eqnarray}
\dot{P} = \frac{(2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~ |\mathcal{M}|^2}{8 E_1 E_2 E_3 \times V^2} ~~ \widetilde{dk_1^\prime} ~ \widetilde{dk_2^\prime}
~,
\end{eqnarray}
where $\widetilde{dk_i} \equiv d^3k_i / (2 \pi)^3 2 E_i$~. ``$\sigma$" is defined such that $\sigma \equiv \dot{P} / \text{flux}$~. Therefore,
\begin{eqnarray}
\sigma = \frac{\dot{P}}{(v_1 / V)(v_2 / V)}
~,
\end{eqnarray}
and hence
\begin{eqnarray}
\sigma v^2 = \dot{P} ~ V^2 = \frac{(2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~ |\mathcal{M}|^2}{8 E_1 E_2 E_3} ~~ \widetilde{dk_1^\prime} ~ \widetilde{dk_2^\prime}
~.
\end{eqnarray}
In the non-relativistic limit, $E_i \approx m_i$,
\begin{eqnarray}
\sigma v^2 = \frac{1}{8 m_1 m_2 m_3} \int d\text{LIPS}_2 ~ |\mathcal{M}|^2
~,
\end{eqnarray}
where $d\text{LIPS}_2 \equiv (2 \pi)^4 ~ \delta^4 (k_\text{in} - k_\text{out}) ~~ \widetilde{dk_1^\prime} ~ \widetilde{dk_2^\prime}$~. Also in the non-relativistic limit, the phase space integral is evaluated to be
\begin{align}
d\text{LIPS}_2 = \frac{d \cos{\theta}}{16 \pi ~ (m_1 + m_2 + m_3)^2} ~ \Big[ (m_1 + m_2 + m_3)^4 - 2 (m_1 + m_2 + m_3)^2 ~ (m_{1^\prime}^2 + m_{2^\prime}^2 ) + (m_{1^\prime}^2 - m_{2^\prime}^2 )^2 \Big]^{1/2}
~. \nonumber \\
\end{align}
Therefore, we find
\begin{eqnarray}
\sigma v^2 = \frac{\Big[ (m_1 + m_2 + m_3)^4 - 2 (m_1 + m_2 + m_3)^2 ~ (m_{1^\prime}^2 + m_{2^\prime}^2 ) + (m_{1^\prime}^2 - m_{2^\prime}^2 )^2 \Big]^{1/2}}{S \times 128 \pi ~ m_1 m_2 m_3 ~ (m_1 + m_2 + m_3)^2} \int_{-1}^1 d \cos{\theta} \, |\mathcal{M}|^2 ~, \nonumber \\
\end{eqnarray}
where $S$ is a symmetry factor for identical outgoing states. In the limit that all incoming and outgoing particles are mass degenerate, $m_{1,2,3} = m_{1^\prime, 2^\prime} = m$, this reduces to
\begin{eqnarray}
\frac{\sqrt{5}}{S \times 384 \pi ~ m^3} \int_{-1}^1 d \cos{\theta} ~ |\mathcal{M}|^2
~,
\end{eqnarray}
in agreement with that presented in Ref.~\cite{Choi:2015bya}.
\end{appendix}
|
1,116,691,500,498 | arxiv | \section{Introduction}
\label{sec:intro}
The latest set of gravitational--wave (GW) observations released by the LIGO Scientific \citep{LIGOScientific:2014pky}, Virgo \citep{VIRGO:2014yos} and KAGRA \citep{Aso:2013eba} Collaboration (LVK) as part of The third Gravitational-wave Transient Catalog (GWTC-3) catalog \citep{LIGOScientific:2018mvr,LIGOScientific:2020ibl,LIGOScientific:2021usb,LIGOScientific:2021djp} contains 69 confident binary black hole (BBH) detections as well as both confident detections for binary neutron star and neutron star black hole mergers. As a consequence, the increasing size of gravitational wave catalogs has allowed for in-depth studies of the binary black hole population properties \citep{LIGOScientific:2018jsj,LIGOScientific:2020kqk,LIGOScientific:2021psn}, cosmic expansion history \citep{LIGOScientific:2019zcs,LIGOScientific:2021aug} as well as tests of general relativity in the strong field regime \citep{LIGOScientific:2019fpa,LIGOScientific:2020tif,LIGOScientific:2021sio} including a search for gravitational wave lensing signatures \citep{LIGOScientific:2021izm}.
When gravitational waves propagate and interact with intervening matter such as galaxies or dense galaxy clusters, there is a change for strong gravitational lensing and for multiply lensed GW images to be produced with time delays ranging from minutes to months \citep{Takahashi:2003ix,Haris:2018vmn,Dai:2016igl}. Over the upcoming years, ground-based GW detectors such as Advanced LIGO, Advanced Virgo and KAGRA are expected to find 0.1 to 1 pairs of strongly lensed GW signals per year originating from binary black hole mergers at their corresponding design sensitivities \citep{Ng:2017yiu,LIGOScientific:2021izm,Xu:2021bfn,Caliskan:2022wbh,Mukherjee:2020tvr}. In fact, the first search for signatures of lensing (including strongly lensed pairs) was performed in \cite{Hannuksela:2019kle} using the 10 BBH events of the GWTC-1 catalog \citep{LIGOScientific:2018mvr}. No conclusive evidence for a strongly lensed pair was found, however, the pair with the highest evidence favoring the lensing hypothesis was GW170104/GW170814 as pointed in \cite{Hannuksela:2019kle,McIsaac:2019use}. Subsequent studies followed up the pair with a fully Bayesian joint parameter estimation study over the lensed images and arrived at similar conclusions disfavoring the lensing hypothesis \citep{Liu:2020par,Dai:2020tpj}. The most comprehensive study to date using the first half of LIGO-Virgo's third observation run observations has also yielded no substantial evidence for lensing \citep{LIGOScientific:2021djp,LIGOScientific:2021izm}
Alternative metric theories of gravity predict up to six distinct polarization modes for GW emission, besides the two tensorial modes allowed by general relativity \citep{Isi:2017equ,Chatziioannou:2012rf}. In order to probe the presence (or lack off) for these alternative polarizations, a network of six linearly independent detectors is needed. Future ground based detector networks will allow for some statements about the relative amplitudes for each mode, however, discerning the full polarization content would be difficult for most systems \cite{Chatziioannou:2021mij}. The most recent observational results using the full GWTC-3 catalog have placed stringent constrains on alternative polarizations being present \citep{LIGOScientific:2019fpa,LIGOScientific:2020tif,LIGOScientific:2021sio}. The strongest of such constraints disfavour the presence of vector or scalar modes being present individually when compared to the expected GR tensor modes. However, the presence of tensor modes as well as either vector or scalar modes (or both) as a fully mixed model has yet to be constrained strongly.
In this work, we explore constraints on alternative GW polarizations with simulated pairs of strongly lensed GW signals. We parameterize the GW model as a fully mixed tensor, vector and scalar mode model with up to 5 degrees of freedom allowing us to make statements about the relative amplitudes for each mode. The difference in arrival times for each strongly lensed image probes the same GW signal arriving at different times. The rotation of the Earth imposes the time dependence of the antenna beam pattern functions, allowing us to see a different projection for the GW signal at each detector (essentially doubling the number of detectors for a pair of lensed events) \citep{Goyal:2020bkm}. We measure the relative amplitudes for each mode by jointly fitting the detected lensed image pairs using the framework described in \citep{Liu:2020par,Lo:2021nae} and show that for some systems the polarization mode amplitude degeneracies can be broken with a single pair of lensed events.
This paper is organized as follows. In Section 2 we describe the alternative (non-tensorial) polarization modes for gravitational-wave signals. In Section 3, we summarize the effect of strong lensing in detected GW signals, focusing on pairs of lensed events. In Section 4, we present the main results of this paper and in Section 5, we provide a summary of this work. We use the Planck 2015 cosmological model \citep{Planck:2015fie} throughout this paper, that is, $H_0 = 67.8 \ \rm km/s/Mpc$, $\Omega_{m} = 0.308$, $\Omega_{\Lambda} = 0.692$ and set $\Omega_{k,0} = 0$.
\section{Nontensor polarizations}
\label{sec:nongr}
\newcommand{\ensuremath{+}}{\ensuremath{+}}
\newcommand{\ensuremath{\times}}{\ensuremath{\times}}
\newcommand{\ensuremath{x}}{\ensuremath{x}}
\newcommand{\ensuremath{y}}{\ensuremath{y}}
\newcommand{\ensuremath{b}}{\ensuremath{b}}
\newcommand{\ensuremath{l}}{\ensuremath{l}}
\newcommand{\ensuremath{s}}{\ensuremath{s}}
\newcommand{\ensuremath{h_+}}{\ensuremath{h_+}}
\newcommand{\ensuremath{h_\times}}{\ensuremath{h_\times}}
\newcommand{\ensuremath{h_x}}{\ensuremath{h_x}}
\newcommand{\ensuremath{h_y}}{\ensuremath{h_y}}
\newcommand{\ensuremath{h_b}}{\ensuremath{h_b}}
\newcommand{\ensuremath{h_l}}{\ensuremath{h_l}}
\newcommand{\ensuremath{h_s}}{\ensuremath{h_s}}
Alternative metric theories of gravity (beyond general relativity) may allow up to six distinct polarization modes on the GW waveform, including the two tensor $\ensuremath{+}$ and $\ensuremath{\times}$ modes expected in general relativity (GR). These additional polarization modes are the two vector modes $\ensuremath{x}$ and $\ensuremath{y}$, as well as two scalar modes $\ensuremath{b}$ and $\ensuremath{l}$ (breathing and longitudinal respectively). The GW perturbation can thus be written as,
\begin{equation} \label{eq:h_A}
h_{ij} = \sum_A h_A\, e^A_{ij} \, ,
\end{equation}
where $e^A_{ij}$ is the polarization tensor for mode $A$ and $h_A$ are the corresponding polarization mode amplitudes. The GW perturbation is thus a linearly independent weighted sum over modes, the most generic case corresponding to $A \in \{\ensuremath{+},\ensuremath{\times}, \ensuremath{x}, \ensuremath{y}, \ensuremath{b}, \ensuremath{l} \}$.
In general, GW interferometers measure the projection of the perturbation given by Eq.~\eqref{eq:h_A} onto the detector arms. Thus the measured GW strain at detector $I$ can be written as,
\begin{equation}
h_I(t) = \sum_A F_I^A(\alpha, \delta, \psi, t)\, h_A(t)\, ,
\end{equation}
with antenna beam pattern functions $F_I^A \equiv D_I^{ij} e^A_{ij}$ defined with respect to the detector tensor $D_I^{ij}$ which encodes the geometry of the GW detector. The antenna pattern functions are in general functions of time and depend on the sky location of the GW source defined by its right ascension $\alpha$ and declination $\delta$ as well as the polarization angle $\psi$. It is worth noting that the breathing and longitudinal mode antenna pattern functions are identical (up to a constant) so that $F_\ensuremath{b} = - F_\ensuremath{l}$. This degeneracy, makes each scalar mode contribution difficult to disentangle unless a specific modified theory of gravity is chosen a-priori, leading to model dependent constraints. Following convention we pick the breathing mode as the scalar mode of interest, thus the sum over linearly independent modes in Eq.~\eqref{eq:h_A} reduces to a sum over five polarization modes, $A \in \{\ensuremath{+},\ensuremath{\times}, \ensuremath{x}, \ensuremath{y}, \ensuremath{s}\}$ where we denoted the breathing mode ($\ensuremath{b}$) by ($\ensuremath{s}$) for convenience. For a detailed discussion on GW polarizations and the various polarization angle conventions we refer the reader to \citep{Isi:2017equ,Isi:2022mbx}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{g_iota.pdf}
\caption{\label {fig:g_iota} We show the values for $g(\iota)$ as a function of the inclination angle $\iota$ for each of the six polarization modes. We note that for face-on systems ($\iota=0$), the vector and scalar modes will not be present in the GW strain data even if emitted. For edge-on systems ($\iota=\pi/2$), $\times$-mode and the vector-$x$ will not be detectable. The optimal inclination angle for which we maximize over the presence of all modes in the GW data corresponds to $\iota_\text{opt} \approx 0.87$. }
\end{figure}
For gravitational waves produced by a compact binary merger such as a pair of merging binary black holes, there is an additional inclination angle dependence for each polarization mode \citep{Chatziioannou:2012rf,Takeda:2020tjj}. We define this dependence via the function $g_A(\iota)$, so that $g_\ensuremath{+}(\iota) = (1+\cos^2\iota)/2$, $g_\ensuremath{\times}(\iota) = \cos\iota$, $g_\ensuremath{x}(\iota) = \sin{2\iota}$, $g_\ensuremath{y}(\iota) = \sin\iota$ and $g_{\ensuremath{b},\ensuremath{l}}(\iota) = \sin^2\iota$ where $\iota$ is the inclination angle of the binary. We can thus write the gravitational wave strain at detector $I$ as,
\begin{equation} \label{eq:h_measured}
h_I(t) = \sum_A F_I^A(\alpha, \delta, \psi, t)\, g_A(\iota)\, h_A(t)\,.
\end{equation}
From the above expression, we can see that the inclination angle dependence on the polarization modes is important since for a face-on system ($\iota=0$) only the tensor modes will be present in the data while for an edge-on system ($\iota=\pi/2)$, only the cross polarization mode vanishes but all other modes are present. The inclination angle dependence is critical for 2nd generation ground based detectors since we expect most mergers to be near the face-on limit. In Fig. \ref{fig:g_iota}, we plot the dependence on inclination for the mode amplitudes. Clearly, if all other parameters are fixed, then the optimal inclination would be $\iota_\text{opt} \approx 0.87$.
\section{Strong Gravitational Wave Lensing}
As gravitational waves propagate, there is a chance for strong gravitational lensing to occur due to intervening galaxies or larger cosmic structures such as galaxy clusters. The strong lensing of gravitational waves can give rise to multiple images of the same GW transient each with its own absolute magnification factor $\mu_k$. When the GW images are detected, each will arrive at a different time $t_c^{(k)}$ and each might have a frequency independent phase shift (Morse phase) $\Delta\phi_k = -\pi n_k /2$ with index $n_k = 0, 1, 2$ defining Type-I, Type-II and Type-III images respectively. The gravitational wave waveform for each lensed image is then given by,
\begin{equation} \label{eq:h_lens}
h_L(f,\theta,\mu_k, t_c^{(k)},\Delta\phi_k) = \sqrt{\mu_k}\exp \left(if\Delta\phi_k\right)h_U(f, \theta, t_c^{(k)})
\end{equation}
where $h_U$ is the waveform without any strong lensing effects (unlensed) and $\theta = \{m_1, m_2, a_1, a_2, \iota, \alpha, \delta, \psi\}$ where $m_1$ and $m_2$ are the primary and secondary masses of the binary in the source frame, $a_1$ and $a_2$ are the (aligned) component spin magnitudes. The set of parameters $\theta$ is common across all lensed images, including the sky location of the GW source due to expected order of arcsecond deflection angles for each image being much smaller than the typical localization regions for 2G detectors \cite{Takahashi:2003ix}.
Now, for a source at luminosity distance $D_L$, the lensed images are magnified (de-magnified) by their corresponding magnification factors $\sqrt{\mu_k}$ as in Eq.~\eqref{eq:h_lens} so that the the observed distances correspond to,
\begin{equation}
D^{(k)}_{\text{obs}}=D_L/\sqrt{\mu_k}\,,
\end{equation}
clearly showing the degeneracy between the luminosity distance to the source and the absolute magnification factors for each lensed image.
For a pair of lensed images, it is convenient to define the relative magnification factor $\mu$ as,
\begin{equation}
\mu = \left( \frac{D^{(1)}_{\text{obs}}}{D^{(2)}_{\text{obs}}} \right)^2 = \frac{\mu_2}{\mu_1} \,,
\end{equation}
where we label the signal that is detected first by $(1)$ and consequently the later arriving signal by $(2)$. Finally we can also define the lensing time delay for the pair of lensed images as $\Delta t = t_c^{(2)} - t_c^{(1)}$ which is always greater than zero.
The most important effect due to strong lensing and the production of multiple images is the different times of arrival for each lensed image. Due to Earth's rotation the location of the network of detectors will change as a function of time relative to the sky location of the lensed signals. This means that the antenna pattern functions for each polarization mode will probe the polarization content of the arriving GW signal differently depending on the arrival time of each lensed image, allowing us to constrain the relative amplitudes for each polarization mode \cite{Goyal:2020bkm}. In principle, this leads to effectively doubling the number of detectors in the network for a pair of strongly lensed images. In order to illustrate this point we show the antenna pattern functions $F_A^2(t)$ in Fig. \ref{fig:antenna} at a fixed polarization angle $\psi=0$ for two different sky locations over a period of two days. As mentioned in Section \ref{sec:intro}, the expected time delay between a pair of lensed events by an intervening galaxy could range from hours to months. With respect to probing the polarization amplitudes for each mode, the expected time delay is not important but the relative time delay corresponding to the rotation of the Earth over a day.
\begin{figure*}[htb]
\includegraphics[width=0.45\textwidth]{antenna_00.pdf}
\includegraphics[width=0.45\textwidth]{antenna_radec.pdf}
\caption{\label {fig:antenna} We show the values for $F_A^2(t)$ for the six polarization modes over the span of two days where we have fixed the polarization angle to $\psi=0$ for convenience. In the left panel we show an example where the sky location of the source is fixed at $(\alpha,\delta) = (0,0)$. Similarly, we show another example but with $(\alpha,\delta) = (1.375,-1.211)$ to illustrate the complex behavior of the antenna beam pattern functions in terms of sky location and time.}
\end{figure*}
\section{Joint Parameter Estimation}
\label{sec:pe}
Since strongly lensed systems leave the frequency evolution of the gravitational-wave binary unchanged and thus only induce an overall amplitude and phase difference amongst the detected images. We are thus able to jointly fit the lensed events by taking into account the predicted strong lensing effects on the GW-waveform. We provide a summary for the joint parameter estimation below. For the full derivations and detailed discussion of the framework, see \citep{Liu:2020par,Lo:2021nae}.
Under the assumption that we have a confidently detected pair of strongly lensed GW events. We can perform joint parameter estimation by considering the strong lensing waveform model for each detected image in Eq.~\eqref{eq:h_lens} and use the model with alternative polarizations as defined in Eq.~\eqref{eq:h_measured} as the definition for $h_U(t,\theta)$. Additionally we parameterize each polarization mode amplitude by a set of relative amplitude parameters $\{\epsilon_A\}$ which must satisfy the following constraint $\sum_A \epsilon_A = 1$. For GR, we must have $\epsilon_\ensuremath{+} = \epsilon_\ensuremath{\times} = 0.5$ while the vector and scalar mode contributions are all zero.
Under the lensing hypothesis, for a pair of lensed events with measured strains $d_1$ and $d_2$, we jointly infer the binary parameters $\theta$, the lensing observables $\{\mu, \Delta t\}$ (in this work we set $\Delta\phi_k=0$ for all images) as well as the relative amplitudes for each polarization mode $\{\epsilon_A\}$ in terms of the observed distance and time of arrival of the first image,
\begin{align} \label{eq:likelihood}
\begin{split}
&\mathcal{L}(d_1, d_2 |\theta, D^{(1)}_{\text{obs}}, t_c^{(1)}, \mu, \delta t, \{\epsilon_A\} ) \\
&=\mathcal{L}(d_1|\theta, D^{(1)}_{\text{obs}}, t_c^{(1)}, \{\epsilon_A\} ) \mathcal{L}(d_2 |\theta, \mu, \delta t, \{\epsilon_A\} ),
\end{split}
\end{align}
where $\mathcal{L}(d_1, d_2 | \ldots)$ is referred to as the strong lensing joint likelihood. We note that this can be generalized to an arbitrary number of lensed images and we refer the reader to \citep{Lo:2021nae} for more details. To obtain the posterior distribution over the parameters describing the joint likelihood function we use Bayes theorem and defer the details of our choice for the prior distribution to Section \ref{sec:results}.
\section{Results}\label{sec:results}
We perform joint parameter estimation to estimate the posterior distribution on the parameters defined through the strong lensing joint likelihood function as defined in Eq.~\eqref{eq:likelihood}. As an example, we simulate a pair of lensed GW images from a non-spinning binary black hole merger with the following intrinsic parameters: $m_1^{\text{det}} = 36 M_\odot$, $m_2^{\text{det}} = 29 M_\odot$ and $a_1 = a_2 = 0$. The extrinsic parameters for the simulated system are $\psi=2.659$, $\phi_c=2.9$, $\alpha=1.375$, $\delta=-1.2108$ and $\iota=\pi/4$. We have chosen a sky location for the merger consistent with the beam pattern functions as shown in the right panel of Fig. \ref{fig:antenna} and an inclination angle close to the value of $\iota_{\text{opt}}$ in order to not suppress the extra polarization modes through the inclination dependence introduce via the $g_A(\iota)$ factors.
As discussed in \ref{sec:pe}, we sample over the observed distance to the first event $D^{(1)}_{\text{obs}}$ and the relative amplification factor for the pair $\mu$. The first lensed pair has $D^{(1)}_{\text{obs}} = 1000 \ \text{Mpc}$ and $\mu=2$ (corresponding to $D^{(2)}_{\text{obs}} = 500 \ \text{Mpc}$) with a time delay $\Delta t = 6 \ \text{hours}$. We set the Morse index for both images to zero (both Type-I) for simplicity. For the polarization mode amplitudes we choose, $\epsilon_+ = \epsilon_\times = 0.35$, $\epsilon_x = \epsilon_y = 0.15$ and $\epsilon_s=0.05$.
We consider two examples, a 2-detector network composed of LIGO Hanford and LIGO Livingston (HL) and a 4-detector network, with Advanced Virgo and KAGRA as additional detectors (HLVK), all at their corresponding design sensitivities. We generate the GW waveform using the \texttt{TaylorF2}\cite{Damour:2000zb} waveform model for simplicity and inject the two lensed GW signals into simulated data streams with Gaussian noise and sample over the strong lensing joint likelihood using \texttt{Bilby} \citep{Ashton:2018jfp,Romero-Shaw:2020owr}. For the 4-detector network, we show in Fig. \ref{Fig:postHLVK_amp} the marginalized posterior distribution on the relative polarization mode amplitudes, inclination angle, relative magnification factor and the observed distance to the first image (See Appendix \ref{sec:fullpe} for our prior choices as well as full parameter estimation results in Fig. \ref{Fig:postHLVK} for the HLVK case and in Fig. \ref{Fig:postHL} for the HL case). It is evident from the posterior distribution shown in Fig. \ref{Fig:postHLVK_amp} that the relative polarization mode amplitudes can be measured with a single pair of lensed events using a 4-detector network at design sensitivity. For the example with a 2-detector network observing the same system, the polarization mode amplitudes cannot be fully constrained due to the lack of linearly independent detectors (in principle four but both the Hanford and Livingston detectors are nearly co-aligned).
\begin{figure}[htb]
\includegraphics[width=0.47\textwidth]{pol_hlvk_amplitude.png}
\caption{Marginalized posterior distribution for the pair of lensed images as described in \ref{sec:results} observed by four detectors (HLVK) on the relative amplitudes for each polarization mode, inclination angle, relative magnification factor and the observed distance of the first image. The simulated system has $\epsilon_+ = \epsilon_\times = 0.35$, $\epsilon_x = \epsilon_y = 0.15$ and $\epsilon_s=0.05$ for the polarization mode amplitude, $D^{(1)}_{\text{obs}} = 1000 \ \text{Mpc}$, $\mu=2$ and $\iota=\pi/4$ (shown in orange) with a relative time delay of six hours. }
\label{Fig:postHLVK_amp}
\end{figure}
\section{Discussion}
In this work we have performed Bayesian joint parameter estimation on pairs of strongly lensed GW events in order to constrain the relative amplitudes for alternative polarization modes using simulated data. We have used a simplified signal model as a proxy for the signal morphology for the additional polarization modes, and have also made sure to include the expected inclination angle dependence for each mode for GWs emitted by a merging binary. We have shown that the relative amplitudes as well as the amplitude relevant parameters such as the observed distance, inclination angle and relative amplification factor for the lensed pair can be measured, since the additional data from the same astrophysical system provides enough independent detectors to measure the aforementioned parameters.
Strongly lensed pairs of GW signals for binary black hole mergers are expected to be detected as early as O4 but more likely in O5. Once a confident detection has been established, the joint parameter estimation framework described in this work can be applied to a real lensed pair of GW signals. However, we do mention that a proper treatment of real GW data will involve the strong lensing joint likelihood with a model independent framework to describe the GW signal morphology as explored in \cite{Chatziioannou:2021mij} which used \texttt{bayeswave} to model the GW signal morphology using sine gaussians. Given that, the results of this paper can be seen as being slightly pessimistic than what they would be if any alternative polarization modes are present in the data with significantly different signal morphology. The varying morphology should allow for the relative mode amplitude degeneracy to be broken, however, using a specific modified gravity model that predicts additional polarization modes for the Bayesian inference would make the results model dependent.
\section*{Acknowledgements}
The author would like to thank Virginia d'Emilio, Jolien Creighton, Soichiro Morisaki and Anarya Ray for useful comments and feedback throughout this work. IMH is supported by the NSF Graduate Research Fellowship Program under grant DGE-17247915. This work was supported by NSF awards PHY-1912649. The author is grateful for computational resources provided by the Leonard E Parker Center for Gravitation, Cosmology and Astrophysics at the University of Wisconsin-Milwaukee. We thank LIGO and Virgo Collaboration for providing the data for this work. This research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org/), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO Laboratory and Advanced LIGO are funded by the United States National Science Foundation (NSF) as well as the Science and Technology Facilities Council (STFC) of the United Kingdom, the Max-Planck-Society (MPS), and the State of Niedersachsen/Germany for support of the construction of Advanced LIGO and construction and operation of the GEO600 detector. Additional support for Advanced LIGO was provided by the Australian Research Council. Virgo is funded, through the European Gravitational Observatory (EGO), by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale di Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by institutions from Belgium, Germany, Greece, Hungary, Ireland, Japan, Monaco, Poland, Portugal, Spain. This material is based upon work supported by NSF's LIGO Laboratory which is a major facility fully funded by the National Science Foundation. This article has been assigned LIGO document number LIGO-P2200329.
|
1,116,691,500,499 | arxiv | \section{Introduction}
Deep neural networks have achieved significant results for a variety of real-world tasks such as image processing~\cite{He2015a,Zhu2017}, natural language processing~\cite{Devlin2018}, and game playing~\cite{Silver2017}.
Their successes depend on hard-coded prior knowledge, such as translation invariance in image recognition~\cite{LeCun1998} and the manifold hypothesis in data modeling~\cite{Rifai2011}.
The prior knowledge guarantees a desirable property of the learned function.
The Hamiltonian neural network (HNN)~\cite{Greydanus2019} implements the Hamiltonian structure
on a neural network and thereby produces the energy conservation law in physics.
After its great success, neural networks specifically designed for physical phenomena have received much attention. They have been intensively extended to various forms, such as the Hamiltonian systems with additional dissipative terms~\cite{Zhong2020a}.
Meanwhile, most previous studies aimed to model continuous-time differential equations and employed numerical integrators (typically, an explicit Runge--Kutta method) to integrate the neural network models for learning and computing the dynamics~\cite{Chen2018e,Chen2020a,Greydanus2019,Zhong2020}.
Surprisingly, our numerical experiments reveal that a higher-order numerical integrator with adaptive time-stepping is quite often inferior in performance as compared to a quantitatively lower order but qualitatively superior numerical integrator.
This is because higher-order integrators aim to reproduce continuous-time dynamics while practical learning and computation are in discrete time.
In this case, the qualitative features that the integrators equipped with could be actually essential.
From this point of view, this study proposes a \textit{deep energy-based discrete-time physical model}, which combines neural networks and discrete-time energy-based modeling.
The key ingredient is the structure-preserving integrators, in particular, the discrete gradient method along with the newly-developed automatic discrete differentiation.
In addition, our framework unifies and also extends the aforementioned previous studies.
The main contributions include:
\paragraph*{Applicable to general energy-based physical models.}
Our framework is applicable to general physical phenomena modeled by the energy-based theory, such as Hamiltonian mechanics, the Landau theory, and the phase field modeling.
Our target class includes a Hamiltonian system composed of position and momentum (a so-called natural system, such as a mass-spring system), a natural system with friction, a physical system derived from free-energy minimization (e.g., phase transitions), and a Hamiltonian partial differential equation (PDE) (e.g., the Korteweg--de Vries (KdV) equation and the Maxwell equation).
All equations can be written as a geometric equation.
Most studies have focused on one of the first two systems~\cite{Greydanus2019,Zhong2020a,Zhong2020} under special conditions~\cite{Chen2020a,Saemundsson2020,Tong2020}, or they are too general to model the conservation and dissipation laws~\cite{Chen2018e,Raissi2018}.
The details of the proposed framework along with the target class of the equations and the geometric aspects are described in Section~\ref{sec:physical_systems}.
\begin{wrapfigure}{r}{0.57\textwidth}
\vspace*{-7mm}
\includegraphics[scale=0.4,page=1]{fig/figs.pdf}
\vspace*{-5mm}
\caption{Modeling based on energy-based theories.}\label{fig:modeling_concept}
\vspace*{-3mm}
\end{wrapfigure}
\paragraph*{Equipping with the laws of physics in discrete time.}
Previous models interpolate the discrete-time data using numerical integrators for learning and computing~\cite{Chen2020a,Greydanus2019,Saemundsson2020,Tong2020,Zhong2020a,Zhong2020}. The discretization may destroy the geometrical structure from which the laws of physics follow (see the lower part of Fig.~\ref{fig:modeling_concept}).
Conversely, our approach, in principle, learns a discrete-time model from the discrete-time data without the time-consuming interpolation and discretization error (see the upper part).
Using the \emph{discrete gradient}, our approach admits the important laws of physics, particularly the energy conservation or dissipation law and the mass conservation law in discrete time.
We demonstrate this property theoretically in Section~\ref{sec:discrete_gradient} and experimentally in Section~\ref{sec:experiments}.
\paragraph*{Easy-to-use.}
Our approach is based on the discrete gradient method~\cite{Furihata1999,Gonzalez1996,Quispel1996}.
Most discrete gradients require the explicit form of the function (see the middle part of Fig.~\ref{fig:modeling_concept}); hence, they are unavailable for neural networks (see Appendix~\ref{appendix:GI} for reference).
We propose an automatic discrete differentiation algorithm, which automatically obtains the discrete gradient of the neural networks composed of linear and nonlinear operations.
The proposed algorithm can be implemented in a similar way to the current automatic differentiation algorithm~\cite{Griewank2008}; we provide it as a PyTorch library~\cite{Paszke2017}%
\footnote{\url{https://github.com/tksmatsubara/discrete-autograd}}.
We introduce the detailed algorithm in Section~\ref{sec:add}.
\section{Related Work}
\paragraph{Neural Networks for Differential Equations.}
Since the 1990s, many studies have attempted to approximate ordinary differential equations (ODEs) and PDEs by applying neural networks ~\cite{Anastassi2014,Cichock1992,Lagaris1998,Raissi2018,Ramuhalli2005,Rudd2014}.
Recent advances in the automatic differentiation algorithm~\cite{Griewank2008} have enabled us to build more complicated neural network architectures.
Neural ODE (NODE)~\cite{Chen2018e} has re-established neural networks for modeling ODEs.
NODE treats the output of a time-dependent neural network as the time derivative of the input; thereby, defining an ODE in a general way.
Moreover, NODE employs numerical integrators to train and integrate the neural network model.
Several studies attempted to model a PDE system using regularization terms to mimic the conservation laws~\cite{Raissi2018,Wu2020a}.
They were insufficient to ensure the conservation laws in physical systems.
The HNN approximates an energy function $H$ from the data using a neural network, and thereby, builds a Hamiltonian system~\cite{Greydanus2019}.
The time-derivative of the states $({\vec{q}},{\vec{p}})$ is given using the gradient $\nabla H$ of the energy $H$, which is called the Hamiltonian, specifically, $\d{\vec{q}}/\d t=\nabla_{\vec{p}} H$ and $\d{\vec{p}}/\d t=-\nabla_{\vec{q}} H$, where ${\vec{q}}$ and ${\vec{p}}$ denote the position and momentum, respectively.
Following the HNN, the symplectic ODE-Net integrates an HNN-like model using a Runge--Kutta method; thus, enabling learning from the discrete-time data~\cite{Zhong2020}.
The dissipative SymODEN generalized it to a model with friction and input~\cite{Zhong2020a}.
We summarized the previous studies in Table~\ref{tab:comparison}.
\begin{table}[t]
\centering\small
\caption{Comparison with Other Studies}\label{tab:comparison}
\begin{tabular}{lcccccccc}
\toprule
\textbf{} & HNN & SymODEN & Dissipative & SRNN/VIN & DGNet \\
\textbf{} & \cite{Greydanus2019} & \cite{Zhong2020} & \cite{Zhong2020a} & \cite{Chen2020a,Saemundsson2020,Tong2020} & (this paper) \\
\midrule
Hamiltonian system & yes & yes & yes & yes & yes \\
Dissipative ODE & & & yes & & yes \\
\midrule
Hamiltonian PDE & & & & & yes \\
Dissipative PDE & & & & & yes \\
\midrule
Learning from finite difference & & approx.$^*$ & approx.$^*$ & approx.$^*$ & yes \\
Strict conservation law in discrete-time & & & & approx.$^{**}$ & yes \\
Strict dissipation law in discrete-time & & & & & yes \\
\bottomrule
\multicolumn{6}{l}{$^*$ Interpolating by numerical integrators. $^{**}$ Conserving only the ``shadow'' Hamiltonian.} \\
\end{tabular}
\vspace*{-3mm}
\end{table}
\paragraph{Structure-Preserving Numerical Methods.}
Most differential equations that arise as models for physical phenomena admit some laws of physics, e.g., the energy and other conservation laws of the Hamilton equation and the mass conservation law and energy dissipation properties of the equations for phase-transition phenomena.
Numerical integrators that reproduce those properties are called structure-preserving integrators or geometric integrators~\cite{Hairer2006}.
The aforementioned studies mainly employed classical Runge--Kutta methods for numerical integration, which in general destroy these properties~\cite{Hairer2006}.
Several recent studies have employed symplectic integrators, which conserve an approximated energy called a ``shadow'' Hamiltonian in discrete time~\cite{Chen2020a,Saemundsson2020,Tong2020}.
These studies considered only the systems of which the Hamiltonian $H$ is separable, i.e., expressible as the sum of the potential and kinetic energies.
This is quite restrictive; in fact, most of the important Hamiltonian PDEs (e.g., the shallow water equations and the nonlinear Schr\"odinger equation) are not in this class.
Moreover, structure-preserving integrators for dissipative systems have never been employed.
This is because these integrators are often based on the discrete gradient method; however, no efficient discrete gradient has been available for neural networks.
Several studies have focused on Lagrangian mechanics~\cite{Cranmer2020,Saemundsson2020}.
Lagrangian mechanics can be expressed using the time derivative of the position, while the Hamiltonian mechanics requires conjugate momentum.
The main drawback is that it is not obviously extendable to general dissipative systems.
We consider it out of scope of this study, but the proposed method is extendable to it~\cite{Yaguchi2013}.
\section{Methods}\label{sec:method}
\subsection{General Form of Energy-Based Dynamical Systems}\label{sec:physical_systems}
We focus on the following formulation of the models by the energy-based theories, which expresses a wide variety of physical systems described by ODEs and discretized PDEs~\cite{Furihata1999, Quispel1996a}.
The system has a state ${\vec{u}}\in{\mathbb{R}}^N$ and an energy function $H:{\mathbb{R}}^N\rightarrow {\mathbb{R}}$.
The time evolution is expressed as
\begin{equation}
\textstyle\frac{\d {\vec{u}}}{\d t}=G({\vec{u}}) \nabla H({\vec{u}}),\label{eq:gradient_flow}
\end{equation}
where $G\in{\mathbb{R}}^N\times{\mathbb{R}}^N$ is a matrix, which can be state-dependent, and $\nabla H({\vec{u}})$ is the gradient of the system energy $H$ with respect to the state ${\vec{u}}$.
Systems of this form arise as differential geometric equations on Riemannian or symplectic manifolds.
See Appendix \ref{appendix:geometry} for reference.
The system $(H,G,{\vec{u}})$ has the following laws of physics.
\begin{thm}\label{thm:continuous_conservation_dissipation}
The system has the energy dissipation law if $G\leq O$ and the energy conservation law if $G$ is skew-symmetric.
\end{thm}
See Appendix~\ref{appendix:proofs} for the proofs of the theorems for this study.
Note that $G\le O$ denotes that the matrix $G$ is negative semi-definite, with which ${\vec{x}}^\top G {\vec{x}}\le 0$ for any vector ${\vec{x}}$.
A matrix $G$ is skew-symmetric if $G^\top = - G$, and then ${\vec{x}}^\top G {\vec{x}}=0$ for any vector ${\vec{x}}$.
\begin{thm}\label{thm:mass_conservation}
The system has the mass conservation law in the sense that $\d (\sum_k u_k) /\d t = 0$ if the vector $\vec{1} = (1,1,\ldots,1)$ is in the left kernel of $G$ (i.e., $\vec{1} G = \vec{0}$).
\end{thm}
Thus, we can design the neural network models with the above laws of physics by defining the models $(H,G,{\vec{u}})$, where $G$ satisfies the required conditions for the laws of physics shown in the above theorems and $H$ is designed by a neural network.
\begin{rmk}
The models $(H,G,{\vec{u}})$ with $H$ represented by neural networks widely extend the scope of the previous studies.
In particular, the discretized-in-space PDEs (e.g., the KdV equation~\cite{KdV1895} and {\color{cyan} the} Cahn--Hilliard equation~\cite{Cahn1958}) have not been treated like this before.
This is a significant contribution in this study.
\end{rmk}
A natural system is a Hamiltonian system associated to a Hamiltonian function $H$ that is the sum of the potential and kinetic energies.
This is expressed as the system $(H,G=S,{\vec{u}})$ for the matrix
\begin{equation}
S =
{\scriptsize\setlength\arraycolsep{1pt}
\renewcommand{\arraystretch}{.4}
\begin{pmatrix}
O & I_n \\
-I_n & O
\end{pmatrix}
},\label{eq:hamiltonian_system}
\end{equation}
where $2n=N$ and $I_n$ denotes an $n$-dimensional identity matrix.
The first $n$ elements of the state ${\vec{u}}$ denote the position ${\vec{q}}$ and the remaining denotes the momentum ${\vec{p}}$.
The matrix $S$ is skew-symmetric, and the system $(H,G=S,{\vec{u}})$ conserves the system energy $H$.
A pendulum, a mass-spring system, and N-body problems are expressible by this form.
Besides, the system $(H,G=S-R,{\vec{u}})$ expresses a natural system with friction when $S$ is the one shown above and
\begin{equation}
R = \mathrm{diag}(0 \dots 0\ g_{1} \dots g_{n}), \label{eq:dissipative_system}
\end{equation}
where $g_k\ge 0$ is a friction term that dampens the momentum $p_k$; thus, dissipating the system energy $H$ because $(S-R)\le O$.
Most previous studies focused on these two types of systems~\cite{Chen2020a,Greydanus2019,Saemundsson2020,Zhong2020a,Zhong2020}.
From a geometric point of view, the matrix $G$ in the above form means that the systems are defined on cotangent bundles, while the following approach is formulated on general symplectic or Riemannian manifolds, enabling our method to handle the various PDE systems~\cite{Hairer2006}.
In fact, the formulation $(H, G, {\vec{u}})$ can express the discretized PDE systems.
For example, PDEs under the periodic boundary condition can be discretized by using the central difference operators, of which the matrix representations are as follows.
\begin{equation}
D=\frac{1}{2\Delta x}
{\scriptsize\setlength\arraycolsep{1pt}
\renewcommand{\arraystretch}{.4}
\begin{pmatrix}
0 & 1 & & & -1 \\
-1 & 0 & 1 \\
& & \ddots \\
& & -1 & 0 & 1 \\
1 & & & -1 & 0 \\
\end{pmatrix}
},\quad
D_2=\frac{1}{(\Delta x)^2}
{\scriptsize\setlength\arraycolsep{1pt}
\renewcommand{\arraystretch}{.4}
\begin{pmatrix}
-2 & 1 & & & 1 \\
1 & -2 & 1 \\
& & \ddots \\
& & 1 & -2 & 1 \\
1 & & & 1 & -2 \\
\end{pmatrix}
},\label{eq:first_second_order_difference}
\end{equation}
where $\Delta x$ is the space mesh size.
The matrices $D$ and $D_2$
represent first--order and second--order central differences, respectively.
The $k$-th element $u_k$ of the state ${\vec{u}}$ corresponds to the mass at the position $x=k \Delta x$, and the systems $(H,G=D,{\vec{u}})$ and $(H,G=D_2,{\vec{u}})$ admit the mass conservation law.
For suitable discretization of general differential operators, see Appendix \ref{appendix:semi-discretization}.
The system $(H,G=D,{\vec{u}})$ is a Hamiltonian PDE, which includes the shallow water equations such as the KdV equation, the advection equation, and the Burgers equation~\cite{Burgers1948}.
The matrix $D$ is skew-symmetric; hence, the system $(H,G=D,{\vec{u}})$ conserves the energy $H$.
The system $(H,G=D_2,{\vec{u}})$ expresses a physical system derived from the Landau free-energy minimization including the Cahn--Hilliard equation and the phase--field model for the phase transitions and the pattern formulations.
The energy $H$ dissipates because $D_2 \leq O$.
Other target equations include the equations with complex state variables, such as the Schr\"odinger equation and the Ginzburg--Landau equation.
See \cite{Furihata1999} for details.
\subsection{Discrete Gradient for Energetic-Property-Preserving Integration}\label{sec:discrete_gradient}
The discrete gradient is defined as the following vector-valued function~\cite{Furihata1999,Gonzalez1996,Itoh1988,Quispel1996}.
\begin{dfn}\label{def:discrete_gradient}
For $H:{\mathbb{R}}^N\rightarrow {\mathbb{R}}$, ${\overline{\nabla}} H:{\mathbb{R}}^N \times {\mathbb{R}}^N \to {\mathbb{R}}^N$ that satisfies the following conditions is called a discrete gradient of $H$:
\begin{equation}\label{eq:discrete_gradient}
\textstyle H({\vec{u}}) - H({\vec{v}}) = {\overline{\nabla}} H({\vec{u}}, {\vec{v}}) \cdot ({\vec{u}} - {\vec{v}}), \quad {\overline{\nabla}} H({\vec{u}}, {\vec{u}}) = \nabla H({\vec{u}}),
\end{equation}
where $\cdot$ denotes an inner product.
\end{dfn}
The first condition corresponds to the chain-rule $\d H(\Delta{\vec{u}};{\vec{u}}) = \nabla H({\vec{u}})\cdot\Delta{\vec{u}}$ for the Fr\'echet derivative $\d H(\cdot;{\vec{u}})$ of $H$ at ${\vec{u}}$ and an infinitesimal change $\Delta{\vec{u}}$ of ${\vec{u}}$.
The second condition verifies that the discrete gradient ${\overline{\nabla}} H$ is certainly an approximation of the gradient $\nabla H$.
The inner product is typically the standard Hermitian inner product for ODEs and the discrete $L^2$ inner product $\langle {\vec{u}}, {\vec{v}} \rangle_{L^2_{\mathrm{d}}} := \sum u_k v_k \Delta x$ for discretized PDEs.
With the discrete gradient ${\overline{\nabla}} H$, a discrete analogue of the system in Eq.~\eqref{eq:gradient_flow} is expressed as follows.
\begin{equation}
\frac{{\vec{u}}^{(n+1)} - {\vec{u}}^{(n)}}{t^{(n+1)}-t^{(n)}}
= {\overline{G}}({\vec{u}}^{(n+1)}, {\vec{u}}^{(n)}) {\overline{\nabla}} H({\vec{u}}^{(n+1)}, {\vec{u}}^{(n)}),\label{eq:discrete_system}
\end{equation}
where ${\vec{u}}^{(n)}$ denotes the state ${\vec{u}}$ at time $t^{(n)}$.
The matrix ${\overline{G}}$ is an approximation to $G$ that satisfies the conditions of Theorem \ref{thm:continuous_conservation_dissipation} and/or \ref{thm:mass_conservation} required by the target system.
\begin{thm}\label{thm:discrete_conservation_dissipation}
The discrete system in Eq.~\eqref{eq:discrete_system} has the discrete energy dissipation law if ${\overline{G}} \leq O$ and the discrete energy conservation law if ${\overline{G}}$ is skew-symmetric.
In particular, if the system is dissipative, the amount of energy dissipation is an approximation of that of the continuous system.
The system has the discrete mass conservation law if the vector $\vec{1} = (1,1,\ldots,1)$ is in the left kernel of ${\overline{G}}$.
\end{thm}
A discrete gradient ${\overline{\nabla}} H$ is not uniquely determined; hence, several methods have been proposed so far~\cite{Celledoni2012}.
However, most methods are inapplicable to neural networks because they require a manual deformation of the system equation~\cite{Furihata1999}.
See Appendix \ref{appendix:GI} for details.
A conceptual comparison between discrete gradient methods and symplectic integrators~\cite{Chen2020a,Saemundsson2020,Zhong2020} is summarized in Appendix~\ref{appendix:comparison_with_symplectic}.
\subsection{Automatic Discrete Differentiation Algorithm}\label{sec:add}
To obtain a discrete gradient ${\overline{\nabla}} H$ of the neural networks, we propose the \emph{automatic discrete differentiation} algorithm as an extension of the automatic differentiation algorithm~\cite{Griewank2008}.
Preparatorily, we introduce a discrete differential ${\overline{\d}} H$, which is a discrete counterpart of the Fr\'echet derivative $\d H$~\cite{Celledoni2014};
\begin{dfn}\label{def:discrete_differential}
A discrete differential ${\overline{\d}} H : {\mathbb{R}}^N\times{\mathbb{R}}^N\times{\mathbb{R}}^N\rightarrow{\mathbb{R}}^M$ of a function $H:{\mathbb{R}}^N\rightarrow{\mathbb{R}}^M$ is a function that satisfies the following conditions;
\begin{equation}
{\overline{\d}} H(a{\vec{x}};{\vec{v}},{\vec{u}})=a{\overline{\d}} H({\vec{x}};{\vec{v}},{\vec{u}}),\ H({\vec{v}})-H({\vec{u}})={\overline{\d}} H({\vec{v}}-{\vec{u}};{\vec{v}},{\vec{u}}),\ {\overline{\d}} H(\cdot;{\vec{u}},{\vec{u}})=\d H(\cdot;{\vec{u}}),
\end{equation}
for a scalar value $a$ and the Fr\'echet derivative $\d H(\cdot;{\vec{u}})$ of $H$ at ${\vec{u}}$.
\end{dfn}
For a discrete differential ${\overline{\d}} H$ of a function $H:{\mathbb{R}}^N\rightarrow{\mathbb{R}}$, there exists a discrete gradient ${\overline{\nabla}} H$ such that ${\overline{\nabla}} H({\vec{v}},{\vec{u}})\cdot {\vec{w}}={\overline{\d}} H({\vec{w}};{\vec{v}},{\vec{u}})$.
This relationship is obvious from Definitions~\ref{def:discrete_gradient} and \ref{def:discrete_differential}, and it is a discrete analogue of the chain-rule $\nabla H({\vec{u}})\cdot {\vec{w}}=\d H({\vec{w}};{\vec{u}})$.
Our proposal is to obtain a discrete differential ${\overline{\d}} H$ of the neural network model $H$ using the automatic discrete differentiation algorithm, and thereby, a discrete gradient ${\overline{\nabla}} H$.
The automatic differentiation algorithm depends on the chain rule, product rule, and linearity.
For the functions $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ and $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$, it holds that
\begin{equation}
\textstyle \frac{\partial}{\partial x}(f\circ g)=\frac{\partial f}{\partial g}\frac{\partial g}{\partial x},\ \
\frac{\partial}{\partial x}(fg)=g\frac{\partial f}{\partial x}+f\frac{\partial g}{\partial x},\ \
\frac{\partial}{\partial x}(f+ g)=\frac{\partial f}{\partial x}+\frac{\partial g}{\partial x}.
\label{eq:rules}
\end{equation}
\begin{thm}\label{thm:discrete_differential_rules}
For any $x_1,x_2,\Delta x\in{\mathbb{R}}$ and functions $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ and $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$, the chain-rule, product rule, and linearity for the discrete differential are respectively expressed as
\begin{equation}
\begin{split}
{\overline{\d}} (f\circ g)(\Delta x;x_1,x_2)
&= {\overline{\d}} f(\cdot;g(x_1),g(x_2))\circ {\overline{\d}} g(\Delta x;x_1,x_2),\\
{\overline{\d}} (fg)(\Delta x;x_1,x_2) &\textstyle=\frac{g(x_1)+g(x_2)}{2}{\overline{\d}} f(\Delta x;x_1,x_2)+\frac{f(x_1)+f(x_2)}{2}{\overline{\d}} g(\Delta x;x_1,x_2),\\
{\overline{\d}} (f+g)(\Delta x;x_1,x_2)&={\overline{\d}} f(\Delta x;x_1,x_2)+{\overline{\d}} g(\Delta x;x_1,x_2).
\end{split}
\end{equation}
\end{thm}
For any linear operations such as the fully-connected and convolution layers, a discrete differential is equal to the Fr\'echet derivative because of the linearity.
For an element-wise nonlinear activation function $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$, we employed the following discrete differential~\cite{Gonzalez1996}.
\begin{equation}
{\overline{\d}} f(\Delta x;x_1,x_2)=\begin{cases}
\frac{f(x_1)-f(x_2)}{x_1-x_2}\Delta x & \mbox{if } x_1\neq x_2 \\
\d f(\Delta x;\frac{x_1+x_2}{2}) & \mbox{otherwise}.
\end{cases}\label{eq:discrete_differential_of_activation}
\end{equation}
The product rule is applicable to bilinear operations such as attention, graph convolution, transformer, and metric function~\cite{Devlin2018,Faghri2018}.
Given the above, we propose the automatic discrete differentiation algorithm.
With the algorithm, one can automatically obtain a discrete differential ${\overline{\d}} H$ of a neural network $H$ given two arguments, which is then converted to a discrete gradient ${\overline{\nabla}} H$.
The computational cost is no more than twice of the ordinary automatic differentiation.
The algorithm is applicable to any computational graph such as convolutional neural network~\cite{Greydanus2019} and graph neural network~\cite{Desai2020}, and thereby one can handle extended tasks or further improve the modeling accuracy.
For reference, we introduce the case with a neural network that is composed of a chain of functions in Algorithm~\ref{alg:add} in Appendix~\ref{app:add}.
We call a neural network obtaining a discrete gradient ${\overline{\nabla}} H$ by using the automatic discrete differentiation algorithm \emph{DGNet}, hereafter.
\subsection{Learning and Computation by the Discrete-Time Model}
Using DGNet, we propose a deep energy-based discrete-time physical model
that can learn from the discrete-time data directly as follows.
Given a time series, DGNet accepts two state vectors $u^{(n)}$ and $u^{(n+1)}$ at time steps $n$ and $n+1$, and then it outputs two scalar system energies $H({\vec{u}}^{(n)})$ and $H({\vec{u}}^{(n+1)})$.
The discrete gradient ${\overline{\nabla}} H({\vec{u}}^{(n+1)}, {\vec{u}}^{(n)})$ is obtained by the automatic discrete differentiation algorithm.
The model is trained to minimize the squared error between the left- and right-hand sides of Eq.~\eqref{eq:discrete_system};
\begin{equation}
\textstyle\mbox{minimize\ }
\sum_n
\|
\frac{{\vec{u}}^{(n+1)} - {\vec{u}}^{(n)}}{t^{(n+1)}-t^{(n)}}
- {\overline{G}}({\vec{u}}^{(n+1)}, {\vec{u}}^{(n)}) {\overline{\nabla}} H({\vec{u}}^{(n+1)}, {\vec{u}}^{(n)})\|_2^2\label{eq:discrete_differential_objective}
\end{equation}
Then, the error is back-propagated through the computational graphs including the neural network model, the discrete gradient, and the matrix ${\overline{G}}$ by the ordinary automatic differentiation algorithm.
For training, the computational cost of the proposed scheme in Eq.~(6) is no more than twice of the HNN with the Euler method and typically tens times smaller than that with the adaptive Dormand--Prince method.
Through this learning process, DGNet potentially restores the true gradient $\nabla H$ from the sampled data because the discrete gradient ${\overline{\nabla}} H$ is equal to the true gradient $\nabla H$ when two arguments are equal by Definition~\ref{def:discrete_gradient}.
For a time-series prediction, DGNet predicts the next state implicitly by solving the implicit scheme in Eq.~\eqref{eq:discrete_system} and conserves the energy strictly.
The proposed discrete gradient ${\overline{\nabla}} H$ is time-symmetric, which implies that the proposed method is at least a second--order method~\cite{Quispel1996}.
Higher-order methods can be designed using the composition method (using multiple sub-steps) and the higher-order temporal difference (using multiple steps) as introduced in \cite{Furihata2010}.
Indeed, the training and prediction can be performed in a different manner.
After learning from the finite differences, DGNet provides the gradient $\nabla H$ so it is available for an explicit numerical method, which can be more computationally efficient and be preferable when the learned models are used in existing physics simulators (e.g., Matlab).
When the true time-derivative is known, DGNet can learn it as the previous models did.
Then, it can predict the next step using the discrete gradient implicitly while conserving energy.
\section{Learning of Partial and Ordinary Differential Equations}\label{sec:experiments}
\paragraph{Comparative Models.}
We examined the proposed DGNet and comparative methods.
NODE is a neural network that outputs the time-derivative of the states in a general way~\cite{Chen2018e}.
The HNN is a neural network where the output represents the system energy $H$, and its gradient with respect to the input state ${\vec{u}}$ is used for the time-derivative~\cite{Greydanus2019}.
In our experiments, they were trained from a finite difference between two successive time steps using a numerical integrator, which is similar to some previous studies~\cite{Chen2020a,Saemundsson2020,Tong2020, Zhong2020a, Zhong2020}.
For numerical integrators, we employed the explicit midpoint method (RK2) and the Dormand--Prince method with adaptive time-stepping (ada.~DP); they are second-- and fourth--order explicit Runge--Kutta methods.
Then, the output error was back-propagated through all stages~\cite{Chen2018e}.
In terms of applying the HNN to the discretized PDEs, we generalized it by using the formulation in Section~\ref{sec:physical_systems} and denoted it as the HNN\raisebox{.2\height}{\scalebox{.8}{++}}.
DGNet was trained to minimize the objective in Eq.~\eqref{eq:discrete_differential_objective}; for simplicity, the matrix $G$ of the system was assumed to be known, and we used ${\overline{G}}=G$.
We also employed explicit numerical integrators for DGNet's prediction to reduce the computational cost from the implicit scheme in Eq.~\eqref{eq:discrete_system}.
\paragraph{Hamiltonian PDE.}
We evaluated the models on a Hamiltonian PDE, namely the KdV equation, which is a famous model that has soliton solutions~\cite{Furihata1999,Furihata2001}.
Of the discretized 1-dimensional KdV equation, the system energy $H$ and time evolution are expressed as follows.
\begin{equation}
\textstyle H({\vec{u}})=\Delta x\textstyle \sum_k (-\frac{1}{6}\alpha u_k^3-\frac{1}{2}\beta(D {\vec{u}})_k^2),\
\textstyle \frac{\partial {\vec{u}}}{\partial t}=\textstyle D\nabla H({\vec{u}}) = D(-\frac{1}{2}\alpha({\vec{u}}\odot{\vec{u}})+\beta(D_2 {\vec{u}})),\label{eq:kdv}
\end{equation}
where the subscript $_k$ denotes the $k$-th element, $D$ and $D_2$ denote the first-- and second--order central differences in Eq.~\eqref{eq:first_second_order_difference}, and $\odot$ denotes the element-wise multiplication.
The coefficients $\alpha$ and $\beta$ determine the spatio-temporal scales.
We set $\alpha=-6$, $\beta=1$, the spatial size to 10 space units, and the space mesh size $\Delta x$ to 0.2 .
At $t=0$, we set two solitons, each of which were expressed as $-\frac{12}{\alpha} \kappa^2 \mathrm{sech}^2(\kappa (x - d))$.
$\kappa$ denotes the size randomly drawn from $\mathcal U(0.5,2)$, and $d$ denotes its initial location randomly, which is determined to stay 2.0 space units away from each other.
We employed the discrete gradient method in \cite{Furihata2001} to ensure the energy conservation law.
We simulated the equation with a time step size of $\Delta t = 0.001$ for 500 steps and obtained 100 time series (90 for training and 10 for the test).
Every experiment in this section was done with double precision.
We employed a neural network composed of a 1-dimensional convolution layer followed by two fully-connected layers.
A convolution layer with a kernel size of 3 is enough to learn the central difference.
The matrix $G=D$ was implemented as a 1-dimensional convolution layer with the kernel of $(-1/2\Delta x,0,1/2\Delta x)$ and periodic padding.
Following the study on the HNN~\cite{Greydanus2019}, the activation function was the hyperbolic tangent, the number of hidden channels was 200, and each weight matrix was initialized as a random orthogonal matrix.
Each network was trained using the Adam optimizer~\cite{Kingma2014b} with a batch size of 200 and a learning rate of 0.001 for 10,000 iterations.
\begin{table}[t]
\centering\small
\caption{Results on the PDE datasets.}\label{tab:pde_score}
\begin{tabular}{lllrrrrrr}
\toprule
& \mcb{Integrator} & \mcc{KdV equation} & \mcc{Cahn--Hilliard equation} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-6}\cmidrule(lr){7-9}
\textbf{Model} & \mca{Training} & \mca{Prediction} & \mca{Deriv.} & \mca{Energy} & \mca{Mass} & \mca{Deriv.} & \mca{Energy} & \mca{Mass} \\
\midrule
\multirow{2}{*}{NODE~\cite{Chen2018e}} & RK2 & RK2 & >10000 & >10000 & 2857.81 & 791.25 & >10000 & 914.72 \\
& ada.~\!DP & ada.~\!DP & >10000 & >10000 & 2836.45 & 790.48 & >10000 & 913.96 \\
\midrule
\multirow{2}{*}{HNN\raisebox{.2\height}{\scalebox{.8}{++}}} & RK2 & RK2 & 36.32 & 6.32 & 0.70 & 344.23 & >10000 & 87.55 \\
& ada.~\!DP & ada.~\!DP & \underline{23.27} & 3.01 & 0.34 & \underline{33.03} & 4.89 & 0.80 \\
\midrule
& \upl{4mm} & RK2 & \up{3mm} & 1.84 & 0.28 & \up{2mm} & >10000 & 821.58 \\
DGNet & Eq.~\eqref{eq:discrete_system} & ada.~\!DP & \textbf{17.48} & \textbf{1.60} & \textbf{0.25} & \textbf{7.14} & \textbf{0.34} & \textbf{0.07} \\
& \downl{4mm} & Eq.~\eqref{eq:discrete_system} & \down{3mm} & \textbf{1.60} & \textbf{0.25} & \down{2mm} & \textbf{0.34} & \textbf{0.07} \\
\bottomrule
\multicolumn{9}{L{13cm}}{\footnotesize The best and second best results are emphasized by bold and underlined fonts, respectively. Multiplied by $10^0$ for Deriv.~and by $10^{\!-\!6}$ for Energy of the Cahn--Hilliard equation, and by $10^{\!-\!3}$ for the others.}
\end{tabular}
\vspace*{-3mm}
\end{table}
\begin{wrapfigure}{r}{2.4in}
\vspace*{-3mm}
\includegraphics[scale=1.0]{fig/kdv.pdf}
\vspace*{-7mm}
\caption{KdV equation. (left) Predicted state $u$. (right) Error ($\times20$).}\label{fig:kdv}
\vspace*{-3mm}
\end{wrapfigure}
After training, we examined the mean squared error (MSE) of the time-derivative; we provided an average over 15 trials on Table~\ref{tab:pde_score} (see the column ``Deriv.'').
We omitted the outliers and standard deviations for readability (see Appendix~\ref{appendix:datasets_results} for the full results).
DGNet restored the true time-derivative well.
The HNN\raisebox{.2\height}{\scalebox{.8}{++}}\ employed the adaptive Dormand--Prince method, but it suffered from the gap between the time-derivative and the finite difference.
Nonetheless, the application of the HNN to a PDE system is one of the contributions of this study.
NODE failed to model the equation.
For evaluating the long-term consistency, we predicted the test time series from the initial state ${\vec{u}}^{(0)}$ and obtained the MSE of the total energy and local mass (see the columns ``Energy'' and ``Mass'').
We also visualized the prediction result for each model with the best integrator, which is depicted in Fig.~\ref{fig:kdv}.
DGNet also conserved energy the best with all integrators.
Even though the implicit scheme in Eq.~\eqref{eq:discrete_system} is computationally expensive, DGNet provided the time-derivative for explicit numerical integrators, and it was enough for conserving energy in the present experiment scale (for a longer case, see Appendix~\ref{appendix:datasets_results}).
This result implies that the discrete gradient method provides a good framework for learning from the finite difference; to the best of our knowledge, this is the first time to confirm such contribution of the discrete gradient.
In addition, one might say that the implicit scheme in Eq.~\eqref{eq:discrete_system} is as powerful as the fourth--order integrator with adaptive time stepping even though it is a second--order method.
\paragraph{Dissipative PDE.}
We evaluated the models on a dissipative PDE, namely the Cahn--Hilliard equation.
This equation is derived from free-energy minimization and it describes, for example, the phase separation of copolymer melts~\cite{Furihata1999,Furihata2001}.
The system energy $H$ and time evolution of the discretized 1-dimensional Cahn--Hilliard equation are expressed as follows.
\begin{equation}
\textstyle H({\vec{u}})=\Delta x\sum_k (\frac{1}{4}(u_k^2-1)^2+\gamma\frac{1}{2}(D {\vec{u}})_k^2), \
\textstyle\frac{\partial {\vec{u}}}{\partial t}=D_2\nabla H({\vec{u}})=D_2(({\vec{u}}\!\odot\!{\vec{u}}\!-\!\vec{1})\!\odot\!{\vec{u}}-\gamma D_2 {\vec{u}}),\label{eq:ch}
\end{equation}
where the coefficient $\gamma>0$ denotes the mobility of the monomers.
The mass $u_k$ has an unstable equilibrium at $u_k=0$ (totally melted) and stable equilibria at $u_k=-1$ and $u_k=1$ (totally separated).
We set $\gamma$ to 0.0005, the spatial size to 1, the space mesh size $\Delta x$ to 0.02, the time step size $\Delta t$ to 0.0001, and the initial state $u_k$ to a random sample from $\mathcal U(-0.05,0.05)$.
The other conditions are the same as the case with the KdV equation.
\begin{wrapfigure}{r}{2.4in}
\vspace*{-7mm}
\includegraphics[scale=1.0]{fig/ch.pdf}
\vspace*{-7mm}
\caption{Cahn--Hilliard equation. (left) Predicted state $u$. (right) Error ($\times3$).}\label{fig:ch}
\vspace*{-5mm}
\end{wrapfigure}
We summarized the results in Table~\ref{tab:pde_score} and visualized the prediction result for each model with the best integrator in Fig.~\ref{fig:ch}.
DGNet outperformed the HNN\raisebox{.2\height}{\scalebox{.8}{++}}\ by a large margin.
The Cahn--Hilliard equation is ``stiff''; this implies that the state can change drastically and an explicit integrator requires a much smaller time step size.
The adaptive Dormand--Prince method evaluated the HNN\raisebox{.2\height}{\scalebox{.8}{++}}~50--100 times per time step in the training phase and consumed the proportional computational cost.
However, it did not learn the discrete-time dynamics well; the HNN\raisebox{.2\height}{\scalebox{.8}{++}}\ underestimated the diffusion as shown in Fig.~\ref{fig:ch}.
Conversely, DGNet can estimate the dissipative term well, as expected in Theorem~\ref{thm:discrete_conservation_dissipation}.
\begin{table}[t]
\centering\small
\caption{Results for the ODE datasets}\label{tab:ode_score}
\tabcolsep=1.2mm
\begin{tabular}{lllrrrrrrrr}
\toprule
& \mcb{Integrator} & \mcb{Mass-Spring} & \mcb{Pendulum} & \mcb{2-Body} & \mcb{Real Pendulum} \\
\cmidrule(lr){2-3}\cmidrule(lr){4-5}\cmidrule(lr){6-7}\cmidrule(lr){8-9}\cmidrule(lr){10-11}
\textbf{Model} & \textbf{Training} & \textbf{Prediction} & \mca{Deriv.} & \mca{Energy} & \mca{Deriv.} & \mca{Energy} & \mca{Deriv.} & \mca{Energy} & \mca{Diff.} & \mca{Energy} \\
\midrule
\multirow{2}{*}{NODE} & RK2 & RK2 & 52.68 & 570.32 & 56.67 & 4602.57 & 20.81 & >10000 & \underline{1.38} & 0.62 \\
& ada.~\!DP & ada.~\!DP & 55.74 & 574.06 & 55.40 & 4624.66 & 20.71 & >10000 & \textbf{1.37} & 0.59 \\
\midrule
\multirow{2}{*}{HNN~\cite{Greydanus2019}} & RK2 & RK2 & \textbf{38.22} & 61.25 & 42.49 & 404.24 & \underline{5.39} & 93.88 & 1.42 & 2.86 \\
& ada.~\!DP & ada.~\!DP & 39.92 & 1.74 & 40.88 & 16.55 & 6.21 & 81.84 & 1.41 & 3.44 \\
\midrule
SRNN~\cite{Chen2020a} & leapfrog & leapfrog & 39.47 & 0.69 & \textbf{39.24} & \underline{11.24} & \textbf{4.36} & \textbf{40.37} & (1.38) & (9.63) \\
\midrule
& \upl{4mm} & RK2 & \up{3mm} & 61.26 & \up{3mm} & 743.42 & \up{2mm} & 81.07 & \up{2mm} & 0.86 \\
DGNet & Eq.~\eqref{eq:discrete_system} & ada.~\!DP & \underline{38.50} & \textbf{0.62} & \underline{39.30} & 16.06 & 7.80 & 81.04 & \underline{1.38} & \textbf{0.49} \\
& \downl{4mm} & Eq.~\eqref{eq:discrete_system} & \down{3mm} & \textbf{0.62} & \down{3mm} & \textbf{10.79} & \down{2mm} & \underline{81.03} & \down{2mm} & \underline{0.50} \\
\bottomrule
\multicolumn{11}{L{13cm}}{The best and second best results are emphasized by the bold and underlined fonts, respectively. Multiplied by $10^{\!-\!6}$ for the 2-body dataset and by $10^{\!-\!3}$ for the others.}
\end{tabular}
\vspace*{-5mm}
\end{table}
\paragraph{Hamiltonian Systems.}
We employed Hamiltonian systems that were examined in the original study of the HNN~\cite{Greydanus2019}, namely a mass-spring system, a pendulum system, and a 2-body system.
Because they are natural systems, we used the matrix $G=S=({\tiny\begin{smallmatrix} 0 & I_n \\ -I_n & 0 \end{smallmatrix}})$.
Instead of the time-derivative, we used the finite difference for training like the cases above.
Moreover, we unified the time step size for training and test (see Appendix~\ref{appendix:datasets_results} for details).
The other experimental settings were the same as the original experiments~\cite{Greydanus2019} and the cases above.
Every experiment of ODEs was done with single precision.
Following the symplectic recurrent neural network (SRNN)~\cite{Chen2020a}, we employed the leapfrog integrator and a pair of networks of the same size to represent the potential energy $V({\vec{q}})$ and kinetic energy $T({\vec{p}})$.
The leapfrog integrator is typically applicable to this class.
We summarized the results in Table~\ref{tab:ode_score}.
DGNet sometimes obtained a worse time-derivative error but it always achieved better prediction errors than the HNN; DGNet learned the contour lines of the Hamiltonian $H$ rather than the time-derivative.
DGNet achieved the best results on the long-term predictions in the mass-spring and pendulum datasets and the second-best result in the 2-body dataset.
The SRNN achieved a remarkable result in the 2-body dataset because its network and integrator are specially designed for the separable Hamiltonian, which is a powerful assumption in general.
DGNet for the separable Hamiltonian is a possible future study.
\paragraph{Physical System with Friction.}
We evaluated the models on the real pendulum dataset that were obtained by \citet{Schmidt2009} following the study on the HNN ~\cite{Greydanus2019}.
This dataset contains the angle and angular momentum readings of a pendulum bob.
Since the real pendulum has friction, we used the matrix $G=S-R=(\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix})-(\begin{smallmatrix} 0 & 0 \\ 0 & g \end{smallmatrix})$, where $g$ is an additional parameter that represents the friction and it was initialized to zero.
Solved by a Runge--Kutta method, this model can be regarded as the dissipative symODEN without the control input~\cite{Zhong2020a}.
\begin{wrapfigure}{r}{3.9cm}%
\centering%
\vspace*{-3mm}
\hspace*{-3mm}
\includegraphics[scale=1.0]{fig/real.pdf}
\vspace*{-6mm}
\caption{Results for the real pendulum dataset.}\label{fig:real}
\vspace*{-4mm}
\end{wrapfigure}
We evaluated the MSE of the finite difference (i.e., 1-step prediction, see the column ``Diff.'') and the MSE of the energies in long-term predictions.
The results are summarized in Table~\ref{tab:ode_score} and Fig.~\ref{fig:real}.
While all methods achieved similar errors in the 1-step prediction, the HNN achieved the worst error in the long-term prediction; the HNN overestimated the friction while DGNet estimated the friction term well, as expected in Theorem~\ref{thm:discrete_conservation_dissipation}.
The energy $H$ derived only from the angle and momentum of the pendulum bob does not monotonically decrease because the other components (e.g., the pendulum rod) are ignored.
DGNet estimated the alternative energy dissipating monotonically, and it predicted the states well.
NODE, which can approximate a general ODE, also worked better than the HNN.
For reference, we confirmed that the SRNN failed in modeling the real pendulum dataset because of the lack of a friction term.
\section{Conclusion}
We proposed a discrete-time energy-based physical model.
The approach unified and widely extended the scope of neural networks for physics simulations.
Using the automatic discrete differentiation algorithm, a neural network is able to use the discrete gradient method, learn from the discrete-time data accurately, and admit the important laws of physics in discrete time.
\section*{Broader Impact}
\paragraph{Novel paradigm of mathematical modeling.}
For computing the physical phenomena, one has to build a difference equation in discrete time.
Mathematical models for physics are typically given as differential equations, and they are discretized using numerical integrators (see the lower part of Fig.~\ref{fig:modeling_concept}).
This discretization may destroy the geometrical structure from which the laws of physics follow.
Most previous studies on neural networks for physical phenomena employ this approach~\cite{Chen2020a,Greydanus2019,Saemundsson2020,Tong2020,Zhong2020a,Zhong2020}.
The discrete gradient method is a discrete-time approximation of a continuous-time structure (see the middle part of Fig.~\ref{fig:modeling_concept})~\cite{Celledoni2012,Celledoni2014,Furihata1999,Furihata2001,Furihata2010,Gonzalez1996,Itoh1988,Quispel1996a,Quispel1996}.
It admits the laws of physics in discrete time, but it suffers from the discretization error, too.
This method has been inapplicable in neural networks until this study. We addressed this issue by introducing the automatic discrete differentiation algorithm.
Our approach is defined in discrete time and it learns discrete-time dynamics directly from discrete-time data (see the upper part of Fig.~\ref{fig:modeling_concept}).
As a result, it never suffers from the discretization error even though the modeling error matters.
In this sense, this study provides a novel paradigm for mathematical modeling.
\paragraph{Novel framework of scientific machine learning.}
The proposed approach combines neural networks and geometric integration, in particular, the discrete gradient method that is derived by the automatic discrete differentiation algorithm.
As far as we know, the proposed framework is the first approach that unifies mathematical modeling from the first principles, data-driven modeling, and energetic-property-preserving numerical computations.
From the viewpoint of scientific computing, the latter two may significantly accelerate scientific simulations.
In practical simulations, modeling and numerical computations have been performed separately, while these must be unified because the results of the simulations often require modification of the mathematical models, and vice versa.
In addition, as implemented by PyTorch, our programming codes for the proposed framework are naturally parallelized.
This implementation is the first numerical library that provides parallelized numerical simulations while using the discrete gradient method, which widely accelerates the computation in scientific simulations.
\begin{ack}
Funding in direct support of this work: JST CREST Grant Number JPMJCR1914, JST PRESTO Grant Number JPMJPR16EC, JSPS KAKENHI Grant Number 19K20344, 20K11693 and scholoarship by Yoshida Scholarship Foundation.
\end{ack}
{\small
|
1,116,691,500,500 | arxiv | \section{Introduction and Summary }
A double Wick rotation on the string world-sheet was proposed in
\cite{Arutyunov:2007tc} and now it is well known that it is a very
useful tool for the study of AdS/CFT correspondence, for review see
\cite{Arutyunov:2009ga}. It is crucial that double Wick rotation has
sense in case of the light cone gauge fixed string since only in
this case the double Wick rotation transforms string world-sheet
theory non-trivially. In fact, the thermodynamics of this different
two dimensional quantum field theory gives the solution of the
spectral problem of AdS/CFT when the spectrum of the string on
$AdS_5\times S^5$ is computed with the help of the thermodynamic
Bethe ansatz for this mirror model
\cite{Arutyunov:2009zu,Arutyunov:2009ur,Bombardelli:2009ns,Gromov:2009bc}.
Due to the significance of the double Wick rotation one can ask the
question whether this transformation has more physical meaning.
Such a question was firstly posed
in \cite{Arutyunov:2014cra} and it was shown there that the double Wick
rotation on the world-sheet of the gauge fixed bosonic string is
equivalent to the particular transformation of the metric and NS-NS
two form. This analysis was then extended to the case of the uniform gauge fixed
Green-Schwarz superstring in \cite{Arutyunov:2014jfa} where it was
shown that the double Wick rotation is equivalent to the particular
transformation of metric, NS-NS two form and
dilaton and Ramond-Ramond fields. It was also
shown there that these transformations can be interpreted from a
target space perspective as a combination of $T$ dualities and
analytic continuition, following A. Tsytlin's suggestion.
Due to the fact that the idea that the double Wick rotation could
have deeper physical meaning is very attractive we applied it for
another two dimensional theory which is a low energy effective
action for D1-brane in \cite{Kluson:2015saa}. In this paper we
firstly determined uniform gauge fixed action for D1-brane in
general background. Then we applied double Wick rotation for this
theory and we found that the double Wick rotated action has the same
form as the original one when the background gravitational and NS-NS
two form fields transform as in \cite{Arutyunov:2014cra} while we
found that dilaton and Ramond-Ramond fields transform differently.
We suggested that this discrepancy could be explained by the fact
that D1-brane behaves differently under T-duality transformations
but we leaved the detailed analysis of this issue for future. The
aim of this paper is to answer this question. More precisely, we
would like to see how suggested combinations of T-dualities and Wick
rotation in the target space-time can be performed on the
world-sheet of the fundamental string and D1-brane action. Due to
the fact that the gauge fixing is important for the given procedure
and since this gauge fixing is performed when we fix one of the
components of momenta rather than target space coordinate we perform
T-duality transformation on the level of the Hamiltonian formalism.
In fact, the interpretation of T-duality transformation as canonical
transformation was given in \cite{Alvarez:1994wj} \footnote{For
review, see \cite{Alvarez:1994dn}.} in case of gauge fixed string
action. We generalize this analysis to the case of the Nambu-Gotto
(NG) action which is invariant under two dimensional diffeomorphism
and we obtain celebrated Buscher's rules for the T-duality
transformations of the background metric and NS-NS two form field
\cite{Buscher:1987sk,Buscher:1987qj} \footnote{We would like to
stress that given procedure could be equivalently performed with
Polyakov form of the string action. Explicitly, the Hamiltonian has
the same form as in case of NG string with subtle difference that
now the primary constraints are the momenta conjugate to the
components of the world-sheet metric. Then the requirement of their
preservations gives the secondary constraints $\mathcal{H}_\tau,\mathcal{H}_\sigma$.
Note that these constraints are the primary constraints of the NG
string.}.
Using this result we perform
sequence of T-duality transformations and Wick rotation in the
target space-time and we show that when we impose the uniform gauge
in the resulting Hamiltonian we obtain that the Hamiltonian for the
physical degrees of freedom is the same as in case of the double
Wick rotated gauge fixed theory with an important exception that the
sequence of T-duality transformation and Wick rotation do not
generate a change $B_{\mu\nu}\rightarrow -B_{\mu\nu}$ where
$B_{\mu\nu}$ are components of NS-NS two form fields that are
transverse to the directions where corresponding T-duality and Wick
rotations were performed. On the other hand it is well known that
the transformation $B\rightarrow -B$ is the symmetry of supergravity
equations of motion with the absence of RR fields. So that whenever
string propagates on background with $g,B$ that solves the
supergravity equations of motion it can propagates on the background
with $g,-B$. So that without lost of consistency we can augment the
sequence T-duality and Wick rotation with an additional operation
$B\rightarrow -B$. Then this extended sequence of operation is
equivalent to the double Wick rotation of the uniform gauge fixed
string theory\footnote{The discussion is more subtle in case of
non-zero RR fields and for detailed discussion see
\cite{Arutyunov:2014jfa}.}.
As the next step we extend this analysis to the case of D1-brane
effective action which consists of Dirac-Born-Infeld (DBI) action
and Wess-Zummino (WZ) term. We firstly apply the canonical
transformations for this theory and we derive how the background
fields transform. However we should stress that we are able to
perform such an analysis when we presume that the electric flux is
fixed. This is a natural requirement since it is well known that the
electric flux is proportional to the number of the fundamental
strings and we would like to compare two actions when the number of
fundamental strings is the same. We find that under this canonical
transformation the dilation and Ramond-Ramond fields transform
differently than we should expect from T-duality rules which has
very simple explanation. We argue that the final action corresponds
to the fundamental string action moving in T-dual background when
however the components of the background fields that appear in
Buscher's rules correspond to the S-dual background when we use the
equivalence of the D1-brane action (with constant electric flux) and
fundamental string action in S-dual background
\cite{Tseytlin:1996it}.
Due to this
fact we can now explain the discrepancy that we found in the
previous paper \cite{Kluson:2015saa}. Explicitly, we perform the
sequence of canonical transformations and Wick rotation in case of
Hamiltonian for D1-brane. Then we perform the uniform gauge fixing
and we again find that the Hamiltonian for the physical degrees of
freedom implies the transformation rules for the target space fields
that have the same form as in case of the double Wick rotation
performed on the uniform gauge fixed D1-brane effective action again
with an exception that components of Ramond-Ramond two forms that
are transverse to the directions of duality transformations do not
transform. On the other hand we can again argue for the existence of
the symmetry $C^{(2)}\rightarrow -C^{(2)}, B\rightarrow -B$ of the
solutions of the supergravity equations of motion so that we find
exact equivalence between double Wick rotation of the uniform gauge
fixed D1-brane action and sequence of "canonical
transformation-target space double Wick rotation-canonical
transformation-$(C^{2}\rightarrow -C^{(2)} \ , B\rightarrow -B)$".
Let us outline results derived in given paper. We generalize the
canonical description of T-duality transformations for the string
with no gauge fixing imposed. Then we show that the sequence of T
dualities and Wick rotation in target space-time together with the
uniform gauge fixing leads to the theory that is equivalent to the
double Wick rotated gauge fixed theory. On the other hand we show
that given procedure when applied to D1-brane action again gives
theory that is equivalent to the double Wick rotated uniform gauge
fixed theory which however cannot be interpreted as T-duality
transformations.
This paper is organized as follows. In the next section
(\ref{second}) we determine T-duality transformations of the
background fields as a
canonical transformation of the Nambu-Gotto action for the
fundamental string.
In section (\ref{third}) we perform sequence of T-duality
transformations and Wick rotation in the Hamiltonian formulation of
given theory and discuss its relation with the double Wick rotated
uniform gauge fixed action.
Section (\ref{fourth}) is devoted to the
generalization of given procedure to the case of D1-brane and
finally in section (\ref{fifth}) we perform the combinations of
T-duality transformations and Wick rotation in case of Hamiltonian
formulation of D1-brane theory.
\section{T-duality for Fundamental String in Canonical
Formalism}\label{second}
We would like to perform the analysis of
T-duality as the canonical transformation of the NG action,
following \cite{Alvarez:1994wj}.
We start with the Nambu-Gotto action for the fundamental string
\begin{equation}\label{SNG}
S_{NG}=-\frac{1}{2\pi\alpha'}\int d\tau d\sigma \left[ \sqrt{-\det
g} +\frac{1}{2}\epsilon^{\alpha\beta}B_{MN}\partial_\alpha
x^M\partial_\beta x^N\right] \ ,
\end{equation}
where $g_{\alpha\beta}=G_{MN}\partial_\alpha x^M
\partial_\beta x^N \ , \epsilon^{\tau\sigma}=-\epsilon^{\sigma\tau}=1$ and where $x^M,
M=0,\dots,d$ label the embedding coordinates of the string. Further,
$G_{MN}(x),B_{MN}(x)$ are components of the background gravitational
and NS-NS fields respectively.
As the first step we proceed to the Hamiltonian formulation of the
action (\ref{SNG}). From (\ref{SNG}) we obtain the conjugate momenta
\begin{eqnarray}\label{pMNG}
p_M=-\frac{1}{2\pi\alpha'}G_{MN}\partial_\alpha x^N g^{\alpha\tau}
\sqrt{-\det g }-\frac{1}{2\pi\alpha'}B_{MN}\partial_\sigma x^N \ .
\nonumber \\
\end{eqnarray}
If we now define
$\Pi_M=p_M+\frac{1}{2\pi\alpha'}B_{MN}\partial_\sigma x^N$ we obtain
from (\ref{pMNG}) following primary constraint
\begin{eqnarray}\label{mHtau}
\mathcal{H}_\tau=(2\pi\alpha')\Pi_M
G^{MN}\Pi_N+\frac{1}{(2\pi\alpha')}G_{MN}\partial_\sigma x^M
\partial_\sigma x^N=0
\nonumber \\
\end{eqnarray}
together with the spatial diffeomorphism constraint
\begin{equation}\label{mHsigma}
\mathcal{H}_\sigma=p_M\partial_\sigma x^M \ .
\end{equation}
Then it can be shown that the bare Hamiltonian density defined as
$H_B=p_M\partial_\tau x^M-\mathcal{L}$ vanishes identically and the extended
Hamiltonian density is a sum of the primary constraints of the
theory
\begin{equation}
\mathcal{H}=\lambda_\tau \mathcal{H}_\tau+\lambda_\sigma \mathcal{H}_\sigma \ ,
\end{equation}
where $\lambda_\tau,\lambda_\sigma$ are Lagrange multipliers
corresponding to the constraints $\mathcal{H}_\tau\approx 0 \ , \mathcal{H}_\sigma
\approx 0$. It can be further shown that $\mathcal{H}_\tau,\mathcal{H}_\sigma$ are
the first class constraints that are generators of two dimensional
diffeomorphism of the world-sheet.
Let us presume that
there is a direction in the target space-time that is invariant
under constant shift
\begin{equation}
\theta\rightarrow \theta+\epsilon \ , \quad \epsilon=\mathrm{const}
\ .
\end{equation}
In other words all background fields do not depend on $\theta$. Our
goal is to perform the canonical transformation from $\theta$ to
$\tilde{\theta}$. Let us presume that this generating function has the form
\begin{equation}\label{defG}
G(\theta,\tilde{\theta})=\frac{1}{4\pi\alpha'}\int d\sigma (
\partial_\sigma \theta \tilde{\theta}-\theta \partial_\sigma\tilde{\theta}) \ ,
\end{equation}
where we presume that $\theta$ has canonical dimension
$[\theta]=\mathrm{length}$. Let us denote the momentum conjugate to
$\tilde{\theta}$ as $p_{\tilde{\theta}}$. From the definition of the canonical
transformations we derive following relations between canonical
momenta $p_\theta$ and $p_{\tilde{\theta}}$
\begin{eqnarray}
p_{\tilde{\theta}}&=&-\frac{\delta G}{\delta
\tilde{\theta}}=-\frac{1}{2\pi\alpha'}\partial_\sigma \theta \ , \nonumber
\\
p_\theta&=&\frac{\delta G}{\delta \theta}= -\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\theta} \ .
\nonumber \\
\end{eqnarray}
Now we obtain canonically dual Hamiltonian when we replace
$\partial_\sigma \theta$ with $-(2\pi\alpha') p_{\tilde{\theta}}$ and
$p_\theta$ with $-\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\theta}$ in
$\mathcal{H}_\tau$ and $\mathcal{H}_\sigma$ given in (\ref{mHtau}) and
(\ref{mHsigma}).
Explicitly, we find
\begin{eqnarray}\label{mHtaudual}
\tilde{\mH}_\tau&=&
(2\pi\alpha') \hat{\Pi}_\mu G^{\mu\nu}\hat{\Pi}_\nu-2(2\pi\alpha') \hat{\Pi}_\mu
G^{\mu\nu}B_{\nu\theta} p_{\tilde{\theta}}+ (2\pi\alpha')
p_{\tilde{\theta}}B_{\mu\theta}G^{\mu\nu}B_{\nu\theta}p_{\tilde{\theta}}-\nonumber
\\
&-&2\hat{\Pi}_\mu G^{\mu\theta}\partial_\sigma \tilde{\theta}+2 \hat{\Pi}_\mu
G^{\mu\theta} B_{\theta \nu}\partial_\sigma x^\nu +
2B_{\mu\theta}p_{\tilde{\theta}}G^{\mu\theta}
\partial_\sigma \tilde{\theta}-
2B_{\mu\theta} p_{\theta} G^{\mu\theta}
B_{\theta\nu}\partial_\sigma x^\nu+\nonumber \\
&+&\frac{1}{2\pi\alpha'}G^{\theta\theta}(\partial_\sigma
\tilde{\theta})^2-2\frac{1}{2\pi\alpha'}
\partial_\sigma \tilde{\theta} G^{\theta\theta}B_{\theta\mu}\partial_\sigma
x^\mu+ \frac{1}{(2\pi\alpha')} B_{\theta\mu}\partial_\sigma x^\mu
G^{\theta\theta}B_{\theta\nu}\partial_\sigma x^\nu+\nonumber \\
&+&(2\pi\alpha')G_{\theta\theta}
p_{\tilde{\theta}}^2-2 G_{\mu \theta}\partial_\sigma x^\mu
p_{\tilde{\theta}}+\frac{1}{(2\pi\alpha')}G_{\mu\nu}\partial_\sigma x^\mu
\partial_\sigma x^\nu \ , \nonumber \\
\tilde{\mH}_\sigma&=& p_\mu\partial_\sigma x^\mu+p_{\tilde{\theta}}\partial_\sigma
\tilde{\theta} \ , \nonumber \\
\end{eqnarray}
where
\begin{equation}
\hat{\Pi}_\mu=p_\mu+\frac{1}{2\pi\alpha'}B_{\mu\nu}\partial_\sigma x^\nu
\ ,
\end{equation}
$\mu,\nu=0,2,\dots,d$ . We see from (\ref{mHtaudual}) that it is
very difficult to identify the theory in dual picture. To do this it
is more instructive to proceed to the Lagrangian formulation of the
theory. Explicitly, with the help of (\ref{mHtaudual}) we derive
following relations
\begin{eqnarray}
\partial_\tau x^\mu&=&\pb{x^\mu,H}=
2\lambda_\tau[(2\pi\alpha')G^{\mu\nu}\hat{\Pi}_\nu-(2\pi\alpha')
G^{\mu\nu}B_{\nu\theta}p_{\tilde{\theta}}-G^{\mu\theta}\partial_\sigma
\tilde{\theta}+G^{\mu \theta}B_{\theta\nu}\partial_\sigma
x^\nu]+\lambda_\sigma \partial_\sigma x^\mu \ , \nonumber \\
\partial_\tau \tilde{\theta}&=&\pb{\tilde{\theta},H}=
2\lambda_\tau[-(2\pi\alpha')\Pi_\mu
G^{\mu\nu}B_{\nu\theta}+(2\pi\alpha')B_{\mu\theta}G^{\mu\nu}B_{\nu\theta}
p_{\tilde{\theta}}+B_{\mu\theta}G^{\mu\theta}\partial_\sigma
\tilde{\theta}-\nonumber \\
&-&B_{\mu\theta}G^{\mu\theta}B_{\theta\nu}\partial_\sigma
x^\nu+(2\pi\alpha')G_{\theta\theta}p_{\tilde{\theta}}- G_{\theta
\nu}\partial_\sigma x^\nu]+\lambda_\sigma
\partial_\sigma \tilde{\theta} \ . \nonumber \\
\end{eqnarray}
Then after some algebra we find
\begin{eqnarray}\label{gPi}
g^{\mu\nu}\hat{\Pi}_\nu&=&\frac{1}{2(2\pi\alpha')\lambda_\tau}(\mathbf{X}^\mu+2\lambda_\tau
\mathbf{V}^\mu+ 2(2\pi\alpha')\lambda_\tau
G^{\mu\nu}B_{\nu\theta}p_{\tilde{\theta}}) \ , \nonumber \\
p_{\tilde{\theta}}&=&\frac{1}{2(2\pi\alpha')\lambda_\tau G_{\theta\theta}}(
\Theta+ B_{\mu\theta}\mathbf{X}^\mu
+2\lambda_\tau G_{\mu\theta}\partial_\sigma x^\mu) \ ,
\nonumber \\
\end{eqnarray}
where
\begin{equation}\label{defXmu}
\mathbf{V}^\mu=G^{\mu\theta}\partial_\sigma \tilde{\theta}- G^{\mu\theta}
B_{\theta\nu}\partial_\sigma x^\nu \ , \quad \mathbf{X}^\mu=\partial_\tau
x^\mu-\lambda_\sigma \partial_\sigma x^\mu \ , \quad
\Theta=\partial_\tau \tilde{\theta}-\lambda_\sigma \partial_\sigma \tilde{\theta}
\ .
\end{equation}
In order to express $\hat{\Pi}_\mu$ from (\ref{gPi}) we have to find the
inverse matrix to $G^{\mu\nu}$. Recall that by definition
\begin{equation}\label{defgMN}
G_{MN}G^{NK}=\delta_M^K \ , \quad G_{\mu M}G^{N\nu}= G_{\mu
\rho}G^{\rho N}+G_{\mu\theta}G^{\theta\nu}=\delta_\mu^\nu
\end{equation}
and hence we see that $G_{\mu\nu}$ is not inverse to $G^{\mu\nu}$.
It turns out that given matrix has the form
\begin{equation}
h_{\mu\nu}=G_{\mu\nu}-\frac{G_{\mu
\theta}G_{\nu\theta}}{G_{\theta\theta}}
\end{equation}
as can be easily seen from (\ref{defgMN})
\begin{eqnarray}
h_{\mu\nu}G^{\nu\rho}
=\delta_\mu^\rho \ , \quad
G^{\theta\mu}h_{\mu\nu}=-\frac{G_{\theta\nu}}{G_{\theta\theta}} \ .
\nonumber \\
\end{eqnarray}
Then we obtain
\begin{equation}
\hat{\Pi}_\mu=\frac{1}{2(2\pi\alpha')\lambda_\tau}h_{\mu\nu}
(\mathbf{X}^\nu+2\lambda_\tau \mathbf{V}^\nu+ 2(2\pi\alpha')\lambda_\tau
G^{\nu\rho}B_{\rho\theta}p_{\tilde{\theta}}) \
\end{equation}
and after some algebra we find the Lagrangian for dual theory in the
form
\begin{eqnarray}\label{LTdual}
\tilde{\mathcal{L}}&=&p_{\tilde{\theta}}\partial_\tau \tilde{\theta}+p_\mu\partial_\sigma
x^\mu-
\lambda_\tau \tilde{\mH}_\tau-\lambda_\sigma \tilde{\mH}_\sigma= \nonumber \\
&=&\frac{1}{4(2\pi\alpha')\lambda_\tau}\left(\tilde{g}_{\tau\tau}+2\lambda_\sigma
\tilde{g}_{\tau\sigma}+\lambda_{\sigma}^2
\tilde{g}_{\sigma\sigma}\right)-\frac{1}{2\pi\alpha'}
\lambda_\tau \tilde{g}_{\sigma\sigma} -\nonumber \\
&-&\frac{1}{2\pi\alpha'}\partial_\tau x^\mu
\tilde{B}_{\mu\nu}\partial_\sigma x^\nu
-\frac{1}{2\pi\alpha'}\partial_\tau \tilde{\theta}
\tilde{B}_{\tilde{\theta}\mu}\partial_\sigma x^\mu- \frac{1}{2\pi\alpha'}
\partial_\tau x^\mu \tilde{B}_{\mu\tilde{\theta}}\partial_\sigma \tilde{\theta}
\ , \nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\tilde{g}_{\alpha\beta}=\tilde{G}_{\tilde{\theta}\ttheta}\partial_\alpha
\tilde{\theta}\partial_\beta \tilde{\theta}+ \tilde{G}_{\tilde{\theta}\mu}\partial_\alpha
\tilde{\theta}\partial_\beta x^\mu+\tilde{G}_{\mu \tilde{\theta}}\partial_\alpha x^\mu
\partial_\beta \tilde{\theta}+\tilde{G}_{\mu\nu}\partial_\alpha x^\mu
\partial_\beta x^\nu \ ,
\nonumber \\
\end{eqnarray}
and where we defined T-dual components of the metric and NS-NS two
form
\begin{eqnarray}
\tilde{G}_{\mu\nu}&=&h_{\mu\nu}+\frac{B_{\mu\theta}B_{\nu\theta}}{G_{\theta\theta}}
\ ,
\tilde{G}_{\mu\tilde{\theta}}=\tilde{G}_{\tilde{\theta}\mu}=\frac{B_{\mu\theta}}{G_{\theta\theta}}
\ , \tilde{G}_{\tilde{\theta}\ttheta}=\frac{1}{G_{\theta\theta}}\nonumber \\
\tilde{B}_{\mu\nu}&=& B_{\mu\nu}+\frac{B_{\mu\theta}G_{\theta
\nu}}{G_{\theta\theta}} -\frac{G_{\mu\theta} B_{\theta
\nu}}{G_{\theta\theta}} \ , \quad
\tilde{B}_{\tilde{\theta}\mu}=-\tilde{B}_{\mu\tilde{\theta}}=-\frac{G_{\theta\mu}}{G_{\theta\theta}}
\nonumber
\\
\end{eqnarray}
that coincide with Buscher's transformations
\cite{Buscher:1987sk,Buscher:1987qj}.
In order to have more familiar form of the Lagrangian density we
solve the equations of motion for $\lambda_\tau$ and
$\lambda_\sigma$ that follow from (\ref{LTdual}). Explicitly, the
equation of motion for $\lambda_\sigma$ has the form
\begin{equation}\label{eqlambdasigma}
\tilde{g}_{\tau\sigma}+\lambda_\sigma \tilde{g}_{\sigma\sigma}=0
\end{equation}
while the equation of motion for $\lambda_\tau$ takes the form
\begin{equation}
-\frac{1}{4\lambda^2_\tau}[\tilde{g}_{\tau\tau}+2\lambda_\sigma
\tilde{g}_{\tau\sigma}+\lambda_{\sigma}^2
\tilde{g}_{\sigma\sigma}]+\tilde{g}_{\sigma\sigma}=0
\end{equation}
that together with (\ref{eqlambdasigma}) implies
\begin{equation}
\lambda_\tau=\frac{\sqrt{\tilde{g}_{\tau\sigma}^2-
\tilde{g}_{\tau\tau}\tilde{g}_{\sigma\sigma}}}{2\tilde{g}_{\sigma\sigma}} \ .
\end{equation}
Inserting these expressions into the Lagrangian density
(\ref{LTdual}) we obtain the final result
\begin{equation}
\tilde{\mathcal{L}}=-\frac{1}{2\pi\alpha'}\left(\sqrt{-\det \tilde{g}}
+\frac{1}{2}\epsilon^{\alpha\beta}\tilde{b}_{\alpha\beta}\right)
\end{equation}
that is the Lagrangian density for Nambu-Gotto string in T-dual
background.
\section{T-duality and Double Wick rotation}\label{third}
We have shown in the previous section that canonical transformation
in the Hamiltonian formulation of NG string gives the action for the
string in T-dual background. Using this result we focus in this
section on the relation between T-duality and double Wick rotation.
We presume that the background has the form
\cite{Arutyunov:2013ega,Arutyunov:2014ota,Arutyunov:2014cra}
\begin{eqnarray}\label{lightconemetric}
ds^2&=&G_{MN}dx^M
dx^N=G_{tt}dt^2+G_{\varphi\varphi}d\varphi^2+G_{\mu\nu}dx^\mu dx^\nu \ , \nonumber \\
B&=&B_{MN}dX^M dX^N=B_{\mu\nu}dx^\mu dx^\nu \ , \nonumber \\
\end{eqnarray}
where $\mu,\nu$ denote the transverse directions.
Following
\cite{Arutyunov:2013ega,Arutyunov:2014ota} we introduce light cone
coordinates
\begin{equation}\label{defxplus}
x^-=\varphi-t \ , \quad x^+=(1-a)t+a\varphi \
\end{equation}
with inverse relations
\begin{equation}
t=x^+-ax^- \ , \quad \varphi=x^++(1-a)x^- \ .
\end{equation}
Then corresponding metric components have the form
\begin{equation}\label{Gplusplus}
G_{++}=G_{tt}+G_{\varphi\varphi} \ , \quad
G_{--}=G_{tt}a^2+(1-a)^2G_{\varphi\varphi} \ , \quad G_{+-}= -a
G_{tt}+(1-a)G_{\varphi\varphi} \
\end{equation}
with inverse
\begin{eqnarray}\label{Gplusplusin}
G^{++}&=&
\frac{G_{tt}a^2+(1-a)^2G_{\varphi\varphi}}{G_{tt}G_{\varphi\varphi}}
\ , \quad G^{--}=
\frac{G_{tt}+G_{\varphi
\varphi}}{G_{tt}G_{\varphi\varphi}} \ ,
\nonumber \\
G^{+-}&=&
\frac{aG_{tt}-(1-a)G_{\varphi\varphi}}{G_{tt}G_{\varphi\varphi}} \
.
\nonumber \\
\end{eqnarray}
In the light cone coordinates the Hamiltonian and diffeomorphism
constraints have the form
\begin{eqnarray}\label{mHtaustring}
\mathcal{H}_\tau&=&(2\pi\alpha')p_+
G^{++}p_++2(2\pi\alpha')p_+G^{+-}p_-+(2\pi\alpha')p_-G^{--}p_-+
\nonumber \\
&+& \frac{1}{2\pi\alpha'}[G_{++}(\partial_\sigma x^+)^2+2G_{+-}
\partial_\sigma x^+\partial_\sigma x^-+G_{--}(\partial_\sigma
x^-)^2]+\mathcal{H}_x \ , \nonumber \\
\mathcal{H}_\sigma&=&p_+\partial_\sigma x^++p_-\partial_\sigma x^-+p_\mu
\partial_\sigma x^\mu \ , \nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{H}_x=(2\pi\alpha')\Pi_\mu G^{\mu\nu}\Pi_\nu+\frac{1}{2\pi\alpha'}
G_{\mu\nu}\partial_\sigma x^\mu \partial_\sigma x^\nu \ , \quad
\Pi_\mu=p_\mu+\frac{1}{2\pi\alpha'}B_{\mu\nu}\partial_\sigma x^\nu \
. \nonumber
\\
\end{eqnarray}
Now we are ready to study the relation between T-duality and double
Wick rotation. Let us presume that the background does not depend on
$x^-$. Our goal is to perform the canonical transformation from
$x^-$ to $\psi$, where, following discussion presented in previous
section, the generating function has the form
\begin{equation}
G(\theta,\psi)=\frac{1}{4\pi\alpha'}\int d\sigma (
\partial_\sigma x^- \psi-x^- \partial_\sigma \psi) \ .
\end{equation}
Let us denote the
momentum conjugate to $\psi$ as $p_{\psi}$. Then we obtain
\begin{eqnarray}
p_{\psi}=-\frac{1}{2\pi\alpha'}\partial_\sigma x^- \ , \quad p_{-}=
-\frac{1}{2\pi\alpha'}\partial_\sigma \psi \
\nonumber \\
\end{eqnarray}
so that we obtain T-dual Hamiltonian when we replace
$\partial_\sigma x^-$ with $-(2\pi\alpha')p_{\psi}$ and $p_-$ with
$-\frac{1}{2\pi\alpha'}\partial_\sigma \psi$ and hence
\begin{eqnarray}\label{mHtaustringTdual}
\tilde{\mH}_\tau&=& (2\pi\alpha')
p_+G^{++}p_+-2p_+G^{+-}\partial_\sigma\psi
+\frac{1}{2\pi\alpha'}G^{--}(\partial_\sigma
\psi)^2+\frac{1}{2\pi\alpha'}
\left(G_{++}(\partial_\sigma x^+)^2+\right.\nonumber \\
&-& \left. 4\pi\alpha' G_{+-}\partial_\sigma x^+
p_{\psi}+(2\pi\alpha')^2 G_{--}(p_\psi)^2\right)+ \mathcal{H}_x \ ,
\nonumber \\
\tilde{\mH}_\sigma &=&p_+\partial_\sigma x^++p_\psi\partial_\sigma
\psi+p_\mu
\partial_\sigma x^\mu \ . \nonumber \\
\end{eqnarray}
Then following \cite{Arutyunov:2014cra} we perform analytic
continuation in the target space-time
\begin{equation}\label{analcon}
(x^+,\psi)\rightarrow (i\tilde{\psi}, -i\tilde{x}^+)
\ , \quad (p_+,p_\psi)\rightarrow (-i\tilde{p}_\psi, i\tilde{p}_+) \ .
\end{equation}
so that we obtain
\begin{eqnarray}
\tilde{\mH}_\tau&=& -(2\pi\alpha')\tilde{p}_\psi G^{++}\tilde{p}_\psi+2 \tilde{p}_\psi
G^{+-}\partial_\sigma \tilde{x}^+
-\frac{1}{2\pi\alpha'}G^{--}(\partial_\sigma \tilde{x}^+)^2 +\nonumber \\
&+&\frac{1}{2\pi\alpha'} \left(-G_{++}(\partial_\sigma \tilde{\psi})^2+
4\pi\alpha' G_{+-}\partial_\sigma \tilde{\psi}
\tilde{p}_{+}-(2\pi\alpha')^2G_{--}(\tilde{p}_+)^2\right)+ \mathcal{H}_x \ ,
\nonumber \\
\tilde{\mH}_\sigma &=&p_+\partial_\sigma x^++\tilde{p}_\psi\partial_\sigma
\psi+p_\mu
\partial_\sigma x^\mu \ . \nonumber \\
\end{eqnarray}
Note that the way how the conjugate momenta $p_+,p_\psi$ transform
under the analytic continuation is given by the requirement that
all terms in $\tilde{\mH}_\sigma$ come with $+$ sign since we demand that
the string theory is invariant under world-sheet diffeomorphism and
hence Wick rotated $\tilde{\mH}_\sigma$ should have the same form as the
original one.
Finally we perform T-duality transformation along $\tilde{\psi}$ direction
that gives
\begin{equation}
\tilde{p}_{\psi}=-\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\phi} \ , \quad
p_{\tilde{\phi}}=-\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\psi} \
\end{equation}
so that we obtain the final form of the Hamiltonian and
diffeomorphism constraints
\begin{eqnarray}
\tilde{\mH}_\tau&=& -\frac{1}{2\pi\alpha'}(\partial_\sigma \tilde{\phi})^2 G^{++}
-\frac{2}{2\pi\alpha'}\partial_\sigma \tilde{\phi} G^{+-}\partial_\sigma
\tilde{x}^+
-\frac{1}{2\pi\alpha'}G^{--}(\partial_\sigma \tilde{x}^+)^2 +\nonumber \\
&+&(2\pi\alpha') \left(-G_{++}p_{\tilde{\phi}}^2- 2G_{+-}p_{\tilde{\phi}}
\tilde{p}_{+}-G_{--}(\tilde{p}_+)^2\right)+ \mathcal{H}_x \ ,
\nonumber \\
\tilde{\mH}_\sigma &=&p_+\partial_\sigma x^++\tilde{p}_\psi\partial_\sigma
\psi+p_\mu
\partial_\sigma x^\mu \ . \nonumber \\
\end{eqnarray}
In order to find the Hamiltonian for the physical degrees of freedom
we have to fix the gauge. It turns out that it is natural to use
uniform gauge fixing
\begin{equation}
p_{\tilde{\phi}}=\frac{1}{2\pi\alpha'} \ , \quad x^+=\tau \ .
\end{equation}
Then from $\tilde{\mH}_\sigma=0$ we find
\begin{equation}
\partial_\sigma \tilde{\phi}=-(2\pi\alpha')(p_\mu\partial_\sigma x^\mu)
\end{equation}
and hence the Hamiltonian constraint is equal to
\begin{eqnarray}\label{mHtaufixed}
\tilde{\mH}_\tau&=& -(2\pi\alpha')(\partial_\sigma x^\mu p_\mu)^2 G^{++}
-\frac{1}{2\pi\alpha'}G_{++}- 2G_{+-} \tilde{p}_{+}-(2\pi\alpha')
\tilde{p}_+G_{--}\tilde{p}_++ \mathcal{H}_x=0 \ .
\nonumber \\
\end{eqnarray}
Note that due to the gauge fixing the constraints
$\tilde{\mH}_\tau,\tilde{\mH}_\sigma$ vanish strongly and hence (\ref{mHtaufixed})
serves as the quadratic equation for $\tilde{p}_+$. In fact, $-\tilde{p}_+$
should be identified as the Hamiltonian density for the physical
degrees of freedom after gauge fixing.
Now we would like to compare the equation (\ref{mHtaufixed}) with
the equation that defines the Hamiltonian density for the physical
degrees of freedom for of the uniform gauge fixed string. Note that
this gauge is imposed in the Hamiltonian formulation of the string
we identify $p_-=\frac{1}{2\pi\alpha'}$. Equivalently we can impose
given gauge in T-dual theory when we identify $\psi$ with $\sigma$
\cite{Kruczenski:2004cn}. This construction has an advantage since
it does not require to go to the Hamiltonian formulation of given
theory which could be extremely difficult in case of Green-Schwarz
action. However in our case we can either choose
the Hamiltonian constraint (\ref{mHtaustring}) and impose the
gauge $p_-=\frac{1}{2\pi\alpha'},x^+=\tau$ or use T-dual Hamiltonian
constraint (\ref{mHtaustringTdual}) with the following gauge fixing
functions
\begin{equation}\label{Tdualfix}
\psi=\sigma \ , x^+=\tau \ .
\end{equation}
We choose the second possibility and using (\ref{Tdualfix}) in
$\mathcal{H}_\sigma$ we obtain $p_\psi=-(\partial_\sigma x^\mu p_\mu)$ that
together with (\ref{Tdualfix}) implies that (\ref{mHtaustringTdual})
has the form
\begin{eqnarray}\label{HfixTdual}
\mathcal{H}_\tau= (2\pi\alpha')p_+G^{++}p_+-2p_+G^{+-}
+\frac{1}{2\pi\alpha'} G^{--} +(2\pi\alpha')
G_{--}(p_\mu\partial_\sigma x^\mu)^2+ \mathcal{H}_x=0 \ .
\nonumber \\
\end{eqnarray}
This is quadratic equation for $p_+$. Comparing (\ref{HfixTdual})
with (\ref{mHtaufixed}) we see that they have the same form when we
define metric components
\begin{equation}
\tilde{G}^{++}=-G_{--} \ , \quad \tilde{G}^{--}=-G_{++} \ , \quad \tilde{G}^{+-}=
G_{+-} \ .
\end{equation}
In other words, the sequence of T-dualities and analytic
continuation in target space-time implies the transformation of the
components of the target metric that has the same form as the double
Wick rotation in the uniform gauge fixed bosonic string
\cite{Arutyunov:2014cra}. On the other hand we also see that
components of NS-NS two form that are transverse to the directions
where these dualities were performed do not transform. At this place
we see the difference with the double Wick rotation of the gauge
fixed action \cite{Arutyunov:2014cra} since in this case these
components change the sign. In order to resolve this issue we can
extend the sequence of T-duality transformation and Wick rotation
with an additional transformation $B\rightarrow -B$ since as we
argued in the introduction whenever the background field $B$ is
solution of the supergravity equations of motion so $-B$ is too. In
other words we have the equivalence between world-sheet double Wick
rotation and the sequence: "T-duality-target space Wick
rotation-T-duality-$B\rightarrow -B$.
\section{D1-brane and Duality Transformation}\label{fourth}
In this section we apply the same ideas to the case of DBI and WZ
action for D1-brane. Let us start with D1-brane action
\begin{eqnarray}\label{SDbrane}
S&=&-T_{D1}\int d\tau d\sigma e^{-\Phi} \sqrt{-\det
(g_{\alpha\beta}+b_{\alpha\beta}+
(2\pi\alpha')F_{\alpha\beta})}+\nonumber \\
&+& T_{D1}\int d\tau d\sigma[
C^{(0)}(b_{\tau\sigma}+(2\pi\alpha')F_{\tau\sigma})+C_{\tau\sigma}^{(2)}]
\ ,
\nonumber \\
\end{eqnarray}
where
\begin{equation}
g_{\alpha\beta}=G_{MN}\partial_\alpha x^M\partial_\beta x^N \ ,
\quad b_{\alpha\beta}=B_{MN}\partial_\alpha x^M\partial_\beta x^N \
, \quad C^{(2)}_{\tau\sigma}=C_{MN}^{(2)}\partial_\tau
x^M\partial_\sigma x^N \ ,
\end{equation}
and where $x^M(\tau,\sigma)$ are embedding coordinates for D1-brane
in given background. Further, $F_{\alpha\beta}=\partial_\alpha
A_\beta-
\partial_\beta A_\alpha$ is the field strength of the world-volume
gauge field $A_\alpha,\alpha=\tau,\sigma$. Finally $T_{D1}$ is
D1-brane tension $T_{D1}=\frac{1}{2\pi\alpha'}$.
Before we proceed to the Hamiltonian formulation of the action
(\ref{SDbrane}) it is useful to use following formula
\begin{equation}
\det (g_{\alpha\beta}+b_{\alpha\beta}+(2\pi\alpha')
F_{\alpha\beta})= \det g+(b_{\tau\sigma}+(2\pi\alpha')
F_{\tau\sigma})^2 \
\end{equation}
that holds in two dimensions only. Then from the action
(\ref{SDbrane}) we find momenta conjugate to $x^M,A_\sigma$ and
$A_\tau$ respectively
\begin{eqnarray}
p_M&=&T_{D1} \frac{e^{-\Phi}}{\sqrt{-\det g
-((2\pi\alpha')F_{\tau\sigma}+
b_{\tau\sigma})^2}}\left(G_{MN}\partial_\alpha x^N g^{\alpha
\tau}\det
g+\right.\nonumber \\
&+&\left.((2\pi\alpha')F_{\tau\sigma}+b_{\tau\sigma})B_{MN}\partial_\sigma
x^N\right)+T_{D1}(C^{(0)}B_{MN}\partial_\sigma x^N+C^{(2)}_{MN}\partial_\sigma x^N) \ , \nonumber \\
\pi^\sigma&=&\frac{e^{-\Phi}T_{D1}(2\pi\alpha')((2\pi\alpha')F_{\tau\sigma}+
b_{\tau\sigma})}{\sqrt{-\det g -((2\pi\alpha')F_{\tau\sigma}+
b_{\tau\sigma})^2}}+T_{D1}(2\pi\alpha')C^{(0)}\ , \quad
\pi^\tau\approx 0 \ . \nonumber \\
\end{eqnarray}
Using these relations we find that the bare Hamiltonian is equal to
\begin{eqnarray}
H_B=\int d\sigma (p_M\partial_\tau x^M+\pi^\sigma \partial_\tau
A_\sigma-\mathcal{L})= \int d\sigma \pi^\sigma\partial_\sigma A_\tau
\end{eqnarray}
while we have three primary constraints
\begin{eqnarray}
\mathcal{H}_\sigma&\equiv& p_M\partial_\sigma x^M\approx 0 \ , \quad \pi^\tau\approx 0 \ , \nonumber \\
\mathcal{H}_\tau &\equiv & (2\pi\alpha')\Pi_M
G^{MN}\Pi_N+\frac{1}{2\pi\alpha'}\left(e^{-2\Phi}+\left(\pi^\sigma-
C^{(0)}\right)^2\right)G_{MN}\partial_\sigma x^M\partial_\sigma x^N
\ ,
\nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\Pi_M&\equiv &
p_M-\frac{\pi^\sigma}{2\pi\alpha'}B_{MN}\partial_\sigma x^N
-\frac{1}{2\pi\alpha'}C^{(2)}_{MN}\partial_\sigma x^N \ ,
\nonumber \\
\end{eqnarray}
and where we used the fact that $T_{D1}=\frac{1}{2\pi\alpha'}$.
According to the standard treatment of the constraint systems we
introduce the extended Hamiltonian with all primary constraints
included
\begin{equation}
H=\int d\sigma (\lambda_\tau\mathcal{H}_\tau+\lambda_\sigma
\mathcal{H}_\sigma-A_\tau\partial_\sigma\pi^\sigma+v_\tau \pi^\tau) \ ,
\end{equation}
where $\lambda_{\tau,\sigma}$ and $v_\tau$ are Lagrange multipliers
corresponding to the constraints $\mathcal{H}_\tau,\mathcal{H}_\sigma $ and
$\pi^\tau$.
Now the requirement of the preservation of the primary constraint
$\pi^\tau\approx 0$ implies the secondary constraint
\begin{equation}
\mathcal{G}=\partial_\sigma \pi^\sigma\approx 0 \ .
\end{equation}
Then it can be shown that $\mathcal{H}_\tau,\mathcal{H}_\sigma$ are the first class
constraints, for details, see \cite{Kluson:2014uaa}.
Now we can formally proceed to the discussion of the canonical
transformation as in the case of fundamental string.
Let us presume that
there is a direction that is invariant under constant shift
\begin{equation}
\theta\rightarrow \theta+\epsilon \ , \quad \epsilon=\mathrm{const}
\ .
\end{equation}
Then we again consider the generating function into the form
\begin{equation}\label{canD1}
G(\theta,\tilde{\theta})=\frac{1}{4\pi\alpha'}\int d\sigma (
\partial_\sigma \theta \tilde{\theta}-\theta \partial_\sigma\tilde{\theta})
\end{equation}
that implies the relation between momenta $p_\theta$ and
$p_{\tilde{\theta}}$ respectively
\begin{eqnarray}
p_{\tilde{\theta}}=-\frac{1}{2\pi\alpha'}\partial_\sigma \theta \ , \quad
p_\theta= -\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\theta} \ .
\nonumber \\
\end{eqnarray}
Then we obtain dual Hamiltonian when we replace $\partial_\sigma
\theta$ with $-(2\pi\alpha')p_{\tilde{\theta}}$ and $p_\theta$ with
$-\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\theta}$ so that
\begin{eqnarray}\label{tildemH}
\tilde{\mathcal{H}}_{\tau}&=&(2\pi\alpha')\Pi_\theta
G^{\theta\theta}\Pi_\theta+ 2(2\pi\alpha')\Pi_\theta G^{\theta
\mu}\Pi_\mu+(2\pi\alpha')\Pi_\nu
G^{\mu\nu}\Pi_\nu+\nonumber \\
&+&\frac{1}{2\pi\alpha'}\left(e^{-2\Phi}+\left(\pi^\sigma
-C^{(0)}\right)^2\right)G_{\mu\nu}\partial_\sigma
x^\mu\partial_\sigma x^\nu-\nonumber \\
&-&2\left(e^{-2\Phi}+\left(\pi^\sigma -C^{(0)}\right)^2\right)G_{\mu
\theta}\partial_\sigma x^\mu
p_{\tilde{\theta}}+\nonumber \\
&+& (2\pi\alpha')\left(e^{-2\Phi}+ \left(\pi^\sigma
-C^{(0)}\right)^2\right)G_{\theta\theta}p_{\tilde{\theta}}p_{\tilde{\theta}}
\ ,
\nonumber \\
\tilde{\mH}_\sigma&=&p_\mu\partial_\sigma x^\mu+p_{\tilde{\theta}}\partial_\sigma
\tilde{\theta} \ . \nonumber \\
\end{eqnarray}
In order to see how the background fields transform under this
duality transformation we should find corresponding Lagrangian.
Before we proceed to this question we should stress one important
point. In principle the electric flux that is given in
(\ref{tildemH}) is off-shell.
However we know that this electric
flux is proportional to the number of the fundamental strings. Our
goal is to compare actions where this number is the same so that we
consider canonical transformations for D1-brane theory where we fix
the gauge symmetry so that $\pi^\sigma=\mathrm{const}$. In fact, if
$\pi^\sigma$ were the dynamical variable we would get very
complicated form of the Lagrangian density due to the fact that now
the Hamiltonian contains term like $(\pi^\sigma)^2 p_{\tilde{\theta}}^2$.
In order to proceed to the Lagrangian formulation we introduce
following notations
\begin{eqnarray}
\Pi_\mu&=&
\hat{\Pi}_\mu +\mathbf{V}_\mu p_{\tilde{\theta}} \
, \quad \mathbf{V}_\mu=\pi^\sigma B_{\mu\theta}+C^{(2)}_{\mu\theta} \ , \nonumber \\
\hat{\Pi}_\mu&=&
p_\mu-\frac{\pi^\sigma}{2\pi\alpha'}B_{\mu\nu}\partial_\sigma x^\nu
-\frac{1}{2\pi\alpha'}C^{(2)}_{\mu\nu}\partial_\sigma x^\nu \ ,
\quad
\nonumber \\
\Pi_\theta&=&-\frac{1}{2\pi\alpha'}(\partial_\sigma \tilde{\theta} +\mathbf{V}_\mu \partial_\sigma
x^\mu)
\ , \quad
\mathbf{X}=e^{-2\Phi}+(\pi^\sigma-C^{(0)})^2 \ . \nonumber \\
\end{eqnarray}
so that we can write $\tilde{\mH}_\tau$ in the form
\begin{eqnarray}
\tilde{\mH}_\tau&=& (2\pi\alpha')
p_{\tilde{\theta}} (\mathbf{V}_\mu G^{\mu\nu}\mathbf{V}_\nu +
\mathbf{X} G_{\theta\theta})p_{\tilde{\theta}}+\nonumber \\
&+&(2\pi\alpha')\hat{\Pi}_\mu G^{\mu\nu}\hat{\Pi}_\nu +\hat{\Pi}_\mu
(-2\partial_\sigma \tilde{\theta} G^{\theta\mu}-2\mathbf{V}_\nu
\partial_\sigma x^\nu G^{\theta\mu})+ 4\pi\alpha'\hat{\Pi}_\mu
G^{\mu\nu}\mathbf{V}_\nu p_{\tilde{\theta}}+\nonumber\\
&+&p_{\tilde{\theta}}(-2\partial_\sigma \tilde{\theta} G^{\theta\mu}\mathbf{V}_\mu
-2\mathbf{V}_\nu
\partial_\sigma x^\nu G^{\theta\mu}\mathbf{V}_\mu
+2\mathbf{X} G_{\mu \theta}\partial_\sigma
x^\mu)+\nonumber \\
&+&\frac{1}{2\pi\alpha'}\left(G^{\theta\theta}(\partial_\sigma
\tilde{\theta})^2 -2\mathbf{V}_\mu
\partial_\sigma x^\mu G^{\theta\theta}\partial_\sigma \tilde{\theta}+
\partial_\sigma x^\mu (\mathbf{V}_\mu
G^{\theta\theta}\mathbf{V}_\nu)\partial_\sigma x^\nu +
\mathbf{X} G_{\mu\nu}\partial_\sigma
x^\mu\partial_\sigma x^\nu\right) \nonumber \\
\end{eqnarray}
and hence we obtain
\begin{eqnarray}
\partial_\tau \tilde{\theta} &=&\pb{\tilde{\theta},H}=4\pi\alpha'
\lambda_\tau (\mathbf{V}_\mu G^{\mu\nu}\mathbf{V}_\nu+\mathbf{X}) G_{\theta\theta}
p_{\tilde{\theta}}+\nonumber \\
&+&4\pi\alpha' \hat{\Pi}_\mu G^{\mu\nu}\mathbf{V}_\nu -2 (\partial_\sigma
\tilde{\theta} G^{\theta\mu}\mathbf{V}_\nu+ \mathbf{V}_\sigma\partial_\sigma x^\sigma
G^{\theta\mu}\mathbf{V}_\mu-
\mathbf{X} G_{\mu\theta}\partial_\sigma x^\mu)+\lambda_\sigma \partial_\sigma \theta \ , \nonumber \\
\partial_\tau x^\mu&=&\pb{x^\mu,H}=2\lambda_\tau((2\pi\alpha') G^{\mu\nu}\hat{\Pi}_\mu
-(\partial_\sigma \tilde{\theta} +\mathbf{V}_\nu\partial_\sigma
x^\nu)G^{\theta\mu}+(2\pi\alpha')G^{\mu\nu}\mathbf{V}_\nu
p_{\tilde{\theta}})+\lambda_\sigma
\partial_\sigma x^\mu \ \ .
\nonumber \\
\end{eqnarray}
Now from the last equation we get
\begin{equation}
\hat{\Pi}_\mu=
\frac{1}{2\lambda_\tau}h_{\mu\nu}\mathbf{X}^\nu-\frac{G_{\mu\theta}}{G_{\theta\theta}}
(\partial_\sigma \tilde{\theta}+\mathbf{V}_\nu\partial_\sigma x^\nu) -\mathbf{V}_\mu
p_{\tilde{\theta}} \ ,
\end{equation}
where
\begin{equation}
p_{\tilde{\theta}}=\frac{1}{4\pi\alpha'\lambda_\tau G_{\theta\theta}\mathbf{X}}
(\Theta+\mathbf{X}^\mu \mathbf{V}_\mu+2\lambda_\tau \mathbf{X}
G_{\mu\theta}\partial_\sigma x^\mu) \ ,
\end{equation}
and where $\Theta,\mathbf{X}^\mu$ are defined in (\ref{defXmu}). If we then
proceed in the same way as in case of the fundamental string we
derive final form of the dual Lagrangian density
\begin{eqnarray}\label{defmL}
\tilde{\mathcal{L}}&=&\partial_\tau x^\mu p_\mu+\partial_\tau \tilde{\theta}
p_{\tilde{\theta}}-H=
\nonumber \\
&=&T_{D1}\frac{1}{4\lambda_\tau}\left(\tilde{g}_{\tau\tau}-2\lambda_\sigma
\tilde{g}_{\tau\sigma}+ \lambda^2_\sigma
\tilde{g}_{\sigma\sigma}\right)-T_{D1}\lambda_\tau \mathbf{X}
\tilde{g}_{\sigma\sigma}+\nonumber \\
&+&T_{D1} \left(\pi^\sigma
B_{\mu\nu}+C_{\mu\nu}^{(2)}-\frac{G_{\mu\theta}}{G_{\theta\theta}}(\pi^\sigma
B_{\mu\theta}+C^{(2)}_{\nu\theta})+
\frac{G_{\nu\theta}}{G_{\theta\theta}}(\pi^\sigma
B_{\mu\theta}+C^{(2)}_{\mu\theta})\partial_\tau
x^\mu \partial_\sigma x^\nu+\right.\nonumber \\
&+&\left.\frac{G_{\mu\theta}}{G_{\theta\theta}}\partial_\tau
\tilde{\theta}\partial_\sigma x^\mu -\frac{G_{\mu\theta}}{
G_{\theta\theta}}\partial_\tau x^\mu \partial_\sigma \tilde{\theta}\right)
\ ,
\nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\tilde{g}_{\tau\tau}&=&\partial_\tau x^\mu \left(h_{\mu\nu}+\frac{\mathbf{V}_\mu
\mathbf{V}_\nu} {G_{\theta\theta}\mathbf{X}}\right)\partial_\tau
x^\nu+\frac{1}{G_{\theta\theta}\mathbf{X}}(\partial_\tau
\tilde{\theta})^2+\frac{2}{G_{\theta\theta}\mathbf{X}}
\partial_\tau \tilde{\theta} \partial_\tau x^\mu \mathbf{V}_\mu \ , \nonumber \\
\tilde{g}_{\tau\sigma}&=&\partial_\tau x^\mu
\left(h_{\mu\nu}+\frac{\mathbf{V}_\mu \mathbf{V}_\nu}
{G_{\theta\theta}\mathbf{X}}\right)\partial_\sigma x^\nu+
\frac{1}{G_{\theta\theta}\mathbf{X}}\partial_\tau \tilde{\theta}
\partial_\sigma \tilde{\theta}+\frac{2}{G_{\theta\theta}\mathbf{X}}
\partial_\tau \tilde{\theta} \partial_\sigma x^\mu \mathbf{V}_\mu
+\frac{2}{G_{\theta\theta}\mathbf{X}}
\partial_\sigma \tilde{\theta} \partial_\tau x^\mu \mathbf{V}_\mu
\ , \nonumber \\
\tilde{g}_{\sigma\sigma}&=&\partial_\sigma x^\mu
\left(h_{\mu\nu}+\frac{\mathbf{V}_\mu \mathbf{V}_\nu}
{G_{\theta\theta}\mathbf{X}}\right)\partial_\sigma x^\nu
+\frac{2}{G_{\theta\theta}\mathbf{X}}
\partial_\tau \tilde{\theta} \partial_\sigma x^\mu \mathbf{V}_\mu
+\frac{1}{G_{\theta\theta}\mathbf{X}}(\partial_\sigma \tilde{\theta})^2
\ . \nonumber \\
\end{eqnarray}
Finally we eliminate $\lambda_\tau,\lambda_\sigma $ using
corresponding equations of motion
\begin{eqnarray}
\lambda_\sigma=-\frac{\tilde{g}_{\tau\sigma}}{\tilde{g}_{\sigma\sigma}} \ ,
\quad
\lambda_\tau=
\frac{1}{2\sqrt{\mathbf{X}} \tilde{g}_{\sigma\sigma}}
\sqrt{\tilde{g}_{\tau\sigma}^2-\tilde{g}_{\tau\tau}\tilde{g}_{\sigma\sigma}} \ .
\nonumber
\\
\end{eqnarray}
Inserting back to the Lagrangian (\ref{defmL}) we obtain final form
of the dual Lagrangian density
\begin{eqnarray}\label{LD1dual}
\mathcal{L}&=&-T_{D1}\sqrt{e^{-2\Phi}+(\pi-C^{(0)})^2} \sqrt{-\det
\tilde{g}_{\alpha\beta}}+\nonumber \\
&+&T_{D1}\left((\pi \tilde{B}_{\mu\nu}+\tilde{C}^{(2)}_{\mu\nu})
\partial_\tau x^\mu \partial_\sigma x^\nu+
\frac{G_{\mu\theta}}{G_{\theta\theta}}\partial_\tau \tilde{\theta}
\partial_\sigma x^\mu-\frac{G_{\mu\theta}}{G_{\theta\theta}}\partial_\tau
x^\mu\partial_\sigma \tilde{\theta}\right) \ ,
\nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\tilde{G}_{\mu\nu}&=&
G_{\mu\nu}-\frac{G_{\mu\theta}G_{\nu\theta}}{G_{\theta\theta}}
+\frac{(\pi B_{\mu\theta}+C_{\mu\theta}^{(2)})(\pi B_{\nu\theta}+
C^{(2)}_{\nu\theta})}{G_{\theta\theta}(e^{-2\Phi}+(\pi-C^{(0)})^2)} \ , \nonumber \\
\tilde{G}_{\tilde{\theta}\ttheta}&=&\frac{1}{G_{\theta\theta}
(e^{-2\Phi}+(\pi-C^{(0)})^2)} \ , \nonumber \\
\tilde{G}_{\tilde{\theta}\mu}&=&\tilde{G}_{\mu\tilde{\theta}}=\frac{1}{G_{\theta\theta}(e^{-2\Phi}+(\pi-C^{(0)})^2)}
(\pi B_{\mu\theta}+C^{(2)}_{\mu\theta}) \ , \nonumber \\
\tilde{B}_{\mu\nu}&=&
B_{\mu\nu}-\frac{G_{\mu\theta}B_{\nu\theta}}{G_{\theta\theta}}
+\frac{G_{\nu\theta}B_{\mu\theta}}{G_{\theta\theta}} \ . \nonumber
\\
\tilde{C}^{(2)}_{\mu\nu}&=&C^{(2)}_{\mu\nu}-\frac{G_{\mu\theta}}{G_{\theta\theta}}
C_{\nu\theta}^{(2)}+\frac{G_{\nu\theta}}{G_{\theta\theta}}C^{(2)}_{\mu\theta}
\ , \quad
\nonumber \\
\end{eqnarray}
We would like to give physical interpretation of given
transformation rules. To begin with note that Type IIB theory is
invariant under $SL(2,Z)$ symmetry
\begin{eqnarray}
\hat{G}_{MN}&=&e^{\frac{1}{2}(\hat{\Phi}-\Phi)}G_{MN} \ , \quad
\hat{\tau}=
\frac{p\tau+q}{r\tau+s} \ , \nonumber \\
\hat{B}_{MN}&=&sB_{MN}-r C_{MN}^{(2)} \ , \quad \hat{C}^{(2)}_{MN}
=pC_{MN}-q B_{MN} \
, \nonumber \\
\end{eqnarray}
where $\tau=C^{(0)}+ie^{-\Phi}$ and where $ps-qr=1$.
Let us presume
that $\pi=-|\pi|$ and then choose
following values of the parameters $p,q,r,s$
\cite{Tseytlin:1996it} :
\begin{equation}
p=0 \ , q=-1 \ , r=1 \ , s=|\pi|
\end{equation}
so that we explicitly obtain
\begin{eqnarray}\label{rulesS}
\hat{C}^{(0)}&=&-\frac{C^{(0)}+|\pi|}{(C^{(0)}+|\pi|)^2+e^{-2\Phi}}
\ , \quad
e^{-\hat{\Phi}}=\frac{e^{-\Phi}}{(C^{(0)}+|\pi|)^2+e^{-2\Phi}} \ ,
\nonumber \\
\hat{G}_{MN}&=&\sqrt{(C^{(0)}+|\pi|)^2+e^{-2\Phi}}G_{MN} \ , \nonumber \\
\hat{B}_{MN}&=&|\pi| B_{MN}-C^{(2)}_{MN} \ , \quad \hat{C}^{(2)}_{MN}=B_{MN} \ . \nonumber \\
\end{eqnarray}
We see that when we combine the square root
$\sqrt{(C^{(0)}+|\pi|)^2+e^{-2\Phi}}$ with $\sqrt{-\det \tilde{g}}$ and
use (\ref{rulesS}) we obtain that the Lagrangian density
(\ref{LD1dual}) has the form
\begin{eqnarray}
\mathcal{L}=-\frac{1}{2\pi\alpha'}
\sqrt{-\det\bar{g}}+\frac{1}{2\pi\alpha'}\left( \bar{B}_{\mu\nu}
\partial_\tau x^\mu \partial_\sigma x^\nu+\bar{B}_{\tilde{\theta}\mu}\partial_\tau \tilde{\theta}\partial_\sigma
x^\mu+\bar{B}_{\mu\tilde{\theta}}\partial_\tau x^\mu
\partial_\sigma\tilde{\theta}\right) \ ,
\end{eqnarray}
where
\begin{equation}
\bar{g}_{\alpha\beta}=\bar{G}_{\mu\nu}\partial_\alpha
x^\mu\partial_\beta x^\nu+\bar{G}_{\mu\tilde{\theta}}\partial_\alpha
x^\mu\partial_\beta \tilde{\theta}+\bar{G}_{\tilde{\theta}\nu}\partial_\alpha
\tilde{\theta} \partial_\beta x^\nu
+\bar{G}_{\tilde{\theta}\ttheta}\partial_\alpha \tilde{\theta}
\partial_\beta\tilde{\theta} \ ,
\end{equation}
and where the background fields $\bar{G}_{MN},\bar{B}_{MN}$ have
explicit form
\begin{eqnarray}
\bar{G}_{\mu\nu}&=&
\hat{G}_{\mu\nu}-\frac{\hat{G}_{\mu\theta}\hat{G}_{\nu\theta}}{\hat{G}_{\theta\theta}}
+\frac{\hat{B}_{\mu\theta}\hat{B}_{\nu\theta}}{\hat{G}_{\theta\theta}} \ , \nonumber \\
\bar{G}_{\tilde{\theta}\ttheta}&=&\frac{1}{\hat{G}_{\theta\theta}} \ ,
\quad
\bar{G}_{\tilde{\theta}\mu}=\bar{G}_{\mu\tilde{\theta}}=\frac{1}{\hat{G}_{\theta\theta}}
\hat{B}_{\mu\theta} \ , \nonumber \\
\bar{B}_{\mu\nu}&=&
\hat{B}_{\mu\nu}-\frac{\hat{G}_{\mu\theta}\hat{B}_{\nu\theta}}{\hat{G}_{\theta\theta}}
+\frac{\hat{G}_{\nu\theta}\hat{B}_{\mu\theta}}{\hat{G}_{\theta\theta}}
\ , \nonumber
\\
\bar{B}_{\tilde{\theta}\mu}&=&-\hat{B}_{\mu\tilde{\theta}}=\frac{\hat{G}_{\mu\theta}}{\hat{G}_{\theta\theta}}
\ .
\nonumber \\
\end{eqnarray}
Now the physical interpretation of the canonical transformation
(\ref{canD1}) on the world-volume of D1-brane is clear. It
corresponds to the T-duality rules for the fundamental string moving
in the S-dual background (\ref{rulesS}).
\section{Canonical Transformations and Double Wick Rotation on D1-brane}\label{fifth}
In this section we perform the same analysis as in section
(\ref{third}) in order to find the relation between sequence of
canonical transformations and Wick rotation with the double Wick
rotation on the world-volume of gauge fixed D1-brane action. We
again consider the metric and NS-NS two form field in the form
(\ref{lightconemetric}) and we introduce the light cone coordinates
as in (\ref{defxplus}) so that the metric components are given in
(\ref{Gplusplus}) and (\ref{Gplusplusin}).
Further, using the relation between light-cone coordinates and the
original ones we obtain following components of Ramond-Ramond two
form
\begin{eqnarray}
C^{(2)}_{+-}&=& -C^{(2)}_{-+}=C^{(2)}_{t\varphi} \ ,
\nonumber \\
C^{(2)}_{+\mu}&=&-C_{\mu+}= C_{t\mu}^{(2)}+C_{\varphi\mu}^{(2)} \ ,
\nonumber \\
C_{-\mu}^{(2)}&=&-C^{(2)}_{\mu -}=
-C_{t\mu}^{(2)}+(1-a)C_{\varphi\mu}^{(2)} \ . \nonumber \\
\end{eqnarray}
In the light cone coordinates the Hamiltonian and diffeomorphism
constraints have the form
\begin{eqnarray}
\mathcal{H}_\tau&=&(2\pi\alpha')\Pi_+G^{++}\Pi_++2(2\pi\alpha')
\Pi_+G^{+-}\Pi_-+(2\pi\alpha')\Pi_-G^{--}\Pi_- +\nonumber \\
&+& \frac{1}{2\pi\alpha'}\mathbf{X} \left(G_{++}(\partial_\sigma x^+)^2+
2G_{+-}\partial_\sigma x^+\partial_\sigma
x^-+G_{--}(\partial_\sigma x^-)^2\right)+ \mathcal{H}_x \ ,
\nonumber \\
\mathcal{H}_\sigma &=&p_+\partial_\sigma x^++p_-\partial_\sigma x^-+p_\mu
\partial_\sigma x^\mu \ , \nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\mathcal{H}_x &\equiv & (2\pi\alpha') \Pi_\mu G^{\mu\nu}\Pi_\nu
+ \frac{1}{2\pi\alpha'}\mathbf{X}
\partial_\sigma x^\mu G_{\mu\nu}
\partial_\sigma x^\nu \ ,
\nonumber \\
\end{eqnarray}
and where $\mathbf{X}=\left(e^{-2\Phi}+\left(C^{(0)}\right)^2\right)$. Note
that for simplicity we presume that there is no electric flux so
that $\pi^\sigma=0$. Now we are ready to study the relation between
sequence of canonical transformation, target space Wick rotation and
world-sheet double Wick rotation in the same way as in section
(\ref{third}). Let us presume that the background does not depend on
$x^-$ and perform the canonical transformation from $x^-$ to $\psi$
so we obtain the relation
\begin{eqnarray}
p_{\psi}=-\frac{1}{2\pi\alpha'}\partial_\sigma x^- \ , \quad p_{-}=
-\frac{1}{2\pi\alpha'}\partial_\sigma \psi \
\nonumber \\
\end{eqnarray}
and hence we find
\begin{eqnarray}
\Pi_+&=& p_+-\frac{1}{2\pi\alpha'} C^{(2)}_{+\mu}\partial_\sigma
x^\mu+C^{(2)}_{+-}p_{\psi} \ , \nonumber \\
\Pi_-&=&-\frac{1}{2\pi\alpha'}(\partial_\sigma \psi +
C_{-+}^{(2)}\partial_\sigma x^+)-\frac{1}{2\pi\alpha'}
C^{(2)}_{-\mu}\partial_\sigma x^\mu \ , \nonumber \\
\Pi_\mu&=&p_\mu-
\frac{1}{2\pi\alpha'}(C^{(2)}_{\mu\nu}\partial_\sigma x^\nu+
C^{(2)}_{\mu +}
\partial_\sigma x^+)+C_{\mu -}^{(2)}p_{\psi} \ .
\nonumber \\
\end{eqnarray}
As the next step we perform analytic continuation
\begin{equation}
(x^+,\psi)\rightarrow (i\tilde{\psi}, -i\tilde{x}^+)
\ , \quad (p_+,p_\psi)\rightarrow (-i\tilde{p}_\psi, i\tilde{p}_+)
\end{equation}
that implies
\begin{eqnarray}
\Pi_+ &\rightarrow &
-i(\tilde{p}_\psi-T_{D1}C_{+-}^{(2)}\tilde{p}_+)-\frac{1}{2\pi\alpha'}C^{(2)}_{+\mu}
\partial_\sigma x^\mu
\ , \nonumber
\\
\Pi_-&\rightarrow & i(\partial_\sigma
\tilde{x}^+-T_{D1}C_{-+}^{(2)}\partial_\sigma
\tilde{\psi})-\frac{1}{2\pi\alpha'}C^{(2)}_{-\mu}\partial_\sigma x^\mu \ , \nonumber \\
\Pi_\mu &\rightarrow& p_\mu -T_{D1}C_{\mu\nu}^{(2)}\partial_\sigma
x^\nu -iT_{D1}C_{\mu+}^{(2)}\partial_\sigma \tilde{\psi}+i
T_{D1}C^{(2)}_{\mu -}\tilde{p}_+ \ , \nonumber \\
\end{eqnarray}
so that the Hamiltonian and spatial diffeomorphism constraints
take the form
\begin{eqnarray}
& &\tilde{\mH}_\tau=-(2\pi\alpha')
(\tilde{p}_\psi-T_{D1}C_{+-}^{(2)}\tilde{p}_+)^2 G^{++} +\nonumber \\
&+& 2(\tilde{p}_\psi-T_{D1}C_{+-}^{(2)}\tilde{p}_+)G^{+-} (\partial_\sigma
\tilde{x}^+-T_{D1}C_{-+}^{(2)}\partial_\sigma \tilde{\psi})
-\frac{1}{2\pi\alpha'}(\partial_\sigma
\tilde{x}^+-T_{D1}C_{-+}^{(2)}\partial_\sigma \tilde{\psi})^2 G^{--}
+\nonumber \\
&+& \frac{1}{2\pi\alpha'}\mathbf{X}(-G_{++}(\partial_\sigma \tilde{\psi})^2+
4\pi\alpha'G_{+-}\partial_\sigma \tilde{\psi} \tilde{p}_+-(2\pi\alpha')^2
G_{--}(\tilde{p}_+)^2+ G_{\mu\nu}
(\partial_\sigma x^\mu \partial_\sigma x^\nu)+\nonumber \\
&+&(p_\mu -\frac{1}{2\pi\alpha'}(C_{\mu\sigma}^{(2)}\partial_\sigma
x^\sigma -iC^{(2)}_{\mu+}\partial_\sigma \tilde{\psi} +i(2\pi\alpha')C_{\mu
-}^{(2)} \tilde{p}_+))G^{\mu\nu}\times \nonumber \\
&\times & (p_\nu -\frac{1}{2\pi\alpha'}(
C_{\nu\rho}^{(2)}\partial_\sigma x^\rho
-iC^{(2)}_{\nu+}\partial_\sigma \tilde{\psi} +i(2\pi\alpha')C_{\nu
-}^{(2)} \tilde{p}_+)) \ , \nonumber \\
& & \tilde{\mH}_\sigma=\tilde{\psi} \partial_\sigma \tilde{p}_{\psi}+
\tilde{x}^+\partial_\sigma
\tilde{p}_++p_\mu \partial_\sigma x^\mu \ . \nonumber \\
\end{eqnarray}
As the third step we perform canonical transformation along $\tilde{\psi}$
direction. We denote dual coordinate as $\tilde{\phi}$ so that we have
\begin{equation}
\partial_\sigma \tilde{\psi}= -(2\pi\alpha')p_{\tilde{\phi}} \ , \quad
\tilde{p}_\psi=-\frac{1}{2\pi\alpha'}\partial_\sigma \tilde{\phi} \ .
\end{equation}
Inserting these relations to $\tilde{\mH}_\tau,\tilde{\mH}_\sigma$ given above we
derive the final form of the Hamiltonian and diffeomorphism
constraints
\begin{eqnarray}
& & \tilde{\mH}_\tau=-(2\pi\alpha')
(\frac{1}{2\pi\alpha'}\partial_\sigma\tilde{\phi}+
C_{+-}^{(2)}\tilde{p}_+)^2 G^{++} -\nonumber \\
&-& \frac{2}{2\pi\alpha'}
(\partial_\sigma \tilde{\phi}+(2
\pi\alpha')C_{+-}^{(2)}\tilde{p}_+)G^{+-} (\partial_\sigma
\tilde{x}^++(2\pi\alpha')C_{-+}^{(2)} p_{\tilde{\phi}}) -\frac{1}{2\pi\alpha'}
(\partial_\sigma \tilde{x}^++(2\pi\alpha')C_{-+}^{(2)} p_{\tilde{\phi}})^2 G^{--}
+\nonumber \\
&+& (2\pi\alpha')\mathbf{X}(-G_{++}(p_{\tilde{\phi}})^2- 2G_{+-}p_{\tilde{\phi}}
\tilde{p}_+-G_{--}(\tilde{p}_+)^2+ \frac{1}{(2\pi\alpha')^2}G_{\mu\nu}
(\partial_\sigma x^\mu \partial_\sigma x^\nu))+\nonumber \\
&+&(p_\mu -\frac{1}{2\pi\alpha'} C_{\mu\sigma}^{(2)}\partial_\sigma
x^\sigma +iC^{(2)}_{\mu+}p_{\tilde{\phi}} +iC_{\mu -}^{(2)}
\tilde{p}_+)G^{\mu\nu}
(p_\nu
-\frac{1}{2\pi\alpha'}C_{\nu\rho}^{(2)}\partial_\sigma x^\rho
+iC^{(2)}_{\nu+}p_{\tilde{\phi}} +iC_{\nu
-}^{(2)} \tilde{p}_+) \ , \nonumber \\
& & \tilde{\mH}_\sigma =\partial_\sigma \tilde{\phi} p_{\tilde{\phi}}+
\partial_\sigma \tilde{x}^+
\tilde{p}_++p_\mu \partial_\sigma x^\mu \ . \nonumber \\
\end{eqnarray}
Finally we fix the gauge by imposing the constraints
\begin{equation}
p_{\tilde{\phi}}=\frac{1}{2\pi\alpha'} \ , \quad x^+=\tau \ .
\end{equation}
Then we obtain that the Hamiltonian for the physical degrees of
freedom should be identified as
\begin{equation}
\mathcal{H}_{fix}=-p_+ \ .
\end{equation}
From $\tilde{\mH}_\sigma$ we obtain
$\partial_\sigma\tilde{\phi}=-(2\pi\alpha')\partial_\sigma x^\mu p_\mu$.
Further we see that in order to have real Hamiltonian we have to
demand that $C^{(2)}_{\mu +}=C^{(2)}_{\mu -}=0$. Then the strong
constraint $\tilde{\mH}_\tau=0$ is equal to
\begin{eqnarray}
\mathcal{H}_\tau&=&-(2\pi\alpha') (p_\mu \partial_\sigma x^\mu-
C_{+-}^{(2)}\tilde{p}_+)^2 G^{++} +\nonumber \\
&+&2
(\partial_\sigma x^\mu p_\mu-C_{+-}^{(2)}\tilde{p}_+)G^{+-}C^{(2)}_{-+}
-\frac{1}{2\pi\alpha'} G^{--}(C^{(2)})^2_{-+}
+\nonumber \\
&+& (2\pi\alpha')\mathbf{X}(-G_{++}\frac{1}{(2\pi\alpha')^2}-
\frac{2}{2\pi\alpha'}G_{+-} \tilde{p}_+-G_{--}(\tilde{p}_+)^2)+ \mathcal{H}_x \ .
\nonumber \\
\end{eqnarray}
This equation can be solved for $\tilde{p}_+$ and we obtain
\begin{eqnarray}\label{p+TWT}
\tilde{p}_+=\frac{\tilde{G}^{+-}}{2\pi\alpha'\tilde{G}^{++}}-\frac{2}{2\pi\alpha'\tilde{G}^{++}}\sqrt{\mathbf{K}}
+ p_\mu\partial_\sigma x^\mu
\tilde{C}^{(2)}_{+-} \ , \nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\mathbf{K}=
(\tilde{G}^{+-})^2-\tilde{G}^{++}\tilde{G}^{--}-(2\pi\alpha')^2\tilde{G}^{++}\tilde{G}_{--}\mathbf{X}(p_\mu
\partial_\sigma x^\mu)^2
-(2\pi\alpha')\tilde{G}^{++}\mathcal{H}_x \ , \nonumber \\
\end{eqnarray}and
where we identified the new background fields
\begin{eqnarray}\label{backfieldD1}
\tilde{C}_{+-}^{(2)}&=&-\frac{C_{+-}^{(2)}G^{++}}{G_{--}\mathbf{X}+(C_{+-}^{(2)})^2
G^{++}} \ , \nonumber \\
\tilde{G}^{++}&=&-\mathbf{X} G_{--}-(C_{+-}^{(2)})^2 G^{++} \ , \nonumber \\
\tilde{G}^{+-}&=&G_{+-}\mathbf{X}-(C^{(2)}_{+-})^2G^{+-} \ , \nonumber \\
\tilde{G}^{--}&=&-G^{--}(C^{(2)}_{+-})^2-\mathbf{X} G_{++} \ . \nonumber \\
\end{eqnarray}
As in section (\ref{third}) we compare this result with the uniform
gauge fixed D1-brane when we consider canonical dual Hamiltonian
constraint
\begin{eqnarray}
\mathcal{H}_\tau&=&(2\pi\alpha')\Pi_+G^{++}\Pi_++2(2\pi\alpha')
\Pi_+G^{+-}\Pi_-+(2\pi\alpha')\Pi_-G^{--}\Pi_- +\nonumber \\
&+& \frac{1}{2\pi\alpha'} \mathbf{X} \left(G_{++}(\partial_\sigma x^+)^2-
4\pi\alpha'G_{+-}\partial_\sigma x^+
p_{\psi}+(2\pi\alpha')^2G_{--}(p_\psi)^2\right)+ \mathcal{H}_x \ ,
\nonumber \\
\mathcal{H}_\sigma &=&p_+\partial_\sigma x^++p_\psi\partial_\sigma
\psi+p_\mu
\partial_\sigma x^\mu \ , \nonumber \\
\end{eqnarray}
where
\begin{eqnarray}
\Pi_+=p_++C^{(2)}_{+-}p_{\psi} \ , \quad
\Pi_-=-\frac{1}{2\pi\alpha'}(\partial_\sigma \psi +
C_{-+}^{(2)}\partial_\sigma x^+)\ , \quad \Pi_\mu=p_\mu-
\frac{1}{2\pi\alpha'}C^{(2)}_{\mu\nu}\partial_\sigma x^\nu\ .
\nonumber \\
\end{eqnarray}
Then we perform the gauge fixing
\begin{equation}
x^+=\tau \ , \psi=\sigma \
\end{equation}
and we obtain
\begin{equation}
\Pi_-=-\frac{1}{2\pi\alpha'} \ , \quad p_{\psi}= -p_\mu
\partial_\sigma x^\mu
\end{equation}
and hence the Hamiltonian constraint is equal to
\begin{eqnarray}
0&=&\mathcal{H}_\tau=(2\pi\alpha')\Pi_+G^{++}\Pi_+-2
\Pi_+G^{+-}+\frac{1}{2\pi\alpha'}G^{--} +\nonumber \\
&+& (2\pi\alpha')\mathbf{X} G_{--}(p_\mu\partial_\sigma x^\mu)^2+ \mathcal{H}_x \
\nonumber \\
\end{eqnarray}
that can be solved for $p_+$ as
\begin{eqnarray}\label{p+}
& &p_+=\frac{G^{+-}}{2\pi\alpha'G^{++}}-
+C_{+-}^{(2)}(p_\mu\partial_\sigma x^\mu)
\nonumber \\
&-&\frac{2}{2\pi\alpha'G^{++}} \sqrt{((G^{+-})^2-G^{++}G^{--})
-(2\pi\alpha')^2G^{++}G_ {--} \mathbf{X}(p_\mu
\partial_\sigma x^\mu)^2- (2\pi\alpha')
G^{++}\mathcal{H}_x} \ . \nonumber \\
\end{eqnarray}
Now comparing (\ref{p+}) with (\ref{p+TWT}) we see that these two
gauge fixed Hamiltonians have the same form when we perform the
identification of the background fields as was given in
(\ref{backfieldD1}). It is important to stress that the
transformation rules (\ref{backfieldD1}) coincide with the rules
that were derived in \cite{Kluson:2015saa} when the double Wick
rotation was performed on the world-volume of uniform gauge fixed
D1-brane action. We also see that $\mathbf{X}$ does not transform again
with agreement with \cite{Kluson:2015saa}. Finally using the
arguments given in section (\ref{fourth}) we can argue that
(\ref{backfieldD1}) are in agreement with \cite{Arutyunov:2014cra}
when $C_{+-}^{(2)}=0$. Then we can rewrite (\ref{backfieldD1}) into
the form
\begin{eqnarray}\label{backfieldD1red}
\frac{1}{\sqrt{(C^{(0)})^2+e^{-2\Phi}}}
\tilde{G}^{++}&=&- \sqrt{e^{-2\Phi}+(C^{(0)})^2} G_{--} \ , \nonumber \\
\frac{1}{\sqrt{(C^{(0)})^2+e^{-2\Phi}}}\tilde{G}^{+-}&=&G_{+-}
\sqrt{(C^{(0)})^2+e^{-2\Phi}} \ , \nonumber \\
\frac{1}{\sqrt{(C^{(0)})^2+e^{-2\Phi}}}
\tilde{G}^{--}&=&\sqrt{(C^{(0)})^2+e^{-2\Phi}} G_{++} \ . \nonumber \\
\end{eqnarray}
We immediately see that these metric components correspond to the
S-dual metric when we used the equivalence between D1-brane action
and the fundamental string action in S-dual background. Hence
(\ref{backfieldD1red}) precisely correspond to the rules derived in
\cite{Arutyunov:2014cra} when are applied for the string moving in
S-dual background.
In summary, we have
shown that there is a equivalence between sequence of canonical
transformations and target space Wick rotation on one side and the
double Wick rotation on the gauge fixed D1-brane action on the
another side. On the other hand we also see that the components of
the two form field $C^{(2)}_{\mu\nu}$ that are transverse to the
directions where the canonical transformations were performed do not
transform while their change the sign in case of double Wick
rotation of the uniform gauge fixed D1-brane action. On the other
hand we can argue as in section (\ref{second}) that the
transformation $C^{2}\rightarrow -C^{(2)}$ maps one solutions of the
supergravity equations of motion to another one (together with
$B\rightarrow -B$) and hence we see that there is an equivalence
between double Wick rotation on the world-volume of uniform gauge
fixed D1-brane and sequence of transformations: "canonical
transformation-target space double Wick rotation-canonical
transformation-$C^{(2)}\rightarrow -C^{(2)},B\rightarrow -B$).
\vskip .5in
\noindent {\bf Acknowledgement:}
This work was
supported by the Grant agency of the Czech republic under the grant
P201/12/G028. \vskip 5mm
|
1,116,691,500,501 | arxiv | \section{Introduction}
The aim of this letter is to provide a coherent description of the impact of scalar ($J^{PC}=0^{++}$) and tensor ($J^{PC}=2^{++}$) mesons in tau decays with three pions in the final state.
The four targets of this theoretical analysis are
\begin{itemize}
\item{\bf Chiral invariance and (partial) axial-vector current conservation: }
the chiral invariant Lagrangian framework considered in this letter ensures the right QCD symmetries and leads to
a hadronic matrix element which is transverse ($\partial_\mu J_A^\mu =0$)
in the chiral limit $m_q\to 0$ and where longitudinal corrections come naturally suppressed by $m_q$.
In addition, as isospin is a subgroup of the chiral symmetry, our chiral invariant
Lagrangian approach yields the right relation between the $\pi^0\pi^0\pi^-$ and $\pi^-\pi^-\pi^+$
tau decay form-factors, prescribed by isospin symmetry~\cite{Girlanda:1999fu}, without any further
requirement. Likewise, we will be always assuming the other symmetries of QCD,
parity and charge conjugation.~\footnote{These assumptions also imply $G$-parity conservation, which is a
combination of charge conjugation and isospin symmetry.}
\item{\bf Low-energy limit:}
the construction of a general chiral invariant Lagrangian that includes the chiral pseudo-Goldstones and the
meson resonances ($1^{++}$ axial-vector, $2^{++}$ tensor, etc.)
ensures the right low-energy structure and the possibility to match the low-energy effective field theory (EFT)
of QCD, Chiral Perturbation Theory ($\chi$PT).
\item{\bf On-shell description}:
previous works, in spite of neglecting the previous principles,
have performed a fine work in describing the decays through axial-vector and tensor resonances
when their intermediate momenta are near their mass shell~\cite{Castro:2011zd,CLEO:1999}.
Our outcome reproduces these previous results
when the momentum $k$ flowing through the intermediate resonance propagator becomes on-shell, this is,
when $k^2\approx M_R^2$ (for the corresponding $k$ and $M_R$). The chiral invariant Lagrangian ensures
that the previous properties are fulfilled also off-shell ($k^2\neq M_R^2$).
\item{\bf High-energy limit:}
by imposing high-energy conditions and demanding the behaviour prescribed by QCD
for the form-factors at short-distances we will constrain the resonance parameters.
Implementing these QCD principles will make our theoretical determination
phenomenologically predictive.
\end{itemize}
This resonance chiral theory (R$\chi$T) approach to the $3\pi$ tau decay was considered in the past
taking into account the impact of the vector and axial-vector
resonances~\cite{Dumm:2009va}. The corresponding current has been implemented into the Monte Carlo event generator Tauola~\cite{Shekhovtsova:2012ra}.
The comparison with the unfolded distributions from the preliminary BaBar Collaboration analysis~\cite{Nugent:2013ij}
for the three-prong mode has demonstrated
the mismatch in the low-energy part of the two-pion spectrum~\cite{Shekhovtsova:2012ra}
and was associated with the lack of the scalar meson multiplet in the original R$\chi$T current~\cite{Dumm:2009va}.
The scalar resonance contribution was later added
to the three pion current phenomenologically in Ref.~\cite{Nugent:2013hxa}.
However, the corresponding part does not obey
isospin symmetry~\cite{Finkemeier:1996,Girlanda:1999fu}
and, as a result, does not reproduce the proper chiral low-energy
behaviour (see the discussion in Sec.~\ref{sec:general} and App.~\ref{app.ChPT}).
This letter focuses on the impact of the lowest scalar ($\sigma$ and $f_0(980)$) resonances and the isosinglet tensor
$f_2(1270)$, which may be directly produced from the $W^-$ or generated via an intermediate pion or
an $a_1$ state. Also we discuss the implementation of the associated currents into Tauola
and present
an estimate of tensor and scalar contributions to the three-pion partial width.
In Sec.~\ref{sec:general}, one finds the general formulae for the three-pion
axial-vector form-factor (AFF): the Lorentz structure decomposition
and the isospin relation between $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$ channels.
In order to avoid any possible double-counting we have separated the contributions to the three-pion AFF
in the following way:
1) previous $3\pi$-AFF computations~\cite{Dumm:2009va,Shekhovtsova:2012ra} incorporate
the diagrams including vector resonance exchanges and non-resonant
contributions from the $\cO(p^2)$ $\chi$PT Lagrangian~\cite{rcht};
2) Sec.~\ref{sec:S} provides the contribution to the $3\pi$-AFF from diagrams with scalar exchanges;
3) the contribution due to spin--2 resonance exchanges is discussed in Sec.~\ref{sec:T}.
Sec.~\ref{sec:Tauola} is dedicated to the implementation in the Monte Carlo generator Tauola and some basic numerical
results. We provide the conclusions in Sec.~\ref{sec:conclusions} and some technical details
have been relegated to the Appendices.
\section{Axial-vector form-factor into three pions: general formulae}
\label{sec:general}
The matrix element of the tau-decay into the three pions is determined
in terms of the transverse form-factors $\mF_1$, $\mF_2$ and $\mF_3$
and a longitudinal one
$\mF_P$:
\bear
\bra 3\pi |\bar{d}\gamma^\mu\gamma_5 u|0\ket &=& H^{3\pi}(q^2,s_1,s_2)^\mu
\nn\\
&=& i \, P_T^{\mu\nu} (q) \bigg[
\mF_1(s_1,s_2,q^2)\,\, (p_1 - p_3)_\nu \,\,+\,\, \mF_2(s_1,s_2,q^2) \,\, (p_2-p_3)_\mu
\nn\\
&& +\,\, \mF_3(s_1,s_2,q^2) \,\, (p_1-p_2)_\mu \bigg]\,\,
+ \,\, i\, q_\mu \,\, \mF_P(s_1,s_2,q^2) \, , \label{eq.hadr-curr-3pions}
\eear
with $q=p_1+p_2+p_3$, $s_1=(p_2+p_3)^2$, $s_2=(p_3+p_1)^2$ and $s_3=(p_1+p_2)^2$,
and $P_T(q)^{\mu\nu}= g^{\mu\nu} -q^\mu q^\nu /q^2$.
The three transverse form-factors are linearly dependent and we will leave
only $\mF_1$ and $\mF_2$ as our basis.
The longitudinal form-factor $\mF_P$ vanishes in the chiral limit and is suppressed
by $m_\pi^2/q^2$~\cite{Dumm:2009va}.
Our formulae for the hadronic form-factors will be calculated in the isospin limit.
We will take $m_\pi = (m_{\pi^0} + 2 m_{\pi^+})/3$
and, in general, apply the relation $q^2 = s_1 + s_2 + s_3 -3m_\pi^2$
to express the form-factors in terms of the three independent kinematic variables $q^2,s_1,s_2$.
Bose symmetry implies that
\bear
\mF_1(s_1,s_2,q^2) &=& \mF_2(s_2,s_1,q^2) \, ,
\nn\\
\mF_P(s_1,s_2,q^2) &=&\, \mF_P(s_2,s_1,q^2) \, ,
\eear
and therefore there are only two independent form-factors, e.g., $\mF_1$ and $\mF_P$.
Isospin symmetry relates the matrix elements with $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$
final states~\cite{Girlanda:1999fu}:~\footnote{Isospin violation effects were found to be very suppressed in this decay, of the order of $0.4\%$
and $10^{-3}\%$, respectively for the $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$
channels~\cite{Mirkes:1997ea}.
}
\bear
H_\mu^{--+}(p_1,p_2,p_3) &=& H_\mu^{00-}(p_3,p_2,p_1) +H_\mu^{00-}(p_3,p_1,p_2) \, .
\label{eq.isospin-rel1}
\eear
Thus, the form-factors for $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$ are related in the form
\bear
\hspace*{-0.85cm} \mF_1^{\pi^-\pi^-\pi^+}(s_1,s_2,q^2) &=&
\mF_1^{\pi^0\pi^0\pi^-}(s_1,s_3,q^2) - \mF_1^{\pi^0\pi^0\pi^-}(s_2,s_3,q^2)
- \mF_1^{\pi^0\pi^0\pi^-}(s_3,s_2,q^2) \, ,
\label{eq.isospin-rel2A}
\\
\hspace*{-0.85cm} \mF_P^{\pi^-\pi^-\pi^+}(s_1,s_2,q^2) &=&
\mF_P^{\pi^0\pi^0\pi^-}(s_1,s_3,q^2)
+ \mF_P^{\pi^0\pi^0\pi^-}(s_2,s_3,q^2) \, .
\label{eq.isospin-rel2}
\eear
It is also possible to revert this expressions and to express the $\pi^0\pi^0\pi^-$
matrix element
in terms of the $\pi^-\pi^-\pi^+$ (App.~\ref{app.relations})
but for sake of simplicity, from now on, we will always refer to the $\pi^0 \pi^0\pi^-$ form-factors
and assume Eqs.~(\ref{eq.isospin-rel2A}) and (\ref{eq.isospin-rel2})
whenever the $\pi^- \pi^-\pi^+$
one is needed. The advantage of our chiral Lagrangian approach is that it implements by default
this isospin relation (and Bose symmetry, of course), as isospin is a subgroup
of the chiral group.
It is worth to stress that the $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$ hadronic currents are
in general not the same~\cite{Finkemeier:1996,Pais:1960zz,CLEO-isospin}.
The diagrams with intermediate vector and axial-vector resonances give the same
$\mF_1(s_1,s_2,q^2)$
form-factor up to a global sign difference~\cite{Dumm:2009va}.
However, on the contrary to the approach therein,
tensor and scalar resonances generate contributions
to the $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$ hadronic currents
with a different kinematical structure (determined by
Eqs.~(\ref{eq.isospin-rel2A}) and (\ref{eq.isospin-rel2})).
For further details on the isospin relation between channels see
Refs.~\cite{Girlanda:1999fu,Finkemeier:1996,Pais:1960zz} and App.~\ref{app.relations}.
In the next Sections we will focus on the three-pion tree-level production via
intermediate scalar and tensor resonances, which will be dressed with appropriate widths
when compared to data. Apart from this, we will not incorporate other one-loop contributions
like, e.g, the non-resonant triangular topologies with three internal propagators
(with the mesons $KKK^*$, $\pi\pi\rho$, etc.) and the external pions and $W$ connected
at the vertices.
\section{The decay $\tau \to \pi\pi\pi \nu_\tau$ through scalar resonances}
\label{sec:S}
We first consider the three-pion production via an intermediate state with a scalar $S$
and a pion.
If isospin and C-parity are conserved then G-parity
requires that the scalar resonance has isospin fulfilling $(-1)^I=+1$
--i.e., even isospin--, which in our case implies $I=0$.
The hadronic matrix element for the transition from an axial-vector current
into an isosinglet scalar $S$ and a pion has the general Lorentz structure~\cite{rcht-FFs}
\bear
\bra S_{I=0}(k) \pi^-(p)|\bar{d}\gamma^\alpha \gamma_5 u|0\ket &=&
- 2 i P_T(q)^{\alpha\nu}\, p_\nu \, \mF^a_{S\pi}(q^2;k^2)
\,\, +\,\, i\, q^\alpha \, \mH^a_{S\pi}(q^2;k^2)\, ,
\eear
where $q = k+ p$ and the scalar function $\mF^a_{S\pi}(q^2)$
provides AFF into $S\pi$ in the chiral limit, as $\mH^a_{S\pi}$ is suppressed by $m_\pi^2$
due to the partial conservation of the axial-vector current.
Here the
isosinglet scalar $S_{I=0}$
refers to the resonance without $s\bar{s}$ component,
$S_{I=0}\sim u\bar{u}+d\bar{d}$, which we will relate with
the lightest scalar isoscalar resonance, the $f_0(500)$ or $\sigma$.
We leave the discussion of the properness of this approach for a next Section:
here we will just assume the large-$N_C$
framework~\cite{tHooft:1973alw,tHooft:1974pnl,Witten:1979kh}
and the phenomenological implementation will be later worked out.
In Fig.~\ref{fig.diagr}, we show the three relevant diagrams that must be taken into account
in the $S\pi$ production at large $N_C$ (and analogously later
in the production of a tensor resonance $T$ and a pion):
a) the direct production $W^- \to S \pi^-$;
b) the intermediate $\pi^-$ production $W^- \to \pi^- \to S \pi^-$;
c) and the scalar production through an intermediate axial-vector resonance,
$W^- \to a_1 \to S \pi^-$.
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 0.7\textwidth]{diagrams2.eps}
\end{center}
\caption{{\small
Relevant diagrams for the hadronic tau decays into an isosinglet scalar $S$ and a pion
and its corresponding AFF
(similar to those for the decay into a isosinglet tensor $T$ and a pion).
Single straight lines stand for pions and the wavy line for the external axial-vector source
(from an incoming $W^-$). }
}
\label{fig.diagr}
\end{figure}
\subsection{The R$\chi$T Lagrangian for scalar
fields}
The resonance Lagrangian has the generic structure
\bear
\mL_{\rm R\chi T} &=& \mL_{\rm non-R} \, +\, \sum_R \mL_R \,+\,
\sum_{R,R'} \mL_{R\, R'}\, +\, ...
\eear
which respectively contains operators without resonances, operators with one resonance field,
terms with two resonance fields, etc.
In the case of the tau decay into three pions through an intermediate scalar production,
the relevant chiral invariant Lagrangian consists of three parts:
\begin{itemize}
\item Operators with one resonance field~\cite{rcht}:
\bear
\mL_A &=& \Frac{F_A}{2\sqrt{2}} \bra A_{\mu\nu} f_-^{\mu\nu} \ket\, ,
\nn\\
\mL_S &=& c_d\bra S u_\mu u^\mu\ket + c_m\bra S\chi_+\ket\, ,
\eear
\item Operators with an axial-vector and a scalar field (which provides the $AS\pi$ vertex in
diagram c) in Fig.~\ref{fig.diagr})~\cite{rcht-FFs}:
\bear
\mL_{AS} &=& \lambda_1^{AS} \bra \{\nabla_\mu S , A^{\mu\nu}\} u_\nu\ket \, .
\label{eq.scalar-lagr}
\eear
Operators of the $\mL_{AS}$ Lagrangian that do not contribute
to the $AS\pi$ vertex are not shown here~\cite{rcht-FFs}.
\item
Operators without resonance fields~\cite{Gasser:1983yg,Gasser:1984gg,rcht}:
\bear
\mL^{(2)}_{\rm non-R} &=& \Frac{F^2}{4}\bra u_\mu u^\mu +\chi_+\ket\, ,
\label{eq.non-R-lagr}
\eear
This non-resonant $O(p^2)$ Lagrangian generates the $W^-\to \pi^-$ transition vertex
in Fig.~\ref{fig.diagr}.b.
It also provides an $O(p^2)$ contribution without intermediate resonances to the $\pi\pi\pi$ AFF
which was accounted in previous analyses~\cite{Dumm:2009va}. Thus,
in order to avoid double counting,
we will not consider these non-resonant $\pi\pi\pi$ AFF diagrams.
\end{itemize}
For the axial-vector field $A_{\mu\nu}=A_{\mu\nu}^a \lambda^a/\sqrt{2}$
we have used the antisymmetric tensor representation~\cite{rcht,op6rxt}, with
\bear
A_{\mu\nu} &=&\left(\begin{array}{ccc}
0& a_1^+ & 0\\ a_1^- &0&0 \\ 0&0&0
\end{array} \right)_{\mu\nu} \,\,\,+\,\,\, ...
\eear
with the dots standing for the other axial-vector resonances of the multiplet,
which will not be relevant in the present study.
For the chiral tensors containing the
light pseudoscalars, the masses and the external vector and axial-vector source fields we used~\cite{chpt,rcht}
\bear
& U = u^2 =\exp\{ \pi^a \lambda^a/ F\}\, , \qquad
D_\mu U = \partial_\mu U - i r_\mu U + i U \ell_\mu\, , \qquad
u_\mu = i u^\dagger (D_\mu U) u^\dagger\, , &
\nn\\
& \chi_{\pm}= u^\dagger \chi u^\dagger \pm u \chi^\dagger u\, , \qquad
f_{\pm}^{\mu\nu} = u F_L^{\mu\nu} u^\dagger \pm u^\dagger F_R^{\mu\nu} u\, , \qquad
\nabla_\mu \cdot = \partial_\mu \cdot + [\Gamma_\mu , \cdot ]\, , &
\nn\\
& \Gamma_\mu =\Frac{1}{2} \left\{ u^\dagger (\partial_\mu - i r_\mu ) u + u(\partial_\mu - i \ell_\mu) u^\dagger
\right\} \, ,&
\eear
with the scalar-pseudoscalar source $\chi=2B_0$diag$(m_u,m_d,m_s)+...$
(the dots stand for terms not relevant for this calculation)
and $F_L^{\mu\nu}$ and $F_R^{\mu\nu}$ the field strength tensors
of the left and right sources,
respectively $\ell_\alpha$ and $r_\alpha$.
If we are only interested in the $W^\pm$ currents one takes
$\ell_\alpha = \frac{g}{\sqrt{2}} (W_\alpha^+ T_+ + {\rm h.c.})$ and $r_\alpha =0$,
with $T_+ = V_{ud} (\lambda^1+i\lambda^2)/2 + V_{us} (\lambda^4+i\lambda^5)/2$.
The $\pi^a$ generically refer to the $SU(3)$ chiral pseudo-Goldstones ($a=1...8$).
At large $N_C$ (and for the non-strange current) this process only occurs for
the isosinglet scalar $S_{I=0}\sim u \bar{u} +d\bar{d}$,
with no $s\bar{s}$ strange quark component:
\bear
S&=& \left(\begin{array}{ccc}
\Frac{S_{I=0}}{\sqrt{2}} & 0& 0 \\ 0 & \Frac{S_{I=0}}{\sqrt{2}} & 0 \\ 0&0&0 \end{array}\right)\,\,\, +\,\,\, ...
\eear
where the dots stand for other resonances in the multiplet not relevant for the present work.
\subsection{AFF into $S\pi^-$}
Our chiral invariant Lagrangian leads to the AFF prediction,~\footnote{
There was a typo in the sign of the $F_A\lambda_1^{SA}$ term
of $\mF^{a}_{S\pi}$ in Table A.2, App.~A in Ref.~\cite{rcht-FFs}.
It has been corrected in Eq.~(\ref{eq.Spi-AFF}).
The same applies to the later high-energy constraint~(\ref{eq.Spi+HE})
(the final constrained form-factor~(\ref{eq.Spi-AFF+HE})
remains nevertheless the same as in Ref.~\cite{rcht-FFs}). }
\bear
\mF^a_{S\pi}(q^2 ;k^2 )&=& \Frac{2c_d}{ F_\pi }
\quad + \quad
\Frac{\sqrt{2} F_A \lambda_1^{AS}}{ F_\pi }\Frac{q^2}{M_A^2-q^2}
\, ,
\label{eq.Spi-AFF}
\\
\mH^a_{S\pi}(q^2 ;k^2 )&=&
\Frac{4}{F_\pi} \, \Frac{ m_\pi^2}{q^2 (q^2-m_\pi^2)}
\, \left[c_d (qp) + c_m q^2 \right] \, ,
\label{eq.Spi-AFF-2}
\eear
with $(q p)= (q^2+m_\pi^2-k^2)/2$, being $k^2=M_S^2$ for an on-shell scalar
(later, when this scalar is considered off-shell and decaying
in two pions with momenta $p_i$ and $p_j$ it will take
the value $k^2=(p_i+p_j)^2$).
The $c_m$ operator contributes through the $s$-channel pion exchange
to the longitudinal form-factor in Eq.~(\ref{eq.Spi-AFF-2}).~\footnote{
There is an indirect large-$N_C$
contribution to these form-factors
through the pion-wave function renormalization proportional to $m_\pi^2$
induced by the scalar
Lagrangian~\cite{SanzCillero:2004sk}.
This effectively amounts to a replacement of $F$ by $F_\pi$,
as shown in~(\ref{eq.Spi-AFF}) and (\ref{eq.Spi-AFF-2}).
A similar thing happens in the other form-factors studied in the next Sections, where this pion-wave
function renormalization due to the scalars~\cite{SanzCillero:2004sk} is taken into account
in a similar way.}
\subsection{$3\pi$-AFF through an intermediate scalar resonance}
\label{sec:3pi-from-S}
Considering not only the $S\pi$ production but also the
subsequent decay $S\to \pi \pi$ one obtains the corresponding contribution to the $\pi\pi\pi$-AFF.
Using the Lagrangian in Eqs.~(\ref{eq.scalar-lagr})--(\ref{eq.non-R-lagr}),
we obtain the contribution from scalar resonance exchanges
to the $\pi^0\pi^0\pi^-$ AFFs defined in~(\ref{eq.hadr-curr-3pions}),
\bear\label{eq.ff-scal}
\mF_1^{\pi^0\pi^0\pi^-}(s_1,s_2,q^2)\bigg|_S &=& \Frac{2}{3} \mF_{S\pi}^a(q^2 ;s_3 )\, \mG_{S \pi\pi}(s_3)\, ,
\\
\mF_P^{\pi^0\pi^0\pi^-}(s_1,s_2,q^2)\bigg|_S &=& \mH^a_{S\pi}(q^2 ;s_3 ) \, \mG_{S\pi\pi}(s_3)\, ,
\eear
with $qp_j = (m_\pi^2 +q^2 -s_j)/2$. The $AS\pi$ form-factor is the previous one
in Eq.~(\ref{eq.Spi-AFF})
whereas propagation of the isosinglet $S$ and its decay into $\pi\pi$ gives
\bear
\mG_{S\pi\pi}(s_3) &=&
\Frac{\sqrt{2}}{ F_\pi^2 } \,\Frac{1 }{M_S^2 - s_3 }
\, [ c_d(s_3-2m_\pi^2) + 2 c_m m_\pi^2] \, .
\eear
Notice that we are giving the full result, including pion mass corrections
produced by our Lagrangian in Eqs.~(\ref{eq.scalar-lagr})--(\ref{eq.non-R-lagr}).
\footnote{
The function $\mG_{S\pi\pi}(s_3)$ is not the scalar form-factor
and, therefore, does not need to obey
asymptotic high-energy behaviour prescribed by QCD~\cite{Brodsky:1973kr}.
Notice that only on-shell hadron matrix elements are well-defined
and the off-shell behaviour is ambiguous as it can be modified through
field redefinitions in the hadronic generating functional~\cite{Gasser:1983yg,Gasser:1984gg}.
$\mG_{S\pi\pi}(s_3)$ just provides a) the on-shell decay $S\to \pi\pi$ (through its residue at $s_3=M_S^2$)
and b) the contribution to the $\pi\pi\pi$ AFF from topologies with an intermediate scalar
--either on-shell or off-shell--.
}
Requiring that the contribution to the transverse component of
the $\Pi_{AA}^{\mu\nu}(q)$ spectral function vanishes
implies that $\mF^a_{S\pi}(q^2) \longrightarrow 0 $ for $q^2\to\infty$ (see App.~\ref{app.optical-theorem}),
giving the constraint~\cite{rcht-FFs}
\bear
F_A\lambda_1^{AS} &=& \sqrt{2} c_d\, ,
\label{eq.Spi+HE}
\eear
and the form-factor prediction
\bear
\mF^a_{S\pi}(q^2 ;s_3 ) &=& \Frac{2c_d}{ F_\pi } \Frac{M_A^2}{M_A^2-q^2}
\, .
\label{eq.Spi-AFF+HE}
\eear
This high-energy constraint is similar to the asymptotic form-factor high-energy behaviour
prescribed by Brodsky-Lepage quark-counting rules~\cite{Brodsky:1973kr},
which imply, for instance, that the pion vector form-factor vanishes like $\sim 1/q^2$ at infinite momentum transfer~\cite{rcht,Brodsky:1973kr}.
The subsequent decay of the scalar into $\pi\pi$ is given by $\mG_{S\pi\pi}(s_3)$ and would provide
the absorptive $\pi\pi\pi$ contribution to Im$\Pi_{AA}^{\mu\nu}$. However,
in the narrow-width limit for $S$, the three-pion phase-space integral yields a delta function $\delta(s_3-M_S^2)$
that sets the $s_3$ value to $M_S^2$. Thus, the integral is factorized into the two-body integration of $|\mF^a_{S\pi}(q^2)|^2$
over the $S\pi^-$ phase-space and a constant angular integration over the phase-space of the two pions produced by the
scalar. Therefore, in this limit, the large $q^2$ behaviour of this three-pion contribution
to the spectral function is ruled by the form-factor $\mF^a_{S\pi}(q^2)$ in the way dictated
by Eq.~(\ref{eq.Spi-spectral-function}) (up to a global constant factor).
We will use this theoretical large--$N_C$ information and use it to constrain our form-factor even if we will later model it
in order to include important subleading effects in $1/N_C$ such as the $\sigma$ width.
\footnote{ Phenomenologically, in order to study the $a_1$ meson finite size effects,
Ref.~\cite{CLEO:1999}
considered an additional {\it ad hoc} exponential suppression factor
$\exp\{ - R^2 |\vec{p}_{\pi^-}|^2 /2\}$
in addition to the analogous $\mG_{S\pi\pi}(s_3)$ functions.
However, the fit to the experimental data did not show an essential difference between a zero
and non-zero value of $R$. As a result
of this, the nominal fit shown therein was the one with $R = 0$
(for details see Section VI of \cite{CLEO:1999}).
Moreover,
these exponential factors do not have the right analytical structure
in the whole complex plane and add an exponentially divergent behaviour
for some complex directions at $|q^2|\to \infty$.
Likewise, this functional dependency may not come from a perturbative Lagrangian computation like the one
worked out in this article and will not be incorporated to our diagrammatic results. }
The $S\pi$ AFF is then ruled by the $c_d$ coupling in the limit $m_\pi^2 \ll q^2$.
Even though its precise experimental value is still unclear,
most analyses agree on a value $c_d\sim 30$~MeV
(see~\cite{Escribano:2010wt} and references therein).
For a discussion on its numerical impact on the spectral distributions,
see Sec.~\ref{sec:Tauola}.
\subsection{Scalar resonance widths}
\label{sec:scalar-width}
The lightest isoscalar particle is the broad scalar $\sigma$, with
$M_\sigma^{\rm pole}= 441^{+16}_{-\, 8}$~MeV,
$\Gamma_\sigma^{\rm pole}=544^{+18}_{-25}$~MeV~\cite{CCL-sigma}.
It is thought to contain mostly just $u$ and $d$ quark components, where the two--pion channel is its only kinematically allowed decay.
On the other hand,
as it follows from its predominant decay into $K\bar{K}$,
the next scalar isosinglet, the $f_0(980)$,
is considered to have a large strange quark component,
being its $n\pi$ decay modes are suppressed.
However, for sake of completeness we will include both isoscalars into consideration.
A first approach to the physical QCD case is provided by the inclusion of
a $\sigma$--$f_0(980)$ splitting
through the substitution~\cite{Escribano:2006mb,Escribano:2010wt},
\begin{equation}
\Frac{1}{M_S^2\,-\,s} \qquad \longrightarrow \qquad
\Frac{\cos^2\phi_S}{
M_\sigma^2\,-\,s
} \, \,+\,\, \Frac{\sin^2\phi_S}{M_{f_0}^2\,-\,s} \, ,
\label{eq.MS-splitting}
\end{equation}
where $\phi_S$ is the scalar mixing angle.
For the $\sigma-f_0$ mixing we will use the numerical value
$\phi_S=-8^\circ$~\cite{Escribano:2006mb}.
Due to the $\sin^2\phi_S$ suppression the $f_0(980)$ produces a clearly
subdominant effect with respect to the impact of the broad~$\sigma$. However, the comparison of the
modified R$\chi$T spectra~\cite{Nugent:2013hxa}~\footnote{
By \textit{modified} we mean a phenomenological approach
proposed in Sec.~II of~\cite{Nugent:2013hxa} to include the $\sigma$-meson
in the hadronic form-factors. }
with the unfolded distributions~\cite{Nugent:2013ij} from the preliminary BaBar Collaboration $\tau\to \nu_\tau \pi\pi\pi$~analysis
has shown a statistically significant mismatch:
the $\pi^+\pi^-$ experimental spectral function is well reproduced up to 1~GeV except for a small sharp bump concentrated at 980~MeV
which differs from the $f_0$-absent theoretical R$\chi$T expression by a few percent.
The inclusion of the $f_0$ and its occurrence here via the $\sigma-f_0$
mixing in Eq.~(\ref{eq.MS-splitting}) is expected to improve the phenomenological description of the data.
\subsubsection{Incorporating the $\sigma$ meson width}
\label{sec:sigma-width}
So far in previous Sections we have carried on a large-$N_C$ computation
where one had an intermediate exchange of narrow-width scalars.
This approximation seems to be suitable for the $f_0(980)$. However,
the $\sigma$ meson is a broad resonance and the effect of its width
is non-negligible.
It is not our intention to enter here in the discussion of the $\sigma$ nature
but, rather,
to propose an improved parametrization of its effect
on the $\tau\to\nu\pi\pi\pi$ decay that incorporates the features described in the introduction.
For this, we follow the
successful analysis of subleading $1/N_C$ effects in scalar exchanges
in the $\eta'\to \eta \pi\pi$ process~\cite{Escribano:2010wt}:
after considering the scalar splitting
in~(\ref{eq.MS-splitting}),
we incorporate the ``dressed'' $\sigma$ propagator in a similar way
by performing the substitution
\begin{equation}
\label{eq.S-propagator2}
\Frac{1}{M_\sigma^2\,-\,s} \qquad\longrightarrow\qquad \Frac{1}{ M_\sigma^2\,-\,s\, -\, f_{\sigma}(s)\, -\, i M_\sigma \Gamma_\sigma(s) }\, ,
\end{equation}
with
\bear
f(s)&=&
c_\sigma s^k\,{\rm Re} \overline{B}_0(s,m_\pi^2,m_\pi^2) \,=\,
\Frac{c_\sigma \, s^k}{16\pi^2}\left[2-\rho_\pi(s)\ln{\frac{\rho_\pi(s)+1}{1-\rho_\pi(s)}} \right]\, ,
\nn\\
M_\sigma \Gamma_\sigma(s)
&=&
c_\sigma s^k \, {\rm Im} \overline{B}_0(s,m_\pi^2,m_\pi^2)
\,=\,
\Frac{c_\sigma \,\rho_P(s)\, s^k }{16\pi}\, ,
\label{eq.S-self-energy}
\eear
in the fashion of Gounaris and Sakurai~\cite{GS-rho}
and the Chew and Mandelstam dispersive integral~\cite{Chew:1960iv}.
We will use the parameters
$M_\sigma$ and $c_\sigma$ tuned such that one recovers the
right position for the $\sigma$ pole,
$M_\sigma^{\rm pole}= 441^{+16}_{-\, 8}$~MeV,
${ \Gamma_\sigma^{\rm pole}=544^{+18}_{-25} }$~MeV~\cite{CCL-sigma}.
The function,
\bear
\overline{B}_0(s,m_P^2,m_P^2)
&=&\frac{1}{16\pi^2}\left[2-\rho_P(s)\ln{\frac{\rho_P(s)+1}{\rho_P(s)-1}} \right]
\nn\\
&&= \frac{1}{16\pi^2}\left[2-\rho_P(s)\ln{\frac{\rho_P(s)+1}{1-\rho_P(s)}}
\, +\, i \pi \rho_P(s) \right]
\, ,
\eear
is \ \ \ the \ \ \ subtracted \ \ \ two--point \ \ \ Feynman \ \ \ integral \ \ \ ($\overline{B}_0(0,m_P^2,m_P^2)=0$), \ \ \ with
${ \rho_P(s)\equiv \lambda(s,m_P^2,m_P^2)^{\frac{1}{2}}/ q^2 = \sqrt{1-4 m_P^2/s} }$.
One of the crucial points of the parametrization~\cite{Escribano:2010wt}
employed here is that it incorporates the real part of the logarithm
that comes along with the imaginary part $-i M_\sigma \Gamma_\sigma(s)$
on the basis of analyticity. In the case of narrow-width resonances, these real logs
are essentially negligible and can be dropped. However, if their corresponding
imaginary part is large one naturally expect the appearance of equally large
real logarithms. Moreover, any attempt to match NLO $\chi$PT
at low-energies must incorporate
both the real and imaginary parts of the logs.
Even though our simple approach~\cite{Escribano:2010wt}
can be further refined, it already
contains some of the basic ingredients that makes this matching possible.
Other works that incorporate the real and imaginary parts of the logarithm
in other observables can be found in Refs.~\cite{ND,Sdecays}.
The power behaviour $k=0$ produces an unphysical bound state
in the first Riemann sheet very close below the $\pi\pi$
threshold, which
unnaturally enhanced the amplitude
in the $\eta'\to\eta\pi\pi$~\cite{Escribano:2010wt}, leading in that work to a very small $S\pi\pi$ coupling $c_d=9.9$~MeV.
This case seems to be clearly
disfavoured from the phenomenological point of view and
was
discarded in the analysis of Ref.~\cite{Escribano:2010wt}.
For $k=1$, the amplitude produces just one pole and its correct position
$\sqrt{s^\sigma_{\rm pole}}= [(441^{+16}_{-\,8})\,-\, i (544^{+18}_{-25})/2] $~MeV~\cite{CCL-sigma}
is recovered for the parameter values $M_\sigma= 806.4$~MeV and $c_\sigma=76.12$.~\footnote{
These are the corresponding central values.
Errors are not discussed in this article. A more detailed numerical analysis
is postponed for a future work. Nonetheless, one may observe that alternative $\sigma$
pole determinations like, e.g.,
$\sqrt{s^\sigma_{\rm pole}}= [(457^{+14}_{-13})\, -\, i ( 558^{+22}_{-\,14})/2]$~MeV~\cite{GarciaMartin:2011jx},
yield similar central value determinations $M_\sigma=804.1$~MeV and $c_\sigma=70.96$.
This variation gives a preliminary estimate of the expected uncertainties in these quantities. }
Power behaviours with $k\geq 2$
are unable to generate the $\sigma$ pole at the right position.
For its closest position, the pole mass is slightly larger and the pole width
is roughly 100 MeV smaller. Likewise, some spurious poles are produced far from
the physical energy range of the problem under study.
For the numerical inputs we will take the $s^k$ scaling with $k=1$
in Eq.~(\ref{eq.S-self-energy})
and the values $M_\sigma= 806.4$~MeV and $c_\sigma=76.12$.
In these expressions the constants $M_\sigma$ and $c_\sigma$ that appear in the denominator are
parameters set
to agree with the
central value of the $\sigma$ pole position
$s_\sigma^{\rm pole}= (M_\sigma^{\rm pole}- i\Gamma_\sigma^{\rm pole}/2)^2$
from Ref.~\cite{CCL-sigma}.
Our estimate of the rescattering of the $\pi\pi$ system related to the isosinglet scalar
is obviously model dependent, as we have introduced an {\it ad hoc}
splitting and self-energy for the scalar multiplet.
The splitting can be easily introduced through the corresponding
terms in the Lagrangian, studied in Ref.~\cite{mass-split}.
On the other hand, while the $1/N_C$ counting would strictly lead to zero-width resonances,
finite widths are needed to regularize the $\tau$ decay phase space integrals
and compare to data. Hence, they need to be taken into account and analyticity requires
the presence of the real logarithm counterparts in the self-energy.
However, if these provide a large contribution,
it seems that $1/N_C$ corrections provide a significant effect
in contradiction with the hypothesis
of neglecting, e.g., resonance-mediated loops.
There is no clear and definitive answer to this issue yet and
one of goals of this work is to explore the raised problem.
In this article, we assume that
this is the only subleading contribution in $1/N_C$ which is numerically relevant
for the current precision of the analysis.
As noticed in Refs.~\cite{RChT-width,RGE},
the resummation of subleading $1/N_C$ corrections can be well defined in perturbation theory
and become crucial even for the $\rho(770)$.
Following previous scalar resonance studies in this line~\cite{Escribano:2010wt},
we consider this resummation of the one-loop $\pi\pi$ self-energy
is also justified, even for the broad $\sigma$:
higher order effects absent in the resummation (multimeson channels) are completely negligible
below 1~GeV and the one-loop amplitude seems to provide
the crucial information in our physical range.
Notwithstanding,
this $\pi\pi$ final state interaction
must be appropriately resummed
in the neighbourhood of the resonance pole,
as noted in Refs.~\cite{RChT-width,RGE}.
Alternatively one might incorporate the $s$--wave rescattering via unitarization
procedures~\cite{Escribano:2010wt,ND} and related dispersion relations
(see, e.g., the semileptonic $B$ decay analysis~\cite{Kang:2013jaa}).
It is important to point out, however, that even in this robust method
only the $\pi\pi$ absorptive corrections are incorporated in the analysis (and the most relevant inelastic intermediate
channels in some cases).
\subsubsection{Incorporating the $f_0$ meson width}
One can take also into account the $f_0(980)$ width in a similar way. Due the $\sin^2\phi_S$ suppression in~(\ref{eq.MS-splitting}), the $f_0(980)$ produces a clearly
subdominant effect with respect to the impact of the broad~$\sigma$.
The important piece of the self-energy
is its imaginary part, being the real part of its corresponding logarithm
almost negligible in comparison with the leading contribution $M_S^2- s$.
In the case of the narrow $f_0$ resonance,
the location of its pole near the $K\overline{K}$ threshold
will modify the $f_0$ propagator into the well-known Flatt\'e form~\cite{Flatte:1976xu}
\bear
\Frac{1}{M_{f_0}^2\,-\,s} \quad \longrightarrow \qquad \Frac{1}{M_{f_0}^2\,-\,s\, -\, i M_{f_0} \Gamma_{f_0}(s) } \, ,
\label{eq.f0-propagator}
\eear
with
\bear
M_{f_0} \Gamma_{f_0}(s) &=& \Frac{ c_{f_0} M_{f_0}^2 \rho_K(s)}{16\pi}\, ,
\eear
which is indeed the near threshold expression of the self-energy
at lowest order in the non-relativistic expansion
in powers of the kaon three-momentum $|\vec{p}_K|\sim \rho_K(s)$~\cite{Braaten:2007dw,Meng:2014ota}.
As the self-energy is only relevant
for $s\approx M_{f_0}^2$, one does not need to consider different
$c_{f_0} s^{k}$ scalings
for the loop corrections as we did for the $\sigma$
meson and the
different values of $k$ amount just for differences
at higher order in the non-relativistic expansion in $\rho_K(s)$.
For $s_{f_0}^{\rm pole} =(M_{f_0}^{\rm pole}-i\Gamma_{f_0}^{\rm pole}/2)^2
=(990 - i 70/2)^2$~MeV$^2$~\cite{pdg}~\footnote{ We take the central PDG values here. }
this implies the parameters $M_{f_0}=1024$~MeV and
$c_{f_0} = 17.7$.
The best estimate, based on Roy equations, gives the value
$\sqrt{s_{f_0}^{\rm pole}}=(996^{\,+ 4}_{-14}) - i (24^{+11}_{\,- 3})$~MeV~\cite{Moussallam:2011zg}.
This deviates by less than 1\% from the PDG central value we will use
in Sec.~\ref{sec:Tauola}. We do not expect any difference for our numerical result.
Likewise, in spite of the fact that we have used the average kaon mass $m_K=496$~MeV,
the latter result is not very sensitive to the precise position of the $K\overline{K}$ threshold, with $M_{f_0}$
and $c_{f_0}$
changing by $\pm 0.5\%$ and $\pm 7\%$, respectively, when $m_K$ is varied between the charged and neutral kaon mass values. By far the largest
effect would be the uncertainty in the $f_0$ mass and width with
errors
of $\pm 20$~MeV and $\pm 30$~MeV, respectively~\cite{pdg}.
Therefore, for the numerical inputs we will take
$M_{f_0}=1024$~MeV
and $c_{f_0} = 17.7$.~\footnote{
We remind that the parameter $M_{f_0}$ is not the pole mass $M_{f_0}^{\rm pole}$.}
\section{The decay $\tau \to \pi\pi\pi \nu_\tau$ through tensor resonances}
\label{sec:T}
In this section we focus on tau decay into three pions through an intermediate tensor resonance
($J^{PC}=2^{++}$) in the cascade decay $\tau\, \to \,\nu_\tau \, \pi^- \,T(\to \pi\pi)$.
Our study reproduces the prediction for the tau decay into a tensor resonance and
a chiral pseudo-Goldstone~\cite{Castro:2011zd}
and expands then for the case of the off-shell tensor resonance.
$G$-parity conservation implies that for the non-strange axial-vector current
(with $G=-1$)
the tensor resonance produced in combination with a pion
must have $G=(-1)^I=+1$ and, hence, even isospin.
As a consequence of this, it must
be an isosinglet in the case of $q\bar{q}$ multiplets
($T=f_2(1270)$, $f_2(1430)$, $f_2'(1525),\, f_2(1565)...$). In this article we study the impact of the lightest
tensor, $f_2(1270)$, which dominantly decays into $\pi\pi$~\cite{pdg}.
The $f_2'(1525)$ mainly goes into $K\overline{K}$ and
has a negligible decay into $\pi\pi$~\cite{pdg}.
Our analysis is then restricted to the lowest tensor resonances.
We discarded not so well established resonances
such as the $f_2(1430)$ and $f_2(1565)$, whose $\pi\pi$ partial width
are not determined in any of the references quoted by PDG~\cite{pdg}.
In addition, we would like to stress that,
the contribution from the $f_2(1270)$
is found to be highly suppressed in our later numerical analysis,
as it is placed near the $\pi^0\pi^0$ spectrum end point
(or the $\pi^+\pi^-$ spectrum
for $\tau\to\nu_\tau \pi^-\pi^-\pi^+$),
$M_{\pi\pi}^{\rm end}=M_\tau-m_{\pi^\pm}\simeq 1637$~MeV.
Thus, heavier $f_2$ resonances should have even stronger phase-space
suppressions. In particular the $f_2(1640)$ and further tensors
lie beyond $M_{\pi\pi}^{\rm end}$.
\subsection{The R$\chi$T Lagrangian for tensor fields}
The relevant part of the chiral invariant Lagrangian for the pion-tensor
production (Fig~\ref{fig.diagr}) consists in this case of
\begin{itemize}
\item Operators with one resonance field~\cite{rcht,Zauner:2007},~\footnote{
There are two more operators for $\mL_T$ in Ref.~\cite{Zauner:2007}
allowed by chiral symmetry
but they contain the trace $T_{\,\, \alpha}^\alpha$~\cite{Zauner:2007}:
$\Delta \mL_T|_{ \mbox{\tiny off-shell} }=
\bra T_{\,\, \alpha}^\alpha \left( \beta u^\mu u_\mu +\gamma \chi_+ \right) \ket$.
Since they are proportional to the equations of motion
of the tensor, which on-shell require it to be transverse ($\nabla^\alpha T_{\alpha\beta}=0$)
and traceless ($T_{\,\,\, \alpha}^\alpha=0$), they can be removed through meson field redefinitions
and we will not discuss them in the present work.
}
\bear
\mL_A &=& \Frac{F_A}{2\sqrt{2}} \bra A_{\mu\nu} f_-^{\mu\nu} \ket\, ,
\nn\\
\mL_T &=& g_T\bra T_{\mu\nu} \{ u^\mu,u^\nu\}\ket\, .
\label{eq.LT}
\eear
\item Operators with an axial-vector and a tensor field
(which provides the $AT\pi$ vertex in
diagram c) in Fig.~\ref{fig.diagr}),
\bear
\mL_{AT\pi} &=&
\lambda_1^{AT}\, \bra\{T_{\mu\nu}, A^{\nu\alpha} \} h_\alpha^\mu\ket
+ \lambda_2^{AT} \bra\{ A_{\alpha\beta} , \nabla^\alpha T^{\mu\beta} \} u_\mu\ket
\, , \label{eq.tensor-lagr}
\eear
with $h_{\alpha\mu}=\nabla_\alpha u_\mu +\nabla_\mu u_\alpha$~\cite{rcht}.
Only the independent operators from $\mL_{AT}$ that contribute to the $AT\pi$
vertex are shown here. We construct here the general
chiral invariant operators at lowest order in derivatives, $\cO(p^2)$,
that may contribute to the $AT\pi$ vertex.~\footnote{
There are also two more $AT\pi$ operators allowed by symmetry
but they contain the trace $T_{\,\, \alpha}^\alpha$ or the contraction
$\nabla^{\alpha} T_{\alpha\beta}$:
$\Delta \mL_{AT\pi}|_{ \mbox{\tiny off-shell} } =
\beta_{AT\pi} \bra \{ A_{\alpha\beta} , \nabla^\alpha T_{\,\, \mu}^\mu \} u^\beta \ket
+ \gamma_{AT\pi} \bra \{ A_{\alpha\beta} , \nabla_\mu T^{\mu\alpha}\} u^\beta \ket$.
They do not propagate the tensor meson and can be removed from the generating functional
through appropriate field redefinitions.
}
\item Operators without resonance fields~\cite{Zauner:2007}:
in addition to~(\ref{eq.non-R-lagr}) we have
\bear
\mL_{\rm non-R}^{(4)} &=&
L_1^{SD}\bra u^\mu u_\mu \ket ^2\,
\,+\, L_2^{SD} \bra u^\mu u^\nu \ket \,\bra u_\mu u_\nu \ket
\, + \, L_3^{SD}\bra (u^\mu u_\mu)^2 \ket \, ,
\label{eq.non-R-lagr+Op4}
\eear
with~\cite{Zauner:2007}
\bear
L_2^{SD}=2 L_1^{SD}= - \Frac{L_3^{SD}}{2} = -\Frac{g_T^2}{M_T^2}\, .
\label{eq.LjSD}
\eear
The appearance of $\mL_{\rm non-R}^{(4)} $ was explained in ~\cite{Zauner:2007}:
in order to reproduce the correct short-distance behaviour
for the forward $\pi\pi$ scattering
--prescribed by the Froissart bound~\cite{Froissart:1961ux}--
one must add non-resonant
$\cO(p^4)$ terms with appropriate $L_{1,2,3}^{SD}$.
As a consequence this, new non-resonant diagrams generated by $L_{1,2,3}^{SD}$
(Fig.~\ref{fig.diagr-nonR}) have to be included in the calculation of the $3\pi$-AFF.
Additional details from Ref.~\cite{Zauner:2007} are provided in App.~\ref{app.ChPT}.
This problem did not appear in the scalar and vector resonance case~\cite{rcht},
i.e. the introduction of the scalar and vector resonance
interaction, $\mL_S$ and $\mL_V$~\cite{Dumm:2009va}, did not spoil the high-energy behaviour
of the forward pion scattering and no additional $\cO(p^4)$ terms were required~\cite{rcht}.
\end{itemize}
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 0.4\textwidth]{diagrams-nonR.eps}
\end{center}
\caption{{\small
New diagrams due to the short-distance
$\cO(p^4)$ operators $L_{1,2,3}^{SD}$. For a more detailed explanation, see the text.
The vertices from $\mL_{\rm non-R}^{(4)}$ ($\mL_{\rm non-R}^{(2)}$)
are represented by squares (circles).
The straight lines are pions and the wavy ones correspond to the incoming $W^-$.
}}
\label{fig.diagr-nonR}
\end{figure}
We will assume the ideal mixing in the tensor nonet
$T_{\mu\nu}=T^a_{\mu\nu} \lambda^a/\sqrt{2}$
and that the $f_2(1270)$ resonance is the pure
$u\bar{u}+d\bar{d}$ component:
\bear
T^{\mu\nu} &=& \left(\begin{array}{ccc}
\Frac{f_2^{\mu\nu}}{\sqrt{2}} &0&0 \\ 0 & \Frac{f_2^{\mu\nu}}{\sqrt{2}} & 0 \\
0&0&0
\end{array}\right)\quad +\quad ...
\eear
\subsection{AFF into $T\pi^-$}
\label{sec:AFF}
The general possible structure for the hadronic matrix element into a tensor and a pion is given by
three independent form-factors~\cite{Castro:2011zd}, which can be arranged in the form
\bear
\bra f_2(k,\epsilon)\, \pi^-( p_3 )|\,\bar{d}\gamma^\alpha\gamma_5 u\, |0\ket &=&
\epsilon_{\mu\nu}^* H^{\alpha,\, \mu\nu}_{ T \pi}
\label{eq.general-f2pi-AFF}
\\
&& \hspace*{-5.cm}
= i \,\epsilon_{\mu\nu}^* \,
\left[ P_T(q)^{\alpha\rho} \, p_3^\nu
\, \left(\, g_\rho^\mu \, \mF^a_{T\pi}(q^2 ; k^2 ) \,\, +\,\,
p_{3\, \rho} p_3^\mu \,\mG^a_{T\pi}(q^2; k^2)
\,\right)
\,\, +\,\,
p_3^\mu p_3^\nu q^\alpha \, \mH^a_{T\pi}(q^2; k^2)
\right]\, ,
\nn
\eear
with $q= p_3 +k$ and $\epsilon_{\mu\nu}$ the polarization of the outgoing tensor~\cite{Castro:2011zd,Zauner:2007}.
Due to the partial conservation of the axial-vector current, the
$\mH^a_{T\pi}(q^2;k^2)$ form-factor is suppressed by $m_\pi^2$.
Here the tensor resonance has been assumed to be the asymptotic final state with polarizations fulfilling the on-shell constraints~\cite{Zauner:2007}
\bear
\epsilon_{\mu\nu}=\epsilon_{\nu\mu}\, , \qquad k^\mu \epsilon_{\mu\nu}=0\, ,
\qquad g^{\mu\nu} \epsilon_{\mu\nu}=0\, .
\eear
We used the completeness relation~\cite{Zauner:2007,LopezCastro:1997im}
\bear
\!\!\!\!\!\!\!\!\mP(k)^{\mu\nu,\alpha\beta}=\sum_{\epsilon}\epsilon_{\mu\nu}\epsilon^*_{\alpha\beta}
&=&\Frac{1}{2}\left(P(k)^{\mu\alpha}P(k)^{\nu\beta}+P(k)^{\nu\alpha}P(k)^{\mu\beta}\right)-\Frac{1}{3}P(k)^{\mu\nu}P(k)^{\alpha\beta} \,\,
\label{eq.tens-deno}
\eear
with $P(k)_{\mu\nu}= \left.P_T(k)^{\mu\nu}\right|_{k^2=M_T^2} =g_{\mu\nu}
-k_\mu k_\nu/M_{T}^2$.
The hadronic Lagrangian from Eqs.~(\ref{eq.LT}) and~(\ref{eq.tensor-lagr}) leads to the
determination
\bear
\mF^a_{T\pi}(q^2 ;k^2 ) &=& - \Frac{8g_T}{ F_\pi }
+ \Frac{4\sqrt{2} F_A\lambda_1^{AT}}{F_\pi} \Frac{ (q p_3) }{M_A^2-q^2}
- \Frac{2\sqrt{2} F_A\lambda_2^{AT}}{F_\pi} \Frac{ (qk) }{M_A^2-q^2}
\, ,
\nn\\
\mG^a_{T\pi}(q^2 ;k^2 ) &=&
- \Frac{4\sqrt{2} F_A\lambda_1^{AT}}{ F_\pi } \Frac{1 }{M_A^2-q^2}
- \Frac{2\sqrt{2} F_A\lambda_2^{AT}}{ F_\pi } \Frac{1 }{M_A^2-q^2}
\, ,
\nn\\
\mH^a_{T\pi}(q^2 ;k^2 ) &=& 0 \, ,
\eear
with $(qp_3)=(q^2+m_\pi^2-k^2)/2$ and $(qk)=(q^2-m_\pi^2+k^2)$. Even
though $k^2=M_T^2$ when the tensor resonance is on-shell we have kept the off-shell momentum dependence
stemming from our R$\chi$T Lagrangian.
The $m_\pi^2$ chiral suppressed form-factor $\mH^a_{T\pi}(q^2)$
is exactly zero in our approach as we are considering
a resonance Lagrangian with the lowest number of derivatives
(this is, two derivatives, $\cO(p^2)$)
and the Lorentz structure corresponding to $\mH^a_{T\pi}(q^2;k^2)$ carries three powers of external momenta.
If one imposes a vanishing behaviour for the contribution of the $T\pi$ absorptive cut to the
axial-vector correlator at $q^2\to \infty$ one finds that the form-factors vanish at large momentum transfer like
$\mF^a_{T\pi}(q^2;M_T^2)\stackrel{q^2\to \infty}{\longrightarrow} \cO(1/q^2)$ and
$\mG^a_{T\pi}(q^2;M_T^2)\stackrel{q^2\to \infty}{\longrightarrow} \cO(1/q^4)$ or faster
(see App.~\ref{app.optical-theorem} for details).
Demanding this to the previous R$\chi$T form-factors
$\mF^a_{T\pi}$ and $\mG^a_{T\pi}$
yields, respectively, the constraints (taking into account $k^2 = M_T^2$
for the on-shell resonance),
\bear
4 \sqrt{2} g_T \, +\, 2 F_A\lambda_1^{AT} \, -\, F_A\lambda_2^{AT} &=& 0\, ,
\qquad\qquad
2\lambda_1^{AT} \, +\, \lambda_2^{AT} \,=\, 0 \, .
\label{eq.constraints1}
\eear
This leads to the resonance coupling relations
\bear
F_A\lambda_2^{AT} &=& - 2 F_A\lambda_1^{AT} \,=\, 2\sqrt{2} g_T\,
\label{eq.constraints2}
\eear
and the form-factors
\bear
\mF^a_{T\pi}(q^2 ;k^2 ) &=& - \Frac{8g_T}{ F_\pi } \Frac{ M_A^2 }{M_A^2-q^2}\, ,
\nn\\
\mG^a_{T\pi}(q^2 ;k^2 ) &=& 0\, .
\label{eq.TP-AFF+constraints}
\eear
This result agrees with that in Ref.~\cite{Castro:2011zd} near the axial-vector resonance.
Furthermore, in the chiral limit, if one requires the same fall-off for the form-factors
therein
one has an agreement in the full energy range.
Additional details can be found in App.~\ref{app.Castro-comparison}.
\subsection{$3\pi$ AFF through an intermediate tensor resonance}
\label{sec:3pi-from-T}
The three possible decay mechanisms involving the tensor resonance are drawn in Fig.~\ref{fig.diagr}.
We present here some useful intermediate results.
The $\pi^0\pi^0\pi^-$ production with the neutral pions mediated by a tensor resonance is provided
by three ingredients:
\begin{itemize}
\item
The transition $W^{-\mu}(q) \to f_2(k)^* \pi^0(p_3)$ taking into account the three diagrams
is given by
\bear
&&\bra f_2^*(k,\epsilon)
\pi^- (p_3) | \bar{d}\gamma^\mu \gamma_5 u|0\ket \,=\, \epsilon_{\alpha\beta}^{*} H^{\mu,\, \alpha\beta}_{T\pi}
\label{eq.tens_axvec_pion}
\\
&&\qquad=\,
\frac{-4\sqrt{2}\, i}{F_\pi }p_{3}^\alpha\epsilon^{\star \alpha\beta}\Bigl[
\sqrt{2}g_T \left(g_{\beta\mu} -\frac{q_\beta q_\mu}{q^2 -m_\pi^2}\right)
\nn\\
&&\qquad\qquad\qquad\qquad\qquad
-
F_A \frac{\left[
\lambda_1^{AT} (qp_3 g_{\beta\mu} - q_\beta p_{3\mu})
\, -\,\frac{1}{2}
\lambda_2^{AT} (qk g_{\beta\mu} - q_\beta k_{\mu})
\right] }{M_A^2 -q^2}
\Bigr]
\, .
\nn
\eear
After imposing the high-energy constraints~(\ref{eq.constraints2}), this expression gets greatly simplified into
\bear
H^{\mu,\, \alpha\beta}_{T\pi}
&=&
\frac{-8\, i g_T}{ F_\pi }p_{3}^\alpha
\Bigl[ \frac{M_A^2 }{M_A^2 -q^2} \, P_T(q)^{\beta\mu}
\, -\,
\Frac{m_\pi^2 q_\beta q_\mu}{q^2 (q^2-m_\pi^2)}
\Bigr]
\, .
\label{eq.tens_axvec_pion+SD}
\eear
We remark that we have not used the on-shell conditions in Eqs.~(\ref{eq.tens_axvec_pion}) and (\ref{eq.tens_axvec_pion+SD})
above.
\item The tensor propagator~\cite{Zauner:2007}:
\bear
\Delta_T(k)^{\mu\nu,\alpha\beta}&=& \Frac{i\, \mP(k)^{\mu\nu,\alpha\beta}}{M_T^2-k^2}\, .
\eear
\item The decay amplitude $\mM( f_2^*(k) \to \pi^0(p_1)\pi^0(p_2))= \epsilon^{\alpha\beta}\Gamma_{\alpha\beta} $ is given by
\bear
\Gamma_{\alpha\beta} &=&
\frac{-i \sqrt{2} g_T}{F_\pi^2} \Bigl[
k^\alpha k^\beta
- \Delta p^\alpha\, \Delta p^\beta
\Bigr] \, , \label{eq.tens_two_pions}
\eear
with $\Delta p^\rho=p_1^\rho-p_2^\rho$ and $k^2=s_3$.
No on-shell condition has been assumed in the expression above.
The term $k^\alpha k^\beta$ becomes zero when contracted with the $\epsilon^{\alpha\beta}$
polarization of an external on-shell tensor resonance.
\end{itemize}
The $\pi^0\pi^0\pi^-$ AFF is then given by
\bear
H^\mu &=&
H_{(0)}^\mu \, +\,
H_{T\pi}(k,p_3)^{\mu,\, \alpha\beta} \,\,\, \Delta_T(k)_{\alpha\beta,\rho\sigma} \,\,\, \Gamma(p_1,p_2)^{\rho\sigma}
\label{eq.Htens}
\\
&&\,\,\,=\,\,\,
H^\mu_{(0)}\,\,\,+ \,\,\,
H^\mu_{(1)} + \frac{H^\mu_{(2)}}{M_{T}^2 - s_3}
\, . \nn
\eear
The first term, $H_{(0)}^\mu$, comes from the non-resonant diagrams in Fig.~\ref{fig.diagr-nonR} generated
by the short-distance terms $L_{1,2,3}^{SD}$ in Eqs.~(\ref{eq.non-R-lagr+Op4}) and (\ref{eq.LjSD}). The
second and third ones, $H_{(1)}^\mu$ and $H_{(2)}^\mu$, respectively, are produced by the
diagrams with tensor resonance exchanges (Fig.~\ref{fig.diagr}).
$H_{(1)}^\mu$ comes from the $k^\alpha k^\beta$ term in the
$\Gamma[T(k)_{\alpha\beta} \to\pi^0(p_1)\pi^0(p_2)]$ vertex function
and does not contribute to the on-shell decay $T\to\pi^0\pi^0$.
For sake of this,
the contribution with $H_{(1)}^\mu$ does not propagate the tensor resonance
and has no pole at $s_3=M_T^2$.
The contribution to the three-pion AFF from the remaining part of the $T\pi^0\pi^0$ vertex is encoded
in $H_{(2)}^\mu$.
The value of these two types of contributions are
\bear
H^\mu_{(0)} &=& \Frac{8\sqrt{2} i g_T^2}{3F_\pi^3 M_T^2} P_T(q)^{\mu\nu}
\big[(s_3 - s_2 + 2s_1 -4 m_\pi^2 )(p_1 - p_3)_\nu
\nn\\
&& \qquad\qquad\qquad \qquad\qquad
+ (s_3 - s_1 + 2s_2 -4 m_\pi^2)(p_2 - p_3)_\nu \big]
\label{eq.contri0}
\\
&&
- \Frac{8\sqrt{2} i g_T^2 m_\pi^2}{F_\pi^3 M_T^2 q^2 (q^2-m_\pi^2) }
\, q^\mu\, \left( s_1 s_2 - m_\pi^2 q^2 - m_\pi^4\right)
\nn\\
&& \nn \\
&& \nn \\
H^\mu_{(1)} &=& \Frac{8\sqrt{2} i g_T^2}{F_\pi^3 M_T^2}
\Frac{m_\pi^2}{q^2(q^2-m_\pi^2)}
\, q^\mu\, \left[ (k q) (k p_3) - \frac{s_3}{3} \left( (qp_3)
+ \Frac{ 2 (k q) (k p_3) }{M_T^2} \right)\right]
\label{eq.contri1}
\\
&&
-\, \Frac{8 i g_T}{F_\pi^3 M_T^2} \Frac{M_A^2}{(M_A^2-q^2)} P_T(q)^{\mu\nu} k_\nu
\bigg[
\sqrt{2} g_T
\left( \left(1 -\Frac{2 s_3}{3 M_T^2} \right)
(kp_3) + \Frac{s_3}{3} \right)
\nn\\
&&
\qquad + (F_A\lambda_1^{AT} +\sqrt{2} g_T) \Frac{ q^2 (kp_3)}{M_A^2} \,
\left(\Frac{2 s_3}{3 M_T^2} -1\right)
+ (F_A\lambda_2^{AT} -2\sqrt{2} g_T) \Frac{ q^2 s_3 }{ 6 M_A^2}
\bigg]\, ,
\nn
\\
&& \nn \\
&& \nn
\\
H^\mu_{(2)\,{\rm a_1-pole}} &=&
-\, \Frac{8 i g_T}{F_\pi^3}
\Frac{F_A}{M_A^2-q^2} P_T(q)^{\mu\nu}
\bigg[
\left( \lambda_1^{AT} M_A^2 - \left(\lambda_1^{AT}+\Frac{\lambda_2^{AT}}{2} \right) (kq) \right)
\, (q\Delta p)\, \Delta p_\nu
\nn\\
&&
\hspace*{-1.5cm}
+ \left( \Frac{\lambda_1^{AT} M_A^2 (\Delta p)^2
( kp_3 + M_T^2)}{3 M_T^2}
+ \left(\lambda_1^{AT}+\Frac{\lambda_2^{AT}}{2} \right)
\left( (q\Delta p)^2 - \Frac{(\Delta p)^2 M_A^2}{3} \right)\right) k^\nu \bigg] \, ,
\label{eq.contri2A}
\\
&& \nn
\\
&& \nn \\
H^\mu_{(2)\,{\rm a_1\, no-pole}} &=&
-\, \Frac{2\sqrt{2} i g_T}{F_\pi^3} P_T(q)^{\mu\nu}
\bigg[
- 2\sqrt{2}(F_A\lambda_1^{AT} +\sqrt{2} g_T)
\left( (q\Delta p) \Delta p_\nu + \Frac{(kp_3) (\Delta p)^2 }{3 M_T^2} k_\nu\right)
\nn
\\
&&
+ \sqrt{2}( F_A\lambda_2^{AT} -2\sqrt{2} g_T ) \Frac{(\Delta p)^2 }{3 } k_\nu
\bigg]
\nn\\
&&
-\, \Frac{8\sqrt{2} i g_T^2 m_\pi^2}{ F_\pi^3 q^2 (q^2-m_\pi^2)} \, q^\mu\,
\bigg[ (q\Delta p)^2 + \Frac{(\Delta p)^2}{3 M_T^2}
\left(kq\, kp_3 -qp_3 M_T^2\right)
\bigg]\, ,
\label{eq.contri2B}
\eear
with $(\Delta p)^2= 4 m_\pi^2-s_3$,
$(kq)=(q^2+s_3-m_\pi^2)/2$ and
$(k\Delta p) =0$. From these, one can derive a series of dependent scalars:
$(kp_3)=(qk)-s_3=(q^2-s_3-m_\pi^2)/2$,
$(qp_3)=q^2-(qk)=(q^2-s_3+m_\pi^2)/2$, $(q\Delta p)=(p_3\Delta p)= (s_2-s_1)/2$
and the relation $s_{1,2}= kp_3 + 2m_\pi^2 \mp q\Delta p$.
For convenience we have split $H^\mu_{(2)}$
into its parts with and without the $a_1$ pole.
We also used the relation
$ (q p_3) k^\mu - (q k) p_3^\mu= q^2 P_T(q)^{\mu\nu} k_\nu$.
We now combine $H_{(0)}^\mu$, $H_{(1)}^\mu$ and $H_{(2)}^\mu$ and rewrite
their sum in terms of the Lorentz decomposition~(\ref{eq.hadr-curr-3pions}).
This provides the contribution to the ${\pi^0\pi^0\pi^-}$ AFFs in~(\ref{eq.hadr-curr-3pions})
derived from tensor resonance exchanges:
\bear
\mF_1^{\pi^0\pi^0\pi^-}(s_1,s_2,q^2) \bigg|_T
&=&
\mF_{1,\,\, (0)}^{\pi^0\pi^0\pi^-} (s_1,s_2,q^2) \, +\,
\mF^{\pi^0\pi^0\pi^-}_{1,\,\,\rm (RSD)} (s_1,s_2,q^2)
\label{eq.ff-tens}
\\
&& \nn \\
&& - \Frac{4}{9 F_\pi^3 }\Frac{ g_T}{M_T^2}
\Frac{( F_A\lambda_2^{AT} - 2\sqrt{2} g_T)}{M_A^2-q^2}
\times \bigg[
s_3 q^2
\nn\\
&&\qquad + \Frac{M_T^2}{M_T^2-s_3} \left(
3 (q\Delta p)^2
- 9 (qk) (q\Delta p) - q^2 (\Delta p)^2
\right)
\bigg]
\nn\\
&& - \Frac{8}{3 F_\pi^3} \Frac{g_T}{M_T^2}
\Frac{(F_A\lambda_1^{AT} + \sqrt{2} g_T)}{M_A^2-q^2}
\times
\bigg[
q^2 (kp_3) \left(\Frac{2 s_3}{3 M_T^2} -1\right)
\nn\\
&&\qquad
+\Frac{M_T^2}{M_T^2-s_3}
\left( (q\Delta p)^2 + 3 (q\Delta p) (qp_3)+ \Frac{ q^2 (kp_3) (\Delta p)^2}{3M_T^2} \right)
\bigg]
\nn\\
\mF_P^{\pi^0\pi^0\pi^-}(s_1,s_2,q^2) \bigg|_T &=&
\mF_{P,\,\, (0)}^{\pi^0\pi^0\pi^-} (s_1,s_2,q^2) \,
\label{eq.ff-tensP}
\\ &&
+
\frac{8 \sqrt{2}g_T^2 m_\pi^2}{3 M_T^2 F_\pi^3 q^2 (m_\pi^2 -q^2)}
\times \bigg[
(qp_3) s_3 + (kq) (kp_3) \left( \Frac{2 s_3}{M_T^2} -3 \right)
\nn\\
&&
\qquad +\Frac{M_T^2}{M_T^2-s_3} \left(
3 (q\Delta p)^2 + \left(\Frac{(kq) (kp_3)}{M_T^2} - (qp_3) \right)(\Delta p)^2\right)
\Bigg] \, ,
\nn
\eear
with
\bear
\mF_{1,\,\, (0)}^{\pi^0\pi^0\pi^-} (s_1,s_2,q^2)
&=&
\Frac{8\sqrt{2} g_T^2}{3 F_\pi^3 M_T^2} (
2 s_1 -s_2+s_3
-4 m_\pi^2) \, ,
\label{eq.F1-0}
\\ && \nn \\
\mF_{P,\,\, (0)}^{\pi^0\pi^0\pi^-} (s_1,s_2,q^2) &=&
- \Frac{8\sqrt{2} g_T^2 m_\pi^2}{ F_\pi^3 M_T^2 q^2 (q^2-m_\pi^2) }
\left( s_1 s_2 - m_\pi^2 q^2 - m_\pi^4\right) \, ,
\label{eq.FP-0}
\\ && \nn \\
\mF^{\pi^0\pi^0\pi^-}_{1,\,\, \rm (RSD)} (s_1,s_2,q^2)
&=&
- \Frac{8 \sqrt{2} }{3 F_\pi^3}\Frac{g_T^2}{ M_T^2} \Frac{M_A^2}{M_A^2-q^2}
\bigg[
(kp_3) + \Frac{s_3}{3}\left(1-\Frac{2 (kp_3)}{M_T^2}\right)
\nn\\
&&\qquad
- \Frac{M_T^2}{M_T^2-s_3}\left(
3 (q\Delta p) + \Frac{(\Delta p)^2}{3}
+ \Frac{(kp_3)(\Delta p)^2 }{3 M_T^2}
\right)
\bigg] \, \label{eq.tens_rsd},
\eear
where the contributions $\mF_{1,\,\, (0)}^{\pi^0\pi^0\pi^-}$
and $\mF_{P,\,\, (0)}^{\pi^0\pi^0\pi^-}$
come from the $H_{(0)}^\mu$ part of the matrix element $H^\mu$.
All the results here refer to the $\pi^0\pi^0\pi^-$ AFF.
Isospin symmetry~\cite{Pais:1960zz,Finkemeier:1996,Girlanda:1999fu}
relates them to the $\pi^-\pi^-\pi^+$ form-factors,
which can be obtained by mean of the relations~(\ref{eq.isospin-rel2}).
The expression of the form-factors get greatly simplified
after applying the high-energy constraints
extracted from the analysis of the $T\pi$ AFF in Eq.~(\ref{eq.constraints2}):
\bear
\mF_1^{\pi^0\pi^0\pi^-}(s_1,s_2,q^2) \bigg|_T&=&
\mF_{1,\,\, (0)}^{\pi^0\pi^0\pi^-}(s_1,s_2,q^2)
\, +\,
\mF^{\pi^0\pi^0\pi^-}_{1,\,\, \rm (RSD)}(s_1,s_2,q^2)
\, ,
\eear
while these resonance short-distance conditions do not affect
the longitudinal form-factor $\mF_P(s_1,s_2,q^2) \bigg|_T$, which remains the same as
in~(\ref{eq.ff-tensP}).
The comparison between CLEO's results and ours for the amplitude and the related AFF is given in App.~\ref{app.cleo}. From that, we conclude that the two parametrizations coincide near the resonance energy regions ($s_3 \simeq M_T^2$, $q^2 \simeq M_A^2$). However,
for an arbitrary off-shell momentum
we have a more general momentum structure which ensures
the right low energy behaviour and the transversality of the matrix element in the chiral limit,
allowing a proper matching with $\chi$PT.
\subsection{Tensor resonance width}
\label{sec:tensor-width}
In order to include the effect of the tensor width,
we modify the tensor resonance propagator in the form
\bear
\frac{1}{M_T^2 - s} \qquad \longrightarrow \qquad \frac{1}{M_{f_2}^2 - s - i M_{f_2} \Gamma_{f_2}(s)}\, ,
\eear
with the spin--2 energy-dependent Breit-Wigner width used in CLEO's analysis~\cite{CLEO:1999},
\bear
\Gamma_{f_2}(s) &=& \, \Gamma^{f_2}_{0} \, \Frac{ s^2 }{M_{f_2}^4}
\, \Frac{\rho_\pi(s)^5}{\rho_\pi(M_{f_2}^2)^{\,5} } \,\, .
\eear
For the numerical estimation in the next Section we will take
the PDG central value $\Gamma^{f_2}_0 = 186.7$~MeV
for the $f_2(1270)$ total decay width~\cite{pdg}.
The tensor contribution to the AFF depends on the $g_T$ coupling, which is related to the on-shell
decay width into two pseudo-Goldstones~\cite{Zauner:2007}:
\bear
\Gamma_{f_2\to \pi\pi} &=&\Frac{g_T^2 M_{f_2}^3 \rho_{\pi}(M_{f_2}^2)^{\,5}}{ 40\pi F_\pi^4}\, .
\eear
Using the PDG central values, $\Gamma_{f_2\to\pi\pi}^{\rm exp} =157.2$~MeV, $M_{f_2} =1275.5$~MeV,
$m_\pi=139.57$~MeV and $F_\pi=92.2$~MeV, one obtains
\bear\label{eq.value-gt}
g_T \simeq 28\, \mbox{MeV}\, ,
\eear
which agrees with the estimation in~\cite{Zauner:2007}.
\section{Implementation in Tauola: numerical results}
\label{sec:Tauola}
In the previous sections we described the set
of the three pion form
factor contributions related with the tensor and scalar
intermediate resonances and calculated on the base of the R$\chi$T Lagrangians.
In this section we present
a first numerical estimate with
the updated version of the Monte Carlo (MC) event generator Tauola~\cite{Jadach:1993hs}.
It incorporates
the new scalar and tensor contributions to the AFF computed in this article,
provided in~(\ref{eq.ff-scal}) and (\ref{eq.ff-tens}), respectively.
\footnote{The MC Tauola implementation of these channels was cross-checked with a Mathematica code, which can be provided on demand. }
First, we compare the analytical and Tauola distributions
for the decay width ($d\Gamma^{\pi\pi\pi}/dq^2$)
and repeat the tests on numerical stability
of the MC, as in Sec.~4 of Ref.~\cite{Shekhovtsova:2012ra}~\footnote{
We use the same samples and integration procedure
as in~\cite{Shekhovtsova:2012ra}.
The MC result here corresponds to a number of events $N_{\rm ev.}=6\cdot 10^6$.}
For further details see this reference.
The comparison is presented in Fig.~\ref{fig.tauola_compar}.
We present here only $d\Gamma^{\pi^0\pi^0\pi^-}/dq^2$ spectrum. A
similar result has been obtained for the $\pi^-\pi^-\pi^+$ mode.
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 0.45\textwidth, height = 0.35\textwidth]{qq_pi0pi0pi-.eps}
\includegraphics[width = 0.45\textwidth, height = 0.35\textwidth]{ratio_pi0pi0pi-.eps}
\end{center}
\caption{{\small
Three pion $q^2$ spectrum $d\Gamma^{\pi^0\pi^0\pi^-}/dq^2$
(left) and the ratio
of the MC and the analytical $q^2$ spectrum (right). }
}
\label{fig.tauola_compar}
\end{figure}
In addition we have compared the two- and three-meson
invariant mass distributions for our theoretical result
and the experimental data.
For the $\pi^-\pi^-\pi^+$ channel,
we used preliminary BaBar data~\cite{Nugent:2013ij}
(Fig.~\ref{fig.tauola_theory}, top panels).
Due to our lack of access to the $\pi^0\pi^0\pi^-$ data, they have been 'emulated'
on the basis of the results in Ref.~\cite{CLEO:1999}:
Tauola was run with CLEO's AFF from App.~A.1 of~\cite{CLEO:1999} and nominal
fit parameters specified therein in Table III.~\footnote{
We thank J.~Zaremba for providing the corresponding unnormalized CLEO distributions. }
The comparison of our parametrization to this
`emulation' of
CLEO data is shown in Fig.~\ref{fig.tauola_theory}, bottom panel.
To produce the theoretical distributions the tensor and scalar resonance parameters were fixed
to their value specified in Secs.~\ref{sec:scalar-width} and~\ref{sec:tensor-width}
whereas the vector and axial-vector parameters were fixed to their fit values
in~\cite{Nugent:2013hxa}. All parameters are
summarized in Table~\ref{tab:num_value}
except $c_m$. This coupling,
whose effects are suppressed by $m_\pi^2$ factors, is extracted from the $c_d$ and $F_\pi$ values in Table~\ref{tab:num_value} and the
short-distance constraint $4c_d c_m=F_\pi^2$~\cite{Jamin:2001zq}.
\begin{table}[h!]
\caption{Numerical values of the parameters used to produce the theoretical spectra
in~\ref{fig.tauola_theory}.
All the parameters are in GeV units except for $c_{\sigma}$ and $c_{f_0}$,
which are dimensionless.
}
\label{tab:num_value}
\begin{center}
\begin{tabularx}{\textwidth}{|X|X|X|X|X|X|X|X|}
\hline
$M_{\rho}$ & $M_{\rho'}$ & $\Gamma_{\rho'}$ & $M_{a_1}$ & $M_{\sigma}$ & $M_{f_2}$ & $\Gamma_{f_2}$ &$F_{\pi}$
\\
\hline
$ 0.772 $ & $ 1.35 $ & $ 0.448 $ & $ 1.10 $ & $ 0.8064 $ & $ 1.275 $ & $ 0.185 $ & $0.0922 $
\\
\hline
\end{tabularx}
\\
\begin{tabularx}{\textwidth}{|X|X|X|X|X|X|X|X|}
\hline
$F_{V}$ & $F_{A}$ & $\beta_{\rho}$ & $g_{T}$ & $c_d$ & $c_{\sigma}$ & $M_{f_0}$ & $c_{f_0}$
\\
\hline
$ 0.168 $ & $ 0.131 $ & $ - 0.32 $ & $ 0.028 $ & $ 0.026 $ & $ 76.12 $ & $1.024$ & $17.7$
\\
\hline
\end{tabularx}
\end{center}
\end{table}
These plots in Fig.~\ref{fig.tauola_theory} are an illustration of our model,
which demonstrates that, even without fitting,
the model qualitatively reproduces the experimental
spectra. No large unwanted deviation from data occurs,
being these values an appropriate
starting point for a more detailed study.
The tuning of our model parameters and the fitting to the data
will be done in a future work~\cite{new-paper-fit}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 1.\textwidth]{tauola_3pi_init_3histo_fixf2_2.eps}
\includegraphics[width = 1.\textwidth]{tauola_pi0pi0pi-_init_3histo_fixf2_2.eps}
\end{center}
\caption{{\small Top:
comparison between the BaBar data
and our theoretical
prediction for the $\pi^-\pi^-\pi^+$ decay mode.
Bottom: comparison between the CLEO 'emulated' data
(for details the text)
and our prediction for the $\pi^0\pi^0\pi^-$ decay mode.
}}
\label{fig.tauola_theory}
\end{figure}
In order to understand the impact of the different contributions we
focus our attention in the $\pi^0\pi^0\pi^-$ channel, where the various contributions
are more neatly separated: vectors only resonate in the $s_1$ and $s_2$ spectra,
and scalars and tensors only resonate in the $s_3$ distribution.
The first thing to notice is that all the distributions are dominated
by the vector contribution ``$V$''
(Lagrangian with only chiral Goldstones,
vectors and axial-vectors~\cite{Dumm:2009va,Shekhovtsova:2012ra}).
The scalar resonances (in particular the $\sigma$ meson)
serve to cure the discrepancies with respect to the data that appear
in the low energy regions, $M_{\pi\pi}< M_\rho$~\cite{Nugent:2013hxa}.
In Fig.~\ref{fig:VST-vs-V} we show the ratio of our
theoretical $\sqrt{s_3}$ distribution including
only the vector contribution $V$~\cite{Nugent:2013hxa}) and
its full result ($V+S+T$) in Fig.~\ref{fig.tauola_theory}
(all with the inputs given in Table~\ref{tab:num_value}).
For this set of parameters, we find that the scalar corrections
are smaller than 10\% in the low-energy region. Therefore,
when fitting the experimental data in this range, we will find that small
variations in the vector parameters may compensate
large modifications in the scalar ones,
being highly correlated for this observable.
Finally, the tensor resonance produces in general a negligible effect
in all the distributions except in the $\sqrt{s_3}$ one around 1.25~GeV,
where one can observe the clear emergence of the $f_2(1270)$ structure
in Fig.~\ref{fig:VST-vs-V}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 0.31\textwidth,clip]{ratio_pi0pi0_pi0pi0pi-_9e6_scal+tens+vec_vec_026.ps}
\hspace*{0.2cm}
\includegraphics[width = 0.32\textwidth,clip]{ratio_pi0pi0_pi0pi0pi-_9e6_scal+vec_vec_1.ps}
\hspace*{0.2cm}
\includegraphics[width = 0.31\textwidth,clip]{ratio_pi0pi0_pi0pi0pi-_6e6_tens+vec_vec_1.ps}
\end{center}
\caption{{\small
a) Ratio of the vector+tensor+scalar and only vector
$\sqrt{s_3}=M_{\pi^0\pi^0}$ spectral function for $\tau\to\nu_\tau \pi^0\pi^0\pi^-$;
b) Ratio of vector+scalar and only vector;
c) Ratio of vector+tensor and only vector.
All the plots use the inputs in Table~\ref{tab:num_value} ($c_d=26$~MeV).
}}
\label{fig:VST-vs-V}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 0.35\textwidth,clip]{ratio_pi0pi0_pi0pi0pi-_9e6_scal+tens+vec_vec_078.ps}
\hspace*{0.9cm}
\includegraphics[width = 0.34\textwidth,clip]{ratio_pi0pi0_pi0pi0pi-_9e6_scal+vec_vec_078.ps}
\end{center}
\caption{{\small
a) Ratio of the vector+tensor+scalar and only vector
$\sqrt{s_3}=M_{\pi^0\pi^0}$ spectral function for $\tau\to\nu_\tau \pi^0\pi^0\pi^-$;
b) Ratio of vector+scalar and only vector.
All the plots use the inputs in Table~\ref{tab:num_value}
except for $c_d$, which is set to $78$~MeV.
The ratio vector+tensor/vector is independent of $c_d$
and is provided in Fig.~\ref{fig:VST-vs-V}.c.
}}
\label{fig:VST-vs-V-cd=78}
\end{figure}
All the former analyses in this article are performed for the $S\pi\pi$ coupling
$c_d=26$~MeV in Table~\ref{tab:num_value}.
It is not clear whether this is the most suitable value, as other studies
do not lead to a conclusive estimate, allowing a much higher coupling~\cite{Sdecays}.
Since the scalar contribution to the amplitude is essentially proportional to $c_d^2$,
multiplying the value of $c_d$ by a factor 3 increases the impact of the scalar
in the spectral function by one order of magnitude
(through the interference with the $V$ contribution).
For illustration, in Fig.~\ref{fig:VST-vs-V-cd=78}, we show the same ratio
as in Fig.~\ref{fig:VST-vs-V} but for $c_d=78$~MeV.
The impact of these variations can be as important as small modifications
of the $V$ parameters.
Thus, it is not possible to pin down the scalar couplings without an accurate determination
of the vector ones. A joint fit is mandatory.
Another important numerical issue refers to the relevance
of the real part of the logarithm that is incorporated to the
$\sigma$ propagator \`a la Gounaris-Sakurai.
In Fig.~\ref{fig:GS-vs-BW}.a (Fig.~\ref{fig:GS-vs-BW}.b)
we show the ratio of our
theoretical $\sqrt{s_3}$ distribution neglecting the real part
of the $\sigma$ logs in Eqs.~(\ref{eq.S-propagator2})--(\ref{eq.S-self-energy})
and the full results from these equations
for $c_d=26$~MeV ($c_d=78$~MeV).
For all the other parameters we use the inputs from Table~\ref{tab:num_value}
and take only the vector+scalar contributions for sake of clarity.
Since the scalar contribution is quite small, the impact of the real logs
of the $\sigma$ propagator in the full spectral distributions
is quite suppressed for this $\tau$ decay.
We want to emphasize that although a Breit-Wigner $\sigma$ can provide an equally good
description of the data~\cite{Nugent:2013hxa},
the aim of the present analysis of the $\sigma$ \`a la Gounaris-Sakurai
is rather to improve the theoretical understanding of broad resonances
within a Lagrangian formalism and its matching to $\chi$PT at low energies.
\begin{figure}[!t]
\begin{center}
\includegraphics[width = 0.45\textwidth]{ratio_pi0pi0_pi0pi0pi-_9e6_scal+vec_loop_scal+vec_noloop_26+zoom.ps}
\hspace*{0.2cm}
\includegraphics[width = 0.45\textwidth]{ratio_pi0pi0_pi0pi0pi-_9e6_scal+vec_loop_scal+vec_noloop_78+zoom_2.ps}
\end{center}
\caption{{\small
Plots for the ratios of the $\sqrt{s_3}=M_{\pi^0\pi^0}$ spectral functions
for $\tau\to\nu_\tau \pi^0\pi^0\pi^-$:
a) ratio of the full result and the spectral function without the real part of the logs in
the $\sigma$ propagator for $c_d=26$~MeV;
b) ratio of the full result and the spectral function without the real part of the logs in
the $\sigma$ propagator for $c_d=78$~MeV.
In order to better pin down the impact of the scalar propagator structure we only consider the vector+scalar
contribution, dropping the tensors.
}}
\label{fig:GS-vs-BW}
\end{figure}
\section{Conclusions}
\label{sec:conclusions}
In this article we have computed the contribution of scalar and tensor resonances
to the $\tau\to \pi\pi\pi\nu_\tau$ decay axial-vector form-factors.
We have made use of a chiral invariant Lagrangian including
the relevant axial-vector, scalar and tensor resonances
together with the chiral (pseudo) Goldstones.
As a consequence of this,
the chiral symmetry is automatically incorporated in our
result. This ensures
the proper low-energy matching with $\chi$PT
and that
the currents for $\pi^0\pi^0\pi^-$ and $\pi^-\pi^-\pi^+$ channels
are related as prescribed by isospin symmetry~\cite{Girlanda:1999fu,Finkemeier:1996}.
In addition,
the tensor resonance contribution to the axial-vector current is transverse
in the chiral limit,
improving previous descriptions~\cite{Castro:2011zd}.
A similar thing applies to the scalar contributions.
Chiral symmetry also guaranties the proper low-energy matching with $\chi$PT,
fixing some issues in former parametrizations~\cite{CLEO:1999} (see App.~\ref{app.cleo}).
In addition, the tensor and scalar resonance contributions
to the tau decay are further refined by demanding the
appropriate asymptotic high-energy QCD behaviour for meson form-factors
prescribed by the quark-counting rules~\cite{Brodsky:1973kr}.
As described in Secs.~\ref{sec:3pi-from-S},~\ref{sec:AFF}
and App.~\ref{app.optical-theorem},
these large--$N_C$ short distance conditions constrain the resonance parameters
of the $T\pi$ and $S\pi$ AFFs, which are essentially determined in terms of the $g_T$
and $c_d$ couplings, respectively, and the resonance masses.
We have also studied an alternative approach
to the sigma description incorporating an analytical description of the width
\`a la Gounaris-Sakurai~\cite{GS-rho}:
instead of just the imaginary part $i\rho_\pi(s)$
required by unitarity in the K-matrix formalism or the Breit-Wigner form~\cite{Nugent:2013hxa},
we considered the full logarithm from the analytical Chew-Mandelstam
dispersive integral~\cite{Chew:1960iv}
or the renormalized two-propagator Feynman integral $\overline{B}_0$.
This parametrization of the $\sigma$ propagator provided a successful description
of the $\eta’\to\eta\pi\pi$ data and its $s$--wave $\pi\pi$ rescattering~\cite{Escribano:2010wt}.
Although it requires further refinements,
we find the exploration of this approach for $\tau\to 3\pi \nu_\tau$ worthy,
as it may help to understand whether it is possible or not to use
a Lagrangian formalism based on a perturbative expansion ($1/N_C$ in our case)
for the description of broad resonances.
We would like to note that in this article
we have considered for the first time the axial-vector--tensor interaction within the
Resonance Chiral Theory approach, extending the work
of Ecker and Zauner on tensors~\cite{Zauner:2007}. We plan to
include vector--tensor interactions in a similar way
in a future paper~\cite{new-paper}
dedicated to the study of the $e^+e^-\to a_2\pi$ process.
We have compared our outcome for the $\pi\pi\pi$ AFF with
former parametrizations with CLEO~\cite{CLEO:1999} and Castro-Mu\~noz~\cite{Castro:2011zd}.
While we coincide on the resonance region, our result incorporates an appropriate
low and high-energy behaviour, improving these works in the latter regimes.
As we plan to incorporate these new results in the Tauola generator, which generates
events from the three pion threshold up to roughly the tau mass, it is important to handle as best as possible the
various energy ranges (low, resonant and high). Some first simulations with the
Tauola Monte Carlo have been provided in Sec.~\ref{sec:Tauola}.
This article is only a preliminary illustration of our resonance chiral Lagrangian approach.
A more thorough numerical analysis is postponed for a future work~\cite{new-paper-fit}.
In order to obtain a good fit to the BaBar data, we will probably need not only
the one-dimensional distributions but also the Dalitz plot. A proper tuning
of the Monte Carlo parameters (e.g., the $S\pi\pi$ coupling $c_d$) should be reading before
the beginning of the Belle-II data taking.
To conclude: we
would like to remind that the forthcoming project Belle-II~\cite{Abe:2010gxa}
has a broad program devoted to $\tau$-physics. By 2022, they expect to
record a $50$ times lager data sample than the Belle experiment.
It will give us an opportunity to measure both $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$ decays
and study their intermediate
production mechanisms like, e.g., the tiny contribution from the $f_2\pi^-$ channel. This
will allow us to test our hadronic model and
the isospin symmetry relation between $\pi^-\pi^-\pi^+$ and $\pi^0\pi^0\pi^-$ form factors.
\section*{Acknowledgements}
We are thankful to G. Ecker, G. L\'opez-Castro and P.~Roig
for their helpful comments and feedback on the draft.
We thank J.~Zaremba for providing us the CLEO 'emulated' spectra
and R.~Escribano and T.~Przedzinski for useful discussions.
This work was partly supported
by the Spanish MINECO fund FPA2016-75654-C2-1-P.
|
1,116,691,500,502 | arxiv | \section{Introduction}
The past decade has witnessed an explosive growth of data and the needs for high-speed data processing. A large-scale data often needs to be sorted to enable higher efficiency. Sorting is a key kernel in many applications such as data mining \cite{aggarwal2015data}, robotics \cite{bayindir2016review} and machine learning \cite{devlin2018bert}. To efficiently sort an array into an order, numerous sorting algorithms have been invented in the past, such as merge-sort \cite{goldstine1947planning} or quick-sort \cite{quicksort}. These algorithms can be accelerated using CPUs/GPUs \cite{chhugani2008efficient,zhang2016high,satish2009designing}, FPGAs \cite{chen2017computer,chen2019sorting,samardzic2020bonsai,song2016parallel} and ASICs \cite{norollah2019rths,najafi2018low,lin2017hardware}. However, transferring data between memory and external processing units incurs a long latency and a degraded energy efficiency. Techniques like memory management \cite{stehle2017memory} have been developed to minimize the data movement, but such optimizations do not fundamentally solve the problem.
Memristive in-memory sorting \cite{rram_sort_alam2020,prasad2021memristive} have been proposed recently to tackle this challenge. Memristor-aided logic is developed in \cite{rram_sort_alam2020} to implement compare-and-select blocks in memory. However, a large number of memristor cells are used to implement logic gates with frequent write operations, resulting in a low memory density and a degraded device lifetime. The latest memristive in-memory sorting \cite{prasad2021memristive} uses iterative in-memory min computations with help of a near-memory circuit. The min values are searched by traversing each bit column using column reads (CR) on a 1T1R memristive memory. Frequent write operations in \cite{rram_sort_alam2020} are eliminated; however, the number of CRs is proportional to the number of 1T1R cells in the memristive memory, degrading the latency and energy efficiency.
In this work, we propose a column-skipping algorithm to minimize the number of CRs for improved sorting speed and hardware efficiency. A near-memory circuit is designed to keep track the column read conditions and skip those that are leading 0's or have been processed previously. A multi-bank management is developed to enhance the scalability when sorting a larger array stored in different memristive memory banks. Implemented in a 40nm CMOS technology with 1T1R memristive memory and experimented on a variety of sorting datasets, the length-1024 32-bit column-skipping memristive sorter with state recording of 2 demonstrates up to 4.08$\times$ speedup, 3.14$\times$ area efficiency and 3.39$\times$ energy efficiency, respectively, over the latest memristive in-memory sorting implementation \cite{prasad2021memristive}.
\section{Background}
\subsection{Sorting Applications}
Sorting is a known bottleneck for many applications \cite{aggarwal2015data,bayindir2016review,devlin2018bert}. Here we briefly introduce two representative applications where sorting dominates the execution time: 1) Kruskal's algorithm for minimum spanning tree (MST). In Kruskal's algorithm, all the graph edges need to be sorted from low weight to high weight. Majority of the weights are small numbers with frequent repetitions; 2) MapReduce in distributed systems. In MapReduce, maps need to be sorted before transferring to the reducer stage \cite{dean2008mapreduce}. These maps are typically clustered in a few groups. We use datasets generated from these two applications for benchmarking in Section~\ref{section:evaluation}.
\subsection{Memristive In-Memory Sorting}
Iterative in-memory min computation is proposed in \cite{prasad2021memristive} for memristive in-memory sorting. It uses $N$ iterations to successively search and exclude the min values in a length-$N$ array. Suppose each memristor cell stores a bit in a 1T1R memristive memory. \figurename~\ref{fig:memristive_sorting} shows an example for a length-$N$ ($N = 3$) array of $w$-bit ($w = 4$) numbers, \{8, 9, 10\}.
In each iteration, a $w$-step bit traversal algorithm searches the min value: at step $j$ ($j = w-1 \rightarrow 0$), a near-memory circuit reads an bit column corresponding to the $j$-th bits of all array elements, searches for 1's in that bit column, and exclude the rows that have 1's. When a bit column contains all 0's or 1's, the row exclusion can be skipped. Rows that are corresponding to non-minimum values are excluded step by step until the min value is reached. The row for the min value is then excluded and marked as sorted before moving to the next min search iteration. The near-memory circuit is designed to support two operations, column read (CR) and row exclusion (RE), and their associated control logic. \figurename~\ref{fig:memristive_sorting} shows the steps to sort \{8,9,10\} using memristive in-memory sorting \cite{prasad2021memristive}. Note that the near-memory circuit in \cite{prasad2021memristive} does not keep track the number of remaining elements in the array; therefore it takes $N = 3$ iterations of min search, each contains $w = 4$ CRs. The total sorting latency is $N\times w = 12$ CRs.
\begin{figure}
\centering
\includegraphics[width = 0.95\linewidth]{figures/example-SIM.pdf}
\caption{Memristive in-memory sorting \cite{prasad2021memristive}.}
\label{fig:memristive_sorting}
\end{figure}
\section{Column-Skipping Memristive In-Memory Sorting}
We observe that memristive in-memory sorting in \cite{prasad2021memristive} introduces a large number of redundant CRs which are repeatedly executed on leading 0's or bit columns that have been processed previously. As shown in \figurename~\ref{fig:memristive_sorting}, when searching the 2nd minimum number 9, the first 3 CRs have been processed in the 1st iteration and are repeated in the 2nd iteration. To efficiently skip these redundant CRs, we propose a low-latency column-skipping algorithm. We use unsigned fixed-point number as example, but it can easily be applicable to signed fixed-point and floating-point number formats with small changes as described in \cite{prasad2021memristive}.
\subsection{Low-Latency Column-Skipping Algorithm}
Redundant CRs can happen in two scenarios: 1) array elements may include leading 0's. CRs on these leading 0's can be skipped at the beginning of each iteration; 2) some CRs may have been processed previously for REs, i.e., we do not need to exclude any new rows for those bit columns.
To detect and skip the redundant CRs, we propose to record the $k$ most recent RE states and their corresponding column indexes. The recorded states can be reloaded to skip redundant CRs. \figurename~\ref{fig:flow_chart} summarizes the iterative min computation for a length-$N$ array with proposed column skipping algorithm (where $n = 1 \rightarrow N$): 1) if state records are empty, the $w$-step algorithm \cite{prasad2021memristive} traverses each bit column from MSB ($i = w-1$) to LSB ($i = 0$). $k$ most recent RE states whose bit columns are not all 0's or 1's and their corresponding column indexes are stored in a state controller; 2) if state records are non-empty, we reload the most recent RE state and the corresponding column index $s$ and start from the next bit column $s-1$. CRs are executed on subsequent bit columns until reaching the min value.
\begin{figure}
\centering
\includegraphics[width = 0.80\linewidth]{figures/flow_chart.pdf}
\caption{Iterative min search with proposed column-skipping algorithm}
\label{fig:flow_chart}
\end{figure}
\begin{figure}
\xdef\xfigwd{\textwidth}
\centering
\includegraphics[width = 0.95\linewidth]{figures/example-k=2.pdf}
\caption{Column-skipping memristive in-memory sorting with state recording $k=2$.}
\label{fig:memristive_sorting_k2}
\end{figure}
\figurename~\ref{fig:memristive_sorting_k2} illustrates the proposed column-skipping algorithm with state recording $k = 2$ when sorting the 4-bit array $\{8,9,10\}$. State recording in the first iteration helps to skip the first 3 CRs in searching the 2nd minimum and the first 2 CRs in searching the 3rd minimum. The total latency is reduced to only 7 CRs. The selection of $k$ affects the performance of the proposed column-skipping algorithm. We study the impacts of $k$ on sorting speedup, silicon area and power consumption in Section~\ref{section:evaluation}.
\begin{figure}
\centering
\includegraphics[width = 0.91\linewidth]{figures/1T1R.pdf}
\caption{Near-memory circuit for column-skipping memristive in-memory sorting.}
\label{fig:near_mem_circuit}
\end{figure}
\subsection{Near-Memory Circuit for Column-Skipping}
\figurename~\ref{fig:near_mem_circuit} demonstrates the near-memory circuit connected to a 1T1R memristive memory to implement the proposed column-skipping algorithm. The 1T1R memristive memory stores the binary bits of array elements with MSB on the leftmost column. Similar to \cite{prasad2021memristive}, select lines with sense amplifiers and bitline drivers are used for column reads. The proposed near-memory circuit consists of three modules: 1) a column processor that controls the column states; 2) a row processor that controls wordline (or RE) states; 3) a state controller that stores the RE states and their corresponding column indexes using a $k$-entry table. It also controls signals to execute all the operations.
The near-memory circuit supports the four operations in \figurename~\ref{fig:flow_chart} as following: 1) column read (CR), where the column processor enables the bitline driver of a column and the corresponding bit column is read to the row processor. The column controller generates the next-step column state and the enable signal for column update ($cen$). Sense amplifiers measure the current on each select line to determine if it's 0 or 1; 2) row exclusion (RE), where the row processor checks if the bit column are all 0's or 1's (through row controller) before updating the wordlines (or RE) states. The row controller generates the enable signal for wordline update ($ren$). The wordlines that are connected to 1's are excluded and set to 0; 3) state recording (SR), where RE states and their corresponding column indexes are stored in a $k$-entry table. The recording is enabled ($sen$) if an iteration starts from the MSB and the bit column is not all 0's or 1's; and 4) state loading (SL), where the most recent RE state and the corresponding column index are sent to the row processor and column processor, respectively. The load enable signal ($len$) selects the reloaded states when updating the wordline and column registers. A top-level controller is used to schedule the four operations.
When multiple rows remain unexcluded at the end of an iteration due to repetitions in the array, the column processor stalls to avoid redundant CRs until all repetition elements are excluded successively in the row processor.
\section{Multi-Bank Management}
\label{section:scalability}
\begin{figure}
\centering
\includegraphics[width = 0.95\linewidth]{figures/multi_bank.pdf}
\caption{Multi-bank management to synchronize sub-sorter operations and select sorted output}
\label{fig:scalability}
\end{figure}
The near-memory circuit shown in \figurename~\ref{fig:near_mem_circuit} can be scaled up to support larger array (i.e. larger $N$) or higher precision (i.e. large $w$). However, practical array can be too big to fit in a single memristive memory. To solve this problem, we propose a scalable solution to sort larger array stored in multi-bank memristive memory.
Suppose a length-$N$ array is stored in $C$-bank memristive memory, each bank stores $N/C$ elements and has its own near-memory circuit that forms a length-$N/C$ sub-sorter. To realize length-$N$ sorting using $C$ sub-sorters of length-$N/C$, sub-sorters' operations need to be synchronized and run as a whole. A multi-bank manager is designed to connect the sub-sorters for this synchronization purpose: the judgement about all 0's or 1's needs to be considered globally to synchronize RE and SR operations while CR and SL operations are synchronized through the OR gates.
\figurename~\ref{fig:scalability} shows the multi-bank manager to generate synchronized operation bits $en_{sync}$ based on local operation bits $en_i$ from sub-sorter $i$, where $i \in [1,C]$. In each sub-sorter, the synchronized operation bits $en_{sync}$ are used for replacing the original signals ($en_i$) to realize the corresponding function. The multi-bank manager monitors the sub-sorters' states and select the output from one of the $C$ sub-sorters if existing repetitions. Performance of the proposed multi-bank management are evaluated in Section~\ref{section:evaluation}.
\section{Evaluation and Benchmarking}
\label{section:evaluation}
We evaluate the proposed techniques using statistically distributed datasets (uniform, normal and clustered) and practical datasets (from Kruskal's and MapReduce). We use 32-bit precision: the uniform distribution ranges from 0 to $2^{32}-1$, the normal distribution has a mean of $2^{31}$ and a standard deviation of $2^{31}/3$, and the clustered distribution has 2 clusters centered at $2^{15}$ and $2^{25}$ with identical standard deviation of $2^{13}$. To estimate the silicon area and power consumption, prototype sorters of length-1024 are implemented with 1T1R memristive memory using a 40nm CMOS technology. The RRAM device has two states and the corresponding resistances are 10M$\Omega$ and 100k$\Omega$, respectively. State-of-the-art memristive in-memory sorter \cite{prasad2021memristive} (baseline) and conventional digital merge sorter are implemented for comparison. All prototype sorters run at a 500MHz clock frequency.
\subsection{Sorting Speedup}
The baseline implementation \cite{prasad2021memristive} has a fixed sorting speed of 32 cycles per number for any datasets. The merge sorter outperforms the baseline by 3.2$\times$ in speed. The speed of column-skipping sorter depends on parameter $k$ and dataset distribution. \figurename~\ref{fig:speed} shows the normalized speedup over the baseline on the selected datasets with $N = 1024$, $w = 32$ and varying state recording $k$. When $k$ increases, the min search is more likely to start from a recorded RE state; however, the reloaded starting position ($s$ in \figurename~\ref{fig:flow_chart}) may be further away from the optimal starting position, degrading the speedup due to less number of skipped CRs. We observe that the speedup saturates when $k$ reaches 2 or 3 and then goes down across selected datasets.
The proposed column-skipping algorithm achieves faster sorting speed (up to 2.22$\times$ over the baseline) on clustered dataset than the speedup on uniformly or normally distributed datasets (up to 1.21$\times$ and 1.23$\times$ over the baseline, respectively). This is because clustered elements with small centers signify more leading 0's and redundant CRs. In Kruskal's and MapReduce dataset, majority of the small and repetitive elements lead to much better results for a speedup up to 3.46$\times$ and 4.16$\times$ over the baseline, respectively.
\begin{figure}
\centering
\includegraphics[width = 0.95\linewidth]{figures/speed.pdf}
\caption{Normalized speedup over the baseline on different datasets with $N$ = 1024, $w$ = 32 and varying state recording $k$.}
\label{fig:speed}
\end{figure}
\begin{figure}
\centering
\includegraphics[width = 0.95\linewidth]{figures/area_power.pdf}
\caption{Normalized area and power over the baseline on MapReduce dataset with $N$ = 1024, $w$ = 32 and varying state recording $k$.}
\label{fig:area_power}
\end{figure}
\subsection{Area and Energy Efficiency}
With $N = 1024$ and $w = 32$, the baseline sorter occupies 77.8K \textmu m$^2$ in silicon while the merge sorter occupies 246.1K \textmu m$^2$. The merge sorter demonstrates 1.01$\times$ area efficiency (throughput/area) over the baseline. We further measure the areas of column-skipping sorters with varying state recording $k$. \figurename~\ref{fig:area_power} presents the normalized area and area efficiency over the baseline when sorting the MapReduce dataset. With $k = 1$, column-skipping sorter demonstrates more than 3.2$\times$ area efficiency over the baseline. When $k$ increases, the sorter area increases due to larger state controller to store more RE states; however, the area efficiency goes down, because the speedup starts saturating when $k$ reaches 2 or 3.
\begin{figure}
\centering
\includegraphics[width = \linewidth]{figures/sca.pdf}
\caption{(a) Implementation summary using 40nm CMOS technology and 1T1R memristive memory (K \textmu m$^2$ for area, Num/ns/mm$^2$ for area efficiency, mW for power, Num/\textmu J for energy efficiency);(b) Normalized area and power (for MapReduce dataset) with varying sub-sorter length $N_s$ for $N$ = 1024, $w$ = 32 and $k = 2$ ($N_s = 1024$ is the baseline).}
\label{fig:performance}
\end{figure}
We measured power using Ansys PowerArtist considering switching activities when sorting MapReduce dataset. The baseline sorter and the merge sorter consume 319.7 mW and 825.9 mW, respectively. The merge sorter demonstrates 1.24$\times$ energy efficiency (throughput/power) over the baseline. Column-skipping sorter consumes more power with increasing $k$, but the energy efficiency reaches the peak at $k = 2$ as shown in \figurename~\ref{fig:area_power}, outperforming the baseline by 3.39$\times$. The area and power consumption of 1T1R array are orders of magnitude less than the near-memory circuit. One can select the parameter $k$ based on target dataset for optimized speed, area and energy efficiency.
\subsection{Multi-Bank Management}
To evaluate multi-bank management, we build a column-skipping sorter of $N = 1024$ using sub-sorters of length $N_s$ = 64, 256, 512. Multi-bank management does not change the speedup brought by column-skipping when clock frequency remains unchanged. Further reducing the sub-sorter length results in a degraded clock frequency under 500MHz due to more complex multi-bank manager. \figurename~\ref{fig:performance}(a) demonstrates the normalized area and power of multi-bank management over the original $N=1024$ sorter. We observe that the area and power of the near-memory circuit in sub-sorters decreases super-linearly when $N_s$ decreases. Even with an extra multi-bank manager, the total area and power for multi-bank management goes down with smaller sub-sorter length. Using 16 sub-sorters of length $N_s = 64$, the area and power reduction can be up to 14\% and 9\% compared to the original $N = 1024$ sorter. \figurename~\ref{fig:performance}(b) summarizes the implementation results for different sorters.
\section{Conclusions}
We present a fast and scalable memristive in-memory sorting that employs a column-skipping algorithm and a multi-bank management. Near-memory circuit with state recording is designed to efficiently skip redundant column reads for improved sorting speed and hardware efficiency. The multi-bank manager enables column-skipping for dataset stored in different banks of memristive memory. Prototype sorters are implemented using 40nm CMOS technology and 1T1R memristive memory. Experimented on a variety of sorting datasets with array length-1024, data precision 32-bit and state recording of 2, the speed, area efficiency and energy efficiency are 4.08$\times$, 3.14$\times$ and 3.39$\times$, respectively, than the state-of-the-art memristive in-memory sorting.
\bibliographystyle{IEEEtran}
|
1,116,691,500,503 | arxiv | \section{Introduction}
\label{intro}
Infrared (soft and collinear) singularities appear in the calculation
of multiparton QCD matrix elements.
Although the singularities cancel in the evaluation of inclusive cross
sections, their factorization properties are at the basis
of many important tools in perturbative QCD applications to
hard-scattering processes~[\ref{book}].
At the {\em leading} order in the QCD coupling, $\as$,
the structure of the infrared singularities is
well known to be
universal.
It is embodied in process-independent factorization formulae
of tree-level [\ref{AP}--\ref{antenna}] and one-loop [\ref{GG}--\ref{BDKrev}]
amplitudes. These factorization formulae have played an essential r\^ole
in the setting up of completely general algorithms
[\ref{CSdipole},\ref{GG},\ref{GGK},\ref{submeth}]
to handle and cancel infrared singularities, when
combining tree-level and one-loop contributions in the evaluation of jet cross
sections at the next-to-leading order (NLO) in perturbation theory.
The extension of these general algorithms at the next-to-next-to-leading
order (NNLO) is at present one of the main goals to improve and precisely
quantify the theoretical accuracy of perturbative QCD predictions.
To this purpose we need to compute two-loop matrix elements
[\ref{gonsalves}--\ref{bern}] and to understand the structure of the infrared
singularities of two-loop, one-loop and tree-level amplitudes
at ${\cal O}(\as^2)$.
The singular behaviour of two-loop QCD amplitudes has been
discussed in Ref.~[\ref{sing2loop}].
The soft and collinear limits of one-loop amplitudes have been derived in
Refs.~[\ref{1loopeps}, \ref{1loopepskos}].
The soft, collinear and soft--collinear singularities of tree-level amplitudes
have been studied in Refs.~[\ref{bgdsoft},\ref{sdsoft}],
[\ref{glover},\ref{lett}] and [\ref{glover}], respectively.
The purpose of this paper is twofold. We consider {\em tree-level}
matrix elements and present general techniques to compute their infrared
singularities and to derive infrared-factorization formulae to {\em any}
order in $\as$. We apply these techniques to the explicit calculation
of all the relevant infrared factors at ${\cal O}(\as^2)$.
Our general method exploits the universality properties of soft
and collinear emission and consists in directly computing
{\em process-independent} Feynman subgraphs in a physical gauge. We use
power-counting arguments [\ref{collpc},\ref{jetcalc}] and the eikonal
approximation [\ref{BCM}] to treat the collinear and soft limits, respectively.
We show how the coherence properties of QCD radiation [\ref{coher}]
can be used to
deal with the mixed soft--collinear limit in terms of the collinear and soft
factorization formulae.
Most of the explicit results at ${\cal O}(\as^2)$ presented in this paper
were first obtained by Campbell and Glover [\ref{glover}]. The strategy
followed in Ref.~[\ref{glover}] was to take universal factorization for granted
and thus to extract the ${\cal O}(\as^2)$-singular factors by performing the
corresponding limits of a set of known matrix elements. We confirm their
calculations by using a completely independent method. We also extend their
results by considering the emission of a soft fermion pair and
by fully taking into account spin (azimuthal) correlations in the collinear
limit. The extension to azimuthal correlations is essential to
apply some general methods to perform exact NNLO calculations of jet cross
sections. For instance,
the subtraction method [\ref{CSdipole}, \ref{submeth}] works by regularizing
the infrared singularities of the tree-level matrix element by identifying
and subtracting a proper {\em local} counterterm. Thus, the study
of the azimuthally {\em averaged} collinear limit [\ref{glover}]
is not sufficient for this purpose.
The knowledge of the infrared structure of multiparton amplitudes is also
important for other perturbative QCD applications.
The leading-logarithmic (LL) parton showers, which are implemented in Monte
Carlo event generators [\ref{book}] to describe the exclusive structure of
hadronic final states, are based on the ${\cal O}(\as)$-factorization
formulae supplemented with `jet calculus' techniques [\ref{KUV}] and
colour-coherence properties [\ref{BCM}, \ref{coher}].
Analytical techniques
to perform all-order resummation of logarithmically enhanced contributions
at next-to-leading logarithmic (NLL) accuracy [\ref{softrev}] rely on the
factorization properties of soft and collinear emission.
The results on infrared factorization presented in this paper can be useful
to improve parton-shower algorithms and resummed calculations beyond their
present logarithmic accuracy.
The outline of the paper is as follows. We start in Sect.~\ref{seccoll} by
studying the collinear behaviour. After reviewing the known
factorization formulae at ${\cal O}(\as)$, we discuss the kinematics of the
triple collinear limit. Then, in Sect.~\ref{power},
we present our derivation of factorization for the multiple collinear limit
at any perturbative order. Finally, in Sect.~\ref{splitt}, we perform the
explicit calculation of the spin-dependent splitting functions at
${\cal O}(\as^2)$.
Our results for the splitting functions were anticipated in Ref.~[\ref{lett}].
In Sect.~\ref{secsoft} we study the soft behaviour.
We first review the known results at ${\cal O}(\as)$ and then, in
Sect.~\ref{secsoftqq}, we compute the emission of a soft $q{\bar q}$ pair
at ${\cal O}(\as^2)$. Section~\ref{secsoftgg} is devoted to double gluon
emission: we present the corresponding soft current and
obtain a compact expression for its square. Factorization for the
mixed soft--collinear limit at ${\cal O}(\as^2)$ and at higher perturbative
orders is discussed in detail in Sects.~\ref{softcoll} and \ref{multilim}.
In Sect.~\ref{summa} we summarize our results. In general,
soft factorization formulae involve colour correlations. At ${\cal O}(\as^2)$
these correlations cancel in four- and five-parton matrix elements. The
explicit expressions for these particular cases are given in the Appendix.
\section{The collinear behaviour}
\label{seccoll}
\subsection{Notation and collinear factorization at ${\cal O}(\as)$}
\label{notations}
We consider a generic scattering process involving
final-state\footnote{The case of incoming partons can be recovered by
simply crossing the parton indices (flavours, spins and colours) and momenta.}
QCD partons ({\em massless} quarks and gluons) with momenta
$p_1, p_2, \dots$. Non-QCD partons $(\gamma^*, Z^0, W^\pm, \dots)$, carrying
a total momentum $Q$, are always understood. The corresponding {\em tree-level}
matrix element is denoted by
\begin{equation}
\label{meldef}
{\cal M}^{c_1,c_2,\dots;s_1,s_2,\dots}_{a_1,a_2,\dots}(p_1,p_2,\dots) \;\;,
\end{equation}
where $\{c_1,c_2,\dots\}$, $\{s_1,s_2,\dots\}$ and $\{a_1,a_2,\dots\}$ are
respectively colour, spin and flavour indices. The matrix element squared,
summed over final-state colours and spins, will be denoted by
$| {\cal M}_{a_1,a_2,\dots}(p_1,p_2,\dots) |^2$.
If the sum over the spin polarizations of
the parton $a_1$ is not carried out, we define the following
`spin-polarization tensor'
\begin{equation}
\label{melspindef}
{\cal T}_{a_1,\dots}^{s_1 s'_1}(p_1,\dots) \equiv
\sum_{{\rm spins} \,\neq s_1,s'_1} \, \sum_{{\rm colours}}
{\cal M}^{c_1,c_2,\dots;s_1,s_2,\dots}_{a_1,a_2,\dots}(p_1,p_2,\dots) \,
\left[ {\cal M}^{c_1,c_2,\dots;s'_1,s_2,\dots}_{a_1,a_2,\dots}(p_1,p_2,\dots)
\right]^\dagger
\;\;.
\end{equation}
We work in $d=4 - 2\epsilon$ space-time
dimensions and consider two helicity states for massless quarks and
$d-2$ helicity states for gluons. This defines the conventional
dimensional-regularization (CDR) scheme of both ultraviolet [\ref{cdruv}]
and infrared [\ref{cdrir}] divergences.
Thus, the fermion spin indices are $s=\pm 1$,
while to label the gluon spin it is convenient to use the corresponding Lorentz
index $\mu=1, \dots, d$. The $d$-dimensional average of the matrix element
over the polarizations of a parton $a$ is obtained by means of the factors
\begin{equation}
\label{ferav}
\frac{1}{2} \delta_{ss'}
\end{equation}
for a fermion, and (the gauge terms are proportional either to $p^\mu$
or to $p^\nu$)
\begin{equation}
\label{gluav}
\frac{1}{d-2} d_{\mu \nu}(p) = \frac{1}{2(1-\epsilon)} ( - g_{\mu \nu} +
{\rm gauge \; terms} )
\end{equation}
with
\begin{equation}
\label{dprop}
- g^{\mu \nu} d_{\mu \nu}(p) = d-2 \;, \;\;\;\;\;
p^\mu \,d_{\mu \nu}(p) = d_{\mu \nu}(p) \,p^\nu = 0 \;,
\end{equation}
for a gluon with on-shell momentum $p$.
The
singular collinear limit at ${\cal O}(\as)$ is approached when the momenta
of two partons, say $p_1$ and $p_2$, become parallel. This limit can
be precisely defined as follows:
\begin{eqnarray}
\label{clim}
&&p_1^\mu = z p^\mu + k_\perp^\mu - \frac{k_\perp^2}{z}
\frac{n^\mu}{2 p\cdot n} \;\;, \;\;\; p_2^\mu =
(1-z) p^\mu - k_\perp^\mu - \frac{k_\perp^2}{1-z} \frac{n^\mu}{2 p\cdot n}\;\;,
\nonumber \\
&&s_{12} \equiv 2 p_1 \cdot p_2 = - \frac{k_\perp^2}{z(1-z)} \;\;,
\;\;\;\;\;\;\;\; k_\perp \rightarrow 0 \;\;.
\end{eqnarray}
In Eq.~(\ref{clim}) the light-like ($p^2=0$) vector $p^\mu$ denotes the
collinear direction, while $n^\mu$ is an auxiliary light-like vector, which
is necessary to specify the transverse component $k_\perp$ ($k_\perp^2<0$)
($k_\perp \cdot p = k_\perp \cdot n = 0$) or, equivalently, how the collinear
direction is approached.
In the small-$k_\perp$ limit (i.e.\ neglecting terms that are less singular
than $1/k_\perp^2$), the square of the matrix element in Eq.~(\ref{meldef})
fulfils the following factorization formula [\ref{book}]:
\begin{eqnarray}
\label{cfac}
| {\cal M}_{a_1,a_2,\dots}(p_1,p_2,\dots) |^2 \simeq \frac{2}{s_{12}} \;
4 \pi \mu^{2\epsilon} \as
\;{\cal T}_{a,\dots}^{s s'}(p,\dots) \;
{\hat P}_{a_1 a_2}^{s s'}(z,k_{\perp};\epsilon) \;\;,
\end{eqnarray}
where $\mu$ is the dimensional-regularization scale.
The spin-polarization tensor ${\cal T}_{a,\dots}^{s s'}(p,\dots)$
is obtained by replacing the partons $a_1$ and $a_2$ on the right-hand side
of Eq.~(\ref{melspindef}) with a single parton denoted by $a$.
This parton carries the quantum numbers of the
pair $a_1+a_2$ in the collinear limit. In other words, its momentum is
$p^\mu$ and its other quantum numbers (flavour, colour) are obtained according
to the following rule: anything~+~gluon gives anything, and
quark~+~antiquark gives gluon.
The kernel ${\hat P}_{a_1 a_2}$ in Eq.~(\ref{cfac}) is the $d$-dimensional
Altarelli--Parisi splitting function [\ref{AP}].
It depends not only on the momentum
fraction $z$ involved in the collinear splitting $a \rightarrow a_1 + a_2$, but also on
the transverse momentum $k_{\perp}$ and on the helicity of the parton $a$ in the
matrix element ${\cal M}_{a,\dots}^{c,\dots;s,\dots}(p,\dots)$.
More precisely, ${\hat P}_{a_1 a_2}$ is in general a matrix
acting on the spin indices $s,s'$ of the parton $a$ in the
spin-polarization tensor ${\cal T}_{a,\dots}^{s s'}(p,\dots)$.
Because of these {\em spin correlations}, the spin-average square
of the matrix element ${\cal M}_{a,\dots}^{c,\dots;s,\dots}(p,\dots)$
cannot be simply factorized on the right-hand side of Eq.~(\ref{cfac}).
The explicit expressions of ${\hat P}_{a_1 a_2}$,
for the splitting processes
\begin{equation}
\label{sppro}
a(p) \rightarrow a_1(zp + k_{\perp} + {\cal O}(k_{\perp}^2)) +
a_2((1-z) p - k_{\perp} + {\cal O}(k_{\perp}^2)) \;\;,
\end{equation}
depend on the flavour of the partons $a_1, a_2$ and are given
by\footnote{The $\epsilon$ dependence on the right-hand side of
Eqs.~(\ref{hpqqep})--(\ref{hpggep}) refers to the
CDR
scheme used throughout the paper. A detailed
discussion of the regularization-scheme dependence of the collinear splitting
functions at ${\cal O}(\as)$, including the corresponding explicit expressions,
can be found in Ref.~[\ref{schemedep}].}
\begin{eqnarray}
\label{hpqqep}
{\hat P}_{qg}^{s s'}(z,k_{\perp};\epsilon) = {\hat P}_{{\bar q}g}^{s s'}(z,k_{\perp};\epsilon)
= \delta_{ss'} \;C_F
\;\left[ \frac{1 + z^2}{1-z} - \epsilon (1-z) \right] \;\;,
\end{eqnarray}
\begin{eqnarray}
\label{hpqgep}
{\hat P}_{gq}^{s s'}(z,k_{\perp};\epsilon) = {\hat P}_{g{\bar q}}^{s s'}(z,k_{\perp};\epsilon)
= \delta_{ss'} \;C_F
\;\left[ \frac{1 + (1-z)^2}{z} - \epsilon z \right] \;\;,
\end{eqnarray}
\begin{eqnarray}
\label{hpgqep}
{\hat P}_{q{\bar q}}^{\mu \nu}(z,k_{\perp};\epsilon)
= {\hat P}_{{\bar q}q}^{\mu \nu}(z,k_{\perp};\epsilon)
= T_R
\left[ - g^{\mu \nu} + 4 z(1-z) \frac{k_{\perp}^{\mu} k_{\perp}^{\nu}}{k_{\perp}^2}
\right] \;\;,
\end{eqnarray}
\begin{equation}
\label{hpggep}
{\hat P}_{gg}^{\mu \nu}(z,k_{\perp};\epsilon) = 2C_A
\;\left[ - g^{\mu \nu} \left( \frac{z}{1-z} + \frac{1-z}{z} \right)
- 2 (1-\epsilon) z(1-z) \frac{k_{\perp}^{\mu} k_{\perp}^{\nu}}{k_{\perp}^2}
\right] \;\;,
\end{equation}
where the $SU(N_c)$ QCD colour factors are
\begin{equation}
\label{colofac}
C_F = \frac{N_c^2 -1}{2N_c} \;, \;\;\; C_A = N_c \;,
\;\;\; T_R = \frac{1}{2} \;,
\end{equation}
and the spin indices of the parent parton $a$ have been denoted by $s,s'$
if $a$ is a fermion and $\mu,\nu$ if $a$ is a gluon.
Note that when the parent parton is a fermion (see Eqs.~(\ref{hpqqep}) and
(\ref{hpqgep})) the splitting function is proportional to the unity matrix
in the spin indices. Thus, in the factorization formula (\ref{cfac}),
spin correlations are effective only in the case of the collinear splitting
of a gluon. Owing to the $k_{\perp}$-dependence of the gluon splitting
functions in Eqs.~(\ref{hpgqep}) and (\ref{hpggep}), these spin correlations
produce a non-trivial azimuthal dependence with respect to the directions
of the other momenta in the factorized matrix element.
Equations (\ref{hpqqep})--(\ref{hpggep}) lead to the more familiar form of the
$d$-dimensional splitting functions only after average over the polarizations
of the parton $a$. The $d$-dimensional average is obtained by means of the
factors in Eqs.~(\ref{ferav}) and (\ref{gluav}).
Denoting by $\langle {\hat P}_{a_1 a_2} \rangle$
the average of ${\hat P}_{a_1 a_2}$ over the polarizations of the parent
parton $a$, we have:
\begin{eqnarray}
\label{avhpqq}
\langle {\hat P}_{qg}(z;\epsilon) \rangle \, = \, \langle {\hat P}_{{\bar q}g}(z;\epsilon) \rangle \,
= C_F\;\left[ \frac{1 + z^2}{1-z} - \epsilon (1-z) \right] \;\;,
\end{eqnarray}
\begin{eqnarray}
\label{avhpqg}
\langle {\hat P}_{gq}(z;\epsilon) \rangle \, = \, \langle {\hat P}_{g{\bar q}}(z;\epsilon) \rangle \,
= C_F \;\left[ \frac{1 + (1-z)^2}{z} - \epsilon z \right] \;\;,
\end{eqnarray}
\begin{eqnarray}
\label{avhpgq}
\langle {\hat P}_{q{\bar q}}(z;\epsilon) \rangle \, = \, \langle {\hat P}_{{\bar q}q}(z;\epsilon) \rangle
\, = T_R \left[ 1 - \frac{2 z(1-z)}{1-\epsilon} \right] \;\;,
\end{eqnarray}
\begin{equation}
\label{avhpgg}
\langle {\hat P}_{gg}(z;\epsilon) \rangle \, = \, 2C_A
\;\left[ \frac{z}{1-z} + \frac{1-z}{z}
+ z(1-z) \right] \;\;.
\end{equation}
In the rest of this section we are mainly interested in the collinear behaviour
of the tree-level matrix element ${\cal M}(p_1,\dots)$ in Eq.~(\ref{meldef})
at ${\cal O}(\as^2)$. At this order there are two different collinear
limits to be considered [\ref{glover}].
The first limit is approached when {\em two pairs}
of parton momenta, say $\{ p_1,p_2 \}$ and $\{ p_3,p_4 \}$, become parallel
independently. In this case collinear factorization
follows from the straightforward iteration of Eq.~(\ref{cfac}): the ensuing
factorization formula simply contains the product of the two splitting functions
${\hat P}_{a_1 a_2}^{s_{12}s_{12}^\prime}$ and
${\hat P}_{a_3 a_4}^{s_{34}s_{34}^\prime}$.
In the second limit, three parton momenta can simultaneously become
parallel. This triple collinear limit is discussed in the following subsections.
\subsection{Kinematics in the triple collinear limit}
\label{kin}
We denote by $p_1, p_2$ and $p_3$ the momenta of the three collinear partons.
The most general parametrization of these collinear momenta is
\begin{equation}
\label{kin3}
p_i^\mu = x_i p^\mu +k_{\perp i}^\mu - \frac{k_{\perp i}^2}{x_i}
\frac{n^\mu}{2p \cdot n} \;, \;\;\;\;\;i=1,2,3 \;,
\end{equation}
where, as in Eq.~(\ref{clim}), the light-like vector $p^\mu$ denotes
the collinear direction and the auxiliary light-like vector $n^\mu$
specifies how the collinear direction is approached
$(k_{\perp i} \cdot p = k_{\perp i} \cdot n = 0)$.
Note that no other
constraint (in particular $\sum_i x_i \neq 1$ and $\sum_i k_{\perp i} \neq 0$)
is imposed on the longitudinal and transverse variables $x_i$ and $k_{\perp i}$.
Thus, we can easily consider any (asymmetric) collinear limit at once.
Note, however, that the triple collinear limit is invariant
under longitudinal boosts along the direction of the total momentum
$p_{123}^\mu = p_1^\mu + p_2^\mu + p_3^\mu$. Thus, the relevant kinematical
variables are the following boost-invariant quantities
\begin{eqnarray}
\label{zvar}
z_i &=& \frac{x_i}{\sum_{j=1}^3 \,x_j} \;\;,\\
\label{kvar}
{\widetilde k}_i^\mu &=& k_{\perp i}^\mu - \frac{x_i}{\sum_{k=1}^3 \,x_k} \;
\sum_{j=1}^3 k_{\perp j}^\mu \;\;.
\end{eqnarray}
Note that these variables automatically satisfy the constraints
$\sum_{i=1}^3 z_i = 1$ and $\sum_{i=1}^3 {\widetilde k}_i = 0$, so that only four of
them are actually independent.
In terms of the longitudinal and transverse variables introduced so far,
the two-particle sub-energies $s_{ij}$ are written as
\begin{equation}
\label{sijvar}
s_{ij}=(p_i+p_j)^2=- x_i x_j \left( \f{k_{\perp j}}{x_j}
-\f{k_{\perp i}}{x_i}\right)^2
= - z_i z_j \left( \f{{\widetilde k}_j}{z_j} -\f{{\widetilde k}_i}{z_i}\right)^2 \;\;.
\end{equation}
It is also convenient to define the following variable $t_{ij,k}$
\begin{equation}
\label{tvar}
t_{ij,k} \equiv 2 \;\f{z_i s_{jk}-z_j s_{ik}}{z_i+z_j} +
\f{z_i-z_j}{z_i+z_j} \,s_{ij} \;\;.
\end{equation}
\subsection{Power counting and tree-level factorization at any order}
\label{power}
In the triple-collinear limit, the matrix element squared
$| {\cal M}_{a_1,a_2,a_3,\dots}(p_1,p_2,p_3,\dots) |^2$ has the
singular behaviour
$| {\cal M}_{a_1,a_2,a_3,\dots}(p_1,p_2,p_3,\dots) |^2 \sim 1/(s s')$, where
$s$ and $s'$ can be either two-particle $( s_{ij} = (p_i + p_j)^2 )$
or three-particle $( s_{123} = (p_1 + p_2 + p_3)^2 )$ sub-energies.
To define the collinear limit more precisely, we can rescale the transverse
momenta $k_{\perp i}$ by an overall factor $\lambda$:
\begin{equation}
\label{kscale}
k_{\perp i} \rightarrow \lambda \; k_{\perp i} \;,
\end{equation}
and then perform the limit $\lambda \rightarrow 0$. In this limit the matrix element
squared behaves as
\begin{equation}
\label{mscale}
| {\cal M}_{a_1,a_2,a_3,\dots} |^2 \rightarrow {\cal O}(1/\lambda^4) + \dots \;,
\end{equation}
where the dots stand for less singular contributions when $\lambda \rightarrow 0$.
We are interested in explicitly evaluating the dominant singular term
${\cal O}(1/\lambda^4)$.
To study this singular behaviour we use power-counting
arguments and the universal factorization properties of collinear
singularities. The method [\ref{jetcalc}] is completely general: it is
{\em process-independent} and can be applied to any {\em multiple} collinear
limit $a \rightarrow a_1 \dots a_m$ at the tree level (i.e. at ${\cal O}(\as^{m-1})$).
Thus, we shall discuss the most general case with $m$ collinear partons.
We shall show that in the multiple collinear limit $a \rightarrow a_1 \dots a_m$,
the matrix element squared
$| {\cal M}_{a_1,\dots,a_m,\dots}(p_1,\dots,p_m,\dots) |^2$
still fulfils a factorization formula analogous to Eq.~(\ref{cfac}), namely
\begin{eqnarray}
\label{ccfacm}
| {\cal M}_{a_1,\dots,a_m,\dots}(p_1,\dots,p_m,\dots) |^2 \simeq
\left( \frac{8 \pi \mu^{2\epsilon} \as}{s_{1 \dots m}}\right)^{m-1}
\;{\cal T}_{a,\dots}^{s s'}(xp,\dots) \;
{\hat P}_{a_1 \dots a_m}^{s s'}
\;\;,
\end{eqnarray}
where $s_{1 \dots m} = (p_1+\dots+p_m)^2$ is the $m$-particle sub-energy
and $x = \sum_{i=1}^m x_i$.
As in Eq.~(\ref{cfac}), the spin-polarization tensor
${\cal T}_{a,\dots}^{s s'}(xp,\dots)$ is obtained by replacing the partons
$a_1, \dots, a_m$ with a single parent parton, whose flavour $a$ is
determined by flavour conservation in the splitting process.
More precisely, $a$ is a quark (antiquark) if the set $\{a_1, \dots, a_m\}$
contains an odd number of quarks (antiquarks), and $a$ is a gluon otherwise.
The factorization formula (\ref{ccfacm}) takes into account all the dominant
singular contributions in the multiple collinear limit, that is, all the
contributions that have the scaling behaviour $(1/\lambda^2)^{m-1}$ under the
scale transformation in Eq.~(\ref{kscale}). Relative corrections of
${\cal O}(\lambda)$ are systematically neglected on the right-hand side
of Eq.~(\ref{ccfacm}).
The $m$-parton splitting functions ${\hat P}_{a_1 \dots a_3}$ are
dimensionless functions of the parton momenta $p_1, \dots, p_m$ and
generalize the Altarelli--Parisi splitting functions in Eq.~(\ref{cfac}).
Owing to their invariance under longitudinal boosts along
the collinear direction, the splitting functions can depend in a non-trivial
way only on the sub-energy ratios $s_{ij}/s_{1 \dots m}$ and on the
longitudinal- and transverse-momentum variables $z_i$ and ${\widetilde k}_i$ defined
by the generalization of Eqs.~(\ref{zvar}) and~(\ref{kvar}) to the $m$-parton
case.
The spin
correlations produced by the collinear splitting are taken into account by
the splitting functions in a universal way, i.e. independently of the
specific matrix element on the right-hand side of Eq.~(\ref{ccfacm}).
In the case of the splitting processes that involve a fermion as parent parton,
we find that spin correlations are absent. We can thus write the
corresponding spin-dependent splitting function in terms of its average
$\langle {\hat P}_{a_1 \dots a_m} \rangle$ over the polarizations of the parent fermion $a$:
\begin{equation}
\label{qmaver}
{\hat P}^{ss'}_{a_1 \dots a_m} = \delta^{ss'} \,
\langle {\hat P}_{a_1 \dots a_m} \rangle \;\;.
\end{equation}
This feature is completely analogous to the
${\cal O}(\as)$ case and follows from helicity conservation in
the quark--gluon vector coupling.
In the case of collinear decays of a parent gluon, however,
spin correlations are highly non-trivial.
Note also that the splitting functions for the collinear decay of an antiquark
can be simply obtained by charge-conjugation invariance from those of the
corresponding quark decay process, i.e.
$\langle {\hat P}_{a_1 \dots a_m} \rangle = \langle {\hat P}_{{\bar a}_1 \dots {\bar a}_m} \rangle$.
The method used to derive these results
exploits the basic observation [\ref{collpc}] that
interfering Feynman diagrams obtained by squaring the amplitude
${\cal M}(p_1,\dots,p_m,\dots)$ are collinearly suppressed when computed
in a physical gauge. Thus, in the evaluation of the multiple
collinear limit we can write
\begin{eqnarray}
| {\cal M}_{a_1,\dots,a_m,\dots}(p_1,\dots,p_m,\dots) |^2 &\simeq&
\left[ {\cal M}_{a,\dots}^{(n)}(p_1+\dots+p_m,\dots)
\right]^{\dagger} \;{\cal V}_{a_1\dots a_m}^{(n)}(p_1,\dots,p_m) \nonumber \\
\label{fcollgen}
&\cdot& {\cal M}_{a,\dots}^{(n)}(p_1+\dots+p_m,\dots) + \dots \;\;.
\end{eqnarray}
The first term on the right-hand side of Eq.~(\ref{fcollgen}) corresponds
to the non-interfering Feynman diagrams in Fig.~\ref{relevant},
while the dots stand for subdominant contributions coming from interferences
(see e.g. the diagram in Fig.~\ref{irrelevant}). The superscripts $(n)$
denote that the various terms are evaluated in a physical gauge.
To simplify the calculation it is convenient to choose the axial gauge
$n\cdot A = 0$, where the gauge vector $n^\mu$ coincides with the auxiliary
light-like vector used in Eq.~(\ref{kin3}) to parametrize the collinear
kinematics. The corresponding gluon polarization tensor $d^{\mu \nu}_{(n)}$
is
\begin{equation}
\label{dgauge}
d^{\mu \nu}_{(n)}(q) = - g^{\mu \nu}
+ \frac{n^\mu q^\nu + q^\mu n^\nu}{n\cdot q} \;\;,
\end{equation}
where $q$ is the gluon momentum.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=9truecm
\epsffile{fact.eps}\\
\end{tabular}
\end{center}
\caption{\label{relevant}{\em Dominant diagrams for the multiple collinear
limit at ${\cal O}(\as^{m-1})$ in a physical gauge.}}
\end{figure}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=9truecm
\epsffile{irr.eps}\\
\end{tabular}
\end{center}
\caption{\label{irrelevant}{\em An interference diagram for the triple
collinear limit $a\rightarrow a_1a_2a_3$.
}}
\end{figure}
The summation over spin and colour indices is understood on
the right-hand side of Eq.~(\ref{fcollgen}).
The function ${\cal V}_{a_1 \dots a_m}$ in Eq.~(\ref{fcollgen}) is the
$m$-parton dispersive contribution to the two-point function of
the parent parton $a$. Being a two-point function, it is proportional to
the unity matrix in the colour indices of the parton $a$. Thus, we can sum
over the colours of the partons in the tree-level amplitudes, and we can
rewrite Eq.~(\ref{fcollgen}) in terms of the spin-polarization tensor
${\cal T}_{a,\dots}$ introduced in Eq.~(\ref{melspindef}):
\begin{equation}
\label{fcollimp}
| {\cal M}_{a_1,\dots,a_m,\dots}(p_1,\dots,p_m,\dots) |^2 \simeq
\left( \frac{8 \pi \mu^{2\epsilon} \as}{s_{1 \dots m}}\right)^{m-1}
\;{\cal T}_{a,\dots}^{(n)}(p_1+\dots+p_m,\dots) \;
V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)
\;.
\end{equation}
The function $V_{a_1 \dots a_m}^{(n)}$ is simply obtained from
${\cal V}_{a_1 \dots a_m}^{(n)}$ in Eq.~(\ref{fcollgen}) by performing the
average over the colours of the parent parton $a$ and extracting the factor
in the round bracket on the right-hand side of Eq.~(\ref{fcollimp}).
Thus the tree-level function
$V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)$ does not contain
any additional power of the QCD coupling $\as$. Note also that the spin tensor
${\cal T}_{a,\dots}^{(n)}(p_1+\dots+p_m,\dots)$ is not yet exactly
the physical polarization tensor of Eq.~(\ref{ccfacm}). In fact,
the momentum of the parton $a$ is off-shell ($(p_1+\dots+p_m)^2 = s_{1 \dots m}
\neq 0$) and, thus,
${\cal T}_{a,\dots}^{(n)}(p_1+\dots+p_m,\dots)$ is gauge-dependent.
To proceed, we should consider separately the two cases in which
the parton $a$ is either a quark (or antiquark) or a gluon.
\bigskip
\noindent {\it Quark splitting processes}
\noindent It is convenient to include the spin-polarization matrices
${\slash p}_1+ \dots+ {\slash p}_m$ of the decaying quark $a=q$ in the
definition of the Dirac matrix $V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)$.
The most general decomposition of $V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)$ is
\begin{equation}
\label{aqdec}
V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m) \sim \sum
\Bigl( {\rm scalar \; amplitude} \Bigr) \cdot
\Bigl( {\rm string \;of \;gamma \;matrices} \Bigr) \;\;.
\end{equation}
Any string of gamma matrices is obtained by multiplying an arbitrary
number of terms ${\slash v}_l$ with $l=1, \dots,m+1$, where ${\slash v}$ can
be either ${\slash v}_i={\slash p}_i$ or
${\slash v}_{m+1} = {\slash n}s_{1 \dots m}/n \cdot(p_1+\dots+p_m)$.
The matrices ${\slash v}_l$, like
$V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)$, are homogeneous functions of $n^\mu$
with vanishing homogeneity degree. Thus, by Lorentz covariance,
the amplitudes on the right-hand side
of Eq.~(\ref{aqdec}) are scalar functions of the sub-energies $s_{ij}$ and the
longitudinal-momentum fractions $z_i= n \cdot p_i /n \cdot(p_1+\dots+p_m)$.
Moreover, they are rational functions of the variables $s_{ij}, z_i$ and thus,
by dimensional analysis, the corresponding strings
can contain only an {\em odd} number of gamma matrices.
We can now exploit the hermiticity properties of
$V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)$. Since the scalar amplitudes are real,
the strings of gamma matrices appear in the form
\begin{equation}
\label{strings}
\left( \frac{1}{s_{1 \dots m}} \right)^{(k-1)/2}
\left[ \;
{\slash v}_{i_1} {\slash v}_{i_2}\cdots {\slash v}_{i_k} +
{\slash v}_{i_k} \cdots {\slash v}_{i_2} {\slash v}_{i_1} \; \right] \;\;,
\end{equation}
where the normalization by the overall power of $1/s_{1 \dots m}$ has been
introduced to make the scalar amplitudes on the right-hand side
of Eq.~(\ref{aqdec}) dimensionless. Owing to the fact that
$k$ is odd, the terms with $k=3,7,11, \ldots$ in Eq.~(\ref{strings})
can in turn be reduced to strings that contain $k=1,5,9, \ldots$ gamma matrices
by using the anticommuting properties of the Dirac algebra.
This is the simplest form in which we can write the general decomposition of
Eq.~(\ref{aqdec}).
We can now discuss separately the cases that involve the collinear decay
of less or more than four partons.
From the previous discussion we conclude that, when $m \leq 3$, the
functions $V_{a_1 \dots a_m}^{(n)}$
can be written as follows
\begin{equation}
\label{aqdecfin}
V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m) = \sum_{i=1}^m
A_i^{(q)}(\{s_{jl}, z_j\}) \;{\slash p}_i + B^{(q)}(\{s_{jl}, z_j\})
\;\frac{{\slash n} \;s_{1 \dots m}}{n \cdot(p_1+\dots+p_m)} \;, \;\;(m \leq 3)
\, ,
\end{equation}
while, when $m= 4$, we have
\begin{eqnarray}
\label{aqdecfin4}
V_{a_1 \dots a_4}^{(n)}(p_1,\dots,p_4) &=& \sum_{i=1}^4
A_i^{(q)}(\{s_{jl}, z_j\}) \;{\slash p}_i + B^{(q)}(\{s_{jl}, z_j\})
\;\frac{{\slash n} \;s_{1 \dots 4}}{n \cdot(p_1+\dots+p_4)} \\
&+& C^{(q)}(\{s_{jl}, z_j\})
\;\frac{{\slash p}_1 {\slash p}_2 {\slash p}_3 {\slash p}_4 {\slash n}
+ {\slash n} {\slash p}_4 {\slash p}_3 {\slash p}_2 {\slash p}_1}{s_{1 \dots 4}
\; n \cdot(p_1+\dots+p_4)} \;.
\end{eqnarray}
Then we can proceed to single out the dominant singular behaviour of
Eqs.~(\ref{aqdecfin}) and (\ref{aqdecfin4}) in the multiple collinear limit.
Since the scalar functions $A_i^{(q)}(\{s_{jl}, z_j\}),
B^{(q)}(\{s_{jl}, z_j\})$ and
$C^{(q)}(\{s_{jl}, z_j\})$ are dimensionless, they are
invariant under the scale transformation (\ref{kscale}). Moreover,
since we can write
\begin{equation}
\label{piki}
p_i^\mu=z_i (p_1+ \dots +p_m)^\mu+{\widetilde k}_i^\mu+{\cal O}(k_{\perp}^2) \;\;,
\end{equation}
by rescaling
the transverse momenta as in Eq.~(\ref{kscale}) we obtain the
following scaling behaviour
\begin{equation}
\label{aqscale}
V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m) = ( {\slash p}_1 +\dots + {\slash p}_m )
\; \sum_{i=1}^m z_i \, A_i^{(q)}(\{s_{jl}, z_j\})
\left[ 1 + {\cal O}(\lambda) \right]
\;, \;\;\;(m \leq 4) \,.
\end{equation}
Thus, inserting Eq.~(\ref{aqscale}) into Eq.~(\ref{fcollimp}), we can use the
spin polarization factor ${\slash p}_1 +\dots + {\slash p}_m$ to reconstruct
the matrix element squared $| {\cal M}_{q,\dots}^{(n)}(p_1+\dots+p_m,\dots) |^2$.
Having already factorized the dominant singular term, we can now replace
$p_1+\dots+p_m \rightarrow xp$ in $| {\cal M}_{q,\dots}^{(n)} |^2$, so that its gauge
dependence disappears, and we obtain the factorization formula (\ref{ccfacm}).
Moreover, we also obtain an explicit expression for the quark splitting function
in terms of the scalar amplitudes $A_i^{(q)}$ in Eqs.~(\ref{aqdecfin}) and
(\ref{aqdecfin4}):
\begin{equation}
\label{qmaver4}
{\hat P}^{ss'}_{a_1 \dots a_m} = \delta^{ss'} \,
\sum_{i=1}^m z_i \, A_i^{(q)}(\{s_{jl}, z_j\}) \;, \;\;\;\; (m \leq 4) \;.
\end{equation}
This argument to prove collinear factorization is based on the fact that a
single spin structure (see Eq.~(\ref{aqscale})) dominates the collinear limit
of the quark decay function $V_{a_1 \dots a_m}^{(n)}$. In particular, this
implies that spin correlations are
absent from the collinear decay of a fermion, independently of the number of
its spin polarizations.
However, the argument works only for the cases with $m \leq 4$.
When $m > 4$ collinear factorization still applies but, as shown below,
spin correlations cancel only if we use a dimensional-regularization
prescription in which the massless fermion has two spin polarizations.
According to our definition, the scalar amplitudes on the right-hand side of
Eq.~(\ref{aqdec}) are dimensionless and, hence, they are invariant
under the scale transformation (\ref{kscale}). The collinear limit of
Eq.~(\ref{aqdec}) is thus determined by that of the strings of gamma matrices
in Eq.~(\ref{strings}). Using Eq.~(\ref{piki}) and rescaling
the transverse momenta as in Eq.~(\ref{kscale}), we see that the strings that
dominate in the multiple collinear limit are those of the form
\begin{eqnarray}
&&\left( \frac{1}{s_{1 \dots m}} \right)^{k} \,\left[
\;({\slash p}_1 + \dots {\slash p}_m) \, {\slash {\widetilde k}}_{i_1}
\, {\slash {\widetilde k}}_{i_2} \cdots {\slash {\widetilde k}}_{i_{2k}}
+ {\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1} \,({\slash p}_1 + \dots {\slash p}_m) \right] \nonumber \\
\label{stringscoll}
&&= \left( \frac{1}{s_{1 \dots m}} \right)^{k} \; x
\,\left[ {\slash p}
\, {\slash {\widetilde k}}_{i_1} \, {\slash {\widetilde k}}_{i_2} \cdots
{\slash {\widetilde k}}_{i_{2k}} +
{\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1} \,{\slash p} \right]
\;\left[ 1 + {\cal O}(\lambda) \right] \;\;,
\end{eqnarray}
where the dots stand for the product of ${\slash {\widetilde k}}_i$ factors. We can
now multiply Eq.~(\ref{stringscoll}) by unity in the form
$1 = ({\slash p} {\slash n} + {\slash n}{\slash p})/(2p\cdot n)$, and,
using $\{ {\slash p} , {\slash {\widetilde k}}_i \} =0$ and
${\slash p}^2 =0$,
we obtain
\begin{eqnarray}
&&\left( \frac{1}{s_{1 \dots m}} \right)^{k} \; x \left[ {\slash p}
\, {\slash {\widetilde k}}_{i_1} \, {\slash {\widetilde k}}_{i_2} \cdots
{\slash {\widetilde k}}_{i_{2k}}
\;\frac{{\slash p} {\slash n} + {\slash n} {\slash p}}{2p\cdot n}
+ \frac{{\slash p} {\slash n} + {\slash n} {\slash p}}{2p\cdot n}
{\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1} \,{\slash p} \right] \nonumber \\
\label{stringdom}
&&= \left( \frac{1}{s_{1 \dots m}} \right)^{k} \; x {\slash p}
\; \frac{{\slash {\widetilde k}}_{i_1} \, {\slash {\widetilde k}}_{i_2} \cdots
{\slash {\widetilde k}}_{i_{2k}} \,{\slash n} +
{\slash n} \,{\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1} }{2xp\cdot n} \; x{\slash p} \;\;.
\end{eqnarray}
Denoting by $\chi_s(p)$ the spinor of an on-shell fermion with momentum
$p$ and spin polarization $s$, we then replace the polarization matrices
$x{\slash p}$ in Eq.~(\ref{stringdom}) by using the identity
$x{\slash p} = \sum_s \chi_s(xp) {\overline \chi}_s(xp)$ and we can rewrite
the string in Eq.~(\ref{stringdom}) as follows
\begin{equation}
\label{stringspin}
\sum_{s,s^\prime} \left[ \;\chi_s(xp) \;{\overline \chi}_{s^\prime}(xp)
\;\right]
\; \left( \frac{1}{s_{1 \dots m}} \right)^{k}
{\overline \chi}_s(xp) \;\frac{{\slash {\widetilde k}}_{i_1} \, {\slash {\widetilde k}}_{i_2}
\cdots {\slash {\widetilde k}}_{i_{2k}} \,{\slash n} + {\slash n} \,
{\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1}}{2xp\cdot n} \;\chi_{s^\prime}(xp)
\;\;.
\end{equation}
When inserted in Eqs.~(\ref{aqdec}) and (\ref{fcollimp}),
the factor in the square bracket reconstructs the polarization matrix of the
decaying quark and, thus, the spin-polarization tensor
${\cal T}_{q,\dots}^{s s^\prime}(xp,\dots)$ in the factorization formula
(\ref{ccfacm}). The remaining factor in Eq.~(\ref{stringspin}) gives the
contribution of the string in Eq.~(\ref{stringscoll}) to the
quark splitting function ${\hat P}^{s s^\prime}_{a_1 \dots a_m}$.
By explicit construction we see that in general the splitting function
${\hat P}^{s s^\prime}_{a_1 \dots a_m}$ is not diagonal with respect to the
spin indices. Nonetheless, the spin correlations are absent within the
dimensional-regularization prescription used throughout the paper. Since we
are considering only two helicity states for massless quarks, we have
$\chi_{s=\pm 1}(p) = \frac{1}{2}(1 \pm \gamma_5) \chi(p)$,
where $\chi(p)$ is a generic Dirac spinor. Thus, using the general properties
of the Dirac algebra, the contribution of Eq.~(\ref{stringspin}) to the
splitting function can straightforwardly be recast in a form that explicitly
shows the cancellation of the spin correlations:
\begin{eqnarray}
&&\left( \frac{1}{s_{1 \dots m}} \right)^{k}
{\overline \chi}_s(xp) \;\frac{{\slash {\widetilde k}}_{i_1} \, {\slash {\widetilde k}}_{i_2}
\cdots {\slash {\widetilde k}}_{i_{2k}} \,{\slash n} + {\slash n} \,
{\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1}}{2xp\cdot n} \;\chi_{s^\prime}(xp) \nonumber \\
\label{stringhel}
&&= \delta^{s s^\prime} \left( \frac{1}{s_{1 \dots m}} \right)^{k}
{\rm Tr} \left[ \frac{{\slash n} \left( {\slash p} \, {\slash {\widetilde k}}_{i_1}
\, {\slash {\widetilde k}}_{i_2} \cdots {\slash {\widetilde k}}_{i_{2k}} +
{\slash {\widetilde k}}_{i_{2k}} \cdots {\slash {\widetilde k}}_{i_2} \,
{\slash {\widetilde k}}_{i_1} \, {\slash p} \right)}{4p\cdot n} \right]
\;\;, \;\;\;s,s^\prime=\pm 1 \;\;,
\end{eqnarray}
where ${\rm Tr}$ denotes the trace of the Dirac matrices.
The identity in Eq.~(\ref{stringhel}) relies on the definition and the
properties of the chiral projectors $\frac{1}{2}(1 \pm \gamma_5)$ and
the absence of spin correlations ultimately follows from helicity conservation
in the quark--gluon vector coupling.
This method to derive collinear factorization also provides us with
an expression of the (spin-averaged)
{\em quark} splitting function in terms of the dispersive
part $V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)$ of the two-point quark amplitude.
From Eqs.~(\ref{aqdec}) and (\ref{stringspin}), we find
\begin{equation}
\label{pqvsa}
\langle {\hat P}_{a_1 \dots a_m} \rangle
= {\rm Tr} \left( \frac{{\slash n} \;
V_{a_1 \dots a_m}^{(n)}(p_1,\dots,p_m)}{4 \,n \cdot(p_1+\dots+p_m) }
\right) \;\;.
\end{equation}
This equation is useful for a straightforward evaluation of the splitting
function for the multiple collinear decay of a quark.
\bigskip
\noindent {\it Gluon splitting processes}
\noindent Unlike the quark case, it is convenient to define the gluon
two-point function $V_{a_1 \dots a_m}^{(n)}$
without including in it the spin-polarization tensors
$d^{\mu \nu}_{(n)}(p_1+\dots+p_m)$ of the two external gluons.
Because of Lorentz covariance and the vanishing degree of homogeneity with
respect to $n^\mu$, the spin tensor $V_{a_1 \dots a_m}^{\nu \rho \, (n)}$
can be decomposed in terms of dimensionless scalar amplitudes as
\begin{eqnarray}
&&\!\! \!\! \! \!\!\! \!\! \!\!\! \!\!
V_{a_1 \dots a_m}^{\mu \nu \,(n)}(p_1,\dots,p_m) =
A^{(g)}(\{s_{jl}, z_j\})
\; g^{\mu \nu} + \sum_{i,j=1}^m B_{i,j}^{(g)}(\{s_{kl}, z_k\}) \;
\frac{p_i^\mu p_j^\nu}{s_{1 \dots m}} \nonumber \\
\label{agdec}
&\!\!+& \sum_{i=1}^m C_{i}^{(g)}(\{s_{jl}, z_j\}) \;
\frac{p_i^\mu n^\nu + n^\mu p_i^\nu}{n \cdot(p_1+\dots+p_m)}
+ D^{(g)}(\{s_{jl}, z_j\}) \;
\frac{n^\mu n^\nu \, s_{1 \dots m}}{(n \cdot(p_1+\dots+p_m))^2}
\;.
\end{eqnarray}
Then, we have to multiply $V_{a_1 \dots a_m}^{\nu \rho \, (n)}$
by the gluon polarization tensors as follows
\begin{equation}
\label{agdecfin}
d^{\mu}_{\nu \,(n)}(p_1+\dots+p_m)
\;V_{a_1 \dots a_m}^{\nu \rho \,(n)}(p_1,\dots,p_m)
\; d^{\sigma}_{\rho \,(n)}(p_1+\dots+p_m) \;\;.
\end{equation}
Inserting Eq.~(\ref{agdec}) into Eq.~(\ref{agdecfin}), we immediately see
that the scalar amplitudes $C_{i}^{(g)}(\{s_{jl}, z_j\})$ and
$D^{(g)}(\{s_{jl}, z_j\})$ give a vanishing contribution because the gauge
vector $n^\mu$ is orthogonal to the polarization tensors.
As for the second term on the right-hand side of Eq.~(\ref{agdec}), we can
extract its dominant collinear contribution by simply performing
the replacement $p_i^\mu \rightarrow {\widetilde k}_i^\mu$. Indeed, using Eq.~(\ref{piki})
and
\begin{equation}
(p_1+\dots+p_m)_\mu d_{(n)}^{\mu\nu}(p_1+\dots+p_m)={\cal O}(s_{1 \dots m})
\;\;,
\end{equation}
we have
\begin{equation}
d^{\mu}_{\nu \,(n)}(p_1+\dots+p_m) \; p_i^\nu \; p_j^\rho
\; d^{\sigma}_{\rho \,(n)}(p_1+\dots+p_m) = {\widetilde k}_i^\mu \;{\widetilde k}_i^\sigma
+{\cal O}(\lambda^3) \;\;,
\end{equation}
so that the longitudinal component of $p_i^\mu$ is suppressed in the
multiple collinear limit $\lambda \rightarrow 0$.
We can now safely perform the collinear limit and we obtain
the factorization formula (\ref{ccfacm}) and an explicit expression
of the gluon splitting function in terms of the scalar amplitudes in
Eq.~(\ref{agdec}):
\begin{equation}
\label{pggen}
{\hat P}^{\mu \nu}_{a_1 \dots a_m} =
A^{(g)}(\{s_{jl}, z_j\})
\; g^{\mu \nu} + \sum_{i,j=1}^m B_{i,j}^{(g)}(\{s_{kl}, z_k\}) \;
\frac{{\widetilde k}_i^\mu {\widetilde k}_j^\nu}{s_{1 \dots m}} \;\;.
\end{equation}
The splitting function
can be averaged over the spin polarizations of the parent
gluon according to Eq.~(\ref{gluav}), and we obtain
\begin{equation}
\label{gsfav}
\langle {\hat P}_{a_1 \dots a_m} \rangle \, \equiv \frac{1}{2 (1 - \epsilon)} \,d_{\mu \nu}(p)
\; {\hat P}^{\mu\nu}_{a_1 \dots a_m}
= - A^{(g)}(\{s_{jl}, z_j\}) - \, \frac{1}{2 (1 - \epsilon)}
\sum_{i,j=1}^m B_{i,j}^{(g)}(\{s_{kl}, z_k\}) \;
\frac{{\widetilde k}_i \cdot {\widetilde k}_j}{s_{1 \dots m}} \;,
\end{equation}
where
\begin{equation}
2 {\widetilde k}_i \cdot {\widetilde k}_j = s_{ij}
- \sum_{k=1}^m \left( z_i s_{jk} + z_j s_{ik} \right) + 2 z_i z_j s_{1 \dots m}
\;\;.
\end{equation}
Note that, since $2 {\widetilde k}_i \cdot {\widetilde k}_j =
2 p_i \cdot d_{(n)}(p_1+\dots+p_m) \cdot d_{(n)}(p_1+\dots+p_m) \cdot p_j$,
the spin-averaged splitting function can also be expressed in terms
of the Lorentz trace of Eq.~(\ref{agdecfin}):
\begin{equation}
\langle {\hat P}_{a_1 \dots a_m} \rangle \, = - \frac{1}{2 (1 - \epsilon)} \;
d^{\mu}_{\nu \,(n)}(p_1+\dots+p_m)
\;V_{a_1 \dots a_m}^{\nu \rho \,(n)}(p_1,\dots,p_m)
\; d_{\rho \mu \,(n)}(p_1+\dots+p_m) \;\;.
\end{equation}
In the following subsection we present the explicit calculation of
the quark and gluon splitting functions in the triple collinear limit.
\subsection{Collinear splitting functions at ${\cal O}(\as^2)$}
\label{splitt}
The list of (non-vanishing) splitting processes that we have to consider is
as follows:
\begin{eqnarray}
\label{qqqprime}
&& q\rightarrow {\bar q}^\prime_1 + q^\prime_2 + q_3 \;\;,
\;\;({\bar q} \rightarrow {\bar q}^\prime_1 + q^\prime_2 + {\bar q}_3 ) \;\;, \\
\label{qqq}
&& q\rightarrow {\bar q}_1 + q_2 + q_3 \;\;,
\;\;({\bar q} \rightarrow {\bar q}_1 + q_2 + {\bar q}_3 )\;\;, \\
\label{ggq}
&& q\rightarrow g_1 + g_2 + q_3 \;\;,
\;\;({\bar q} \rightarrow g_1 + g_2 + {\bar q}_3 ) \;\;, \\
\label{gqq}
&& g\rightarrow g_1 + q_2 + {\bar q}_3 \;\;, \\
\label{ggg}
&& g \rightarrow g_1 + g_2 + g_3 \;\;.
\end{eqnarray}
The superscripts in $q^\prime, {\bar q}^\prime$ denote fermions with different
flavour with respect to $q, {\bar q}$. As already mentioned in
Sect.~\ref{power}, the splitting functions for the
processes in parenthesis in Eqs.~(\ref{qqqprime}) and (\ref{qqq}) can be simply
obtained by charge-conjugation invariance, i.e.
${\hat P}_{{\bar q}^\prime_1 q^\prime_2 {\bar q}_3} =
{\hat P}_{q^\prime_1 {\bar q}^\prime_2 q_3}$ and
${\hat P}_{{\bar q}_1 q_2 {\bar q}_3} =
{\hat P}_{q_1 {\bar q_2} q_3}$.
In summary, we have to compute five independent splitting functions.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=5truecm
\epsffile{qqqnid.eps}\\
\end{tabular}
\end{center}
\caption{\label{qqqnid}{\em The diagram for the collinear decay
$q\rightarrow {\bar q^\prime}_1q^\prime_2q_3$. }}
\end{figure}
To illustrate our calculation,
we first consider the process in Eq.~(\ref{qqqprime}), that is,
the case in which a quark--antiquark pair ${\bar q^\prime}_1q^\prime_2$
and a quark $q_3$ with different flavour become collinear.
This is the simplest case, because the two-point function
$V_{{\bar q}^\prime_1 q^\prime_2 q_3}^{(n)}(p_1,p_2,p_3)$
for the corresponding splitting process is obtained by
squaring the sole Feynman diagram in Fig.~\ref{qqqnid}.
According to the definition in Eqs.~(\ref{fcollimp}) and (\ref{aqdec}), we
extract the overall factor $\left( 8 \pi \mu^{2\epsilon} \as/s_{123} \right)^2$
and, performing the average over the colours of the decaying quark $q$,
we find
\begin{eqnarray}
V_{{\bar q}^\prime_1 q^\prime_2 q_3}^{(n)}(p_1,p_2,p_3)
&=& \f{1}{2} \,
C_F T_R \,\f{s_{123}}{s_{12}} \left\{ \left[ \f{2z_3}{z_1+z_2}
- \left( \f{t_{12,3}^2}{s_{12}^2} + 1 - 2\epsilon \right) \f{s_{12}}{s_{123}}
\right] ( {\slash p}_1 + {\slash p}_2 + {\slash p}_3 ) \right. \nonumber \\
\label{prelimit}
&+& \left. \f{2}{z_1+z_2}{\slash p}_3 + (1-2\epsilon) ({\slash p}_1 + {\slash p}_2)
+ \f{z_1 - z_2}{z_1+z_2} ({\slash p}_1 - {\slash p}_2) \right. \\
&+& \left. \f{2t_{12,3}}{(z_1+z_2)s_{12}} (z_1{\slash p}_2 - z_2{\slash p}_1)
+ \f{1}{z_1+z_2} \left( \f{s_{12}}{s_{123}} -1 \right)
\f{ {\slash n} \;s_{123}}{n\cdot(p_1+p_2+p_3)} \right\} \nonumber
\;\;,
\end{eqnarray}
where $t_{12,3}$ is the kinematical variable defined in Eq.~(\ref{tvar}).
Note that Eq.~(\ref{prelimit}) has the general structure obtained in
Eq.~(\ref{aqdecfin}). Using Eq.~(\ref{pqvsa}) to compute the splitting function
$\langle {\hat P}_{{\bar q}^\prime_1 q^\prime_2 q_3} \rangle$,
the last two terms on the right-hand side of
Eq.~(\ref{prelimit}) give a vanishing contribution and we obtain
the final result:
\begin{equation}
\label{qqqprimesf}
\langle {\hat P}_{{\bar q}^\prime_1 q^\prime_2 q_3} \rangle \, = \f{1}{2} \,
C_F T_R \,\f{s_{123}}{s_{12}} \left[ - \f{t_{12,3}^2}{s_{12}s_{123}}
+\f{4z_3+(z_1-z_2)^2}{z_1+z_2}
+ (1-2\epsilon) \left(z_1+z_2-\f{s_{12}}{s_{123}}\right)
\right] \;\;.
\end{equation}
The calculation of the splitting functions for the other processes
in Eqs.~(\ref{qqq})--(\ref{ggg}) can be performed exactly in the same manner,
by using the general procedure discussed in Sect.~\ref{power}.
We first compute the corresponding two-point functions
$V_{a_1 a_2 a_3}^{(n)}(p_1,p_2,p_3)$ and then, using Eqs.~(\ref{pqvsa}) and
(\ref{pggen}), we evaluate the splitting functions. Since the intermediate
expressions for the two-point functions are quite cumbersome, in the following
we limit ourselves to showing the relevant Feynman diagrams and to presenting
the final results for the splitting functions.
The calculation of the splitting function for the case of final-state fermions
with identical flavour involves a diagram analogous to that in
Fig.~\ref{qqqnid} plus its crossed diagram (see Fig.~\ref{qqqid}). Thus,
the result can be written in terms of that in Eq.~(\ref{qqqprimesf}),
as follows
\begin{equation}
\label{qqqsf}
\langle {\hat P}_{{\bar q}_1q_2q_3} \rangle \, =
\left[ \langle {\hat P}_{{\bar q}^\prime_1q^\prime_2q_3} \rangle \, + \,(2\leftrightarrow 3) \,\right]
+ \langle {\hat P}^{({\rm id})}_{{\bar q}_1q_2q_3} \rangle \;\;,
\end{equation}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=8.5truecm
\epsffile{qqqid.eps}\\
\end{tabular}
\end{center}
\caption{\label{qqqid}{\em The diagrams for the collinear decay
$q\rightarrow {\bar q}_1q_2q_3$. }}
\end{figure}
where the interference contribution is given by
\begin{eqnarray}
\label{idensf}
\langle {\hat P}^{({\rm id})}_{{\bar q}_1q_2q_3} \rangle \,
&=& C_F \left( C_F-\f{1}{2} C_A \right)
\Biggl\{ (1-\epsilon)\left( \f{2s_{23}}{s_{12}} - \epsilon \right)\nonumber\\
&+& \f{s_{123}}{s_{12}}\Biggl[\f{1+z_1^2}{1-z_2}-\f{2z_2}{1-z_3}
-\epsilon\left(\f{(1-z_3)^2}{1-z_2}+1+z_1-\f{2z_2}{1-z_3}\right)
- \epsilon^2(1-z_3)\Biggr] \nonumber\\
&-& \f{s_{123}^2}{s_{12}s_{13}}\f{z_1}{2}\left[\f{1+z_1^2}{(1-z_2)(1-z_3)}-\epsilon
\left(1+2\f{1-z_2}{1-z_3}\right)
-\epsilon^2\right] \Biggr\} + (2\leftrightarrow 3) \;\;.
\end{eqnarray}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=13truecm
\epsffile{qgg.eps}\\
\end{tabular}
\end{center}
\caption{\label{qgg}{\em The diagrams for the collinear decay
$q\rightarrow g_1g_2q_3$. }}
\end{figure}
The splitting function of the remaining quark-decay subprocess is obtained
by squaring the diagrams in Fig.~\ref{qgg}. It can be
decomposed according to the different colour coefficients:
\begin{equation}
\label{qggsf}
\langle {\hat P}_{g_1 g_2 q_3} \rangle \, =
C_F^2 \, \langle {\hat P}_{g_1 g_2 q_3}^{({\rm ab})} \rangle \,
+ \, C_F C_A \, \langle {\hat P}_{g_1 g_2 q_3}^{({\rm nab})} \rangle \;\;,
\end{equation}
and the abelian and non-abelian contributions are
\begin{eqnarray}
\label{qggabsf}
\langle {\hat P}_{g_1 g_2 q_3}^{({\rm ab})} \rangle \,
&=&\Biggl\{\f{s_{123}^2}{2s_{13}s_{23}}
z_3\left[\f{1+z_3^2}{z_1z_2}-\epsilon\f{z_1^2+z_2^2}{z_1z_2}-\epsilon(1+\epsilon)\right]\nonumber\\
&+&\f{s_{123}}{s_{13}}\Biggl[\f{z_3(1-z_1)+(1-z_2)^3}{z_1z_2}+\epsilon^2(1+z_3)
-\epsilon (z_1^2+z_1z_2+z_2^2)\f{1-z_2}{z_1z_2}\Biggr]\nonumber\\
&+&(1-\epsilon)\left[\epsilon-(1-\epsilon)\f{s_{23}}{s_{13}}\right]
\Biggr\}+(1\leftrightarrow 2) \;\;,
\end{eqnarray}
\begin{eqnarray}
\label{qggnabsf}
\langle {\hat P}_{g_1 g_2 q_3}^{({\rm nab})} \rangle \,
&=&\Biggl\{(1-\epsilon)\left(\f{t_{12,3}^2}{4s_{12}^2}+\f{1}{4}
-\f{\epsilon}{2}\right)+\f{s_{123}^2}{2s_{12}s_{13}}
\Biggl[\f{(1-z_3)^2(1-\epsilon)+2z_3}{z_2}\nonumber\\
&+&\f{z_2^2(1-\epsilon)+2(1-z_2)}{1-z_3}\Biggr]
-\f{s_{123}^2}{4s_{13}s_{23}}z_3\Biggl[\f{(1-z_3)^2(1-\epsilon)+2z_3}{z_1z_2}
+\epsilon(1-\epsilon)\Biggr]\nonumber\\
&+&\f{s_{123}}{2s_{12}}\Biggl[(1-\epsilon)
\f{z_1(2-2z_1+z_1^2) - z_2(6 -6 z_2+ z_2^2)}{z_2(1-z_3)}
+2\epsilon\f{z_3(z_1-2z_2)-z_2}{z_2(1-z_3)}\Biggr]\nonumber\\
&+&\f{s_{123}}{2s_{13}}\Biggl[(1-\epsilon)\f{(1-z_2)^3
+z_3^2-z_2}{z_2(1-z_3)}
-\epsilon\left(\f{2(1-z_2)(z_2-z_3)}{z_2(1-z_3)}-z_1 + z_2\right)\nonumber\\
&-&\f{z_3(1-z_1)+(1-z_2)^3}{z_1z_2}
+\epsilon(1-z_2)\left(\f{z_1^2+z_2^2}{z_1z_2}-\epsilon\right)\Biggr]\Biggr\}
+(1\leftrightarrow 2) \;\;.
\end{eqnarray}
As discussed in Sect.~\ref{power},
in the case of collinear decays of a gluon (see Eqs.~(\ref{gqq}, \ref{ggg})),
spin correlations are highly non-trivial.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=13truecm
\epsffile{qqg.eps}\\
\end{tabular}
\end{center}
\caption{\label{qqgfig}{\em The diagrams for the collinear decay
$g\rightarrow g_1q_2{\bar q}_3$. }}
\end{figure}
To compute the splitting function for the decay into a $q{\bar q}$ pair plus a
gluon, we have to evaluate the square of the diagrams in Fig.~\ref{qqgfig}.
The colour-factor decomposition of the splitting function is
\begin{equation}
\label{gqqsf}
{\hat P}^{\mu\nu}_{g_1 q_2 {\bar q}_3} \, =
C_F T_R \, {\hat P}_{g_1 q_2 {\bar q}_3}^{\mu\nu \,({\rm ab})} \,
+ \, C_A T_R\, {\hat P}_{g_1 q_2 {\bar q}_3}^{\mu\nu \,({\rm nab})} \;\;,
\end{equation}
where the abelian and non-abelian terms are given by
\begin{eqnarray}
\label{gqqabsf}
{\hat P}^{\mu\nu \,({\rm ab})}_{g_1q_2{\bar q}_3} &=&
-g^{\mu\nu}\Biggl[ -2
+ \f{2 s_{123} s_{23} + (1-\epsilon) (s_{123} - s_{23})^2}{s_{12}s_{13}}\Biggr]\nonumber\\
&+& \f{4s_{123}}{s_{12}s_{13}}\left({\widetilde k}_{3}^\mu
{\widetilde k}_{2}^\nu+{\widetilde k}_{\hspace{.1mm} 2}^\mu
{\widetilde k}_{3}^\nu-(1-\epsilon){\widetilde k}_{\hspace{.1mm} 1}^\mu
{\widetilde k}_{1}^\nu \right)
\;\;,
\end{eqnarray}
\begin{eqnarray}
\label{gqqnabsf}
{\hat P}^{\mu\nu \,({\rm nab})}_{g_1q_2{\bar q}_3} &=& \f{1}{4}
\,\Biggl\{ \f{s_{123}}{s_{23}^2}
\Biggl[ g^{\mu\nu} \f{t_{23,1}^2}{s_{123}}-16\f{z_2^2z_3^2}{z_1(1-z_1)}
\left(\f{{\widetilde k}_2}{z_2}-\f{{\widetilde k}_3}{z_3}\right)^\mu
\left(\f{{\widetilde k}_2}{z_2}-\f{{\widetilde k}_3}{z_3}\right)^\nu \,\Biggr]\nonumber\\
&+& \f{s_{123}}{s_{12}s_{13}} \Biggl[ 2 s_{123} g^{\mu\nu}
- 4 ( {\widetilde k}_2^\mu {\widetilde k}_3^\nu + {\widetilde k}_3^\mu {\widetilde k}_2^\nu
- (1-\epsilon) {\widetilde k}_1^\mu {\widetilde k}_1^\nu ) \Biggr] \nonumber\\
&-& g^{\mu\nu} \Biggl[ - ( 1 -2 \epsilon) + 2\f{s_{123}}{s_{12}}
\f{1-z_3}{z_1(1-z_1)} + 2\f{s_{123}}{s_{23}}
\f{1-z_1 + 2 z_1^2}{z_1(1-z_1)}\Biggr]\nonumber\\
&+& \f{s_{123}}{s_{12}s_{23}} \Biggl[ - 2 s_{123} g^{\mu\nu}
\f{z_2(1-2z_1)}{z_1(1-z_1)} - 16 {\widetilde k}_3^\mu {\widetilde k}_3^\nu
\f{z_2^2}{z_1(1-z_1)}
+ 8(1-\epsilon) {\widetilde k}_2^\mu {\widetilde k}_2^\nu \nonumber\\
&+& 4 ({\widetilde k}_2^\mu {\widetilde k}_3^\nu + {\widetilde k}_3^\mu {\widetilde k}_2^\nu )
\left(\f{2 z_2 (z_3-z_1)}{z_1(1-z_1)}+ (1-\epsilon) \right)
\Biggr] \Biggr\} + \left( 2 \leftrightarrow 3 \right) \;\;.
\end{eqnarray}
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=12truecm
\epsffile{ggg.eps}\\
\end{tabular}
\end{center}
\caption{\label{gggfig}{\em The diagrams for the collinear decay
$g\rightarrow g_1g_2g_3$. }}
\end{figure}
In the case of gluon decay into three collinear gluons we have to consider
the diagrams in Fig.~\ref{gggfig}. Note that the contribution of the
four-gluon vertex cannot be neglected. The result for the splitting function
is quite involved. Its expression can be simplified by exploiting the complete
symmetry under the six permutations of the gluon momenta. We obtain
\begin{eqnarray}
\label{gggsf}
{\hat P}^{\mu\nu}_{g_1g_2g_3} &=& C_A^2
\,\Biggl\{\f{(1-\epsilon)}{4s_{12}^2}
\Biggl[-g^{\mu\nu} t_{12,3}^2+16s_{123}\f{z_1^2z_2^2}{z_3(1-z_3)}
\left(\f{{\widetilde k}_2}{z_2}-\f{{\widetilde k}_1}{z_1}\right)^\mu
\left(\f{{\widetilde k}_2}{z_2}-\f{{\widetilde k}_1}{z_1}\right)^\nu \;\Biggr]\nonumber\\
&-& \f{3}{4}(1-\epsilon)g^{\mu\nu}+\f{s_{123}}{s_{12}}g^{\mu\nu}\f{1}{z_3}
\Biggl[\f{2(1-z_3)+4z_3^2}{1-z_3}-\f{1-2z_3(1-z_3)}{z_1(1-z_1)}\Biggr]\nonumber\\
&+& \f{s_{123}(1-\epsilon)}{s_{12}s_{13}}\Biggl[2z_1\left({\widetilde k}^\mu_2
{\widetilde k}^\nu_2\hspace{.1mm}\f{1-2z_3}{z_3(1-z_3)}+
{\widetilde k}^\mu_3{\widetilde k}^\nu_3\hspace{.1mm}
\f{1-2z_2}{z_2(1-z_2)}\right)\nonumber\\
&+& \f{s_{123}}{2(1-\epsilon)} g^{\mu\nu}
\left(\f{4z_2z_3+2z_1(1-z_1)-1}{(1-z_2)(1-z_3)}
- \f{1-2z_1(1-z_1)}{z_2z_3}\right)\nonumber\\
&+& \left({\widetilde k}_2^\mu{\widetilde k}_3^\nu
+{\widetilde k}_3^\mu{\widetilde k}_2^\nu\right)
\left(\f{2z_2(1-z_2)}{z_3(1-z_3)}-3\right)\Biggr]\Biggr\}
+ (5\mbox{ permutations}) \;\;.
\end{eqnarray}
The splitting functions in Eqs.~(\ref{gqqabsf})--(\ref{gggsf})
can be averaged over the spin polarizations of the parent
gluon according to Eq.~(\ref{gsfav}):
\begin{equation}
\langle {\hat P}_{a_1 a_2 a_3} \rangle \, \equiv \frac{1}{2 (1 - \epsilon)} \,d_{\mu \nu}(p)
\; {\hat P}^{\mu\nu}_{a_1 a_2 a_3} \;\;.
\end{equation}
Performing the average, we obtain
\begin{eqnarray}
\label{gqqabsfav}
\langle {\hat P}^{({\rm ab})}_{g_1q_2{\bar q}_3} \rangle \,&=&
-2-(1-\epsilon)s_{23}\left(\f{1}{s_{12}}+\f{1}{s_{13}}\right)
+ 2\f{s_{123}^2}{s_{12}s_{13}}\left(1+z_1^2-\f{z_1+2z_2 z_3}{1-\epsilon}\right)
\nonumber\\
&-&\f{s_{123}}{s_{12}}\left(1+2z_1+\epsilon-2\f{z_1+z_2}{1-\epsilon}\right)
- \f{s_{123}}{s_{13}}\left(1+2z_1+\epsilon-2\f{z_1+z_3}{1-\epsilon}\right)
\;,
\end{eqnarray}
\begin{eqnarray}
\label{gqqnabsfav}
\langle {\hat P}^{({\rm nab})}_{g_1q_2{\bar q}_3} \rangle
\,&=&\Biggl\{-\f{t^2_{23,1}}{4s_{23}^2}
+\f{s_{123}^2}{2s_{13}s_{23}} z_3
\Biggl[\f{(1-z_1)^3-z_1^3}{z_1(1-z_1)}
-\f{2z_3\left(1-z_3 -2z_1z_2\right)}{(1-\epsilon)z_1(1-z_1)}\Biggr]\nonumber\\
&+&\f{s_{123}}{2s_{13}}(1-z_2)\Biggl[1
+\f{1}{z_1(1-z_1)}-\f{2z_2(1-z_2)}{(1-\epsilon)z_1(1-z_1)}\Biggr]\nonumber\\
&+&\f{s_{123}}{2s_{23}}\Biggl[\f{1+z_1^3}{z_1(1-z_1)}
+\f{z_1(z_3-z_2)^2-2z_2z_3(1+z_1)}
{(1-\epsilon)z_1(1-z_1)}\Biggr] \nonumber\\
&-&\f{1}{4}+\f{\epsilon}{2}
-\f{s_{123}^2}{2s_{12}s_{13}}\Biggl(1+z_1^2-\f{z_1+2z_2z_3}{1-\epsilon}
\Biggr) \Biggr\}
+ (2\leftrightarrow 3) \;\;,
\end{eqnarray}
\begin{eqnarray}
\label{gggsfav}
\langle {\hat P}_{g_1g_2g_3} \rangle \,&=& C_A^2\Biggl\{\f{(1-\epsilon)}{4s_{12}^2}
t_{12,3}^2+\f{3}{4}(1-\epsilon)+\f{s_{123}}{s_{12}}\Biggl[4\f{z_1z_2-1}{1-z_3}
+\f{z_1z_2-2}{z_3}+\f{3}{2} +\f{5}{2}z_3\nonumber\\
&+&\f{\left(1-z_3(1-z_3)\right)^2}{z_3z_1(1-z_1)}\Biggr]
+\f{s_{123}^2}{s_{12}s_{13}}\Biggl[\f{z_1z_2(1-z_2)(1-2z_3)}{z_3(1-z_3)}
+z_2z_3 -2 +\f{z_1(1+2z_1)}{2}\nonumber\\
&+&\f{1+2z_1(1+z_1)}{2(1-z_2)(1-z_3)}
+\f{1-2z_1(1-z_1)}{2z_2z_3}\Biggr]\Biggr\}
+ (5\mbox{ permutations}) \;\;.
\end{eqnarray}
The ${\cal O}(\as^2)$-collinear limit of tree-level QCD
amplitudes has been independently considered by Campbell and Glover
[\ref{glover}]. They have computed only the spin-averaged splitting
functions. The comparison with their results has been discussed in detail in
Ref.~[\ref{lett}] and we do not repeat it here.
Our results agree with those of Ref.~[\ref{glover}].
Since the methods used by the two groups are completely different
(cf. the discussion in Sect.~\ref{intro}),
this agreement can be regarded as an important cross-check of the calculations.
The expressions of the spin-dependent splitting functions
${\hat P}^{s s'}_{a_1a_2a_3}$ derived in this section refer to the CDR scheme.
Other dimensional-regularization schemes can be used. We mention two of them,
which differ from CDR only by the number of spin-polarizations of quarks and
gluons.
The dimensional-reduction (DR) scheme [\ref{dimred}] works by considering two
spin-polarization states for quarks and two for gluons. The corresponding
spin-dependent splitting functions ${\hat P}^{s s'}_{a_1a_2a_3}$ are simply
obtained from those in the CDR scheme by setting $\epsilon = 0$.
The `toy' dimensional-regularization (TDR) scheme introduced in
Ref.~[\ref{schemedep}] considers $d-2=2(1-\epsilon)$ spin-polarization states
for quarks as for gluons. Its practical implementation is very simple.
When computing traces of gamma matrices, we should use the relation
${\rm Tr} \;{\bom 1}= 4(1-\epsilon)$, where ${\bom 1}$ is the unity matrix in
the spinor space. The corresponding
spin-dependent splitting functions ${\hat P}^{s s'}_{a_1a_2a_3}$ are
obtained from those in the CDR scheme by the simple replacement
$T_R \rightarrow T_R(1-\epsilon)$.
The QCD results presented in this section can also be extended in a
straightforward way to the abelian and supersymmetric cases.
In the case of QED, we have to perform the replacement $\as \rightarrow \alpha$ in the
factorization formula (\ref{ccfacm}), and the relevant splitting functions,
${\hat P}^{(QED)}_{a_1a_2a_3}$, for the triple collinear limit are obtained
from the QCD splitting functions as
\begin{eqnarray}
{\hat P}^{(QED)}_{{\bar q}^{\prime}_1 q^{\prime}_2 q_3} &=&
e_q^2 e_{q^\prime}^2 \Bigl( {\hat P}_{{\bar q}^{\prime}_1 q^{\prime}_2 q_3}
\Bigr)_{\rm {ab.}} \;\;, \nonumber \\
{\hat P}^{(QED)}_{{\bar q}_1 q_2 q_3} &=& e_q^4 \Bigl(
{\hat P}_{{\bar q}_1 q_2 q_3} \Bigr)_{\rm {ab.}} \;\;,\\
{\hat P}^{(QED)}_{\gamma_1 \gamma_2 q_3} &=& e_q^4 \Bigl(
{\hat P}_{g_1 g_2 q_3} \Bigr)_{\rm {ab.}} \;\;, \nonumber \\
{\hat P}^{(QED)}_{\gamma_1 q_2 {\bar q}_3} &=& e_q^4 \Bigl(
{\hat P}_{g_1 q_2 {\bar q}_3} \Bigr)_{\rm {ab.}} \;\;, \nonumber
\end{eqnarray}
where $e_q$ is the quark electric charge and the notation
$\Bigl( \dots \Bigr)_{\rm {ab.}}$ means that we have to set $C_F=T_R=1$ and
$C_A=0$ in the QCD expression inside the round bracket.
The supersymmetric version of QCD, namely $N=1$ supersymmetric Yang--Mills
theory, is obtained by replacing the quark with the gluino $\tilde g$,
a Majorana fermion in the adjoint representation of the gauge group.
To obtain the corresponding splitting functions, we have to change the colour
factors accordingly, and we have to identify $q={\bar q}=\tilde g$ after having
summed over the different permutations of the final-state fermions. We have
\begin{eqnarray}
{\hat P}_{{\tilde g}_1 {\tilde g}_2 {\tilde g}_3} &=& \Bigl(
{\hat P}_{{\bar q}_1 q_2 q_3} +
{\hat P}_{q_1 {\bar q}_2 q_3} + {\hat P}_{q_1 q_2 {\bar q}_3}
\Bigr)_{\rm {SQCD}} \;\;, \nonumber \\
{\hat P}_{g_1 g_2 {\tilde g}_3} &=& \Bigl(
{\hat P}_{g_1 g_2 q_3} \Bigr)_{\rm {SQCD}} \;\;, \\
{\hat P}_{g_1 {\tilde g}_2 {\tilde g}_3} &=& \Bigl(
{\hat P}_{g_1 q_2 {\bar q}_3} + {\hat P}_{g_1 {\bar q}_2 q_3}
\Bigr)_{\rm {SQCD}} \;\;, \nonumber
\end{eqnarray}
where the notation $\Bigl( \dots \Bigr)_{\rm {SQCD}}$ means that we have to set
$C_F=2 T_R= C_A$ in the QCD expression inside the round bracket.
Gluino and gluon amplitudes are related by supersymmetry transformations.
In the collinear limit, these transformations relate
the total splitting functions ${\hat P}^{s s'}_{{\tilde g} \rightarrow 3}$
and ${\hat P}^{\mu \nu}_{g \rightarrow 3}$ for gluino and gluon decays, which are
defined as
\begin{equation}
{\hat P}^{s s'}_{{\tilde g} \rightarrow 3} \equiv
{\hat P}^{s s'}_{{\tilde g}_1 {\tilde g}_2 {\tilde g}_3}
+ \left[ {\hat P}^{s s'}_{g_1 g_2 {\tilde g}_3} + (3\!\leftrightarrow
\! 1)+(3\!\leftrightarrow\! 2) \right] = \delta^{s s'}
\;\langle {\hat P}_{{\tilde g} \rightarrow 3} \rangle \;\;,
\end{equation}
\begin{equation}
{\hat P}^{\mu \nu}_{g \rightarrow 3} \equiv
{\hat P}^{\mu \nu}_{g_1 g_2 g_3}
+ \left[ {\hat P}^{\mu \nu}_{g_1 {\tilde g}_2 {\tilde g}_3}
+ (1\!\leftrightarrow \! 2)+(1\!\leftrightarrow\! 3) \right] \;\;.
\end{equation}
In the four-dimensional supersymmetric theory, gluon and gluino have the same
decay probability.
Provided supersymmetry is not broken by the dimensional-regularization
procedure, we thus
have the following supersymmetric Ward identity:
\begin{equation}
\label{susywi}
{\hat P}^{\mu \nu}_{g \rightarrow 3} = - g^{\mu \nu}
\;\langle {\hat P}_{{\tilde g} \rightarrow 3} \rangle \;\;.
\end{equation}
Note that the Ward identity holds for the spin-dependent splitting
functions. Since spin correlations are absent in the gluino splitting function,
they cancel in the right-hand side of Eq.~(\ref{susywi}), and
${\hat P}^{ss'}_{{\tilde g} \rightarrow 3}$ and ${\hat P}^{\mu \nu}_{g \rightarrow 3}$
differ only by the overall spin-factors $\delta^{ss'}$ and $-g^{\mu\nu}$.
As is well known, the splitting functions
${\hat P}^{ss'}_{{\tilde g} \rightarrow 2}$ and ${\hat P}^{\mu \nu}_{g \rightarrow 2}$
are related by a similar Ward identity at ${\cal O}(\as)$.
The identity (\ref{susywi}) is violated in the CDR scheme,
because gluinos and gluons have a different number of spin-polarization
states. The Ward identity is recovered in the $\epsilon\rightarrow 0$ limit or,
equivalently, in the DR scheme, which is known to explicitly preserve
supersymmetry. Our results for the spin-dependent splitting functions fulfil
Eq.~(\ref{susywi}), and this is an important check of our calculation.
As pointed out in Ref.~[\ref{schemedep}], the Ward identity at ${\cal O}(\as)$
is fulfilled also in the TDR scheme.
We have verified that this remains true at ${\cal O}(\as^2)$, as expected from
the fact that in the TDR scheme
the number of gluino states is the same as
the number of gluon states.
\section{The soft behaviour}
\label{secsoft}
The tree-level matrix elements ${\cal M}(p_1,p_2,\dots)$ are singular not only
when parton momenta become collinear but also when one or more of them become
soft. In QCD calculations of physical cross sections at NLO, the soft limit
is approached when the momentum of a single gluon vanishes. At NNLO we have to
consider three different types of soft configurations:
\begin{itemize}
\item the emission of a soft quark--antiquark pair,
\item the emission of two soft gluons,
\item the emission of a soft gluon and a pair of collinear partons.
\end{itemize}
The behaviour of the tree-level matrix elements in these
singular limits is considered in this section. We also discuss
the generalization of the corresponding factorization formulae
to higher perturbative orders.
\subsection{Colour correlations and eikonal current at ${\cal O}(\as)$}
\label{secsoftlo}
The emission of a soft gluon does not affect the kinematics (momenta and spins)
of the radiating partons. However, it does affect their colour because
the gluon always carries away some colour charge, no matter how soft it is.
Unlike the case of soft-photon emission in QED, soft-gluon emission thus
does not factorize exactly and leads to colour correlations.
To take into account the colour structure (as well as the spin and flavour
structures), it is useful to introduce a basis
$\{ \ket{c_1,...,c_n} \otimes \ket{s_1,...,s_n} \}$
in colour + helicity space in such a way that the tree-level matrix element
in Eq.~(\ref{meldef}) with $n$ final-state partons can be written as
\begin{equation}
\label{cmmdef}
{\cal M}_{a_1,\dots,a_n}^{c_1,\dots,c_n; s_1,\dots,s_n}(p_1,\dots,p_n) \equiv
\Bigl( \bra{c_1,\dots,c_n} \otimes \bra{s_1,\dots,s_n} \Bigr) \;
\ket{{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)} \;.
\end{equation}
Thus $\ket{{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)}$
is a vector in colour + helicity space.
According to this notation, the matrix element squared (summed
over final-state colours and spins) $|{\cal M}|^2$ can be written as
\begin{equation}
|{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)|^2 =
\langle \, {\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \,
| \, {\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \, \rangle \;.
\end{equation}
To describe the colour correlations produced by soft-gluon emission,
it is convenient to associate a colour charge ${\bom T}_i$
with the emission of a gluon from each parton $i$. If the emitted gluon
has colour index $c$ ($c= 1, ...,$ $N_c^2-1$), the colour-charge operator is:
\begin{equation}
{\bom T}_i \equiv \bra{c} \;T_i^c
\end{equation}
and its action onto the colour space is defined by
\begin{equation}
\bra{c_1,\dots, c_i, \dots, c_m, c} {\bom T}_i
\ket{b_1, \dots, b_i, \dots, b_m} = \delta_{c_1 b_1} ...
T_{c_i b_i}^c ...\delta_{c_m b_m} \;\;,
\end{equation}
where $T_{c b}^a \equiv i f_{cab}$ (colour-charge matrix
in the adjoint representation) if the emitting particle $i$
is a gluon and $T_{\alpha \beta}^a \equiv t^a_{\alpha \beta}$
(colour-charge matrix in the fundamental representation with
$\alpha, \beta =1,\dots,N_c$)
if the emitting particle $i$ is a quark (in the case of an emitting
antiquark $T_{\alpha \beta}^a \equiv {\bar t}^a_{\alpha \beta}
= - t^a_{\beta \alpha }$).
The colour-charge algebra is\footnote{More details on the colour algebra
and useful colour-matrix relations can be found in Appendix~A of
Ref.~[\ref{CSdipole}].}:
\begin{equation}
T_i^c \, T_j^c \equiv
{\bom T}_i \cdot {\bom T}_j ={\bom T}_j \cdot {\bom T}_i \;\;\;\;{\rm if}
\;\;i \neq j; \;\;\;\;\;\;{\bom T}_i^2= C_i,
\end{equation}
where $C_i$ is the Casimir operator, that is,
$C_i=C_A=N_c$ if $i$ is a gluon and $C_i=C_F=(N_c^2-1)/2N_c$ if $i$ is a quark
or antiquark.
Note that, by definition, each vector $\ket{{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)}$
is a colour-singlet state. Therefore colour conservation is simply
\begin{equation} \label{cocon}
\sum_{i=1}^n {\bom T}_i \; \ket{{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)} = 0 \;.
\end{equation}
Let us now consider the tree-level matrix element
${\cal M}_{g,a_1,\dots,a_n}(q,p_1,\dots,p_n)$ in the limit where the
momentum $q$ of the gluon becomes soft. Denoting by
$c$ and $\mu$ the colour and spin indices of the soft gluon,
the matrix element fulfils the following factorization formula~[\ref{BCM}]
\begin{equation}
\label{eikfac}
\langle c; \mu |\,{\cal M}_{g,a_1,\dots,a_n}(q,p_1,\dots,p_n) \rangle
\simeq g_S \mu^\epsilon
J^{c;\mu}(q)
\; |\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle \;,
\end{equation}
where $|\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle$
is obtained from the original matrix by simply removing the soft gluon $q$.
The factor ${\bom J}^\mu(q)$ is the eikonal current
\begin{equation}
\label{eikcur}
{\bom J}^\mu(q)=\sum_{i=1}^{n} {\bom T}_i\,\f{p_i^\mu}{p_i\cdot q} \;,
\end{equation}
which depends on the momenta and colour charges of the
hard partons in the matrix element on the right-hand side of
Eq.~(\ref{eikfac}). The symbol `$\simeq$' means that on the right-hand side
we have neglected contributions that are less singular than $1/q$ in the soft
limit $q \rightarrow 0$. Note that Eq.~(\ref{eikfac}) is valid in any number
$d=4-2\epsilon$ of space-time dimensions, and the sole dependence on $d$ is in the
overall factor $\mu^\epsilon$.
The factorization formula (\ref{eikfac}) can be derived in a simple way
by working in a physical gauge and using the following
{\em soft-gluon insertion rules}. The coupling of the gluon to any
{\em internal} (i.e. highly off-shell) parton in the amplitude
${\cal M}_{g,a_1,\dots,a_n}(q,p_1,\dots,p_n)$ is not singular in the soft limit;
it can thus be neglected. The soft-gluon coupling to any {\em external}
or, in general, {\em nearly on-shell} parton with colour charge $\bom T$
and momentum $p$ can be factorized by extracting the contribution
$g_S \mu^\epsilon 2 p^\mu {\bom T}$ for the vertex and the contribution
$1/(p+q)^2 \simeq 1/(p^2 + 2p\cdot q)$ for the propagator.
An important property of the eikonal current is current conservation.
Multiplying Eq.~(\ref{eikfac}) by $q^\mu$, we obtain
\begin{equation}
\label{eikcons}
q_\mu {\bom J}^\mu(q) = \sum_{i=1}^{n} {\bom T}_i \;,
\end{equation}
and thus
\begin{equation}
q_\mu {\bom J}^\mu(q) |\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle
= \sum_{i=1}^{n} {\bom T}_i \;|\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle
= 0 \;\;,
\end{equation}
where the last equality follows from colour conservation
as in Eq.~(\ref{cocon}).
Although Eq.~(\ref{eikfac}) is most easily derived in a physical gauge,
the conservation of the eikonal current implies the gauge invariance
of the squared amplitude summed over the soft-gluon polarizations.
Squaring the eikonal current and introducing the gluon
polarization tensor $d_{\mu \nu}(q) = ( - g_{\mu \nu} + {\rm gauge \; terms})$
in Eq.~(\ref{gluav}), we have
\begin{equation}
\label{eikonal2}
\left[ {\bom J}^{\mu}(q) \right]^\dagger \;d_{\mu \nu}(q) \;{\bom J}^{\nu}(q) =
- \sum_{i,j=1}^n \;{\bom T}_i \cdot {\bom T}_j
\;\frac{p_i \cdot p_j}{(p_i \cdot q) (p_j \cdot q)} + \dots \;\;,
\end{equation}
where we have used the fact that the gauge terms in $d_{\mu \nu}(q)$ are due to
longitudinal polarizations proportional either to $q^\mu$ or to $q^\nu$. Thus,
the dots on the right-hand side stand for gauge-dependent contributions that
are proportional to the total colour charge $\sum_{i=1}^{n} {\bom T}_i$ and,
hence, that cancel when they are inserted in
$|\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle$.
Using Eq.~(\ref{eikonal2}), the soft-gluon factorization formula at
${\cal O}(\as)$ for the squared amplitude~is
\begin{eqnarray}
\label{ccfact}
| {\cal M}_{g,a_1,\dots,a_n}(q,p_1,\dots,p_n) |^2 \simeq
- 4 \pi \as \mu^{2\epsilon} \sum_{i,j=1}^n\, {\cal S}_{ij}(q)
\;| {\cal M}^{(i,j)}_{a_1,\dots,a_n}(p_1,\dots,p_n) |^2 \;\;,
\end{eqnarray}
where the scalar eikonal function ${\cal S}_{ij}(q)$ for the emission of
a single gluon can be written in terms of two-particle sub-energies
$s_{ij}=(p_i + p_j)^2$ as follows
\begin{equation}
\label{eikfun}
{\cal S}_{ij}(q) = \f{p_i \cdot p_j}{(p_i \cdot q)\, (p_j\cdot q)}
= \f{ 2 s_{ij}}{s_{iq} \,s_{jq}} \;\;.
\end{equation}
The colour correlations produced by soft-gluon emission are taken into
account by the square of the colour-correlated tree-amplitude
$| {\cal M}^{(i,j)} |^2$ on the right-hand side. This is defined by
\begin{eqnarray}
\label{colam}
| {\cal M}^{(i,j)}_{a_1,\dots,a_n}(p_1,\dots,p_n) |^2 \!\!\!&\equiv&\!\!\!
\langle \,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \,|
\,{\bom T}_i \cdot {\bom T}_j
\,|\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle
\\
\!&=&\!\!\! \left[
{\cal M}_{a_1,\dots,a_n}^{c_1.. b_i ... b_j ... c_n}(p_1,\dots,p_n) \right]^*
\; T_{b_id_i}^c \, T_{b_jd_j}^c
\; {\cal M}_{a_1,\dots,a_n}^{c_1.. d_i ... d_j ... c_n}(p_1,\dots,p_n) \;,
\nonumber
\end{eqnarray}
where the sum over the spin indices is understood.
The soft-gluon factorization formula is often presented [\ref{glover}]
in an equivalent way by decomposing the matrix element in terms of
colour subamplitudes [\ref{mangano}]. In this formalism, the eikonal function
${\cal S}_{ij}(q)$ in Eq.~(\ref{eikfun}) controls the factorization properties
of the square of the colour-connected subamplitudes.
\subsection{Emission of a soft $q{\bar q}$-pair}
\label{secsoftqq}
We now consider the tree-level matrix element
${\cal M}_{q,{\bar q},a_1,\dots,a_n}(q_1,q_2,p_1,\dots,p_n)$ when the momenta
$q_1$ and $q_2$ of the quark $q$ and the antiquark ${\bar q}$ become soft
($q_1,q_2\rightarrow 0$ at
fixed $q_1/q_2$). In this limit the matrix element squared has the
dominant behaviour:
\begin{equation}
| {\cal M}_{q,{\bar q},a_1,\dots,a_n}(q_1,q_2,p_1,\dots,p_n) |^2\sim
\f{1}{(q_1 \cdot q_2)\, [p_i \cdot (q_1+q_2)]\, [p_j \cdot (q_1+q_2)]} \;\;.
\end{equation}
When integrated over the phase space of
the quark-antiquark pair, this behaviour gives rise to a single-logarithmic
soft singularity, in addition to possible single- and double-logarithmic
collinear singularities.
The soft singularity arises when the $q{\bar q}$-pair is produced by the decay
of a gluon that carries the soft momentum $q_1+q_2$ (Fig.~\ref{figsoftqq}).
Thus, using the soft-gluon insertion rules described in Sect.~\ref{secsoftlo},
we can straightforwardly derive the following factorization formula:
\begin{eqnarray}
\label{insqq}
&&| {\cal M}_{q,{\bar q},a_1,\dots,a_n}(q_1,q_2,p_1,\dots,p_n) |^2 \simeq
(4 \pi \mu^{2\epsilon} \as)^2\nonumber\\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\cdot \;
\langle \,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \,|
\; {\bom I}_{(q{\bar q})}(q_1,q_2) \;
| \, {\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \,\rangle
\;,
\end{eqnarray}
where
\begin{eqnarray}
\label{ioper1}
{\bom I}_{(q{\bar q})}(q_1,q_2)&=&
\left[ {\bom J}^\lambda(q_1 + q_2) \right]^\dagger d_{\lambda \mu}(q_1 + q_2)
\; \Pi^{\mu \nu}(q_1,q_2) \; d_{\nu \rho}(q_1 + q_2)
\;{\bom J}^\rho(q_1 + q_2) \\
\label{ioper2}
&=& \left[ {\bom J}_\mu(q_1 + q_2) \right]^\dagger
\; \Pi^{\mu \nu}(q_1,q_2) \;{\bom J}_\nu(q_1 + q_2) +~{\dots} \;\;.
\end{eqnarray}
The insertion operator ${\bom I}_{(q{\bar q})}(q_1,q_2)$ depends on the
colour charges of the fast partons $a_1,\dots,a_n$ and it is given in terms
of the soft-gluon current ${\bom J}^\mu(q_1 + q_2)$ in Eq.~(\ref{eikcur})
and of $\Pi^{\mu\nu}(q_1,q_2)$, which is the $q{\bar q}$-contribution
to the discontinuity of the gluon propagator:
\begin{equation}
\Pi^{\mu\nu}(q_1,q_2) = \frac{T_R}{(q_1 \cdot q_2)^2} \;
\Bigl\{ \;- g^{\mu\nu} q_1 \cdot q_2 + q_1^\mu q_2^\nu + q_2^\mu q_1^\nu
\; \Bigr\} \;\;.
\end{equation}
The dots on the right-hand side of Eq.~(\ref{ioper2}) denote
the gauge-dependent contribution to the insertion operator
${\bom I}_{(q{\bar q})}(q_1,q_2)$.
This term is due to the longitudinal
polarizations (proportional to $(q_1 + q_2)^\alpha$ or $(q_1 + q_2)^\beta$)
of the polarization tensors $d_{\alpha \beta}(q_1 + q_2)$ in
Eq.~(\ref{ioper1}). Since $\Pi^{\mu\nu}(q_1,q_2)$ is transverse
$(q_{1 \mu} \Pi^{\mu\nu}(q_1,q_2) = 0)$ and the soft current
${\bom J}^\mu(q_1 + q_2)$ is conserved (see Eq.~(\ref{eikcons})),
the contribution of the longitudinal
polarizations is either vanishing or proportional to the total colour charge
of the fast partons. Because of colour conservation (see Eq.~(\ref{cocon})),
we thus conclude that the gauge-dependent part of
${\bom I}_{(q{\bar q})}(q_1,q_2)$ does not contribute to the
factorization formula (\ref{insqq}).
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=10truecm
\epsffile{qbarqsoft.eps}\\
\end{tabular}
\end{center}
\caption{\label{figsoftqq}{\em Soft-gluon insertion diagram for the emission
of a soft $q{\bar q}$ pair.
}}
\end{figure}
Inserting Eq.~(\ref{ioper2}) into Eq.~(\ref{insqq}) and performing the
Lorentz algebra, we obtain the final factorization formula
\begin{equation}
\label{qqsoftfac}
| {\cal M}_{q,{\bar q},a_1,\dots,a_n}(q_1,q_2,p_1,\dots,p_n) |^2 \simeq
(4 \pi \mu^{2\epsilon} \as)^2 \,T_R
\sum_{i,j=1}^{n} {\cal I}_{ij}(q_1,q_2) \,
| {\cal M}_{a_1,\dots,a_n}^{(i,j)}(p_1,\dots,p_n) |^2 ,
\end{equation}
where
$| {\cal M}_{a_1,\dots,a_n}^{(i,j)}|^2$ is the colour-correlated tree-amplitude
of Eq.~(\ref{colam}) and the soft function ${\cal I}_{ij}(q_1,q_2)$ is given by
\begin{eqnarray}
\label{Iij1}
{\cal I}_{ij}(q_1,q_2)&=& \f{(p_i \cdot q_1)\, (p_j \cdot q_2)
+ (p_j \cdot q_1)\, (p_i \cdot q_2) - (p_i \cdot p_j)
\,(q_1 \cdot q_2)}{(q_1 \cdot q_2)^2
\,[p_i\cdot (q_1+q_2)]\, [p_j \cdot (q_1+q_2)]} \\
\label{Iij2}
&=& - \;\f{2 (p_i \cdot p_j) \,(q_1 \cdot q_2)
+ [p_i \cdot (q_1 - q_2)]\, [p_j \cdot (q_1 - q_2)]}{2 (q_1 \cdot q_2)^2
\,[p_i\cdot (q_1+q_2)]\, [p_j \cdot (q_1+q_2)]} + \dots \,.
\end{eqnarray}
Note that both the expressions (\ref{Iij1}) and (\ref{Iij2}) can equivalently
be used to compute Eq.~(\ref{qqsoftfac}). The difference between the two
expressions, denoted by the dots on the right-hand side of Eq.~(\ref{Iij2}),
gives a vanishing contribution to the factorization formula (\ref{qqsoftfac})
because of colour conservation (see Eq.~(\ref{cocon})).
\subsection{Soft current for double gluon emission}
\label{secsoftgg}
The limit of QCD tree-amplitudes when the momenta
of two gluons simultaneously become soft was independently
studied by Berends and Giele [\ref{bgdsoft}] and by one of the authors
[\ref{sdsoft}]. The singular behaviour of the matrix elements
can be described in terms of factorization formulae given in terms
of process-independent two-gluon currents acting either on
colour-ordered subamplitudes [\ref{bgdsoft}] or on the colour space of the
hard partons [\ref{sdsoft}].
The formalism of the colour subamplitudes was used in Ref.~[\ref{glover}]
to derive explicit soft-gluon factors for the square of
colour-connected and colour-unconnected subamplitudes. In this section
we recall the formalism and the results of Ref.~[\ref{sdsoft}] and we present
the corresponding factorization formula for the square of the matrix elements.
We consider the tree-level matrix element
${\cal M}_{g,g,a_1,\dots,a_n}(q_1,q_2,,p_1,\dots,p_n)$ when
the momenta $q_1$ and $q_2$ of the two gluons become soft.
The limit is precisely defined by rescaling the gluon momenta by an
overall factor $\lambda$:
\begin{equation}
q_1\rightarrow \lambda q_1 \,, \;\;\;\;\;\; q_2\rightarrow \lambda q_2 \,,
\end{equation}
and then performing the limit $\lambda\rightarrow 0$.
The matrix element thus behaves as
\begin{equation}
{\cal M}_{g,g,\dots}\rightarrow {\cal O}(1/\lambda^2)+\dots
\end{equation}
where the dots stand for less singular contributions as $\lambda\rightarrow 0$.
We are interested in explicitly evaluating the dominant singular term
${\cal O}(1/\lambda^2)$.
Note that the double soft limit is more general (accurate) than the soft limit
in the strong-ordering approximation, that is, when $p_i \gg q_1 \gg q_2$.
The strongly-ordered limit describes only the double-logarithmic
soft singularity of the matrix elements. The double soft limit
reproduces consistently the double-logarithmic behaviour and correctly
evaluates also the single-logarithmic soft singularity.
We denote by $a_1, a_2$ and $\mu_1, \mu_2$ the colour and Lorentz indices
of the two gluons, respectively. In the double soft limit the matrix element
fulfils the following factorization formula~[\ref{sdsoft}]
\begin{equation}
\label{softff2}
\bra{a_1,a_2;\mu_1, \mu_2} \,
{\cal M}_{g,g,a_1,\dots,a_n}(q_1,q_2,p_1,\dots,p_n)\rangle
\simeq g_S^2 \mu^{2\epsilon} \, J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)\,
|\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \rangle \;,
\end{equation}
where the two-gluon soft current $J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)$
is the generalization of the eikonal current in Eq.~(\ref{eikcur}).
The explicit expression of the soft current is~[\ref{sdsoft}]
\begin{eqnarray}
\label{dsoftcur}
J_{a_1a_2}^{\mu_1\mu_2}(q_1,q_2)&=&\sum_{i \neq j}
T^{a_1}_i\f{p_i^{\mu_1}}{p_i \cdot q_1}\;
T_j^{a_2}\f{p_j^{\mu_2}}{p_j\cdot q_2}+\nonumber\\
&+&\sum_i\Bigg[\left(\delta^{a_1a}\, T_i^{a_2}\, \f{p_i^{\mu_2}}{p_i\cdot q_2}
-if^{a_2a_1a}\f{q_1^{\mu_2}}{q_1\cdot q_2}\right)\, T_i^a\,
\f{p_i^{\mu_1}}{p_i\cdot (q_1+q_2)}+\nonumber\\
&+&\left(\delta^{a_2a}\, T_i^{a_1}\, \f{p_i^{\mu_1}}{p_i\cdot q_1}
-if^{a_1a_2a}\f{q_2^{\mu_1}}{q_1\cdot q_2}\right)\,
T_i^a\, \f{p_i^{\mu_2}}{p_i\cdot (q_1+q_2)}+\nonumber\\
&+&\f{1}{2}if^{aa_1a_2}T_i^a\f{g^{\mu_1\mu_2}}{q_1\cdot q_2}
\;\f{p_i\cdot (q_2-q_1)}{p_i\cdot (q_2+q_1)}\Bigg].
\end{eqnarray}
It can be derived by working
in a physical gauge and using the soft-gluon insertion technique described in
Sect.~\ref{secsoftlo}. We have to consider the diagrams in Fig.~\ref{figdsoft}.
The contribution on the first line of Eq.~(\ref{dsoftcur})
comes from the eikonal emission of the two soft gluons from two
different external partons (diagrams $(\rm a)$ in Fig. \ref{figdsoft}).
The first term on
the second and third lines come from the eikonal emission of the two gluons
from the same external parton (diagrams $(\rm b)$ in Fig.~\ref{figdsoft}).
The remaining contributions in Eq.~(\ref{dsoftcur}) are proportional
to $f_{aa_1a_2}$ and originate from the
non-abelian diagrams of Fig.~\ref{figdsoft}~(c). Note that the three-gluon
vertex has to be treated exactly, without introducing any soft approximation.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=14truecm
\epsffile{dsoft.eps}\\
\end{tabular}
\end{center}
\caption{\label{figdsoft}{\em Soft-gluon insertion diagrams used to evaluate
the two-gluon current $J_{a_1a_2}^{\mu_1\mu_2}(q_1,q_2)$.}}
\end{figure}
The two-gluon current in Eq.~(\ref{dsoftcur}) can be recast in the following
equivalent form
\begin{eqnarray}
\label{dsoftcurac}
&&\!\!\!\!\!\!\!\! J_{a_1a_2}^{\mu_1\mu_2}(q_1,q_2)= \frac{1}{2} \left\{
J_{a_1}^{\mu_1}(q_1) \;, J_{a_2}^{\mu_2}(q_2) \right\} \\
&+& i f_{a_1a_2a} \sum_{i=1}^{n} T_i^a \left\{
\frac{p_i^{\mu_1} q_1^{\mu_2} - p_i^{\mu_2}
q_2^{\mu_1}}{(q_1\cdot q_2) \,[p_i\cdot (q_1+q_2)]}
- \frac{p_i\cdot (q_1-q_2)}{2 [p_i\cdot (q_1+q_2)]}
\left[ \frac{p_i^{\mu_1} p_i^{\mu_2}}{(p_i \cdot q_1) (p_i \cdot q_2)}
+ \f{g^{\mu_1\mu_2}}{q_1\cdot q_2} \right] \right\} \;, \nonumber
\end{eqnarray}
where the first term on the right-hand side is the colour anticommutator
of the single-gluon eikonal currents of Eq.~(\ref{eikcur}). This is the
only contribution that survives in the abelian case, where it reduces itself
to the product of two independent single-gluon currents. The second term
on the right-hand side is typical of the non-abelian theory.
Note that, as in the single-gluon case, the expressions (\ref{dsoftcur})
and (\ref{dsoftcurac}) for the two-gluon current do not explicitly depend
on the number $d=4-2\epsilon$ of space-time dimensions. However, because
of the contribution proportional to $g^{\mu_1\mu_2}$ in
$J_{a_1a_2}^{\mu_1\mu_2}$, an explicit dependence on the number
$d-2=2(1-\epsilon)$ of gluon polarizations appears (see Eqs.~(\ref{eik22})
and (\ref{dsoftfun}))
by squaring the factorization formula~(\ref{softff2}).
The current $J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)$ fulfils the following
properties.
\begin{itemize}
\item It is symmetric under the exchange of the two soft gluons,
\begin{equation}
J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)=J^{a_2a_1}_{\mu_2\mu_1}(q_2,q_1) \;.
\end{equation}
\item Its divergence is proportional to the total colour charge
of the hard partons:
\begin{equation}
q_1^{\mu_1} J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2) =
\left( J^{a_2}_{\mu_2}(q_2) \;\delta_{a_1 a} + \frac{i}{2}
f_{a_1a_2a} \frac{q_1^{\mu_2}}{q_1\cdot q_2} \right)
\sum_{i=1}^{n} T_i^a \;\;,
\end{equation}
\begin{equation}
q_2^{\mu_2} J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2) =
\left( J^{a_1}_{\mu_1}(q_1) \;\delta_{a_2 a} + \frac{i}{2}
f_{a_2a_1a} \frac{q_2^{\mu_1}}{q_1\cdot q_2} \right)
\sum_{i=1}^{n} T_i^a \;\;.
\end{equation}
This property is analogous to Eq.~(\ref{eikcons}) for the single-gluon
emission and implies that the two-gluon current is conserved
when it acts on a colour singlet state:
\begin{equation}
q_1^{\mu_1} J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)|\,{\cal M}_{a_1,\dots}(p_1,\dots)
\rangle= q_2^{\mu_2} J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)|
\,{\cal M}_{a_1,\dots}(p_1,\dots)\rangle=0.
\end{equation}
Thus, the factorization formula (\ref{softff2}) is gauge-invariant.
\item In the strong-ordered limit $q_2 \ll q_1$, the third and fourth
lines in Eq.~(\ref{dsoftcur}) give subleading contributions
and the current becomes
\begin{equation}
J_{a_1a_2}^{\mu_1\mu_2}(q_1,q_2) \rightarrow
\left( J_{a_2}^{\mu_2}(q_2) \;\delta_{a_1 a} +
i f_{a_1a_2a} \frac{q_1^{\mu_2}}{q_1\cdot q_2} \right)
J_{a}^{\mu_1}(q_1) \;\;.
\end{equation}
Thus, the current correctly factorizes into the product of the two eikonal
currents corresponding to the iterative application of the leading-order
factorization formula~(\ref{eikfac}).
\end{itemize}
The double soft limit of
$| {\cal M}_{g,g,a_1,\dots,a_n}(q_1,q_2,,p_1,\dots,p_n) |^2$
is obtained by squaring Eq.~(\ref{softff2}) and by summing over the soft-gluon
polarizations. The square of the two-gluon current involves a quite cumbersome
colour algebra. Nonetheless, we find that the final result can be recast in
a relatively simple form:
\begin{eqnarray}
\!\!\! \left[J^{a_1a_2}_{\mu\rho}(q_1,q_2)\right]^\dagger
\,d^{\mu\nu}(q_1)\,d^{\rho\sigma}(q_2)\, J^{a_1a_2}_{\nu\sigma}(q_1,q_2)
&=& \f{1}{2}\left\{{\bom J}^2(q_1) \;, {\bom J}^2(q_2) \right\} \nonumber \\
\label{eik22}
&-& \,C_A\, \sum_{i,j=1}^{n} {\bom T}_i\cdot {\bom T}_j
\;{\cal S}_{ij}(q_1,q_2) + \dots \;,
\end{eqnarray}
where, as in Eq.~(\ref{eikonal2}), the dots stand for gauge-dependent terms.
These are proportional to the total colour charge of the hard partons and,
thus, give a vanishing contribution when
inserted on $|\,{\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n)\rangle$.
The first term on the right-hand side of Eq.~(\ref{eik22}) is the only one that
survives in the abelian case. It is given in terms of the colour anticommutator
of the squares of the single-gluon currents in Eq.~(\ref{eikonal2}).
The second term is proportional to $C_A$ and, hence, is purely non-abelian.
It is given in terms of the two-gluon soft function ${\cal S}_{ij}(q_1,q_2)$:
\begin{eqnarray}
{\cal S}_{ij}(q_1,q_2) &=& \f{(1-\epsilon)}{(q_1 \cdot q_2 )^2} \;
\f{p_i \cdot q_1 \,p_j \cdot \,q_2 + p_i \cdot q_2 \,p_j \cdot \,q_1}
{p_i\cdot (q_1+q_2) \; p_j\cdot (q_1+q_2)} \nonumber \\
\label{dsoftfun}
&-& \f{(p_i \cdot p_j)^2}{2 p_i\cdot q_1 \; p_j\cdot q_2 \;
p_i\cdot q_2 \; p_j\cdot q_1}
\left[ 2 - \f{p_i \cdot q_1 \,p_j \cdot \,q_2 + p_i \cdot q_2 \,p_j \cdot \,q_1}
{p_i\cdot (q_1+q_2) \; p_j\cdot (q_1+q_2)} \right] \\
&+& \f{p_i\cdot p_j}{2 q_1 \cdot q_2}
\left[ \f{2}{p_i\cdot q_1 \,p_j \cdot \,q_2} +
\f{2}{p_j\cdot q_1 \,p_i \cdot \,q_2} \right. \nonumber \\
&-& \left.
\f{1}{p_i\cdot (q_1+q_2) \; p_j\cdot (q_1+q_2)}
\left( 4 +
\f{(p_i \cdot q_1 \,p_j \cdot \,q_2 + p_i \cdot q_2
\,p_j \cdot \,q_1)^2}{p_i\cdot q_1 \; p_j\cdot q_2 \;
p_i\cdot q_2 \; p_j\cdot q_1}
\right) \right] \;\;. \nonumber
\end{eqnarray}
Expression (\ref{dsoftfun}) can also be written as
\begin{eqnarray}
{\cal S}_{ij}(q_1,q_2) &=& {\cal S}_{ij}^{{\rm (s.o.)}}(q_1,q_2)
+ \f{p_i \cdot q_1 \,p_j \cdot \,q_2 + p_i \cdot q_2 \,p_j \cdot \,q_1}
{p_i\cdot (q_1+q_2) \; p_j\cdot (q_1+q_2)}
\left[ \f{(1-\epsilon)}{(q_1 \cdot q_2 )^2} - \f{1}{2}
\;{\cal S}_{ij}^{{\rm (s.o.)}}(q_1,q_2) \right] \nonumber \\
&-&
\f{2 p_i\cdot p_j}{q_1 \cdot q_2 \;p_i\cdot (q_1+q_2) \; p_j\cdot (q_1+q_2) }
\;\;,
\end{eqnarray}
where ${\cal S}_{ij}^{{\rm (s.o.)}}$ is the approximation of the
soft function ${\cal S}_{ij}(q_1,q_2)$ in the strong-ordering
limit (either $q_1 \ll q_2$ or $q_2 \ll q_1$):
\begin{eqnarray}
{\cal S}_{ij}^{{\rm (s.o.)}}(q_1,q_2) =
\f{p_i\cdot p_j}{q_1 \cdot q_2}
\left( \f{1}{p_i\cdot q_1 \,p_j \cdot \,q_2} +
\f{1}{p_j\cdot q_1 \,p_i \cdot \,q_2} \right)
- \f{(p_i \cdot p_j)^2}{p_i\cdot q_1 \; p_j\cdot q_2 \;
p_i\cdot q_2 \; p_j\cdot q_1} \;\;.
\end{eqnarray}
Using Eq.~(\ref{eik22}), we can write the
soft-gluon factorization formula for the square of the matrix element
as follows:
\begin{eqnarray}
\label{dsoftm2}
&&\!\!\!\!\!\!\!\!
| {\cal M}_{g,g,a_1,\dots,a_n}(q_1,q_2,p_1,\dots,p_n) |^2 \simeq
\left( 4 \pi \as \mu^{2\epsilon} \right)^2 \\
&&\!\!\!\!\! \cdot \left[ \f{1}{2} \sum_{i,j,k,l=1}^{n} {\cal S}_{ij}(q_1) \;
{\cal S}_{kl}(q_2) \;
| {\cal M}_{a_1,\dots,a_n}^{(i,j)(k,l)}(p_1,\dots,p_n) |^2
- \,C_A\, \sum_{i,j=1}^{n} {\cal S}_{ij}(q_1,q_2)
| {\cal M}_{a_1,\dots,a_n}^{(i,j)} |^2
\right] \nonumber \;\;,
\end{eqnarray}
where ${\cal S}_{ij}(q)$ is the soft function in Eq.~(\ref{eikfun}) and
$| {\cal M}_{a_1,\dots,a_n}^{(i,j)} |^2$ is the colour-correlated amplitude
in Eq.~(\ref{colam}). We can see that the double soft limit
involves colour correlations that are more cumbersome than those
appearing in the case of single-gluon emission. Indeed, the amplitude
$| {\cal M}_{a_1,\dots,a_n}^{(i,j)(k,l)} |^2$ on the right-hand side
of Eq.~(\ref{dsoftm2}) is defined by
\begin{equation}
| {\cal M}_{a_1,\dots,a_n}^{(i,j)(k,l)}(p_1,\dots,p_n) |^2
\equiv
\langle {\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) |
\left\{{\bom T}_i \cdot {\bom T}_j \;, {\bom T}_k \cdot {\bom T}_l \right\}
| {\cal M}_{a_1,\dots,a_n}(p_1,\dots,p_n) \rangle \;,
\end{equation}
and leads to irreducible correlations among four different hard partons.
The results discussed in this subsection can be
presented in a different manner by using the colour subamplitude formalism.
Considering the projection of Eq.~(\ref{softff2}) onto colour-ordered
subamplitudes, it is straightforward to check that the colour current
$J_{a_1a_2}^{\mu_1\mu_2}(q_1,q_2)$ leads to the colourless current derived by
Berends and Giele (see Eqs.~(3.11) and (3.18) in Ref.~[\ref{bgdsoft}]).
The square of this colourless current is denoted by $S_{iq_1q_2j}$ in
Sect.~5.3 of Ref.~[\ref{glover}] and is related to the soft function
${\cal S}_{ij}(q_1,q_2)$ in Eq.~(\ref{dsoftfun}). More
precisely, using the following relation
\begin{equation}
\label{dsoftvscg}
\sum_{i,j=1}^{n} {\bom T}_i\cdot {\bom T}_j \left[
{\cal S}_{ij}(q_1,q_2) + {\cal S}_{ij}(q_1) \;{\cal S}_{ij}(q_2) \right]
= \f{1}{2} \sum_{i,j=1}^{n} {\bom T}_i\cdot {\bom T}_j \;S_{iq_1q_2j}
+ \dots \;\;,
\end{equation}
the second term on the right-hand side of Eqs.~(\ref{eik22})
and (\ref{dsoftm2}) can equivalently be written in terms
of $S_{iq_1q_2j}$. The contribution denoted by the dots on the right-hand side
of Eq.~(\ref{dsoftvscg}) is proportional to the total colour charge of the hard
partons and, thus, it vanishes when inserted in the factorization formula
(\ref{dsoftm2}).
\subsection{Soft--collinear limit at ${\cal O}(\as^2)$ and at higher orders}
\label{softcoll}
We now consider the tree-level matrix element
${\cal M}_{g,a_1,\dots,a_n}(q,p_1,\dots,p_n)$
in the limit where the momentum $q$ of the gluon becomes soft $( q \rightarrow 0)$
and, at the same time, two partons, say $p_1$ and $p_2$, become
collinear. The collinear region is parametrized as in Eq.~(\ref{clim})
and we are interested in the limit $k_\perp \rightarrow 0$.
Studying this soft--collinear limit we can
neglect $i)$ contributions that are uniformly of ${\cal O}(q)$ when $q \rightarrow 0$,
and $ii)$ contributions that are uniformly of ${\cal O}(k_\perp)$ when
$k_\perp \rightarrow 0$. The terms in class $i)$ are not singular in the soft limit
and their contribution
in the collinear limit can thus be taken into account by supplementing the
results
of this section with the ${\cal O}(\as)$-collinear factorization discussed in
Sect.~\ref{notations}.
Analogously, the terms in class $ii)$ are not
singular in the collinear limit and their contribution in the soft limit can be
taken into account by supplementing the results
of this section with the soft-gluon factorization
formula at ${\cal O}(\as)$ presented in Sect.~\ref{secsoftlo}.
This comment can be summarized in a formal manner by writing the square
of the matrix element as
\begin{equation}
| {\cal M}_{g,a_1,a_2,\dots,a_n}(q,p_1,p_2,\dots,p_n) |^2 =
\frac{1}{s_{12} s_{1q} s_{2q}} \;F(q,p_1,p_2,\dots,p_n) \;\;.
\end{equation}
The first factor on the right-hand side contains the correct scaling
behaviour in the soft and collinear regions. Thus, the soft--collinear
limit is defined by the soft $(q \rightarrow 0)$ and collinear $(k_\perp \rightarrow 0)$
approximations of the function $F(q,p_1,p_2,\dots,p_n)$ at
fixed ratio $q/k_\perp^2$.
To compute the soft--collinear limit we perform first soft approximations and
then collinear approximations.
The singular behaviour of the matrix element in the soft limit (and at fixed
$q/k_\perp^2$) is given by a factorization formula analogous to
Eq.~(\ref{eikfac}), namely
\begin{equation}
\label{coleikfac}
\langle c; \mu |\,{\cal M}_{g,a_1,a_2,\dots,a_n}(q,p_1,p_2,\dots,p_n) \rangle
\simeq g_S \mu^\epsilon
J_{(12)}^{c;\mu}(q)
\; |\,{\cal M}_{a_1,a_2,\dots,a_n}(p_1,p_2,\dots,p_n)\rangle \;,
\end{equation}
but now the soft current $J_{(12)}^{c;\mu}(q)$ is no longer equal to the
eikonal
current in Eq.~({\ref{eikcur}). In fact, since $p_1$ and $p_2$ can become
collinear, the internal partonic line with momentum $p_1+p_2$ in
${\cal M}_{a_1,a_2,\dots,a_n}(p_1,p_2,\dots,p_n)$ is close to the mass shell
$( \,(p_1+p_2)^2 = s_{12} \rightarrow 0)$. Near the mass shell, soft-gluon radiation
from this internal line leads to soft singularities and it cannot be neglected.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=10truecm
\epsffile{softcol.eps}\\
\end{tabular}
\end{center}
\caption{\label{figsc}{\em Soft-gluon insertion diagrams for the soft--collinear
limit.
}}
\end{figure}
The explicit expression of the gluon current $J_{(12)}^{c;\mu}(q)$
can be derived by working in a physical gauge and
using the soft-gluon insertion rules described in
Sect.~\ref{secsoftlo} (Fig.~\ref{figsc}). We find
\begin{eqnarray}
\label{eikcolcur}
{\bom J}_{(12)}^\mu(q) &=& \sum_{i=3}^{n} {\bom T}_i\,\f{p_i^\mu}{p_i\cdot q}
+ \frac{(p_1+p_2)^2}{(p_1+p_2+q)^2}
\left[ {\bom T}_1\,\f{p_1^\mu}{p_1 \cdot q}
+{\bom T}_2\,\f{p_2^\mu}{p_2 \cdot q} \right] \nonumber \\
&+& \left( {\bom T}_1 + {\bom T}_2 \right) \f{2(p_1^\mu+p_2^\mu)}{(p_1+p_2+q)^2}
\;.
\end{eqnarray}
We discuss the three contributions on the right-hand side in turn.
The first contribution comes from the usual
eikonal insertions on the external parton lines $i=3,\dots,n$ (the diagrams
$(\rm a)$ in Fig.~\ref{figsc}).
The second contribution comes from the soft-gluon
emission from the external partons $p_1$ and $p_2$ (diagrams
$(\rm b)$ and $(\rm c)$ in Fig.~\ref{figsc}). The factor in the square bracket
is the usual
contribution from the eikonal vertices and propagators of the
lines $p_1+q$ and $p_2+q$. The factor in front of the square bracket
has the following origin. In Eq.~(\ref{coleikfac}) we have already factorized
the tree amplitude ${\cal M}_{a_1,a_2,\dots,a_n}(p_1,p_2,\dots,p_n)$, which
contains the propagator factor $1/(p_1+p_2)^2$. In diagrams $(\rm b)$ and
$(\rm c)$ of
Fig.~\ref{figsc}, this propagator is instead absent, and it is replaced by the
propagator $1/(p_1+p_2+q)^2$ of the internal line with momentum $p_1+p_2+q$.
Thus, the rescaling propagator factor $(p_1+p_2)^2/(p_1+p_2+q)^2$ has
to be applied to the contribution to the current.
The third contribution is the eikonal factor due to the soft emission
from the nearly on-shell internal line $p_1+p_2+q$ (diagram
$(\rm d)$ in Fig.~\ref{figsc}).
Note that we have neglected diagrams in which $p_1$ and $p_2$ are not produced
by a single line with momentum $p_1+p_2$.
These diagrams are not collinearly singular (see the discussion in
Sect.~\ref{power}) in the physical gauge we are working on.
Note also that, as the eikonal current in Eq.~(\ref{eikcons}), the
soft current in Eq.~(\ref{eikcolcur}) satisfies the property
$q_\mu {\bom J}_{(12)}^\mu(q) = \sum_{i=1}^{n} {\bom T}_i$. The ensuing
current conservation, which follows from Eq.~(\ref{cocon}), guarantees
the gauge invariance of the factorization formula (\ref{coleikfac}).
Using Eq.~(\ref{coleikfac}) we could now perform the collinear limit
of the tree-level matrix element
${\cal M}_{a_1,a_2,\dots,a_n}(p_1,p_2,\dots,p_n)$
on the right-hand side. However, since we are eventually interested
in the soft--collinear limit of the square of the matrix element
${\cal M}_{g,a_1,\dots,a_n}(q,p_1,\dots,p_n)$,
this procedure is not convenient for two reasons. First, we have to introduce
collinear splitting functions for the various colour subamplitudes that
contribute to the colour vector
$|\,{\cal M}_{a_1,a_2,\dots,a_n}(p_1,p_2,\dots,p_n)\rangle$. These
splitting functions differ from the Altarelli--Parisi splitting functions
of Sect.~\ref{notations}
(roughly speaking, the former are the square root
of the latter) and, although they are well known [\ref{mangano}],
we shall show that they are not really necessary for the final result.
Secondly, the colour-charge transformation produced by the soft current
in Eq.~(\ref{coleikfac}) implies a non-trivial relation between the
colour-subamplitude decomposition of the matrix element on the left-hand side
and the corresponding decomposition for the matrix element on the right-hand
side. This non-trivial relation complicates the colour structure and
leads to mixed soft--collinear splitting functions [\ref{glover}], whose
introduction can instead be avoided or, at least, simplified.
In other words, if we square the right-hand side of Eq.~(\ref{coleikfac}),
the soft current ${\bom J}_{(12)}$ produces non-trivial colour correlations
of the type ${\bom T}_1 \cdot {\bom T}_i$ or ${\bom T}_2 \cdot {\bom T}_i$
(with $i=3,\dots,n$) between
${\cal M}(p_1,p_2,\dots,p_n)$ and ${\cal M}^\dagger(p_1,p_2,\dots,p_n)$.
Thus, we cannot perform the collinear limit $k_\perp \rightarrow 0$ by simply using the
known ${\cal O}(\as)$ results of Sect.~\ref{notations}
for the colour-summed
squared amplitude $|{\cal M}(p_1,p_2,\dots,p_n)|^2$.
The whole procedure can be simplified by exploiting the QCD {\em coherence}
properties of soft-gluon emission. We rewrite Eq.~(\ref{eikcolcur})
by splitting the soft current in two terms as follows:
\begin{equation}
\label{eikcoh}
{\bom J}_{(12)}^\mu(q) =
\sum_{i=3}^{n} {\bom T}_i\,\f{p_i^\mu}{p_i\cdot q}
+ \left( {\bom T}_1 + {\bom T}_2 \right) \f{p_1^\mu+p_2^\mu}{(p_1+p_2) \cdot q}
+ \delta {\bom J}_{(12)}^\mu(q) \;,
\end{equation}
where
\begin{equation}
\label{delj}
\delta {\bom J}_{(12)}^\mu(q) =
\frac{(p_1+p_2)^2}{(p_1+p_2+q)^2}
\left[ {\bom T}_1\,\f{p_1^\mu}{p_1 \cdot q}
+{\bom T}_2\,\f{p_2^\mu}{p_2 \cdot q}
- \left( {\bom T}_1 + {\bom T}_2 \right) \f{p_1^\mu+p_2^\mu}{(p_1+p_2) \cdot q}
\right] \;.
\end{equation}
The two terms, $\delta {\bom J}_{(12)}$ and the other contribution
on the right-hand side of Eq.~(\ref{eikcoh}), are separately conserved
and, thus, the decomposition in Eq.~(\ref{eikcoh}) does not spoil the gauge
invariance.
Then we note that each of the two terms in Eq.~(\ref{eikcoh}) has the correct
scaling behaviour of ${\cal O}(1/q)$ when $q \rightarrow 0$. Their collinear behaviour
is nonetheless quite different. Performing the limit $k_\perp \rightarrow 0$ at fixed
$k_\perp^2/q$ in Eq.~(\ref{delj}), the propagator factor
$(p_1+p_2)^2/(p_1+p_2+q)^2$ is of ${\cal O}(1)$ but the term in the
square bracket is of ${\cal O}(k_\perp/q)$. Thus, the contribution of
$\delta {\bom J}_{(12)}$ to the soft current ${\bom J}_{(12)}$ is suppressed
by a relative factor of ${\cal O}(k_\perp)$ in the collinear region, and
it can be neglected in the soft--collinear limit.
We conclude that in the factorization formula (\ref{coleikfac}) we can
consistently use the following approximation for the soft current
in Eq.~(\ref{eikcolcur}):
\begin{equation}
\label{eikcohfin}
{\bom J}_{(12)}^\mu(q) \simeq \sum_{i=3}^{n} {\bom T}_i\,\f{p_i^\mu}{p_i\cdot q}
+ {\bom T}_{(12)} \f{p_1^\mu+p_2^\mu}{(p_1+p_2) \cdot q} \;\;,
\end{equation}
where ${\bom T}_{(12)} = {\bom T}_1 + {\bom T}_2$. The subdominant effect of
$\delta {\bom J}_{(12)}$ is due to the cancellation between the different
contributions
in the square bracket on the right-hand side of Eq.~(\ref{delj}). The
cancellation is a typical consequence of colour coherence. When the parton
momenta $p_1$ and $p_2$ become collinear, they radiate soft gluon in
a coherent way, i.e. as a single parton with momentum $p_1+p_2$
and colour charge ${\bom T}_{(12)} = {\bom T}_1 + {\bom T}_2$ (see the last
term in Eq.~(\ref{eikcohfin})).
The expression in Eq.~(\ref{eikcohfin}) is certainly simpler than that in
Eq.~(\ref{eikcolcur}). More importantly, it depends on the colour charge
${\bom T}_{(12)}$ rather than separately on the colour charges
${\bom T}_1$ and ${\bom T}_2$. This implies that, when we square the amplitude
in Eq.~(\ref{coleikfac}), the partons $p_1$ and $p_2$ are no longer
colour-correlated, and the collinear limit $k_\perp \rightarrow 0$ can be performed by
using the collinear factorization formula (\ref{cfac}). We obtain the final
soft--collinear factorization formula:
\begin{eqnarray}
\label{scfac}
&&\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
| {\cal M}_{g,a_1,a_2,\dots,a_n}(q,p_1,p_2,\dots,p_n) |^2 \simeq
- \frac{2}{s_{12}} \; (4 \pi \mu^{2\epsilon} \as)^2 \nonumber \\
&& \;\;\; \;\;\;\cdot \;
\langle \,{\cal M}_{a,\dots,a_n}(p,\dots,p_n) \,|
\, {\hat {\bom P}}_{a_1 a_2} \,
\left[{\bom J}_{(12) \mu}^{\dagger}(q) \,{\bom J}_{(12)}^{\mu}(q) \right]
| \, {\cal M}_{a,\dots,a_n}(p,\dots,p_n) \,\rangle
\;,
\end{eqnarray}
where the matrix elements on the right-hand side are obtained by removing
the soft gluon $q$ and by replacing the partons $a_1$ and $a_2$ by the single
parton $a$ that leads to the collinear splitting process $a \rightarrow a_1 + a_2$.
Since these matrix elements are vectors in the colour+helicity space,
both spin and colour correlations are present in Eq.~(\ref{scfac}).
The spin correlations are exactly the same as in Eq.~(\ref{cfac}).
The spin indices $s, s'$ of the parent parton $a$ are correlated by the
Altarelli--Parisi splitting functions
${\hat {\bom P}}_{a_1 a_2} \equiv {\hat P}_{a_1 a_2}^{ss'}(z,k_\perp;\epsilon)$
in Eqs.~(\ref{hpqqep})--(\ref{hpggep}).
The colour correlations affect all the partons and are
analogous to those in Eq.~(\ref{ccfact}). They
are produced by the square of the soft current:
\newpage
\begin{eqnarray}
\label{j12s}
{\bom J}_{(12) \mu}^{\dagger}(q) \,{\bom J}_{(12)}^{\mu}(q) &=&
\sum_{i,j=3}^n \;{\bom T}_i \cdot {\bom T}_j
\;\frac{p_i \cdot p_j}{(p_i \cdot q) (p_j \cdot q)}
+ 2 \sum_{i=3}^n \; {\bom T}_i \cdot {\bom T}_{(12)}
\;\frac{p_i \cdot (p_1+p_2)}{(p_i \cdot q) (p_1+p_2) \cdot q} \nonumber \\
&+& {\bom T}_{(12)}^2 \;\frac{(p_1+p_2)^2}{((p_1+p_2)\cdot q)^2} \\
\label{j12sapp}
&\simeq& \sum_{i,j=3}^n \;{\bom T}_i \cdot {\bom T}_j \;{\cal S}_{ij}(q)
+ 2 \sum_{i=3}^n \; {\bom T}_i \cdot {\bom T}_{(12)}
\;{\cal S}_{i \,(12)}(q) \;\;.
\end{eqnarray}
In Eq.~(\ref{j12sapp}) we have neglected the last term on the right-hand side
of Eq.~(\ref{j12s}), because it is not collinearly singular.
We have also introduced the eikonal functions ${\cal S}_{ij}(q)$
of Eq.~(\ref{eikfun}) and the analogous eikonal function
${\cal S}_{i \,(12)}(q)$,
\begin{equation}
\label{eikfunc}
{\cal S}_{i \,(12)}(q) = \frac{2 (s_{i1}+ s_{i2})}{s_{iq} (s_{1q}+ s_{2q})}
\;\;.
\end{equation}
From Eqs.~(\ref{scfac}) and (\ref{j12sapp}) we can see that the soft--collinear
limit at ${\cal O}(\as^2)$ is simply and fully described in terms of the same
factors, namely, soft eikonal functions and Altarelli--Parisi splitting
functions, which control the soft and collinear limits at ${\cal O}(\as)$,
respectively.
The soft--collinear limit at ${\cal O}(\as^2)$ was first studied by Campbell
and Glover [\ref{glover}]. They neglected spin correlations and considered
the singular behaviour of the colour subamplitudes. This behaviour, which was
extracted by directly performing the singular limit of known squared matrix
elements, was given in terms of two different factors. The first factor
(see Sect.~4.4 in Ref.~[\ref{glover}]) refers to subamplitudes in which the
collinear partons are not colour-connected and it corresponds exactly to the
${\cal S}_{ij}(q)$-term in Eq.~(\ref{j12sapp}). The second factor regards
the subamplitudes in which the collinear partons $p_1$ and $p_2$ are
colour-connected. This factor is given in Sect.~5.2 of Ref.~[\ref{glover}]
and it can be written as
\begin{eqnarray}
S_{i; \,q12} &=& \f{2(s_{i1}+s_{i2})}{s_{iq} \;s_{1q}}
\left[ z + \f{s_{1q} + zs_{12}}{s_{12q}} \right] \; \\
\label{sscid}
&=&\f{2(s_{i1}+s_{i2})}{s_{iq}}
\left\{ \f{2}{s_{1q}+s_{2q}}+\left(1+\f{s_{12}}{s_{12q}}\right)
\left[\f{z}{s_{1q}}-\f{1}{s_{1q}+s_{2q}}\right]\right\} \\
\label{sscapp}
&\simeq& \f{4(s_{i1}+s_{i2})}{s_{iq} (s_{1q}+s_{2q})} \;\;.
\end{eqnarray}
Since $z$ is the longitudinal momentum fraction carried by $p_1$ in the
collinear region, the term in the square bracket of
Eq.~(\ref{sscid}) vanishes in the collinear limit and Eq.~(\ref{sscapp})
follows. This simplification, which is due to colour coherence, was not
performed in Ref.~[\ref{glover}]. Taking it into account, we have
$S_{i;\,q12} \simeq 2 {\cal S}_{i \,(12)}(q)$, which, when inserted in
Eq.~(\ref{j12sapp}), shows the equivalence of our results with those of
Ref.~[\ref{glover}].
Our derivation of the soft--collinear factorization formula (\ref{scfac})
can straightforwardly be extended to higher orders. We can consider
the limit where a single gluon with momentum $q$ becomes soft and,
at the same time, $m$ partons, say $p_1, \dots , p_m$,
become simultaneously collinear (see Sect.~\ref{power}). In this limit the
factorization formula is
\begin{eqnarray}
\label{smcfac}
&&\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!
| {\cal M}_{g,a_1,\dots,a_m,\dots,a_n}(q,p_1,\dots,p_m,\dots,p_n) |^2 \simeq
- \;4 \pi \mu^{2\epsilon} \as
\left( \frac{8 \pi \mu^{2\epsilon} \as}{s_{1 \dots m}}\right)^{m-1}
\nonumber \\
&& \!\!\! \!\!\!\cdot \;
\langle \,{\cal M}_{a,\dots,a_n}(xp,\dots,p_n) \,|
\, {\hat {\bom P}}_{a_1 \dots a_m} \,
\left[{\bom J}_{(1 \dots m) \mu}^{\dagger}(q)
\,{\bom J}_{(1 \dots m)}^{\mu}(q) \right]
| \, {\cal M}_{a,\dots,a_n}(xp,\dots,p_n) \,\rangle
\;,
\end{eqnarray}
where ${\hat {\bom P}}_{a_1 \dots a_m} \equiv
{\hat P}_{a_1 \dots a_m}^{s s'}$ is the spin-dependent splitting
function in Eq.~(\ref{ccfacm}), and the soft current
${\bom J}_{(1 \dots m)}(q)$ is:
\begin{equation}
\label{eikmcohfin}
{\bom J}_{(1 \dots m)}^\mu(q) \simeq
\sum_{i=m+1}^{n} {\bom T}_i\,\f{p_i^\mu}{p_i\cdot q}
+ {\bom T}_{(1 \dots m)} \f{p_1^\mu+\dots p_m^\mu}{(p_1+\dots +p_m) \cdot q}
\;\;,
\end{equation}
with ${\bom T}_{(1 \dots m)} = {\bom T}_1 + \dots + {\bom T}_m$.
Squaring the soft current as in Eqs.~(\ref{j12s}) and (\ref{j12sapp}), we obtain
\begin{equation}
\label{j1ms}
{\bom J}_{(1 \dots m) \mu}^{\dagger}(q) \,{\bom J}_{(1 \dots m)}^{\mu}(q)
\label{j12smapp}
\simeq \sum_{i,j=m+1}^n \;{\bom T}_i \cdot {\bom T}_j \;{\cal
S}_{ij}(q)
+ 2 \sum_{i=m+1}^n \; {\bom T}_i \cdot {\bom T}_{(1 \dots m)}
\;{\cal S}_{i \,(1 \dots m)}(q) \;\;,
\end{equation}
where ${\cal S}_{i \,(1 \dots m)}(q) =
2 (s_{i1}+ \dots + s_{im})/[ s_{iq} (s_{1q}+ \dots + s_{mq})]$.
The proof of these results is very simple.
The soft current ${\bom J}_{(1 \dots m)}(q)$ is derived by using the soft-gluon
insertion rules as in Eq.~(\ref{eikcolcur}). Then, the coherence argument
used in Eqs.~(\ref{eikcoh}) and (\ref{delj}) can iteratively be applied to
any vertex in the $m$-parton dispersive amplitude
${\cal V}_{a_1 \dots a_m}$ of Eq.~(\ref{fcollgen}). This leads to the
expression in Eq.~(\ref{eikmcohfin}).
\subsection{Multiple soft and soft--collinear limits}
\label{multilim}
In Sects.~(\ref{secsoftlo}) and (\ref{secsoftgg}) we have discussed in
detail single and double soft-gluon emission. The generalization to multiple
soft-gluon radiation is straightforward. If we consider the matrix
element ${\cal M}_{g,\dots,g,a_1,\dots,a_n}(q_1,\dots,q_k,p_1,\dots,p_n)$
when the $k$ gluons with momenta $q_1,\dots,q_k$ become soft
simultaneously, we can still write a factorization formula similar to
Eq.~(\ref{softff2}) by performing the simple replacement
\begin{equation}
\label{softgen}
g_S^2 \mu^{2\epsilon} \, J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)
\rightarrow
\left( g_S \mu^{\epsilon} \right)^k \,
J^{a_1\dots a_k}_{\mu_1\dots \mu_k}(q_1,\dots,q_k) \;\;,
\end{equation}
where $a_1, \dots, a_k$ and $\mu_1, \dots, \mu_k$ denote the colour and
Lorentz indices of the soft gluons.
As the two-gluon current $J^{a_1a_2}_{\mu_1\mu_2}(q_1,q_2)$
in Eq.~(\ref{dsoftcur}), the multigluon current on the right-hand side of
Eq.~(\ref{softgen}) is obtained by working in a physical gauge and using
the soft-gluon insertion rules described in Sect.~\ref{secsoftlo}. Of course,
the explicit expression of ${\bom J}(q_1,\dots,q_k)$ turns out to be quite
involved, because all the possible interactions between the soft gluons have to
be included without using any soft approximation.
It is also clear that the soft-gluon insertion rules can be used to
derive a factorization formula analogous to Eq.~(\ref{coleikfac}) for the
multiple soft--collinear limit in which $k$ gluons are soft and
$m$ partons become collinear simultaneously. More importantly, it is worth while
pointing out that the {\em coherence} argument leading to Eqs.~(\ref{scfac})
and (\ref{smcfac}) still applies.
Thus, the factorization formula can be written as
\begin{eqnarray}
\label{scfacgen}
&&\!\!\!\!\!\!\!\!\!\!
| {\cal M}_{g,\dots,g,a_1,\dots,a_m,\dots,a_n}(q_1,\dots,q_k,p_1,\dots,p_m,
\dots,p_n) |^2 \simeq
\left( - 4 \pi \mu^{2\epsilon} \as\right)^k
\left( \frac{8 \pi \mu^{2\epsilon} \as}{s_{1 \dots m}}\right)^{m-1}
\\
&& \!\!\! \!\!\!\!\!\!\!\cdot \;
\langle \,{\cal M}_{a,\dots,a_n}(xp,..,p_n) \,|
\, {\hat {\bom P}}_{a_1 \dots a_m} \,
\left[{\bom J}_{(1 \dots m)}^{\dagger}(q_1,\dots,q_k)
\,{\bom J}_{(1 \dots m)}(q_1,\dots,q_k) \right]
| \, {\cal M}_{a,\dots,a_n}(xp,..,p_n) \,\rangle \nonumber
\;.
\end{eqnarray}
Equation (\ref{scfacgen}) does not involve any additional factor with respect
to those that are necessary to deal with the multiple collinear and multiple
soft limits separately. The spin-dependent splitting function
${\hat {\bom P}}_{a_1 \dots a_m} \equiv {\hat P}_{a_1 \dots a_m}^{s s'}$
is exactly the same as that in Eq.~(\ref{ccfacm}). The current
${\bom J}_{(1 \dots m)}(q_1,\dots,q_k)$ is completely analogous to the soft
current ${\bom J}(q_1,\dots,q_k)$ in Eq.~(\ref{softgen}). As the latter,
the former is constructed by inserting the soft gluons only on the
$(1+n-m)$ {\em external} parton lines with momenta
$p_1+\dots+p_m, p_{m+1}, \dots, p_n$, and each insertion on the collinear
parton $p_1+\dots+p_m$ is taken into account by the simple eikonal factor
$({\bom T}_1 + \dots + {\bom T}_m) (p_1+\dots+p_m)^\mu/(p_1+\dots+p_m)\cdot q$,
despite the fact that $(p_1+\dots+p_m)^2 \neq 0$ (see e.g.
Eqs.~(\ref{eikmcohfin})).
In the most general case, the infrared singularities of the tree-level
amplitudes
are produced by the multiple collinear decay of hard partons and by the
associated radiation of soft gluons and $q{\bar q}$-pairs. The corresponding
factorization formula can
be constructed in a straightforward manner by using the rules derived and
illustrated throughout the paper. Using a shorthand symbolic notation, we
have~(Fig.~\ref{general})
\begin{equation}
|{\cal M}|^2\simeq \langle {\cal M}_{hard}|
\, \left(\prod_i {\hat {\bom P}}_i\right) \,{\bom S}\, |{\cal M}_{hard}\rangle \;\;.
\end{equation}
Here ${\cal M}_{hard}$ denotes the factorized amplitude that depends only on the
momenta of the hard partons. The factor $\prod_i {\hat {\bom P}}_i=\prod_i
{\hat P}_i^{s_is_i'}$ is the product of the spin-dependent splitting functions
for the collinear decay of $i=1,\dots,l$ hard partons. The factor ${\bom S}$ is
a colour matrix that takes into account the radiation of soft partons. It has
to be computed exactly at the tree level, but its external {\em gluon} lines
are coupled to the hard partons by using the eikonal approximation as in the
calculation of the current
${\bom J}_{(1 \dots m)}(q_1,\dots,q_k)$ in Eq.~(\ref{scfacgen}). Note that spin
and colour correlations are factorized independently. This decoupling follows
from colour coherence.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{c}
\epsfxsize=10truecm
\epsffile{gen1.eps}\\
\end{tabular}
\end{center}
\caption{\label{general}{\em General structure of infrared factorization
at any perturbative order.
}}
\end{figure}
\section{Summary}
\label{summa}
In this paper we have studied the infrared structure of tree-level QCD
amplitudes in all the possible soft and collinear limits.
We have first considered the collinear behaviour.
We have shown that, in the limit in which $m$ partons become parallel
simultaneously, the singularities are given by
the universal factorization formula (\ref{ccfacm}) and are controlled by
process-independent splitting functions that generalize the customary
Altarelli--Parisi splitting functions. These splitting functions fully
take into account the azimuthal correlations produced in the collinear
decay. We have presented a recipe to compute the splitting functions at
any perturbative order and we have performed their explicit calculation
at ${\cal O}(\as^2)$.
Then we have studied the soft behaviour and shown how to construct
soft factorization formulae at any order in $\as$. We have considered
the limit in which a $q{\bar q}$ pair becomes soft and we have computed
the corresponding singularity at ${\cal O}(\as^2)$ in terms of a simple
universal insertion factor. We have then recalled the known results about
the limit in which two gluons become soft. This limit is controlled by an
${\cal O}(\as^2)$ soft current that is tensor in colour space and
generalizes the eikonal current at ${\cal O}(\as)$. We have obtained
a compact expression for the square of the two-gluon current that, in
particular, shows the absence of colour correlations in the case of
four- and five-parton amplitudes.
Finally, we have studied the mixed soft--collinear limit and pointed out
that its description does not require the introduction of new infrared
factors. Exploiting the coherence property of soft gluon radiation,
we have been able to show that
the singularities are given by a factorization formula written only in terms
of the soft currents and of the splitting functions that control the soft
and collinear limits, respectively.
These results are one of the necessary ingredients to extend QCD predictions
at higher perturbative orders. In particular, our calculation of the
${\cal O}(\as^2)$ singular factors is relevant to setting up general methods to
compute QCD jet cross sections at NNLO.
\noindent {\bf Acknowledgements}. \\
\noindent
We would like to thank Daniel de Florian and Zoltan Kunszt for discussions.
One of us (M.G.) would like to thank the
Fondazione `Angelo della Riccia' and the INFN for financial support at
earlier stages of this work.
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\section*{Appendix: Soft limits of four- and five-parton amplitudes}
\label{appendixa}
In general,
soft factorization formulae involve colour correlations.
As shown in Eqs.~(\ref{qqsoftfac}), (\ref{dsoftm2}) and (\ref{j12sapp}), at
${\cal O}(\as^2)$ the correlations are completely given in terms of
products of colour-charge factors ${\bom T}_i\cdot{\bom T}_j$.
In this appendix we collect the factorization formulae for the ${\cal O}(\as^2)$
soft (and soft--collinear) limits of the square of the
matrix elements with four and five partons plus an arbitrary number of
colourless particles. In these particular cases, using
colour conservation, it is possible (see e.g. the Appendix~A in
Ref.~[\ref{CSdipole}]\, ) to express the products
${\bom T}_i\cdot{\bom T}_j$ in terms of the Casimir invariants $C_i$
($C_i=C_F$ if $i=q,{\bar q}$ and $C_i=C_A$ if $i=g$)
of the hard partons. Thus the colour algebra completely
factorizes and colour correlations cancel.
Note that two of the hard partons in the four- and five-parton amplitudes
necessarily form a particle--antiparticle pair $a, {\bar a}$. This further
simplifies the combinations of Casimir invariants that appear in the
factorization formulae.
\noindent {\bf Emission of a soft $q{\bar q}$-pair}
We consider the four-parton amplitude
${\cal M}_{q,{\bar q},a,{\bar a}}(q_1,q_2,p_1,p_2)$ in the limit in
which
\newline
$q_1,q_2\rightarrow~0$. From Eq.~(\ref{qqsoftfac}), we get
\begin{equation}
\label{qq4}
\!\!\!\!\!\! |{\cal M}_{q,{\bar q},a,{\bar a}}\,(q_1,q_2,p_1,p_2)|^2 \simeq
(4 \pi\mu^{2\epsilon} \as)^2\, T_R\; C_a
\; \Bigg( \;{\cal I}_{11} +
{\cal I}_{22} -2\,{\cal I}_{12} \,\Bigg) \;
|{\cal M}_{a,{\bar a}}\,(p_1,p_2)|^2 \;.
\end{equation}
In the case of five partons we get
\begin{eqnarray}
\label{qq5}
|{\cal M}_{q,{\bar q},a,{\bar a},a_3}\,(q_1,q_2,p_1,p_2,p_3)|^2
&\simeq& (4 \pi\mu^{2\epsilon} \as)^2\,T_R\;
\Bigg[C_a \left( \,{\cal I}_{11}+{\cal I}_{22} - 2 {\cal I}_{12} \right) \\
&+& C_{a_3} \left( \,{\cal I}_{33} + {\cal I}_{12} - {\cal I}_{13}
- {\cal I}_{23}\right) \Bigg]
\;|{\cal M}_{a,{\bar a},a_3}\,(p_1,p_2,p_3)|^2 \,. \nonumber
\end{eqnarray}
The soft function ${\cal I}_{ij}={\cal I}_{ij}(q_1,q_2)$ is given in
Eq.~(\ref{Iij1}).
\noindent {\bf Emission of two soft gluons}
We consider the amplitude ${\cal M}_{g,g,a,{\bar a}}\,(q_1,q_2,p_1,p_2)$ in
the limit in which the two gluons become soft. Using Eq.~(\ref{dsoftm2}),
we get
\begin{eqnarray}
\label{gg4}
|{\cal M}_{g,g,a,{\bar a}}\,(q_1,q_2,p_1,p_2)|^2 \!&\simeq&\!\!
(4 \pi \mu^{2\epsilon} \as)^2\, C_a\; \Bigg[4\,C_a\,{\cal S}_{12}(q_1)
\,{\cal S}_{12}(q_2) + C_A\,\Big( 2\,{\cal S}_{12} - {\cal S}_{11}
-{\cal S}_{22} \Big)\Bigg]\nonumber\\
&\cdot& |{\cal M}_{a,{\bar a}}\,(p_1,p_2)|^2 \,.
\end{eqnarray}
In the case of five partons we get
\begin{eqnarray}
\label{gg5}
&&\!\!\!\!\! \!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!
|{\cal M}_{g,g,a,{\bar a},a_3}\,(q_1,q_2,p_1,p_2,p_3)|^2
\simeq (4 \pi \mu^{2\epsilon} \as)^2 \;
|{\cal M}_{a,{\bar a},a_3}\,(p_1,p_2,p_3)|^2 \nonumber\\
\;\;\;\;\;\;&\cdot&\!\!\!\!\! \Bigg\{ \;\Bigg[
\left( 2 C_a - C_{a_3} \right) {\cal S}_{12}(q_1)
+ C_{a_3} \left( {\cal S}_{13}(q_1) + {\cal S}_{23}(q_1)
\right) \Bigg] \nonumber\\
\;\;\;\;\;\;\;\;\; \;\;&\cdot&
\Bigg[ \left( 2 C_a - C_{a_3} \right) {\cal S}_{12}(q_2)
+ C_{a_3} \left( {\cal S}_{13}(q_2) + {\cal S}_{23}(q_2)
\right) \Bigg] \\
\;\;\;\;\;\;&+&\!\!\!C_A\,\Bigg[\,C_a \left( 2 {\cal S}_{12} - {\cal S}_{11}
-{\cal S}_{22} \right) + C_{a_3} \left( {\cal S}_{13}+{\cal S}_{23}
- {\cal S}_{33} - {\cal S}_{12}
\right) \Bigg] \Bigg\}\,. \nonumber
\end{eqnarray}
The soft functions ${\cal S}_{ij}(q)$ and ${\cal S}_{ij}={\cal S}_{ij}(q_1,q_2)$
are given in Eqs.~(\ref{eikfun}) and (\ref{dsoftfun}), respectively.
\newpage
\noindent {\bf Soft--collinear limit}
We consider the amplitude ${\cal M}_{g,a_1,a_2,a_3}\,(q,p_1,p_2,p_3)$ in the
limit in which $q\rightarrow 0$ and $s_{12}\rightarrow 0$. Using Eq.~(\ref{scfac}), we get
\begin{equation}
\label{sc4}
|{\cal M}_{g,a_1,a_2,a_3}(q,p_1,p_2,p_3)|^2
\simeq \f{4}{s_{12}} (4 \pi \mu^{2\epsilon} \as)^2\,C_{a_3}\,{\cal S}_{3(12)}(q)
\;{\hat P}_{a_1a_2}^{ss'} \;{\cal T}_{aa_3}^{ss'}(xp,p_3) \;\;.
\end{equation}
In the case of five partons we get
\begin{eqnarray}
\label{sc5}
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
|{\cal M}_{g,a_1,a_2,a_3,a_4}(q_1,p_1,p_2,p_3,p_4)|^2
\simeq \f{2}{s_{12}} (4 \pi \mu^{2\epsilon} \as)^2
\;{\cal T}_{aa_3a_4}^{ss'}(xp,p_3,p_4) \;{\hat P}^{ss'}_{a_1a_2} \\
&&\!\!\!\!\!\!\!\!\!\cdot \Bigg[ (C_{a_3}+C_{a_4}-C_{a})\,{\cal S}_{34}(q)
+(C_{a}+C_{a_3}-C_{a_4})\,{\cal S}_{3(12)}(q)
+(C_{a}+C_{a_4}-C_{a_3})\,{\cal S}_{4(12)}(q)\Bigg] \;. \nonumber
\end{eqnarray}
Here $a$ denotes the parton that decays collinearly, $a \rightarrow a_1 a_2$,
${\cal T}_{a \dots}^{ss'}(xp,\dots)$ is the spin-polarization tensor in
Eq.~(\ref{melspindef}) and ${\hat P}^{ss'}_{a_1a_2}$ is the spin-dependent
splitting function in Eq.~(\ref{cfac}).
The soft functions ${\cal S}_{ij}(q)$ and ${\cal S}_{i(12)}(q)$
are given in Eqs.~(\ref{eikfun}) and (\ref{eikfunc}), respectively.
\section*{References}
\def\ac#1#2#3{Acta Phys.\ Polon.\ #1 (19#3) #2}
\def\ap#1#2#3{Ann.\ Phys.\ (NY) #1 (19#3) #2}
\def\ar#1#2#3{Annu.\ Rev.\ Nucl.\ Part.\ Sci.\ #1 (19#3) #2}
\def\cpc#1#2#3{Computer Phys.\ Comm.\ #1 (19#3) #2}
\def\ib#1#2#3{ibid.\ #1 (19#3) #2}
\def\np#1#2#3{Nucl.\ Phys.\ B#1 (19#3) #2}
\def\pl#1#2#3{Phys.\ Lett.\ #1B (19#3) #2}
\def\pr#1#2#3{Phys.\ Rev.\ D #1 (19#3) #2}
\def\prep#1#2#3{Phys.\ Rep.\ #1 (19#3) #2}
\def\prl#1#2#3{Phys.\ Rev.\ Lett.\ #1 (19#3) #2}
\def\rmp#1#2#3{Rev.\ Mod.\ Phys.\ #1 (19#3) #2}
\def\sj#1#2#3{Sov.\ J.\ Nucl.\ Phys.\ #1 (19#3) #2}
\def\zp#1#2#3{Z.\ Phys.\ C#1 (19#3) #2}
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R.K.\ Ellis, W.J.\ Stirling and B.R.\ Webber, {\it QCD and collider
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G.\ Altarelli and G.\ Parisi, \np{126}{298}{77}.
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A.\ Bassetto, M.\ Ciafaloni and G.\ Marchesini, \prep{100}{201}{83};
Yu.L.~Dokshitser, V.A.\ Khoze, A.H.\ Mueller and S.I. Troian,
{\it Basics of Perturbative QCD} (Editions Fronti\`eres, Gif-sur-Yvette, 1991)
and references therein.
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M.L.\ Mangano and S.J.\ Parke, \prep{200}{301}{91}
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S.\ Catani and M.H.\ Seymour, \np{485}{291}{97}
(Erratum {\it ibid.} B510 (1998) 503).
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D.A.\ Kosower, \pr{57}{5410}{98}.
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W.T. Giele and E.W.N. Glover, \pr{46}{1980}{92}.
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Z.\ Kunszt, A.\ Signer and Z. Tr\'ocs\'anyi, \np{420}{550}{94}.
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S.\ Catani and M.H.\ Seymour, \pl{378}{287}{96}.
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Z.\ Bern, L.\ Dixon and D.A.\ Kosower, \ar{46}{109}{96} and references therein.
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W.T. Giele, E.W.N. Glover and D.A. Kosower, \np{403}{633}{93};
S.\ Keller and E.\ Laenen, \pr{59}{114004}{99}.
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Z.\ Kunszt and D.E.\ Soper, \pr{46}{192}{92};
S.\ Frixione, Z.\ Kunszt and A.\ Signer, \np{467}{399}{96};
Z. Nagy and Z. Tr\'ocs\'anyi, \np{486}{189}{97};
S.\ Frixione, \np{507}{295}{97}.
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R.J.\ Gonsalves, \pr{28}{1542}{83};
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T.\ Matsuura, S.C.\ van der Marck and W.L.\ van Neerven, \np{319}{570}{89}.
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V.A. Smirnov, hep-ph/9905323;
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See, for instance, Z.\ Bern, J.S.\ Rozowsky and B.\ Yan, \pl{401}{273}{97};
C.\ Anastasiou, E.W.N.\ Glover and C.\ Oleari, preprint DTP/99/80
(hep-ph/9907494); and references therein.
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S.\ Catani, \pl{427}{161}{98}.
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Z.\ Bern, V.\ Del Duca and C.R.\ Schmidt, \pl{445}{168}{98};
Z.\ Bern, V.\ Del Duca, W.B. Kilgore and C.R.\ Schmidt,
preprint BNL-HET-99-6 (hep-ph/9903516).
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D.A.\ Kosower, \np{552}{319}{99};
D.A.\ Kosower and P.\ Uwer, preprint SACLAY-SPHT-T99-032
(hep-ph/9903515).
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S.\ Catani, in Proceedings of the Workshop on {\it New Techniques for
Calculating Higher Order QCD Corrections}, report ETH-TH/93-01, Zurich (1992).
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J.M.\ Campbell and E.W.N. Glover, \np{527}{264}{98}.
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S.\ Catani and M.\ Grazzini, \pl{446}{143}{99}.
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D.\ Amati, R.\ Petronzio and G.\ Veneziano, \np{140}{54}{78},
\np{146}{29}{78}; R.K.\ Ellis, H.\ Georgi, M.\ Machacek, H.D.\ Politzer and
G.G.\ Ross, \pl{78}{281}{78}, \np{152}{285}{79}.
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J.\ Kalinowski, K.\ Konishi and T.R.\ Taylor, \np{181}{221}{81};
J.\ Kalinowski, K.\ Konishi, P.N.\ Scharbach and T.R.\ Taylor,
\np{181}{253}{81}; J.F.\ Gunion, J.\ Kalinowski and L.\ Szymanowski,
\pr{32}{2303}{85}.
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B.I.\ Ermolaev and V.S.\ Fadin, JETP Lett. 33 (1981) 269.
\item \label{KUV}
K.\ Konishi, A.\ Ukawa and G.\ Veneziano, \np{157}{45}{79}.
\item \label{softrev}
G.\ Sterman, in Proc. {\it 10th Topical Workshop on Proton-Antiproton
Collider Physics}, eds. R.\ Raja and J.\ Yoh (AIP Press, New York, 1996),
p.~608;
S.\ Catani,
in Proc. of the {\it 32nd Rencontres de Moriond: QCD and High-Energy
Hadronic Interactions}, ed. J. Tran Than Van (Editions Fronti\`eres, Paris,
1997), p.~331 and references therein.
\item \label{cdruv}
G.\ 't Hooft and M.\ Veltman, \np{44}{189}{72};
G.\ Bollini and J.J. Giambia\-gi, Nuovo Cimento 12B (1972) 20;
J.F.\ Ashmore, Nuovo Cimento Lett. 4 (1972) 289;
G.M. Cicuta and E.\ Montaldi, Nuovo Cimento Lett. 4 (1972) 329.
\item \label{cdrir}
R.\ Gastmans and R.\ Meuldermans, \np{63}{277}{73}.
\item \label{schemedep}
S.\ Catani, M.H.\ Seymour and Z.\ Tr\'ocs\'anyi, \pr{55}{6819}{97}.
\item \label{dimred}
W.\ Siegel, \pl{84}{193}{79}; Z.\ Bern and D.A.\ Kosower, \np{379}{451}{92}.
\end{enumerate}
\end{document}
=====
As far as tree level calculations are concerned QCD with one flavour can
be made supersymmetric by setting $C_F=C_A=2T_R$.
However in the regularization scheme used in this paper SUSY is broken because
quark and gluon have different numbers of helicity states.
Nonetheless SUSY is recovered in the $\epsilon\rightarrow 0$ limit and
AP splitting functions fulfil in this limit the following $N=1$ SUSY identity
\begin{equation}
\label{apsusy}
{\hat P}_{q_1g_2}+(1\!\leftrightarrow\! 2)=2{\hat P}_{q_1{\bar q}_2}+{\hat P}_{g_1g_2}
\end{equation}
This
identity means that the total quark and gluon decay probabilities are
the same. Notice that the identity holds without taking the average over
polarizations.
This means that spin correlations cancel out in the right-
hand side of eq. (\ref{apsusy}) and
the identity holds when we identify
$\delta^{ss'}$ with $-g^{\mu\nu}$.
One expects a similar identity to hold for ${\hat P}_{a_1 a_2 a_3}^{s s'}$.
In fact when $\epsilon=0$ we find:
\begin{equation}
\Big[{\hat P}_{{\bar q}_1q_2q_3}+(1\!\leftrightarrow
\! 2)+(1\!\leftrightarrow\! 3)\Big]\! +\!\Big[{\hat
P}_{q_3g_1g_2}+(1\!\leftrightarrow\! 3)+(2\!\leftrightarrow\! 3)\Big]
={\hat P}_{g_1g_2g_3}+\Big[{\hat
P}_{g_3{\bar q}_1q_2}+5\mbox{ permutations}\Big]
\end{equation}
As before the total quark and gluon decay probabilities are
the same and spin-correlations cancel out in the gluon channel. This result
represent an important check of our calculation.
A further check of our results is provided by the strong-ordered limit.
In this limit the three partons become collinear sequentially and, as expected,
${\hat P}_{a_1 a_2 a_3}^{s s'}$ factorize into the product of two
AP splitting functions.
|
1,116,691,500,504 | arxiv | \section{Introduction}
\label{sec:intro}
The possibility of generation of short light pulses by locking the
longitudinal modes of a laser was discussed only a few years after the
development of the laser in 1960. Mode-locking techniques can be classified
into two major classes: (i) active mode-locking, based on an external
modulation at a frequency close to the cavity free spectral range and (ii)
passive mode-locking where an intracavity nonlinear component reduces losses
for pulsed operation with respect to the those of continuous-wave (CW)
regime. A standard theoretical approach to study the properties of
mode-locked devices is based on direct integration of the so-called
{\em traveling wave equations} describing space-time evolution of the electric
field and carrier density in the laser sections \cite%
{Tromborg94,Avrutin00,Bandelow01,Bandelow06}. Another, much simpler approach
limited to small gain and loss approximation was developed by Haus in \cite{Haus00}.
To overcome this limitation the third modelling approach was suggested in
\cite{VT04,Vladimirov,VT05} based on the lumped element method that allows
to derive a delay differential equation (DDE) model for the temporal
evolution of the optical field at some fixed position in the cavity. DDE
models successfully describe the dynamics of multi-section mode-locked
semiconductor lasers \cite%
{VMVB06,RVBHK06,vladimirov2010dynamical,arkhipov,Marconi,ViktorovCoherence},
frequency swept light sources \cite{Slepneva,Slepneva2,pimenovprl},
optically injected lasers \cite{rebrova,pimenov2014}, semiconductor lasers
with feedback \cite{Kelleher,otto,Jaurigue17}, as well as some other
multimode laser devices \cite{ViktorovOL}.
In this work, we propose and study a DDE model for nonlinear amplifying loop mirror (NALM) mode-locked laser.
Nonlinear mirror laser as a device for ultrafast light processing was proposed in \cite{doran1988nonlinear}.
Mode-locked pulse formation mechanism in this laser is based on
the asymmetric nonlinear propagation in a waveguide loop, where two counterpropagating waves aquire different intensity-dependent phase shifts caused by the Kerr nonlinearity.
As a result of the interference of these waves the loop acts as a nonlinear mirror with the reflectivity dependent on the intensity of incident light. Such nonlinear mirror plays a role of a saturable absorber that leads to the appearance of a pulsed laser operation.
We develop a DDE laser model by assuming dispersion-free unidirectional operation inside
the cavity and symmetrical beam splitting of the field into two
counter-propagating fields at the entrance of the NALM. When the material variables are adiabatically eliminated, one obtains a single DDE for the complex electric field envelope, which, despite of its simplicity, gives a good insight on the laser dynamics. Using this equation we find different mode-locked regimes of laser operation including square waves, ultrashort optical pulses and their harmonics. The mode-locked pulses are always bistable with the non-lasing state, and, hence, can also be considered as
temporal cavity solitons or nonlinear localized structures of light \cite{leo2010temporal,Marconi}. The linear stability analysis of CW solutions corresponding to different longitudinal modes of the laser reveals modulational and Turing instabilities, as well as an instability leading to the emergence of square waves.
We demonstrate analytically the mode-locked pulses are asymmetric with exponential decay of the pulse trailing tail in positive time and faster-than-exponential (super-exponential) decay of the leading tail in negative time. We find that the repulsive interaction of such asymmetric pulses leads to a harmonic mode-locking regime. In order to explain the experimentally observed square wave generation recently reported in \cite{Aadhi} in the figure-of-eight laser, we construct a one-dimensional map exhibiting a period doubling route to chaos.
\section{Model equation and CW solutions}
A schematic of a
NALM laser with a gain and a spectral filter in an unidirectional cavity coupled to a bidirectional loop with a second gain medium and a nonlinear element, is shown in Fig.~\ref{fig:scheme}. This scheme corresponds to experimentally implemented setups of mode-locked lasers with a high-Q microring resonator \cite{Kues2017} and an integrated waveguide \cite{Aadhi} acting as nonlinear elements. Using the approach of \cite{Vladimirov,VT05} we write the following DDE model for the time evolution of the electric field amplitude $E$ and saturable gain $g$ in a laser shown in Fig.~\ref{fig:scheme}:
\begin{equation}
\gamma^{-1}\frac{dE}{dt}+E=\sqrt{\kappa_1}e^{g/2+i\theta}{\tilde f}\left[|E(t-T)|^{2},G\right]E(t-T),
\label{eq:A}
\end{equation}
\begin{equation}
\gamma_{g}^{-1}\frac{dg}{dt}=g_{0}-g-\left(e^{g}-1\right)|E(t-T)|^{2}.
\label{eq:B}
\end{equation}
Here $\gamma $ is the spectral filter bandwidth, $\kappa_1$ is the linear
attenuation factor describing nonresonant cavity losses in the unidirectional part of the laser cavity, $g_{0}$ is the pump parameter, $\gamma_g$ is the relaxation rate of the amplifying medium, $\theta$ describes the detuning between the central frequency of the filter and one of the cavity modes, and the delay parameter $T$ is equal to the cold cavity round trip time.
The function ${\tilde f}\left(|E|^2,G\right)$ in Eq.~\eqref{eq:A} describes the reflectivity and the phase shift introduced by the NALM. It is given by ${\tilde f}\left(|E|^2,G\right)=\sqrt{\kappa_2}e^{G/2}\left(e^{i\phi_1}-e^{i\phi_2}\right)/2$, where $G$ and $\kappa_2$ and describe amplification and linear losses inside the loop, while $\phi_1=\eta |E|^2/2$ and $\phi_2=\eta e^G|E|^2/2$ are the phase shifts of the clockwise and counter-clockwise propagating waves indicated by arrows in Fig.~\ref{fig:scheme}. Since the counter-clockwise propagating wave is amplified before passing through the nonlinear element its phase shift $\phi_2$ is $e^G$ times larger than that the clockwise propagating wave, which is amplified after passing through the nonlinear element. For the gain $G$ an equation similar to Eq.~\eqref{eq:B} can be written. Here, however, we assume for simplicity that the electric field intensity $|E|^2$ is small and gain medium inside the NALM operates far below the saturation regime. In this case $G$ can be considered to be a constant parameter. Furthermore, for $\phi_1=\eta |E|^2/2\ll1$ we have $e^{\phi_1}\approx 1$ and the function ${\tilde f}$ can be rewritten in the form
\begin{equation*}
{\tilde f}(|E|^{2},G)=\sqrt{\kappa_2}e^{G/2}f(|E|^{2}),
\end{equation*}%
with
\begin{equation}
f(|E|^{2})=\frac{1}{2}\left(1-e^{ia|E|^{2}}\right), \label{eq:f}
\end{equation}
where $a=\eta (e^G-1)/2$. Finally, replacing $e^g-1$ with $g$ in Eq.~\eqref{eq:B}, eliminating the gain $g$ adiabatically, $g=g_0/\left[1+|E(t-T)|^2\right]$, and substituting this expression into Eq.~\eqref{eq:A} we get
\begin{equation}
\gamma ^{-1}\frac{dE}{dt}+E=\sqrt{\kappa }%
e^{g_{0}/[2(1+|E(t-T)|^{2})]+i\theta}f(|E(t-T)|^{2})E(t-T), \label{eq:model}
\end{equation}%
where $\kappa=\kappa_1\kappa_2e^{G/2}<1$, which means that the linear gain in the NALM is compensated by the linear cavity losses, and the nonlinear function $f(|E|^2)$ is defined by Eq.~\eqref{eq:f}. In the following
we will show that despite being very simple, the model equation (\ref%
{eq:model}) is capable of reproducing such experimentally observed
behaviours of nonlinear mirror lasers as mode-locking and square wave generation \cite{Aadhi}.
\begin{figure}[tbp]
\includegraphics[width=0.45\textwidth]{ModelSchemeNALM} \centering
\caption{Schematic view of a ring laser with a saturable gain (SG) medum, bandpass filter (F), optical isolator (ISO), output coupler (OC), and linear gain (LG) together with a
non-linear element (NE) in a Sagnac interferometer forming a NALM.
Arrows show different propagation directions inside the
interferometer. }
\label{fig:scheme}
\end{figure}
The simplest solution of Eq.~(\ref{eq:model}) is that corresponding to laser
off state, $E=0$. Linear stability analysis of this trivial non-lasing
solution indicates that it is always stable, which means that the laser is
non-self-starting for all possible parameter values. Non-trivial CW
solutions are defined by the relation
\begin{equation}
E=\sqrt{R}e^{i\Omega t},\label{eq:CW}
\end{equation}
where $R=|E|^{2}>0$ is the intensity and $\Omega $ is the frequency detuning of CW
regime from the reference frequency coinciding with the central frequency of
the spectral filter. $R$ and $\Omega$ are the solutions of a system of two transcendental equations
\begin{equation}
\kappa e^{\frac{g_{0}}{1+R}}\sin ^{2}\left( \frac{aR}{2}\right) =1+\frac{%
\Omega ^{2}}{\gamma ^{2}}, \label{eq:CW1}
\end{equation}%
\begin{equation}
\tan \left( \frac{aR^{2}}{2}-T\Omega +\theta\right) +\frac{\gamma }{\Omega }=0.
\label{eq:CW2}
\end{equation}%
Multiple solutions of these equations corresponding to different longitudinal modes of the laser are illustrated in the left panel of
Fig~\ref{fig:CWbranches}, where CW regimes correspond to the intersections of black closed curves obtained by solving Eq.~\eqref{eq:CW1} and thin gray lines calculated from Eq.~\eqref{eq:CW2}. Black curves in right panel of Fig.~\ref{fig:CWbranches} show the branches of non-trivial CW
solutions as functions of the pump parameter $g_{0}$, while gray lines are defined by the condition
\begin{equation}
g_{0}=\tilde{g}\equiv a(1+R)^{2}\cot \left( \frac{aR}{2}\right).
\label{eq:SN}
\end{equation}%
The intersections of the gray lines with black CW branches indicate fold bifurcation points, each of which separates the corresponding branch into two parts with the lower part being always unstable.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{modes2}\quad
\includegraphics[width=0.45\textwidth]{modes1}
\centering
\caption{Left: CW solutions at fixed pump parameter $g_0=5.0$. CW solutions lie on the intersections of the closed black curves [solutions of Eq.~(\ref{eq:CW1})] with thin gray lines [solutions of Eq.~(\ref{eq:CW2})].
Other parameter values: $a=2$, $\gamma=1$, $\kappa=0.3$, $\theta=0$, and $T=20$.
Right: Branches of CW solutions corresponding to different longitudinal
laser modes (black lines). Two gray lines are defined by the condition (\ref{eq:SN}). They intersect CW branches at the fold bifurcation points. Lower parts of the CW branches lying below the fold bifurcation points are always unstable, while upper parts can be either stable or unstable.
\label{fig:CWbranches}}
\end{figure*}
\section{CW stability in the large delay limit}
To study linear stability the CW solutions of Eq.~\eqref{eq:model} we apply the large delay limit approach described in Ref.~\cite{Yanchuk2010a}. By linearizing Eq.~\eqref{eq:model} on a CW solution given by Eq.~\eqref{eq:CW} we obtain the characteristic equation in the form:
\begin{equation}
c_2Y^2+2c_1(\lambda)Y+c_0(\lambda)=0,\label{eq:characteristic}
\end{equation}
where $Y=e^{-\lambda T}$ and $c_{0,1,2}$ are given in the Appendix. In the large delay limit, $\gamma T\gg 1$, the eigenvalues belonging to the so-called {\em pseudo-continuous spectrum} can be represented in the form:
$$\lambda= i \lambda_0 + \frac{\lambda_1}{T}+{\cal O}\left(\frac{1}{T^2}\right),\quad \lambda_1=\lambda_{11}+i\lambda_{12},$$
with real $\lambda_0$, $\lambda_{11}$, and $\lambda_{12}$. Therefore, using the approximate relations $c_{0,1}(\lambda)\approx c_{0,1}(i\lambda_0)$ and $Y\approx e^{-i \lambda_0 T-\lambda_1}$ we can solve the characteristic equation to express $\lambda_{11}$ as a function of $\lambda_0$ \cite{Yanchuk2010a}
\begin{eqnarray}
\lambda_{11}^{\pm}=\Re{\lambda_1^{\pm}}=\Re{\ln\left(Y_{\pm}^{-1}\right)},\nonumber\\ Y_{\pm}=\frac{-c_1(i\lambda_0)\pm\sqrt{c_1(i\lambda_0)^2-c_0(i\lambda_0)c_2}}{c_2}.\label{eq:pseudo}
\end{eqnarray}
Two solutions $\lambda_{11}^{\pm}(\lambda_0)$ given by Eq.~\eqref{eq:pseudo} define two branches of pseudo-continuous spectrum shown in the left panel of Fig.~\ref{fig:stability}. Due to the phase shift symmetry of the model equation \eqref{eq:model} $E\to Ee^{i\varphi}$ with arbitrary constant $\varphi$ one of these branches is tangent to the $\lambda_{11}=0$ axis on the ($\lambda_0$,$\lambda_{11}$)-plane at the point $\lambda_0=0$, i.e., $Y_{-}^{-1}\rvert_{\lambda_0=0}=1$. The right panel of Fig.~\ref{fig:stability} shows the stability diagram of the CW solutions on the plane of two parameters: normalized frequency offset $\Omega/\gamma$ of a CW solution and pump rate $g_0$. CW solutions are stable (unstable) in the dark (light) gray domains. Black curve indicates the fold bifurcation, where two CW solutions merge and disappear. Below this curve calculated from Eqs.~\eqref{eq:CW1} and \eqref{eq:SN} there are no CW solutions while above it a pair of CW solutions is born with one them corresponding to smaller intensity $R$ being always unstable. Similarly to the case of Eckhaus instability \cite{tuckerman1990bifurcation,wolfrum2006eckhaus} the upper branch of CW solutions can be stable only within the so-called Busse balloon (region labelled ``1'' in Fig.~\ref{fig:stability}) limited from below by modulational instability curve shown by blue line. The modulational instability curve is defined by the condition that one of the two branches of the pseudo-continuous spectrum, which satisfies the relation $\left(\lambda_{11}^{-}\right)_{\lambda_0=0}=\left(d\lambda_{11}^{-}/d\lambda_0\right)_{\lambda_0=0}=0$, changes the sign of its curvature at the point $\lambda_0=0$:
\begin{equation}
\left(\frac{d^2\lambda_{11}^{-}}{d\lambda_0^2}\right)_{\lambda_0=0}=0.\label{eq:MI}
\end{equation}
This instability is illustrated sub-panels 2 and 6 of the left panel of Fig.~\ref{fig:stability}. Explicit expression for the modulational instability condition \eqref{eq:MI} in terms of the model equation parameters is given in the Appendix.
It is seen from the right panel of Fig.~\ref{fig:stability} that the modulational instability curve is tangent to the fold bifurcation curve at $\Omega=0$ and that this curve becomes asymmetric with respect to the axis $\Omega=0$ sufficiently far away from the tangency point.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{spectr_br.png}\quad
\includegraphics[width=0.45\textwidth]{plane2.png}
\caption{Left: Two branches of pseudo-continuous spectrum. Different numbers illustrate qualitative behaviour of the branches in numbered parameter domains of the right panel.
Right: Bifurcation diagram on the plane of two parameters: CW frequency offset $\Omega$ and pump parameter $g_0$, obtained in the limit of large delay time.
Black -- fold bifurcation curve defined by Eq.~\eqref{eq:SN}, blue -- modulational instability curve, which serves as a boundary of the Busse balloon and is defined by Eq.~\eqref{eq:modulational} in the Appendix, green -- flip instability leading to square wave appearance and defined by Eq.~\eqref{eq:SW}, red -- Turing-type (wave) instability. $C_{\pm}$ are codimension-two points defined by Eqs.~\eqref{eq:PD} and \eqref{eq:CTpm}. CW solutions are stable (unstable) in dark gray region labeled ``1'' (light gray regions). $\gamma=2$ and $T\to\infty$. Other parameters are as for Fig.~\ref{fig:CWbranches}}\label{fig:stability}
\end{figure*}
The upper boundary of the CW stability domain shown in the right panel of Fig.~\ref{fig:stability} consists of two parts separated by codimension-two point C$_+$. The right part of this boundary lying between the points C$_+$ and C is indicated by red line and corresponds to the so-called Turing-type (wave) instability \cite{Yanchuk2010a}, where one of the two branches of pseudo-continuous spectrum cross the $\lambda_{11}=0$ axis at two symmetric points with $\lambda_0\neq 0$ i.e., $\lambda_{11}^+=0$, at $\lambda_0=\pm\lambda_0^*$ with $\lambda_0^*>0$, see sub-panel 5 of the left panel of Fig.~\ref{fig:stability}. The left part of the stability boundary lying between two symmetric codimension-two points C$_{\pm}$ corresponds to a flip instability leading to the appearance of square waves with the period close to $2T$. In the right panel of Fig.~\ref{fig:stability} the flip instability curve is shown by green line defined by the condition
\begin{equation}
Y_+^{-1}\rvert_{\lambda_0=0}=-1, \label{eq:SW}
\end{equation}
which can be rewritten in the form:
\begin{equation}
2-\frac{g_0 R}{(1+R)^2}+aR \cot\left(\frac{aR}{2}\right)=0.\label{eq:PD}
\end{equation}
The codimension-two points C$_{\pm}$ are defined by Eq.~\eqref{eq:PD} together with additional conditions $\left(d^2\lambda_{11}^{\pm}/d\lambda_0^2\right)_{\lambda_0=0}=0$. Using the relation \eqref{eq:PD} the additional conditions can be rewritten as
\begin{equation}
\Omega^2\pm\gamma \Omega a R =\gamma^2.\label{eq:CTpm}
\end{equation}
An implicit equation for the coordinates of C$_{\pm}$ on the ($\Omega$,$g_0$)-plane are obtained by solving Eq.~\eqref{eq:CTpm} for $R$ and substituting the resulting solution into Eq.~\eqref{eq:PD}. It follows from Eq.~\eqref{eq:CTpm} that the codimension-two points C$_{\pm}$ shown in the right panel of Fig~\eqref{fig:stability} are symmetric with respect to $\Omega=0$ axis. It is seen from the right panel of Fig.~\ref{fig:stability} that the central longitudinal mode with $\Omega=0$ is the first one undergoing a flip instability with the increase of the pump parameter $g_0$. The larger is the frequency offset of the mode, the higher is the flip bifurcation threshold for this mode. When, however, positive (negative) frequency offset is sufficiently large the mode is already unstable with respect to Turing (modulational) instability at the flip instability point. In this case the flip bifurcation results in the appearance of unstable square waves.
Note, that for any finite values of delay time, $\gamma T<\infty$, it is an Andronov-Hopf bifurcation of a CW solution that leads to the emergence of square waves. However, in the limit of infinite delay the period of square wave regime tends to infinity, which means that the imaginary parts of complex eigenvalues crossing the imaginary axis at the Andronov-Hopf bifurcation tend to zero. Hence, in the limit of infinite delay we refer to this bifurcation as a flip instability.
Let us consider the CW solution of Eq.~\eqref{eq:model} corresponding to the central longitudinal mode with zero detuning from the central frequency of the spectral filter, $\Omega=0$. For this solution the two quantities $Y_{\pm}$ in Eq.~\eqref{eq:pseudo} take the form
\begin{eqnarray}
Y_{-}\rvert_{\Omega=0}&=&1-i\frac{\lambda_0}{\gamma},\label{eq:Ym}\\
Y_{+}\rvert_{\Omega=0}&=&\frac{1-i \lambda_0/\gamma }{1+a R \cot {\left(\frac{a R}{2}\right)}-\frac{g_0 R}{(1+R)^2}}.\label{eq:Yp}
\end{eqnarray}
From Eq.~\eqref{eq:Ym} we get the relations $Y_{-}^{-1}\rvert_{\Omega=0,\lambda_0=0}=1$ and $\left|Y_{-}^{-1}\right|_{\Omega=0,\lambda_0\neq0}<1$ meaning that the first branch of the pseudo-continuous spectrum of the central longitudinal mode is always stable and is tangent to the imaginary axis at $\lambda_0=0$. On the other hand, from Eq.~\eqref{eq:Yp} we see that the fold bifurcation of the central longitudinal mode defined by the condition $Y_{+}^{-1}\rvert_{\Omega=0,\lambda_0=0}=1$ coincides with Eq.~\eqref{eq:SN}. Similarly the flip instability responsible for the emergence of square-waves is defined by the condition $Y_{+}^{-1}\rvert_{\Omega=0,\lambda_0=0}=-1$ coinciding with Eq.~\eqref{eq:PD}.
As it will be shown in the next section, the condition \eqref{eq:PD} defines also the period doubling bifurcation of a 1D map, which we construct in the next section to study the square wave formation in the DDE model Eq.~\eqref{eq:model}.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{map.png}\quad
\includegraphics[width=0.47\textwidth]{bifmap}
\caption{Left: Graph of the function $h$ defined by Eq.~\eqref{eq:1dmap}. Two period one fixed points of the map correspond to the intersections of the black curve with straight gray line $R_{n+1}=R_n$. These fixed pointa correspond to the CW solutions of Eq.~\eqref{eq:model} lying on the upper and lower part of the CW branch with zero frequency offset $\Omega=0$. $a=2$, $\kappa=0.3$, and $g_0=3.8$. Right: Bifurcation diagram illustrating period-doubling route to chaos in the map \eqref{eq:1dmap} with $a=2$ and $\kappa=0.3$. }\label{fig:doubling}
\end{figure*}
\section{1D map and square waves}
The existence of stable square wave solutions in Eq.~\eqref{eq:model} can be demonstrated by constructing a 1D map \cite{chow1983singularly} that exhibits a period doubling bifurcation corresponding to the emergence of square waves in the DDE model \eqref{eq:model}. To this end we rescale the time $\tau=t/T$ in Eq.~\eqref{eq:model} and obtain
\begin{equation}
\varepsilon\frac{dE}{d\tau}+E=\sqrt{\kappa}e^{g_{0}/[2(1+|E(\tau-1)|^{2})]}f\left[|E(\tau-1)|^{2}\right]E(\tau-1),
\label{eq:model1}
\end{equation}
where in the large delay limit we have $\varepsilon\equiv 1/\gamma T\ll 1$. By discarding the time derivative term, which is proportional to the small parameter $\varepsilon$, we transform this equation into a complex map, which describes the transformation of the electric field envelope $E$ after a round trip in the cavity and resembles the well known Ikeda map \cite{ikeda1979multiple}. Then, taking modulus square of both sides we obtain a 1D map for the intensity $R$:
\begin{equation}\label{eq:1dmap}
R_{n+1}=h(R_n),\quad h(R_n)=\kappa e^{\frac{g_0}{1+R_n}}\sin^2\left(\frac{aR_n}{2}\right)R_n,
\end{equation}
where $R_n\equiv R(n)$ ($n=0,1,2\dots$) with fixed points $R_n=R^*$ satisfying the condition $R^*=h(R^*)$:
\begin{equation}
\kappa e^{\frac{g_{0}}{1+R^*}}\sin^{2}\left(\frac{aR^*}{2}\right)=1.\label{eq:fixedpoint}
\end{equation}
Graphical representation of the function $h$ is given in the left panel of Fig.~\ref{fig:doubling}. Note that since Eq.~\eqref{eq:fixedpoint} is equivalent to Eq.~\eqref{eq:CW1} taken at $\Omega=0$ and $\Omega=0$ is a solution of Eq.~\eqref{eq:CW2}, fixed points of the map \eqref{eq:1dmap} have the intensity $R$ coinciding with that of the central longitudinal mode, i.e. the CW solution of Eq.~\eqref{eq:model} with zero frequency offset $\Omega=0$ from the central frequency of the spectral filter. Furthermore, for sufficiently large $g_0$ a stable fixed point of the map \eqref{eq:1dmap} exhibits a period doubling bifurcation which is defined by
\begin{equation}
1-\frac{g_0 R^*}{(1+R^*)^2}+aR^* \cot\left(\frac{aR^*}{2}\right)=-1, \label{eq:PDmap}
\end{equation}
together with \eqref{eq:fixedpoint}. Since the relations \eqref{eq:fixedpoint} and \eqref{eq:PDmap} are equivalent to \eqref{eq:CW1} and \eqref{eq:SW} evaluated at $\Omega=0$, the period doubling bifurcation point of the map \eqref{eq:1dmap} coincides in the limit of large delay with the flip instability to square waves of the central longitudinal mode having zero frequency offset, $\Omega=0$.
It is seen from right panel of Fig.~\ref{fig:doubling} that after the first period doubling bifurcation the 1D map \eqref{eq:1dmap} demonstrates a period doubling transition to chaos.
\begin{figure*}
\begin{centering}
\includegraphics[width=0.45\textwidth]{bifDDE1}\quad
\includegraphics[width=0.48\textwidth]{SW.png}
\end{centering}
\caption{Left: Bifurcation diagram obtained by numerical integration of Eq. (\ref{eq:model})
with $\kappa=0.3$, $a=2$, $T=20$, and $\gamma=5$. Right: Square waves calculated numerically for $g_{0}=4.0$. \label{fig:squarewaves}}
\end{figure*}
Period-doubling route to chaos obtained by numerical integration of the DDE model \eqref{eq:model} is illustrated in left panel of Fig.~\ref{fig:squarewaves}, where local maxima of the electric field intensity time-trace are plotted versus increasing values of the pump parameter $g_0$. It is seen that the diagram in this figure is very similar to that obtained with the 1D map \eqref{eq:1dmap}, cf. Fig.~\ref{fig:doubling}. Note, however, that the period doubling threshold is slightly higher in Fig.~\ref{fig:squarewaves} than in Fig.~\ref{fig:doubling}. This can be explained by taking into consideration that in the DDE model \eqref{eq:model} the threshold of the flip instability leading to the emergence of square waves increases with the absolute value of the frequency detuning $\Omega$. Therefore, we can conclude that in Fig.~\ref{fig:squarewaves} the CW solution undergoing the period-doubling cascade must have a small non-zero frequency detuning, $\Omega\neq 0$. The first period doubling bifurcation in the left panel of Fig.~\ref{fig:doubling} is responsible for the formation of square waves shown in the right panel of Fig.~\ref{fig:doubling}, while further period doublings give rise to more complicated square wave patterns with larger periods. Finally, we note that with the increase of the gain parameter $g_0$ new pairs of fixed points of the map \eqref{eq:1dmap} appear in saddle-node bifurcations. For example, the second pair of fixed points corresponds to the second (right) maximum of the function $h$ shown in the left panel of Fig.~\ref{fig:doubling} and to additional branches of CW solutions visible in the upper right part of the right panel of Fig.~\ref{fig:CWbranches}.
Although the linear stability analysis performed in the previous section is applicable to these additional high intensity CW solutions as well, here we restrict our consideration to moderate pumping levels.
\begin{figure*}
\includegraphics[width=0.45\textwidth]{ppNOMDDE}
\includegraphics[width=0.46\textwidth]{ml.png}
\caption{Left: Bifurcation diagram obtained by numerical integration of Eq. (\ref{eq:model}). Right: fundamental mode-locked regime with the repetition period close to the cavity round trip time calculated for $g_0=5.0$. Other parameters are same as for Fig. \ref{fig:squarewaves}. }
\label{fig:full}
\end{figure*}
\section{Mode-locking}
Bifurcation diagram in the left panel of Fig.~\ref{fig:full} was obtained by numerical integraton of Eq.~\eqref{eq:model}. It is similar to that shown in Fig.~\ref{fig:squarewaves}, but spans a larger range of the pump parameter values. It follows from this diagram that with the increase of the pump parameter $g_0$ after a chaotic regime associated with the period doubling cascade the phase trajectory of the system jumps to a pulsed solution with time periodic laser intensity. This solution corresponding to a fundamental mode-locked regime with the repetition period close to the cavity round trip time, $T$, is illustrated in right panel of Fig.~\ref{fig:full}, where it is seen that the leading edge the pulses is much steeper than the trailing edge. In the following, we will show that unlike the trailing tail of the pulses, which decays exponentially, their leading tail demonstrates faster-than-exponential (super-exponential) decay in negative time.
Let us consider a mode-locked solution of Eq.~(\ref{eq:model}) with the period $T_{0}$ close to the cavity round trip time $T$. For this solution satisfying the condition $E(t)=E(t+T_{0})e^{i\Delta}$ we can write
\begin{equation}
E(t-T)=E(t-T+T_{0})e^{i\Delta}\equiv E(t+\delta)e^{i\Delta}\label{eq:periodic}
\end{equation}
where $\delta=T_{0}-T>0$ is the small difference between the solution period and the delay time. Substituting \eqref{eq:periodic} into \eqref{eq:model} we get a time advance equation
\begin{equation}
\gamma^{-1}\frac{dE}{dt}+E=\frac{\sqrt{\kappa}}{2}e^{g_{0}/[2(1+|E(t+\delta)|^{2})]+i\Theta}f(|E(t+\delta)|^{2})E(t+\delta),\label{eq:delta}
\end{equation}
where $\Theta=\theta+\Delta$. Note, that unlike the original DDE model \eqref{eq:model}, which has stable trivial solution $E=0$, the trivial solution of Eq.~\eqref{eq:delta} is a saddle with one stable and an infinite number of unstable directions. The stable direction determines the decay rate of the trailing tail of mode-locked pulses, while unstable directions are responsible for the decay of the leading tail in negative time. To perform linear stability analysis of the trivial solution of Eq.~\eqref{eq:delta} we write the following equation linearized at $E=0$:
\begin{equation}
\gamma^{-1}\frac{dE}{dt}+E=\epsilon E(t+\delta)e^{i\Theta},
\label{eq:lin}
\end{equation}
where linear time advance term in the right hand side proportional to small perturbation parameter $\epsilon$ describes an imperfection introduced by a slight asymmetry of the coupler between the laser cavity and the NALM.
The spectrum of Eq.~(\ref{eq:lin}) is defined by
\begin{equation}
\lambda_{k}=-\gamma\left[1+\frac{W_{k}(-\epsilon\gamma\delta e^{-\gamma\delta+i\Theta})}{\gamma\delta}\right],
\end{equation}
where $W_{k}$ denotes the $k$th branch of multivalued Lambert function. In particular, in
the limit $\epsilon\to0$ the eigenvalue with the index $k=0$ is
the only stable and negative one $\lambda_{0}\to-\gamma<0$. This eigenvalue determines the decay rate of the trailing tail of mode-locked pulses. The remaining eigenvalues with $k\neq 0$ have positive real parts diverging in the limit $\epsilon\to0$, $\Re\lambda_{k}\to+\infty$. Among these unstable eigenvalues, depending on the value of $\Theta$, one of the two eigenvalues $\lambda_{\pm 1}$ with $k=\pm 1$ has the smallest real
part. All other eigenvalues with $|k|>1$ have larger real parts.
Hence, for small nonzero $\epsilon$ generically one of the two eigenvalues $\lambda_{\pm 1}$ determines the decay rate the pulse leading tail in negative time. The fact that this eigenvalue tends to infinity as $\epsilon\to0$ suggests that this decay is super-exponential.
Finally let us discuss briefly the interaction of asymmetric mode-locked pulses shown in the right panel of Fig.~\ref{fig:full}. When integrating the model equation \eqref{eq:model} numerrically it is possible to seed two or more non-equidistant pulses in the laser cavity as an initial condition. Then the pulses start to interact locally via their decaying tails. The asymmetric nature of these tails suggests that similarly to the case discussed in \cite{vladimirov2018effect} local interaction of non-equidistant pulses will be very asymmetric as well. This can be is seen in Fig.~\ref{fig:interaction} illustrating an interaction of two asymmetric pulses of Eqs.~\eqref{eq:model} on the time-round trip number plane. We see that two initially non-equidistant pulses repel each other and tend to be equidistantly spaced in the long time limit. Furthermore, when the two pulses are sufficiently close to one another, exponentially decaying trailing tail of the left pulse repels noticeably the right pulse, while super-exponentially decaying leading tail of the right pulse almost does not affect the position of the left pulse. When the pulses become equidistant the forces acting on a pulse from opposite directions balance each other and a stable harmonic mode-locking regime with two pulses per cavity round trip time is established.
\begin{figure}
\begin{centering}
\includegraphics[scale=0.6]{interaction.png}
\caption{Interaction of two mode-locked pulses leading to a harmonic mode-locked regime with two pulses per cavity round trip. A common drift of the two interacting pulses is eliminated. $T=10$. The time axis spans the interval $2T$. Other parameters are same as for Fig.~\ref{fig:squarewaves}. }
\label{fig:interaction}
\end{centering}
\end{figure}
\section{Conclusion}
We have considered a simple DDE model of unidirectional class-A ring NALM mode-locked laser. Linear stability analysis in the large delay limit revealed that similarly to the well-known Eckhaus instability only those CW solutions, which belong to the Busse balloon can be stable. This balloon is limited from below by the modulational instability boundary. We demonstrated that with the increase of the pump parameter CW regimes loose their stability either via a Turing-type instability, or through a flip instability leading to a formation of stable square waves. We have constructed a 1D map which describes the transition to square waves and their secondary bifurcations giving rise to a more complicated square wave patterns with larger and larger periods. We have shown that mode-locked pulses, which appear after a chaotic square-wave dynamics, have exponentially decaying trailing tail and a leading tail, which decays super-exponentially in negative time. When two or more pulses circulate in the laser cavity, the interaction of these pulses is repulsing and very asymmetric leading to a formation of harmonic mode-locked regimes. Noteworthy, that the mode-locked regime considered here is always non-self-starting, which means that the pulses are sitting on a stable laser-off solution. Hence, these pulses can be viewed as temporal cavity solitons having similar properties to spatial and temporal localized structures of light observed in bistable optical systems.
\bigskip
Authors thank D. Turaev for useful discussions. A.V.K. and E.A.V. acknowledge the support by Government of Russian Federation (Grant 08-08). A.G.V. acknowledges the support of the F{\'e}d{\'e}ration Doeblin CNRS and SFB 787 of the DFG, project B5
\section{Appendix}
The coefficients $c_0(\lambda)$, $c_1(\lambda)$, and $c_2$ in the characteristic equation \eqref{eq:characteristic} are defined by
\begin{widetext}
\begin{eqnarray}
c_0(\lambda)&=&(\gamma -\lambda )^2+\Omega ^2,\nonumber\\
c_1(\lambda)&=&-\frac{a \lambda R (R+1)^2 \Omega +\gamma (\gamma -\lambda ) [R (2 R+4-g_0)+2]+\Omega ^2 [R (2
R+4-g_0)+2]}{(R+1)^2}-a R \cot \left(\frac{a R}{2}\right) \left(\gamma ^2-\gamma \lambda +\Omega ^2\right),\nonumber\\
c_2&=&\frac{\gamma ^2 \kappa e^{\frac{g_0}{R+1}}}{2
(R+1)^2} \{a R (R+1)^2 \sin (a R)-[R (R+2-g_0)+1] [\cos (a R)-1]\}.\nonumber
\end{eqnarray}
The modulational instability condition \eqref{eq:MI} of the CW solutions of Eq.~\eqref{eq:model} can be rewritten in the following form:
\begin{eqnarray}
\frac{\left[a(1+R)^{2}\Omega+\gamma(\tilde g-g_{0})\right]^{2}}{\left(\gamma^{2}+\Omega^{2}\right)^{2}}+\frac{2(1+R)^{2}\Omega^{2}\left[ g_0^2+2a^2(1+R)^4-g_0^2 \cos{(aR)}-2ag_0(1+R)^2\sin{(aR)}\right] }{R(\tilde g-g_{0})\left(\gamma^{2}+\Omega^{2}\right)^{2}\left[1-\cos\left(aR\right)\right]}=0.
\label{eq:modulational}
\end{eqnarray}
\end{widetext}
|
1,116,691,500,505 | arxiv | \section*{Code Availability}
Our codes are available at: https://github.com/weiT1993/CutQC.
\section*{Acknowledgements}
This work is partly funded by EPiQC, an NSF Expedition in Computing, under grants CCF-1730082/1730449.
This work is partly based upon work supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704.
\bibliographystyle{unsrt}
\section*{ACKNOWLEDGMENT}
We thank A, B, and C. This work was supported in part by a grant from XYZ.
\subsection{Dynamic Definition Query}\label{sec:dynamic_definition_results}
\setlength{\belowcaptionskip}{-5pt}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/dd_runtime.pdf}
\caption{CutQC executes benchmark circuits mapped to quantum devices with up to $\frac{3}{4}$ of the number of qubits in the input circuits.
The horizontal axis shows the number of qubits in the input circuit.
The vertical axis shows the postprocessing runtime of $1$ DD recursion with a definition of $2^{30}$ bins.
GPU is the fastest backend as expected.}
\label{fig:dd_expand}
\end{figure}
We used DD to efficiently sample quantum circuits of which the full Hilbert space is too large to even store.
NISQ devices will gradually improve in fidelity and sizes to allow evaluating subcircuits beyond the classical simulation limit.
CutQC will then allow the use of those NISQ devices to efficiently evaluate even larger quantum circuits.
We cut and executed circuits of up to $100$ qubits and used DD query to sample their blurred probability landscape with a definition of $2^{30}$ bins in one recursion.
Figure~\ref{fig:dd_expand} shows the runtime of cutting and mapping circuits to quantum computers with up to $\frac{3}{4}$ of the qubits.
The classical post-processing overhead in FIgure~\ref{fig:dd_expand} is hence the classical `cost' to expand the reach of QPUs by at least a quarter more of the qubits available.
Certain benchmarks, such as BV, almost double the number of qubits possible via CutQC.
Furthermore, the novel incorporation of GPUs makes such cost minimal
to gain the huge benefit of significantly expanding the reach of the underlying quantum and classical platforms alone.
In fact, GPU provides up to two orders of magnitude runtime improvements in benchmarks that are harder to cut and hence require more classical post-processing,
such as \textit{AQFT} and \textit{Supremacy}.
This is all without the need for either a large quantum computer or vast classical computing resources.
Note that neither the CPU or the GPU backends used in the experiments alone is capable of running any of the benchmark circuits in Figure~\ref{fig:dd_expand}.
\section{EXPERIMENT RESULTS}\label{sec:results}
\input{text/DD_Results.tex}
\input{text/Device_Results.tex}
\section{BACKGROUND}\label{sec:background}
\begin{figure}[t]
\centering
\includegraphics[width=1\linewidth]{figures/eval_modes.pdf}
\caption{Different quantum circuit evaluation modes.
(a) Purely classical simulation produces the ground truth to verify other evaluation outputs.
(b) Purely quantum evaluation on a single QPU.
Multiple vendors provide cloud access to their devices.
(c) Our hybrid mode, which is orders of magnitude faster than (a),
produces much less noisy outputs than (b),
and evaluates much larger circuits than (a) and (b).}
\label{fig:eval_modes}
\end{figure}
This section introduces quantum circuits and explains the differences between several quantum circuit evaluation modes.
Quantum programs are expressed as circuits that consist of a sequence of single- and multiqubit gate operations.
Quantum circuits can be evaluated by using classical simulations, on quantum computers, or in a hybrid mode as explored in this paper.
Figure~\ref{fig:eval_modes} provides an overview of the different evaluation modes.
State vector simulation (Figure~\ref{fig:eval_modes}a) is an idealized noiseless simulation of a quantum circuit.
All quantum operations are represented as unitary matrices.
N-qubit operations are $2^N\times2^N$ unitary matrices.
State vector simulation executes circuits by sequentially multiplying each gate's corresponding unitary matrix with the current state vector.
This yields an error-free output represented as complex amplitudes,
which cannot be obtained on quantum computers.
This evaluation mode scales exponentially and serves to provide the ground truth and runtime baselines for benchmarking NISQ devices for small quantum circuits.
We use this evaluation mode as a baseline to verify the output of modes (b) and (c) in Figure~\ref{fig:eval_modes} and to compute the $\chi^2$ metric to quantify the noise and quality of quantum circuit executions.
QC evaluation (Figure~\ref{fig:eval_modes}b) physically executes quantum circuits on NISQ computers using a shot-based model.
Quantum algorithms are first compiled to satisfy device-specific characteristics such as qubit connectivity, native gate set, noise, and crosstalk.
A real NISQ device then executes the compiled quantum circuit thousands of times (``shots'') in quick succession.
At the end of each shot, all qubits are measured; and the output, a classical bit string, is recorded.
After all shots are taken, a distribution of probabilities over the observed states is obtained.
This paper explores the CutQC evaluation (Figure~\ref{fig:eval_modes}c) that combines both quantum and classical platforms.
Section~\ref{sec:results} demonstrates the runtimes of the CutQC evaluation (Figure~\ref{fig:eval_modes}c) for large quantum circuits on smaller QPUs.
We also compare the execution fidelities of the QC evaluation (Figure~\ref{fig:eval_modes}b) and the hybrid evaluation (Figure~\ref{fig:eval_modes}c) modes.
\section{Framework Overview}\label{sec:framework}
\setlength{\belowcaptionskip}{-5pt}
\begin{figure}[t]
\centering
\includegraphics[width=.6\linewidth]{figures/cutting_mode.pdf}
\caption{Framework overview of CutQC. A mixed-integer programming (MIP) cut searcher
automatically finds optimal cuts given an input quantum circuit.
The small subcircuits resulting from the cuts are then evaluated by using quantum devices. The reconstructor then reproduces the probability distributions of the original circuit.}
\label{fig:cutting_mode}
\end{figure}
Figure~\ref{fig:cutting_mode} summarizes the key components of our framework.
CutQC is built on top of IBM's Qiskit package in order to use IBM's quantum devices for the experiments on fidelity,
but we note that the hybrid approach works with any gate-based quantum computing platforms.
Given a quantum circuit specified as an input,
the first step is to decide where to make cuts.
We propose the first automatic scheme that uses mixed-integer programming to find optimal cuts for arbitrary quantum circuits.
The backend for the MIP cut searcher is implemented in the Gurobi solver.
Multiple QPUs then evaluate the different combinations of the subcircuits.
Eventually, a reconstructor running on either CPUs or GPUs postprocesses the subcircuit outputs
and reproduces the original full circuit outputs from the Kronecker products.
\input{text/MIP.tex}
\input{text/Full_Definition.tex}
\input{text/Dynamic_Definition.tex}
\subsection{Circuit Cutting: Example}\label{sec:cc_example}
\begin{figure*}[t]
\centering
\includegraphics[width=.95\textwidth]{figures/cutting_example.pdf}
\caption{Example of cutting a $5$-qubit circuit into two smaller subcircuits
of $3$ qubits each. The subcircuits are produced by cutting the
$q_2$ wire between the first two $cZ$ gates.
The three variations of $subcircuit_1$ and four variations of $subcircuit_2$ can then be evaluated on a 3-qubit QPU, instead of a 5-qubit QPU.
The classical postprocessing involves summing over $4$ Kronecker products between the two subcircuits for the one cut made.}
\label{fig:cutting_example}
\end{figure*}
Consider the quantum circuit example in Figure~\ref{fig:cutting_example}.
One cut separates a 5-qubit quantum circuit into 2 subcircuits of 3 qubits each.
Time goes from left to right in quantum circuit diagrams,
and each row represents a qubit wire.
CutQC performs vertical cuts on qubit wires, in other words, timewise cuts.
The qubit states across the cutting point are then decomposed into their Pauli bases.
With a proper selection of the cutting points, a large quantum circuit can be divided into smaller isolated subcircuits.
Without cutting, the circuit in Figure~\ref{fig:cutting_example} at least requires a $5$-qubit QPU with good enough qubits to execute all the quantum gates before too many errors accumulate.
Circuit cutting divides this quantum circuit and produces two smaller subcircuits, each with both fewer qubits and fewer gates.
Now multiple less powerful 3-qubit QPUs can run these independent subcircuits in parallel.
The quantum interactions among the subcircuits are substituted by classical post-processing,
which are analogues to the communication cost paid in classical parallel computing.
In general, a $n$ qubit quantum circuit undergoes $K$ cuts to divide into $n_C$ completely separated subcircuits $C = \left\{C_1,\ldots,C_{n_C}\right\}$.
A complete reconstruction of the quantum interactions requires each cut to permute each of the Pauli $\{I,X,Y,Z\}$ bases, for a total of $4^K$ combinations.
Depending on the Pauli basis assigned to each cut, the subcircuits are initialized and measured slightly differently to produce a distinct entry.
We use $p_{i,k}$ to represent the output of subcircuit $i$ in the $k$th edge bases assignment, where $i\in\{1,\ldots,n_C\}$ and $k\in\{1,\ldots,4^K\}$.
The physics theory dictates that the output of the original circuit is given by
\begin{equation}
P=\sum_{k=1}^{4^K}\otimes_{i=1}^{n_C}p_{i,k}\in\mathbb{R}^{2^n}\label{eq:CutQC}
\end{equation}
where $\otimes$ is the tensor product between two subcircuit output vectors.
\section{Conclusion}\label{sec:conclusion}
This paper demonstrates how to leverage both quantum and classical computing platforms together to execute quantum algorithms of up to $100$ qubits while simultaneously improving the fidelity of the output.
Our results are significantly beyond the reach of current quantum or classical methods alone, and our work pioneers pathways for scalable quantum computing.
Even as NISQ machines scale to larger sizes and as fault-tolerant QPUs emerge,
CutQC's techniques for automatically cutting and efficiently reconstructing quantum circuit executions offer a practical strategy for hybrid quantum/classical advantage in QC applications.
\section{Related Work}\label{sec:related_works}
Many quantum compilation techniques have been developed to improve the performance of NISQ devices.
However, these focus on improving a purely quantum computing approach and are intrinsically limited by the size and fidelity of NISQ devices.
Specifically, our experiments used the noise adaptive compiler~\cite{murali2019noise} in both CutQC and QC evaluations.
The improved fidelity we demonstrate is in addition to that given by the compiler.
Furthermore, previous compilers do not allow executions of circuits beyond quantum computer sizes at all.
Our approach can work in concert with any compilers to execute circuits both larger in size and better in fidelity.
Previous works on classical simulation require massive computing resources,
or only simulate very few output states at a time~\cite{liu2021closing}.
Many small-scale quantum circuit cutting demonstrations exist for chemical molecule simulations~\cite{eddins2021doubling}
and variational quantum solvers~\cite{yuan2021quantum}.
\section{INTRODUCTION}\label{sec:introduction}
QC has emerged as a promising computational approach with
the potential to benefit numerous scientific fields.
For example, some of the earliest QC work shows that
quantum algorithms for factoring~\cite{shor1999polynomial} can be exponentially faster than their classical counterparts.
However, these quantum algorithms assume the existence
of large-scale, fault-tolerant, universal quantum computers.
Instead, today's quantum computers are noisy intermediate-scale quantum (NISQ) devices.
Major challenges limit their effectiveness.
Noise can come from limited coherence time,
frequency selection for individual qubits,
crosstalk among qubits,
and limited control bandwidth.
Because of these and other issues,
the difficulty of building reliable quantum computers increases dramatically with increasing number of qubits.
More fundamentally, such intermediate-scale quantum devices are hard limited by their qubit count.
Currently, only small quantum circuits can be run on small quantum computers.
The largest superconducting quantum computers available today have $127$ qubits,
and their relatively poor fidelity further limits the size of circuits that can be reliably run.
Both the noise and the intermediate-scale characteristics of NISQ devices present significant obstacles to their practical applications.
On the other hand, the alternative for quantum circuits evaluation---classical simulations of quantum circuits---produces noiseless output but is not tractable in general.
For example, state-of-the-art classical simulations of quantum circuits of $100$ qubits require $42$ million cores~\cite{liu2021closing}.
This work uses circuit cutting to
expand the reach of small quantum computers with partitioning and post-processing techniques
that augment small QPU platforms with CPUs and GPUs.
CutQC is an end-to-end hybrid approach that automatically locates efficient cut positions to cut a
large quantum circuit into smaller subcircuits that are each independently executed by QPUs with less quality and size requirements.
Via scalable post-processing techniques,
the output of the original circuit can then be reconstructed or sampled efficiently from the subcircuit outputs with classical computing.
To evaluate the performance of CutQC,
we benchmarked four different quantum circuits that represent a general set of circuits for gate-based QC platforms and promising near-term applications.
We demonstrate executing quantum circuits of up to 100 qubits on existing NISQ devices and classical computing.
This is significantly beyond the current reach of either quantum or classical methods alone.
Our contributions include the following:
\begin{enumerate}
\item \textbf{Expanding the size} of quantum circuits that can be run on NISQ devices and classical simulation by combining the two.
Our method allows executions of quantum circuits more than twice the size of the available quantum computer backend and much beyond the classical simulation limit.
\item \textbf{Improving the fidelity} of quantum circuit executions on NISQ devices.
We show an average of $21\%$ to $47\%$ improvement to $\chi^2$ loss for different benchmarks by using CutQC with small QPUs, as compared with direct executions on large QPUs.
\end{enumerate}
\subsection{MIP Cut Searcher}\label{sec:MIP}
Unlike the manual example in Section~\ref{sec:cc_example},
CutQC's cut searcher uses mixed-integer programming (MIP) to automate the identification of cuts that require the least amount of classical postprocessing.
Our problem instances are solved by the Gurobi mathematical optimization solver~\cite{gurobi}.
The framework assumes that the input quantum circuit is fully connected.
That is, all qubits are connected via multiqubit gates either directly or indirectly through intermediate qubits.
A quantum circuit that is not fully connected can be readily separated into fully connected subcircuits without cuts,
and does not need the classical postprocessing techniques to sew together.
We hence focus on the more difficult general cases where cutting and reconstruction are needed.
We adopt the public MIP solver from~\cite{tang2021cutqc},
which solved the constrained partition problem by predicting the post-processing to directly compute Equation~\ref{eq:CutQC}.
Besides the input quantum circuit,
the MIP cut searcher also requires the user to specify
(1) the maximum number of qubits allowed per subcircuit,
and (2) the maximum number of subcircuits allowed.
(1) is just the size of the quantum devices available to the user.
(2) is set to $5$ in this paper.
Locating the cut points is equivalent to clustering the multi-qubit gates in the input quantum circuit.
A quantum circuit can be modeled as a directed acyclic graph (DAG).
Quantum operations are always applied sequentially to the qubits.
The single-qubit gates are ignored during the cut-finding process,
since they do not affect the connectivity of the quantum circuit.
The multi-qubit quantum gates are then modeled as the vertices,
and the qubit wires are modeled as the edges.
Choosing which edges to cut in order to split the circuit into subcircuits is equivalent to clustering the vertices.
The corresponding cuts required to produce the clustering are hence the cross-cluster edges.
We seek to minimize the classical postprocessing overhead required to reconstruct a circuit from its subcircuits.
Therefore, the objective is set to be the number of floating-point multiplications involved in computing Equation~\ref{eq:CutQC},
given by:
\begin{equation}\label{eq:MIP_obj}
L\equiv4^K\sum_{c=2}^{n_C}\prod_{i=1}^c2^{n_i}.
\end{equation}
where $K$ is the number of cross-cluster edges, i.e. the number of cuts.
$n_C$ is the number of subcircuits,
and $n_i$ is the number of qubits in subcircuit $i$.
This cost objective accurately captures the bulk of the computation when we aim
to build the full $2^n$ probabilities for an $n$-qubit uncut circuit, under the
full definition CutQC mode (discussed in Section~\ref{sec:FD_post_processing}).
However, there is a prohibitive memory requirement for storing the $2^n$
probabilities as floating-point numbers when circuits get larger.
Section~\ref{sec:DD_post_processing} introduces a novel dynamic definition
method to efficiently sample very large circuits with a much lower
postprocessing overhead. Nevertheless, we chose to minimize
Equation~\ref{eq:MIP_obj} during cut search as a positively correlated
objective.
\subsection{Dynamic Definition Post-Processing}\label{sec:DD_post_processing}
Quantum circuits can be loosely categorized into two groups.
The first group produces sparse output probabilities,
where just a few ``solution'' states have very high probabilities
and the ``non-solution'' states have low or zero probabilities.
Most known quantum algorithms fall into this category,
such as Bernstein--Vazirani algorithm~\cite{bernstein1997quantum} and the Quantum Fourier Transform (QFT)~\cite{cooley1965algorithm}.
This is where QC shows promise over classical computing by efficiently locating the ``solution'' states.
The second group of circuits produces dense output probabilities,
where many states have nonzero probabilities.
For this type of circuit,
even with access to QPUs large enough to execute the circuits directly,
querying the FD probability output quickly becomes impossible.
The reasons are that
(1) an exponentially increasing amount of memory is required to store the probabilities
and (2) an exponentially increasing number of shots are required on a QPU before the probabilities converge.
Fortunately, knowing the FD probabilities of all states simultaneously is usually not of interest.
Instead, users are interested in the distribution itself.
DD query allows us to find the ``solution'' states
or sample dense probability distributions efficiently with very large quantum circuits,
even when storing the full-state probability is not tractable.
DD query produces a probability distribution that merges certain states into one bin and maintains the sum of their probabilities instead of the individual states within.
Algorithm~\ref{alg:dynamic_definition} presents the DD algorithm.
In each recursion, DD runs the subcircuits to produce the merged subcircuit outputs before post-processing.
The \textit{active} qubits in each recursion determine the number of bins,
the \textit{merged} qubits determine which states are merged into the same bin,
and the \textit{zoomed} qubits indicate the qubit states that have been fixed.
Each subsequent recursion zooms into the bin with the largest sum of probability from the previous recursions,
improving the `definition' of the states contained in the bin.
This lets DD recursively obtain more fine-grained outputs for the input circuit.
\begin{algorithm}[t]
\DontPrintSemicolon
\SetAlgoVlined
\caption{Dynamic Definition}\label{alg:dynamic_definition}
\KwIn{Subcircuits from cutting.\;
Max number of qubits that fit in the memory per recursion $M$.\;
Max number of recursions $R$.}
Initialize an empty list $L$\;
$r\gets0$\;
\While{$r<R$}{
\uIf{$r=0$}{
Choose a maximum of $M$ qubits to label as ${\color{red}active}$\;
}
\uElse{
Fix the quantum states of the ${\color{red}active}$ qubits in $bin$\;
Label the qubits as ${\color{blue}zoomed}$\;
}
Label the rest of the qubits as ${\color{orange}merged}$\;
\textbf{QPUs} : Run the subcircuits to produce the sum of probabilities for the subcircuit bins
by grouping shots with the same ${\color{red}active}$ qubits quantum states together\;
Reconstruct the $2^{\#{\color{red}active}}$ probability output for the ${\color{red}active}$ qubits\;
Append the $R$ largest bins still with ${\color{orange}merged}$ qubits to $L$\;
Sort and truncate $L$ to keep the largest $R$ bins\;
Pop $bin$ from $L$\;
$r\gets r+1$\;
}
\end{algorithm}
For sparse outputs, DD recursively pinpoints the ``solution'' states and their probabilities.
To do so, DD query follows a DFS-like search strategy to recursively choose the $bin$ with higher probabilities to zoom in on.
By recursively locating the $active$ qubits in their most probable $zoomed$ states,
``solution'' states can be easily located after just a few recursions.
For an $n$-qubit full circuit, the number of recursions needed is $\mathcal{O}(n)$.
For dense outputs, DD builds a ``blurred'' probability landscape of the exact FD probability distribution,
with the ability to arbitrarily ``zoom in'' on any region of the state space.
To do so, DD query follows a BFS-like strategy to choose the $bin$ with higher probabilities to zoom in on.
This is equivalent to efficient sampling of very large circuits on less powerful QPUs and less memory.
\subsection{Full Definition Post-Processing}\label{sec:FD_post_processing}
We developed two types of classical postprocessing algorithms: a full-definition (FD) query and a dynamic-definition (DD) query algorithms.
The difference in these methods lies in whether the entire $2^n$ full-state probability output of the uncut circuit is reconstructed.
The reconstruction step (computing Equation~\ref{eq:CutQC}) is essentially taking vector-vector tensor products.
Previous work~\cite{tang2021cutqc} used Intel CPUs as the classical backends,
and demonstrated significant runtime advantages of hybrid computation over classical simulations in the full state setting.
Since GPUs are particularly suitable for inter vector tensor products,
this paper runs the classical post-processing on a single GPU via Tensorflow.
\subsection{Circuit Cutting: Challenges}\label{sec:cc_challenges}
The first challenge is to find cut locations.
While quantum circuits can always be split into smaller ones,
finding the optimal cut locations is crucial in order to minimize the classical postprocessing overhead.
In general, large quantum circuits may require more than one cuts in order to be separated into subcircuits.
In this case, the cutting scheme evaluates all possible measurement-initialization combinations.
The resulting number of Kronecker products is $4^K$, where $K$ is the number of edges cut.
For general quantum circuits with $n$ quantum edges,
this task faces an $\mathcal{O}(2^n)$ combinatorial search space.
Section~\ref{sec:MIP} addresses this problem with mixed-integer programming.
Our work shows that with only a few cuts, many useful applications can be tractably mapped to NISQ devices currently available.
The second challenge is to scale the classical postprocessing.
Large quantum circuits have exponentially increasing state space that quickly becomes intractable to even store the full-state probabilities.
Section~\ref{sec:DD_post_processing} addresses this problem with a dynamic definition algorithm to efficiently locate the ``solution'' states
or sample the full output distribution for large quantum circuits beyond the current QC and classical simulation limit.
\section{Methodology}\label{sec:methodology}
This section introduces the various backends, metrics and benchmarks for the experiments.
\subsection{Backends}
We test our approach by running post-processing and classical simulation benchmarks on both CPUs and GPUs.
The CPU backend comprises of Intel(R) Xeon(R) Platinum 8260 CPUs at 2.40GHz, with $256$ GB allocated memory.
We tested on two single-node CPU settings, one with $16$ CPUs and another with $64$ CPUs.
The GPU backend is a single Nvidia A100 GPU.
\subsection{Metrics}
The CutQC runtime is the end-to-end runtime except the QPU time in Algorithm~\ref{alg:dynamic_definition}.
This is because the NISQ QPUs nowadays are small, slow and too noisy for any practical purposes.
The applications of CutQC to useful algorithms at large scales requires medium sized reliable QPUs instead.
It is hence irrelevant to profile the NISQ QPU runtime now.
Furthermore, we expect that the QPU runtime will be negligible as compared to the other parts of the toolflow because
(1) QPUs operate at much faster timescales than post-processing on CPUs and GPUs,
and (2) multiple small QPUs can be used in parallel to reduce the runtime.
In addition, the runtime advantage of QPUs over CPUs will be even more significant for larger circuits.
We expect CutQC to offer more significant advantages over purely classical methods as larger and more reliable QPUs become available.
In addition, we profile the output fidelity of CutQC with IBM's 5-qubit Bogota device to compare the fidelity with directly executing the circuits on IBM's 20-qubit Johannesburg device.
As NISQ devices improve, CutQC can be applied to larger devices to produce useful executions on larger scales.
To quantify the noise behaviors, we used $\chi^2$ loss
\begin{equation}
\chi^2=\sum_{i=0}^{2^n-1}\frac{(a_i-b_i)^2}{a_i+b_i}\label{eq:chi2},
\end{equation}
where $a_i$ are elements of circuit execution probability distributions (from Figure~\ref{fig:eval_modes}b,~\ref{fig:eval_modes}c)
and $b_i$ are elements of the ground truth probability distributions (from Figure~\ref{fig:eval_modes}a).
The smaller the $\chi^2$ is, the better the execution results are.
\subsection{Benchmarks}
We used the following circuits as benchmarks.
\begin{enumerate}
\item \textit{Bernstein--Vazirani} (\textit{BV}).
This quantum algorithm solves the hidden string problem more efficiently than classical algorithms do~\cite{bernstein1997quantum}.
\item \textit{Adder}.
Adder is a quantum ripple-carry adder with one ancilla and linear depth.
It is an important subroutine in quantum arithmetic involving summing two quantum registers of the same width;
hence only even numbers of qubits are valid.
\item \textit{Approximate Quantum Fourier Transform} (\textit{AQFT}).
QFT is a common subroutine in many quantum algorithms that promise speedup over classical algorithms.
AQFT has been proposed to yield better results than QFT on NISQ devices by truncating small angle rotations~\cite{barenco1996approximate}.
\item \textit{Supremacy}.
This is a type of 2-D random circuit with dense probability output.
It was used by Google to demonstrate quantum advantage~\cite{google2019quantum}.
The circuit depth is $10$ in our experiments.
We verified that the rectangular shapes (such as $2*10$) are much easier to be cut and require little postprocessing.
We therefore focused only on the more difficult near-square shapes, with the two dimensions differing by up to $2$ qubits (such as $4*5$).
Hence not all numbers of qubits are valid.
\end{enumerate}
The benchmark circuits represent a general set of circuits for gate-based QC platforms and promising near-term applications.
\subsection{Summary of Experiments}
Previous work has demonstrated significant runtime advantages of the CPU implementations over classical simulations in the FD settings~\cite{tang2021cutqc},
we hence focus on comparing the performance of GPUs versus CPUs in the DD settings for large circuits.
We tested DD query for circuits up to $100$ qubits,
significantly beyond the current classical and quantum limit.
Because no backends are capable of producing accurate circuit executions on this scale,
we used random numbers as the subcircuit output to focus on studying the runtime.
In addition, we tested running circuits in the FD mode on a $5$-qubit IBM QPU,
and compared the output fidelity against direct QC evaluations on a $20$-qubit IBM QPU.
\subsection{Full Definition Query}\label{sec:full_definition_results}
\setlength{\belowcaptionskip}{-5pt}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/fd_runtime.pdf}
\caption{CutQC executes benchmark circuits mapped to quantum devices with up to $\frac{3}{4}$ of the number of qubits in the input circuits.
The horizontal axis shows the number of qubits in the input circuit.
The vertical axis shows the postprocessing runtime of the full definition reconstruction.
CutQC enables FD query almost always faster than classical simulations do.
CutQC offers an average of 60X to 8600X runtime speedup over classical simulation alternatives for different benchmarks.
GPU is the fastest backend as expected.}
\label{fig:fd_expand}
\end{figure}
The size of quantum devices serves as the baseline to
demonstrate CutQC's ability to expand the size of quantum circuits.
The Qiskit runtime of classically simulating the benchmark circuits serves as the
baseline to demonstrate CutQC's ability to speed up quantum circuit evaluations.
The experiments in Figure~\ref{fig:fd_expand} show the effect of different benchmarks, quantum circuit sizes, and quantum computer sizes on postprocessing runtime.
We used 10-, 15-, 20-, and 25-qubit quantum computers and ran benchmark circuits larger than the devices in FD query using 16 compute nodes for postprocessing.
We achieve an average of 60X to 8600X runtime speedup over classical simulation for our benchmarks.
The type of benchmarks, quantum circuit sizes, and available quantum computer sizes are all important contributors to runtime.
First, some benchmarks are harder to cut and require more postprocessing overhead.
Specifically, \textit{Supremacy} \textit{AQFT} are more densely connected circuits and generally require more postprocessing.
Second, larger quantum circuits generally require more postprocessing.
The reason is that executing quantum circuits that significantly exceed the available quantum resources has to rely more on classical computing resources.
In some cases, the classical postprocessing incurred outweighs any benefit from having quantum computers,
and the resulting runtime is longer than classical simulation.
Third, having larger quantum computers generally improves the runtime.
However, having larger quantum computers faces diminishing returns.
The postprocessing overhead eventually plateaus when the quantum computer is large enough to support an efficient partitioning of the circuit.
For example, the 5*7 \textit{Supremacy} circuit is cut into 2 subcircuits with 5 cuts on both 20- and 25-qubit computers and has similar runtime.
\subsection{Real QC Runs}\label{sec:device_results}
\setlength{\belowcaptionskip}{-5pt}
\begin{figure}[t]
\centering
\includegraphics[width=.9\linewidth]{figures/chi2_reductions_ibmq_bogota.pdf}
\caption{Comparison of the 20-qubit Johannesburg quantum computer versus the 5-qubit Bogota device with CutQC.
For each benchmark we find the ideal output distribution via statevector simulation.
We then use this ideal distribution to compute the $\chi^2$ metric for two execution modes:
QC evaluation on the Johannesburg device ($\chi^2_J$) and CutQC evaluation utilizing the Bogota device ($\chi^2_B$).
The reported $\chi^2$ percentage reduction is computed as $100*(\chi^2_J - \chi^2_B) / \chi^2_{J}$.
A distribution that is close to ideal will have a small $\chi^2$ value,
and therefore a positive $\chi^2$ percentage reduction indicates improved performance.
Only the AQFT workloads experience a negative reduction and are omitted.
CutQC achieves an average of $21\%$ to $47\%$ $\chi^2$ reduction for different benchmarks.}
\label{fig:chi2}
\end{figure}
To study the effect of device noise on our toolchain,
we ran experiments on IBM's real quantum devices.
Figure~\ref{fig:chi2} compares the circuit output obtained from
(a) directly executing circuits on the state-of-the-art 20-qubit Johannesburg device
and (b) executing circuits with more than 5 qubits on the 5-qubit Bogota device with CutQC.
We show that CutQC evaluation with small quantum computers produces a lower $\chi^2$ loss and hence outperforms QC evaluation with large quantum computers.
CutQC reduces $\chi^2$ loss by nearly $60\%$ in the best cases.
The experiments stop at $12$ qubits because QC evaluation beyond this point succumbs to the effects of noise and fails to produce meaningful output.
Among the benchmarks, only the AQFT circuits experienced a negative reduction.
This is because AQFT compiled for the current NISQ devices is much deeper than the other benchmarks.
Therefore both QC and CutQC on AQFT have accuracy too low for meaningful comparisons.
As NISQ devices improve in noise and connectivity, we expect AQFT to improve.
Despite requiring more subcircuits and readout,
CutQC evaluates circuits with better fidelity.
The main reason for such improvements is that CutQC runs subcircuits that are both smaller and shallower than the uncut circuit run by the QC mode.
Furthermore, CutQC substitutes the noisy quantum entanglement across subcircuits by noise-free classical postprocessing.
Not only does CutQC need smaller quantum computers,
it also produces better outputs.
Therefore, combined with CutQC, building small but reliable quantum computers becomes much more useful than merely increasing qubit counts at the cost of degrading fidelity.
\section{CIRCUIT CUTTING}
While we refer the readers to~\cite{peng2020simulating} for a proof of the physics theory behind cutting quantum circuits,
this section provides an intuitive understanding of the cutting process and its challenges via an illustrative example.
\input{text/CC_Example.tex}
\input{text/CC_Challenges.tex} |
1,116,691,500,506 | arxiv | \section{INTRODUCTION}
Understanding the structure and kinematics
of spiral galaxies, in particular explaining their rotation curves at
large galactic radii, remains one of the pressing open questions in astrophysics.
Optical observations of galactic rotation curves find that rather than falling off as one
would expect from galaxy models where the mass traces the observed
light, the rotational velocities remain constant at large
radii. These findings are further borne out
by radio observations of the
21 cm line, from HI gas in the outer parts of the
galactic disk. An overview of rotation curves is provided in \cite{Rubin01}, \cite{Persic96} and
\cite{Salucci01}.
There is strong evidence from the CMB data for considerable amounts of
non-baryonic dark matter on the cosmological scale \citep{WMAP03}. While is it
enticing to imagine that the dark matter problems on the galactic and cosmological
scales have a common resolution, this need not necessarily be the case. Our focus
in this paper will be on the galactic dark matter problem as manifested in the
ubiquitous observation of flat rotation curves.
A number of ideas have been put forth to account for flat rotation
curves. All require some form of new physics. These ideas include:
\begin{enumerate}
\item{} New particles: Missing mass in the form of galactic dark matter, most likely non-baryonic \citep{Alcock00}.
\item{} New interactions: Non-gravitational long-range couplings might exist, or gravitational physics might be subject to revisions over large distances.
\item{} New dynamics: scenarios such as Modified Newtonian Dynamics (MOND) in which
gravity from visible matter is the only force acting but the system's response takes on new aspects.
\end{enumerate}
The community consensus at present prefers the dark matter hypothesis, but galactic dark matter has thus far evaded all attempts to detect it. We should strive to test, whenever possible, alternatives to the dark matter scenario.
\subsection{The MOND Approach to the Rotation Curve Puzzle: Novel Dynamics}
Motivated by the observed spiral galaxy rotation curves,
\citet{Milgrom83} proposed a modification of the dynamics of
non-relativistic matter. This modified behavior, termed
MOND for Modification of Newtonian Dynamics, is conjectured to
arise only in the regime of low accelerations. MOND is a proposed
modification to an object's acceleration under an applied force,
such that $a=g/\mu(x)$ where $g$ is the acceleration expected under
Newtonian physics, and $x=a/a_0$ depends upon the MOND acceleration
scale $a_0 \sim 1.2 \times 10^{-10} m/s^2$, with $\mu(x \gg 1) \simeq
1, \mu(x \ll 1) \simeq x$. A commonly adopted form is $\mu(x) =
{{x}/{\sqrt{1+x^2}}}$. In general a gravitating system's behavior
under MOND can be described by taking the Newtonian description and
replacing $G$, the coupling constant, by $G/\mu$, with the
understanding that the dynamics is being altered rather than the
nature of the gravitational interaction.
In this scenario the mass of a galaxy resides in the ordinary
astronomical components that we can detect by their emission or
absorption of electromagnetic radiation, and the galaxy's light
distribution traces out its mass distribution. A review of MOND as
an alternative to dark matter is presented in \citet{Sanders02}.
In the MOND model the response of a test particle to an applied
force depends upon the magnitude of its absolute acceleration
relative to a preferred frame, taken to be the local rest frame of
the microwave background. ``Overacceleration'' at low values of $x$
then produces the observed rotation curves of spiral galaxies. MOND
thereby eliminates the need for dark matter, at the expense of novel
dynamics at low accelerations. MOND does a remarkably good job of
fitting the rotation curves of galaxies across a wide range of
surface brightness, with $a_0$ as the single free parameter
\citep{Sanders02}.
The MOND idea was recently placed on a more formal footing \citep{Beckenstein04},
but our approach will be cast in the original phenomenological framework,
in terms of $\mu(x)$. From this standpoint, the formulation described
above suggests an observational test for the self-consistency of MOND.
Because $\mu(x)$ depends on the (scalar) magnitude of a particle's
total acceleration, comparing the vertical and rotational dynamics of test
particles in the disk of a spiral galaxy provides a means to test for self-consistency.
This paper proposes a framework for carrying out such a test and
illustrates the technique with recent data from M33 \citep{Ciardullo04}.
Other recent efforts to investigate the viability of the MOND
hypothesis using kinematics include using galaxy clusters
\citep{Silk05} and globular clusters of stars \citep{Baum05}. Our
approach differs in that, as discussed in the following section, we
are checking the self-consistency of MOND rather than comparing
the observed kinematics to a prediction.
\section{USING 3-d DISK KINEMATICS TO TEST SELF-CONSISTENCY}
We will adopt the common model for a galactic disk as a mass distribution with a
volume matter density $\rho(r,z)$ given by
\begin{equation}
\rho(r,z)= \rho_0 e^{-r/R_0} sech^2(z/z_0)
\end{equation}
\noindent
where $R_0$ and $z_0$ are characteristic scale lengths in the radial
and vertical directions. In the MOND scenario, the rotational speed
of a thin galaxy with a mass structure described by equation (1)
obeys (adapting the expression for Newtonian physics from
\citet{Pad02})
\noindent
\begin{equation}
v^2(r) = a(r)~ r = {g(r) \over \mu_r(r)} r = {4 \pi G \Sigma_0 R_0 \over \mu_r(r)}
y^2 [I_0(y)K_0(y) - I_1(y)K_1(y)]
\end{equation}
\noindent
where $a$ and $g$ are the MOND and Newtonian accelerations respectively,
$G$ is Newton's gravitational constant,
$R_0$ is the disk scale length,
$\Sigma_0$ is the surface mass density at $r=0$,
$y=r/2R_0$ is a dimensionless radius variable, $I_n(y)$ and
$K_n(y)$ are $n^{th}$ order modified Bessel functions,
and $\mu_r$ gives rise to an object's
modified response in the radial direction.
The vertical kinematics of the objects in the disk are described by
$\sigma_z^2(r) = 2 \pi G \Sigma(r) z_0/\mu_z(r)$, where $\Sigma(r)$ is the
local surface mass density,
$z_0$ is the vertical scale height, and $\mu_z$
accounts for MONDian behavior in the vertical direction.
Given these two expressions and the exponential radial form for $\Sigma(r)$ given in equation (1),
we can construct the dimensionless ``Consistency Parameter" ratio $CP(r)=\mu_r(r)/
\mu_z(r)$.
{\em Regardless of the local dynamical law that describes a star's
response to feeble forces, $CP(r)$ should be
unity, at all radii. This is true even if the system makes a transition from the Newtonian to the
MOND regime.}
The expressions given above allow us to write $CP(r)$ as
\begin{equation}
CP(r) = {\frac {\mu_r(r)} { \mu_z(r)}} =
2{\frac{R_0}{z_0(y)}} {\frac{\sigma^2_z(y)}{v^2_c(y)}}
[y^2 e^{2y} (I_0(y)K_0(y) - I_1(y)K_1(y))].
\end{equation}
For a disk-only system, the combination of the observables on the right side should
equal to unity at all radii in both the
Newtonian (where $\mu$=1) and MONDian (where $\mu <$ 1) regimes. $CP(r)$
can be understood as a parameter testing whether the tracer material
is behaving in a self-consistent fashion. Note that this expression
is independent of both the mass-to-light ratio of the disk material
and of the central surface density of the disk.
Ideally, one would obtain both face-on and edge-on observations of a
single galaxy. This would then allow the measurement of the rotation curve, of the
vertical velocity dispersion, and of the scale lengths $R_0$ and $z_0$. In practice, this is of course not possible.
There is, nevertheless, a realistic possibility of measuring the
{\it radial dependence} of $CP(r)$, using kinematic information alone. A nearly face-on bulgeless spiral galaxy
would provide a powerful testbed. The apparent (projected) rotation curve
would be suppressed by an unknown $sin (i) $ factor, where $i$ is the
inclination angle, giving an observed $v_{obs}=v_c sin(i)$. This would simply
rescale $CP$ by an overall
multiplicative factor, so that while still radius--independent it will differ from
unity by $1/sin^2(i)$. Given that typical rotational velocities are a
few 100 km/s while velocity dispersions are tens of km/s, a tenfold
suppression of the rotation curve is quite tolerable. This corresponds
to using systems with inclination angles as small as a few degrees. The advantage
of using a face-on galaxy is that the line-of-sight velocity
dispersion is a clean measure of vertical velocity dispersion,
uncontaminated by other components of the velocity ellipsoid within
the galactic disk.
The main observational challenge is obtaining high
signal-to-noise measurements and then extracting both the circular velocity field and the
vertical velocity dispersion.
This is a tractable problem. The projected circular velocity produces a Doppler-shifted
centroid of a spectral feature. The velocity dispersion can be determined from either the
broadening of spectral lines, or by the width of the velocity distribution of a set of individual
resolved objects.
We emphasize the fact that as long as the disk scale height $z_0$ is independent of $r$,
and the self-supporting stellar disk dominates $\rho(r,z)$, only velocity data are needed to test
for variation in $CP(r)$.
By selecting nearly face-on galaxies for this test, we lose the ability to
measure their vertical scale height and must instead appeal to a statistical argument that
invokes measurements of edge-on analogous systems. Measurements of the light distribution of edge-on galaxies in the near infrared, using images from the 2Mass survey \citep{Bizyaev02} indicate that for typical galaxies the vertical scale height {\em is}
independent of galactic radius, with typical values of $z_0/R_0$ varying between 0.1 and 0.4.
We will therefore adopt the working hypothesis that
$z_0$ in equation (3) is independent of $r$.
\section{AN ILLUSTRATIVE KINEMATIC SELF-CONSISTENCY TEST USING M33}
There are a few instances where a galaxy disk's vertical velocity
dispersion has been measured (e.g.~\citet{Bottema93}). A recent data set on the
kinematics of M33 \citep{Ciardullo04} provides an interesting test case. These
authors obtained line-of-sight velocity data on 140 planetary
nebulae in M33. This archival data set provides an opportunity for a concrete
example of the $CP$-violation test outlined above.
\subsection{The Kinematic Properties of M33}
This local group galaxy is inclined at 56 degrees to the plane of the sky.
Various determinations (using multiple techniques) yield a distance modulus of
24.8 $\pm$ 0.1 {\it mag}. The radial scale lengths for light are
$R_0^V=$ 2.5 kpc and $R_0^K=$ 1.56 kpc in the V and K bands, respectively.
This implies
(taking the 2Mass-derived typical values for $R_0/z_0$) a likely vertical
scale height $z_0$ in M33 of a few hundred parsecs.
The M33 inclination angle of 56 degrees is not optimal for
our purposes, but Ciardullo et al (2004) provide their best
estimates of M33's rotational velocity and vertical velocity
dispersion as a function of galactocentric distance. Their
results are presented in Table 1. The M33 circular velocity
at $r$=10 kpc implies, in the MOND scenario, that objects at
that radius should experience a threefold increase in their
radial (and hence vertical) acceleration, relative to the
Newtonian value. The typical acceleration component normal
to the disk is $a_{z}\sim \sigma^2_z/z_0$, well into in the
MOND regime. It is therefore the radial component that
determines the kinematics of the objects in the disk. An
extended rotation curve for M33 is presented in
\citet{Corbelli00}.
\begin{deluxetable}{rrrrr}
\tabletypesize{\scriptsize}
\tablecaption{Kinematic Properties of M33, from \citet{Ciardullo04}.
The
first column lists galactocentric distance, the second and third the
rotational velocity and vertical rms velocity dispersion, and the fourth column
shows the inferred value of $\mu_r$, the radial MOND dynamical parameter, based
on the measured circular velocity and radial distance. The circular velocity has
been corrected for projection effects and represents the best estimate for the actual rotation
curve of the galaxy.}
\tablewidth{0pt}
\tablehead{
\colhead{ R(kpc)} & \colhead{$v_c~(km/s)$} & \colhead{$\sigma_z$ (km/s)} &\colhead{~~$\mu_{radial}$~~}
}
\startdata
0.5 & 40 & 21~~ & 0.66 \\
1 & 55 & 18.7 & 0.64 \\
2 & 80 & 17~~ & 0.66 \\
3 & 90 & 14~~ & 0.60 \\
4 & 98 & 12.5 & 0.55 \\
5 & 100 & 10.5 & 0.49 \\
6 & 105& 10~~ & 0.45 \\
7 & 106& 8~~ & 0.41 \\
8 & 107& 7.5 & 0.37 \\
9 & 108& 5.5 & 0.34 \\
10 & 109 & 5~~ & 0.31 \\
\enddata
\label{tab:Table1}
\end{deluxetable}
The observational data in Table 1, in conjunction with equation (3), provide us with the opportunity to map out the radial dependence of the consistency parameter, $CP$, across the face of M33.
\subsection{A Mass-Traces-Light MOND Consistency Analysis}
In the MONDian view, the ordinary astronomical inventory of M33, plus
novel dynamics, produce the observed rotation curve. The light
distribution across M33 should then trace the galaxy's mass
distribution. We have evaluated the radial dependence of $CP$ using
values of $R_0$ obtained from visible and near-IR wavelengths.
The measurements of M33's kinematics from Table 1 were used in
conjunction with equation (3) to determine $CP(r)$, for different
values of the structural parameters $R_0$ and $z_0$. One choice we
made was to set $R_0=$ 1.56 kpc and $z_0=$ 0.4 kpc, corresponding to
roughly the midpoint of the 2Mass aspect ratio distribution. The
resulting $CP(r)$ values are shown in Figure 1. Since for the M33 data
$R_0$ is better constrained than $z_0$ we also explored values of
$z_0$ that produced the $CP(r)$ curve closer to unity, by fitting for
$z_0$ while minimizing the sum $\Sigma(1-CP(r))^2$ for the radii
listed in Table 1. With $R_0$ fixed at 1.56 and 2.5 kpc, the best-fit
values of $z_0$ were 0.23 and 0.079 kpc, respectively, corresponding
to values of $z_0/R_0$ of 0.14 and 0.03. The best-fit value of $z_0$
for the larger disk scale length is only 80 pc, and corresponds an
unreasonably thin disk. The corresponding $CP(r)$ curves for these
cases are also shown in Figure 1.
The curves in Figure 1 indicate that, as shown in equation (3), the value of $R_0$ determines the {\it shape} of the $CP(r)$ curve, and the $z_0$ parameter only provides an overall multiplicative scaling that can be adjusted to drive the average $CP$ value towards unity.
In this mass-traces-light analysis, the $CP$ parameter changes by a factor of 3--10 (depending upon the fit used) between the inner and outer regions of the M33 disk. In this illustrative example the $CP(r)$ behavior appears inconsistent with the MOND scenario. The sense of the discrepancy, with $CP$ values less than one, corresponds to $\mu_z>\mu_r$, so that matter appears to be overaccelerating in the radial direction more than in the vertical direction.
\begin{figure}
\plotone{f1.eps}
\caption{Values of the dimensionless Consistency Parameter, $CP$, in M33 vs. galactic radius, for light-traces-mass scenarios. The plot shows the radial dependence of $CP=\mu_r/\mu_z$, the ratio of the MOND overacceleration parameter in the radial and vertical directions. The curves show $CP(r)$ for galactic structural parameters (in kpc) of ($R_0, z_0$) equal to
(1.56, 0.23)=solid, (1.56, 0.4) = dotted, and (2.5, 0.079) = dashed. A self-consistent MOND galaxy would have $CP=1$ at all radii.}
\end{figure}
\subsection{Relaxing the Mass-Traces-Light Constraint: Allowing a Wider Range of $R_0$ and $z_0$}
The exponential scale length of the M33 disk emission depends upon the passband used to measure the surface brightness, ranging from 1.56 kpc in the K band to 2.5 kpc in the V band. Stepping back from the light-traces-mass approach, is interesting to explore what combinations of $R_0$ and $z_0$ would produce a $CP(r)$ closest to unity.
Allowing both $R_0$ and $z_0$ as free parameters, with no constraints and with no assumption about the galaxy's mass-to-light ratio, the values that best match $CP=1$ are $R_0=5.4$ kpc and $z_0=0.035$ kpc. This corresponds to a remarkably thin disk, with a vertical scale height of only 35 pc.
The $CP(r)$ profile from this more general fit is shown in Figure 2. This provides a better fit to the kinematic observations, but $CP$ still varies by over a factor of two across the face of M33. Also, the value of $z_0/R_0=0.006$ is over an order of magnitude less than typical aspect ratios.
Although we can achieve improved MOND-inspired fits to the kinematic data, these models imply strong radial and vertical gradients in the galaxy's mass-to-light ratio. This is at variance with the elegant what-you-see-is-all-there-is MOND scenario.
\begin{figure}
\plotone{f2.eps}
\caption{Consistency Parameter $CP$ vs. radius after relaxing the light-traces-mass constraint. The solid line corresponds to $(R_0, z_0)=$ (5.4, 0.035) kpc, the best fit to $CP=1$ when both are allowed as unconstrained free parameters. This kinematic fit is improved over those shown in Figure 1, but still exhibits significant variation with radius. Also the resulting aspect ratio is at variance with that seen in other similar galaxies.}
\end{figure}
\section{DISCUSSION}
Our objective is to propose a general technique for testing the self-consistency of MOND, using the existing M33 data as an illustrative example. The vertical and circular motions of a galaxy can be jointly used for this test. Potential weaknesses in the argument presented above include i) the assertion that the vertical scale height of galaxies is radius-independent, ii) modeling the galaxy with the form shown in equation (1), and iii) the implicit assertion that either objects overaccelerate, or they don't. The first issue can be addressed with better observations and more statistics, and the second by a more comprehensive treatment of the system's kinematics.
The third concern, namely the isotropy of MONDian dynamics, is an
interesting issue. If MONDian behavior arises from a modification of
inertia \citep{Milgrom05}, then this scalar quantity will determine an
object's response to {\it any} applied force, and it will exhibit the
same modified dynamics in all directions. On the other hand one might
imagine that MOND only applies component by component, with a modified
response only to those forces that would give rise to accelerations
below the $a_0$ threshold. This could produce a difference in the
radial and vertical dynamics and could perhaps account for a ratio of
$\mu_{radial}/\mu_{vertical}$ that differs from unity. In this
circumstance however a terrestrial Cavendish experiment conducted at
the North or South pole should see differing effective values of $G$ in
different regimes of $\mu$.
Sensible next steps to obtaining observations that are optimally
suited to the test we propose include 1) assessing the relative merits
of planetary nebulae vs. integrated starlight as probes of vertical
velocity dispersion, 2) selecting a favorable list of target galaxies,
and 3) carrying out a set of appropriate observations. It is sensible
to include, as a control, examples of high surface brightness disk
galaxies which should have their inner regions in the Newtonian
disk-dominated regime where $\mu$ =1, to verify that $CP$ is constant
and equal to unity for these systems. H$\alpha$ and 21 cm observations
of the velocity field might also contribute to this technique.
\acknowledgments
We are grateful to J. Beckenstein, G. Bothun, J. Dalcanton, and K. Cook for
interesting conversations about MOND as an alternative to dark
matter. J. Battat, A. Miceli, D. Sherman and the thoughtful students in Harvard's Fall 2005 freshman seminar on the Hidden Universe provided important encouragement. We thank Harvard University and the Department of Energy Office of Science for their support.
|
1,116,691,500,507 | arxiv | \section{Introduction}
Exclusive electroproduction of vector mesons
was suggested in \cite{knnz} as an effective tool in
search for color transparency (CT) \cite{prp}.
The key idea is based upon
absence of strict correlation between the photon's
energy and virtuality, typical for reactions of
quasielastic scattering. Data from the E665 experiment
\cite{e665} nicely confirms the predicted value of
the effect \cite{knnz}. The statistical confidence of the
observed growth of nuclear transparency with the photon
virtuality $Q^2$ is, however, quite modest and new experiments
are planned at lower energies (HERMES, TJNAF, ELFE).\\
Although CT phenomenon is most naturally interpreted in
the quark-gluon representation,
the calculations are easy only at energies $\nu \gg Q^2R_A/2$,
when size of the photon fluctuations is frozen during
propagation through the nucleus. At lower energies the produced
colorless wave packet is developing while it propagates through
the nucleus. Accordingly, absorption in nuclear matter varies
and the expected CT effect may be substantially reduced.
Such an evolution is controlled by the so-called formation time $t_f$,
\beq
\frac{2\nu}{Q^2} < t_f < \frac{2\nu}{m_{V'}^2 - m_V^2}\ .
\label{1}
\eeq
The bottom limit
corresponds to most quickly expanding states of small size,
$r_T^2 \propto 1/Q^2$. In order to observe a full effect of
CT, one should make this time , $t^{min}_f \approx 1/m_Nx_B \gg R_A$,
where $x_B$ is the Bjorken variable.
The upper limit in (\ref{1}) corresponds to a
long evolution of a rather large-size
wave packet consisted mostly from the two lightest
states, $V$ and $V'$. This time $t_f^{max}$ controls the onset of CT.\\
There is another phenomenon, which follows from
the quantum-mechanical uncertainty for the production time of
the final wave packet, which is usually called coherence time,
\beq
t_c \approx \frac{2\nu}{Q^2+m_V^2}
\label{2}\ .
\eeq
This uncertainty may be interpreted as a lifetime of
hadronic fluctuations of the photon. If this time
substantially exceeds the nuclear size, $t_c \gg R_A$,
one deals with a virtual
hadronic, rather than with photonic beam. Correspondingly,
nuclear attenuation increases \cite{hkn2,hkn1,kn1}. \\
Two methods for correct calculation of the wave packet evolution
are known. One was developed in \cite{kz}
using the quark-gluon representation and Feynman path integral technique.
However, as far as the coherence time effects are involved, the
path integrals approach becomes unreasonably complicate, and no
solution is still found.\\
Another approach uses the hadronic basis for
the photon fluctuations, which is dual
to the quark-qluon basis, provided that completeness takes place.
An exact solution, incorporating
both coherence and formation time effects
is found and the results are presented in this talk.
\section{Coherence time}
Coherence (interference) of the vector meson waves produced
at different longitudinal coordinates is important both for
coherent (the nucleus remains intact) and incoherent (the nucleus
breaks up) electroproduction.
Effect of coherence time exists even in Glauber
approximation, although a correct formula for incoherent
electroproduction was derived only recently \cite{hkn2}.
Vector mesons produced at different points
separated by longitudinal
distance $\Delta z$ have
a relative phase shift $q_c\Delta z$, where
$q_c = (Q^2 + m_V^2)/2\nu$ is the
longitudinal momentum transfer in
$\gamma^*N \to VN$,
$Q^2$ and $\nu$ are
the virtuality and energy
of the photon, respectively.
Taking this into account one arrives at the following
expression \cite{hkn2} for nuclear transparency defined as
$Tr=\sigma_A/A\sigma_N$,
\beqn
& &Tr_{inc} =
\frac{\sigma_{tot}^{VN}}{2A\sigma_{el}^{VN}}(\sigma_{in}^{VN}-\sigma_{el}^{VN})
\int d^2b\ \int_{-\infty}^{\infty} dz_2\ \rho(b,z_2)
\int_{-\infty}^{z_2} dz_1\ \rho(b,z_1)\
\nonumber\\
&\times &
e^{iq_c(z_2-z_1)}\
\exp\left[ -{1\over 2}\sigma_{tot}^{VN}\ \int_{z_1}^{z_2}dz
\rho(b,z)\right]
\exp\left[ -\sigma_{in}^{VN}\ \int_{z_2}^{\infty} dz\
\rho(b,z)\right]\
\nonumber\\
&+&
\frac{1}{A\sigma_{in}^{VN}}
\int d^2b \left
[1-e^{-\sigma_{in}^{VN} T(b)}\right ]
-Tr_{coh}\ ,
\label{3}
\eeqn
\beq
Tr_{coh} = \frac{(\sigma_{tot}^{VN})^2}{4A\sigma_{el}^{VN}}
\int d^2b\left |\int_{-\infty}^{\infty} dz\
\rho(b,z)\
e^{iq_cz}
\exp\left[-{1\over 2}\sigma_{tot}^{VN}\int_z^{\infty}
dz'\rho(b,z')\right
]\right |^2\ ,
\label{4}
\eeq
where $Tr_{coh}$
corresponds to the coherent case \cite{bauer,hkn2,hkn1}.
$Tr_{inc}$, in contrast to the coherent case \cite{kn1},
decreases with energy from $Tr_{inc}=
\sigma_{in}^{VA}/A\sigma_{in}^{VN}$
($q_c \gg 1/R_A$) down to
$Tr_{inc}=\sigma_{qel}^{VA}/A\sigma_{el}^{VN}$
($q_c \ll 1/R_A$).
Numerical examples are presented in \cite{hkn1,kn1,hkn2}.\\
Variation of $t_c$
may be caused either by its $\nu$-
or $Q^2$-dependence. In the latter case $t_c$ decreases
with $Q^2$ and the nuclear transparency
grows, what is usually expected to be a signature of CT
\cite{knnz}.
Our results for incoherent
electroproduction of $\rho$-meson on lead are
shown by dashed curves in Fig.~1 (more examples are
in \cite{hkn1,kn1,hkn2}). The predicted $Q^2$-dependence
is so steep that makes it quite problematic to observe a
signal of CT on such a background.
\section{Formation time}
Inclusion of excited states of the vector meson into the multiple
scattering series is known as Gribov's inelastic corrections \cite{gribov}.
CT corresponds to a special tuning of these corrections, when
the diagonal and off diagonal amplitudes cancel in
final state interaction at high $Q^2$. The amplitudes
we use satisfy such a condition,
since we calculate the photoproduction
amplitudes projecting to the $Q^2$-dependent
$q\bar q$ component of the
photon wave function.\\
In the case of incoherent production one has to sum over
all final states of the nucleus. Therefore, the
the wave packet should be
described by density matrix $P_{ij} =
\sum_{A^*}|\psi_i\rangle|\psi_j\rangle^+$.
Wave function $|\psi_i\rangle$,
has components
$\gamma^*$, $V$, $V'$, etc.
The evolution equation for the density matrix reads \cite{hk1},
\beq
i\frac{d}{dz}\widehat P =
\widehat Q \widehat P -
\widehat P \widehat Q^+ -
{i\over 2}\sigma^{V N}_{tot}
\left(\widehat T \widehat P +
\widehat
P \widehat T^+\right) +
i\sigma^{VN}_{el}
\widehat T \widehat P
\widehat T^+\ ,
\label{11}
\eeq
\beq
\widehat Q =
\left(\begin{array}
{cccc}0&0&0&...\\0&q&0&...\\0&0&q'&...
\\.&.&.&...
\end{array}\right)\ ;\ \ \ \
\widehat T =
\left(\begin{array}
{cccc}0&0&0&...\\\lambda&1&\epsilon&...
\\\lambda R&\epsilon&r&...\\
.&.&.&...
\end{array}\right)\ ,
\label{8}
\eeq
where $\widehat Q$ and $\widehat T$ are the $(n+1)\times(n+1)$
matrices, and $n$ is the number
of states involved into consideration.
$q,\ q'\ ...$
are the transferred longitudinal momenta,
$q(q') = (m^2_{V(V')} + Q^2)/2\nu$.
For other parameters we use notations from
\cite{hk}, $r=\sigma_{tot}^{V'N}/\sigma_{tot}^{VN}$,
$\epsilon = f(VN \to V'N)/f(VN \to VN)$ and
$R= f(\gamma N \to V'N)/f(\gamma N \to VN)$.
The value of parameter $\lambda =
f(\gamma N \to V'N)/f(VN \to VN)$
is inessential, since it cancels in nuclear transparency.
The boundary condition for the density matrix is
$P_{ij}(z\to -\infty) = \delta_{i0}\delta_{j0}$.
Note, eq.~(\ref{11}) reproduces
(\ref{3}) if $\epsilon=0$ and eq.~(\ref{4})
if $\sigma_{el}^{VN} = 0$.\\
We calculated the $Q^2$-dependence of nuclear transparency for incoherent
electroproduction of $\rho$-meson on lead
in two-channel approximation, using the parameters in (\ref{8})
as in \cite{kn1} (more examples, including coherent production,
radial excitations and other flavours are in \cite{kn1,hk1}).
The two channels should well reproduce
the onset of CT, while at higher energies, $1/x_B \ll m_NR_A$
the full CT may develop only after inclusion the the higher excitation states.
The results for different energies are shown by solid curves
in Fig.~1.
\begin{figure}[tbh]
\special{psfile=fig1.ps
angle=0. voffset=-250. hoffset=-20.
hscale=45. vscale=45.}
\special{psfile=fig2.ps angle=0. voffset=-250. hoffset=160.
hscale=45. vscale=45.}
\vspace{6.5cm}
\parbox{14cm}
{\caption[Delta]
{\it $Q^2$-dependence of nuclear transparency for
$\rho$-meson electroproduction on lead at
photon energies $\nu = 5,\ 10, 20$ and $30\ GeV$.
Dashed curves correspond to Glauber approximation,
solid curves are calculated with the evolution
equation eq.~(\ref{11}).}
\label{fig1}}
\parbox{14cm}
{\caption[Delta]
{\it The same as in Fig.~1, but with fixed
$t_c=2\nu_{min}/m_{\rho}^2$.}
\label{fig2}}
\end{figure}
Although the growth of nuclear transparency is steeper than what we expect
in Glauber approximation, the difference is too small to be used as a signature
of CT. Even the present state of art of Glauber-model calculations
leaves enough freedom to fit in such a narrow corridor in nuclear
transparency.\\
At this point we would like to soften our pessimism and
suggest a method for unambiguous detection of onset of CT.
The key idea is quite straitforward \cite{hk1}: as soon as
the variation of the coherence time may mock the CT effects,
one should fix $t_c$. This can by done by means of
a special selection
of events with different $Q^2$ and $\nu$.
Starting from minimal energy
$\nu_{min} = t_cQ^2/2$ with real photoproduction one
should increase both $Q^2$ and $\nu$, while $t_c = const$,
in accordance with
(\ref{2}). Our predictions for $Q^2$-dependence of the lead
transparency at different values of minimal energy (or $t_c$)
are depicted in Fig.~2 in comparison with $Q^2$-independent
expectations of Glauber approximation.
Concluding,
we have developed a multichannel
approach to incoherent exclusive electroproduction of
vector mesons off nuclei, which incorporates the effects of
coherent and formation times, as well as CT.
Variation of the coherence time with the photon energy and $Q^2$
causes substantial changes of the nuclear transparency and may mock
onset of CT. We suggest such a mapping
of $\nu$ and $Q^2$ values, which keeps the coherence time constant.
This helps to single out an unambiguous signal of CT at medium energies.
|
1,116,691,500,508 | arxiv | \section{Introduction}
As locally complicated spaces naturally appear in mathematics (examples:
boundaries of groups, limits under Gromov-Hausdorff convergence)
there is an effort to extend homotopy-theoretical concepts to such spaces.
This paper is devoted to a theory of coverings by locally path-connected
spaces. Zeeman's example \cite[6.6.14 on p.258]{HilWyl} demonstrates
difficulty in constructing a theory of coverings by non-locally path-connected
spaces (that example amounts to two non-equivalent classical coverings
with the same image of the fundamental groups).
For coverings in the uniform category see \cite{BP3} and \cite{BDLM1}.
To simplify exposition let us introduce the following concepts:
\begin{definition}\label{PeanoSpacesDef}
A topological space $X$ is an {\bf lpc-space}
if it is locally path-connected.
$X$ is a {\bf Peano space}
if it is locally path-connected and connected.
\end{definition}
Fischer and Zastrow \cite{FisZas} defined {\bf generalized regular coverings}
of $X$
as functions $p\colon \bar X\to X$ satisfying the following conditions
for some normal subgroup $H$ of $\pi_1(X)$:
\begin{itemize}
\item[R1.] $\bar X$ is a Peano space.
\item[R2.] The map $p\colon \bar X\to X$ is a continuous surjection
and $\pi_1(p)\colon \pi_1(\bar X)\to \pi_1(X)$ is a monomorphism onto $H$.
\item[R3.] For every Peano space $Y$, for
every continuous function $f\colon (Y, y)\to (X, x_0)$ with $f_\ast(\pi_1(Y, y)) \subset H$, and for
every $\bar x \in \bar X$ with $p(\bar x) = x_0$, there is a unique continuous
$g\colon (Y,y)\to (\bar X,\bar x))$ with $p \circ g = f$.
\end{itemize}
Our view of the above concept is that of being universal in a certain class of maps
and we propose a different way of defining covering maps
between Peano spaces in Section \ref{SECTION Peano-coverings}.
\par Our first observation is that each path-connected space
$X$ has its universal Peano space $P(X)$, the set $X$ equipped with new topology,
such that the identity function $P(X)\to X$ corresponds to a generalized regular
covering for $H=\pi_1(X)$. That way quite a few results in the literature
can be formally deduced from earlier results for Peano spaces.
\par The way the projection $P(X)\to X$ is characterized in \ref{LPCExistsThm} generalizes to the concept
of {\bf Peano maps} in Section \ref{SECTION Peano-coverings}
and our {\bf Peano covering maps} combine Peano maps with two classical concepts:
Serre fibrations and unique path lifting property.
Peano covering maps possess several properties analogous to the classical
covering maps \cite{Lim} (example: local Peano covering maps are Peano covering maps).
One of them is that they are all quotients $\widehat X_H$ of
the universal path space $\widetilde X$ equipped with the topology
defined in the proof of Theorem 13 on p.82 in \cite{Spa} and used successfully
by Bogley-Sieradski \cite{BogSie} and Fischer-Zastrow \cite{FisZas}.
It turns out the endpoint projection $\widehat X_H\to X$ is a Peano covering
map if and only if it has the uniqueness of path lifts property (see \ref{ProjIsPeanoCMCharThm}).
In an effort to unify Peano covering maps with uniform covering maps of
\cite{BP3} and \cite{BDLM1} (we will explain the connection in \cite{BDLM2})
we were led to a new topology on $\widetilde X_H$ (see Section \ref{SECTION: BS topology}). Its main advantage is that there is a necessary and sufficient condition
for $\widetilde X_H\to X$ to have the unique path lifting property in case $H$ is a normal
subgroup of $\pi_1(X)$. It is $H$ being closed in $\pi_1(X)$.
That explains Theorem 6.9 of \cite{FisZas} as the basic groups there turn out
to be closed in $\pi_1(X)$. As an application of our approach we
show existence of a universal Peano covering map over a given path-connected space.
\par
We thank Sasha Dranishnikov for bringing the work of Fischer-Zastrow \cite{FisZas}
to our attention. We thank Greg Conner, Katsuya Eda, Ale\v s Vavpeti\' c, and Ziga Virk for helpful comments.
\section{Constructing Peano spaces}\label{SECTION: UPeanoSpace}
The purpose of this section is to discuss various ways of constructing
new Peano spaces.
\subsection{Universal Peano space}
In analogy to the universal covering spaces we introduce the following notion:
\begin{definition}\label{LPCspacesDef}
Given a topological space $X$ its {\bf universal lpc-space}
$lpc(X)$ is an lpc-space together with a continuous map (called the {\bf universal Peano map})
$\pi\colon lpc(X)\to X$ satisfying the following universality condition:
\par For any map $f\colon Y\to X$ from an lpc-space $Y$
there is a unique continuous lift $g\colon Y\to lpc(X)$ of $f$ (that means $\pi\circ g=f$).
\end{definition}
\begin{theorem}\label{LPCExistsThm}
Every space $X$ has a universal lpc-space. It is homeomorphic to the set $X$ equipped
with a new topology, the one generated by all path-components of all open subsets of the existing topology of $X$.
\end{theorem}
{\bf Proof. } Let $U$ be an open set in $X$ containing the point $x$ and $c(x,U)$ be the path component of $x$ in $U$.
Since $z\in c(x,U)\cap c(y,V)$ implies $c(z,U\cap V)\subset c(x,U)\cap c(y,V)$,
the family $\{c(x,U)\}$, where $U$ ranges over all open subsets of $X$ and $x$
ranges over all elements of $U$, forms a basis.
Given a map $f\colon Y\to X$ and given an open set $U$ of $X$ containing $f(y)$
one has $f(c(y,f^{-1}(U)))\subset c(f(y),U)$. That proves $f\colon Y\to lpc(X)$
is continuous if $Y$ is an lpc-space. It also proves $lpc(X)$ is locally path-connected
as any path in $X$ induces a path in $lpc(X)$.
\hfill \qed
\begin{remark}\label{ReftoFisZas4.17}
The topology above was mentioned in Remark 4.17 of \cite{FisZas}.
After the first version of this paper was written we were informed by Greg Conner
of his unpublished preprint \cite{ConFea} with David Fearnley, where that topology
is discussed and its properties (compactness, metrizability) are investigated.
\end{remark}
If $X$ is path-connected, then $lpc(X)$ is a {\bf universal Peano space} $P(X)$
in the following sense: given a map $f\colon Z\to X$ from a Peano space $Z$
to $X$ there is a unique lift $g\colon Z\to P(X)$ of $f$.
In the remainder of this section we give sufficient conditions for a function
on an lpc-space to be continuous. Those conditions are in terms of maps
from basic Peano spaces: the arc in the first-countable case and {\bf hedgehogs}
(see Definition \ref{GeneralizedHEarrings})
in the arbitrary case.
\begin{proposition}\label{MapsFromPeanoToFirstCountable}
Suppose $f\colon Y\to X$ is a function from a first-countable lpc-space $Y$. $f$ is continuous if $f\circ g$ is continuous for every path $g\colon I\to Y$
in $Y$.
\end{proposition}
{\bf Proof. } Suppose $U$ is open in $X$. It suffices to show that for each $y\in f^{-1}(U)$
there is an open set $V$ in $Y$ containing $y$ such that the path component
of $y$ in $V$ is contained in $f^{-1}(U)$. Pick a basis of neighborhoods $\{V_n\}_{n\ge 1}$
of $y$ in $Y$ and assume for each $n\ge 1$ there is a path $\alpha_n$ in $V_n$
joining $y$ to a point $y_n\notin f^{-1}(U)$. Those paths can be spliced to one path
$\alpha$
from $y$ to $y_1$ and going through all points $y_n$, $n\ge 2$.
$f\circ \alpha$ starts from $f(y)$ and goes through all points $f(y_n)$, $n\ge 1$.
However, as $U$ is open, it must contain almost all of them, a contradiction.
\hfill \qed
The construction of the topology on $lpc(X)$ in \ref{LPCExistsThm} can be
done in the spirit of the finest topology on $X$ that retains
the same continuous maps from a class of spaces.
\begin{proposition}\label{UniversalArcLiftingConstruction}
Suppose $X$ is a path-connected topological space
and $\mathcal{P}$ is a class of Peano spaces.
The family $\mathcal{T}$ of subsets $U$ of $X$ such that
$f^{-1}(U)$ is open in $Z\in\mathcal{P}$ for any map $f\colon Z\to X$ in the original topology,
is a topology and $\mathcal{P}(X):=(X,\mathcal{T})$ is a Peano space.
\end{proposition}
{\bf Proof. } Since $f^{-1}(U\cap V)=f^{-1}(U)\cap f^{-1}(V)$, $\mathcal{T}$
is a topology on $X$. Suppose $U\in\mathcal{T}$ and $C$ is a path component
of $U$ in the new-topology.
Suppose $f\colon Z\to X$ is a map and $f(z_0)\in C$. As $f^{-1}(U)$ is open,
there is a connected neighborhood $V$ of $z_0$ in $Z$ satisfying $f(V)\subset U$.
As $f(V)$ is path-connected, $f(V)\subset C$ and $C\in \mathcal{T}$.
\hfill \qed
In case of first-countable spaces $X$ we have a very simple characterization of the universal
Peano map of $X$:
\begin{corollary}\label{PeanoForFirstCountable}
If $X$ is a first-countable path-connected topological space, then a map $f\colon Y\to X$
is a universal Peano map if and only if $Y$ is a Peano space, $f$ is a bijection, and $f$
has the path lifting property.
\end{corollary}
{\bf Proof. } Consider $\mathcal{A}(X)$ as in \ref{UniversalArcLiftingConstruction},
where $\mathcal{A}$ consists of the unit interval.
Notice the identity function $P(X)\to \mathcal{A}(X)$ is continuous as $P(X)$ is first-countable
(use \ref{MapsFromPeanoToFirstCountable}). Since the topology on $\mathcal{A}(X)$ is finer
than that on $P(X)$, $P(X)=\mathcal{A}(X)$.
Since $f$ induces a homeomorphism from $\mathcal{A}(Y)$ to $\mathcal{A}(X)$ (due to the uniqueness of
path lifting property of $f$),
the composition $\mathcal{A}(Y)\to \mathcal{A}(X)\to P(X)$ is a homeomorphism and $f\colon Y\to P(X)$
must be a homeomorphism (its inverse is $P(X)\to \mathcal{A}(Y)\to Y$).
\hfill \qed
The construction in \ref{UniversalArcLiftingConstruction} can be used
to create counter-examples to \ref{PeanoForFirstCountable} in case $X$ is not first-countable.
\begin{example}\label{ExampleOfNonFirstCountable}
Let $X$ be the cone over an uncountable discrete set $B$.
Subsets of $X$ that miss the vertex $v$ are declared open if and only if
they are open in the CW topology on $X$. A subset $U$ of $X$ that contains $v$ is declared open
if and only if $U$ contains all but countably many edges of the cone
and $U\setminus\{v\}$ is open in the CW topology on $X$ (that means $X$ is a hedgehog
if $B$ is of cardinality $\omega_1$ - see \ref{GeneralizedHEarrings}).
Notice $\mathcal{A}(X)$ is $X$ equipped with the CW topology,
the identity function $\mathcal{A}(X)\to X$ has the path lifting property but is not a homeomorphism.
\end{example}
{\bf Proof. } Notice every subset of $X\setminus\{v\}$ that meets each edge in at most one point
is discrete. Hence a path in $X$ has to be contained in the union of finitely many
edges. That means $\mathcal{A}(X)$ is $X$ with the CW topology.
\hfill \qed
We generalize \ref{ExampleOfNonFirstCountable} as follows:
\begin{definition}\label{GeneralizedHEarrings}
A {\bf generalized Hawaiian Earring} is the wedge
\par\noindent
$(Z,z_0)=\bigvee\limits_{s\in S} (Z_s,z_s)$ of pointed Peano spaces indexed by a directed set $S$ and equipped
with the following topology (all wedges in this paper are considered with that particular
topology):
\begin{enumerate}
\item $U\subset Z\setminus\{z_0\}$ is open if and only if
$U\cap Z_s$ is open for each $s\in S$,
\item $U$ is an open neighborhood of $z_0$ if and only if
there is $t\in S$ such that $Z_s\subset U$ for all $s > t$
and $U\cap Z_s$ is open for each $s\in S$.
\end{enumerate}
A {\bf hedgehog} is a generalized Hawaiian Earring
$(Z,z_0)=\bigvee\limits_{s\in S} (Z_s,z_s)$ such that each $(Z_s,z_s)$
is homeomorphic to $(I,0)$.
\end{definition}
Our definition of generalized Hawaiian Earrings is different from the definition of Cannon and Conner~\cite{CanCon Big}. Also, the preferred terminology in \cite{CanCon Big}
is that of a {\bf big Hawaiian Earring}.
Observe each generalized Hawaiian Earring is a Peano space.
\begin{lemma}\label{BasicHedgeHogLemma}
Let $S$ be a basis of neighborhoods of $x_0$ in $X$ ordered by inclusion
(i.e., $U \leq V$ means $V\subset U$).
If, for each $U\in S$, $\alpha_U\colon I\to U$ is a path in $U$ starting
from $x_0$, then their wedge
$$\bigvee\limits_{U\in S}\alpha_U\colon \bigvee\limits_{U\in S}(I_U,0_U)\to (X,x_0)$$
is continuous, where $(I_U,0_U)=(I,0)$ for each $U\in S$.
\end{lemma}
{\bf Proof. } Only the continuity of $g=\bigvee\limits_{U\in S}\alpha_U$
at the base-point of the hedgehog $\bigvee\limits_{U\in S}(I_U,0_U)$
is not totally obvious. However, if $V$ is a neighborhood of $x_0$ in $X$,
then $g^{-1}(V)$ contains all $I_U$ if $U\subset V$ and $g^{-1}(V)\cap I_W$
is open in $I_W$ for all $W\in S$.
\hfill \qed
\begin{proposition}\label{MapsFromPeanoToArbitrary}
Suppose $f\colon Y\to X$ is a function from an lpc-space $Y$.
$f$ is continuous if $f\circ g$ is continuous for every map $g\colon Z\to Y$
from a hedgehog $Z$ to $Y$.
\end{proposition}
{\bf Proof. } Assume $U$ is open in $X$ and $x_0=f(y_0)\in U$.
Suppose for each path-connected neighborhood $V$ of $y_0$ in $Y$ there is a path
$\alpha_V\colon (I,0)\to (V,y_0)$ such that $\alpha_V(1)\notin f^{-1}(U)$.
By \ref{BasicHedgeHogLemma} the wedge $g=\bigvee\limits_{V\in S}\alpha_V$
is a map $g$ from a hedgehog to $Y$ (here $S$ is the family of all path-connected neighborhoods
of $y_0$ in $Y$). Hence $h=f\circ g$ is continuous
and there is $V\in S$ so that $I_V\subset h^{-1}(U)$.
That means $f(\alpha_V(I))\subset U$, a contradiction.
\hfill \qed
\subsection{Basic topology on $\widetilde X$}
The philosophical meaning of this section is that many results can be reduced
to those dealing with Peano spaces via the universal Peano space construction. Let us illustrate this point of view by discussing
a topology on $\widetilde X$.
Suppose $(X,x_0)$ is a pointed topological space.
Consider the space $\widetilde X$ of homotopy classes of paths
in $X$ originating at $x_0$.
It has an interesting topology (see the proof of Theorem 13 on p.82 in \cite{Spa})
that has been put to use in \cite{BogSie} and \cite{FisZas}. Its basis consists of
sets $B([\alpha],U)$ ($U$ is open in $X$, $\alpha$ joins $x_0$
and $\alpha(1)\in U$) defined as follows: $[\beta]\in B([\alpha],U)$ if and only if there is a path $\gamma$
in $U$ from $\alpha(1)$ to $\beta(1)$ such that $\beta$ is homotopic rel. endpoints to
the concatenation $\alpha\ast\gamma$.
$\widetilde X$ equipped with the above topology will be denoted by
$\widehat X$ as in \cite{BogSie}.
Both \cite{BogSie} and \cite{FisZas} consider quotient spaces $\widehat X/H$,
where $H$ is a subgroup of $\pi_1(X,x_0)$. We find it more convenient to follow
\cite[pp.82-3]{Spa}:
\begin{definition}\label{BSModHDef}
Suppose $H$ is a subgroup of $\pi_1(X,x_0)$.
Define $\widetilde X_H$ as the set of equivalence classes of paths
in $X$ under the relation $\alpha\sim_H\beta$ defined via $\alpha(0)=\beta(0)=x_0$,
$\alpha(1)=\beta(1)$ and $[\alpha\ast\beta^{-1}]\in H$
(the equivalence class of $\alpha$ under the relation $\sim_H$ will be denoted by $[\alpha]_H$).
\end{definition}
To introduce a topology on $\widetilde X_H$ we define sets $B_H([\alpha]_H,U)$
(denoted by $<\alpha,U>$ on p.82 in \cite{Spa}), where $U$ is open in $X$, $\alpha$ joins $x_0$
and $\alpha(1)\in U$, as follows: $[\beta]_H\in B_H([\alpha]_H,U)$ if and only if there is a path $\gamma$
in $U$ from $\alpha(1)$ to $\beta(1)$ such that $[\beta\ast (\alpha\ast\gamma)^{-1}]\in H$ (equivalently, $\beta\sim_H \alpha\ast\gamma$).
$\widetilde X_H$ equipped with the topology (which we call the
{\bf basic topology on $\widetilde X_H$})
whose basis consists of $B_H([\alpha]_H,U)$, where $U$ is open in $X$, $\alpha$ joins $x_0$
and $\alpha(1)\in U$, is denoted by $\widehat X_H$
in analogy to the notation $\widehat X$ in \cite{BogSie} that corresponds
to $H$ being trivial.
Given a path $\alpha$ in $X$ and a path $\beta$ in $X$ from $x_0$ to $\alpha(0)$ one can define
a {\bf standard lift} $\hat \alpha$ of it to $\widehat X_H$ originating at $[\beta]_H$ by the formula
$\hat \alpha(t)=[\beta \ast\alpha_t]_H$, where $\alpha_t(s)=\alpha(s\cdot t)$
for $s,t\in I$ (see \cite[Proposition 6.6.3]{HilWyl}).
Let us extract the essence of the proof of \cite[Theorem 13 on pp.82--83]{Spa}:
\begin{lemma}\label{SemiSCLemma}
Suppose $X$ is a path-connected space and $H$ is a subgroup of $\pi_1(X,x_0)$.
An open set $U\subset X$ is evenly covered by $p_H\colon \widehat X_H\to X$
if and only if $U$ is locally path-connected and the image of $h_\alpha\colon \pi_1(U,x_1)\to \pi_1(X,x_0)$
is contained in $H$ for any path $\alpha$ in $X$ from $x_0$ to any $x_1\in U$.
\end{lemma}
{\bf Proof. } Recall that $U$ is {\bf evenly covered} by $p_H$ (see \cite[p.62]{Spa}) if $p_H^{-1}(U)$
is the disjoint union of open subsets $\{U_s\}_{s\in S}$ of $\widehat X_H$
each of which is mapped homeomorphically onto $U$ by $p_H$.
Also, recall $h_\alpha\colon \pi_1(U,x_1)\to \pi_1(X,x_0)$ is given by
$h_\alpha([\gamma])=[\alpha\ast\gamma\ast\alpha^{-1}]$.
Suppose $U$ is evenly covered, $\gamma$ is a loop in $(U,x_1)$,
and $\alpha$ is a path from $x_0$ to $x_1$.
If $[\alpha]_H\ne [\alpha\ast\gamma]_H$, then they belong
to two different sets $U_u$ and $U_v$, $u,v\in S$. However, there is a path from
$[\alpha]_H$ to $[\alpha\ast\gamma]_H$ in $p_H^{-1}(U)$ given by the standard lift of $\gamma$, a contradiction.
Thus $[\alpha]_H= [\alpha\ast\gamma]_H$ and
$[\alpha\ast\gamma\ast\alpha^{-1}]\in H$.
To show that $U$ is locally path-connected, take a point $x_1\in U$, pick
a path $\alpha$ from $x_0$ to $x_1$ and select the unique $s\in S$
so that $[\alpha]_H\in U_s$. There is an open subset $V$ of $U$ satisfying
$B_H([\alpha]_H,V)\subset U_s$. As $p_H|U_s$ maps $U_s$ homeomorphically
onto $U$, $p_H(B_H([\alpha]_H,V))$ is an open neighborhood of $x_1$ in $U$
and it is path-connected.
Suppose $U$ is locally path-connected and the image of $h_\alpha\colon \pi_1(U,x_1)\to \pi_1(X,x_0)$
is contained in $H$ for any path $\alpha$ in $X$ from $x_0$ to any $x_1\in U$.
Pick a path component $V$ of $U$ and notice sets $B_H([\beta]_H,V)$,
$\beta$ ranging over paths from $x_0$ to points of $V$, are either identical or disjoint.
Observe $p_H|B_H([\beta]_H,V)$ maps $B_H([\beta]_H,V)$ homeomorphically
onto $V$. Thus each $V$ is evenly covered and that is sufficient to conclude $U$
is evenly covered.
\hfill \qed
\par As in \cite[p.81]{Spa}, given an open cover $\mathcal{U}$ of $X$,
$\pi(\mathcal{U},x_0)$ is the subgroup of $\pi_1(X,x_0)$
generated by elements of the form $[\alpha\ast\gamma\ast\alpha^{-1}]$,
where $\gamma$ is a loop in some $U\in\mathcal{U}$
and $\alpha$ is a path from $x_0$ to $\gamma(0)$.
Here is our improvement of \cite[Theorem 13 on p.82]{Spa} and \cite[Theorem 6.1]{FisZas}:
\begin{theorem}\label{BasicThmOnWidehatX}
If $X$ is a path-connected space and $H$ is a subgroup of $\pi_1(X,x_0)$,
then the endpoint projection $p_H\colon \widehat X_H\to X$
is a classical covering map if and only if $X$ is a Peano space and there is an open covering
$\mathcal{U}$ of $X$ so that $\pi(\mathcal{U},x_0)\subset H$.
\end{theorem}
{\bf Proof. } Apply \ref{SemiSCLemma}.
\hfill \qed
\begin{proposition}\label{UniversalCoveringForXAndPX}
$\widehat {P(X)}_H$
is naturally homeomorphic to $\widehat X_H$ if $X$ is path-connected.
\end{proposition}
{\bf Proof. } Since continuity of $f\colon (Z,z_0)\to (P(X),x_0)$, for any Peano space $Z$,
is equivalent to the continuity of $f\colon (Z,z_0)\to (X,x_0)$, paths in $(P(X),x_0)$
correspond to paths in $(X,x_0)$. Also, $\pi_1(P(X),x_0)\to \pi_1(X,x_0)$
is an isomorphism so $H$ is a subgroup of both $\pi_1(P(X),x_0)$ and $\pi_1(X,x_0)$,
and the equivalence classes of relations $\sim_H$ are identical in both spaces
$\widetilde {P(X)}$ and $\widetilde X$. Notice that basis open sets are identical in $\widehat {P(X)}_H$ and $\widehat X_H$.
\hfill \qed
\begin{remark}\label{ApplyingPXRem}
In view of \ref{UniversalCoveringForXAndPX} some results in \cite{FisZas} dealing
with maps $f\colon Y\to X$, where $Y$ is Peano, can be derived formally from corresponding
results for $f\colon Y\to P(X)$. A good example is Lemma 2.8
in \cite{FisZas}:
\par $p\colon \tilde X \to X$ has the unique path lifting property if and only if
$\tilde X$ is
simply connected.
\par It follows formally from Corollary 4.7 in \cite{BogSie}:
\par The universal endpoint projection $p\colon \hat Z \to Z$ for a connected and locally
path-connected space $Z$ has the unique path lifting property if and only if $\hat Z$ is simply connected.
\end{remark}
When working in the pointed topological category the space $\widehat X_H$
is equipped with the base-point $\widehat x_0$ equal to the equivalence class
of the constant path at $x_0$.
Let us illustrate $\widehat X_H$
in the case of $H=\pi_1(X,x_0)$.
\begin{proposition}\label{CaseHBeingWholeGroup}
If $H=\pi_1(X,x_0)$, then
\begin{itemize}
\item[a.] The endpoint projection
$p_H\colon (\widehat X_H,\widehat x_0)\to (X,x_0)$
is an injection and
\par\noindent
$p_H(B([\alpha]_H,U))$ is the path component of $\alpha(1)$ in $U$,
\item[b.] $\widehat X_H$
is a Peano space,
\item[c.] Given a map $g\colon (Z,z_0)\to (X,x_0)$ from a pointed Peano space to
$(X,x_0)$, there is a unique lift $h\colon (Z,z_0)\to (\widehat X_H,\widehat x_0)$ of $g$
($p_H\circ h=g$).
\end{itemize}
\end{proposition}
{\bf Proof. } a). Clearly, $p_H(B_H([\alpha]_H,U))$ equals path component of $\alpha(1)$ in $U$.
If $[\beta_1]_H$ and $[\beta_2]_H$ map to the same point $x_1$,
then $\beta_1(1)=\beta_2(1)$ and $\gamma=\beta_1\ast\beta_2^{-1}$ is a loop.
Hence $[\gamma]\in H$ and $[\beta_2]_H=[\gamma\ast\beta_2]_H=[\beta_1]_H$
proving $p_H$ is an injection.
\par b) is well-established in both \cite{BogSie} and \cite{FisZas}. Notice it follows from a).
\par c). For each $z\in Z$ pick a path $\alpha_z$ from $z_0$ to $z$ in $Z$.
Define $h(z)$ as $[\alpha_z]_H$ and notice $h$ is continuous as
$h^{-1}(B_H([\alpha_z]_H,U))$ equals the path component of $g^{-1}(U)$ containing $z$
(use Part a)). As $p_H$ is injective, there is at most one lift of $g$.
\hfill \qed
In view of \ref{CaseHBeingWholeGroup} we have a convenient definition
of a universal Peano space in the pointed category:
\begin{definition}\label{PeanoSpacePointedDef}
By the {\bf universal Peano space} $P(X,x_0)$ of $(X,x_0)$
we mean the pointed space $(\widehat X_H,\widehat x_0)$, $H=\pi_1(X,x_0)$,
and the {\bf universal Peano map of $(X,x_0)$}
is the endpoint projection $P(X,x_0)\to (X,x_0)$.
Equivalently, $P(X,x_0)$ is $(P(C),x_0)$, where $C$ is the path component of $x_0$ in $X$.
\end{definition}
Due to standard lifts the endpoint projection
$p_H\colon \widehat X_H\to (X,x_0)$ always has the path lifting property.
Thus the issue of interest is the uniqueness of path lifting property of $p_H$.
Here is a necessary and sufficient condition for $p_H$ to have the
unique path lifting property (compare it to \cite[Theorem 4.5]{BogSie} for
Peano spaces):
\begin{proposition}\label{ProjHasUPLP}
If $X$ is a path-connected space and $x_0\in X$, then the following conditions are equivalent:
\begin{itemize}
\item[a.] $p_H\colon (\widehat X_H,\widehat x_0)\to (X,x_0)$ has the unique path lifting property,
\item[b.] The image of $\pi_1(p_H)\colon \pi_1(\widehat X_H,\widehat x_0)\to \pi_1(X,x_0)$ is contained in $H$.
\end{itemize}
\end{proposition}
{\bf Proof. } a)$\implies$b). Given a loop $\alpha$ in $\widehat X_H$
it must equal the standard lift of $\beta=p_H(\alpha)$. For the standard lift
of $\beta$ to be a loop in $\widehat X_H$ one must have $[\beta]\in H$.
\par
b)$\implies$a) Given a lift $\bar \alpha$ of a path $\alpha$ in $(X,x_0)$
it suffices to show $\bar \alpha(1)=[\alpha]_H$ as that implies $\bar \alpha$
is the standard lift of $\alpha$ (use $\alpha|[0,t]$ instead of $\alpha$).
Pick a path $\beta$ satisfying $\hat\alpha(1)=[\beta]_H$
and let $\hat\beta$ be its standard lift. As $\bar\alpha\ast(\hat\beta)^{-1}$
is a loop in $\widehat X_H$, its image $\gamma=p_H(\bar\alpha\ast(\hat\beta)^{-1})$
generates an element $[\gamma]$ of $H$.
Hence $\alpha\sim \gamma\ast\beta$ and $\bar\alpha(1)=[\beta]_H=[\alpha]_H$.
\hfill \qed
\section{A new topology on $\widetilde X$}\label{SECTION: BS topology}
We do not know how to characterize subgroups $H$ of $\pi_1(X,x_0)$
for which $p_H\colon \widehat X_H\to X$ has the unique path lifting property.
Therefore we will create a new topology on $\widetilde X_H$ for which
analogous question has a satisfactory answer in the case $H$ being a normal subgroup.
Given an open cover $\mathcal{U}$ of $X$, a subgroup $H$ of
$\pi_1(X,x_0)$, a path $\alpha$ in $X$
originating at $x_0$, and $V\in \mathcal{U}$ containing $x_1=\alpha(1)$
define $B_H([\alpha]_H,\mathcal{U},V)\subset \widetilde X_H$ as follows: $[\beta]_H\in B_H([\alpha]_H,\mathcal{U},V)$
if and only if there is
a path $\gamma_0$ in $V$ originating at $x_1=\alpha(1)$
and a loop $\lambda$ at $x_1$ such that
$[\lambda]\in\pi(\mathcal{U},x_1)$ and
$\beta \sim_H \alpha \ast\lambda\ast\gamma_0$.
\par Observe $[\beta]_H\in B_H([\alpha]_H,\mathcal{U},V)$ implies
$B_H([\alpha]_H,\mathcal{U},V)=B_H([\beta]_H,\mathcal{U},V)$
and
\par\noindent
$B_H([\alpha]_H,\mathcal{U}\cap\mathcal{V},V_1\cap V_2)\subset B_H([\alpha]_H,\mathcal{U},V_1)\cap B_H([\alpha]_H,\mathcal{V},V_2)$, so the family
of sets $\{B_H([\alpha]_H,\mathcal{U},V)\}$ forms a basis of a new topology
on $\widetilde X_H$. When we consider $\widetilde X_H$
as a topological space, then we use precisely that topology.
In the particular case of $H=\{1\}$, the trivial subgroup of $\pi_1(X,x_0)$,
we simplify $\widetilde X_H$ to $\widetilde X$.
Observe that, as $\pi_1(X,x_0)$ is the fiber of the endpoint
projection $p\colon \widetilde X\to X$, any subgroup $G$ of $\pi_1(X,x_0)$
can be considered as a subspace of $\widetilde X$ and we may consider it
as a topological space that way.
Notice the identity function $\widehat X_H\to\widetilde X_H$
is continuous. Indeed, $B_H([\alpha]_H,V)\subset B_H([\alpha]_H,\mathcal{U},V)$
for any $V\in \mathcal{U}$ containing $\alpha(1)$.
When dealing with the pointed topological category the space $\widetilde X_H$
is equipped with the base-point $\widetilde x_0$ equal to the equivalence class
of the constant path at $x_0$.
Let us prove a basic functorial property of our construction.
\begin{proposition}\label{FunctorialityOfWidetilde}
Suppose $f\colon (X,x_0)\to (Y,y_0)$ is a map of pointed topological
spaces. If $H$ and $G$ are subgroups of $\pi_1(X,x_0)$ and $\pi_1(Y,y_0)$,
respectively, such that $\pi_1(f)(H)\subset G$,
then $f$ induces a natural continuous function
$\tilde f\colon(\widetilde X_H,\widetilde x_0)\to(\widetilde Y_G,\widetilde y_0)$.
\end{proposition}
{\bf Proof. } Put $\tilde f([\alpha]_H)=[f\circ \alpha]_G$ and notice
$$\tilde f(B_H([\alpha]_H,f^{-1}(\mathcal{U}),f^{-1}(V)))\subset
B_G(\tilde f([\alpha]_H),\mathcal{U},V)$$ for any open covering
$\mathcal{U}$ of $Y$ and any neighborhood $V$ of $\alpha(1)$.
\hfill \qed
In connection to \ref{BasicThmOnWidehatX} let us prove the following:
\begin{proposition}\label{DiscreteFibersOfWidetilde}
If $X$ is a path-connected space and $H$ is a subgroup of $\pi_1(X,x_0)$,
then the following conditions are equivalent:
\begin{itemize}
\item[a)] A fiber of the endpoint projection $p_H\colon \widetilde X_H\to X$
has an isolated point,
\item[b)] The endpoint projection $p_H\colon \widetilde X_H\to X$
has discrete fibers,
\item[c)] There is an open covering
$\mathcal{U}$ of $X$ so that $\pi(\mathcal{U},x_0)\subset H$,
\item[d)] $\widetilde X_H$ is a Peano space and $p_H\colon \widetilde X_H\to P(X)$
is a classical covering map.
\end{itemize}
\end{proposition}
{\bf Proof. } a)$\implies$c). Suppose $[\alpha]_H\in p_H^{-1}(x_1)$ is isolated. There is an open covering $\mathcal{U}$ of $X$ and $V\in \mathcal{U}$ containing $x_1$
such that $B_H([\alpha]_H,\mathcal{U},V)\cap p_H^{-1}(x_1)=\{[\alpha]_H\}$.
Given $\gamma$ in $\pi(\mathcal{U},x_0)$, the homotopy class
$[\alpha^{-1}\ast\gamma\ast\alpha]_H$ belongs to $\pi(\mathcal{U},x_1)$, so
$[\alpha\ast \alpha^{-1}\ast\gamma\ast\alpha]_H=[\gamma\ast\alpha]_H$
belongs to
$B_H([\alpha]_H,\mathcal{U},V)\cap p_H^{-1}(x_1)$.
Hence $[\gamma\ast\alpha]_H=[\alpha]_H$ and
$[\gamma]\in H$.
\par c)$\implies$d).
Suppose there is an open covering
$\mathcal{U}$ of $X$ so that $\pi(\mathcal{U},x_0)\subset H$
and $W$ is a path component of $U\in\mathcal{U}$.
Notice $B_H([\alpha]_H,\mathcal{U},U)$ is mapped by $p_H$
bijectively onto $W$ and that is sufficient for d).
d)$\implies$b) and b)$\implies$a) are obvious.
\hfill \qed
Applying \ref{DiscreteFibersOfWidetilde} to $H$ being trivial one gets
the following (see \cite{Fab} for analogous result in case of
a different topology on the fundamental group):
\begin{corollary}\label{DiscretePiOne}
If $X$ is a path-connected space, then $\pi_1(X,x_0)$
is discrete if and only if $X$ is semilocally simply connected.
\end{corollary}
\begin{proposition}\label{ComparisonOfBSAndOursForHTotal}
If $\pi(\mathcal{V},x_0)\subset H$ for some open cover
$\mathcal{V}$ of $X$, then the identity function $\widehat X_H\to \widetilde X_H$
is a homeomorphism.
\end{proposition}
{\bf Proof. } Let us show $B_H([\alpha]_H,\mathcal{U},W)=B_H([\alpha]_H,W)$
if $\mathcal{U}$ is an open cover of $X$ refining $\mathcal{V}$ and
$W$ is an element of $\mathcal{U}$ containing $\alpha(1)$.
Clearly, $B_H([\alpha]_H,W)\subset B_H([\alpha]_H,\mathcal{U},W)$,
so assume $[\beta]_H\in B_H([\alpha]_H,\mathcal{U},W)$.
There are $h\in H$, $[\lambda]\in\pi(\mathcal{U},\alpha(1))$,
and a path $\gamma$ in $W$ such that $[\beta]=[h\ast\alpha\ast\lambda\ast\gamma]$.
Choose $h_1\in H$ so that $[h_1\ast\alpha]=[\alpha\ast\lambda]$
($h_1=[\alpha\ast \lambda\ast\alpha^{-1}]\in\pi(\mathcal{U},x_0)\subset H$).
Now $[\beta]=[h\ast\alpha\ast\lambda\ast\gamma]=[h\ast h_1\ast\alpha\ast\gamma]$
and $[\beta]_H\in B_H([\alpha]_H,W)$.
\par Now we can show the identity function $\widehat X_H\to \widetilde X_H$
is open: given an open cover $\mathcal{W}$ of $X$ and given
a path $\alpha$ from $x_0$ to $x_1$ pick an element $W$ of $\mathcal{U}=\mathcal{W}\cap\mathcal{V}$ containing $x_1$ and notice $B_H([\alpha]_H,\mathcal{U},W)\subset B_H([\alpha]_H,W)$.
\hfill \qed
\begin{lemma}\label{ProjectionIsOpenLemma}
If $G\subset H$ are subgroups of $\pi_1(X,x_0)$, then the projection $p\colon \widetilde X_G\to \widetilde X_H$ is open.
\end{lemma}
{\bf Proof. } It suffices to show
$p(B_G([\alpha]_G,\mathcal{U},V))=B_H([\alpha]_H,\mathcal{U},V)$.
Clearly,
\par\noindent
$p(B_G([\alpha]_G,\mathcal{U},V))\subset B_H([\alpha]_H,\mathcal{U},V)$,
so suppose $[\beta]_H\in B_H([\alpha]_H,\mathcal{U},V)$
and $[\beta]=[h\ast \alpha\ast \lambda\ast\gamma]$,
where $[\lambda]\in\pi(\mathcal{U},\alpha(1))$ and $\gamma$
is a path in $V$ originating at $\beta(1)$. Observe
$[\beta]_H=[\alpha\ast\lambda\ast\gamma]_H=p([\alpha\ast\lambda\ast\gamma]_G)$.
\hfill \qed
We arrived at the fundamental result for the new topology on $\widetilde X_H$:
\begin{theorem}\label{TopActionTheorem}
Suppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
If $G$ is normal in $\pi_1(X,x_0)$, then $H/G$, identified with the fiber $p^{-1}([\tilde x_0]_H)$
of the projection $p\colon \widetilde X_G\to \widetilde X_H$,
is a topological group and
acts continuously on $\widetilde X_G$
so that
\begin{itemize}
\item[a)] The natural map $(H/G)\times \widetilde X_G\to \widetilde X_G\times \widetilde X_G$ defined by $([\alpha]_G,[\beta]_G)\mapsto ([\alpha*\beta]_G,[\beta]_G)$ is an embedding,
\item[b)] The quotient map from $\widetilde X_G$ to the orbit space corresponds
to the projection $p\colon\widetilde X_G\to \widetilde X_H$.
\end{itemize}
\end{theorem}
{\bf Proof. } The fiber $F$
of the projection $p\colon \widetilde X_G\to \widetilde X_H$
is the set of classes $[\alpha]_G$ such that $[\alpha]\in H$, so it corresponds
to $H/G$. Define $\mu\colon F\times \widetilde X_G\to \widetilde X_G$ as follows: given $[\alpha]_G\in F$ and given $[\beta]_G\in \widetilde X_G$
put $\mu([\alpha]_G,[\beta]_G)= [\alpha\ast\beta]_G$.
To see $\mu$ is well defined assume $[\gamma_1], [\gamma_2]\in G$.
Now $[\gamma_1\ast \alpha\ast\gamma_2\ast\beta]_G[(\alpha\ast\gamma_2\ast\alpha^{-1})\ast (\alpha\ast\beta)]_G=[\alpha\ast\beta]_G$
as $[\alpha\ast\gamma_2\ast\alpha^{-1}]\in G$ due to normality of $G$ in $H$.
\par Suppose $\mathcal{U}$ is an open cover of $X$, $V,V_1\in\mathcal{U}$, and
\begin{enumerate}
\item $[\alpha]_G\in F$, $[\beta]_G\in \widetilde X_G$,
\item $[\alpha_1]_G\in B_G([\alpha]_G,\mathcal{U},V_1)\cap F$,
and $[\beta_1]_G\in B_G([\beta]_G,\mathcal{U},V)$.
\end{enumerate}
Thus $[\alpha_1]=[g_1\ast \alpha\ast \lambda_1]$ for some
$[\lambda_1]\in\pi(\mathcal{U},x_0)$ and $[g_1]\in G$.
Similarly, $[\beta_1]= [g_2\ast \beta\ast\lambda_2\ast\gamma]$,
where $[g_2]\in G$, $[\lambda_2]\in\pi(\mathcal{U},\beta(1))$,
and $\gamma$ is a path in $V$.
Now,
$$[\alpha_1^{-1}\ast\beta_1]_G=[\lambda_1^{-1}\ast \alpha^{-1}\ast g_1^{-1}\ast g_2\ast\beta\ast\lambda_2\ast\gamma]_G=$$
$$[(\lambda_1^{-1}\ast \alpha^{-1}\ast g_1^{-1}\ast g_2\ast\alpha\ast\lambda_1)\ast\lambda_1^{-1}\ast \alpha^{-1}\ast\beta\ast\lambda_2\ast\gamma]_G=$$
$$[\lambda_1^{-1}\ast \alpha^{-1}\ast\beta\ast\lambda_2\ast\gamma]_G=[(\alpha^{-1}\ast\beta)\ast (\beta^{-1}\ast\alpha\ast \lambda_1^{-1}\ast \alpha^{-1}\ast\beta)\ast\lambda_2\ast\gamma]_G \in
B_G([\alpha^{-1}\ast\beta]_G,\mathcal{U},V)$$
as $[\lambda_1^{-1}\ast \alpha^{-1}\ast g_1^{-1}\ast g_2\ast\alpha\ast\lambda_1]\in G$
and
$[\beta^{-1}\ast\alpha\ast \lambda_1^{-1}\ast \alpha^{-1}\ast\beta]\in \pi(\mathcal{U},(\alpha^{-1}\ast\beta)(1))$.
\par The above calculations amount to
$$\rho((F\cap B_G(x,\mathcal{U},V_1))\times B_G(y,\mathcal{U},V))\subset B_G(\rho(x,y),\mathcal{U},V),$$
where $\rho(x,y):=\mu(x^{-1}, y)$,
which implies the following
\begin{enumerate}
\item $F$ is a topological group,
\item $\mu$ is continuous,
\item $(x,y)\to (\mu(x^{-1},y),y)$
from $F\times \widetilde X_G$ onto its image is open.
\end{enumerate}
As the map in (3) is injective,
it is an embedding. Hence $(x,y)\to (\mu(x,y),y)$ is an embedding.
\par To see b) use \ref{ProjectionIsOpenLemma} or check it directly.
\hfill \qed
\section{Path lifting}\label{SECTION: PathLifting}
\begin{definition}\label{PathLiftingDef}
A pointed map $f\colon (X,x_0)\to (Y,y_0)$ has the {\bf path lifting property}
if any path $\alpha\colon (I,0)\to (Y,y_0)$ has a lift $\beta\colon (I,0)\to (X,x_0)$.
\par A surjective map $f\colon X\to Y$ has the {\bf path lifting property}
if for any path $\alpha\colon I\to Y$ and any $y_0\in f^{-1}(\alpha(0))$ there is a lift $\beta\colon I\to X$ of $\alpha$ such that $\beta(0)=y_0$.
\end{definition}
\begin{definition}\label{UniquenessOfPathLiftsDef}
A pointed map $f\colon (X,x_0)\to (Y,y_0)$ has the {\bf uniqueness of path lifts property}
if any two paths $\alpha,\beta\colon (I,0)\to (X,x_0)$ are equal if $f\circ\alpha=f\circ \beta$.
\par A pointed map $f\colon (X,x_0)\to (Y,y_0)$ has the {\bf unique path lifting property}
if it has both the path lifting property and the uniqueness of path lifts property.
\par A map $f\colon X\to Y$ has the {\bf uniqueness of path lifts property}
if any two paths $\alpha,\beta\colon I\to X$ are equal if $f\circ\alpha=f\circ \beta$
and $\alpha(0)=\beta(0)$.
\par A surjective map $f\colon X\to Y$ has the {\bf unique path lifting property}
if it has both the path lifting property and the uniqueness of path lifts property.
\end{definition}
\begin{corollary}\label{UniquePathLiftingForHandG}
Supppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
If $G$ is normal in $\pi_1(X,x_0)$, then the following conditions
are equivalent:
\begin{itemize}
\item[a)] The natural map $\widetilde X_G\to \widetilde X_H$
has the uniqueness of path lifts property,
\item[b)] $\pi_0(H/G)=H/G$, i.e. $H/G$ has trivial path components.
\end{itemize}
\end{corollary}
{\bf Proof. } a)$\implies$b). If $H/G$ has a non-trivial path component, then there is a non-trivial lift of the constant path at the base-point of $\widetilde X_H$.
\par b)$\implies$a). Suppose $\alpha$ and $\beta$ are two lifts of
the same path $\gamma$ in $\widetilde X_H$ and $\alpha(0)=\beta(0)$. By
\ref{TopActionTheorem} there is a path
$\lambda$ in $H/G$ with the property $\lambda(t)\cdot \alpha(t)=\beta(t)$
for each $t\in I$. As $\lambda(0)=1\in H/G$ and $H/G$ has trivial path components,
$\lambda(t)=1\in H/G$ for all $t\in I$ and $\alpha=\beta$.
\hfill \qed
\begin{proposition}\label{FibersAreT2Lemma}
Supppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
If $G$ is normal in $\pi_1(X,x_0)$, then the following conditions are equivalent:
\begin{itemize}
\item[a)] $H/G$ is a $T_0$-space,
\item[b)] $H/G$ is Hausdorff,
\item[c)] Fibers of the projection $p\colon \widetilde X_G\to \widetilde X_H$ are $T_0$,
\item[d)] Fibers of the projection $p\colon \widetilde X_G\to \widetilde X_H$ are Hausdorff,
\item[e)] For each $h\in H-G$ there is
a cover $\mathcal{U}$ such that $(G\cdot h)\cap \pi(\mathcal{U},x_0)=\emptyset$,
\item[f)] $G$ is closed in $H$.
\end{itemize}
\end{proposition}
{\bf Proof. } In view of \ref{TopActionTheorem}, a)$\equiv$c) and b)$\equiv$d).
\par
a)$\implies$e). Assume $H/G$ is $T_0$ and $h\in H- G$.
Since $[\beta]_G\in B_G([\alpha]_G,\mathcal{U},V)$
is equivalent to $[\alpha]_G\in B_G([\beta]_G,\mathcal{U},V)$,
there is an open cover $\mathcal{U}$ and $V\in\mathcal{U}$ containing $x_0$
such that $G\cdot h\notin B_G(G\cdot 1,\mathcal{U},V)$.
That means precisely there is no $\lambda\in\pi(\mathcal{U},x_0)$
such that $G\cdot h=G\cdot \lambda$, hence $(G\cdot h)\cap \pi(\mathcal{U},x_0)=\emptyset$.
\par b)$\equiv$d) and a)$\equiv$c) follow from \ref{TopActionTheorem}.
\par e)$\implies$d).
Suppose $\alpha,\beta$ are two paths in $(X,x_0)$
so that $[\alpha]_H=[\beta]_H$ but $[\alpha]_G\ne [\beta]_G$.
choose $h\in H- G$ satisfying $[h\cdot \alpha]=[\beta]$.
Pick an open cover $\mathcal{U}$ of $X$ satisfying
$G\cdot h\cap \pi(\mathcal{U},x_0)=\emptyset$ and let $V\in\mathcal{U}$ contain $\alpha(1)$.
Suppose $[\gamma]_G\in B_G([\alpha]_G,\mathcal{U},V)\cap B_G([\beta]_G,\mathcal{U},V)$
and $[\gamma]_H=[\alpha]_H$. Let $h_0\in H$ satisfy $[h_0\cdot \alpha]=[\gamma]$.
Choose $\lambda_1,\lambda_2\in\pi(\mathcal{U},\alpha(1))$
such that $G\cdot [h_0\cdot \alpha]=G\cdot \alpha\cdot \lambda_1$ and
$G\cdot [h_0\cdot \alpha]=G\cdot [h\cdot \alpha]\cdot \lambda_2$.
As $G$ is normal in $H$,
$G\cdot h=h\cdot G=G\cdot (\alpha\cdot \lambda_1\cdot\lambda_2^{-1}\alpha^{-1})$,
a contradiction as $\alpha\cdot \lambda_1\cdot\lambda_2^{-1}\cdot \alpha^{-1}\in \pi(\mathcal{U},x_0)$.
b)$\implies$a) is obvious.
e)$\equiv$f). $G$ being closed in $H$ means existence, for each $h\in H- G$, of an open cover $\mathcal{U}$ such that $G\cap B(h,\mathcal{U},V)=\emptyset$
for some $V\in\mathcal{U}$ containing $x_0$.
That, in turn, is equivalent to non-existence of $\lambda\in\pi(\mathcal{U},x_0)$
satisfying $h\cdot\lambda\in G$, i.e. $(G\cdot h^{-1})\cap \pi(\mathcal{U},x_0)=\emptyset$.
\hfill \qed
\begin{corollary}\label{GroupIsClosed}
Suppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
If $G$ is a normal subgroup of $\pi_1(X,x_0)$, then
the following conditions are equivalent:
\begin{itemize}
\item[a.] $H/G$ has trivial components,
\item[b.] $H/G$ has trivial path components,
\item[c.] $G$ is closed in $H$.
\end{itemize}
\end{corollary}
{\bf Proof. } b)$\implies$c). Suppose $H/G$ has trivial path components. In view of
\ref{FibersAreT2Lemma} it suffices to show $H/G$ is $T_0$
to deduce $G$ is closed in $H$.
If there are two points $u$ and $v$ of $H/G$ such that any open subset
of $H/G$ either contains both of them or contains none of them,
then any function $I\to \{u,v\}\subset H/G$ is continuous. Hence $u=v$
as $H/G$ does not contain non-trivial paths.
\par c)$\implies$a). \par
{\bf Claim.} If $h_1,h_2\in H$ and $G\cdot f\in B_H(G\cdot h_1,\mathcal{U},V)\cap B_H(G\cdot h_{2},\mathcal{U},V)\cap (H/G)$ for some open cover $\mathcal{U}$ of $X$
and some $V\in\mathcal{U}$ containing $x_0$, then
$G\cdot h_1^{-1}\cdot h_2\subset G\pi(\mathcal{U},x_0)$.
\par {\bf Proof of Claim:}
$G\cdot f=G\cdot h_1\cdot \lambda_1$ and $G\cdot f=G\cdot h_2\cdot \lambda_2$ for some $\lambda_1,\lambda_2\in\pi(\mathcal{U},x_0)$.
Now $h_1\cdot G=h_2\cdot G\cdot (\lambda_2\cdot \lambda_1^{-1})$
and $(h_1^{-1}\cdot h_2)\cdot G\subset G\cdot (\lambda_1\cdot \lambda_2^{-1})
\subset G\pi(\mathcal{U},x_0)$.
\hfill \qed
Suppose $G$ is closed in $H$ and $h\in H- G$.
By \ref{FibersAreT2Lemma} there is
a cover $\mathcal{U}$ such that $(G\cdot h)\cap \pi(\mathcal{U},x_0)=\emptyset$.
If there is a connected subset $C$ of $H/G$ containing $G\cdot h_1h$ and $G\cdot h_1$ for some $h_1\in H$, we consider the open cover $\{C\cap B_G(z,\mathcal{U},V)\}_{z\in C}$
of $C$ and define the equivalence relation on $C$ determined by that cover
($z\sim z^\prime$ if there is a finite chain $z=z_1,\ldots,z_k=z^\prime$ in $C$
such that $B_G(z_i,\mathcal{U},V)\cap B_G(z_{i+1},\mathcal{U},V)\cap C\ne\emptyset$
for all $i<k$). Equivalence classes of that relation are open, hence closed and must equal $C$.
Thus there is a finite chain $h_1,\ldots,h_k=h_1\cdot h$ in $H$
such that $B_G([h_i]_G,\mathcal{U},V)\cap B_G([h_{i+1}]_G,\mathcal{U},V)\cap (H/G)\ne\emptyset$
for all $i<k$. By Claim there are elements $g_i\in G$ ($i < k$) so that
$g_i\cdot h_i^{-1}\cdot h_{i+1}\in \pi(\mathcal{U},x_0)$. By normality of $G$ in $H$
there is $g\in G$ satisfying $g\cdot \prod\limits_{i=1}^{k-1}h_i^{-1}\cdot h_{i+1}
=g\cdot h\in \pi(\mathcal{U},x_0)$, a contradiction.
\hfill \qed
\begin{theorem}\label{PiOneOfXHAndPathLifting}
If $G$ is a normal subgroup of $\pi_1(X,x_0)$,
then the following conditions are equivalent:
\begin{itemize}
\item[a.] The endpoint projection
$p_G\colon(\widetilde X_G,\widetilde x_0)\to (X,x_0)$ has the unique
path lifting property,
\item[b.] $G$ is closed in $\pi_1(X,x_0)$,
\item[c.] $\pi_1(p_G)\colon \pi_1(\widetilde X_G,\widetilde x_0)\to\pi_1(X,x_0)$
is a monomorphism and its image equals $G$.
\end{itemize}
\end{theorem}
{\bf Proof. } Put $H=\pi_1(X,x_0)$ and observe $\widetilde X_H$
is the Peanification of $(X,x_0)$ by \ref{CaseHBeingWholeGroup}.
a)$\equiv$b). By \ref{UniquePathLiftingForHandG} the group $H/G$ has trivial path components. Use \ref{GroupIsClosed}.
a)$\implies$c). Given a loop in $(\widetilde X_G,\widetilde x_0)$ we may assume
it is a canonical lift of a loop $\alpha$ in $(X,x_0)$.
For that lift to be a loop we must have $[\alpha]\in G$.
Thus the image of $\pi_1(p_G)\colon \pi_1(\widetilde X_G,\widetilde x_0)\to\pi_1(X,x_0)$
equals $G$ (canonical lifts of elements of $G$ show that the image
contains $G$). If $\alpha$ is null-homotopic in $(X,x_0)$,
then its canonical lift is null-homotopic as well.
Thus $\pi_1(p_G)\colon \pi_1(\widetilde X_G,\widetilde x_0)\to\pi_1(X,x_0)$
is a monomorphism.
c)$\implies$a). If $H/G$ has a non-trivial path component (we use~\ref{UniquePathLiftingForHandG}),
then there is a path from the base-point to a different point $[\alpha]_G$
of $H/G$. Concatenating the canonical lift of $\alpha$ with the reverse of that path gives
a loop in $(\widetilde X_G,\widetilde x_0)$ whose image in $\pi_1(X,x_0)$ is $[\alpha]\notin G$,
a contradiction.
\hfill \qed
\begin{proposition}\label{ClosureOfSubgroups}
Suppose $(X,x_0)$ is a pointed topological space and $H$ is a subgroup of $\pi_1(X,x_0)$.
The closure of $H$ in $\pi_1(X,x_0)$
consists of all elements $g\in \pi_1(X,x_0)$ such that for each open
cover $\mathcal{U}$ of $X$ there is $h\in H$ and $\lambda\in\pi(\mathcal{U},x_0)$
satisfying $g=h\cdot \lambda$. If $H$ is a normal subgroup of $\pi_1(X,x_0)$,
then so is its closure.
\end{proposition}
{\bf Proof. } Suppose $g\in \pi_1(X,x_0)$ and for each open
cover $\mathcal{U}$ of $X$ there is $h\in H$ and $\lambda\in\pi(\mathcal{U},x_0)$
satisfying $g=h\cdot \lambda$. Notice $B(g,\mathcal{U})$ contains $h$, so $g$
belongs to the closure of $H$. If $H$ is normal, then
$k\cdot g\cdot k^{-1}=(k\cdot h\cdot k^{-1})\cdot (k\cdot \lambda\cdot k^{-1})$ also belongs to the closure of $H$.
\hfill \qed
\begin{corollary}\label{ExamplesOfClosedSubgroupsOne}
The closure of the trivial subgroup of $\pi_1(X,x_0)$ in $\pi_1(X,x_0)$ equals
$\bigcap\limits_{\mathcal{U}\in COV}\pi(\mathcal{U},x_0)$, where $COV$ stands
for the family of all open covers of $X$.
\end{corollary}
\begin{example}\label{HarmonicArchipelago}
The Harmonic Archipelago $HA$ of Bogley and Sieradski \cite{BogSie}
is a Peano space such that $\pi_1(X,x_0)$ equals
$\bigcap\limits_{\mathcal{U}\in COV}\pi(\mathcal{U},x_0)$. Hence $\pi_1(X,x_0)$
is the only closed subgroup of $\pi_1(X,x_0)$.
$HA$ is built by stretching disks $B(2^{-n},2^{-n-2})$ to form cones
over its boundary with the vertices at height $1$ in the $3$-space.
\end{example}
\begin{corollary}\label{ExamplesOfClosedSubgroupsTwo}
Suppose $(X,x_0)$ is a pointed topological space.
The following subgroups of $\pi_1(X,x_0)$ are closed:
\begin{itemize}
\item[a)] Subgroups $H$ containing $\pi(\mathcal{U},x_0)$ for some open cover $\mathcal{U}$ of $X$,
\item[b)] $\bigcap\limits_{\mathcal{U}\in S}\pi(\mathcal{U},x_0)$
for any family $S$ of open covers of $X$,
\item[c)] The kernel of $\pi_1(f)\colon \pi_1(X,x_0)\to\pi_1(Y,y_0)$
for any map $f\colon (X,x_0)\to (Y,y_0)$ to a pointed semilocally simply connected space.
\item[d)] The kernel of the natural homomorphism $\pi_1(X,x_0)\to\check\pi_1(X,x_0)$
from the fundamental group to the \v Cech fundamental group.
\end{itemize}
\end{corollary}
{\bf Proof. } a) Any subgroup containing $\pi(\mathcal{U},x_0)$ is open. Any open subgroup of a topological group is closed.
\par b) easily follows from a).
\par c) follows from \ref{DiscretePiOne} and \ref{FunctorialityOfWidetilde}
as $\pi_1(f)\colon \pi_1(X,x_0)\to\pi_1(Y,y_0)$ is continuous and
$\pi_1(Y,y_0)$ is discrete.
\par d) follows from c). Indeed $\check\pi_1(X,x_0)$ is defined (see \cite{DydSeg}
or \cite{MarSeg}) as the inverse limit of an inverse system $\{\pi_1(K_s,k_s)\}_{s\in S}$,
where each $K_s$ is a simplicial complex and there are maps
$f_s\colon (X,x_0)\to (K_s,k_s)$ so that for $t > s$ the map $f_s$ is homotopic
to the composition of $f_t$ and the bonding map $(K_t,k_t)\to (K_s,k_s)$.
That means the kernel of the natural homomorphism $\pi_1(X,x_0)\to\check\pi_1(X,x_0)$
is the intersection of kernels of all $\pi_1(f_s)$, $s\in S$.
\hfill \qed
The concept of a space $X$ being {\bf homotopically Hausdorff} was introduced
by Conner and Lamoreaux \cite[Definition 1.1]{ConLam}
to mean that for any point $x_0$ in $X$ and for any non-homotopically trivial
loop $\gamma$ at $x_0$ there is a neighborhood $U$ of $x_0$ in $X$
with the property that no loop in $U$ is homotopic to $\gamma$ rel.$x_0$
in $X$. Subsequently, Fischer and Zastrow \cite{FisZas}) defined a space $X$ to be {\bf homotopically Hausdorff relative to
a subgroup $H$ of $\pi_1(X,x_0)$} if for any $g\notin H$
and for any path $\alpha$ originating at $x_0$ there is an open neighborhood $U$
of $\alpha(1)$ in $X$ such that no element of $H\cdot g$ can be expressed as $[\alpha\ast\gamma\ast\alpha^{-1}]$ for some loop $\gamma$ in $(U,\alpha(1))$.
We generalize this definition as follows:
\begin{definition}\label{HGHausdorffDef}
Suppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
$X$ is {\bf $(H,G)$-homotopically Hausdorff}
if for any $h\in H\setminus G$ and any path $\alpha$
originating at $x_0$ there is an open neighborhood $U$
of $\alpha(1)$ in $X$ such that none of the elements of $G\cdot h$ can be expressed as $[\alpha\ast\gamma\ast\alpha^{-1}]$ for any loop $\gamma$ in $(U,\alpha(1))$.
\end{definition}
Notice $X$ being homotopically Hausdorff relative to $H$ corresponds
to $X$ being $(\pi_1(X,x_0),H)$-homotopically Hausdorff.
Let us characterize the concept of being $(H,G)$-homotopically Hausdorff
in terms of the basic topology on the fundamental group.
\begin{proposition}\label{HGHausdorffAndBasicTopology}
If $G\subset H$ are subgroups of $\pi_1(X,x_0)$,
then $X$ is $(H,G)$-homotopically Hausdorff if and only if for every path
$\alpha$ in $X$ that terminates at $x_0$ the group
$h_\alpha(G)$ is closed in $h_\alpha(H)$ in the basic topology.
\end{proposition}
{\bf Proof. } $h_\alpha(G)$ being closed in $h_\alpha(H)$ means existence,
for each $h\in H\setminus G$, of a neighborhood U of $x_1=\alpha(0)$
such that $B([\alpha\ast h\ast\alpha^{-1}],U)\cap ([\alpha]\cdot G\cdot [\alpha^{-1}])=\emptyset$. Thus, for every loop $\gamma$ in $U$ at $x_1$, there is no $g\in G$
satisfying $[\alpha\ast h\ast\alpha^{-1}\ast\gamma^{-1}]=[\alpha\ast g\ast\alpha^{-1}]$.
The last equality is equivalent to $[g\ast h]=[\alpha^{-1}\ast\gamma\ast\alpha]$
which completes the proof.
\hfill \qed
\begin{example} Proposition \ref{HGHausdorffAndBasicTopology}
allows for an easy construction of subgroups $H$ of $\pi_1(X,x_0)$
such that $X$ is not homotopically Hausdorff relative to $H$.
Namely, $X=S^1\times S^1\times \ldots$ and $H=\bigoplus Z\subset
\prod Z=\pi_1(X)$.
\end{example}
Let us show $G$ being closed in $H$ (in the new topology) is a stronger condition
than $X$ being $(H,G)$-homotopically Hausdorff.
\begin{lemma}\label{ClosedImpliesHH}
Suppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
If $G$ is closed in $H$, then $X$ is $(H,G)$-homotopically Hausdorff.
\end{lemma}
{\bf Proof. } Given $h\in H\setminus G$ pick an open cover $\mathcal{U}$
and $W\in\mathcal{U}$ containing $x_0$
so that $B(h,\mathcal{U},W)$ does not intersect $G$.
Given a path $\alpha$ in $X$ from $x_0$ to $x_1$ choose $V\in\mathcal{U}$
containing $x_1$. Suppose there is a loop $\gamma$ in $(V,x_1)$
so that $[\alpha\ast\gamma\ast\alpha^{-1}]=g\cdot h$ for some $g\in G$.
Now $[\alpha\ast\gamma^{-1}\ast\alpha^{-1}]\in \pi(\mathcal{U},x_0)$
and $g^{-1}=h\ast [\alpha\ast\gamma^{-1}\ast\alpha^{-1}]\in G\cap B(h,\mathcal{U},W)$, a contradiction.
\hfill \qed
\begin{remark}\label{StronglyHHRemark}
The proof of \ref{ClosedImpliesHH} suggests that the trivial subgroup of
$\pi_1(X,x_0)$ being closed is philosophically related
to the concept of $X$ being {\bf strongly homotopically Hausdorff} (see \cite{RepZas}).
Recall a metric space $X$ is strongly homotopically Hausdorff if
for any non-null-homotopic loop $\alpha$ in $X$ there is an $\epsilon > 0$
such that $\alpha$ is not freely homotopic to a loop of
diameter less than $\epsilon$.
\end{remark}
\begin{lemma}\label{HSmallLemma}
Given subgroups $G\subset H$ of $\pi_1(X,x_0)$ the following conditions are equivalent:
\begin{itemize}
\item[a)] The fibers of the natural projection
$p\colon\widehat X_G\to \widehat X_H$ are $T_0$,
\item[b)] The fibers of the natural projection
$p\colon\widehat X_G\to \widehat X_H$ are Hausdorff,
\item[c)] $X$ is $(H,G)$-homotopically Hausdorff.
\end{itemize}
\end{lemma}
{\bf Proof. } a)$\implies$c). Suppose $h\in H\setminus G$ and $\alpha$
is a path in $X$ from $x_0$ to $x_1$.
As $[h\ast\alpha]_G\ne [\alpha]_G$ belong to the same fiber
of $p$, there is a neighborhood $U$ of $x_1$ so that $[h\ast\alpha]_G\notin
B_G([\alpha]_G,U)$ or $[\alpha]_G\notin
B_G([h\ast\alpha]_G,U)$. Notice $[h\ast\alpha]_G\notin
B_G([\alpha]_G,U)$ is equivalent to $[\alpha]_G\notin
B_G([h\ast\alpha]_G,U)$.
Suppose there is a loop $\gamma$ in $(U,x_1)$
so that $g\cdot h=[\alpha\ast\gamma\ast\alpha^{-1}]$ for some $g\in G$.
Now $[h\ast\alpha]_G=[g\cdot h\ast\alpha]_G=[\alpha\ast\gamma]_G\in B_G([\alpha]_G,U)$,
a contradiction.
c)$\implies$b). Any two different elements of the same fiber of $p$ can be represented
as $[h\ast\alpha]_G\ne [\alpha]_G$ for some path $\alpha$
in $X$ from $x_0$ to $x_1$ and some $h\in H\setminus G$.
Choose a neighborhood $U$ of $x_1$ with the property
that none of the elements of $G\cdot h$ can be expressed as $[\alpha\ast\gamma\ast\alpha^{-1}]$ for any loop $\gamma$ in $(U,x_1)$.
Suppose $[\beta]_G\in (H/G)\cap B_G([\alpha]_G,U)\cap B_G([h\ast\alpha]_G,U)$.
That means existence of loops $\gamma_1,\gamma_2$
in $(U,x_1)$ so that $[\beta]_G=[h\ast\alpha\ast\gamma_1]_G=[\alpha\ast\gamma_2]_G$.
Hence $[h]_G=[\alpha\ast(\gamma_2\ast\gamma_1^{-1})\ast \alpha^{-1}]_G$,
a contradiction.
\hfill \qed
\begin{lemma}\label{NewHMediumLemma}
Supppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$, $G$ is normal in $\pi_1(X,x_0)$,
and $X$ is $(H,G)$-homotopically Hausdorff.
If $\alpha,\beta\colon (I,0)\to (\widehat X_G,\widehat x_0)$ are two
continuous lifts of the same path
$\gamma\colon (I,0)\to (\widehat X_H,\widehat x_0)$, then for every $h\in H$ the set
$$S=\{t\in I | \alpha(t)= h\cdot \beta(t)\}$$ is closed.
\end{lemma}
{\bf Proof. }
Choose paths $u_t,v_t$ in $(X,x_0)$
so that $\alpha(t)=[u_t]_G$ and $\beta(t)=[v_t]_G$ for all $t\in I$.
Assume $[u_t]_G\ne [h\cdot v_t]_G$ for some $t\in I$.
Pick a neighborhood $U$ of $x_1=u_t(1)$ so that
$[v_t\ast u_t^{-1}]\cdot h\cdot G\ne [v_t\ast \gamma\ast v_t^{-1}]\cdot G$
for any loop $\gamma$ in $(U,x_1)$.
There is a neighborhood $V$ of $t$ in $I$ so that
$[u_s]_G\in B_G([u_t]_G,U)$ and $[v_s]_G\in B_G([v_t]_G,U)$
for all $s\in V$. That means $[u_s]=[g_1\ast u_t\ast\gamma_1]$
and $[v_s]=[g_2\ast v_t\ast \gamma_2]$
for some $g_1,g_2\in G$ and some paths $\gamma_1,\gamma_2$
in $U$ joining $x_1$ and $u_1(1)=v_s(1)$.
Put $\gamma=\gamma_1\ast \gamma_2^{-1}$
and notice $[u_s\ast v_s^{-1}]=[g_1\ast u_t\ast v_t^{-1}\ast (v_t\ast \gamma\ast v_t^{-1})\ast g_2^{-1}]$.
As $G$ is normal in $\pi_1(X,x_0)$, there is $g_3\in G$
satisfying
$[g_1\ast u_t\ast v_t^{-1}\ast (v_t\ast \gamma\ast v_t^{-1})\ast g_2^{-1}]=[g_3\ast u_t\ast v_t^{-1}\ast (v_t\ast \gamma\ast v_t^{-1})]$
and that element cannot belong to $G\cdot h$ by the choice of $U$.
\hfill \qed
\begin{corollary}\label{HAndPathLifting}
Supppose $G\subset H$ are subgroups of $\pi_1(X,x_0)$.
If $H/G$ is countable, $G$ is normal in $\pi_1(X,x_0)$,
and $X$ is $(H,G)$-homotopically Hausdorff, then the natural map
$\widehat X_G\to \widehat X_H$ has the uniqueness of path lifts property.
\end{corollary}
{\bf Proof. } Pick representatives $h_i\in H$, $i\ge 1$, of all right cosets of $H/G$
so that $h_1=1$.
If $\alpha$ and $\beta$ are two continuous lifts
in $\widehat X_G$ of the same path in $\widehat X_H$,
then each set $S_i= \{t\in I | \alpha(t)= h_i\cdot \beta(t)\}$ is closed,
they are disjoint, and their union is the whole interval $I$.
Hence only one of them is non-empty and it must be $S_1$.
Thus $\alpha=\beta$.
\hfill \qed
\section{Peano maps}\label{SECTION Peano-maps}
This section is about one of the main ingredients of our theory
of covering maps for lpc-spaces. It amounts to the following generalization of Peano spaces:
\begin{definition}\label{PeanoMapDef}
A map $f\colon X\to Y$ is a {\bf Peano map} if the family of path components
of $f^{-1}(U)$, $U$ open in $Y$, forms a basis of neighborhoods of $X$.
\end{definition}
Notice $X$ is an lpc-space if $f\colon X\to Y$ is a Peano map.
One may reword the above definition as follows:
$X$ is an lpc-space and lifts of short paths in $Y$ are short in $X$.
Indeed, given a neighborhood $U$ of $x_0\in X$ there is a neighborhood $V$
of $f(x_0)$ in $Y$ such that any path $\alpha$ in $(f^{-1}(V),x_0)$
(i.e. $f\circ \alpha$ is contained in $V$, hence short) must be contained in $U$.
\begin{proposition}\label{ProductOfPeanoMaps}
Any product of Peano maps is a Peano map.
\end{proposition}
{\bf Proof. } Suppose $f_s\colon X_s\to Y_S$, $s\in S$, are Peano maps.
Observe $X=\prod\limits_{s\in S}X_s$ is an lpc-space.
Given a neighborhood $U$ of $x=\{x_s\}_{s\in S}\in X$, we find a finite subset $T$ of $S$
and neighborhoods $U_s$ of $x_s$ in $X_s$ such that
$\prod\limits_{s\in S}U_s\subset U$ and $U_s=X_s$ for $s\notin T$.
Choose neighborhoods $V_s$ of $f_s(x_s)$ in $Y_s$, $s\in T$,
so that the path-component of $x_s$ in $f_s^{-1}(V_s)$ is contained in $U_s$.
Put $V_s=X_s$ for $s\notin T$ and observe the path component
of $x$ in $f^{-1}(V)$, $f=\prod\limits_{s\in S}f_s$ and $V=\prod\limits_{s\in S}V_s$,
is contained in $U$.
\hfill \qed
Here is our basic class of Peano maps:
\begin{proposition}\label{EndpointProjectionIsPeano}
If $H$ is a subgroup of $\pi_1(X,x_0)$, then the endpoint projection
$p_H\colon\widehat X_H \to X$ is a Peano map.
\end{proposition}
\begin{proof}
It suffices to show that for any $U$ open in $X$ the path component of any $
[\alpha]_H$ in $p_H^{-1}(U)$ is precisely $B_H([\alpha]_H,U)$. It's straightforward that $
B_H([\alpha ]_H,U)$ is path-connected so suppose $\beta $ is a path in $
p_H^{-1}(U)$ starting at $[\alpha]_H.$ We wish to show that
$\beta ([0,1])\subset B_H([\alpha]_H,U).$ Let $T=\{t:\beta (t)\in B_H([\alpha]_H,U)\}.$ Now $T$ is
nonempty since $\beta (0)=[\alpha]_H$ and open as the inverse image of an open set.
It suffices to prove $[0,t)\subset T$ implies $[0,t]\subset T$.
Set $\beta (t)=[b]_H.$
Now $p_H\beta ([0,1])\subset U$ so in particular $p_H([b]_H)\in U.$ Consider $
B_H([b]_H,U).$ There is an $\varepsilon >0$ such that $\beta (t-\varepsilon
,t]\subset B_H([b]_H,U).$ Pick $s\in (t-\varepsilon
,t)$. Then $\beta (s)=[c_{1}]_H$ and $
[b]_H=[b_{1}]_H$ such that $c_{1}\simeq b_{1}\ast \gamma _{1}$ for some $\gamma
_{1}$ with $\gamma _{1}[0,1]\subset U.$ But $\beta (s)\in B_H([\alpha]_H,U)$ so $
\beta (s)=[c_{2}]_H$ and $[\alpha]_H=[a_{1}]_H$ such that $c_{2}\simeq a_{1}\ast
\gamma _{2}$ for some $\gamma _{2}$ with $\gamma _{2}([0,1])\subset U.$ Then $
b\simeq_H b_{1}\simeq c_{1}\ast \gamma _{1}^{-1}\simeq_H c_{2}\ast \gamma
_{1}^{-1}\simeq a_{1}\ast \gamma _{2}\ast \gamma _{1}^{-1}\simeq_H a\ast
\gamma _{2}\ast \gamma _{1}^{-1}$ and $(\gamma _{2}\ast \gamma
_{1}^{-1})([0,1])\subset U$ so $[b]_H\in B_H([\alpha]_H,U)$ and $t\in T.$ Therefore $
T=[0,1].$
\end{proof}
In analogy to path lifting and unique path lifting properties (see \ref{PathLiftingDef}
and \ref{UniquenessOfPathLiftsDef})
one can introduce the corresponding concepts for hedgehogs:
\begin{definition}\label{HedgehogLiftingDef}
A surjective map $f\colon X\to Y$ has the {\bf hedgehog lifting property}
if for any map $\alpha\colon \bigvee\limits_{s\in S} I_s\to Y$ from a hedgehog and any $y_0\in f^{-1}(\alpha(0))$ there is a
continuous lift $\beta\colon \bigvee\limits_{s\in S} I_s\to X$ of $\alpha$ such that $\beta(0)=y_0$.
\end{definition}
\begin{definition}\label{UniquenessOfHedgehogLiftsDef}
$f\colon X\to Y$ has the {\bf unique hedgehog lifting property}
if it has both the hedgehog lifting property and the uniqueness of path lifts property.
\end{definition}
\begin{theorem}\label{HedgehogPeanoTheorem}
If $f\colon X\to Y$ has the unique hedgehog lifting property, then
\par
\noindent
$f\colon lpc(X)\to Y$ is a Peano map.
\end{theorem}
{\bf Proof. } Assume $U$ is open in $X$ and $x_0\in U$.
Suppose for each neighborhood $V$ of $f(x_0)$ in $X$ there is a path
$\alpha_V\colon (I,0)\to (f^{-1}(V),x_0)$ such that $\alpha_V(1)\notin U$.
By \ref{BasicHedgeHogLemma} the wedge $\bigvee\limits_{V\in S}f\circ\alpha_V$
is a map $g$ from a hedgehog to $Y$ (here $S$ is the family of all neighborhoods
of $f(x_0)$ in $Y$). Its lift must be the wedge $h=\bigvee\limits_{V\in S}\alpha_V$.
However $h^{-1}(U)$ is not open in $lpc(X)$, a contradiction.
\hfill \qed
\begin{definition}\label{PeanoFunctorDef}
Given a map $f\colon X\to Y$ of topological spaces
its {\bf Peano map} $P(f)\colon P_f(X)\to Y$ is $f$ on $X$ equipped with
the topology generated by path components of sets $f^{-1}(U)$,
$U$ open in $Y$.
\end{definition}
Notice that in the case of $f=id_X$ the range $P_{id_X}(X)$ of $P(id_X)$, where $id_X\colon X\to X$
is the identity map, is identical to $lpc(X)$ as defined in
\ref{LPCExistsThm}.
Recall $f\colon X\to Y$ is a {\bf Hurewicz fibration} if
every commutative diagram
$$
\begin{CD}
K\times \{0\} @> \alpha >> X \\
@V VV @VV f V \\
K\times I @> H >> Y
\end{CD}
$$
\noindent
has a filler $G\colon K\times I\to X$ (that means $f\circ G=H$
and $G$ extends $\alpha)$.
If the above condition is satisfied for $K$ being any $n$-cell $I^n$, $n\ge 0$
(equivalently, for any finite polyhedron $K$), then $f$
is called a {\bf Serre fibration}. Notice for $K$ being a point
this is the classical {\bf path lifting property}.
If the above condition is satisfied for $K$ being any hedgehog, then $f$
is called a {\bf hedgehog fibration}.
If the above condition is satisfied for $K$ being any Peano space, then $f$
is called a {\bf Peano fibration}.
We will modify those concepts for maps between pointed spaces as follows:
\begin{definition}\label{SerreFibrationPropDef}
A map $f\colon (X,x_0)\to (Y,y_0)$ is a {\bf Serre $1$-fibration}
if any commutative diagram
$$
\begin{CD}
(I\times \{0\},(\frac{1}{2},0)) @> \alpha >> (X,x_0) \\
@V VV @VV f V \\
(I\times I,(\frac{1}{2},0)) @> H >> (Y,y_0)
\end{CD}
$$
\noindent has a filler $G\colon (I\times I,(\frac{1}{2},0))\to (X,x_0)$ (that means $f\circ G=H$
and $G$ extends $\alpha)$.
\end{definition}
Observe Serre $1$-fibrations have the path lifting property
in the sense that any path in $Y$ starting at $y_0$ lifts to a path in $X$
originating at $x_0$.
\begin{theorem}\label{MainCoveringTheorem}
Suppose $$
\begin{CD}
(T,z_0) @> g_1 >> (X,x_0) \\
@V i VV @VV f V \\
(Z,z_0) @> g >> (Y,y_0)
\end{CD}
$$
is a commutative diagram in the topological category
such that $(Z,z_0)$ is a Peano space and $i$ is the inclusion from a path-connected
subspace $T$ of $Z$.
If $f$ is a Serre $1$-fibration, then there is a continuous lift $h\colon (Z,z_0)\to (P_f(X),x_0)$ of $g$ extending $g_1$ if the image of $\pi_1(g)\colon \pi_1(Z,z_0)\to \pi_1(Y,y_0)$
is contained in the image of $\pi_1(f)\colon \pi_1(X,x_0)\to \pi_1(Y,y_0)$.
\end{theorem}
{\bf Proof. } For each point $z\in Z$ pick a path $\alpha_z$ in $Z$ from $z_0$ to $z$
and let $\beta_z$ be a lift of $g\colon\alpha_z\mapsto Y$. In case of $z=z_0$ we pick the constant paths $\alpha_z$ and $\beta_z$. In case $z\in T$
the path $\alpha_z$ is contained in $T$
and $\beta_z=g_1\circ\alpha_z$.
Define $h\colon (Z,z_0)\to (P_f(X),x_0)$ by $h(z)=\beta_z(1)$.
Given a neighborhood $U$ of $g(z)$ in $Y$, let $V$ be the path component
of $h(z)$ in $f^{-1}(U)$ and let $W$ be the path component
of $g^{-1}(U)$ containing $z$. Our goal is to show $h(W)\subset V$
as that is sufficient for $h\colon (Z,z_0)\to (P_f(X),x_0)$ to be continuous.
For any $t\in W$ choose a path $\mu_t$ in $W$ from $z$ to $t$.
Let $\gamma$ be a loop in $X$ at $x_0$ so that $f(\gamma)$ is homotopic
to $g(\alpha_z\ast\mu_t\ast\alpha_t^{-1})$. Notice $f(\beta_z)$
is homotopic to $f(\gamma\ast \beta_t)$ via a homotopy $H$ so that
$H(\{1\}\times I)\subset U$. By lifting that homotopy to $X$
we get a path in $f^{-1}(U)$ from $h(z)$ to $h(t)$, i.e., $h(t)\in V$.
\hfill \qed
\begin{corollary}\label{PeanoMapsAndFibrations}
A Peano map $f\colon X\to Y$ is a Peano fibration if and only if it is
a Serre $1$-fibration.
\end{corollary}
{\bf Proof. } Assume $f\colon X\to Y$ is a Peano map and a Serre $1$-fibration
(in the other direction \ref{PeanoMapsAndFibrations} is left as an exercise),
$g\colon Z\times\{0\}\to X$ is a map from a Peano space,
and $H\colon Z\times I\to Y$ is a homotopy starting from $f\circ g$.
Pick $z_0\in Z$ and put $x_0=g(z_0,0)$, $y_0=f(x_0)$.
Notice the image of $\pi_1(g)\colon \pi_1(Z\times \{0\}, (z_0,0))\to \pi_1(Y,y_0)$
is contained in the image of $\pi_1(f)$.
Use
\ref{MainCoveringTheorem} to produce an extension $G\colon Z\times I\to X$
of $g$ that is a lift of $H$.
\hfill \qed
\section{Peano covering maps}\label{SECTION Peano-coverings}
\ref{MainCoveringTheorem} suggests the following concept:
\begin{definition}\label{PeanoCoveringDef}
A map $f\colon X\to Y$ is called a {\bf Peano covering map}
if the following conditions are satisfied:
\begin{enumerate}
\item $f$ is a Peano map,
\item $f$ is a Serre fibration,
\item The fibers of $f$ have trivial path components.
\end{enumerate}
\end{definition}
Notice 3) above can be replaced by $f$ having the unique path lifting property
(see \ref{PointedSerrePlusFibersImpliesUPLPLem}).
Also notice that, in case fibers of a Peano map $f\colon X\to Y$ are $T_0$ spaces,
path-components of fibers are trivial. Indeed, two points
in a path-component of a fiber are always in any open set that contains one of them.
\begin{proposition}\label{ProductOfPeanoCoveringMaps}
Any product of Peano covering maps is a Peano covering map.
\end{proposition}
{\bf Proof. }
Suppose $f_s\colon X_s\to Y_S$, $s\in S$, are Peano covering maps.
Put $f=\prod\limits_{s\in S}f_s$, $X=\prod\limits_{s\in S}X_s$,
and $Y=\prod\limits_{s\in S}Y_s$.
By \ref{ProductOfPeanoMaps} $f$ is a Peano map. It is obvious $f$
is a Serre fibration and has the uniqueness of path lifting property.
\hfill \qed
\begin{corollary}\label{MainPropertyOfPeanoCoverings}
Suppose $f\colon (X,x_0)\to (Y,y_0)$ is a Peano covering map.
If $(Z,z_0)$ is a Peano space, then any map $g\colon (Z,z_0)\to (Y,y_0)$ has a unique continuous lift $h\colon (Z,z_0)\to (X,x_0)$ if the image of $\pi_1(g)$
is contained in the image of $\pi_1(f)$.
\end{corollary}
{\bf Proof. } By \ref{MainCoveringTheorem} a lift $h$ exists and is unique by the uniqueness of
path lifting property.
\hfill \qed
Our basic example of Peano covering maps is related to the basic topology:
\begin{theorem}\label{ProjIsPeanoCMCharThm}
If $X$ is a path-connected space and $x_0\in X$, then the following conditions are equivalent:
\begin{itemize}
\item[a.] $p_H\colon (\widehat X_H,\widehat x_0)\to (X,x_0)$ has the unique path lifting property,
\item[b.] $p_H\colon \widehat X_H\to X$ is a Peano covering map.
\end{itemize}
\end{theorem}
{\bf Proof. } a)$\implies$b). In view of \ref{EndpointProjectionIsPeano}
and \ref{PointedSerreImpliesSerreLem} it suffices to show
$p_H\colon (\widehat X_H,\widehat x_0)\to (X,x_0)$ is a Serre fibration.
Suppose $f\colon (Z,z_0)\to (X,x_0)$ is a map from a simply connected Peano space $Z$
(the case of $Z=I^n$ is of interest here). There is a standard lift $g\colon (Z,z_0)\to
\widehat X_H$ of $f$ defined as $g(z)=[\alpha_z]_H$, where $\alpha_z$ is a path
in $Z$ from $z_0$ to $z$. If $T$ is a path-connected subspace of $Z$ containing $z_0$
and $h\colon (T,z_0)\to (\widehat X_H,\widehat x_0)$ is any continuous lift of $f|T$,
then $h=g|T$ due to the uniqueness of the path lifting property of $p_H$.
That proves $p_H$ is a Serre fibration in view of \ref{PointedSerreImpliesSerreLem}.
\par
b)$\implies$a) is obvious.
\hfill \qed
\begin{theorem}\label{CharacterizationOfPeanoCoverings}
If $f\colon X\to Y$ is a map and $X$ is an lpc-space, then the following conditions are
equivalent:
\begin{itemize}
\item[a)] $f$ is a Peano covering map,
\item[b)] $f$ is a Peano fibration and has the uniqueness of path lifting property,
\item[c)] $f$ is a hedgehog fibration and has the uniqueness of path lifting property,
\item[d)] For any $x_0\in X$ and any map $g\colon (Z,z_0)\to (Y,f(x_0))$
from a simply-connected Peano space there is a lift
$h\colon (Z,z_0)\to (X,x_0)$ of $g$ and that lift is unique.
\end{itemize}
\end{theorem}
{\bf Proof. } a)$\implies$b). Suppose $H\colon Z\times I\to Y$ is a homotopy, $Z$
is a Peano space, and $G\colon Z\times\{0\}\to X$ is a lift of
$H|Z\times\{0\}$. Pick $z_0\in Z$, put $x_0=G(z_0,0)$ and $y_0=f(x_0)$,
and notice $im(\pi_1(Z\times I,(z_0,0)))\subset im(\pi_1(f))$.
Using \ref{MainCoveringTheorem} there is a lift of $H$ and that lift
is unique, hence it agrees with $G$ on $Z\times\{0\}$.
\par b)$\implies$c) is obvious.
\par d)$\implies$c) is obvious.
\par a)$\implies$b) follows from \ref{MainCoveringTheorem}.
\par c)$\implies$a). Notice $f$ has the unique hedgehog lifting property
and is a Serre $1$-fibration. By \ref{HedgehogPeanoTheorem} $f$
is a Peano map.
\hfill \qed
\begin{corollary}\label{CompositionOfPeanoCoverings}
Suppose $f\colon X\to Y$ and $g\colon Y\to Z$
are maps of path-connected spaces and $Y$ is a Peano space. If any two of $f$, $g$, $h=g\circ f$ are Peano
covering maps, then so is the third provided its domain is an lpc-space.
\end{corollary}
{\bf Proof. } In view of \ref{CharacterizationOfPeanoCoverings} it amounts
to verifying that the map has uniqueness of lifts of simply-connected Peano spaces,
an easy exercise.
\hfill \qed
\begin{proposition}\label{LocalPeanoCoverings}
Suppose $f\colon X\to Y$ is a map.
\begin{itemize}
\item[a.] If $f\colon X\to Y$ is a Peano covering map, then $f\colon f^{-1}(U)\to U$
is a Peano covering map for every open subset $U$ of $Y$.
\item[b.]
If every point $y\in Y$ has a neighborhood $U$ such that
$f\colon f^{-1}(U)\to U$
is a Peano covering map, then $f$ is a Peano covering map.
\end{itemize}
\end{proposition}
{\bf Proof. } a). $f\colon f^{-1}(U)\to U$ is clearly a Peano map, is a fibration,
and has the unique path lifting property.
\par b). $f$ is a Serre $1$-fibration and path components of fibers are trivial.
If $V$ is an open subset of $Y$ containing $y$ we pick an open subset
$U$ of $X$ containing $f(y)$ such that $f\colon f^{-1}(U)\to U$ is a Peano covering map.
There is an open neighborhood $W$ of $f(y)$ in $U$ so that
the path component of $y$ in $f^{-1}(W)$ is open and is contained in $V\cap f^{-1}(U)$.
That proves $f\colon Y\to X$ is a Peano map.
\hfill \qed
In analogy to regular classical covering maps let us introduce
regular Peano covering maps:
\begin{definition}\label{RegularPeanoCoverings}
A Peano covering map $f\colon X\to Y$ is {\bf regular} if
lifts of loops in $Y$ are either always loops of are always non-loops.
\end{definition}
\begin{corollary}\label{FZCoversArePeano}
Given a map $f\colon X\to Y$ the following conditions
are equivalent if $X$ is path-connected:
\begin{enumerate}
\item[a)] $f$ is a regular Peano covering map,
\item[b)] $f$ is a Peano covering map and the image of $\pi_1(f)$ is a normal subgroup of $\pi_1(Y,f(x_0))$
for all $x_0\in X$,
\item[c)] $f\colon X\to Y$ is a generalized covering map in the sense of Fischer-Zastrow.
\end{enumerate}
\end{corollary}
{\bf Proof. } a)$\implies$b). If the image of $\pi_1(f)$ is not a normal subgroup of
$\pi_1(Y,f(x_0))$
for some $x_0\in X$, then there is a loop $\alpha$ in $Y$ at $y_0=f(x_0)$
that lifts to a loop in $X$ at $x_0$ and there is a loop $\beta$ in $Y$ at $y_0$
such that $\beta\ast\alpha\ast\beta^{-1}$ does not lift to a loop in $X$ at $x_0$.
Let $\gamma$ be a lift of $\alpha$ originating at $x_0$. Let $x_1=\beta(1)$.
Notice the lift of $\alpha$ originating at $x_1$ cannot be a loop,
a contradiction.
b)$\implies$c). As $im(\pi_1(f))$ is a normal subgroup $H$ of $\pi_1(Y,y_0)$,
it does nor depend on the choice of the base-point of $X$ in $f^{-1}(y_0)$.
Using \ref{MainCoveringTheorem} one gets $f$ is a generalized covering map.
c)$\implies$a). Since each hedgehog is contractible, $f$ has the unique hedgehog
lifting property and is a Peano map by \ref{HedgehogPeanoTheorem}.
It is also a Serre fibration, hence a Peano covering map.
Also, as $im(\pi_1(f))$ is a normal subgroup $H$ of $\pi_1(Y,y_0)$,
it does nor depend on the choice of the base-point of $X$ in $f^{-1}(y_0)$.
Hence a loop in $Y$ lifts to a loop in $X$ if and only if it represents
an element of $H$. Thus $f$ is a regular Peano covering map.
\hfill \qed
In the remainder of this section we will discuss the relation of Peano covering maps to
classical covering maps.
\begin{proposition}\label{PeanoCMIsTrivialBundle}
If $f\colon Y\to X$ is a Peano covering map and $U$ is an open
subset of $X$ such that every loop in $U$ is null-homotopic in $X$,
then $f^{-1}(V)\to P(V)$ is a a trivial discrete bundle for every path component
$V$ of $U$.
\end{proposition}
{\bf Proof. } Consider a path component $W$ of $f^{-1}(U)$ intersecting $f^{-1}(V)$.
$f$ maps $W$ bijectively onto $V$ and it is easy to see $f|W\colon W\to V$
is equivalent to $P(V)\to V$.
\hfill \qed
\begin{corollary}\label{PeanoCovsForSemiSimple}
If $X$ is a semilocally simply connected Peano space, then
\par\noindent $f\colon Y\to X$
is a Peano covering map if and only if it is a classical covering map and $Y$ is connected.
\end{corollary}
{\bf Proof. } If $f$ is a classical covering map and $Y$ is connected, then $Y$ is locally
path-connected, $f$ has unique path lifting property and is a Serre $1$-fibration.
Thus it is a Peano covering map.
\par Suppose $f$ is a Peano covering map and $x\in X$. Choose
a path-connected neighborhood $U$ of $x$ in $X$ such that any loop
in $U$ is null-homotopic in $X$. By \ref{PeanoCMIsTrivialBundle}
$U$ is evenly covered by $f$.
\hfill \qed
\begin{corollary}\label{PeanoAndClassicalCovers}
If $f\colon Y\to P(X)$ is a classical covering map, then
$f\colon Y\to X$ is a Peano covering map.
\end{corollary}
{\bf Proof. } By \ref{LocalPeanoCoverings}, $f\colon Y\to P(X)$ is a Peano covering map.
As the identity function induces a Peano covering map $P(X)\to X$,
$f\colon Y\to X$ is a Peano covering map by \ref{CompositionOfPeanoCoverings}.
\hfill \qed
\begin{proposition}\label{CardinalityOfFibers}
If $f\colon Y\to X$ is a Peano covering map and $X$ is path-connected,
then all fibers of $f$ have the same cardinality.
\end{proposition}
{\bf Proof. } Given two points $x_1,x_2\in X$ fix a path $\alpha$ from $x_1$ to $x_2$
and notice lifts of $\alpha$ establish bijectivity of fibers $f^{-1}(x_1)$
and $f^{-1}(x_2)$.
\hfill \qed
The following result has its origins in Lemma 2.3 of \cite{ConLam}
and Proposition 6.6 of \cite{FisZas}.
\begin{proposition}\label{RegularPeanoCMAreCCM}
Suppose $f\colon Y\to X$ is a regular Peano covering map. If $f^{-1}(x_0)$
is countable and $x_0$ has a countable basis of neighborhoods
in $X$, then there is a neighborhood $U$ of $x_0$ in $X$
such that $f^{-1}(V)\to P(V)$ is a classical covering map, where $V$ is the path component of $x_0$ in $U$.
\end{proposition}
{\bf Proof. } Switch to $X$ being Peano by considering $f\colon Y\to P(X)$. Notice $x_0$ has a countable basis of neighborhoods
and $f$ is open. Suppose there is no open subset $U$ of $X$ containing $x_0$
such that $U$ is evenly covered. That means path components of $f^{-1}(U)$
are not mapped bijectively onto their images.
\par
First, we plan to show there is a neighborhood $U$ of $x_0$ in $X$
such that the image of $\pi_1(U,x_0)\to \pi_1(X,x_0)$ is contained
in the image of $\pi_1(f)\colon \pi_1(Y,y_0)\to \pi_1(X,x_0)$.
In particular, there is a lift of $P(U,x_0)\to (Y,y_0)$ of the
inclusion induced map $P(U,x_0)\to (X,x_0)$.
\par Suppose no such $U$ exists.
By induction we will find a basis of neighborhoods $\{U_i\}$ of $x_0$ in $X$
and elements $[\alpha_i]\in \pi_1(U_i,x_0)$ that are not contained
in the image of $\pi_1(U_{i+1},x_0)\to \pi_1(X,x_0)$ and whose lifts
are not loops and end at points $y_i$ such that $y_i\ne y_j$ if $i\ne j$. Given a neighborhood $U_i$ pick a loop $\alpha_i$
in $(U_i,x_0)$ whose lift (as a path) in $(Y,y_0)$ is not a loop
and ends at $y_i\ne y_0$. There is a neighborhood $U_{i+1}$
of $x_0$ in $U_i$ such that the no path components of $f^{-1}(U_{i+1})$
contains both $y_0$ and some $y_{j}$, $j\leq i$. Pick a loop $\alpha_{i+1}$
in $(U_{i+1},x_0)$ whose lift is not a loop.
\par As in \cite{Paw} one can create infinite concatenations
$\alpha_{i(1)}\ast\ldots\ast \alpha_{i(k)}\ast\ldots$ for any increasing sequence
$\{i(k)\}_{k\ge 1}$. By looking at lifts of those infinite concatenations,
there are two different infinite concatenations
$\alpha_{i(1)}\ast\ldots\ast \alpha_{i(k)}\ast\ldots$
and
$\alpha_{j(1)}\ast\ldots\ast \alpha_{j(k)}\ast\ldots$ whose lifts end at the same point
$y\in f^{-1}(x_0)$. Pick the smallest $k_0$ so that $i(k_0)\ne j(k_0)$.
We may assume $i(k_0) < j(k_0)$ and conclude there are
loops $\beta$ in $(U_{k_0+1},x_0)$ and $\gamma$ in $(Y,y_0)$
so that $\alpha_{i(k_0)}\sim f(\gamma)\ast\beta$ i which case the lift
of $\alpha_{i(k_0)}$ in $(Y,y_0)$ ends in the path component of $f^{-1}(U_{i(k_0)+1})$ containing $y_0$, a contradiction.
\par As $f$ is a regular Peano covering map, we can find lifts
$(U,x_0)\to (Y,y)$ of the inclusion map $(U,x_0)\to (X,x_0)$
for any $y\in f^{-1}(x_0)$.
\hfill \qed
\section{Peano subgroups}\label{SECTION Peano-subgroups}
\begin{definition}\label{PeanoSubgroupDef}
Suppose $(X,x_0)$ is a pointed path-connected space.
A subgroup $H$ of $\pi_1(X,x_0)$ is a {\bf Peano subgroup} of $\pi_1(X,x_0)$
if there is a Peano covering map $f\colon Y\to X$ such that $H$
is the image of $\pi_1(f)\colon \pi_1(Y,y_0)\to\pi_1(X,x_0)$
for some $y_0\in f^{-1}(x_0)$.
\end{definition}
\begin{proposition}\label{PeanoSubsAreHH}
If $H$ is a Peano subgroup of $\pi_1(X,x_0)$, then $X$ is homotopically
Hausdorff relative to $H$. In particular, $H$ is closed in $\pi_1(X,x_0)$
equipped with the basic topology.
\end{proposition}
{\bf Proof. } Choose a Peano covering map $f\colon Y\to X$ so that
$im(\pi_1(f))=H$ for some $y_0\in f^{-1}(x_0)$. If $g\in \pi_1(X,x_0)\setminus H$
and $\alpha$ is a path in $X$ from $x_0$ to $x_1$,
then lifts of $\alpha$ and $g\cdot \alpha$ end in two different points $y_1$
and $y_2$ of the fiber $f^{-1}(x_1)$ and there is a neighborhood $U$
of $x_1$ in $X$ such that no path component of
$f^{-1}(U)$ contains both $y_1$ and $y_2$.
Suppose there is a loop $\gamma$ in $(U,x_1)$ with the property
$[\alpha\ast\gamma\ast\alpha^{-1}]\in H\cdot g$. In that case
the lifts of both $\alpha\ast\gamma$ and $g\cdot\alpha$
end at $y_2$. Since the lift of $\alpha$ ends in the same path component
of $f^{-1}(U)$ as the lift of $\alpha\ast\gamma$, both $y_1$ and $y_2$
belong to the same component of $f^{-1}(U)$, a contradiction.
\par Use \ref{HGHausdorffAndBasicTopology} to conclude
$H$ is closed in $\pi_1(X,x_0)$
equipped with the basic topology.
\hfill \qed
\begin{remark}\label{WeakPeanoSubsAreHHRemark}
In case of $H$ being the trivial subgroup, Lemma 2.10 of \cite{FisZas}
seems to imply that $X$ is homotopically
Hausdorff but the proof of it is valid only in a special case.
\end{remark}
\begin{proposition}\label{ConjugateOfPeanoSubs}
If $H$ is a Peano subgroup of $\pi_1(X,x_0)$, then
any conjugate of $H$ is a Peano
subgroup of $\pi_1(X,x_0)$.
\end{proposition}
{\bf Proof. } Choose a Peano covering map $f\colon Y\to X$ so that
$im(\pi_1(f))=H$ for some $y_0\in f^{-1}(x_0)$.
Suppose $G=g\cdot H\cdot g^{-1}$ and choose a loop $\alpha$
in $(X,x_0)$ representing $g^{-1}$. Let $\beta$ be a path in $(Y,y_0)$
that is the lift of $\alpha$. Put $y_1=\beta(1)$ and notice the image of
$\pi_1(f)\colon \pi_1(Y,y_1)\to \pi_1(X,x_0)$ is $G$.
\hfill \qed
\begin{proposition}\label{IntroToBogley-SieradskiTopology}
Suppose $(X,x_0)$ is a pointed path-connected topological space.
If $f\colon (Y,y_0)\to (X,x_0)$ is a Peano covering map
with image of $\pi_1(f)$ equal $H$, then $f$ is equivalent to the projection
$p_H\colon \widehat X_H\to X$.
\end{proposition}
{\bf Proof. }
Define $h\colon (\widehat X_H,\widehat x_0)\to (Y,y_0)$ by choosing a lift $\widehat \alpha$
of every path $\alpha$ in $X$ starting at $x_0$ and declaring $h([\alpha]_H)=\widehat\alpha(1)$. Note $h$ is a bijection.
Given $y_1=\widehat\alpha(1)$ and given a neighborhood $U$ of $y_1$ in $Y$
choose a neighborhood $V$ of $f(y_1)=\alpha(1)$ in $X$ so that the path component
of $f^{-1}(V)$ containing $y_1$ is a subset of $U$. Observe $B_H([\alpha]_H,V)\subset
h^{-1}(U)$ which proves $h$ is continuous.
\par Conversely, given a neighborhood $W$ of $\alpha(1)$ in $X$
the image $h(B_H([\alpha]_H,W))$ of $B_H([\alpha]_H,W)$ equals the path component
of $\widehat\alpha(1)$ in $f^{-1}(W)$ and is open in $Y$.
\hfill \qed
\begin{theorem}\label{BasicPeanoCoveringThm}
If $X$ is a path-connected space, $x_0\in X$, and $H$ is a subgroup of $\pi_1(X,x_0)$, then the following conditions are equivalent:
\begin{itemize}
\item[a.] $H$ is a Peano subgroup of $\pi_1(X,x_0)$,
\item[b.] The endpoint projection $p_H\colon (\widehat X_H,\widehat x_0)\to X$ is a Peano covering map,
\item[c.] The image of $\pi_1(p_H)\colon \pi_1(\widehat X_H,\widehat x_0)\to \pi_1(X,x_0)$ is
contained in $H$,
\item[d.] $p_H\colon (\widehat X_H,\widehat x_0)\to (X,x_0)$ has the unique path lifting property.
\end{itemize}
\end{theorem}
{\bf Proof. } c)$\equiv$d) is done in \ref{ProjHasUPLP}.
b)$\equiv$d) is contained in \ref{ProjIsPeanoCMCharThm}.
\par
a)$\implies$b) follows from \ref{IntroToBogley-SieradskiTopology}.
\par
b)$\implies$a) holds as c) implies the image of $\pi_1(p_H)$ is $H$.
\hfill \qed
Let us state a straightforward consequence of \ref{BasicPeanoCoveringThm}:
\begin{corollary}\label{SimplyPeanoCoveringThm}
If $X$ is a path-connected space and $x_0\in X$, then the following conditions are equivalent:
\begin{itemize}
\item[a.] The endpoint projection $p\colon \widehat X\to X$ is a Peano covering map,
\item[b.] $\pi_1(p)\colon \pi_1(\widehat X,\widehat x_0)\to \pi_1(X,x_0)$ is trivial,
\item[c.] $\widehat X$ is simply connected,
\item[d.] $p\colon (\widehat X,\widehat x_0)\to (X,x_0)$ has the unique path lifting property.
\end{itemize}
\end{corollary}
\begin{corollary}\label{ClosedNormalSubsArePeano}
Closed and normal subgroups of $\pi_1(X,x_0)$ are Peano subgroups of $\pi_1(X,x_0)$.
\end{corollary}
{\bf Proof. } By \ref{PiOneOfXHAndPathLifting} the endpoint projection $p_H\colon (\widetilde X_H,\widetilde x_0)\to X$ has unique path lifting property.
Since $p_H\colon (\widehat X_H,\widehat x_0)\to X$ has path lifting property,
this implies $p_H\colon (\widehat X_H,\widehat x_0)\to X$ has the unique path lifting
property.
\hfill \qed
\begin{corollary}\label{IntersectionArePeano}
If $H(s)$ is a Peano subgroup of $\pi_1(X,x_0)$ for each $s\in S$,
then $G=\bigcap\limits_{s\in S}H(s)$ is a
Peano subgroup of $\pi_1(X,x_0)$.
\end{corollary}
{\bf Proof. } The projection $p_G\colon (\widehat X_G,\widehat x_0)\to (X,x_0)$ factors through
\par\noindent
$p_{H(s)}\colon (\widehat X_{H(s)},\widehat x_0)\to (X,x_0)$ for each $s\in S$.
Therefore $im(\pi_1(p_G))\subset \bigcap\limits_{s\in S}H(s)=G$
and \ref{ProjIsPeanoCMCharThm} (in conjunction with \ref{ProjHasUPLP}) says $G$ is a Peano subgroup of $\pi_1(X,x_0)$.
\hfill \qed
\begin{corollary}\label{UniversalPeanoCM}
For each path-connected space $X$ there is a universal Peano covering map
$p\colon Y\to X$. Thus, for each Peano covering map $q\colon Z\to X$
and any points $z_0\in Z$ and $y_0\in Y$ satisfying $q(z_0)=p(y_0)$,
there is a Peano covering map $r\colon Y\to Z$ so that $r(y_0)=z_0$.
Moreover, the image of $\pi_1(Y)$ is normal in $\pi_1(X)$.
\end{corollary}
{\bf Proof. } Let $H$ be the intersection of all Peano subgroups of $\pi_1(X,x_0)$
by \ref{IntersectionArePeano} and \ref{ConjugateOfPeanoSubs} it is a normal Peano subgroup of $\pi_1(X,x_0)$.
Put $Y=\widehat X_H$ and use \ref{MainPropertyOfPeanoCoverings}.
\hfill \qed
It would be of interest to characterize path-connected spaces $X$
admitting a universal Peano covering that is simply connected (that amounts to
$\widehat X$ being simply connected).
Here is an equivalent problem:
\begin{problem}\label{TrivialPeanoSubsProblem}
Characterize path-connected spaces $X$ so that the trivial group is a Peano subgroup
of $\pi_1(X,x_0)$.
\end{problem}
So far the following classes of spaces
belong to that category:
\begin{enumerate}
\item Any product of spaces admitting simply connected Peano cover (see \ref{ProductOfPeanoCoveringMaps}).
\item Subsets of closed surfaces: it is proved in \cite{FisZasFirst} that if $X$ is any
subset of a closed surface, then $\pi_1(X,x_0)\to\check\pi_1(X,x_0)$ is injective.
\item $1$-dimensional, compact and Hausdorff,
or $1$-dimensional, separable and metrizable: $\pi_1(X,x_0)\to\check\pi_1(X,x_0)$
is injective by \cite[Corollary 1.2 and Final Remark]{EdaKaw}.
It is shown in \cite{Eda} (see proof of Theorem 1.4) that
the projection $\widehat X\to X$ has the uniqueness of path-lifting
property if $X$ is $1$-dimensional and metrizable.
See \cite{CanCon}
for results on the fundamental group of $1$-dimensional spaces.
\item Trees of manifolds: If $X$ is the limit of an inverse system of
closed PL-manifolds of some fixed dimension, whose consecutive terms are obtained
by connect summing with closed PL-manifolds, which in turn are trivialized by the
bonding maps, then X is called a tree of manifolds. Every tree of manifolds is
path-connected and locally path-connected, but it need not be semilocally simplyconnected
at any one of its points. Trees of manifolds arise as boundaries of certain
Coxeter groups and as boundaries of certain negatively curved geodesic spaces
\cite{FisGui}.
It is shown in \cite{FisGui} that if X is a tree of manifolds (with a certain denseness of the
attachments in the case of surfaces), then $\pi_1(X,x_0)\to\check\pi_1(X,x_0)$ is injective.
\end{enumerate}
Notice Example 2.7 in \cite{FisZas} gives $X$ so that $p\colon \widehat X\to X$ does not
have the unique path lifting property (one can construct a simpler example
with $X$ being the Harmonic Archipelago). However, $X$ is not homotopically
Hausdorff.
\begin{problem}\label{HHProblem}
Is there a homotopically Hausdorff space $X$ such that $p\colon \widehat X\to X$
does not have the uniqueness of path lifting property?
\end{problem}
\begin{corollary}\label{CountableIndexNormalSubsOfPeanoArePeano}
Suppose $H$ is a normal subgroup of $\pi_1(X,x_0)$.
If there is a Peano subgroup $G$ of $\pi_1(X,x_0)$ containing $H$ such that $G/H$ is countable,
then $H$ is a
Peano subgroup of $\pi_1(X,x_0)$ if and only if $X$ is homotopically Hausdorff
relative to $H$.
\end{corollary}
{\bf Proof. } By \ref{PeanoSubsAreHH}, $X$ is homotopically Hausdorff
relative to $H$ if $H$ is a
Peano subgroup of $\pi_1(X,x_0)$.
\par Suppose $X$ is homotopically Hausdorff
relative to $H$. Given two lifts in $\widehat X_H$ of the same path in $X$,
their composition with $\widehat X_H\to \widehat X_G$ are the same by
\ref{BasicPeanoCoveringThm}. By
\ref{HAndPathLifting} the two lifts are identical and \ref{BasicPeanoCoveringThm}
says $H$ is a Peano subgroup of $\pi_1(X,x_0)$.
\hfill \qed
\begin{corollary}\label{CountableNormalSubsArePeano}
Suppose $H$ is a normal subgroup of $\pi_1(X,x_0)$.
If $\pi_1(X,x_0)/H$ is countable,
then $H$ is a
Peano subgroup of $\pi_1(X,x_0)$ if and only if $X$ is homotopically Hausdorff
relative to $H$.
\end{corollary}
\section{Appendix: Pointed versus unpointed}\label{SECTION-PointedVsUnpointed}
In this section we discuss relations between pointed and unpointed lifting properties.
\begin{lemma}\label{PointedUPLPImpliesUPLPLem}
If $f\colon (X,x_0)\to (Y,y_0)$ has the uniqueness of path lifts property
and $X$ is path-connected, then $f\colon X\to Y$
has the uniqueness of path lifts property.
\end{lemma}
{\bf Proof. } Given two paths $\alpha$ and $\beta$ in $X$ originating at the same point
and satisfying $f\circ \alpha=f\circ\beta$, choose a path $\gamma$ in $X$
from $x_0$ to $\alpha(0)$. Now $f\circ (\gamma\ast\alpha)=f\circ (\gamma\ast\beta)$,
so $\gamma\ast\alpha=\gamma\ast\beta$ and $\alpha=\beta$.
\hfill \qed
\begin{lemma}\label{PointedUPLPImpliesUPLPLem}
If $f\colon (X,x_0)\to (Y,y_0)$ has the unique path lifting property
and $X$ is path-connected, then $f\colon X\to Y$
has the unique path lifting property.
\end{lemma}
{\bf Proof. } In view of \ref{PointedUPLPImpliesUPLPLem} it suffices to show
$f\colon X\to Y$ is surjective and has the path lifting property.
If $y_1\in Y$, we pick a path $\alpha$ from $y_0$ to $y_1$ and lift it to
$(X,x_0)$. The endpoint of the lift maps to $y_1$, hence $f$ is surjective.
Suppose $\alpha$ is a path in $Y$ and $f(x_1)=\alpha(0)$.
Choose a path $\beta$ in $X$ from $x_0$ to $x_1$ and lift
$(f\circ \beta)\ast\alpha$ to a path $\gamma$ in $(X,x_0)$.
Due to the uniqueness of path lifts property of $f\colon (X,x_0)\to (Y,y_0)$
one has $\gamma(t)=\beta(2t)$ for $t\leq\frac{1}{2}$.
Hence $\gamma(\frac{1}{2})=x_1$ and $\lambda$ defined
as $\lambda(t)=\gamma(\frac{1}{2}+\frac{t}{2})$ for $t\in I$
is a lift of $\alpha$ originating from $x_1$.
\hfill \qed
\begin{lemma}[Lemma 15.1 in \cite{Hu}]\label{PointedSerrePlusFibersImpliesUPLPLem}
If $f\colon X\to Y$
is a Serre $1$-fibration, then $f$ has the unique path lifting property if
and only if path components of fibers of $f$ are trivial.
\end{lemma}
{\bf Proof. } Suppose the fibers of $f$ have trivial path components
and $\alpha,\beta$ are two lifts of the same path in $Y$ that
originate at $x_1\in X$. Let $H\colon I\times I\to Y$ be the
standard homotopy from $f\circ (\alpha^{-1}\ast\beta)$ to the constant
path at $f(x_1)$. There is a lift $G\colon I\times I\to X$ of $H$
starting from $\alpha^{-1}\ast\beta$.
As path components of $f$ are trivial, $\alpha=\beta$ due to the way
the standard homotopy $H$ is defined.
\hfill \qed
\begin{lemma}\label{PointedSerreImpliesSerreLem}
Suppose $n\ge 1$.
If $f\colon (X,x_0)\to (Y,y_0)$ is a Serre $n$-fibration,
both $X$ and $Y$ are path-connected, and $f$ has the uniqueness of path lifts property, then $f\colon X\to Y$
is a Serre $n$-fibration.
\end{lemma}
{\bf Proof. } Suppose $H\colon I^n\times I\to Y$ is a homotopy
and $G\colon I^n\times \{0\}\to X$ is its partial lift.
Choose a path $\alpha$ in $X$ from $x_0$ to $G(b,0)$, where $b$ is the center of $I^n$.
We can extend $G$ to a homotopy $G\colon I^n\times [-1,0]\to X$ starting from the constant map to $x_0$. By splicing $f\circ G$ with original $H$, we can extend $H$
to $H\colon I^n\times [-1,1]\to Y$. That $H$ can be lifted to $X$ and the lift
must agree with $G$ on $I^n\times [-1,0]$ due to the uniqueness of path lifts property of
$f$.
\hfill \qed
|
1,116,691,500,509 | arxiv | \section{\label{introdution}Introduction}
The study of the properties of networks led to the development of many mathematical, statistical and computational tools that can be used to analyze, model, and understand how systems behave in many areas of knowledge such as physics, biology, ecology, and social sciences, to name some of them. Modeling complex systems by networks \cite{AB02, DM03, N10} is a natural strategy to investigate a system from a very basic structure composed of agents and interactions among them, represented, respectively, by vertices (or nodes) and links.
In this work, we are mainly interested in the time evolution of the degree of a given node. Concretely, a vertex can gain and/or lose connections during its dynamics, and we investigate when it achieves a pre-established degree for the first time. This is a particularly relevant issue when agents can not afford an indefinite number of connections and some indication of approaching the maximum capacity of the node \cite{ASBS00, HCG16} is desired. As an instance, it is known that airports (where the links can be assigned to the routes) have constraints that prevent growth without careful planning \cite{UTGR03}.
We map the dynamics of increasing/decreasing degrees into a random walk process, as was introduced in \cite{CGCH17}, and see if and how long it takes for a vertex with degree $k_{0}$ to achieve degree $k$ for the first time. This is a one-dimensional random walk in degree space, where the rules of gaining/losings degree are governed by the dynamics of the network. The random walk is a classical problem \cite{R80, P05} where a particle moves in random directions and one typically inquires about its statistical properties after a long time. Starting from an origin, one possible question that can be formulated concerns the probability of returning to the starting point, and the first-passage process refers to its return for the first time \cite{R07}. First-passage processes are seen in many applications, and examples are present in fluorescence quenching, integrate-and-fire neurons, and triggering of stock options, to cite some of them \cite{KRBN10, R07}. When the random walk is defined on a (hyper)cubic lattice \cite{P21}, and the particle can move in any direction with the same probability, it is known that the mean first-passage time scales as $L^{d}$ \cite{MW65}, where $L$ is the linear size of the $d$-dimensional lattice with periodic boundary conditions; furthermore, in the limit of infinite lattice, this random walk is known to be recurrent (\textit{i.e.}, it returns to the origin with probability $1$) for one-dimensional chains and two-dimensional square lattices, while the process is transient (\textit{i.e}, there is a positive probability of not returning to the origin) for hypercubic lattices with larger dimensions \cite{MW65, KRBN10}.
We investigate how the mean first-passage time of a vertex to reach a pre-established degree scales with the size of the network and other relevant parameters. Our study is based on an extended version of the Watts-Strogatz model \cite{WS98}, which was also introduced in \cite{CGCH17}, but we consider first a dynamical version of the Erd\H os-R\'enyi model \cite{ER59} to illustrate and outline the main steps of analysis. The choice of these two models is justified by a simplification that arises from a property shared between them, which is the time-translational invariance. Systems that do not have this property, like the random recursive tree \cite{NR70} or Barab\'asi-Albert network \cite{BA99}, indicate the need for a different approach, and will be examined elsewhere.
This paper is organized as follows. We define the dynamical version of the Erd\H{o}s-R\'enyi and Watts-Strogatz models in Section II and the general formalism to investigate the moments of the first-passage time is presented in Section III. The results for both models are shown in Section IV and some final comments are given in the last section.
\section{\label{models}Models}
Two models are introduced to test our ideas in this work. Both of them are already well-known in the literature \cite{ER59, WS98}, but were initially defined as static networks.
The first one, which is a minimal model, is the dynamical Erd\H os-R\'enyi network: the dynamics is just a simple addition of edges per time unit, and we monitor the increase of degrees only. The second one, the dynamical Watts-Strogatz model, is the simplest network that contains the process where a vertex can gain and/or lose connections randomly. The usual first-passage process (which is concerned with the return to the starting point) in the latter model corresponds to the so-called Motzkin paths \cite{OvdJ15}.
\subsection{Time-dependent Erd\H{o}s-R\'enyi model}
In the dynamical version of the Erd\H{o}s-R\'enyi model, consider a network with $N$ vertices. At each unitary time step, two vertices are randomly chosen and connected with probability $p$; this includes the possibility of (a) having a self-loop (\textit{i.e}, an edge that connects a vertex to itself) and (b) having more than one connection between the same pair of vertices. Since there is no preferential attachment, the probability of any vertex being chosen is $1/N$.
Defining $p_{s}(k,t)$ as the probability that a vertex $s$ has degree $k$ at time $t$, the dynamics can be represented by the recurrence relation
\begin{align}
\nonumber p_{s}(k,t+1) &= \omega_{\text{\tiny ER}}(k|k-2)p_{s}(k-2,t)+\\
\nonumber & + \omega_{\text{\tiny ER}}(k|k-1)p_{s}(k-1,t) + \\
& + \omega_{\text{\tiny ER}}(k|k)p_{s}(k,t).
\label{rr_ER}
\end{align}
The term $\omega_{\text{\tiny ER}}(k|m)$ is the time-independent transition rate of changing the degree of a vertex from $m$ to $k$; in this time-discrete case with unitary time step, the transition rate coincides numerically to the conditional probability. The right-hand-side of the dynamics \eqref{rr_ER} contemplates three cases:
\renewcommand\labelenumi{(\roman{enumi})}
\begin{enumerate}
\item The degree of vertex $s$ changes from $k-2$ (at time $t$) to $k$ (at time $t+1$). An edge is introduced, with probability $p$ (there should be no confusion with $p_{s}$), the vertex $s$ is chosen twice and is connected to itself; this leads to
\begin{align}
\omega_{\text{\tiny ER}}(k|k-2) = \frac{p}{N^{2}};
\label{w(k-2)_ER}
\end{align}
\item The degree of vertex $s$ changes from $k-1$ (at time $t$) to $k$ (at time $t+1$). An edge is introduced, with probability $p$, and links to two different vertices: the vertex $s$ is just one of them. This situation is described by
\begin{align}
\omega_{\text{\tiny ER}}(k|k-1) = \frac{2p}{N}\left(1 - \frac{1}{N}\right);
\label{w(k-1)_ER}
\end{align}
\item The vertex $s$ already has degree $k$, and one should consider the probability of not changing its degree, \textit{i.e.}, the link is not introduced (with probability $1-p$) or, when the edge joins the network (with probability $p$), it connects two vertices other than $s$ with probability $\left(1-1/N\right)^{2}$. In this case, one has
\begin{align}
\nonumber \omega_{\text{\tiny ER}}(k|k) &= \left(1- p\right) + p\left(1-\frac{1}{N}\right)^{2} \\
&=1 - \frac{2p}{N} + \frac{p}{N^{2}}.
\label{w(k)_ER}
\end{align}
\end{enumerate}
\subsection{\label{sec:level3}Time-dependent Watts-Strogatz model}
In this version of the Watts-Strogatz model, the network has a fixed number $N$ of vertices and
\begin{align}
M := cN
\label{M}
\end{align}
degrees, where $c$ is the mean degree of the network (therefore, the entire graph has $cN/2$ edges). At each time step, an edge end is chosen at random with uniform probability $1/M$ and reconnected with probability $p$ (and no action takes place with probability $1-p$). This scheme does not forbid self-loops.
Defining $p_{s}(k,t)$ as the probability that a vertex $s$ has degree $k$ at time $t$ as before, the dynamics can be represented by
\begin{align}
\nonumber p_{s}(k,t+1) &= \omega_{\text{\tiny WS}} (k|k-1)p_{s}(k-1,t) + \\
\nonumber & + \omega_{\text{\tiny WS}} (k|k+1)p_{s}(k+1,t) + \\
& + \omega_{\text{\tiny WS}} (k|k)p_{s}(k,t),
\label{rr_WS}
\end{align}
where $\omega_{\text{\tiny WS}} (k|m)$ represents the time-independent transition rate of a vertex changing its degree from $m$ to $k$.
There are some different possible scenarios for a given vertex to change its degree from $m$ to $k$ in a single time step:
\renewcommand\labelenumi{(\roman{enumi})}
\begin{enumerate}
\item The degree of vertex $s$ changes from $k-1$ (at time $t$) to $k$ (at time $t+1$). An edge end not connected to $s$ is chosen with probability $1-\frac{k-1}{M}$, rewired with probability $p$ and connects to $s$ with probability $\frac{1}{N}$. This gives
\begin{align}
\omega(k | k-1) = \frac{p}{N}\left(1-\frac{k-1}{M}\right);
\label{w(k-1)_WS}
\end{align}
\item The degree of vertex $s$ changes from $k+1$ (at time $t$) to $k$ (at time $t+1$). An edge end connected to $s$ is chosen with probability $\frac{k+1}{M}$, rewired with probability $p$ and connects to a vertex other than $s$ with probability $1-\frac{1}{N}$, resulting in
\begin{align}
\omega(k | k+1) = \frac{k+1}{M} p \left(1-\frac{1}{N}\right);
\label{w(k+1)_WS}
\end{align}
\item The vertex $s$ has degree $k$ at time $t$ and neither gains or loses connections. This is represented by the sum of some disjoint cases: (a) there is no rewiring at all in the process with probability $1-p$, or (b) an edge end connected to $s$ is chosen with probability $\frac{k}{M}$, rewired with probability $p$ and connected again to $s$ with probability $\frac{1}{N}$; (c) an edge end not connected to $s$ is chosen with probability $1-\frac{k}{M}$ and rewired (with probability $p$) to connect to a vertex othen than $s$ with probability $1-1/N$. The sum of these probabilities results in
\begin{align}
\nonumber \omega(k|k) &= \left(1-p\right) + p\frac{k}{M}\frac{1}{N} + p\left(1-\frac{k}{M}\right)\left(1-\frac{1}{N}\right) \\
&= 1 - \frac{p}{N}\left(1+\frac{kN}{M} - \frac{2k}{M}\right).
\label{w(k)_WS}
\end{align}
\end{enumerate}
\section{\label{rwds}Random walk in degree space}
Considering that vertices, in general, gain or lose connections, one can look at these changes in degree (of a specified vertex) as a one-dimensional random walk in degree space \cite{CGCH17}. Furthermore, the mean time required by a vertex to achieve a certain degree for the first time can be evaluated through a parallel with the first-passage problem of random walks \cite{R07, KRBN10}.
In both models presented in the previous section, there are two important symmetries. Firstly, the particular choice of a vertex $s$ is irrelevant, and this parameter has no role in our work - except for remembering that we are dealing with the time evolution of the degree of a given vertex.
The mean time $\langle t \rangle$ to achieve a certain degree $k$ for the first time (starting from $k_{0}$ at time $t_{0} = 0$) is given by
\begin{align}
\langle t \rangle = \sum_{t=0}^{\infty} tf_{s}(k,t|k_0,0),
\label{<t>}
\end{align}
where $f_{s}(k,t|k_{0},0)$ is the probability of vertex $s$ having degree $k$ for the first time at $t$, given that it had degree $k_{0}$ at time $t_{0}=0$. This probability can be obtained from the discrete time version of the first-passage process equation \cite{R07, KRBN10}, and can be cast as
\begin{align}
p_{s}(k,t|k_{0},0) = \sum_{t^{\prime}=0}^{t}f_{s}(k,t^\prime|k_{0},0) p_{s}(k,t|k,t^\prime),
\label{bare_fp}
\end{align}
which describes the probability $p_{s}(k,t|k_{0},0)$ of the vertex $s$ having degree $k$ at time $t$ (not necessarily for the first time), given that it had degree $k_{0}$ at time $t_{0}=0$. This is a sum of all disjoint probabilities where the degree of the vertex achieves degree $k$ at time $t^{\prime}$ ($\leq t$) for the first time, and then reaches degree $k$ again at instant $t$. The initial condition $p_{s}(k,0|k_{0},0)=\delta_{k,k_{0}}$ is satisfied by assuming $f_{s}(k,0|k_{0},0)=\delta_{k,k_{0}}$ (an extra term in \eqref{bare_fp} associated to the initial condition is not required here as it is in the continuous-time version \cite{R07, KRBN10} of the equation).
The second important symmetry of our models can be seen from the transitions rates $\omega_{\text{\tiny ER}}$ and $\omega_{\text{\tiny WS}}$: they are invariant under time translation. As a consequence, $p_{s}(k,t|k^{\prime},t^{\prime})=p_{s}(k|k^{\prime};t-t^{\prime})$ and $f_{s}(k,t|k^{\prime},t^{\prime})=f_{s}(k|k^{\prime};t-t^{\prime})$ depend on the difference $t-t^{\prime}$ only. Therefore, equation \eqref{bare_fp} can be cast as
\begin{align}
p_{s}(k|k_{0};t) = \sum_{t^{\prime}=0}^{t}f_{s}(k|k_{0};t^{\prime}) p_{s}(k|k;t-t^\prime).
\label{fp}
\end{align}
As usual, the convolution product in \eqref{fp} suggests the introduction of the characteristic function
\begin{align}
p_{s}^{z}(k|k_0;z) = \sum_{t=0}^{\infty} z^{t}p_{s}(k|k_0;t)
\label{characteristic_t}
\end{align}
and a similar definition for the characteristic function of the function $f_{s}$. Then, it is immediate that
\begin{align}
f_{s}^{z}(k|k_0;z) = \frac{p_{s}^{z}(k|k_{0};z)}{p_{s}^{z}(k|k;z)},
\label{fz_pz}
\end{align}
and we can obtain $f_{s}^{z}$ from $p_{s}^{z}$. As stated before, this is a consequence of the time-translation invariance; models that do not have this symmetry (like the random recursive tree \cite{R80, P05} or Barab\'asi-Albert network \cite{BA99}) do not display the form \eqref{fp}.
We are mainly interested in \eqref{fz_pz} because it provides some quantities of interest. The first one is
\begin{align}
\mathcal{A} := \lim_{z\rightarrow 1}f_{s}^{z}(k|k_0;z) = \sum_{t=0}^{\infty} f_{s}(k|k_0;t),
\label{A}
\end{align}
which stands for the arriving probability of a vertex achieving degree $k$, starting from degree $k_{0}$, at some time, while
\begin{align}
\langle t^{n}\rangle = \lim_{z\rightarrow 1}\,\left(z\partial_z\right)^{n}\,f_{s}^{z}(k|k_0;z) = \sum_{t=0}^{\infty}t^{n}f_{s}(k|k_{0};t),
\label{<tn>}
\end{align}
where $\partial_{z}$ stands for the partial derivation in $z$ variable, shows that the quantity $f_{s}^{z}$ is also useful to evaluate any moment of the first-passage time. In this work, we are particularly interested in the first and second moments, $\langle t\rangle$ and $\langle t^{2}\rangle$, respectively; the latter is directly associated to the variance $\sigma^{2}=\langle t^{2}\rangle-\langle t\rangle^{2}$.
Hence, one can also expand \eqref{fz_pz} as
\begin{align}
f_{s}^{z}(k|k_{0};z) = \mathcal{A} + \langle t\rangle\left(z-1\right) + \left[\frac{\langle t^{2}\rangle-\langle t\rangle}{2}\right]\left(z-1\right)^{2} + \cdots
\label{exp_fz}
\end{align}
and obtain the desired quantites ($\mathcal{A}$, $\langle t\rangle$ and $\langle t^{2}\rangle$) through this representation.
\section{\label{results}Results}
In this section, we present the results for the first and second moments of the first-passage time for both models.
\subsection{\label{ER}Time-dependent Erd\H{o}s-R\'enyi model}
The discrete time evolution for the dynamical version of Erd\H{o}s-R\'enyi model, introduced in section \ref{models}, is given by \eqref{rr_ER}. Introducing the characteristic function
\begin{align}
p_{s}^{K}(K; t) = \sum_{k=0}^{\infty}K^{k}p_{s}(k|k_{0}; t)
\label{characteristic_K}
\end{align}
into \eqref{rr_ER} leads to
\begin{align}
p_{s}^{K}(K,t) = \left[ 1 - p + p\,\left(\frac{K}{N} + 1 - \frac{1}{N}\right)^{2} \right]^{t}\,K^{k_{0}},
\label{pK_ER}
\end{align}
where the initial condition $p_{s}(k|k_{0}; 0)=\delta_{k,k_{0}}$ or, equivalently, $p_{s}^{K}(K; 0)=K^{k_{0}}$ was adopted.
From \eqref{characteristic_K}, the probability $p_{s}(k|k_{0}; t)$ is the coefficient of the term $K^{k}$ in the series; therefore, expanding \eqref{pK_ER} and organizing the terms implies
\begin{align}
\nonumber p_{s}(k|k_{0}; t) &= \sum_{m = \left\lceil\frac{\Delta}{2}\right\rceil}^{t} {t \choose m}{2m\choose \Delta}\left(1-p\right)^{t-m}p^{m}\times\\
&\times\left(1-\frac{1}{N}\right)^{2m-\Delta}\frac{1}{N^{\Delta}}.
\label{p_ER}
\end{align}
From \eqref{p_ER}, the function $p_{s}$ depends on the difference $\Delta:=k-k_{0}$ only, and not on the initial and final degrees independently. This property is propagated to the quantities of interest in this work.
Using \eqref{p_ER}, the characteristic function (in time variable) of $p_{s}$ is
\begin{align}
\nonumber \lefteqn{p_{s}^{z}(k|k_{0}; z) = \sum_{t=0}^{\infty}z^{t}p_{s}(k|k_{0}; t)} & \\
\nonumber &= \sum_{m=\left\lceil\frac{\Delta}{2}\right\rceil}^{\infty}{2m\choose\Delta}\left(1-\frac{1}{N}\right)^{2m-\Delta}\frac{1}{N^{\Delta}}\frac{\left(zp\right)^{m}}{\left[1-z\left(1-p\right)\right]^{m+1}}, \\
\label{pz_ER}
\end{align}
from which one can also evaluate $p_{s}^{z}(k|k;z)$ by taking $k_{0}=k$ (or $\Delta=0$). Then, using the relation
\begin{align}
\nonumber \lefteqn{\sum_{m=\left\lceil\frac{\Delta}{2}\right\rceil}^{\infty}{2m\choose\Delta}x^{2m} = \frac{x^{\Delta}}{2}\Big[\left(1-x\right)^{-\Delta-1} +}& \\
\nonumber &+ \left(-1\right)^{\Delta}\left(1+x\right)^{-\Delta-1}\Big]\quad (\Delta\in\mathbb{N},x\in(-1,1)\subset\mathbb{R}), \\
\label{rel1}
\end{align}
which can be seen by combining the expansion of $(1\pm x)^{-\Delta-1}$ for $|x|<1$, it is now possible to obtain the function $f_{s}^{z}(k|k_{0};z)$
\begin{align}
\nonumber f_{s}^{z}(k|k_{0};z) &= \displaystyle\frac{1-\zeta^{2}}{2\left(N-1\right)^{\Delta}}\left[ \frac{\zeta^{\Delta}}{\left(1-\zeta\right)^{\Delta+1}} + \frac{\left(-1\right)^{\Delta}\zeta^{\Delta}}{\left(1+\zeta\right)^{\Delta+1}} \right],\\
\label{fz_ER}
\end{align}
where
\begin{align}
\zeta := \zeta(z) = \left(1-\frac{1}{N}\right)\sqrt{\frac{zp}{1-z\left(1-p\right)}}.
\label{zeta}
\end{align}
Expanding \eqref{fz_ER} as in \eqref{exp_fz} is a tedious, but direct procedure. From this operation, the arrival probability can be obtained as being
\begin{align}
\mathcal{A}_{\text{\tiny ER}} = 1 - \frac{1}{2N}\left[1-\frac{\left(-1\right)^{\Delta}}{\left(2N-1\right)^{\Delta}}\right].
\label{A_ER}
\end{align}
Although the dynamics suggests that the vertex $s$ can achieve any larger degree if one waits a sufficiently long time, the probability \eqref{A_ER} is less than one. However, this odd result is a consequence of the growing rule, which allows a vertex to increase its degree by two units by forming a loop. In this case, the targeted degree, $k$, may be surpassed from $k-1$ to $k+1$ without being. For this reason, the arrival probability is not $1$. Nonetheless, if one evaluates
\begin{align}
\nonumber\lefteqn{\sum_{t=0}^{\infty}f_{s}(\text{degree $\geq k$}|k_{0};t) =}& \\
\nonumber &= \displaystyle\sum_{t=0}^{\infty}\Big[f_{s}(k|k_{0};t) + \omega_{\text{\tiny ER}}(k+1|k-1)p_{s}(k-1,t)\Big], \\
\label{A_corrected}
\end{align}
which is a correction to \eqref{A_ER}, the arrival probability is $1$, as expected. Note that the arrival probability \eqref{A_ER} tends to $1$ with the size of the network, which is expected since the loop becomes rare with the number of vertices. One should also note that this result is valid for any positive probability $p$ (the case $p=0$ is trivial), but does not depend explicitly on this parameter. As shown below, this parameter scales the time elapsed until a vertex reaches some degree for the first time, but it does not have any impact on the probability of reaching the pre-established degree (except the trivial case $p=0$, when $\mathcal{A}=0$ for $\Delta>0$).
The first and second time moments can also be derived from \eqref{fz_ER}. The leading term of the mean first-passage time is
\begin{align}
\langle t \rangle_{\text{\tiny ER}} \simeq \frac{N\Delta}{2p}
\label{<t>_ER}
\end{align}
for $N\gg 1$, while the second moment is
\begin{align}
\langle t^{2} \rangle_{\text{\tiny ER}} \simeq \left(\frac{N}{2p}\right)^{2}\Delta\left(\Delta+1\right).
\label{<t2>_ER}
\end{align}
The variance can also be determined from \eqref{<t>_ER} and \eqref{<t2>_ER}, and depends quadratically on $N/2p$, but linearly on the difference $\Delta:=k-k_{0}$ as $\sigma^{2}_{\text{\tiny ER}}:=\langle t^{2}\rangle_{\text{\tiny ER}}-\langle t\rangle_{\text{\tiny ER}}^{2}\simeq\left(\frac{N}{2p}\right)^{2}\Delta$. The results \eqref{<t>_ER} and \eqref{<t2>_ER} are supported by numerical simulations, as one can see in figure \eqref{fig_er}.
\begin{figure}
\begin{center}
\includegraphics[width=121.6pt]{er_t1image_k0=2_k=5_R=100.png}
\includegraphics[width=121.6pt]{er_t2image_k0=2_k=5_R=100.png}
\caption{\label{fig_er}The mean first (left) and second (right) moments of the first-passage time as a function of the ratio $N/p$ for the dynamical version of the Erd\H os-R\'enyi model with $k_{0}=2$ and $k=5$. The simulations used $100$ samples and compared with the asymptotic results \eqref{<t>_ER} and \eqref{<t2>_ER}; the error bars are smaller than the size of the points.}
\end{center}
\end{figure}
\subsection{\label{sec:level4B}Time-dependent Watts-Strogatz model}
The analysis of the dynamical version of the Watts-Strogatz model is much more intricate than the previous model. To convey better the ideas, all the technical details are presented in the supplemental material, and we will restrict ourselves to highlighting only the important points in this subsection.
The dynamics of this model was already presented in \eqref{rr_WS}, where the transition rates are given in \eqref{w(k-1)_WS}, \eqref{w(k+1)_WS} and \eqref{w(k)_WS}. Introducing a characteritic function that transforms both the degree and time variables (see \eqref{characteristic_K} and \eqref{characteristic_t}) into new ones, the recurrence relation \eqref{rr_WS} can be converted into the differential equation
\begin{align}
\nonumber \lefteqn{\frac{\partial}{\partial K} p_{s}^{Kz}(K,z) =} & \\
\nonumber &= - \frac{M}{p} \left( \frac{1-z^{-1}}{1-K} + \frac{1-p-z^{-1}}{N+K-1} \right) p_{s}^{Kz}(K,z) - \\
& - \frac{Mz^{-1}}{p}\left( \frac{1}{1-K} + \frac{1}{N+K-1} \right) p_{s}^{K}(K,t=0).
\label{ode_WS}
\end{align}
Using the normalization condition $p_{s}^{Kz}(K=1,z)=\frac{1}{1-z}$ and assuming $N\gg 1$, the solution of \eqref{ode_WS} can be cast as
\begin{align}
\nonumber p_{s}^{Kz}(K,z) &= \frac{Mz^{-1}}{p}\left(1-K\right)^{-M\alpha}e^{-c\left(1+\alpha\right)\left(1-K\right)} \times \\
&\times \int_{K}^{1} \text{d}\xi e^{c\left(1+\alpha\right)\left(1-\xi\right)}\left(1-\xi\right)^{M\alpha-1} p_{s}^{z}(\xi,t=0),
\label{sol_ode_WS}
\end{align}
where
\begin{align}
\alpha := \frac{1}{p}\left(z^{-1}-1\right).
\label{alpha}
\end{align}
Then, returning back to the degree variable by inverting the transform \eqref{characteristic_K} leads to
\begin{align}
\nonumber p_{s}^{z}(k|m;z) &= \frac{1}{k!}\frac{\partial^{k}}{\partial K^{k}}\Bigg[ \frac{Mz^{-1}}{p}A_{\alpha}^{-1}(K) \times \\
&\times \int_{K}^{1}\textup{d}\xi\left(1-\xi\right)^{-1}A_{\alpha}(\xi)\xi^{m} \Bigg]_{K\rightarrow 0},
\label{pkz_WS}
\end{align}
for $m\in\{k,k_{0}\}$ and
\begin{align}
A_{\alpha}(K) := e^{c\left(1+\alpha\right)\left(1-K\right)}\left(1-K\right)^{M\alpha}.
\label{A(K)}
\end{align}
The probability \eqref{pkz_WS} is the key function to compute \eqref{fz_pz}, which can be used to evaluate some quantities of interest through \eqref{exp_fz}. This procedure is not direct as it was in the case of the dynamical Erd\H os-R\'enyi model and the technicalities are exposed in the supplemental material. Here, we will show the results only.
The arrival probability in this model is
\begin{align}
\mathcal{A}_{\text{\tiny WS}} = 1,
\label{A_WS}
\end{align}
as expected. No anomalous behavior as seen in the previous model is present here, where the degree changes by a single unit only.
The leading term of the first-passage time is
\begin{align}
\langle t\rangle_{\text{\tiny WS}} \sim \left\{
\begin{array}{ccl}
\displaystyle\frac{N}{p}e^{c}\sum_{n=k_{0}}^{k-1}\frac{\Gamma(n+1,c)}{c^{n}} &,& k>k_{0} \\
& & \\
\displaystyle\frac{N}{p}e^{c}\sum_{n=k}^{k_{0}-1}\frac{\gamma(n+1,c)}{c^{n}} &,& k<k_{0}
\end{array}
\right.,
\label{<t>_WS}
\end{align}
where $\Gamma(\cdot,\cdot)$ and $\gamma(\cdot,\cdot)$ are, respectively, the upper and lower incomplete Gamma functions. It is worth mentioning that this time is also proportional to $N/p$, as in the dynamical Erd\H os-R\'enyi model. The simulation of this model supports the analytical expression \eqref{<t>_WS}, as shown in figure \ref{fig1}.
\begin{figure}
\begin{center}
\includegraphics[width=121.6pt]{t1image_k0=2_k=5_c=4_R=100.png}
\includegraphics[width=121.6pt]{t1image_k0=5_k=2_c=4_R=100.png}
\caption{\label{fig1}The mean first-passage time as a function of the ratio $N/p$. Left: $k_{0}=2$ and $k=5$; right: $k_{0}=5$ and $k=2$. In both graphs, the mean degree of the network is $c=4$ and the results were obtained from $100$ samples; the error bar is smaller than the size of the points. These simulations were compared with the analytical result \eqref{<t>_WS}.}
\end{center}
\end{figure}
On the other hand, the leading contribution to the second moment is given by
\begin{widetext}
\begin{align}
\langle t^{2}\rangle_{\text{\tiny WS}} \sim \left\{
\begin{array}{ccl}
\displaystyle 2\left(\frac{N}{p}\right)^{2}e^{c}\sum_{n=k_{0}}^{k-1}\frac{n!}{c^{n}}\sum_{\ell=0}^{n}\frac{c^{\ell}}{\ell!}\sum_{m=\ell}^{k-1}\frac{\Gamma(m+1,c)}{c^{m}} &,& k>k_{0} \\
& & \\
\displaystyle 2\left(\frac{N}{p}\right)^{2}e^{c}\sum_{n=k}^{k_{0}-1}\frac{n!}{c^{n}}\sum_{\ell=n+1}^{\infty}\frac{c^{\ell}}{\ell!}\sum_{m=k}^{\ell-1}\frac{\gamma(m+1,c)}{c^{m}} &,& k<k_{0} \\
\end{array}
\right.,
\label{<t2>_WS}
\end{align}
\end{widetext}
and is proportional to $\left(N/p\right)^{2}$. The validity of \eqref{<t2>_WS} was tested by comparing to simulation in figure \ref{fig2}. There is an alternative representation of \eqref{<t2>_WS} in the supplemental material, but the form given here seems to be the most compact one. Naturally, \eqref{<t2>_WS} and \eqref{<t>_WS} can be used to compute the variance, which is also proportional to $\left(N/p\right)^{2}$. Since this expression shows no special aesthetic appeal, it will not be presented here.
\begin{figure}
\begin{center}
\includegraphics[width=121.6pt]{t2image_k0=2_k=5_c=4_R=100.png}
\includegraphics[width=121.6pt]{t2image_k0=5_k=2_c=4_R=100.png}
\caption{\label{fig2}The (mean) second moment of the first-passage time as a function of the ratio $N/p$. Left: $k_{0}=2$ and $k=5$; right: $k_{0}=5$ and $k=2$. In both graphs, the mean degree of the network is $c=4$ and the results were obtained from $100$ samples; the error bar is smaller than the size of the points. These simulations were compared with the analytical result \eqref{<t2>_WS}.}
\end{center}
\end{figure}
\section{\label{conclusion}Conclusion}
In this work, we investigated the time needed for a vertex to achieve a pre-established degree for the first time. The main strategy was mapping the problem into a first-passage problem in degree space. The gain/loss of degrees was illustrated by the time-dependent version of the Erd\H os-R\'enyi and Watts-Strogatz models, which display time-translational symmetry. This property was explored and analytical results concerning the first and second moments of the first-passage time were obtained. In both cases, the arrival probability ensured that the pre-established degree is achieved with probability $1$ (with a careful interpretation in the case of the Erd\H os-R\'enyi dynamics). Furthermore, the mean first-passage time is scaled linearly with the ratio $N/p$ for both models in the asymptotic regime of large networks, while this scale is quadratic for the second moment also in both models. On the other hand, these moments depend on the difference $\Delta$ in the Erd\H os-R\'enyi network only. In the dynamical Watts-Strogatz model, both moments depend on the initial and final degrees independently, as can be seen from \eqref{<t>_WS} and \eqref{<t2>_WS}, although the structure of both $\langle t\rangle_{\text{\tiny WS}}$ and $\langle t^{2}\rangle_{\text{\tiny WS}}$ can be compactly represented as a sum of terms involving upper (lower) incomplete Gamma functions when the final degree is larger (smaller) than the initial one.
\section*{\label{acknowledgements}Acknowledgments}
F.A. was financed in part by the Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de N\'ivel Superior - Brasil (CAPES) - Finance Code 001.
|
1,116,691,500,510 | arxiv | \section{Introduction}
The present paper deals with analytic properties of the
{\em partial theta function}
\begin{equation}\label{eqtheta}
\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j~.
\end{equation}
It owes its name to the resemblance between the function
$\theta (q^2,x/q)=\sum _{j=0}^{\infty}q^{j^2}x^j$ and the
{\em Jacobi theta function}
$\Theta (q,x):=\sum _{j=-\infty}^{\infty}q^{j^2}x^j$; ``partial'' refers to the fact
that summation in the case of $\theta$ takes place only from $0$ to $\infty$.
We consider the situation when the variable $x$ and the parameter $q$ are real,
more precisely, when $(q,x)\in (0,1)\times \mathbb{R}$. This function
has been studied also for $(q,x)\in (-1,0)\times \mathbb{R}$
and $(q,x)\in \mathbb{D}_1\times \mathbb{C}$; here $\mathbb{D}_1$ stands for
the open unit disk. For any fixed non-zero value of the parameter $q$
($|q|<1$),
the function
$\theta (q,.)$ is an entire function in $x$ of order~$0$.
The partial theta function finds various applications -- from
Ramanujan type $q$-series
(\cite{Wa}) to the theory
of (mock) modular forms (\cite{BrFoRh}), from asymptotic analysis (\cite{BeKi})
to statistical physics
and combinatorics (\cite{So}). How $\theta$ can be applied to problems
dealing with asymptotics and modularity of partial and false theta
functions and their relationship to representation theory and conformal field
theory is made clear in \cite{CMW} and \cite{BFM}. The place which this
function finds in Ramanujan's lost notebook is explained
in~\cite{AnBe} and~\cite{Wa}.
Its Pad\'e approximants are studied in~\cite{LuSa}.
A recent interest in the partial theta function is connected with the study of
section-hyperbolic polynomials, i.~e. real polynomials with positive coefficients, with
all roots real negative and all whose finite sections (i.e. truncations)
have also this property, see \cite{KoSh}, \cite{KaLoVi} and \cite{Ost};
the cited papers use results of Hardy, Petrovitch and Hutchinson
(see \cite{Ha}, \cite{Pe} and \cite{Hu}). Various analytic properties of the
partial theta function are proved in \cite{Ko2}-\cite{Ko13} and other papers
of the author.
The analytic properties of $\theta$ known up to now, in particular, the
behaviour of its zeros, are discussed in the next section. One of them
is the fact that for any $q\in (0,1)$, all complex conjugate
pairs of zeros of $\theta (q,.)$ remain within the domain
$$\{ {\rm Re}x\in (-5792.7, 0), |{\rm Im}x|<
132\} \cup \{ |x|<18\} ~.$$
For $q\in (-1,0)$, this is true for the domain
$\{ |${\rm Re}$x|<364.2, |${\rm Im}$x|<132\}$, see~\cite{Ko10} and~\cite{Ko8}.
In this sense the complex conjugate zeros of $\theta$ never go too far from the
origin. It is also true that they never enter into the unit disk,
see~\cite{Ko13} (but this property is false if $q$ and $x$ are complex, see
the next section). In the present paper we exhibit a convex domain
which contains the left unit half-disk, which is more than
$7$ times larger than the latter and which is free of zeros
of $\theta$ for any $q\in (0,1)$:
\begin{tm}\label{tmmain}
For any fixed $q\in (0,1)$, the partial theta function has no zeros
in the domain
$\mathcal{D}:=\{ \{ |x|\leq 3\} \cap \{${\rm Re}$x\leq 0\}
\cap \{ |${\rm Im}$x|\leq 3/\sqrt{2}
\} \} \subset \mathbb{C}$ (with $3/\sqrt{2}=2.121320344\ldots$).
\end{tm}
When only the real zeros of $\theta$
are dealt with, one can improve the above theorem:
\begin{prop}\label{propmain}
For any $q\in (0,1)$ fixed, the function $\theta (q,.)$ has no
real zeros~$\geq -5$.
\end{prop}
Before giving comments on these results in the next section we explain
the structure of the paper. Section~\ref{secremind}
reminds certain analytic properties of $\theta$. Proposition~\ref{propmain}
is proved in Section~\ref{prpropmain}. In Section~\ref{secprelim} we prove
some lemmas which are used to prove Theorem~\ref{tmmain}; their proofs
can be skipped at first reading. Section~\ref{secplan} contains a plan
of the proof of Theorem~\ref{tmmain}. The proofs of the proposition and lemmas
formulated in Section~\ref{secplan} can be found in
Section~\ref{secproofs}.
\section{Comments\protect\label{seccomments}}
Throughout the paper we use the following notation:
\begin{nota}\label{notat}
{\rm We define four arcs of the circle centered at $0$ and of radius~$3$:}
$$C_k:=\{ x\in \mathbb{C}||x|=3,
{\rm arg}x\in [\pi /2+(k-1)\pi /4,\pi /2+k\pi /4]\} ~,~~~k=1,~2,~3,~4~.$$
{\rm We set $w:=3/\sqrt{2}=2.121320344\ldots$. The border
$\partial \mathcal{D}$ of the domain $\mathcal{D}$ defined in
Theorem~\ref{tmmain} consists
of the arc $C_2\cup C_3$, the horizontal segments $S_{\pm}:=[-w\pm wi,\pm wi]$
and the vertical segment $S_v:=[-wi,wi]$. We parametrise the segment
$S_+$ by setting $x:=-t+wi$, $t\in [0,w]$.}
\end{nota}
One can make the following observations with regard to Theorem~\ref{tmmain}
and Proposition~\ref{tmmain}:
\vspace{1mm}
(1) It is not clear whether Theorem~\ref{tmmain} should hold true for
the whole of the left half-disk of radius $3$, because
$|\theta (0.71,e^{0.5188451144\pi i})|=0.0141\ldots$, i.~e. one obtains
a very small value of $|\theta |$ for a point of the arc $C_1$. This
might mean that a zero of $\theta$ crosses the arc $C_1$ for $q$ close to
$0.71$.
\vspace{1mm}
(2) The difficulty to prove results as the ones of Theorem~\ref{tmmain} and
Proposition~\ref{propmain} resides in the fact that the
rate of convergence of the series of $\theta$ decreases as $q$
tends to $1^-$,
and for $q=1$, one obtains as limit of $\theta$ the rational (not entire)
function $1/(1-x)$.
It is true that the series
of $\theta$ converges to the function $1/(1-x)$ (which has no zeros at all)
on a domain larger than the unit disk
and containing the domain $\mathcal{D}$, see~\cite{Ka}. Yet one disposes
of no concrete estimations about this convergence, so one cannot deduce
from it the absence of zeros of $\theta$ in the domain~$\mathcal{D}$ for
all $q\in (0,1)$.
\vspace{1mm}
(3) The domain $\mathcal{D}$ contains the left half-disk of radius
$3/\sqrt{2}>2$. The ratio of the surfaces of $\mathcal{D}$ and of the
left unit half-disk is
$(\pi 3^2/4+(3/\sqrt{2})^2)/(\pi /2)=7.364788974\ldots$.
\vspace{1mm}
(4) One knows that for $q=0.3092\ldots$, the function $\theta (q,.)$
has a double
real zero $-7.5032\ldots$, see~\cite{KoSh}. Pictures of the zero set of the
function $\theta$ (see \cite{Ko12}) suggest that for certain values of
$q\in (0,1)$, it has
a zero in the interval $(-7,-6)$, so Proposition~\ref{propmain} cannot be
made much stronger.
\vspace{1mm}
We explain by examples why analogs of
the property of the zeros of $\theta$ to avoid
the domain $\mathcal{D}$ cannot be found in cases other than
$q\in (0,1)$, $x\leq 0$:
\vspace{1mm}
(i) If $q$ is complex, then some of the zeros of $\theta$ can be of modulus
$<1$. Indeed, for $q=\rho e^{3\pi i/4}$, where $\rho \in (0,1)$
is close to $1$, the function $\theta$ has a zero close to
$0.33\ldots +0.44\ldots i$
whose modulus is $0.56\ldots <1$. Similar examples
can be given for any $q$ of the form
$\rho e^{k\pi i/\ell}$, $k$, $\ell \in \mathbb{Z}^*$, see~\cite{Ko13}.
It is true however that $\theta$ has no zeros for $|x|\leq 1/2|q|$, see
Proposition~7 in~\cite{Ko1}.
\vspace{1mm}
(ii) If $q\in (0,1)$, the function $\theta$
has no positive zeros, but $\theta (0.98,.)$ is likely to have a zero close
to $1.209\ldots+0.511\ldots i$ (i.~e. of modulus $1.312\ldots$),
see~\cite{Ko13}. Conjecture: {\em As $q\rightarrow 1^-$, one can find complex
zeros of $\theta (q,.)$ as close to $1$ as possible.} One can check
numerically that for $q$ close to $0.726475$, $\theta$ has a complex
conjugate couple of zeros close to $\pm 2.9083\ldots i$ (which by the way
corroborates the idea that the statement of Theorem~\ref{tmmain} cannot be
extended to the whole of the left half-disk of radius $3$).
Thus a convex domain free of zeros of $\theta$ should belong to the
rectangle $\{ {\rm Re}x\in (0,1)$, $|{\rm Im}x|<2.9083\ldots \}$.
\vspace{1mm}
(iii) For
$q\in (-1,0)$, it is true that the leftmost of the positive zeros of $\theta$
tends to $1^+$ as $q$ tends to $-1^+$, see part (2) of
Theorem~3 in~\cite{Ko12}. The function
$\theta (-0.96,.)$ is supposed to have a couple of conjugate zeros
close to the zeros
$z_{\pm}:=0.824\ldots \pm 1.226\ldots i$ (of modulus $1.478\ldots$)
of its truncation
$\theta _{100}^{\bullet}(-0.96,.)$; when truncating, the first two skipped terms
are of modulus
$6.57\ldots \times 10^{-75}$ and $1.51\ldots \times 10^{-76}$.
As $q\rightarrow -1^+$, the limit of $\theta$ equals $(1-x)/(1+x^2)$.
One can suppose that the zeros, which equal $z_{\pm}$ for $q=-0.96$,
tend to~$\pm i$
as $q\rightarrow -1^+$. One knows that for $q\in (-1,0)$,
complex zeros do not cross the imaginary axis, see Theorem~8 in~\cite{Ko12}.
Hence these zeros of $\theta$ should remain in the right half-plane and close
to~$\pm i$. This means that it is hard to imagine a convex domain in the right
half-plane much larger than the right unit half-disk and free of zeros of
$\theta$.
As for the left half-plane, the truncation $\theta _{100}^{\bullet}(-0.96,.)$ of
$\theta (-0.96,.)$ has conjugate zeros $0.769\ldots \pm 1.255\ldots i$
(of modulus $1.473\ldots$) about which, as about $z_{\pm}$ above,
one can suggest that they tend to $\pm i$ as $q\rightarrow -1^+$.
This could make one think that if one wants to find a
domain in the left half-plane containing the left unit half-disk and free of
zeros of $\theta$, then in this domain
the modulus of the imaginary part should not be larger than $1$.
On the other hand $\theta (-0.7,.)$ has a zero close to $w_0:=-2.69998\ldots$
so the width of
the desired domain should be $<|w_0|$.
\section{Known properties of the partial theta function
\protect\label{secremind}}
In this section we remind first that the Jacobi theta function satisfies the
{\em Jacobi triple product}
$$\sum _{j=-\infty}^{\infty}q^{j^2}x^{2j}=
\Theta (q,x^2)=\prod _{m=1}^{\infty}(1-q^{2m})(1+x^2q^{2m-1})(1+x^{-2}q^{2m-1})$$
from which we deduce the equalities
\begin{equation}\label{eqtriple}
\begin{array}{rcl}
\Theta ^*(q,x):=\Theta (\sqrt{q},\sqrt{q}x)&=&
\sum _{j=-\infty}^{\infty}q^{j(j+1)/2}x^j=
\prod _{m=1}^{\infty}(1-q^m)(1+xq^m)(1+q^{m-1}/x)\\ \\
&=&(1+1/x)\prod_{m=1}^{\infty}((1-q^m)(1+xq^m)(1+q^m/x))~.\end{array}
\end{equation}
It is clear that
\begin{equation}\label{eqthetaG}
\theta =\Theta ^*-G~~~\, {\rm with}~~~\,
G(q,x):=\sum _{j=-\infty}^{-1}q^{j(j+1)/2}x^j~.
\end{equation}
\begin{nota}\label{notaxXt}
{\rm (1) When treating the function $G$ we often change the variable $x$ to
$X:=1/x$. To distinguish the truncations of the function $\theta$ in
the variable $x$ from the ones in the variable $t$
(see Notation~\ref{notat}) we write
$\theta =\theta _k^{\bullet}+\theta _*^{\bullet}$, where
$\theta _k^{\bullet}:=\sum _{j=0}^kq^{j(j+1)/2}x^j$ and
$\theta _*^{\bullet}:=\sum _{j=k+1}^{\infty}q^{j(j+1)/2}x^j$, i.~e. we use the
superscript ``bullet'' when in the variable $x$ (no index $k$ is added to $\theta _*^{\bullet}$).
No superscript is used for the truncations
of $\theta (q,-t+wi)$ and of $G$.
(2) We set $\lambda :=3e^{3\pi i/4}$, $R(q,x):=\prod _{m=1}^{\infty}(1+q^{m-1}/x)$, $M:=|(1+qx)(1+q/x)|$, $M_0:=(1-q)M$ and $M_1(q,t):=M_0(q,-t+wi)$.}
\end{nota}
\begin{rem}
{\rm In the proofs we use the convergence of the series (\ref{eqtheta})
when the
parameter $q$ belongs to an interval of the form $[0,a]$, $a\in (0,1)$.
When we need to deal with intervals of the form $[a,1]$, we use the
equalities (\ref{eqthetaG}) in which the modulus of
the term $\Theta ^*$ tends to $0$ as $q$
tends to $1^-$ while the series of $G$ converges uniformly for
$|x|\in [c,\infty )$ for any fixed $c>1$. When in the proof of a lemma or a proposition we use the fact that a certain function in one variable (mainly a polynomial) is increasing or decreasing, we do not give a detailed proof of this, because in all such cases the proof can be given using elementary methods (computation of derivatives and numerical computation of their real roots). We do not give details when proving the absence of critical points of polynomials in two variables in given rectangles. In this text their degree is never too high and the necessary computations are easily performed using MAPLE.}
\end{rem}
For $q\in (0,1)$, the real zeros of $\theta$ (which are all negative)
and of any of its derivatives
w.r.t. the variable $x$ form a sequence tending to $-\infty$ and
behaving asymptotically as a geometric progression with ratio~$1/q$,
see Theorem~4 in~\cite{Ko1}.
There exists an increasing and tending to $1^-$ sequence of
{\em spectral values}
$\tilde{q}_j$ of $q$ such that $\theta (\tilde{q}_j,.)$ has a multiple
(more exactly double) real zero, see~\cite{KoSh}.
The $6$-digit truncations of the first $6$
spectral values are:
$$0.309249~,~~~0.516959~,~~~0.630628~,~~~
0.701265~,~~~0.749269~,~~~0.783984~.$$
When $q$ passes from $\tilde{q}_j^-$ to $\tilde{q}_j^+$, the rightmost two of
the real zeros of $\theta$ coalesce and then form a complex conjugate pair.
All other real zeros of $\theta$ remain negative and distinct, see Theorem~1
in~\cite{Ko1}. The inverse
(complex couples becoming double and then two distinct real zeros) never
happens. No zeros are born at~$\infty$.
Thus for $q$ fixed, the function $\theta$ belongs to the Laguerre-P\'olya
class $\mathcal{L-P}I$ exactly if $q\in (0,\tilde{q}_1]$. For
$q\in (\tilde{q}_j,\tilde{q}_{j+1}]$, the function $\theta$ is the product of
a real polynomial of degree $2j$ without real zeros and a function of the
class $\mathcal{L-P}I$. See the details in~\cite{Ko1A}.
Spectral values exist also for $q\in (-1,0)$, see~\cite{Ko5}. The existence
of spectral values for complex values of $q$ is proved in~\cite{Ko5}, see
Proposition~8 therein.
At the end of this section we mention the fact that the function
$\theta$ satisfies the two conditions
$$\theta (q,x)=1+qx\theta (q,qx)~~~\, {\rm and}~~~\,
2q\partial \theta /\partial q=2x\partial \theta /\partial x+
x^2\partial ^2\theta /\partial x^2~.$$
\section{Proof of Proposition~\protect\ref{propmain}\protect\label{prpropmain}}
For $q\leq 0.1$, all zeros of $\theta (q,.)$ are real negative and smaller than
$-1/q$, see Proposition~7 in~\cite{Ko1}. Hence they are smaller than $-5$ and
one has $\theta (q,x)>0$ for $x\in [-1/q,0]\supset [-5,0]$.
As $q$ increases, its zeros
depend continuously on $q$. For a spectral value of $q$,
certain zeros coalesce to form then
a complex conjugate pair, but new real zeros are not born,
see the previous section. Therefore it
suffices to show that for $q\in (0,1)$, one has $\theta (q,-5)>0$.
For $q\in (0,0.8]$, one finds numerically that $\theta (q,-5)$ is larger
than $0.04$. To this end one can consider
$\theta _{15}^{\bullet}(q,-5):=\sum _{j=0}^{15}q^{j(j+1)/2}(-5)^j$
which is a polynomial
in $q$ and show that for $q\in (0,0.5]$ (resp. for $q\in [0.5,0.8]$),
it is larger than $0.05$ (resp. larger than $0.16$).
The sum of the moduli
of all skipped terms is smaller than $1.8\times 10^{-30}$ and
$0.07$ respectively.
To prove that $\theta (q,-5)>0$ for $q\in [0.8,1)$, we first show that
$-G(q,-5)>4/25$. This is true, because $-G(q,-5)$ is a Leibniz series
with first two terms $1/5$ and $-q/25$, so its sum is
$\geq 1/5-q/25>1/5-1/25=4/25$.
Now we estimate $|\Theta ^*|$. We set
$$K(q):=(1+qx)(1+q/x)|_{x=-5}=(1-5q)(1-q/5)=1-26q/5+q^2~.$$
Hence $\Theta ^*(q,-5)=(4/5)\prod _{m=1}^{\infty}(1-q^m)
\prod _{m=1}^{\infty}K(q^m)$, see (\ref{eqtriple}).
The graph of the function $|K|$
(for $q\in [0,1]$) is shown in
Fig.~\ref{figmodulusK}.
\begin{figure}[htbp]
\centerline{\hbox{\includegraphics[scale=0.7]{modulusK.eps}}}
\caption{The graph of the function $|K|:=|1-26q/5+q^2|$.}
\label{figmodulusK}
\end{figure}
The function $|K|$ is decreasing on $[0,u']$ and increasing on
$[u',1]$, where $u'=0.2$. One has $|K(0)|=|K(u'')|=1$,
$u''=0.4182575771\ldots$, and $|K(1)|=3.2$.
In Fig.~\ref{figmodulusK} we show the values
$0.1357556939\ldots =:t'<t'':=0.2660119966\ldots$ of $q$ for which
$|K|=1/3.2$, the values
$0.09800079936\ldots =:d'<d'':
=0.3065310118\ldots$ where
$|K(q)|=1/2$ and the value $s:
=0.6609280570\ldots$ of $q$
such that $|K(q)|=2$.
For $a\in (0,1]$, set $\phi (a):=\ln (1/a)/\ln (1/q)$, so
$\phi (a)\geq 0$ and $\phi (a_1)-\phi (a_2)=\phi (a_1/a_2)$.
We remind that
\begin{equation}\label{eq[]}
{\rm for}~~~\, b, c\in \mathbb{R}~,~~~\,
[b-c]=[b]-[c]-\chi (b,c)~,~~~\, {\rm where}~~~\, \chi (b,c)=0
~~~\, {\rm or}~~~\, 1~;\end{equation}
here $[.]$ stands for the
integer part. Denote by $\ell _1$, $\ell _2\in \mathbb{N}$
the maximal indices
$\ell$ for which
one has $q^{\ell}\geq u''$ and $q^{\ell}\geq s$
respectively:
$$\ell _1=[\phi (u'')]~~~\, {\rm and}~~~\, \ell _2=[\phi (s)]~.$$
These are the numbers of terms of the sequence $\{ q^m\}$ belonging
to the intervals $[u'',1)$ and $[s,1)$. For the intervals
$[t',t'')$ and $[d',d'')$, these numbers are equal to
$$m_1:=[\phi (t')]-[\phi (t'')]~~~\, {\rm and}~~~\,
m_2:=[\phi (d')]-[\phi (d'')]~.$$
One computes directly that
$$\ln (t''/t')=0.672\ldots >0.414\ldots =\ln (1/s)~,~~~\,
{\rm thus}~~~\, \ln (t''/t')-\ln (1/s)>0.25~.$$
Moreover, as $q\in [0.8,1)$, one has
$1/\ln (q)\geq 1/\ln (0.8)=4.48\ldots >4$, so
$$\begin{array}{l}
\ln (t''/t')/\ln (1/q)-\ln (1/s)/\ln (1/q)>1~~~\, {\rm hence}~~~\,
[\phi (t'/t'')]>[\phi (s)]~~~\, {\rm and}\\ \\
m_1=[\phi (t')]-[\phi (t'')]=[\phi (t'/t'')]+
\chi (\phi (t'),\phi (t''))
>[\phi (s)]=\ell _2~.\end{array}$$
This means that out of the factors
$|K(q^m)|$ which are present in $|\Theta ^*(q,-5)|$,
the ones which are in the interval $[2,3.2)$ are less than
the ones which are in $[0,1/3.2]$. We denote their sets by
$\Sigma _{[2,3.2)}$ and $\Sigma _{[0,1/3.2]}$ and we write
$\sharp (\Sigma _{[2,3.2)})<\sharp (\Sigma _{[0,1/3.2]})$.
The latter inequality implies
$$\prod _{|K(q^m)|\in \Sigma _{[2,3.2)}\cup \Sigma _{[0,1/3.2]}}|K(q^m)|<1~.$$
Using the above notation and (\ref{eq[]}) one can write
\begin{equation}\label{eqphi}
\begin{array}{lllll}
\sharp (\Sigma _{[1,2)})&=:&\ell _3&=&[\phi (u'')]-
[\phi (s)]\leq [\phi (u''/s)]+1~~~\, {\rm and}\\ \\
\sharp (\Sigma _{(1/3.2,1/2]})&=:&m_3&\geq &[\phi (d')]-
[\phi (t')]+[\phi (t'')]-
[\phi (d'')]-1~.\end{array}
\end{equation}
To prove the latter inequality one has to observe
that the numbers $q^m$
corresponding to factors $|K(q^m)|$ from the set
$\Sigma _{(1/3.2,1/2]}$ belong to the union
$[d',t')\cup (t'',d'']$. The numbers $q^m$ which belong to the
interval
$[d',t')$ are exactly $[\phi (d')]-[\phi (t')]$. The ones that
are in $(t'',d'']$ are not less than
$[\phi (t'')]-[\phi (d'')]-1$ (at most one of them
equals $t''$ and there are exactly
$[\phi (t'')]-[\phi (d'')]$ numbers
$q^m$ in $[t'',d'')$). One finds that
$$
\phi (u''/s)=0.457\ldots <\phi (d'/t')+\phi (t''/{d'}')=0.467\ldots ~,~~~\,
{\rm so}$$
$$[\phi (u''/s)|\leq [\phi (d'/t')]+[\phi (t''/d'')]+2\leq m_3+5$$
(see the equalities and inequalities (\ref{eqphi}) and
(\ref{eq[]})) and one
concludes that $\ell _3\leq m_3+6$. The factors $|K(q^m)|$
which have not been
mentioned up to now are all of modulus $<1$; the corresponding
numbers $q^m$ belong to the intervals $(0,d')$ and $(t'',u'')$. Thus
$\prod _{m=1}^{\infty}|K(q^m)|<2^6$. At the same time
$$\prod_{m=1}^{\infty}(1-q^m)\leq \prod_{m=1}^{\infty}(1-0.8^m)<
7\times 10^{-6}~.$$
This shows that $|\Theta ^*(q,-5)|<10^{-4}<4/25<|-G(q,-5)|$
from which
the proposition follows.
\section{Some technical lemmas\protect\label{secprelim}}
In this section we formulate and prove several lemmas:
\begin{lm}\label{lmG}
For $q\in [0,1]$ and $|x|=a>2$, one has $|G|\geq (a-2)/a(a-1)$.
In particular, for $a=3$, $|G|\geq 1/6$.
\end{lm}
\begin{proof}
Indeed,
$$\begin{array}{cclcl}
|G|&\geq&(1/|x|)(1-\sum _{j=-\infty}^{-2}q^{j(j+1)/2}|x^{j+1}|)&\geq&
(1/|x|)(1-\sum _{j=-\infty}^{-2}|x^{j+1}|)\\ \\
&=&(1/a)(1-\sum _{j=-\infty}^{-2}a^{j+1})&=&(a-2)/a(a-1)~.\end{array}$$
\end{proof}
We set $X:=1/x$ and we represent the function $G$ in the form
$G=G_5+G_*$, $G_5:=X+qX^2+q^3X^3+q^6X^4+q^{10}X^5$,
$G_*:=\sum _{j=5}^{\infty}q^{j(j+1)/2}X^{j+1}$.
\begin{lm}\label{lm147}
For $(q,t)\in [0.6,1]\times [0,w]$, one has $|G_*|<0.0208$ and
$|G_5|>0.147$.
\end{lm}
\begin{proof}
For $(q,t)\in [0.6,1]\times [0,w]$, it is true that
$|X|=1/|-t+wi|\leq 1/w$ and
$$|G_*|\leq \sum _{j=5}^{\infty}|q^{j(j+1)/2}X^{j+1}|\leq
\sum _{j=5}^{\infty}|X^{j+1}|\leq \sum _{j=5}^{\infty}|w^{-j-1}|=
0.02076055760\ldots <0.0208$$
which proves the first claim of the lemma. To prove the second one we
represent the function $G_I:=$Im$(G_5(q,1/(-t+wi)))$ in the form
$$G_I=(3\sqrt{2}/(2t^2+9)^5)G^{\flat}_I~,~~~\, {\rm where}~~~\,
G^{\flat}_I=g_0+g_1q+g_3q^3+g_6q^6+g_{10}q^{10}~,~~~\, {\rm with}$$
$$\begin{array}{cclccl}
g_0&:=&-16t^8-288t^6-1944t^4-5832t^2-6561~,&g_6&:=&64t^5-1296t~,\\ \\
g_1&:=&32t^7+432t^5+1944t^3+2916t~,&&\\ \\
g_3&:=&-48t^6-360t^4-324t^2+1458~,&g_{10}&:=&-80t^4+720t^2-324~.\end{array}$$
Our first goal is to give an upper bound for $G_I$ for $t\in [0,1]$
(hence an upper bound for $G_I^{\flat}$). We use the evident equalities and
inequalities
$$\begin{array}{llll}
288=256+32~,&1944=1512+432~,&5832=3168+1944+720~,&
6561=2916+1458+2187~,\\ \\
32t^6\geq 32t^7q~,&432t^4\geq 432t^5q~,&1944t^2\geq 1944t^3q~,&
2916\geq 2916tq~,\\ \\
64tq^6\geq 64t^5q^6~,&720t^2\geq 720t^2q^{10}~,&1296=64+1232&1458\geq 1458q^3~
\end{array}$$
to obtain an upper bound $G^u$ for $G_I^{\flat}$ in which all coefficients are
negative:
$$\begin{array}{ccl}G^u&:=&-16t^8-256t^6-1512t^4-3168t^2-2187\\ \\
&&-(48t^6+360t^4+324t^2)q^3-1232tq^6-(80t^4+324)q^{10}~.\end{array}$$
For $t\in [0,1]$ fixed, the upper bound of the product
$(3\sqrt{2}/(2t^2+9)^5)G^u$ is attained for $q=0.6$. The function
$(3\sqrt{2}/(2t^2+9)^5)G^u|_{q=0.6}$ is decreasing in $t$ and its value
for $t=0$ is $v_1:=-0.1572756008\ldots$, so $v_1$ is the upper bound of $G_I$
for $t\in [0,1]$, $q\in [0.6,1]$.
Suppose now that $t\in [1,w]$ and $q\in [0.6,1]$. We observe first that
the function $G_I|_{q=1}$ is increasing and
$(G_I|_{q=1})|_{t=w}=-0.1478254790\ldots =:v_2$. Next, we prove that
$$\partial G^{\flat}_I/\partial q=g_1+3qg_3+6q^5g_6+10q^9g_{10}>0~~~\,
{\rm hence}~~~\, \partial G_I/\partial q>0~.$$
The quantities $g_1$, $g_6$ and $g_{10}$ do not change sign for $t\in [1,w]$:
$g_1>0$, $g_6\leq 0$, with equality only for $t=w$, while $g_{10}>0$.
The polynomial $g_3$ is negative for $t>t_1:=1.224744871\ldots$ and
positive for $t\in [1,t_1)$; it vanishes for $t=t_1$. Thus for $t\geq t_1$,
$$\partial G^{\flat}_I/\partial q\geq g_1+3g_3+6g_6+10\times 0.6^{10}g_{10}=:
G^{\ddagger}~.$$
The polynomial $(G^{\ddagger})'$ has a single real root
$t_2:=1.144295977\ldots$. The function $G^{\ddagger}$ is increasing for
$t\geq t_2$, so
for $t\geq t_1>t_2$, one has
$G^{\ddagger}(t)>G^{\ddagger}(t_2)=9.468005\ldots >0$.
For $t\in [1,t_1]$, we minorize the function
$\partial G^{\flat}_I/\partial q$ in each of the four cases
$q\in [0.6,0.7]$, $q\in [0.7,0.8]$, $q\in [0.8,0.9]$ and $q\in [0.9,1]$.
Denote any of these four intervals by $[a,b]$. The minoration is
looked for in the form
$$G_{a,b}:=g_1+3a^2g_3+6b^5g_6+10a^9g_{10}~.$$
Each of the four functions $G_{a,b}$ turns out to be monotone increasing
on $[1,t_1]$, so the four minima are attained for $t=1$. They equal
$4897.5\ldots$, $4096.5\ldots$, $2777.1\ldots$ and $920.4\ldots$
respectively. Hence for $t\in [1,w]$, $\partial G^{\flat}_I/\partial q>0$
and the function $G_I$ is maximal for $q=1$. Hence it is
$\leq v_2$. For $t\in [0,1]$, it is $\leq v_1$, so it is $\leq v_2<-0.147$
for $(q,t)\in [0.6,1]\times [0,w]$ and $|G_I|>0.147$.
\end{proof}
\begin{lm}\label{lmcos}
Consider the factors $|1+q^mx|$ and $|1+q^{m-1}/x|$, $m=1$, $2$, $\ldots$.
(1) For $q\in (0,1)$ fixed and for $x\in C_1\cup C_2$,
these quantities are decreasing functions in $\varphi :={\rm arg}x$.
(2) For $x=3e^{3\pi i/4}$, each factor $|1+q^{m-1}/x|$, $m\geq 2$,
is a decreasing function
in $q\in (0,1)$.
(3) For $q\in [0.5,1]$ and for $x=3e^{3\pi i/4}$, the factor $|1+qx|$ is an
increasing function in~$q$.
\end{lm}
\begin{proof}
The first claim of the lemma follows from the cosine theorem. Indeed,
recall that $1/x=\bar{x}/|x|^2$, so arg$(1/x)=-$arg$x=-\varphi$ and
$\cos ($arg$(1/x))=\cos \varphi$. Hence
$$\begin{array}{ccl}
|q^mx-(-1)|^2&=&q^{2m}|x|^2+(-1)^2-2(-1)q^m|x|\cos \varphi\\ \\ &=&
q^{2m}|x|^2+1+2q^m|x|\cos \varphi~,\\ \\
|(q^{m-1}/x)-(-1)|^2&=&q^{2m-2}/|x|^2+(-1)^2-2(-1)(q^{m-1}/|x|)
\cos \varphi\\ \\ &=&
q^{2m-2}/|x|^2+1+2(q^{m-1}/|x|)\cos \varphi~.
\end{array}$$
For $q$ fixed, these quantities are decreasing in $\varphi$, because
such is $\cos \varphi$. Set $\cos \varphi :=-\sqrt{2}/2$, $|x|:=3$.
The displayed formulas show that
$$\begin{array}{llll}
d(|1+q^{m-1}/x|)/dq&=&((m-1)q^{m-2}/|x|^2)(2q^{m-1}-\sqrt{2}|x|)<0&
{\rm and}\\ \\
d(|1+qx|)/dq&=&m|x|(2q|x|-\sqrt{2})>0&\end{array}$$
from which one deduces the last two claims of the lemma.
\end{proof}
In the proofs we need some properties of the functions $M:=|(1+qx)(1+q/x)|$
and $M_0:=(1-q)M$. We remind that we set $x=-t+wi$, $t\in [0,w]$, $w=3/\sqrt{2}$.
\begin{lm}\label{lmmaxmodulus}
For $t\in [0,w]$ and for $q\in [0.6,1]$ fixed, the quantities $M$ and $M_0$
are maximal
for $t=0$. For $q\in [0.6,0.75]$ fixed and for $t\in [1,w]$, they are maximal for $t=1$.
\end{lm}
\begin{proof}
It suffices to prove the claims of the lemma about the function $M$. One checks directly for the square of $M$ that
$$M^2=(2q^2t^2+9q^2-4qt+2)(2q^2-4qt+2t^2+9)/(2t^2+9)~.$$
One verifies straightforwardly that
$$\begin{array}{l}
M^2-M^2|_{t=0}=-2qtP/(2t^2+9)~,~~~\, {\rm where}\\ \\
P:=36q^2t^2-18qt^3+ 198q^2-149qt+36t^2+ 198~.
\end{array}$$
The discriminant of the trinomial $36(qt)^2-149qt+198$ is negative, so this trinomial is positive-valued. For the remaining terms of $P$, for $t\in [0,w]$ (hence $t^2\leq 9/2$), one obtains
$$-18qt^3+198q^2+36t^2\geq -81qt+198q^2+36t^2$$
which is again a trinomial with negative discriminant. Thus $P>0$ and $M^2-M^2|_{t=0}\leq 0$ with equality only for $t=0$ which proves the first claim of the lemma. To prove its second statement we consider the difference
$$\begin{array}{l}M^2-M^2|_{t=1}=-2q(t-1)(V_2q^2+V_1q+V_0)/11(2t^2+9)~,~~~\, {\rm where}\\ \\
V_2=V_0:=44t^2-8t+234~~~\, {\rm and}~~~\, V_1:=-(t+1)(22t^2+167)~.
\end{array}
$$
The polynomial $V_2q^2+V_1q+V_0$ has no crfitical points for $(q,t)\in [0.6,0.75]\times [1,w]$. Its restrictions to each of the sides of this rectangle (i.~e. its restrictions obtained for $q=0.6$, $q=0.75$, $t=1$ and $t=w$) are positive-valued. Hence the difference $M^2-M^2|_{t=1}$ is negative in the given rectangle which proves the second claim of the lemma.
\end{proof}
\begin{rem}\label{remM}
{\rm For $x=-t+wi$, we represent in Fig.~\ref{twographs} the graph of the function}
$$\begin{array}{ccl}
M_1(q,t)&:=&M_0(q,-t+wi):=(1-q)|(1+qx)(1+q/x)|\\ \\
&=&(1-q)(2q^2t^2+9q^2-4tq+2)^{1/2}
(2q^2-4tq+2t^2+9)^{1/2}/(2(2t^2+9))^{1/2}\end{array}$$
{\rm for two fixed values of $t$, namely
$t=0$ (in solid line) and
$t=1$ (in dashed line). The two functions}
\begin{figure}[htbp]
\centerline{\hbox{\includegraphics[scale=0.7]{modulus.eps}}}
\caption{The graphs of the functions $M_1(q,0)$ (in solid line) and
$M_1(q,1)$ (in dashed line).}
\label{twographs}
\end{figure}
$$\begin{array}{l}
M_1(q,0)=(1-q)(9q^2+2)^{1/2}
(2q^2+9)^{1/2}/3\sqrt{2}~~~\, {\rm and}\\ \\ M_1(q,1)=(1-q)(11q^2-4q+2)^{1/2}
(2q^2-4q+11)^{1/2}/\sqrt{22}\end{array}$$
{\rm are decreasing on $[0,1]$.}
\end{rem}
\section{Plan of the proof of
Theorem~\protect\ref{tmmain}\protect\label{secplan}}
The zeros of $\theta$ depend continuously on $q$ and no zeros are born at
$\infty$. We prove that for $q\in (0,1)$, there is no zero of $\theta$ on the
border $\partial \mathcal{D}$ of the domain $\mathcal{D}$. For $q\in (0,0.5]$,
this follows from the proposition below. We remind that (see Notation~\ref{notat})
$$\partial \mathcal{D}=C_2\cup C_3\cup S_+\cup S_-\cup S_v~.$$
\begin{prop}\label{prop05}
For $q\in (0,0.5]$, the function $\theta (q,.)$ has no zeros in the
closed rectangle
$\Delta :=\{ -3\leq {\rm Re}x\leq 0,~-3\leq {\rm Im}x\leq 3\}$.
\end{prop}
The proof of this proposition and of all lemmas formulated in this
section are given in Section~\ref{secproofs}. The rectangle
$\Delta$ contains the domain $\mathcal{D}$, so for $q\in (0,0.5]$, there
are no zeros of $\theta$ on $\partial \mathcal{D}$. One can observe that
for $q\in (0,\tilde{q}_1]$, $\tilde{q}_1=0.3092\ldots$,
there are no complex conjugate pairs of $\theta$ (see Section~\ref{secremind}),
and for $q\in (\tilde{q}_1,0.5]$, there is exactly one such pair.
From now on we assume that $q\in [0.5,1)$. The next lemma explains why
no zeros of $\theta$ can be found on the arc~$C_2$ hence none on the arc
$C_3$ either.
\begin{lm}\label{lm051}
For $q\in [0.5,1)$ and $x\in C_2$, one has $|G|>|\Theta ^*|$
hence $|\theta |>0$.
\end{lm}
The next lemma states that the function $\theta$ has no zeros on the
segment $S_v$:
\begin{lm}\label{lmnoimaginary}
For $q\in (0,1)$, the function $\theta (q,.)$ has no purely imaginary zeros
of modulus $\leq 2.2$ hence no such zeros of modulus
$\leq 3/\sqrt{2}=2.1\ldots$.
\end{lm}
It remains to show that there are no zeros of $\theta$ on the segments
$S_{\pm}$. It suffices to deal with the segment $S_+$. We consider the
restrictions to $[0.3,0.6]\times [0,w]$
of the functions $\theta _5(q,t):=\sum _{j=0}^5q^{j(j+1)/2}(-t+wi)^j$ and
$\theta _*(q,t)=\sum _{j=6}^{\infty}q^{j(j+1)/2}(-t+wi)^j$.
\begin{lm}\label{lm0306}
For $(q,t)\in [0.3,0.6]\times [0,w]$, one has $|\theta _*(q,t)|\leq 0.018$
and $\theta _I:=${\rm Im}$(\theta _5(q,t))>0.13$. Hence for
$(q,t)\in [0.3,0.6]\times [0,w]$, the
function $\theta$ has no zeros.
\end{lm}
Next we settle the case $q\in [0.75,1)$.
\begin{lm}\label{lm0751}
For $(q,t)\in [0.75,1)\times [0,w]$, the function $\theta$ has no zeros.
\end{lm}
The remaining case $q\in [0.6,0.75]$ will be subdivided in two cases:
\begin{lm}\label{lm1w}
For $(q,t)\in [0.6,0.75]\times [1,w]$, the function $\theta$ has no zeros.
\end{lm}
\begin{lm}\label{lm01}
For $(q,t)\in [0.6,0.75]\times [0,1]$, the function $\theta$ has no zeros.
\end{lm}
\section{Proofs\protect\label{secproofs}}
\begin{proof}[Proof of Proposition~\ref{prop05}]
A) For $q\in [0,0.3]$, all zeros of $\theta (q,.)$ are real, negative and
distinct, see Section~\ref{seccomments}. All these zeros are $<-5<-3$,
see Proposition~\ref{propmain}.
\vspace{1mm}
B) We set $\theta =\theta _4^{\bullet}+\theta _*^{\bullet}$, where
$\theta _4^{\bullet}:=1+qx+q^3x^2+q^6x^4+q^{10}x^4$ and
$\theta _*^{\bullet}:=\sum _{j=5}^{\infty}q^{j(j+1)/2}x^j$. For $x\in \Delta$, one has
$|x|\leq 3\sqrt{2}=4.24\ldots <4.25$, so for $(q,x)\in [0,0.5]\times \Delta$,
one obtains the majoration
\begin{equation}\label{eqestim}
|\theta _*^{\bullet}(q,x)|\leq
\sum _{j=5}^{\infty}0.5^{j(j+1)/2}4.25^j=0.045\ldots <0.046~.
\end{equation}
C) We denote the border of the rectangle $\Delta$ by $\partial \Delta$
and we set $I_0:=[0,0.5]$. We
show that for each $q\in I_0$ fixed, one has
\begin{equation}\label{eqRouche}
|\theta _4^{\bullet}(q,x)|>|\theta _*^{\bullet}(q,x)|>0
\end{equation}
for any $x\in \partial \Delta$. For
$q\in [0,0.01]$, there is no zero of $\theta _4^{\bullet}$ in
$\Delta$. Indeed, one
obtains
$$|\theta _4^{\bullet}|\geq 1-0.01\times 4.25-0.01^3\times 4.25^2-0.01^6\times
4.25^3-0.01^{10}\times 4.25^4=0.95\ldots >0~.$$
The condition (\ref{eqRouche}) is fulfilled for
$x\in \partial \Delta$, so it implies that no zero of $\theta _4^{\bullet}$
may enter $\Delta$
as $q$ increases from $0.01$ to $0.5$. Hence $\theta _4^{\bullet}$
has no zeros in $\Delta$ for $q\in I_0$.
Again from condition (\ref{eqRouche}) and from the Rouch\'e theorem
follows that $\theta$ has no zeros for $(q,x)\in I_0\times \Delta$. So
our aim is to show that condition~(\ref{eqRouche}) holds true.
\vspace{1mm}
D) When proving condition (\ref{eqRouche}) we deal only with the part of
$\partial \Delta$ with Im$x\geq 0$. For Re$x=-3$, we set $x:=-3+it$,
$t\in [0,3]$. Then
$$\begin{array}{ccccl}G_R(q,t)&:=&{\rm Re}(\theta _4^{\bullet})&=&
q^{10}t^4-54q^{10}t^2+81q^{10}+9q^6t^2-27q^6-q^3t^2+9q^3-3q+1~,\\ \\
G_I(q,t)&:=&{\rm Im}(\theta _4^{\bullet})&=&
12q^{10}t^3-108q^{10}t-q^6t^3+27q^6t-6q^3t+qt~.\end{array}$$
We use the fact that
$|\theta _4^{\bullet}|\geq \max (|{\rm Re}(\theta _4^{\bullet})|,
|{\rm Im}(\theta _4^{\bullet})|)=:\mu$.
Neither of the functions $G_R$ and $G_I$ has a critical point with $q\in I_0$,
so $G_R$ (resp. $G_I$) attains its maximal and its
minimal value when one
of the following conditions takes place: $t=0$, $t=3$, $q=0$ or $q=0.5$.
For $q=0$, one has $G_R\equiv 1$, so $\mu \geq 1>0.046$. For $q=0.5$,
one gets
$$G_R=0.0009765625t^4-0.037109375t^2+0.2822265625~,~~~\,
G_I=-0.00390625t^3+0.06640625t$$
and one checks directly that for $t\in [0,1]$ and $t\in [1,3]$, one has
$G_R>0.05>0.046$ and $G_I>0.05>0.046$ respectively.
For
$t=0$, one obtains $G_R=81q^{10}-27q^6+9q^3-3q+1$ which is $>0.2>0.046$
for $q\in I_0$. For $t=3$, it is clear that
$$G_R=-324q^{10}+54q^6-3q+1~~~\, {\rm and}~~~\,
G_I=54q^6-18q^3+3q~,$$
with $G_R>0.2>0.046$ for $q\in [0,0.2]$ and with
$G_I>0.05>0.046$ for $q\in [0.2,0.5]$ respectively.
\vspace{1mm}
E) For Re$x=0$, one sets $x:=i\tau$ to obtain
$$U_R(q,\tau ):={\rm Re}(\theta _4^{\bullet})=
q^{10}\tau ^4-q^3\tau ^2+1~~~\, {\rm and}~~~\,
U_I(q,\tau ):={\rm Im}(\theta _4^{\bullet})=-q\tau (q^5\tau ^2-1)~.$$
Neither of the functions $U_R$ and $U_I$ has a critical point with $q\in I_0$,
so their maximal and minimal values are attained for $\tau =0$, $\tau =3$,
$q=0$ or $q=0.5$. In each of the cases $\tau =0$ and $q=0$ one has $U_R\equiv 1>0.046$.
Suppose that $\tau =3$. Then $U_R>0.05>0.046$ for $q\in [0,0.3]$ and
$U_I>0.05>0.046$ for $q\in [0.3,0.5]$. Finally, if $q=0.5$, then
$$U_R=0.0009765625\tau ^4-0.125\tau ^2+1~~~\, {\rm and}~~~\,
U_I=-0.5\tau (0.03125\tau ^2-1)~,$$
with $U_R>0.05$ for $\tau \in [0,2]$ and with $U_I>0.05$ for
$\tau \in [2,3]$.
\vspace{1mm}
F) Suppose that Im$x=3$. Then we set $x:=u+3i$, $u\in [-3,0]$. Then
$$\begin{array}{ccccl}
S_R(q,u)&:=&{\rm Re}(\theta _4^{\bullet})&=
&q^{10}u^4-54q^{10}u^2+81q^{10}+q^6u^3-27q^6u+
q^3u^2-9q^3+qu+1~,\\ \\
S_I(q,u)&:=&{\rm Im}(\theta _4^{\bullet})&=
&12q^{10}u^3-108q^{10}u+9q^6u^2-27q^6+6q^3u+3q~.
\end{array}$$
The functions $S_R$ and $S_I$ have no critical points for $(q,u)$ inside
the rectangle $I_0\times [-3,0]$, so their maximal and minimal values are
attained on its border. Obviously $S_R|_{q=0}\equiv 1>0.046$, $S_R|_{u=0}=81q^{10}+1\geq 1>0.046$ and
$$S_I|_{q=0.5}=0.01171875u^3+0.140625u^2+0.64453125u+1.078125$$
which is $>0.05>0.046$ for $u\in [-3,0]$. For $u=-3$, one obtains
$$S_R=-324q^{10}+54q^6-3q+1~~~\, {\rm and}~~~\, S_I=54q^6-18q^3+3q~,$$
with $S_R>0.05>0.046$ for $q\in [0,0.2]$ and with $S_I>0.05>0.046$
for $q\in [0.2,0.5]$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lm051}]
It suffices to show that $|\Theta ^*|<1/6$, see Lemma~\ref{lmG} with $a=3$.
By part (1) of Lemma~\ref{lmcos}, it is
sufficient to prove this for $x=\lambda :=3e^{3\pi i/4}$. The modulus
$|R|:=\prod_{m=1}^{\infty}|1+q^{m-1}/x|$ is maximal (see Notation~\ref{notaxXt} and part (2) of Lemma~\ref{lmcos}) when $q=0.5$ in which case one gets
$$|1+q^{m-1}/x|=r_m:=|1+0.5^{m-1}(-\sqrt{2}-\sqrt{2}i)/6|=
((1-0.5^{m-1}\sqrt{2}/6)^2+0.5^{2m-2}/18)^{0.5}$$
and one finds numerically that
$$|R|\leq \prod _{m=1}^{11}r_m=0.6329437509\ldots <0.633~.$$
Next, for $x=\lambda$, the points representing the complex numbers
$u_m:=1+xq^m$ lie on the
straight line passing through the points $1$ and $i$; they lie above the
abscissa-axis. We denote by $m_0\in \mathbb{N}$ the index $m$ for which
Re$(u_m)\leq 0$ (hence $|u_m|\geq 1$) and Re$(u_{m+1})>0$ (hence $|u_{m+1}|<1$).
One has $m_0\geq 1$. Indeed, for $q\in [0.5,1]$,
$$|1+q\lambda |\geq |1+0.5\lambda |=1.062393362\ldots >1$$
(see part (3) of Lemma~\ref{lmcos}). A numerical check shows that one
has $m_0\geq 2$ exactly if $q\geq 0.6865890479\ldots >0.68=:q^{\dagger}$.
For $m\leq m_0$, one has $|u_m|\leq |v_m|$, where
$v_m:=q^m+\lambda q^m=q^m(1+\lambda )$ (with equality only for $q=1$). One finds numerically that
$$|1+\lambda |=2.399449794\ldots <2.4~,$$
so for $m\leq m_0$, $|u_m|<2.4$. For each product $p_m:=(1-q^m)u_m$ one can write
$$|p_m|\leq |(1-q^m)v_m|=|(1-q^m)q^m||1+\lambda |\leq (1/4)\times 2.4=0.6~.$$
Suppose that $m_0\geq 3$. Then
$$\begin{array}{ccl}
|\Theta ^*|&\leq&(\prod _{m=1}^{m_0}|p_m|)\times 0.633\times
(\prod _{m=m_0+1}^{\infty}|u_m|)\times \prod _{m=m_0+1}^{\infty}(1-q^m)\\ \\
&\leq&0.6^{m_0}\times 0.633\leq 0.6^3\times 0.633=0.136728<1/6~.\end{array}$$
Suppose that $m_0=2$. The maximal value of the function
$q(1-q)q^2(1-q^2)=q^3(1-q)^2(1+q)$ for $q\in [0,1]$ is
$0.05579835315\ldots <0.056$; it is attained for $q=0.6286669788\ldots$.
Thus
$$|p_1||p_2|\leq 0.056\times 2.4^2~,~~~\, \prod _{m=3}^{\infty}(1-q^m)<0.78~~~\,
{\rm and}$$
$$\begin{array}{ccl}
|\Theta ^*|&\leq&|p_1||p_2|\times 0.633\times
(\prod _{m=3}^{\infty}|u_m|)\times \prod _{m=3}
^{\infty}(1-q^m)\\ \\
&\leq&0.056\times 2.4^2\times 0.633\times 0.78<0.16<1/6~.\end{array}$$
Suppose that $m_0=1$. Then $0.5\leq q<0.69$. One finds that
$$\begin{array}{l}
\prod_{m=2}^{\infty}(1-q^m)<\prod_{m=2}^{100}(1-0.5^m)=0.5775\ldots <0.5776~~~\, {\rm and}\\ \\
\prod_{m=2}^{\infty}|u_m|<\prod_{m=2}^{30}|u_m|=:g(q)~.
\end{array}$$
The function $g$ is decreasing for $q\in [0.5,0.69]$ and $g(0.5)=0.4254\ldots <0.4255$. Therefore
$$\begin{array}{ccl}|\Theta ^*|&\leq&|p_1|\times 0.633\times g(q)\times 0.5776\\ \\
&<&0.6\times 0.633\times 0.4255\times 0.5776=0.093\ldots <1/6~.
\end{array}$$
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lmnoimaginary}]
Indeed, set $x:=iy$, $y\in \mathbb{R}$. Hence
$$\theta (q,iy)=\theta (q^4,-y^2/q)+iqy\theta (q^4,-qy^2)~.$$The first and the
second summand represent the real and the imaginary part of $\theta$ when
restricted to the imaginary axis. If $iy_0$ is a zero of $\theta (q,.)$,
$y_0\in \mathbb{R}$,
then
$$\theta (q^4,-y_0^2/q)=\theta (q^4,-qy_0^2)=0~.$$
By Proposition~\ref{propmain}, $-y_0^2/q<-5$ and
$-qy_0^2<-5$, so $|y_0|>\sqrt{5}>2.2$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lm0306}]
For $t\in [0,w]$, one has $|-t+wi|\leq 3$, with equality only for $t=w$.
Thus
$$|\theta _*(q,t)|\leq \sum _{j=6}^{\infty}q^{j(j+1)/2}3^j\leq
\sum _{j=6}^{\infty}0.6^{j(j+1)/2}3^j=0.017\ldots <0.018~.$$
One finds by direct computation that
$$\theta _I=(3\sqrt{2}q/8)(20q^{14}t^4-180q^{14}t^2+81q^{14}-16q^{9}t^3+
72q^{9}t+12q^5t^2-18q^5-8q^2t+4)~.$$
By lowercase indices $t$ or $q$ we denote derivations w.r.t. these variables.
We show first that $(\theta _I)_t<0$. Thus for $q\in [0.3,0.6]$ fixed,
$\theta _I$ is maximal for $t=0$ and minimal for $t=w$. One finds that
$$\begin{array}{cclccl}
(\theta _I)_t&=&3\sqrt{2}(10q^{12}t^3-45q^{12}t-6q^7t^2+
9q^7+3q^3t-1)q^3~,&&&\\ \\
(\theta _I)_{tt}&=&9\sqrt{2}q^6(10q^9t^2-15q^9-4q^4t+1)~,&
(\theta _I)_{ttt}&=&36\sqrt{2}q^{10}(5q^5t-1)~.\end{array}$$
For $(q,t)\in [0.3,0.6]\times [0,w]$, one has $(\theta _I)_{ttt}<0$. Hence
$(\theta _I)_{tt}$ is minimal for $t=w$; in this case it equals
$$9\sqrt{2}q^6(30q^9-6q^4\sqrt{2}+1)$$
which is positive for $q>0$. Therefore
$(\theta _I)_t$ is maximal for $t=w$ when it equals
$$3\sqrt{2}(-18q^7+(9\sqrt{2}q^3)/2-1)q^3$$
which is negative for $q>0$.
So $\theta _I$ is minimal for $t=w$ and
$$\theta _I(q,w)=-(3\sqrt{2}q(81q^{14}-9q^5+3\sqrt{2}q^2-1))/2~.$$
The derivative $(\theta _I(q,w))_q$ is negative for $q\geq 0.3$,
this means that $\theta _I$ is minimal for $(q,t)=(0.6,w)$. One has
$\theta _I(0.6,w)=0.1387526518\ldots >0.018$, so for $q\in (0,0.6]$,
$\theta$ has no zeros for $t\in [0,w]$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lm0751}]
For $q\in [0.75,1)$ fixed and for $t\in [0,w]$, the quantity
\begin{equation}\label{eqM1}
M_1:=(1-q)|(1+qx)(1+q/x)||_{x=-t+wi}\end{equation}
is maximal for
$t=0$, see Lemma~\ref{lmmaxmodulus}. The quantity $M_1|_{t=0}$ is maximal for $q=0.75$, see Remark~\ref{remM}. In this case
$$\begin{array}{ccl}
\prod_{m=1}^{\infty}M_1(q^m,t)&\leq&
\prod_{m=1}^{\infty}M_1(0.75^m,0)\\ \\
&<&\prod_{m=1}^{40}M_1(0.75^m,0)=0.1103687051\ldots =:h_1~,\\ \\
|1+1/x|&=&|1+1/(-t+wi)|=|(1-t)+wi|/|-t+wi|\\ \\
&=&f(t):=(((1-t)^2+w^2)/(t^2+w^2))^{1/2}\\ \\ &\leq&
(11/9)^{1/2}=1.105541597\ldots =:h_2
\end{array}$$
and $|\Theta ^*|\leq h_1h_2=0.1220171945\ldots <0.123$, see (\ref{eqthetaG}). (It is easy
to show that the function $f$ is decreasing on $[0,w]$, so
$f(t)\leq f(0)=(11/9)^{1/2}$.)
On the other hand Lemma~\ref{lm147} implies
$$|\theta |\geq |G|-|\Theta ^*|\geq |G_5|-|G_*|-|\Theta ^*|\geq
0.147-0.0208-0.123=0.0032>0~.$$
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lm1w}]
For $q\in [0.6,0.75]$ fixed and for $t\in [1,w]$, the quantity $M_1$ (see (\ref{eqM1}))
is maximal for
$t=1$ (see Lemma~\ref{lmmaxmodulus}). The quantity $M_1|_{t=1}$ is maximal for $q=0.6$, see Remark~\ref{remM}. Therefore it is true that
$$
\prod_{m=1}^{\infty}M_1(q^m,t)\leq
\prod_{m=1}^{\infty}M_1(0.6^m,1)<\prod_{m=1}^{40}M_1(0.6^m,1)=0.1048026086\ldots =:h_3~.
$$
The function $f$ defined in the proof of Lemma~\ref{lm0751} is decreasing and takes its maximal value $h_4:=(9/11)^{1/2}$ for $t=1$. Thus
$|\Theta ^*|\leq h_3h_4=
0.09479752467\ldots <0.095$. Using
Lemma~\ref{lm147}, one deduces that
$$|\theta|\geq |G_5|-|G_*|-|\theta ^*|\geq 0.147-0.0208-0.095=0.0312>0~.$$
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lm01}]
We set $\theta (q,x)=\theta _7^{\bullet}(q,x)+\theta _{*}^{\bullet}(q,x)$, where
$\theta _7^{\bullet}:=\sum _{j=0}^7q^{j(j+1)/2}x^j$ and
$\theta _{*}^{\bullet}:=\sum _{j=8}^{\infty}q^{j(j+1)/2}x^j$. The maximal possible
modulus $|x|$ for $t\in [1,w]$ is obtained for $t=1$. This gives $x= -1+3i/\sqrt{2}$ and in this case
$|x|=2.345207880\ldots <2.346$. This means that
$$|\theta _{*}^{\bullet}|\leq \sum _{j=8}^{\infty}|q|^{j(j+1)/2}|x|^j\leq
\sum _{j=8}^{\infty}0.75^{j(j+1)/2}2.346^j<0.036~.$$
On the other hand when setting $x:=-t+iw$, $t\in [0,1]$, and
$T_I:={\rm Im}(\theta _7^{\bullet}(q,-t+iw))$,
one obtains that
$$\begin{array}{ccl}
T_I&=&(3\sqrt{2}q/16)(56q^{27}t^6-1260q^{27}t^4+3402q^{27}t^2-729q^{27}-
48q^{20}t^5+720q^{20}t^3-972q^{20}t\\ \\
&&+40q^{14}t^4-360q^{14}t^2+162q^{14}-32q^9t^3+144q^9t+24q^5t^2-
36q^5-16q^2t+8)~.
\end{array}$$
We set $T_{I,k}:=\partial ^kT_I/\partial t^k|_{t=0}$. These functions
are equal respectively to:
$$\begin{array}{cclccl}
T_{I,0}&=&-(3\sqrt{2}/16)q(81q^{18}+18q^9-18q^5+4)(9q^9-2)~,&
T_{I,4}&=&-90\sqrt{2}q^{15}(63q^{13}-2)~,\\ \\
T_{I,1}&=&-(3\sqrt{2}/4)q^3(243q^{18}-36q^7+4)~,&
T_{I,5}&=&-1080\sqrt{2}q^{21}~,\\ \\
T_{I,2}&=&(9\sqrt{2}/4)q^6(567q^{22}-60q^9+4)~,&
T_{I,6}&=&7560\sqrt{2}q^{28}~.\\ \\
T_{I,3}&=&18\sqrt{2}q^{10}(45q^{11}-2)~,&&&
\end{array}$$
These derivatives do not change sign on the interval $[0.6,0.75]$, with
sgn$(T_{I,k})=(-1)^k$. The Taylor series of $T_I$ at $t=0$ reads:
$$T_I=(T_{I,0}+tT_{I,1})+(t^2/2)(T_{I,2}+tT_{I,3}/3)+(t^4/24)(T_{I,4}+tT_{I,5}/5)+
T_{I,6}t^6/6!~.$$
One verifies directly that
the following inequalities hold true:
$$
T_{I,0}>|T_{I,1}|+0.02~,~~~\, T_{I,2}>|T_{I,3}|/3~~~\, {\rm and}~~~\,
T_{I,4}>|T_{I,5}|/5~.$$
Hence the Taylor series of $T_I$
takes only values $>0.02$ for $t\in [0,1]$. We need, however,
a stronger result. It is to be checked directly that
(i) $T_{I,0}-|T_{I,1}|/2>0.2$, so for $t\in [0,1/2]$, one has
$T_I>0.2$, $\theta _7^{\bullet}>0.2>0.038>|\theta _{*}^{\bullet}|$ and thus
$|\theta |>0$;
(ii) $T_{I,2}/2-|T_{I,3}|/6>0.1$, so for $t\in [1/2,1]$, one has
$$T_I>(T_{I,0}-|T_{I,1}|)+(1/4)(T_{I,2}/2-|T_{I,3}|/6)>0.02+0.025>0.038$$
and again $|\theta |\geq \theta _7^{\bullet}-|\theta _{*}^{\bullet}|>0$.
\end{proof}
|
1,116,691,500,511 | arxiv | \section{INTRODUCTION}
With recent developments in soft robotics~\cite{c1,c2,c3}, robot arms equipped with soft grippers are able to achieve more human-like tasks such as grasping fragile objects like vegetables and fruits. Soft grippers are made of compliant materials, which have high elasticity and compliance. On the positive side, soft materials enable the robot to perform complex motions during operations. On the negative side, the flexibility makes the overall robot grasping and manipulation problems more challenging. To achieve robust performance in dexterous in-hand manipulation, soft grippers require soft touch sensors. The recent works have been discussed below.
\begin{figure}[http]
\centering
\includegraphics[width=220pt]{Fig1_1.JPG}
\caption{The appearance of the soft sensor.}
\label{fig: Apparence}
\end{figure}
Capacitive sensors~\cite{c10,c11,c12,c13,c14,c15} are compliant and can be layout as sensor array to detect contact force in different positions~\cite{c12}. However, this type of sensors required complex circuits, which is hard to maintain in practice. Optical, ionic, and magnetic sensors have been developed recently. The vision-based sensors were embedded inside soft materials~\cite{c16,c17}. With machine learning algorithms, a model could be established to predict sensors' contact force or deformation. Nevertheless, the vision-based sensor might increase the costs, and the size of the vision sensors limits the size of soft sensors. In addition, Jamone et al.~\cite{c19} reported an idea about embedding magnets and Hall-effect sensors into soft material and formed soft sensors, which were integrated with the anthropomorphic robot hand. Unfortunately, the elasticity of soft sensors will be changed by embedded magnets and Hall-effect sensors inside.
Another type of sensors is resistive sensor. The traditional resistive sensors, such as the strain gauge, are high-cost and easy to fail, so the recent works used self-made resistive sensors. Singh et al.~\cite{c5} presented a design of a 3D printed soft resistive sensor. Hughes et al.~\cite{c6} introduced a soft sensor made by both sandwiched rubber and resistive line sensors, which was made of composite material. The line sensors were spanned in both x-direction and y-direction, and it could detect both deformation and contact positions. Ma et al.~\cite{c7} made a low-cost sensor with carbon powder. The resistive sensor was made of rubber and carbon powder. Shuichi et al.~\cite{c8} proposed a soft displacement method by painted conductive resin ink on a piece of rubber. Nassour et al.~\cite{c9} designed a soft sensor with two layers for the soft gripper. The first layer has multi-sensors that can detect force. The second one is the curvature sensor and was used to sense the curvature of the finger. However, those works might not be used to deal with manipulation tasks, which requires force estimation and contact location detection at the same time.
\begin{figure}[http]
\centering
\includegraphics[width=230pt]{Fig2_1.png}
\caption{Humans' skin contain different kinds of receptors which are able to detect touch force and objects' features. Those receptors are Meissener's, Pacinian, and Merkel's corpuscle.~\cite{c18,c20}}
\label{fig: human skin}
\end{figure}
The recently developed soft sensors usually could only perform a simple function, such as estimating contact force or deformation. The performance of them might not satisfy the requirements of robot grippers to do dexterous manipulation. By contrast, humans' hands are able to do complex grasping tasks without the help of vision because their skin has more than a single function such as force detection, slippery detection, and feature identification. Motivated from that, we designed a novel dual-layer soft sensor that is able to satisfy the need for dexterous manipulation. In~\cite{c6}, similar work presented that a multi-layer sensor, but it is unable to predict contact force. Another related work was proposed~\cite{c9}, but it cannot detect the contact location for further feature identification.
Our soft tactile sensor contains two layers of sensors and can estimate contact force and contact position simultaneously.
Resistive material is the ideal choice for both layers in this work since they are highly sensitive and small size, and the sensing circuit is relatively simple. Due to the analysis of the sensing circuit, the output behavior is linear within a certain sensing range. The linear behavior of the sensor makes it easier to build the model and estimate the contact force accurately. In addition, since it is totally made of soft materials, it works consistently during deformation.
The remainder of this paper is organized as follows. Section II describes the design and fabrication of the soft sensor and the design of the sensing circuit. Section III models the sensor's behavior. Section IV demonstrates the experimental result, and Section V concludes the work.
\section{Design and Fabrication of Sensor}
Good touch sensitivity enables human to grasp objects that have complex geometric shapes even without vision.
Furthermore, when there is slippery, fingers can react within a few miniseconds~\cite{c20} and adjust the force to hold it stably. The main reason is that several types of receptors have different functions, as displayed in Fig.~\ref{fig: human skin}~\cite{c20}. Three of those receptors, including Meissener's, Pacinian, and Merkel's corpuscle, are responsible for light contact, gross contact, and feature identification, respectively. Those receptors enable our hands to perform complex manipulation tasks. The soft sensor should be similar to the skin of humans. Thus, we design the sensor that was inspired by the nature of humans, and will make robot grippers perform human-like tasks.
The sensor was designed to have dual-layers. The first one imitates the Meissener's and Pacinian corpuscles and is responsible for detecting contact force. The second one mimics the Merkel's corpuscle and takes the responsibility for locating the contact positions for further feature identification. For example, if the contact position is located and the contact pattern is identified as point contact, the object may be a sphere. If the contact pattern forms a line, the object might be a cylinder. The detailed design information will be discussed below.
\subsection{Design of the Force-measurement Layer}
The first layer (force-measurement layer) is similar to Meissner's and Pacinian corpuscles and will be applied to estimate the contact force. Like the human's skin, receptors are widely distributed over the finger. To achieve this, we selected conductive fabric for this layer, which could be cut into shapes that fit the possible contact area. The fabric is composed of 95 \% cotton and 5 \% spandex~\cite{c21}. After conductive ink was painted on its surface, the fabric became conductive and served as a soft resistive sensor. The key advantage of the conductive fabric is that it is highly elastic and can sustain large deformation during operations. It is also low-cost and easy to access. In the end, the sensor can be cut into any shape to fit into the sensor's size. In this work, the size of the fabric is 40 mm by 40 mm. It might be possible to be used to design artificial skin for detecting force in a large area in the future.
\begin{figure*}[h]
\centering
\includegraphics[width=1\textwidth]{Fig3.png}
\caption{The fabrication process}
\label{fig: process}
\end{figure*}
For the fabrication process, we soap a piece of fabric, which is made of cotton and spandex, in hot water at 60$^{o}$C for 30 minutes to relieve its residual stress. Then, the fabric will be hung for 3 hours to make it dry. By doing so, the elastic behaviors become more stable.
Cotton and elastic fiber inside the fabric may behave unstable when undergoing cyclic stress because of the hysteresis effect~\cite{c22}. The main cause is the friction and residual stress inside the fabric. Soaping the fabric in hot water, which references the heat treatment for mechanical materials, can release its residual stress~\cite{c23}. The hysteresis effect problem can be reduced, and it is verified by real experiments. By contrast, the sensor behaves unstable when cyclic loads are applied without soaping in hot water. After this heat treatment-like process, the fabric was painted with conductive ink that is made of organic chemical and carbon powder with a ratio of 4:1 in weight. The organic chemical is ethyl alcohol, whose concentration is 95$\%$. Then, the fabric is conductive and becomes a resistive sensor.
When the fabric undergoes elastic deformation, the fabric's fibers will expand their length and reduce their cross-sectional areas. With the conductive ink on the surface, the fabric's resistance will change if elastic deformation occurs. Based on the definition of Ohm's law, the resistance R is defined as~\cite{c25}:
\begin{align}
R = \rho L/A,
\label{eqn: Ohm law}
\end{align}
where A is the cross-sectional area of the material, $\rho$ is the resistivity of the conductor, and L stands for the length. As the fibers of the fabric experience deformation caused by the contact force, the length L increases and cross-sectional A decreases because of the conservation of volume. The resistance of the fabric will increase based on (\ref{eqn: Ohm law}). By using this property, the conductive fabric can be utilized to measure the applied force.
\subsection{Design of the Position-detection Layer}
The second layer~(position-detection layer) is designed as a sensor array to detect the contact location. In order to achieve a high resolution, each sensor component should be small. The conductive fabric we used for the force-measurement layer is not a suitable material for this layer. When the fabric is cut into small pieces, each piece of fabric tends to make the structure loose and may break easily. Traditional resistive sensors, such as the strain gauge or the piezoelectric sensor, are made of metal. They are relatively weak and are easy to break~\cite{c4}. More importantly, when they are embedded inside soft materials, the sensor will become a composite material. Its mechanical properties will change (including Young's modulus) and makes the overall model analysis inaccurate. The metal has a large Young's modulus and will increase the sensor's Young's modulus~\cite{c24}. Thus, the sensor's elasticity and compliance will reduce and might not be able to sustain large deformation. Moreover, the decrease of elasticity and compliance will restrict the operations of the sensor. For example, if the grippers equipped with a low-compliance sensor undergo large deformation, the sensor cannot work consistently.
One intuitive idea is to use soft materials with conductive ink to handle this problem. The soft sensor is made of silicone, Ecoflex{\textregistered} 00-30, and we use the same material for the sensor component in the position-detection layer. A set of molds is designed to make the sensor element. Liquid rubber, Ecoflex{\textregistered} 00-30, was poured into the mold. The liquid rubber was cured in 2 hours and then unmolded from the mold. The size of the sensor element for the current design is 10 mm in diameter and 2 mm in height. After the conductive ink was painted on its surface, the element became conductive rubber. The conductive ink is the same as the one used for making conductive fabric. The conductive rubber is similar to the conductive fabric. As a force is applied, the resistance of the rubbers will change based on the (\ref{eqn: Ohm law}). However, it is discovered that the element only reacts to certain force change~(e.g., 10~20 gw) by experiment results. There is no obvious relationship between the force and the sensing signal. The main difference is that the conductive ink only covers the surface of the rubber. That is, the cross-sectional area of the coating is quite small. If there is a large force, the rubber expands, and the cross-sectional area becomes very small and resistance surges. While this phenomenon happens, the conductive rubber is closed to an open loop, and the further change cannot be measured. That is the reason why it cannot sensitively react to the large normal force.
However, the conductive rubber is suitable to be applied as the position-prediction layer. When a force is applied, the voltage signal increases slightly. As the force is removed, the signal goes back to zero. By the change of increasing and decreasing of the signal, the element can help to detect the contact position. In this paper, we deployed four elements in the position-detection layer. We will use more elements in the future version to increase the resolution. In addition, there are two nodes on each element, which will be connected to the amplifier circuit to get sensing information. The nodes are 3 mm by 2 mm, and its length is 19.5 mm.
\begin{figure}[http]
\centering
\includegraphics[width=230pt]{Fig4.png}
\caption{The Wheatstone bridge for the first layer's sensor can get the sensing signal and the amplifier circuit can amplify the signal to to further process.}
\label{fig: wheatstone bridge for 1st layer}
\end{figure}
\subsection{Fabrication of Soft Sensor}
The terminal goal of the designed sensor is to fit in the robot grippers. In this paper, we primarily focus on its performance. The size of this sensor was determined to be 50 mm by 50 mm, and the height is 8 mm. Concerning the application purposes, the size of this design can be adjusted to meet different requirements.
The manufacturing process is shown in Fig.~\ref{fig: process}. In the beginning, liquid rubber was poured into the mold to form the sensor's bottom layer. After the rubber cured, four elements of the position-detection layer were placed on top of the bottom layer. Then, another component of the mold was stacked on top of the bottom mold, and more liquid rubber was poured into it. Thirdly, a conductive fabric was placed on top of the sensor, and more rubber was poured. After the rubber was cured, the molds were removed, and the sensor is shown in Fig.~\ref{fig: Apparence}. For the parameters of the material we used, the resistance of the conductive fabric is approximately 100 K$\Omega$, and the elements in the second layer range from 1~M$\Omega$ to 2~M$\Omega$.
\subsection{Sensing Circuit Design}
The sensors of both layers are resistive. The commonly used sensing circuit for resistive sensors is the Wheatstone bridge~\cite{c25}. As shown in Fig.~\ref{fig: wheatstone bridge for 1st layer}, $R_x$ denotes the resistance of the sensor, and we need to choose the magnitude of $R_1$, $R_2$, and $R_3$ in order to balance the bridge, which means the current flowing out of the circuit is zero. The resistance will be selected by the rule
\begin{align}
\frac{R1}{R2} = \frac{R3}{Rx}
\label{eqn: Wheatstone brdige}
\end{align}
Choosing all resistors have the same resistance as the sensor is one way to satisfy the above equation. To analysis the Wheatstone bridge, we firstly calculate the Thevenin resistance~\cite{c25} of the bridge,
\begin{align}
R_t = \frac {R_xR_x}{R_x+R_x} + \frac {R_x(R_x+ \Delta R_x)}{R_x+(R_x + \Delta R_x)}
\label{eqn: Wheatstone brdige ana}
\end{align}
where $R_t$ is the Thevenin resistance. The (\ref{eqn: Wheatstone brdige ana}) can be simplified as:
\begin{figure}[http]
\centering
\includegraphics[width=215pt]{Fig5.png}
\caption{The Wheatstone bridge with three extended bridges were designed to get voltage signal from the four elements in the second layer.}
\label{fig: Wheatstone bridge with extended}
\end{figure}
\begin{align}
R_t = \frac {R_x}{2} + \frac {R_x+\Delta R_x}{2R_x+\Delta R_x}{R_x}
\label{eqn: Wheatstone brdige simplified}
\end{align}
Owing to the elasticity of conductive fabric, the variation of $\Delta R_x$ can be up to 35 $\%$. The variation of $R_x$ versus $\Delta R_x$ is almost linear, and the slope change of (\ref{eqn: Wheatstone brdige simplified}) is 0.25$\sim$0.19:
This advantage can be observed via the model of the sensor in Sec III, and the experimental results in Sec IV.
Nevertheless,
the contact force might be small and hard to discover.
We used an INA 126A operational amplifier, which contains two operational amplifiers, to enlarge the signal. The component is precision instrumentation amplifiers and has a low offset voltage, which is able to reduce the output error. The operational amplifier also has a good ability to reject common-mode noise, which may cause by parasitic capacitance effect inside the circuit. Overall, we can obtain an accurate output without harsh noises.
\begin{figure*}[http]
\centering
\includegraphics[width=1\textwidth]{Fig6.png}
\caption{The modeling results of Model 1 to Model 3 can be observed in (a) while the results of Model 4 and Model 5 can be seen in (b). The collected data and averaged errors of training results can be seen in (d).}
\label{fig: The modeling results}
\end{figure*}
We have conducted simulations on the circuit using the simulator SIMPLIS. The results indicate the sensing circuit's output errors with INA 126 amplifier are about 1$\%$. Initially, we used operational amplifier LM 741. The output errors could be up to 6$\%$, which may negatively influence the estimation of the contact force. In addition, the amplifying gain can be adjusted by tuning a single external resistor $R_6$, as shown in Fig~\ref{fig: wheatstone bridge for 1st layer}.
The circuit for the position-detection layer is similar to that of the force-measurement layer. Each element in the second layer is connected with a separate sensing circuit, which is shown in Fig.~\ref{fig: Wheatstone bridge with extended}.
\section{Sensor Modeling}
When a force is applied, the sensing signal can be measured by the sensing circuit. Then the signal is converted to the contact force via a pre-trained model.
A regression method is applied to model the transformation from the sensing signal to the contact force. For the second layer, we only need to find the threshold of each element. The threshold means the smallest force that will trigger the resistance change of the element. The thresholds were discovered by experiments.
\subsection{Collecting Data}
Collecting the data from the sensor is a necessary process to determine the model.
We used a calibration weight set to serve as the contact force since the set comes in different weights and are supposed to be more accurate. The disadvantage is that we could not get continuous data but only the discrete one. At first, we used the F/T sensor as the ground truth. The F/T sensor was placed at the bottom, and the soft sensor was placed on its top. While a force was applying, we were unable to get the correct contact force because of the sensor's weight.
Twelve different weights were used including 5, 10, 20, 25, 35, 45, 50, 55, 65, 75, 85, 100 gw. The 5, 10, 20, 50, and 100 gw are the original weights that the set has. We used two or three weights together to generate the other weights like 25 gw, 35 gw, etc. Each weight was measured eight times. Among them, the 20, 100 gw were measured nine times, and 50 gw was ten times, so we have 100 data. The data has been divided into training and testing sets. The 80 \% data is the training set, while the other 20 \% is the testing set. We utilized the k-Fold Cross Validation method to get the testing set, and k is chosen as five. The shuffled data will be split into five folds. The first fold acted as testing data, and the remaining folds served as training data. By using k-Fold Cross-Validation, the overfitting problem of the following regression could be avoided~\cite{c28}.
\subsection{Modeling}
The common method to build a model for sensors is regression~\cite{c26,c27} since it is simple to implement. Also, the output of the sensing signal has linear behavior, as discussed in Sec II, so the regression method would be able to fit the data and build an accurate model. Before we apply this method, we need to select appropriate models. The five different models were selected to train the data and find the best model. The five models are the first-order polynomial to the fifth-order polynomial equations. The equation is expressed as
$f = a_0v + a_1 v^2 +\dots+ a_n v^{n}$.
The $n$ stands for the order of the polynomial, $f$ is the estimated force, and $v$ is the sensing signal~(voltage). We marked the first-order polynomial equation as Model 1, the second-order polynomial equation as Model 2, and etc.
Assume we have m training data here, and m is 80 here. If the m data is plugged into the equations, there will be m equations. Thus, they are written as matrix form and become $\bm{y}=\bm{A}\bm{x}$.
\begin{align}
\begin{bmatrix}
f_1\\
f_2\\
\vdots\\
f_m
\end{bmatrix}
=
\begin{bmatrix}
1 & v_1^{1} & \dots & v_2^{n}\\
1 & v_2^{1} & \dots & v_2^{n}\\
\vdots & \vdots & \vdots & \vdots\\
1 & v_m^{1} & \dots & v_m^{n}\\
\end{bmatrix}
\begin{bmatrix}
a_0\\
a_1\\
\vdots\\
a_n
\end{bmatrix}
\end{align}
where $\bm{y}$ $\in$ R$^{m\times1}$ is the vector that contains all force information, and $\bm{x}$ $\in$ R$^{n\times1}$ is the vector that includes all constants of the equations. The $\bm{A}$ $\in$ R$^{m\times n}$ matrix contains voltage signals from zero order to n$^{th}$ order. The objective is to minimize the 2-norm of the residual error $\bm{A}\bm{x}-\bm{y}$, i.e.,
$\min_{\substack{x}}
{||\bm{Ax}-\bm{y}||_2}$.
The equation can be solved by times $\bm{A}^{\bm{T}}$ to both side. Then, $(\bm{AA}^{\bm{T}})^{-1}$ is multiplied to both sides. The solution is
$\bm{x}= (\bm{A}\bm{A}^{\bm{T}})^{-1}\bm{A}^{\bm{T}}y$.
We repeated the training process 20 times for each model. After eight times of training, the averaged training errors and testing errors were nearly the same. The model can be seen in Fig~\ref{fig: The modeling results}(a) and (b). The first three models are quite linear, while the last two are slightly nonlinear. It is hard to tell which model is the best by direct observation, so the averaged error was considered and could be found in Fig.~\ref{fig: The modeling results}(c). The averaged errors here are the average of repeated training results. The index used here was the root-mean-square error. The average error of the training set decreases as the model's order increases. Among the five models, the Model 3 has the smallest testing error. The testing error of Model 3 is 0.0984 N. The detailed information can be observed in Table II. Hence, the Model 3 is the best here. In real experiments, we compared the results of Model 1 and Model 3, and no significant differences were observed between those two models. Thus, we utilize Model 1, the linear model, to estimate the contact force for this design.
\subsection{Filter Design}
\begin{table*}[h]
\caption{The averaged training and testing errors of each model}
\label{table_example}
\begin{center}\normalsize%
\begin{tabular}{|l|c|c|}
\hline
& Training error[N] & Testing error[N]\\
\hline
Model 1 (f=-0.0650+0.0889v) & 0.0985 & 0.1010\\
\hline
Model 2 (f=-0.0301+0.0737v+0.0012v$^{2}$) & 0.0974 & 0.1028\\
\hline
Model 3 (f=0.0653-0.0047v+0.0169v$^{2}$-0.000863v$^{3}$) & 0.0950 & 0.0984\\
\hline
Model 4 (f=0.0924-0.0405v+0.0295v$^{2}$-0.0025v$^{3}$+0.0000675v$^{4}$) & 0.0945 & 0.1005\\
\hline
Model 5 (f=0.0603+0.0189v-0.0015v$^{2}$+0.0041v$^{3}$-0.000539v$^{4}$+0.00002v$^{5}$) & 0.0944 & 0.1014\\
\hline
\end{tabular}
\end{center}%
\end{table*}
Based on our observation, the output signal from the circuit does not have severe noises which influence the estimation of force. The reason is that the operational amplifier~(INA126) can reject most noises in this design. Nevertheless, a filter was still designed for fear that there will be unexpected noises and negatively influence the estimation accuracy. The moving average filter is used here. Before the data is input into the filter, it will be converted to estimated force by using the model we built. Then, the signal will be input into the filter. The algorithm of the filter is shown below:
\begin{align}
{\overline{f} = 1/n\sum_{i=1}^{m} f(n+1-i)}
\end{align}
where $\overline{f}$ is the filtered signal, n is the number of sampled data, and $f(n+1-i)$ is the sampled data. The $m$ sampled data was considered, and $m$ is equal to 4 here. Also, each data has equal weight.
\section{Experimental Results}
\subsection{Experimental Setup}
The sensing information from the circuits is processed by using Arduino UNO R3. Arduino UNO R3 is a microcontroller based on the Microchip Atmega 325. The Arduino board is equipped with 14 digital I/O pins and six analog I/O pins. The on-chip ADC is applied to sample information from those pins, and its resolution and sampling frequency are 8 bit and 9.6 Hz, respectively. Among the analog I/O pins, one will be used to read data from the force-measurement layer. The other four will be utilized to read data from four elements from the position-detection layer. Therefore, the information will be processed concurrently, and the sensor can detect contact force and contact location simultaneously.
\begin{figure}[http]
\centering
\includegraphics[width=230pt]{Fig7.png}
\caption{The experimental setup.}
\label{fig: setup}
\end{figure}
While conducting the experiment, the sensor is connected to the circuit, as shown in Fig.~\ref{fig: setup}. The circuit is connected to both the oscilloscope and the Arduino board. The sensing signal would be processed in Arduino. The oscilloscope serves as the reference to check whether the output information from the circuit is correct. After the sensing signal is processed in Arduino, the contact force and contact position would be shown on the computer's monitor.
\subsection{Results}
\subsubsection{Thresholds of Position-detection Layer}
Before we tested the sensor, we did an experiment by using weights to find the thresholds of four elements in the position-prediction layer. These information was programmed into Arduino. Their threshols are about 0.1N. Two of them are about 0.15 to 0.2N.The differences are caused by conductive ink. The ink might not distribute averagely on the rubbers' surface, so their sensitivities are slightly different. Hence, as a force is applied and greater than the threshold of the element in that location, the sensor will simultaneously predict the contact force and contact position.
\subsubsection{Test of the Soft Sensor}
The first experiment is to know the sensor's performance. The experimental results are demonstrated in Fig.~\ref{fig: Experimental result}. The contact locations are divided into four different areas, as shown in Fig.~\ref{fig: Experimental result}(c)
We applied force on each of the areas, and the sensor was able to estimate the force and also detected the location, as shown in Fig.~\ref{fig: Experimental result}(a). In the algorithm, when the contact location was detected, it would be on mode. When the contact force reduces below the threshold of the element, it would be off mode. During the 4.5 to 5.5s in Fig.~\ref{fig: Experimental result}(a), the force was exceeding the sensing range, and the sensor hit the ceiling at 1 N.
\begin{figure*}[http]
\centering
\includegraphics[width=1\textwidth]{Fig8_1.png}
\caption{The experimental results can be seen in (a). The forces were applied at location 1 to 4 successively. The solid line represents the estimated contact force by the first layer, and the dotted line represents the detected contact location from the second layer. We can see from the results that both layer behaves stable in the experiments.
Another experiment was conducted to test the sensor's accuracy. The 20, 50, and 100 gw weights were used, respectively and can be observed in (b). The soft sensor could estimate contact force and location concurrently. The locations were marked and is shown in (c).}
\label{fig: Experimental result}
\end{figure*}
\subsubsection{Accuracy Tests of Force-Measurement Layer}
Another experiment was to test the accuracy of the force-measurement layer. The accuracy of a sensor is an important issue. Designing an accurate soft sensor for robot grippers is one of our objectives. Hence, three weights of calibration set, 20, 50, and 100 gw(0.196, 0.49, and 0.98 N), were used to test its accuracy. Those three weights were placed on the sensor successively during the experiment. The corresponding measured forces of the three weights were 0.27 N, 0.63 N, and 1.01 N. The data was sampled during the period each weight was applied on the sensor. The root-mean-square error is 0.0923 N.
\subsubsection{Sensing Range and Resolution of the Sensor}
During the experiment, the sensing range and resolution of this design were discovered. The current sensor uses the amplifier gain of 41.36, the sensing range is 1~N, and the resolution is 0.05~N. This sensor's maximum sensing range is 1.5~N, as shown in Table II with amplifier gain 22. The resolution is around 0.1~N. If we increase the amplifier gain, the resolution will become better, and the sensing range decreases. If we decrease the amplifier gain, the sensing range will increase, and the resolution becomes worse. For future applications, we can adjust the amplifier gain depends on different requirements. If the sensing range is more important than the resolution, the amplifier gain can be decreased to increases the range. If high resolution is needed, the amplifier gain will be increased to get better resolution.
\begin{table}[!h]
\caption{The sensing range and resolution of the soft sensor}
\label{table_example}
\begin{center}\normalsize%
\begin{tabular}{|c|c|c|}
\hline
Amplifier gain & Sensing ange[N] & Resolution[N]\\
\hline
22 & 1.5 & 0.1\\
\hline
41.36 & 1 & 0.05\\
\hline
\end{tabular}
\end{center}%
\end{table}
\section{CONCLUSIONS}
This paper presents a new soft sensor which is inspired by the receptors inside the fingers of humans. The sensor contains two layers of sensing units, which include a force-measurement layer and a position-detection layer.
The force-measurement layer mimics the Meissener's and Pacinian corpuscles of humans' skin, which can estimate the contact force.
The position-detection layer imitates the Merkel's corpuscle,
and it is able to detect the contact location. The Wheatstone bridge and amplifier circuits are utilized to get sensing information from the sensor. A model between the contact force and signal from the force-measurement layer was pre-trained by a regression method. In addition, a filter method was applied to estimate the force under noisy measurements robustly. Experiments are provided to show the performance.
For the position-detection layer, the threshold of each element was determined by experiments. The model and threshold were coded into Arduino UNO board for real-time processing. Experiment results show that the sensor is capable of measuring the contact force and contact position concurrently, and the root-mean-square error of the force-measurement layer is 0.0923~N.
In the future, more conductive rubbers will be layout in the position-detection layer to increase its resolution and enable it to identify objects' features. The enhanced version will be deployed on a robot gripper to do dexterous manipulation.
\bibliographystyle{ieeetr}
|
1,116,691,500,512 | arxiv | \section{Introduction}
\label{sec:intro}
Widely distributed radar systems are robust and fault-tolerant systems that provide high angular resolution and permit the exploitation of spatial diversity and occlusions avoidance \cite{haimovich2007mimo}. With applications in surveillance, assisted living, and health monitoring, radar systems with widely distributed antennas are expected to play a vital role in emerging sensing paradigms \cite{gurbuz2019radar,gennarelli2019radar}. Under this architecture, the observed targets feature an aspect dependent scattering behavior restricting the employment of conventional imaging methods. The main impediment arises due to the adoption of the isotropic point scattering model of targets, thereby preventing algorithms like back-projection (BP) to provide an adequate imaging performance \cite{moses_wide-angle_2004}.\par
The problem of radar imaging with widely distributed sensors has not received enough attention in the literature. The works \cite{lodhi_coherent_2019,mansour_sparse_2018,mansour2018radar} considered radar imaging with distributed antennas and the related issues due to ambiguity in antenna positions and clocks synchronization. In these works, model-based optimization algorithms are utilized to jointly achieve the imaging task and resolve such issues. However, in these works, an isotropic scattering model, suitable when antennas are closely spaced, is assumed; this is clearly not suitable when widely separated antennas are considered. On the other hand, in wide-angle synthetic aperture radars (WSAR), which bear a close resemblance to a widely distributed architecture, two approaches exist for imaging \cite{ash_wide-angle_2014}. The first one is based on parametric modeling that characterizes the canonical scattering behavior of scatterers \cite{potter1997attributed,liu_efficient_2019,sugavanam_interrupted_2017,yang_robust_2019}. Correspondingly, the scene image is reconstructed through joint processing of the measurements from the whole aperture exploiting the model. Nevertheless, the imaging involves a dictionary search process that is computationally cumbersome \cite{hammond2013sar}. The other approach is composite imaging \cite{hu_video-sar_2017,sanders_composite_2017,wei_wide_2018,xu_accurate_2021} in which the full aperture is divided into sub-apertures within which the point scattering model holds. Respectively, images of each sub-aperture are formed through regularized optimization exploiting specific features such as sparsity. As a final step, individual images are fused to constitute an aggregate image of the scene through simple techniques such as the generalized likelihood ratio test (GLRT). This approach does not fully exploit the information from different aspects where the final image of the scene is only a fused version of the images reconstructed with sub-aperture data.\par
While in this paper we propose a sub-aperture method, unlike composite imaging, we propose to solve the problem of widely distributed radar imaging by directly reconstructing a global image that is introduced as an aggregate view of the scene. Besides, the prior information is only imposed on the global image rather than the local images of individual sensors. Concurrently, the correspondence between the local images and the global one is defined as a constraint to the optimization problem. Our approach allows for better data exploitation by including the global image as a decision variable in the optimization problem. We then provide a solution based on Alternating direction method of multipliers (ADMM) framework \cite{boyd_distributed_2011}. ADMM is a powerful distributed optimization regime suitable for systems that incorporate collection of measurements through a distributed architecture. In \cite{afonso_fast_2010} and \cite{afonso_augmented_2011}, ADMM has been introduced as a fast reconstruction method for generic imaging inverse problems. Further, in \cite{guven_augmented_2016} it is applied to reconstruct complex SAR images with enhanced features in particular, and to perform imaging with undersampled measurements in the presence of phase errors in \cite{guven_autofocused_2017}.\par
While in these works ADMM has been mainly utilized to facilitate the solution of a non-constrained optimization problem by the virtue of variable splitting, we employ its constrained formulation directly in the interest of exploiting the system architecture and implementing parallelizable image reconstruction algorithms. Accordingly, we establish two problem formulations inspired by consensus ADMM (CADMM) and sharing ADMM (SADMM). The first formulation comes as a generalization of our previous work \cite{hu_widely-distributed_2021} in which CADMM is utilized to mitigate the layover artifacts in widely distributed radar imaging by considering sub-aperture measurements from different elevations. In this work, however, we present CADMM formulation to introduce the association between sub-aperture images and the global image, and generally reconstruct the image of the scene without restriction on data viewing angles. Moreover, by stipulating the more relaxed sharing association in the constraints, we introduce the second problem formulation based on SADMM. The different association introduced by SADMM formulation enables another exploitation of the relationship between the data collected by the sub-apertures. Additionally, it provides an alternative realization of the system architecture through the ensuing unalike solution. We provide the solutions as iterative algorithms with a recommendation of a parallel implementation paradigm. Finally, Civilian Vehicles Dome data-set \cite{dungan_civilian_2010} is used to realize three experiments which comprise different practical use cases. Through them, we validate our algorithms and show the performance of CADMM and SADMM, where the latter is found to provide an enhanced imaging performance in most of the scenarios. Our proposed approach can be regarded as a general framework suitable to be implemented on various architectures including WSAR and radar systems with collocated antennas.\par
\section{Signal Model and Background}
\label{sec:SigMod}
In this section, we provide the signal model we adopted for our distributed architecture and provide background about the imaging problem formulation in the state of the art.\par
Throughout this paper, vectors are denoted by lower case bold font, while matrices are in uppercase bold. $\mathbf{I}_{L}$ is the identity matrix of size $L\times L$ and $\mathbf{1}_{N}$ is a vector of all ones of size $N \times 1$. The superscripts $\mathbf{.}^{T}$ and $\mathbf{.}^{H}$ denote respectively the transpose and the complex conjugate transpose of a vector or a matrix. On the other hand, superscripts in parenthesis denote the iteration count. The symbol $\otimes$ is used for the Kronecker product.\par
\begin{figure}[!htbp]
\centering
\includegraphics[width=3.2 in]{figures/SysGeo.pdf}
\caption{Geometry of Distributed Radar System}
\label{fig:Geo}
\end{figure}
Considering the system geometry illustrated in Fig.~\ref{fig:Geo}, a group of red crossed circles constitute a cluster of antenna phase centers (APCs). The figure shows the case we consider in our paper where the $Q$ sensors would each form a single cluster are at identical elevation angles. We consider a mono-static configuration where each sensor receives the reflections due its own illumination of the scene and does not process the reflections induced by transmissions from others. Accordingly, our proposed algorithms can be applied for architectures that be formed either by a real or a synthetic aperture. At each cluster, the isotropic scattering model of the targets in the scene is assumed. This way, the problem of aspect-dependant scattering behavior can be relaxed and $Q$ local images can be formed by processing the measurements of individual clusters.\par
Since our goal is to form a reflectivity image of the scene, we adopt the 2D tomographic radar imaging framework \cite{munson1983tomographic}. Accordingly, the signal received at the $q^\text{th}$ cluster after de-chirping is
\begin{equation}
\label{eq:radon}
{y_{q}}\left( {w,m} \right) = \iint {\tilde{x}_{q}\left({x,y} \right){e^{j \frac{4 \pi f_{w} \cos \varphi_{q}}{c}\left( {x\cos {\theta _m} + y\sin {\theta _m}} \right)}}dx \ dy},
\end{equation}
where $w = {1, \ldots, W}$ is the index of the sampled fast time frequency, $m = {1, \ldots, M}$ is the index of an APC within the cluster, $\tilde{x}_{q}\left( {x,y} \right)$ indicates the complex reflectivity coefficient of a ground target at coordinates $\left({x,y}\right)$ with respect to the $q^\text{th}$ cluster, $f_{w}$ denotes the beat linear frequency, ${\theta _m}$ is the azimuth angle of the $m^\text{th}$ element, and $\varphi_{q}$ is the elevation angle of the $q^\text{th}$ cluster.
Approximating the scene with a uniform grid of $N = N_{x} \times N_{y}$ pixels and stacking the $M$ vectors containing the frequency domain samples received by the APCs in the $q^\text{th}$ sensor, the phase history measurements can be written in a matrix form as
\begin{equation}
\label{eq:ModelMat}
\mathbf{y}_{q}=\mathbf{A}_{q}\tilde{\mathbf{x}}_{q}+\mathbf{n}_{q}\in \mathbb{C}^{WM\times 1},
\end{equation}
where $\mathbf{A}_{q} \in \mathbb{C}^{WM \times N}$ is the system model based forward operator, $\tilde{\mathbf{x}}_{q} \in \mathbb{C}^{N\times 1}$ is the vector containing the complex scattering coefficients of the entire scene with respect to the $q^\text{th}$ cluster, and $\mathbf{n}_{q}\in \mathbb{C}^{WM \times 1}$ summarizes all errors including receiver and measurement noise as well as model imperfections.\par
Composite imaging algorithms obtain local images utilizing the signal received at each cluster and subsequently fuse them into a global image. The scene size is usually much larger than the number of measurements $N>>WM$ and the imaging task is the inverse problem of (\ref{eq:ModelMat}) which, consequently, becomes ill-posed. Compressed sensing methods are commonly used to solve this inverse problem. Particularly, local images are obtained by solving $Q$ regularized least square optimization problems for each cluster of the form
\begin{equation}
\label{eq:MBR}
\mathbf{\hat{x}}_{q}={\text{arg}}\mathop{\min }\limits_{{\tilde{\mathbf{x}}_{q}}}\left\{\left\| \mathbf{y}_{q}-\mathbf{A}_{q}\tilde{\mathbf{x}}_{q} \right\|_{2}^{2}+ h(\tilde{\mathbf{x}}_{q}) \right\},
\end{equation}
where $\mathbf{\hat{x}}_{q}$ is the estimated local image using the measurements $\mathbf{y}_{q}$ for $q=1,\ldots,Q$ and $h(\cdot)$ is a regularization function that imposes apriori information about local images. Different choices of regularization function $h(\cdot)$ exist to enhance some image features such as sparsity and smoothness, among others. when $h(\cdot)$ is a separable function (e.g. $l_1$-norm), the $Q$ problems can be represented as a single optimization problem in $Q$ variables since the least squares term is naturally separable. Explicitly, the problem can be written as
\begin{equation}
\label{eq:OptMod}
\left\{\mathbf{\hat{x}}_{1},\ldots,\mathbf{\hat{x}}_{Q} \right\}= \underset{\tilde{\mathbf{x}}_{1},\tilde{\mathbf{x}}_{2},\cdots ,\tilde{\mathbf{x}}_{Q}}{\min}\sum_{q=1}^{Q} \left\{{ \left\| \mathbf{y}_{q}-\mathbf{A}_{q}\tilde{\mathbf{x}}_{q} \right\| _{2}^{2}} + \left\| {\tilde{\mathbf{{x}}}_{q}} \right\|_1\right\}
\end{equation}
The problem in (\ref{eq:OptMod}) is an unconstrained regularized optimization problem which has been tackled through different optimization techniques in the literature. Finally, the image of the scene is obtained through a fusion step of the $Q$ reconstructed images which can be as simple as a pixel-wise maximization among the $Q$ local images.\par
As mentioned in the introduction, we alternatively reconstruct the global image of the scene by introducing its variable in the objective function and imposing the $l_1$-norm on it directly for a sparsity-driven solution. Simultaneously, the relationship between the global image and local images is defined as a constraint for our optimization problem. In the next section, based on ADMM framework, we provide two alternative problem formulations along with their solutions.
\section{ADMM Framework for Distributed Radar Imaging}
\label{sec:ADMM}
ADMM is a powerful framework that renders itself amenable for optimization problems of distributed nature. It is a suitable tool to be utilized in a distributed radar system especially when the component sensors are equipped with some computation power capabilities. Although this computation power might be limited, it can be exploited to process some information in order to reduce the communication overhead and the computational burden. It also reduces latency as certain operations can already be performed in parallel at the nodes. Here, we first give a brief introduction of general ADMM formulation followed by our proposed reformulations of the problem in (\ref{eq:OptMod}) according to ADMM framework.\par
Consider the following constrained optimization problem with linear constraints over two separable functions in two variables $\mathbf{u}$ and $\mathbf{z}$
\begin{equation}
\begin{gathered}
\label{eq:ADMM}
{\text{arg}}\mathop {\min }\limits_{\mathbf{u},\mathbf{z}} f(\mathbf{u}) + g(\mathbf{z}) \hfill \\
s.t.\,\,\,\,\,\,\,\,\mathbf{G} \mathbf{u} + \mathbf{H} \mathbf{z} = \mathbf{c},
\end{gathered}
\end{equation}
where $\mathbf{G}$, $\mathbf{H}$, and $\mathbf{c}$ are the matrices and vector of appropriate dimensions that establish the constraints on the variables $\mathbf{u}$ and $\mathbf{z}$.
The augmented Lagrangian function of the above problem becomes
\begin{equation}
\label{eq:AugLag}
\mathcal{L}\left(\mathbf{u},\mathbf{z},{\boldsymbol{\sigma}} \right) = \left\{
\begin{gathered}
\begin{aligned} & f(\mathbf{u}) + g(\mathbf{z}) + \left\langle {\boldsymbol{\sigma }},{\mathbf{G}{\mathbf{u}} + \mathbf{H} \mathbf{z} - \mathbf{c}} \right\rangle \\
+ & \frac{\beta }{2}\left\|{\mathbf{G}{\mathbf{u}} + \mathbf{H} \mathbf{z} - \mathbf{c}}\right\|_2^2\hfill \end{aligned} \end{gathered} \right\}
\end{equation}
where ${\boldsymbol{\sigma}}$ is the dual variable, $\beta$ is the augmented Lagrangian parameter, and $\left< \cdot, \cdot \right>$ denotes the inner product of vectors.\par
The ADMM solution to the above problem is obtained by iteratively minimizing the augmented Lagrangian function with respect to both the variables $\mathbf{u}$ and $\mathbf{z}$ in an alternating fashion in addition to updating the dual variable each iteration. Accordingly, after the $k^\text{th}$ iteration, the ADMM variable updates consist of \cite{boyd_distributed_2011}
\begin{equation}
\label{eq:VarUpdt}
\begin{aligned}
\mathbf{u}^{\left(k+1 \right)}:= & {\text{arg}}\mathop {\min }\limits_{\mathbf{u}} \mathcal{L} \left(\mathbf{u},\mathbf{z}^{\left( k \right)},{\boldsymbol{\sigma }}^{\left( k \right)} \right) \\
\mathbf{z}^{\left(k+1 \right)}:= & {\text{arg}}\mathop {\min }\limits_{\mathbf{z}} \mathcal{L} \left(\mathbf{u}^{\left( k \right)},\mathbf{z},{\boldsymbol{\sigma }}^{\left(k \right)} \right)\\
{\boldsymbol{\sigma}}^{\left( k+1 \right)}:= & {\boldsymbol{\sigma }}^{\left ( k \right)} + {\beta} \left(\mathbf{G} \mathbf{x}^{\left( k+1 \right)} + \mathbf{H} \mathbf{z}^{\left( k+1 \right)} - \mathbf{c} \right)
\end{aligned}
\end{equation}
Embracing ADMM framework, we propose two different formulations for (\ref{eq:OptMod}). By introducing a new variable $\mathbf{x_{G}} \in \mathbb{R}^{N \times 1}$ representing the magnitude of the global image, both the formulations will have the same objective function of minimizing the sum of the least square terms with respect to local images, in addition to minimizing the $l_1$ norm of the global image. The formulations differ in the constraints which define the relation between the global and local images. We consider the magnitude of the images as our optimization variables assuming that the phases are estimated in a previous step. Specifically, we assume that we have estimated $\mathbf{\Theta}_{q} \in \mathbb{C}^{N \times N}$, the diagonal matrix containing the phase of all pixels of local image over its diagonal such that $\tilde{\mathbf{x}}_{q}=\mathbf{\Theta}_{q} {\mathbf{x}}_{q}$. For ease of notation, from now on we will consider the matrix $\mathbf{\Theta}_{q}$ included in the measurement matrix $\mathbf{A}_{q}$. The details regarding the estimation of $\mathbf{\Theta}_{q}$ will be discussed in the next section. Accordingly, with reference to (\ref{eq:ADMM}), our first variable is $\mathbf{x} \in \mathbb{R}^{QN \times 1}$ containing the magnitude of all local images $\mathbf{x} = \{\mathbf{x}_{q}\}_{q=1}^{Q}$, and the second variable represents the magnitude of the global image $\mathbf{x_{G}}$. Consequently, our objective function will be $f(\mathbf{x}) = \sum_{q=1}^{Q}{ \left\| \mathbf{y}_{q}-\mathbf{A}_{q}\mathbf{x}_{q} \right\|_{2}^{2}}$ and $g(\mathbf{x_{G}}) = \left\|{\mathbf{{x}_{G}}}\right\|_{1}$.\par
In the sequel, we will provide our proposed aforementioned formulations and their solutions in terms of variable updates according to (\ref{eq:VarUpdt}).
\subsection{Consensus ADMM (CADMM)}
\label{sec:CADMM}
As the name suggests, by posing the problem according to this formulation, we pursue a solution which, at optimum, provides a global image on which all clusters reach a consensus. Consequently, the constraints, in this case, are defined to impose this relationship between the global and local images. Additionally, as mentioned earlier and by following our paper \cite{hu_widely-distributed_2021}, we impose the $l_1$-norm function to promote sparse global image solution. The problem becomes
\begin{equation}
\label{eq:CADMM_problem}
\begin{gathered}
\underset{\mathbf{x},\mathbf{{x}_{G}}}{\text{arg}\mathop{\min}}\,\, \, \, \,\sum_{q=1}^{Q}{ \frac{\mathrm{\mu}}{2}\left\| \mathbf{y}_{q}-\mathbf{A}_{q}\mathbf{x}_{q} \right\| _{2}^{2}} + \lambda \left\|{\mathbf{{x}_{G}}}\right\|_{1}\,\,\,\,\,
\hfill \\
\,\,\,\,\, \, \, \, \, \,\, \, \, \,\, \, \, \, \, \, \,\, \, \, \, s.t.\, \, \,\, \, \, \,\, \, \,\, \, \, \,\, \, \,\, \, \, \,\mathbf{x}_{q}-\mathbf{{x}_{G}}=\mathbf{0}\,\,\,\,\,\,\,\forall q.\,\,\,\,\,\,\,\hfill \\
\end{gathered}
\end{equation}
where $\lambda$ and $\mu$ are positive hyperparameters set to penalize less sparse global image solutions and trade-off the data fidelity term, respectively.
Note that the $Q$ constraints in (\ref{eq:CADMM_problem}) can be written in the form of the constraint in (\ref{eq:ADMM}) by having $\mathbf{G} = \mathbf{I}_{QN}$, $\mathbf{u}=\mathbf{x}$, $\mathbf{H}=-[\mathbf{I}_{N}, \mathbf{I}_{N}, \cdots \mathbf{I}_{N}]^{T}$ of the size $QN \times N$, $\mathbf{z}=\mathbf{x_{G}}$, and $\mathbf{c} = \mathbf{0}$ of size $QN \times 1$.\par
As indicated in (\ref{eq:VarUpdt}), the solution of (\ref{eq:CADMM_problem}) can be obtained by alternately minimizing its associated augmented Lagrangian with respect to $\mathbf{x}$, $\mathbf{{x}_{G}}$, and the dual variable $\boldsymbol{\sigma}$. The augmented Lagrangian is
\begin{equation}
\label{eq:aug_lag_C}
\begin{gathered}
\mathcal{L}\left( {{\mathbf{x}},{\mathbf{{x}_{G}}},{\boldsymbol{\sigma }}} \right) = \\
\left\{
\begin{gathered}
\sum\limits_{q = 1}^{Q}
\begin{gathered}
\left\{
\frac{\mu }{2}\left\|{{{\mathbf{y}}_{q}} - {{\mathbf{A}}_{q}}{{\mathbf{x}}_{q}}} \right\|_2^2 + \left\langle {{{\boldsymbol{\sigma }}_{q}},{{\mathbf{x}}_{q}}-{\mathbf{{x}_{G}}} } \right\rangle
+ \frac{\beta }{2}\left\| { {{\mathbf{x}}_{q}}-{\mathbf{{x}_{G}}} } \right\|_2^2 \right\} \end{gathered}
\\ +\lambda {\left\| {\mathbf{{x}_{G}}} \right\|_1}\end{gathered} \right\},
\end{gathered}
\end{equation}
where ${\boldsymbol{\sigma }}_{q} \in \mathbb{R}^{N \times 1}$ is the block of values of the dual variable ${\boldsymbol{\sigma }} \in \mathbb{R}^{QN \times 1}$ which corresponds to the local image $\mathbf{x}_{q}$.\par
The resulting updates of each variable according to CADMM formulation are provided hereinafter in details.
\\
\subsubsection{Update of $\mathbf{x}$ (Local Images)}
Let $\mathbf{{x}_{G}}^{\left(k\right)}$ and $\boldsymbol{\sigma }^{\left(k\right)}$ denote the values of $\mathbf{{x}_{G}}$ and $\boldsymbol{\sigma }$ after the $k^{\text{th}}$ iteration. Since $\mathcal{L}\left( \mathbf{x},\mathbf{x_{G}},\boldsymbol{\sigma} \right)$ in (\ref{eq:aug_lag_C}) is decomposable with respect to ${\mathbf{x}_{q}}$, the updated ${\mathbf{{x}}}^{\left(k+1\right)}$ can be obtained by updating all local images $\mathbf{{x}}_{q}^{\left(k+1\right)}$ for $q=1,\ldots,Q$ in parallel as
\begin{equation}
\begin{gathered}
\begin{aligned}
\label{eq:updt_x_CADMM}
{\mathbf{x}}_{q}^{\left(k + 1\right)} &= {\text{arg}}\mathop {\min }\limits_{{{\mathbf{x}}_{q}}} {\mathcal{L}}\left( {{{\mathbf{x}}_{q}};{{\mathbf{{x}_{G}}}^{\left(k \right)}},{\boldsymbol{\sigma }}_{q}^{\left(k\right)}} \right) \\
&=
{\text{arg}}\mathop {\min }\limits_{{{\mathbf{x}}_{q}}}
\left\{ \begin{gathered}
\frac{\mu }{2}\left\| {{{\mathbf{y}}_{q}} - {{\mathbf{A}}_{q}}{{\mathbf{x}}_{q}}} \right\|_2^2 + {\boldsymbol{\sigma}_{q}^{\left(k \right)}}^{T} {{\mathbf{x}}_{q}} \\ + \frac{\beta}{2}\left\| {{{\mathbf{x}}_{q}} - {{\mathbf{{x}_{G}}}^{\left(k \right)}}} \right\|_2^2
\end{gathered} \right\},
\end{aligned}
\end{gathered}
\end{equation}
The problem in (\ref{eq:updt_x_CADMM}) is differentiable with respect to $\mathbf{x}_{q}$ and the $(k+1)^{\text{st}}$ update can be obtained in a closed-form by letting $\nabla_{\mathbf{x}_{q}}\mathcal{L} =\mathbf{0}$ resulting in
\begin{equation}
\label{eq:updt_x_CADMM_CF}
{\mathbf{x}}_{q}^{\left(k + 1\right)} = \left( \mu \mathbf{A}_{q}^{H}\mathbf{A}_{q}+\beta \mathbf{I}_{N} \right)^{-1} \left( \mu \mathbf{A}_{q}^{H}\mathbf{y}_{q}+\beta {{\mathbf{{x}_{G}}}^{\left(k \right)}} - {\boldsymbol{\sigma }}_{q}^{\left(k\right)} \right),
\end{equation}
Note that the inverse in (\ref{eq:updt_x_CADMM_CF}) is possible since $\left(\mu \mathbf{A}_{q}^{H}\mathbf{A}_{q}+\beta \mathbf{I}_{N}\right)$ is a positive definite matrix.
\\
\subsubsection{Update of $\mathbf{x_{G}}$ (Global Image)}
For the global image update, following the ADMM framework, we consider
\begin{equation}
\begin{gathered}
\begin{aligned}
\label{eq:updt_xG_CADMM}
{\mathbf{{x}}^{\left(k + 1\right)}_\mathbf{G}} &={\text{arg}}\mathop {\min }\limits_{{\mathbf{x_{G}}}}\mathcal{L}\left(\mathbf{x_{G}};\mathbf{{x}}^{\left(k+1 \right)},\boldsymbol{\sigma }^{\left(k \right)} \right) \hfill \\
&= {\text{arg}}\mathop {\min }\limits_{{\mathbf{{x}_{G}}}} \left\{ \begin{aligned}\lambda \left\| {\mathbf{x_{G}}} \right\|_1 + \sum_{q=1}^{Q} {\boldsymbol{\sigma}_{q}^{\left(k \right)}}^{T} {\mathbf{{x}_{G}}} \hfill\\
+ \frac{\beta}{2} \sum_{q=1}^{Q}\left\| {\mathbf{{x}}_{q}^{\left(k+1 \right)} - {{\mathbf{{x}_{G}}}}} \right\|_2^2
\end{aligned} \right\}.
\end{aligned}
\end{gathered}
\end{equation}
This objective function above involves information from all $Q$ clusters and is not decomposable with respect to $\mathbf{{x}_{G}}$. It further involves a non-differentiable function $\left\| \mathbf{{x}_{G}} \right\|_1$. Thus, it can neither be parallelized nor solved in a closed form like (\ref{eq:updt_x_CADMM}). As a result, it is more suitable for the global image update to be carried out in a central processor after collecting local updates calculated at the distributed clusters. Moreover, for the subsequent update of local images, global image needs to be broadcast to all the clusters. Alternatively, if the global image update were to be carried out at distributed clusters, a fully meshed communication network would be needed to exchange all local updates among the $Q$ clusters. Later in \ref{sec:SolTech}, we will show how to solve (\ref{eq:updt_xG_CADMM}) in the central node.
\\
\subsubsection{Update of ${\boldsymbol{\sigma}}$ (Dual Variable)}
After updating the global image, the dual variable can be updated by
\begin{equation}
\label{eq:updt_sig_CADMM}
\boldsymbol{\sigma }^{\left(k + 1\right)} = \boldsymbol{\sigma}^{\left(k \right)} + \beta \left( {\mathbf{{x}}}^{\left(k + 1\right)} - \mathbf{1}_{Q} \otimes {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k + 1\right)} \right),
\end{equation}
The Kronecker product is used to replicate the global image to the same size of the vectors ${\mathbf{{x}}}$ and $\boldsymbol{\sigma}$. Since (\ref{eq:updt_sig_CADMM}) is decomposable, it can be carried out in parallel as well as local images updates. Instead, it is more convenient for the dual variable to be updated in the central node subsequent to that of global images and broadcast to the distributed clusters for the next update of the local image.
\subsection{Sharing ADMM (SADMM)}
\label{sec:SADMM}
Under this formulation, we impose a different constraint in the optimization problem to explore a different relationship between local images and global image. The constraint is set such as the reconstructed global image is the average of all local images. Accordingly, the problem becomes
\begin{equation}
\label{eq:SADMM}
\begin{gathered}
\underset{\mathbf{x},\mathbf{{x}_{G}}}{\text{arg}\mathop{\min}}\,\, \, \, \,\sum_{q=1}^{Q}{ \frac{\mathrm{\mu}}{2}\left\| \mathbf{y}_{q}-\mathbf{A}_{q}\mathbf{x}_{q} \right\| _{2}^{2}} + \lambda \left\|{\mathbf{{x}_{G}}}\right\|_{1}\,\,\,\,\,
\hfill \\
\,\,\,\,\, \, \, \, \, \,\, \, \, \,\, \, \, \, \, \, \,\, \, \, \, s.t.\, \, \,\, \, \, \,\, \, \,\, \, \, \,\, \, \,\, \, \, \,\bar{\mathbf{x}}-{\mathbf{x}_\mathbf{{G}}}=\mathbf{0}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hfill \\
\end{gathered}
\end{equation}
where $\bar{\mathbf{x}} = \sum_{q=1}^{Q} \mathbf{x}_{q}$ contains the sum of magnitudes of local images. Note that the size of constraints is reduced to the size of a single image instead of $Q$ images in the consensus formulation. The nomenclature stems from the constraint above since the global image is considered a shared combination of all local images. We can again write the constraint of (\ref{eq:SADMM}) in the form of the constraint in (\ref{eq:ADMM}) by having $\mathbf{G} = [\mathbf{I}_{N}, \mathbf{I}_{N}, \cdots \mathbf{I}_{N}]$ of size $N \times QN$, $\mathbf{u} = \mathbf{x}$, $\mathbf{H}=-\mathbf{I}_{N} $, $\mathbf{z} = \mathbf{x_{G}}$, and $\mathbf{c} = \mathbf{0}$.\par
The augmented Lagrangian of (\ref{eq:SADMM}) can then be written as
\begin{equation}
\label{eq:aug_lag_S}
\begin{gathered}
\mathcal{L}\left({{\mathbf{x}},{\mathbf{{x}_{G}}},{\boldsymbol{\sigma }}} \right) = \left\{ \begin{gathered} \sum\limits_{q = 1}^{Q} \frac{\mu }{2}\left\| {{{\mathbf{y}}_{q}} - {{\mathbf{A}}_{q}}{{\mathbf{x}}_{q}}} \right\|_2^2 + {\lambda}{\left\| {\mathbf{{x}_{G}}} \right\|_1} \\ \hfill+\left\langle {{{\boldsymbol{\sigma }}},{\mathbf{\bar{x}}} - {\mathbf{{x}_{G}}}} \right\rangle + \frac{\beta }{2}\left\| {{\mathbf{\bar{x}}} - {\mathbf{{x}_{G}}}} \right\|_2^2 \end{gathered} \right\}.
\end{gathered}
\end{equation}
Note that since the number of constraints are reduced, the dual variable ${\boldsymbol{\sigma}}$ has a size ${N \times 1}$ instead of ${QN \times 1}$ as in CADMM. Next, we provide the variable updates due to SADMM formulation.
\\
\subsubsection{Update of $\mathbf{x}$ (Local Images)}
Unlike the consensus case, the augmented Lagrangian function (\ref{eq:aug_lag_S}) is not directly decomposable into $Q$ terms because of the sum variable $\bar{\mathbf{x}}$ inside the augmented quadratic term. However, we show here that it is still possible to solve for each local image variable $\mathbf{x}_{q}$ in parallel. Similar to (\ref{eq:updt_x_CADMM}), we use the values of ${\mathbf{{x}}^{\left(k \right)}_\mathbf{{G}}}$ and ${{\boldsymbol{\sigma }}_{q}^{\left(k\right)}}$ in order to solve for $\mathbf{x}_{q}$ at the $(k+1)^\text{st}$ iteration. However, since we have also $\bar{\mathbf{x}}$ in (\ref{eq:aug_lag_S}), we fix also all other local image variables to $\mathbf{x}_{i}^{\left(k \right)} \ \forall i \neq q$. Let $ \bar{\mathbf{x}}_{q}^{\left(k \right)} = \sum_{\left(i \neq q\right)} {\mathbf{x}}_{i}^{\left(k \right)} = \bar{\mathbf{x}}^{\left(k \right)} - {\mathbf{x}}_{q}^{\left(k \right)}$. Consequently, the $q^{\text{th}}$ local image update can be obtained by
\begin{equation}
\begin{aligned}
\label{eq:updt_x_SADMM}
{\mathbf{x}}_{q}^{\left(k + 1\right)} &= {\text{arg}}\mathop {\min }\limits_{{{\mathbf{x}}_{q}}} {\mathcal{L}}\left( {{{\mathbf{x}}_{q}};\bar{\mathbf{x}}_{q}^{\left(k \right)},{\mathbf{{x}}^{\left(k \right)}_\mathbf{{G}}},{\boldsymbol{\sigma }}^{\left(k\right)}} \right) \hfill \\
&=
{\text{arg}}\mathop {\min }\limits_{{{\mathbf{x}}_{q}}} \left\{ \begin{gathered}\frac{\mu }{2}\left\| {{{\mathbf{y}}_{q}} - {{\mathbf{A}}_{q}}{{\mathbf{x}}_{q}}} \right\|_2^2 + {\boldsymbol{\sigma}^{\left(k \right)}}^{T} {{\mathbf{x}}_{q}} \\ + \frac{\beta}{2}\left\| {{{\mathbf{x}}_{q}} + \bar{\mathbf{x}}_{q}^{\left(k \right)} - {\mathbf{{x}}^{\left(k \right)}_\mathbf{{G}}}} \right\|_2^2
\end{gathered} \right\}\hfill \\
&=
{\text{arg}}\mathop {\min }\limits_{{{\mathbf{x}}_{q}}} \left\{ \begin{gathered}\frac{\mu }{2}\left\| {{{\mathbf{y}}_{q}} - {{\mathbf{A}}_{q}}{{\mathbf{x}}_{q}}} \right\|_2^2 + {\boldsymbol{\sigma}^{\left(k \right)}}^{T} {{\mathbf{x}}_{q}} \\ + \frac{\beta}{2}\left\| {{{\mathbf{x}}_{q}} + \bar{\mathbf{x}}^{\left(k \right)} - {\mathbf{x}}_{q}^{\left(k \right)} - {\mathbf{{x}}^{\left(k \right)}_\mathbf{{G}}}} \right\|_2^2
\end{gathered} \right\}\hfill
\end{aligned}
\end{equation}
Now similar to (\ref{eq:updt_x_CADMM}), the problem in (\ref{eq:updt_x_SADMM}) is fully differentiable with respect to $\mathbf{x}_{q}$ and the $k+1^\text{st}$ update can be obtained in the closed-form
\begin{equation}
\label{eq:updt_x_SADMM_CF}
{\mathbf{x}}_{q}^{\left(k + 1\right)} = \resizebox{.85\hsize}{!}{$ \left( \mu \mathbf{A}_{q}^{H}\mathbf{A}_{q}+\beta \mathbf{I}_{N} \right)^{-1} \left( \mu \mathbf{A}_{q}^{H}\mathbf{y}_{q}+\beta \left({\mathbf{{x}}^{\left(k \right)}_\mathbf{{G}}}-{\bar{\mathbf{x}}_{q}^{\left(k\right)}} \right) - {\boldsymbol{\sigma }}^{\left(k\right)} \right)$}.
\end{equation}
From (\ref{eq:updt_x_SADMM_CF}), we can observe that in SADMM, the $q^{\text{th}}$ local image update requires the previous state ${\mathbf{x}}_{q}^{\left(k \right)}$ and the sum of all other local images earlier updates $\bar{\mathbf{x}}^{\left(k \right)}$ in addition to the global image update ${{\mathbf{{x}}_{G}}^{\left(k \right)}}$, and the dual variable update ${\boldsymbol{\sigma}_{q}^{\left(k \right)}}$. This suggests the need for extra memory with respect to CADMM to track the previous state at each cluster. Additionally, the distributed clusters will need to receive each other updates. This can be broadcast by the central node subsequent to the update of global image and dual variables. The central node will have such values any way since they are needed for the global image update. Although the exchanged information between the central node and the distributed clusters in SADMM seems to be more than CADMM, the size of those variables to be exchanged is still less than CADMM by a factor of $(Q+2)/3$ due to the reduced size of the dual variable in SADMM. This provides a significant reduction in communication bandwidth requirements between the central node and the sensors especially for a large number of distributed sensors.
\\
\subsubsection{Update of $\mathbf{{x}_{G}}$ (Global Image)}
Similar to the global image update in Section \ref{sec:CADMM}, after collecting the local images updates from the distributed sensors, the global image update for the sharing formulation is obtained by minimizing the augmented Lagrangian with respect to ${\mathbf{{x}_{G}}}$ as follows
\begin{equation}
\begin{gathered}
\label{eq:updt_xG_SADMM}
{\mathbf{{x}}^{\left(k+1 \right)}_\mathbf{{G}}} =\resizebox{.85\hsize}{!}{$ \mathrm{arg}\mathop{\min} \limits_{{\mathbf{x_{G}}}}\left\{ \lambda \left\| {\mathbf{x_{G}}} \right\|_1 + \frac{\beta}{2}\left\| {\mathbf{x_{G}}} - \bar{\mathbf{x}}^{\left(k + 1\right)} \right\|_{2}^{2} +{\boldsymbol{\sigma}^{\left(k \right)}}^{T} {\mathbf{{x}_{G}}}\right\}.$}
\end{gathered}
\end{equation}
Again, we here assume that both the global image and the dual variable is calculated at the central node. As a result, the sum of the local images $\bar{\mathbf{x}}^{\left(k \right)}$ is calculated directly at the central node following the local updates collection needed for the global image update. The solution of (\ref{eq:updt_xG_SADMM}) will be detailed later in the section.
\\
\subsubsection{Update of ${\boldsymbol{\sigma }}$ (Dual variable)}
The dual variable update then is a straight forward step of the ADMM algorithm
\begin{equation}
\label{eq:updt_sig_SADMM}
\boldsymbol{\sigma }^{\left(k + 1\right)} = \boldsymbol{\sigma}^{\left(k \right)} + \beta \left( \bar{\mathbf{x}}^{\left(k + 1\right)} - {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k + 1\right)} \right)
\end{equation}
To summarize, despite the performance of each formulation in terms of image quality which will be examined in the next section, SADMM provides an alternative processing architecture with respect to CADMM in which: 1) extra memory is needed at the distributed clusters for local image updates, 2) communication overhead between the distributed clusters and the central node is reduced by a factor of $(Q+2)/3$ due to the size difference of the constraints. In both CADMM and SADMM, local images can be updated in parallel at each cluster node and communicated back to the central node. The central node in turns updates both the global image and the dual variable. Subsequently, both the global image and dual variable updates are broadcast back to the distributed clusters in the case of CADMM in addition to the sum of the previous local images in case of SADMM in order to calculate the next local image updates. Other comparisons and performance metrics such as image reconstruction quality and convergence rate will be provided later in Section \ref{sec:Perf_Eval}.
\subsection{Solution Techniques}
\label{sec:SolTech}
Considering the above formulations, in this section, we provide the techniques used to solve the sub-problems for local and global images updates. Additionally, we provide the stopping criteria adopted to terminate both algorithms. Lastly, we show how to obtain the phases of complex-valued local images prior to ADMM iterations.
\\
\subsubsection{Variable Updates}
The update of local images employs a matrix inversion in a closed-form solution. However, due to the large size of the problem, it needs to be solved iteratively using a numerical procedure. In particular, we carry out the inversion in the local update using conjugate gradient (CG) method \cite{shewchuk1994introduction}. Being a numerical method, the output of CG is a complex-valued imaged since both the measurements $\mathbf{y}_{q}$ and the forward model $\mathbf{A}_{q}$ are complex. However, the optimization is carried out over the real-valued magnitude of the images where the phase of the images is included in the measurement matrix. Therefore, subsequent to the update of the local image using CG, a projection of the resulting complex image on the real positive orthant is applied to obtain the magnitude of the local image. This projection implicitly states that the phase of the complex-valued output of CG is regarded as a numerical phase error.\par
The global image on the other hand, requires solving a LASSO like optimization problem which can be solved using a proximal gradient method. In our numerical experiments, we used the accelerated proximal gradient \cite{beck2009fast} to calculate the global image updates.
\\
\subsubsection{Stopping Criteria}
Variable updates are repeated until termination which is decided upon comparing the values of the primal and dual residuals with their corresponding feasibility tolerances $\epsilon_{pri}$ and $\epsilon_{dual}$ respectively. Following the definitions of the residuals and the stopping criteria brought up in \cite{boyd_distributed_2011}, let $\boldsymbol{\eta}_{pri}$ and $\boldsymbol{\eta}_{dual}$ denote the primal and dual residuals respectively. The dual residual is defined over the successive updates of global image variable. Hence, it is the same for both CADMM and SADMM. Accordingly, at the $k^{\text{th}}$ iteration, the dual residual for both formulations is given by
\begin{equation}
\label{eq:resid_dual}
\begin{aligned}
\boldsymbol{\eta }^{\left(k\right)}_{dual} = \beta \left( {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k + 1\right)} - {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k \right)}\right).
\end{aligned}
\end{equation}
On the other hand, the primal residual measures the constraint satisfaction and takes a different form in the two formulations. For CADMM, the primal residual is given by
\begin{equation}
\label{eq:resid_pri_C}
\begin{aligned}
&\boldsymbol{\eta }^{\left(k\right)}_{pri}= {\mathbf{{x}}}^{\left(k + 1\right)} - \mathbf{1}_{Q} \otimes {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k + 1\right)}.
\end{aligned}
\end{equation}
Similarly, in SADMM the primal residual is
\begin{equation}
\label{eq:resid_pri_S}
\begin{aligned}
&\boldsymbol{\eta }^{\left(k\right)}_{pri}= \bar{\mathbf{{x}}}^{\left(k + 1\right)} - {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k + 1\right)}.
\end{aligned}
\end{equation}
The feasibility tolerances can be chosen based on an absolute tolerance $\epsilon_{abs}$ and a relative tolerance $\epsilon_{rel}$. Similar to the primal and dual residuals, the feasibility tolerances indicate non-identical definitions depending on the formulation due to the different constraints. In CADMM, they are given by
\begin{equation}
\begin{aligned}
\label{eq:tol_C}
&\epsilon_{pri}= \sqrt{QN} \ \epsilon_{abs} + \epsilon_{rel} \ \text{max} \left\{ \left\| {\mathbf{{x}}}^{\left(k \right)} \right\|_2, \sqrt{Q}\left\| {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k \right)} \right\|_2 \right \}\\
&\epsilon_{dual} = \sqrt{QN} \ \epsilon_{abs} + \epsilon_{rel} \left\| {\boldsymbol{\sigma }}^{\left(k \right)} \right\|_2.
\end{aligned}
\end{equation}
Likewise, for SADMM, feasibility tolerances are
\begin{equation}
\begin{aligned}
\label{eq:tol_S}
&\epsilon_{pri}= \sqrt{N} \ \epsilon_{abs} + \epsilon_{rel} \ \text{max} \left\{ \left\| \bar{\mathbf{{x}}}^{\left(k \right)} \right\|_2, \left\| {{\mathbf{x}}}_{\mathbf{{G}}}^{\left(k \right)} \right\|_2 \right \}\\
&\epsilon_{dual} = \sqrt{N} \ \epsilon_{abs} + \epsilon_{rel} \left\| {\boldsymbol{\sigma }}^{\left(k \right)} \right\|_2.
\end{aligned}
\end{equation}
Lastly, for both CADMM and SADMM, the algorithm is terminated either when a defined maximum number of iterations is reached or when both the ensuing inequalities are satisfied
\begin{equation}
\begin{aligned}
\label{eq:stop}
& \left\| \boldsymbol{\eta }^{\left(k\right)}_{pri} \right\|_2 \leqslant \epsilon_{pri}\\
& \left\| \boldsymbol{\eta }^{\left(k\right)}_{dual} \right\|_2 \leqslant \epsilon_{dual},
\end{aligned}
\end{equation}
where the variables in the above inequalities are calculated according to the definitions of the corresponding formulation.
\\
\subsubsection{Phase Matrix $\mathbf{\Theta}$}
As mentioned earlier, we assume that the phase of local images is already provided prior to carrying out the optimization algorithms. Our proposed imaging methods can be considered partially non-coherent imaging method since the phases are only used within the data-fidelity term in the objective function. Thus, a coarse estimated phase of local images is sufficient for our algorithms to perform satisfactorily. Therefore, we use the phase of the images obtained by back-projection for each cluster as an estimate of the phase of local images. Accordingly, for each cluster $q$, the diagonal matrix containing the phase of all pixels of its local image is constructed as
\begin{equation}
\mathbf{\Theta}_{q} = \text{diag}\left\{\exp{(j[\angle{(\mathbf{A}_{q}^{H} \mathbf{y}_{q})])}}\right\}.
\end{equation}
\section{Performance Evaluation}
\label{sec:Perf_Eval}
In this section, we validate and evaluate the performance of the algorithms proposed in Section \ref{sec:ADMM} to reconstruct radar images using distributed sensor clusters. To achieve this goal, we use the publicly available civilian vehicle dome (CV domes) data-set which offers simulated scattering data of civilian vehicle facet models. Although the data-set is originally intended to simulate circular synthetic aperture radars, a particular configuration of WSAR, it can also be used to simulate a mono-static distributed radar sensors system.\par
First, we give a brief introduction of the data-set and its parameters. Consequently, we define the performance metrics used in our evaluation to compare both algorithms. Finally, we evaluate our algorithms on three different scenarios of practical relevance for several applications. The scenarios are realized by different combinations of full/limited views and full/limited bandwidth measurements as we will show later in this section.
\subsection{Data-set Introduction}
CV Domes data-set contains simulated high-frequency scattering data of ten civilian vehicles. For each model, an X-band electromagnetic mono-static scattering is simulated in a far-field scenario. Scattered waves are simulated with full polarization over an azimuth extent of $360 \degree$ where $16$ viewing angles per degree of azimuth are considered. Similarly, data are simulated over the range of elevation angles from $30\degree$ to $60 \degree$. For each azimuth and elevation viewing angle tuple, $512$ frequency samples of complex-valued scattering coefficients centered at $9.6$ GHz and spanning a bandwidth of approximately $5.35$ GHz are provided. The range information of those frequency measurements is compressed already resulting in what is usually referred to as phase history.\par
\subsection{Performance Metrics}
As discussed in the previous section, the difference in problem formulation between CADMM and SADMM has induced slightly different system implementation features in terms of memory requirements and communication bandwidth. Additionally, to compare the performance
of the proposed algorithms, the following aspects are considered.
\begin{enumerate}
\item Convergence rate: it can be assessed by the number of iterations needed to reach the stopping criteria that is defined identically for both algorithms.
\item Computational complexity: the main computational burden of both algorithms lies in the local image updates (\ref{eq:updt_x_CADMM_CF}) and (\ref{eq:updt_x_SADMM_CF}), where the matrix inversion term is present. Due to their large size, the measurement matrices $\mathbf{A}_{q}$ are realized through matrix operators based on two-dimensional non-uniform Fast Fourier transform (2D NuFFT) \cite{fessler_nonuniform_2003}. Moreover, the inversion step is carried out numerically using CG as mentioned earlier. Thus, while the complexity of both algorithms seems to be equivalent, the convergence of CG highly depends on the other variables in the update formulas (\ref{eq:updt_x_CADMM_CF}) and (\ref{eq:updt_x_SADMM_CF}). As a result, the comparison solely in terms of the number of iterations is not indicative since a single iteration in each of the algorithms may realize a different cost. Accordingly, computational complexity can be measured by calculating the total processing time spent until termination.
\item Image reconstruction quality: the data-set does not contain a reference image with which a comparison can be made in order to evaluate the quality of reconstructed images. Correspondingly, we use the image entropy as a quantitative metric to assess the image quality being a measure of its sharpness or constituent randomness. The smaller the image entropy the sharper is the reconstructed image and vice versa. The entropy of an image is calculated in bits after intensity saturation for the values beyond a desired dynamic range in the dB scale followed by image translation to the gray-scale. Consequently, a randomly generated image would have an entropy equal or close to $8$ bits. Additionally, as a subjective measure of images quality, the images reconstructed utilizing full aperture and full bandwidth measurement could act as a visual reference for the other scenarios when the aperture and/or the bandwidth measurements are reduced.
\end{enumerate}
\subsection{Experiments}
\label{sec:experiments}
\begin{figure*}[t!]
\centering
\includegraphics[width=7 in]{figures/FVFB_ParSw_Mod.pdf}
\caption{Hyperparameters sweep for 'Jeep99' data-set (FVFB); normalized sparsity (top row), image entropy (bottom row)}
\label{fig:FVFB_ParSw}
\end{figure*}
In this subsection, we illustrate the reconstructed images by CADMM and SADMM using the simulated data of two different vehicle models differing in type and geometry. The first is of a Jeep Cherokee (SUV) 'Jeep99', while the second is of a Toyota Tacoma (Pick-up) 'Tacoma'. We consider image reconstruction utilizing the data-set according to the following scenarios:
\begin{enumerate}
\item Full aperture views measurements: for a general validation of both algorithms, the full $360 \degree$ aperture measurements of the entire available bandwidth is considered.
\item Full views and limited bandwidth measurements: limited frequency samples of the full aperture measurements are considered for image reconstruction realizing a typical use case of WSAR imaging.
\item Limited views and limited bandwidth measurements: assuming a distributed system of radar sensors illuminating the scene according to a time division multiplexing (TDM) scheme, limited frequency samples of limited aperture measurements are considered for images reconstruction.
\end{enumerate}
For all experiments, measurements are taken at a fixed elevation angle of $30 \degree$ and with 'HH' polarization. Moreover, all measurements are impaired with a white Gaussian noise realizing a signal to noise ratio (SNR) of $15$ dB. A fine grid of $256$ cells in both range and cross-range directions ($7$ meter-long each) is used resulting in a total number of $N =65536$ pixels. Additionally, images are reconstructed considering an elevated image plane at $1$ meter from the ground level. This renders the projection of layover-ed elements to be mostly contained within vehicles outlines and permits a better visual interpretation.\par
The choice of $\mu$, $\beta$, and $\lambda$ for both CADMM and SADMM is made through a parameter sweep guided by normalized image sparsity and image entropy as performance metrics. The normalized sparsity considered is the percentage of the non-zero pixels in the image. Hyperparameters used to reconstruct the illustrated images throughout this section are those that guarantee a similar degree of sparsity for both CADMM and SADMM images at a lower entropy value. Also, we tried to pick the parameters where $\beta$ is as close as possible in both methods for a fair convergence comparison. Given the considered scene size and typical dimensions of a vehicle, a sparsity level around the range of $5\%-10\%$ is considered in our experiments based on the scenario and the vehicle. Needless to say, parameter sweep analysis is conducted separately for each data-set and each scenario. It is worth mentioning that, for a given $\beta$, the ratio $\lambda/\mu$ can be automatically selected given the desired sparsity range following our method proposed in \cite{murtada_efficient_2021}. However, since for both CADMM and SADMM an empirical search for $\beta$ is needed and at almost exact sparsity levels, a parameter sweep would facilitate finding more accurate parameters for the sake of comparison.
An example of image sparsity and image entropy versus different parameters is shown for the first scenario. For later experiments, such analysis will be omitted for brevity. CADMM and SADMM are run for a maximum number of $100$ iterations while the feasibility tolerances $\epsilon_{abs}$ and $\epsilon_{rel}$ are both set to $10^{-2}$.
Scenario-specific parameters and reconstructed images of all the aforementioned experiments are provided and discussed in the sequel.
\\
\subsubsection{Full Views - Full Bandwidth (FVFB)}
Using the full $360 \degree$ azimuth extent and the full bandwidth of the data-set (approximately equal to $5.35$ GHz), we reconstruct the images of the two vehicles to validate our proposed methods and show their superior reconstruction quality with respect to the conventional BP method. The reconstructed images in this experiment can also be used as a reference for the subsequent scenarios when images are reconstructed using limited views and/or limited bandwidth measurements.
As mentioned previously, to avoid anisotropic scattering over the large angular azimuth extent, it is divided into sub-apertures (clusters) in which point isotropic scattering assumption is valid. The angular width of a cluster has a double-faced effect on the imaging performance trading-off between angular resolution and homogeneous targets scattering. The wider the cluster the better the resolution while the scattering becomes anisotropic. Therefore, in this experiment, the cluster width is chosen to maximize the imaging performance in a balanced manner. We use a cluster width of $5\degree$ which is a good trade-off since it allows for a cross range resolution approximately equivalent to the range resolution provided by the bandwidth while maintaining a minimum degree of scattering anisotropicity of the targets.\par
As anticipated earlier, we perform a hyperparameter sweep over different values of $\beta$ and $\lambda /\mu$ to obtain the values that provide good reconstruction quality given image sparsity level. The sparsity level is decided based on the assumed dimensions of the observed targets relative to the scene dimensions. They are set separately for each scenario due to the dissimilarities of aperture size and signal bandwidth utilized, hence images exhibit a different resolution in each scenario. Fig.~\ref{fig:FVFB_ParSw} shows the normalized sparsity level and entropy of the reconstructed image of the 'Jeep99' data-set versus the varying parameters of $\beta$ and $\lambda/\mu$ for both SADMM and CADMM. The desired sparsity level range is highlighted with a light green color in the figure. As expected, the higher the ratio $\lambda/\mu$, the more sparse are the reconstructed images for both the algorithms and naturally the lower the entropy. This is true except for some values of $\beta$ for which SADMM optimization does not converge to a sparse solution. Such divergence for those parameters can be seen from the entropy values which indicate reconstruction of random images. Moreover, beyond those values of $\beta$, reconstructed images using the same $\lambda/\mu$ ratio exhibit similar sparsity levels and attain close values for entropy as shown in the magnified part of Fig.~\ref{fig:FVFB_ParSw}.\par
\begin{figure}[h!]
\centering
\includegraphics[width=3.7 in]{figures/FVFB_ParComp_Mod.pdf}
\caption{Jeep image reconstruction at different sparsity levels; $12\%$ (left column), $5\%$ (right column)}
\label{fig:FVLB_ParComp}
\end{figure}
To first show the performance of both methods, CADMM and SADMM images reconstructed considering two different sparsity levels are shown for the 'Jeep99' data-set in Fig.~\ref{fig:FVLB_ParComp}. The left column images have a normalized sparsity degree of $12\%$ versus $5\%$ for the images in the right column. For lower sparsity images, CADMM show a sharp high intensity reconstruction of the strong features of the vehicle such as edges and ceiling structure, and a lower intensity reconstruction of the weaker features such as the projection of the tire wheels. On the other hand, SADMM images manifest an averaged intensity of the different parts of the vehicle resulting in less sharp images. While the behavior is maintained for the images at higher sparsity, the weaker features are further suppressed in the images of both methods. Moreover, the images of both methods at a similar sparsity level have similar entropy values.
This behavior is again confirmed by the reconstructed images of 'Tocoma' vehicle. At a sparsity level of $10\%$, the reconstructed images of the two vehicles are shown in Fig~.\ref{fig:FVFB} in addition to the images reconstructed through conventional back-projection averaged over all clusters. The red dots on the images show the angular aperture views which cover $360 \degree$ in this experiment. Note that due to the abundance of the available bandwidth and the full aperture measurements, both CADMM and SADMM images have very high resolution and show the detailed structure of the imaged objects.\par
Additionally, the processing time until termination in SADMM is slightly higher than CADMM. The numerical values of the parameters used in image reconstruction and the corresponding metrics are reported in Table \ref{tab:summary} while the ratio between SADMM and CADMM processing time and number of iterations are reported in Table \ref{tab:proc_time}.\\
\begin{figure}[ht!]
\centering
\includegraphics[width=3.6 in]{figures/FVFB_Mod.pdf}
\caption{FVFB image reconstruction at sparsity level $\approx 10\%$, $Q=72$ clusters, bandwidth $=5.35$ GHz; Jeep Cherokee (left column), Toyota Tacoma (right column)}
\label{fig:FVFB}
\end{figure}
\subsubsection{Full Views - Limited Bandwidth (FVLB)}
\begin{figure}[ht!]
\centering
\includegraphics[width=3.6 in]{figures/FVLB_Mod.pdf}
\caption{FVLB image reconstruction at sparsity level $\approx 10\%$, $Q=72$ clusters, bandwidth $=600$ MHz; Jeep Cherokee (left column), Toyota Tacoma (right column)}
\label{fig:FVLB}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=3.6 in]{figures/FVLB_Tac_E_Mod.pdf}
\caption{Less sparse FVLB reconstruction of Toyota Tacoma (sparsity level $\approx 15\%$).}
\label{fig:FVLB_T_E}
\end{figure}
In practice, the signal bandwidth of a SAR system is usually an order of magnitude less than the available bandwidth of the data-set. Thus, analyzing the performance of our proposed algorithms in this case is of high interest and is a relevant use case in WSAR imaging. To realize a limited bandwidth measurement scenario, we utilized an equivalent of $600$ MHz samples around the center frequency from the samples of phase history. In this experiment as well, SADMM images exhibit the property of higher averaging than that of concentrated intensity resulting from CADMM images as shown in Fig.~\ref{fig:FVLB}. This peculiarity makes it capable of capturing the true structure of the vehicles even with low bandwidth measurements when referring to the images in Fig.~\ref{fig:FVFB} of FVFB experiment. On the other hand, images reconstructed using CADMM have higher intensity around the strong reflectors and weaker or no intensity of the poorly reflective components of the vehicles; this is evident in the reconstructed images of both vehicles in Fig.~\ref{fig:FVLB}. For example, the crossing of the beams in the rear part of 'Jeep99' data-set is localized and well identified with strong intensity in SADMM image, while in CADMM image, the same area exhibits only a strong intensity in a wider region. Similarly, in 'Tacoma' images, CADMM fails to capture the outline of the vehicle at this sparsity level and the image is dominated by the strong trunk while in SADMM image the vehicle outline makes an appearance and even the trunk is better identified. The depicted images have a sparsity level approximately equals to $10\%$. Note that by considering lower sparsity, images of both algorithms will have an increase of the background intensity around the strong features without capturing the general structure of the object differently. An example is shown in Fig.~\ref{fig:FVLB_T_E} where the images of 'Tacoma' vehicle are reconstructed at lower normalized sparsity level of about $15\%$. In summary, although CADMM images have similar entropy values as their SADMM counterparts at the same sparsity level, the latter provides higher capability of capturing the structure of the observed targets given relatively low bandwidth measurements. The exact values of sparsity and entropy for each image are reported in Table \ref{tab:summary}.\par
The superior performance of SADMM comes at the cost of increased computational complexity to reach convergence. This complexity is manifested through the higher number of iterations and the longer processing time required for SADMM to reconstruct the shown images. The ratio of these two quantities for SADMM and CADMM is depicted in Table \ref{tab:proc_time}.\\
\subsubsection{Limited views - Limited Bandwidth (LVLB)}
\begin{figure}[h!]
\centering
\includegraphics[width=3.6 in]{figures/LVLB_O0_Mod.pdf}
\caption{LVLB image reconstruction at sparsity level $\approx 10\%$, $Q=16$ clusters, bandwidth $=600$ MHz; Jeep Cherokee (left column), Toyota Tacoma (right column)}
\label{fig:LVLB_O0}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=3.4 in]{figures/LVLB_O1r_Mod.pdf}
\caption{LVLB reconstruction with different random sensors orientation; Jeep Cherokee (left column), Toyota Tacoma (right column)}
\label{fig:LVLB_O1}
\end{figure}
A mono-static distributed sensing scenario can be realized by considering far-field illumination with limited-narrow views of the full aperture measurements. In this experiment, we consider using the data-set to realize a system of distributed radar sensors in which TDM scheme is used to illuminate the scene of interest where a single cluster transmits at a time. Consequently, in addition to the limited bandwidth measurements of $600$ MHz introduced in the previous experiment, we further consider limited aperture measurements representing the views of distributed sensors. In particular, $16$ clusters of $1 \degree$ width each are uniformly distributed around the scene and considered the viewing angles of the sensors. This number of clusters makes a realistic choice for the number of sensors where they cover only a span of about $4.4\%$ of the full aperture measurements.\par
For this experiment, the reconstructed images of the two vehicles are shown in Fig.~\ref{fig:LVLB_O0}. Similar to the previous experiment, SADMM captures both vehicles' structure better than CADMM. However, due to the limited aperture measurements, the artifacts present in the images of both methods are stronger. Increasing the sparsity would eliminate the artifacts but further limits the reconstruction of the entire outline of the vehicles. Of course, reconstructed images are views dependent. However, the performance of both methods is the same when different orientation of the sensors is considered. For example, the images reconstructed using another random orientation of views are shown in Fig.~\ref{fig:LVLB_O1}. The images confirm the capability of SADMM to retain the original structure of the imaged target while CADMM has a better ability to diminish the artifacts. On another note, processing times of both algorithms are roughly similar in this experiment given a limited amount of measurements. Finally, the parameters used to reconstruct the images in Fig.~\ref{fig:LVLB_O0} and the corresponding values of entropy and sparsity are reported in Table \ref{tab:summary} while processing time and the number of iterations ratios are reported in Table \ref{tab:proc_time}.\par
\begin{table*}[h!]
\centering
\caption{Summary of parameters used in the experiments and corresponding metrics}
\label{tab:summary}
\begin{tabular}{@{}clllllll@{}}
\cmidrule(l){3-8}
\multicolumn{1}{l}{} &
&
\multicolumn{2}{c}{FVFB} &
\multicolumn{2}{c}{FVLB} &
\multicolumn{2}{c}{LVLB} \\ \cmidrule(l){3-8}
\multicolumn{1}{l}{} &
&
\multicolumn{1}{c}{CADMM} &
\multicolumn{1}{c}{SADMM} &
\multicolumn{1}{c}{CADMM} &
\multicolumn{1}{c}{SADMM} &
\multicolumn{1}{c}{CADMM} &
\multicolumn{1}{c}{SADMM} \\ \midrule
\multirow{2}{*}{($\beta$, $\lambda/\mu$)} & Jeep & (30,18) & (30,2.5) & (30,16) & (30,1.4) & (10,1.25) & (20,0.4) \\ \cmidrule(l){2-8}
& Tacoma & (30,25) & (30,2.5) & (30,24) & (50,1.8) & (10,1.75) & (20,0.65) \\ \midrule
\multirow{2}{*}{Sparsity} & Jeep & 0.100 & 0.102 & 0.104 & 0.102 & 0.100 & 0.098 \\ \cmidrule(l){2-8}
& Tacoma & 0.096 & 0.10 & 0.097 & 0.098 & 0.099 & 0.103 \\ \midrule
\multirow{2}{*}{Entropy} & Jeep & 1.24 & 1.25 & 1.27 & 1.26 & 1.25 & 1.22 \\ \cmidrule(l){2-8}
& Tacoma & 1.19 & 1.24 & 1.20 & 1.21 & 1.24 & 1.26 \\ \bottomrule
\end{tabular}
\end{table*}
\begin{table*}[!h]
\centering
\caption{Relative convergence and complexity}
\label{tab:proc_time}
\begin{tabular}{@{}ccccccc@{}}
\toprule
& \multicolumn{2}{c}{FVFB} & \multicolumn{2}{c}{FVLB} & \multicolumn{2}{c}{LVLB} \\ \midrule
\begin{tabular}[c]{@{}c@{}} Ratio\\SADMM/CADMM\end{tabular}&
\begin{tabular}[c]{@{}c@{}}Number of\\ iterations \end{tabular} &
\begin{tabular}[c]{@{}c@{}}Processing\\ time \end{tabular} &
\begin{tabular}[c]{@{}c@{}}Number of\\ iterations \end{tabular} &
\begin{tabular}[c]{@{}c@{}}Processing\\ time \end{tabular} &
\begin{tabular}[c]{@{}c@{}}Number of\\ iterations \end{tabular} &
\begin{tabular}[c]{@{}c@{}}Processing\\ time \end{tabular} \\ \midrule
Jeep & 1.41 & 1.19 & 3.45 & 2.11 & 1.45 & 1.20 \\ \midrule
Tacoma & 1.41 & 1.22 & 3.13 & 2.06 & 1.51 & 1.04 \\ \bottomrule
\end{tabular}
\end{table*}
To summarize, in the first experiment where a plenitude of measurements in both aperture and bandwidth is available, both CADMM and SADMM are capable of reconstructing detailed and super-resolution images of the observed targets far surpassing the conventional methods. On the other hand, in the latter experiments where measurements are limited in aperture and/or bandwidth, SADMM exhibits superior performance over CADMM in terms of capturing the structure of the target and reconstructing smoother images. Although they have similar entropy in all cases, the depicted images reconstructed by both algorithms show a clear visual advantage of SADMM when compared with the images of full measurements. Such higher quality comes at the expense of computational cost. Surprisingly, in terms of convergence and complexity, SADMM fell behind the most in the second experiment where the full aperture measurements with limited bandwidth are considered. This can be owed to the fact that CADMM features a high convergence rate requiring less than a third of SADMM iterations to reach such concentrated intensity images and by having less degrees of freedom.
\section{Conclusion}
\label{sec:conc}
In this paper, a novel approach for widely distributed radar imaging based on ADMM framework is proposed. Sparsity prior has been imposed on a defined global image assumed to represent an aggregate view of the scene. Then, developing on top of our previous work, the problem formulation is tailored to this approach and a new formulation has been introduced. The two formulations named CADMM and SADMM are designed to mathematically stipulate the relationship between the images of individual sensors and the global image. The solutions to the proposed formulations have been provided as iterative algorithms that are flexible and amenable to be implemented on a distributed architecture. We have demonstrated the performance of our proposed algorithms through several experiments and showed their significant edge over conventional methods in terms of reconstructed images quality. Moreover, we showed that SADMM outperforms CADMM by reconstructing images of high resolution that better exhibit the structure and the shape of the observed objects, especially when the measurements are limited in bandwidth and/or sparse in aperture. As we illustrated in our experiments, our proposed algorithms are applicable in many scenarios of distributed radar systems and WSAR imaging. Following our approach, various formulations can be further studied and developed either by imposing different prior on the global image and/or by imposing alternative associations with the images of the individual sensors.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
|
1,116,691,500,513 | arxiv | \section{Introduction}
Graph analysis has been receiving more and more attention on social networks, transportation, protein forecast, etc.
However, directly publishing graph statistics may leak sensitive information about an individual~\cite{RHMS_2}.
Recently, many research works have studied the problem of publishing sensitive graph statistics under differential privacy (DP) \cite{dwork_foundations,li2016differential}.
Compared with previous privacy models (e.g., $k$-anonymity, $l$-diversity, $t$-closeness), differential privacy can resist most private attacks and provide a provable privacy guarantee.
When DP is applied to graph analysis, there are two common variants of DP \cite{task2012guide,li2021private}: Edge Differential Privacy \cite{qian2018publishing, hay2009accurate, karwa2012differentially, proserpio2012workflow} and Node Differential Privacy \cite{day2016publishing, kasiviswanathan2013analyzing, raskhodnikova2016lipschitz, blocki2013differentially}.
Intuitively, Edge Differential Privacy guarantees that a query result does not significantly reveal the sensitive information about a particular edge in a graph; while Node Differential Privacy protects the information about a node and its all adjacent edges.
Obviously, Node Differential Privacy provides much stronger privacy guarantee than Edge Differential Privacy.
Existing works related to Node Differential Privacy are almost in the central (or global) model, where a trusted curator holds the entire graph data before data publishing.
We refer to the above two variants under a central server setting as Edge Central Differential Privacy (Edge-CDP) and Node Central Differential Privacy (Node-CDP), respectively.
However, the assumption about a trusted server may not be practical in many applications (i.e., individual contact lists) due to security reasons, such as privacy leaks and breaches in recent years \cite{yang2020local}.
Local differential privacy (LDP) \cite{duchi2013local, kasiviswanathan2011can} is a promising model that does not require a trusted server to collect user information.
In LDP, each user perturbs its sensitive information by herself and sends perturbed messages to the untrusted data curator; hence it is difficult for the curator to infer sensitive information with high confidence.
We refer to the above two variants of DP without a trusted server as Edge Local Differential Privacy (Edge-LDP) and Node Local Differential Privacy (Node-LDP), respectively.
Although there are many recent studies on publishing statistics under Edge-LDP \cite{AsgLDP, qin2017generating, imola2020locally}, to the best of our knowledge, no existing work in literature attempts to investigate graph statistics release under Node-LDP.
In general, it is very challenging to publish graph statistics under LDP due to the lack of global view and prior knowledge about the entire graph.
The high sensitivity has been identified as the primary challenge when satisfying Node Differential Privacy, whether in central settings or local settings.
Consider querying the node degree in a social graph, if two graphs differ in one node, the results may differ at most $(n-1)$ edges in the worst case, where $n$ is the number of users.
Thus the sensitivity of Node Differential Privacy is $(n-1)$ while that of Edge-DP is 1.
Naively scaling the sensitivity of Edge-LDP for achieving Node-LDP suffers prohibitive utility drop.
Graph projection \cite{kasiviswanathan2013analyzing, blocki2013differentially, day2016publishing} is the key technique to reduce the high sensitivity, but existing projections are just designed for Node-CDP.
When attempting to apply central graph projection for Node-LDP, the main challenges come from the local view and little knowledge of the whole graph.
To be specific, it is difficult for each local user to project its neighboring information with a limited local view.
In central models, with the global view, the server can easily finish the projection by updating the two neighboring node information simultaneously.
However, in local scenarios, each user can only see its own information but not other neighboring information.
On the other hand, it is difficult for local users to obtain an optimal $\theta$ as they have little knowledge about the entire graph.
The projection parameter $\theta$ plays a vital role as upper bound on the maximum degree, as well as significantly influences the utility.
If $\theta$ is too small, a large number of edges will be removed during the projection.
If $\theta$ is too large, the sensitivity will be higher and more noise will be added during the protection.
Central projection methods can easily opt for the desirable projection parameter $\theta$ with some prior knowledge of the whole graph, for instance, the maximum degree, the average degree; while it is harder for local users to achieve it, since they have little prior knowledge about the entire graph.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{figures/framework.pdf}
\caption{Framework of our methods}
\label{fig:framework}
\end{figure}
In this paper, we publish the degree distribution under Node-LDP by introducing a novel local graph projection method that achieves two main goals: (1) How to execute local graph projection privately; (2) How to obtain the optimal graph projection parameter $\theta$.
The general framework is depicted in detail in Fig. \ref{fig:framework}, which includes three phases: (1) selecting the optimal projection parameter $\theta$ (Sec. \ref{sec:projection_parameter}); (2) executing the local graph projection (Sec. \ref{sec:local_projection}); (3) collecting noisy degrees and publishing the degree distribution.
First, to find the optimal projection parameter $\theta$, we make the best of the global view of the data curator.
After collecting the messages from local users, the data curator can help local users find the optimal parameter $\theta$ from the candidate pool based on our designed utility loss function.
But these communication messages may reveal personal information. Here, we design two methods to protect individual messages based on different privacy techniques: PureLDP and Crypto-assisted.
To begin with, we design a naive local graph projection method while satisfying Node-LDP.
The obvious disadvantage is that $K$-round selections bring more noise to true values, significantly influencing the utility.
To improve it, we design a Crypto-assisted parameter selection method that combines LDP with cryptographic primitives for graph statistics.
Here, one key challenge is that different summations of utility loss can be compared while protecting individual sensitive messages including order information.
We first use the order-preserving encryption (OPE) scheme \cite{kamara2020review, tueno2020efficient} to preserve order information for comparing different utility loss values.
Then, we mask the encrypted value with Secure Aggregation \cite{bonawitz2017practical} for protecting the individual order information.
Thus, the optimal parameter $\theta$ can be selected with high accuracy while guaranteeing personal privacy.
Second, we propose a baseline approach based on Node-level granularity, where each node is the minimal unit of a graph and correlations among neighboring users will be ignored roughly.
However, this approach loses too much neighboring information that significantly influences the overall utility (detailed analysis in Sec. \ref{sec:node-level} ).
Then, we propose an improved approach based on Edge-level granularity, where each edge is the minimal unit that is more fine-granularity information.
One main challenge is that sensitive information may be leaked via mutual edges among neighboring nodes.
we carefully design a randomized mechanism based on the Random Response (RR) mechanism \cite{warner1965randomized, dwork_foundations} to perturb each bit of projection operation vectors while satisfying Node-LDP.
Therefore, it is difficult to distinguish whether the current node degree is larger than $\theta$ or smaller than $\theta$.
Our contributions can be summarized as follows:
\begin{itemize}
\item We propose and study the problem of publishing the degree distribution under Node-LDP for the first time.
We give a detailed description of the problem definition and conclude the research gap.
We present an overview of publishing the degree distribution under Node-LDP.
\item We design two local graph projection approaches based on different granularity: Node-level and Edge-level.
The improved Edge-level method preserves more information and provides better utility than the baseline Node-level method.
\item We design two methods to find the optimal projection parameter $\theta$: PureLDP and Crypto-assisted.
Crypto-assisted method combines LDP with cryptographic primitives for graph statistics, which achieves a higher accuracy than the baseline PureLDP method.
\item Extensive experiments on real-world graph datasets validate the correctness of our theoretical analysis and the effectiveness of our proposed methods.
\end{itemize}
\section{Preliminaries and Problem Definition}
\label{sec:preliminaries}
\subsection{Preliminaries}
According to different trusted assumptions, differential privacy can be divided into two types: central differential privacy (CDP) in Definition \ref{def:CDP} and local differential privacy (LDP) in Definition \ref{def:LDP}.
\begin{definition}[CDP]
\label{def:CDP}
A random algorithm $M$: $\mathbb{X}$$^n$ $\rightarrow$ $\mathbb{Z}$ satisfies $\epsilon$-DP, where $\epsilon$ $\geq$ 0, iff for any two neighboring datasets D, D$^{\prime}$ $\in$ $\mathbb{X}$$^n$, any subsets S $\subseteq$ $\mathbb{Z}$,
\begin{center}
$Pr[M(D) \in S] \leq e^{\epsilon} Pr[M(D^{\prime}) \in S]$
\end{center}
\end{definition}
\begin{definition}[LDP]
\label{def:LDP}
A random algorithm $M$: $\mathbb{X}$ $\rightarrow$ $\mathbb{Y}$ satisfies $\epsilon$-LDP, where $\epsilon$ $\geq$ 0, iff for any two input x, x$^{\prime}$ $\in$ $\mathbb{X}$, any output y $\in$ $\mathbb{Y}$,
\begin{center}
$Pr[M(x) = y] \leq e^{\epsilon} Pr[M(x^{\prime}) = y]$
\end{center}
\end{definition}
Since the trusted third party is impractical, LDP has become the de facto standard of privacy protection to protect individual information.
As a graph consists of nodes and edges, there are two definitions when LDP is applied to either of them: edge local differential privacy (Edge-LDP) in Definition \ref{def:edgeLDP} and node local differential privacy (Node-LDP) in Definition \ref{def:nodeLDP}.
Node-LDP is clearly a much stronger privacy guarantee than Edge-LDP since it requires hiding the existence of each node along with its incident edges.
To our knowledge, however, there are few research work that releases graph statistics under Node-LDP. Although \cite{zhang2020differentially} considers Node-DP in the local model where a node represents a software component and an edge represents control flow between components, the directed graphs on the control-flow behavior of different users are mutually independent.
We consider a totally different setting where each node represents a user and each edge represents the correlation between neighboring users.
\begin{definition}[Edge-LDP]
\label{def:edgeLDP}
A random algorithm $M$ satisfies $\epsilon$-edge LDP, iff for any $i \in [n]$, two adjacency bit vectors $B_i$ and \textbf{$B_i^{\prime}$} that differ only in one bit, and any output y $\in range(M)$,
\begin{center}
$Pr[M(B) = y] \leq e^{\epsilon} Pr[M(B^{\prime}) = y]$
\end{center}
\end{definition}
\begin{definition}[Node-LDP]
\label{def:nodeLDP}
A random algorithm $M$ satisfies $\varepsilon$-node LDP, iff for any $i \in [n]$, two adjacency bit vectors $B$ and \textbf{$B^{\prime}$} that differ at most $(n-1)$ bits, and any output $y\in range(M)$,
\begin{center}
$Pr[M(B) = y] \leq e^{\varepsilon} Pr[M(B^{\prime}) = y]$
\end{center}
\end{definition}
There are two kinds of DP, namely, $bounded$ DP and $unbounded$ DP \cite{li2016differential, 10.1145/1989323.1989345}.
In a bounded DP, two neighboring datasets $D$, $D'$ have the same size $n$ and $D\prime$ is obtained from $D$ by changing or replacing one element.
In unbounded DP, $D\prime$ can be derived from $D$ by deleting or adding one element. Here, we use the bounded DP to publish the degree distribution.
That is to say, the size of each adjacency bit vector is equal to $n$, where $n$ is the number of users.
Node-LDP satisfies the post-processing property \cite{dwork_foundations} (Theorem \ref{theorem:post-process}) and the composition property (Theorem \ref{theorem:composition}).
\begin{theorem}[Post-Processing]
\label{theorem:post-process}
If a randomized algorithm $R$ satisfies $\varepsilon$-DP, then for an arbitrary randomized algorithm $S$, $S \circ R$ also satisfies $\varepsilon$-DP.
\end{theorem}
\begin{theorem}[Composition Property]
\label{theorem:composition}
$\forall \varepsilon \geq 0, k\in N$, the family of $\varepsilon$-DP mechanism satisfies $t\varepsilon-DP$ under t-fold adaptive composition.
\end{theorem}
\iffalse
\begin{definition}[global sensitivity]
\label{theorem:global_sensitivity}
let $x\simeq x^\prime$ denote that x and x$^\prime$ are neighboring.
the global sensitivity of a query function $f$ denoted by $\triangle$ is given below
\begin{center}
$\triangle f=\underset{x\simeq x^\prime}{max}|f(x)-f(x^\prime)|$
\end{center}
\end{definition}
\fi
To satisfy DP, one way to add some noise into the query result.
In the Laplace mechanism (Theorem \ref{theorem:laplace}) \cite{dwork_foundations, li2016differential}, given the privacy budget $\varepsilon$ and sensitivity $\delta$, one publishes the result after adding Lap($\frac{\triangle}{\epsilon}$) noise.
\begin{theorem}[Laplace Mechanism]
\label{theorem:laplace}
For any function $f$, the Laplace mechanism $A(D)=f(D)+Lap$ ($\frac{\triangle f}{\varepsilon}$) satisfies $\varepsilon$-DP.
\end{theorem}
Additionally, the randomized response (RR) \cite{warner1965randomized, erlingsson2014rappor} is a common method to guarantee LDP. In specific, each user gives the true answer with the flipping probability $p$ and the opposite answer with probability $1-p$. Some works have proved that RR satisfies $\varepsilon$-LDP if $p=\frac{e^\varepsilon}{1+e^\varepsilon}$ \cite{LF-GDPR, AsgLDP}.
\subsection{Problem Definition}
\label{sec:problem}
In this paper, we consider an undirected graph with no additional attributes on nodes or edges.
An input graph is defined as $G=(V,E)$, where $V=\{v_1,...,v_n\}$ is the set of nodes, and $E\subseteq V \times V$ is the set of edges.
For each user $i$, $B_i=\{b_{i1},b_{i2},...,b_{in}\}$ is its adjacency bit vector, where $b_{ij}=1$ if the edge $(v_i,v_j)\in E$ and $b_{ij}=0$ otherwise.
The number of neighboring edges adjacent to one node $i$ is the node degree $d_i$, namely, $d_i=\sum_{j=1}^n b_{ij}$.
The data curator collects a degree sequence $seq=\{d_1, d_2,..., d_n\}$ from each local user and publishes the degree histogram $hist(G)$.
The degree distribution $dist(G)$ can easily be obtained from $hist(G)$ by counting each degree frequency.
Figure \ref{fig:degree_example} shows an example of degree sequence, degree histogram, and degree distribution, respectively.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{figures/degree_dist_example.pdf}
\caption{Example of degree distribution}
\label{fig:degree_example}
\end{figure}
We use two common measures to assess the accuracy of our algorithms. First, we use the mean squared error (MSE) to estimate the error between noisy histogram ${hist(G)}^\prime$ and original histogram $hist(G)$, as used in \cite{LF-GDPR}. Generally, the MSE can be computed as $MSE(hist(G), hist(G)^\prime)=\frac{1}{n}\sum_{i=1}^n(hist(G)_i- {hist(G)_i}^\prime)^2$, where $n$ is the number of users in a graph. Second, we compute the mean absolute error (MAE)\cite{willmott2005advantages} which can be represented by $MAE(D,D\prime)=\frac{1}{n}\sum_{i=1}^n|hist(G)_i- {hist(G)_i}^\prime)|$.
\section{Overview of Proposed Methods}
\label{sec:overview}
We aim to design a method for publishing the degree distribution that approximates the original distribution as possible while satisfying the strict Node-LDP. Our proposed methods support the following functions:
1) obtaining the optimal graph projection parameter $\theta$;
2) conducting the local graph projection privately;
3) publishing the degree distribution under Node-LDP.
We provide an overview of our solutions in Algorithm~\ref{alg:degree_distribution}.
First, a private parameter selection method is designed to get the optimal projection parameter (Section \ref{sec:projection_parameter}).
The curator collects randomized information from local users and helps them select the optimal projection parameter based on its global view.
To protect sensitive information from local users during communications, we propose one naive approach, PureLDP parameter selection, which adds noise into true values under Node-LDP.
However, this method adds too much noise to destroy the order information among different utility loss values, significantly influencing the selection accuracy.
Then, we propose an improved Crypto-assisted parameter selection method that combines LDP with cryptographic primitives for graph degree distribution.
Concretely, the utility loss of each local user is encrypted by order-preserving encryption (OPE) \cite{popa2013ideal} scheme where the numerical order in the plaintext domain will be preserved in the ciphertext domain.
To prevent the leakage of individual order information, we add one mask into encrypted values with Secure Aggregation technique \cite{bonawitz2017practical}.
Thus the data curator only sees an encrypted sum of each local utility loss values and learns no about individual information.
\begin{algorithm}[t]
\caption{Publishing the degree distribution}
\label{alg:degree_distribution}
\begin{algorithmic}[1]
\REQUIRE
Adjacency bit vectors $\{B_1,..., B_n\}$, \\
\qquad privacy budget $\varepsilon_1, \varepsilon_2, \varepsilon_3$
\ENSURE
A noisy degree distribution $dist(G)^\prime$
\STATE $\theta \leftarrow \mathtt{SelectParameter}$($\{B_1,...,B_N\}, \varepsilon_1$) // Sec. \ref{sec:projection_parameter} \\
/* User side. \qquad \qquad */
\FOR{each user $i\in \{1,2,...,n\}$}
\STATE $\hat{d_i} \leftarrow \mathtt{LocalProjection}$($B_i$, $\theta, \varepsilon_2$) // Sec. \ref{sec:local_projection}
\STATE $d_i^\prime \leftarrow \hat{d_i}$+Lap($\frac{2\theta}{\varepsilon_3}$)
\STATE User $i$ sends $d_i^\prime$ to data curator
\ENDFOR
/* Curator side. \qquad \qquad */
\STATE Curator collects all noisy degree $d_i^\prime$
\RETURN $dist(G)^\prime$
\end{algorithmic}
\end{algorithm}
Second, as soon as the projection parameter is decided, each user can execute the local projection (Section \ref{sec:local_projection}).
Compared with the Node-CDP, it is more difficult for each user to execute the local projection because of the limited local view of the entire graph.
We first give a baseline Node-level approach that is motivated by graph projection \cite{imola2020locally} with Edge-LDP.
In Node-level local projection, the node is the minimal unit and correlations among users are ignored.
It is easy to deploy but loses much information that significantly influences the utility.
Then we design an improved Edge-level local projection where each edge is the minimal unit during the projection.
The key challenge is that information leakage may happen via mutual edges among neighboring users
For example, others may know that the current degree is larger than or less than $\theta$ during the local projection.
We carefully design a randomized mechanism using Random Response \cite{warner1965randomized} to perturb each bit of projection operation vector.
Thus each user can distinguish its neighboring node degree is whether larger than $\theta$ or smaller than $\theta$.
Third, after finishing the local projection, each user perturbs its projected degree using the Laplace mechanism.
Here, the sensitivity is $2\theta$ since any change of one edge will make an effect on two node degrees.
Then, they send the noisy degree to the data curator.
The curator collects the degree sequence and publishes the degree histogram and degree distribution.
\section{Local Projection Methods}
\label{sec:local_projection}
\subsection{Node-level Local Projection}
\label{sec:node-level}
Local scenarios make projection operation more challenging, since no party owns the entire graph and local users cannot easily add or remove edges.
We propose a Node-level projection method where each node is the minimal unit, which is motivated by $\mathtt{GraphProjection}$ under Edge-LDP in \cite{imola2020locally}.
As presented in Algorithm \ref{alg:nodeprojection}, it inputs an adjacency bit vector and projection parameter $\theta$.
Each local user first counts the number of neighboring edges.
If node degree $d_i$ is larger than $\theta$, projected degree $\hat{d_i}$ will be directly set as $\theta$;
otherwise, $\hat{d_i}$ remains the original value.
\begin{algorithm}[h]
\caption{Node-level Local Projection}
\label{alg:nodeprojection}
\begin{algorithmic}[1]
\REQUIRE
Adjacency bit vector $B_i$=$\{b_{i1},...,b_{in}\}$,\\ \qquad projection parameter $\theta$
\ENSURE
$\theta$-bounded node degree $\hat{d_i}$
\STATE $d_i \leftarrow \sum_{j=1}^n b_{i,j}$
\IF{$d_i > \theta$}
\STATE $\hat{d_i} = \theta$
\ELSE
\STATE $\hat{d_i} \leftarrow d_i$
\ENDIF
\RETURN $\hat{d_i}$
\end{algorithmic}
\end{algorithm}
\textbf{Limitations.}
Although Node-level projection is easy to implement, it omits coarsely correlations among neighboring users, influencing the accuracy significantly.
For example, we have a simple graph with five nodes and some edges, as shown in Figure \ref{fig:node_level}.
The original histogram can be represented as $H_1$ = (0, 3, 1, 1, 0).
Assume that the projection parameter $\theta$ = 1, the projected degree sequence becomes $Seq_1$ = (1, 1, 1, 1, 1) and current histogram is $H_2$= (0, 5, 0, 0, 0) after Node-level projection.
We can compute the projection loss: MSE($H_1$, $H_2$)= $\frac{6}{5}$.
If correlations are considered, any change in mutual edges will update two neighboring adjacency bit vectors.
For example, if edge 2 and 3 are removed to bound all degrees, the degree sequence will become $Seq_2$= (1, 1, 1, 0, 1) and the degree histogram will be $H_3$= (1, 4, 0, 0, 0).
The projection loss can be computed: MSE($H_1$, $H_3$)= $\frac{4}{5}$.
Obviously, Node-level method loses more edge information, which significantly affects overall utility.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{figures/example_node-level.pdf}
\caption{Example of degree histogram}
\label{fig:node_level}
\end{figure}
Generally, we assume that the number of users in a graph is $n$, projection parameter is $\theta$, and original degree histogram is $H_1$=($h_1, h_2, ..., h_n$).
If there are $m$ nodes with degree larger than $\theta$, we can get the projected histogram $H_2$=($h_1,h_2,...,h_\theta+m,0,...,0$) using Node-level projection.
On the other hand, if there is a trusted third party that can use the edge removal method, the new histogram becomes $H_3$=($h_1+t_1,h_2+t_2,...,h_\theta+t_m,0,...,0$).
We can easily achieve $m=t_1+t_2+...+t_m$.
Assume that the general metric function $f(H_i,H_j)=\frac{1}{n}(\triangle_1^k+\triangle_2^k+...+\triangle_m^k)$, where $\triangle$ is each utility loss value.
When $k$ is equal to 1, this function is MAE; when $k$ is equal to 2 when it is MSE .
Then we can compute their projection loss, namely, $f(H_1, H_2$)=$\frac{m^k}{n}=\frac{1}{n}(t_1+t_2+...+t_m)^k$ and $f(H_1, H_3$)=$\frac{1}{n}(t_1^k+t_2^k+...+t_m^k)$.
Since $(t_1+t_2+...+t_m)^k \geq (t_1^k+t_2^k+...+t_m^k)$, where $k \geq 1$, we can get $f(H_1,H_2) \geq f(H_1,H_3)$.
Therefore, the result of Node-level projection method is not desirable.
\subsection{Edge-level Local Projection}
Base on above discussion, if we consider the correlation among users, more edge information will be reserved after the projection.
However, unlike Node-CDP where the trusted server can easily finish the projection with the global view, it is difficult for a local user to update the mutual edges.
The key challenge is that any change in the edges may leak individual sensitive information.
For example, if one node degree $d_i$ is larger than $\theta$, it will delete some edges.
At the same time, this user $i$ will send the messages to its neighboring users to update their adjacency bit vectors.
The message itself reveals that the current node degree may be larger than $\theta$.
Therefore, we further perturb the communication messages during the local graph projection.
\textbf{Security Assumptions.}
We assume that 1) the communication between neighboring users is perfectly anonymous, that's to say, the third party (e.g., server or third user) cannot know the communication exists or not;
2) the user does not reveal sensitive neighboring information to other users, for example, B will not tell C that A is one of its friends or not.
\begin{table}[h]
\caption{Randomized projection vector}
\label{tab:projection_probability}
\centering
\setlength{\tabcolsep}{5mm}{
\begin{tabular}{lrr}
\hline
Pr & 0 &1 \\
\hline
$\textless \theta$ & 1-x & x\\
$\geq \theta$ & 1-p & p \\
\hline
\end{tabular}}
\end{table}
\textbf{Algorithm.}
We propose the Edge-level projection based on Random Response (RR) \cite{warner1965randomized} and the edge is a minimal unit during the projection, as shown in Algorithm \ref{alg:edgeprojection}.
Privacy leakage may occur when the local projection is performed.
We carefully design a randomized mechanism to perturb each bit of projection bit vector.
There are two cases: node degree $d_i$ is larger than $\theta$;
$d_i$ is smaller than $\theta$.
The randomized mechanism guarantees that the data curator cannot distinguish current node degree is larger than $\theta$ or smaller than $\theta$.
Assume that the bit vector of the projection operation is $R_i={r_{i1},..., r_{id_i}}$, and the size of $R_i$ is $d_i$.
If $r_{ij}=1$, the corresponding edges in two neighbor lists will be removed; otherwise, they remain the same.
Ideally, we want to flip each bit of the projection bit vector with probability in Table~\ref{tab:projection_probability}, where $p=\frac{d_i-\theta}{d_i}$ and $x=0$. Obviously, when $x=0$, our randomized mechanism cannot satisfy the Node-LDP.
To satisfy the Node-LDP, we have the following inequation:
\begin{algorithm}[t]
\caption{Edge-level Local Projection}
\label{alg:edgeprojection}
\begin{algorithmic}[1]
\REQUIRE
Adjacency bit vector $B_i$=$\{b_{i1},...,b_{in}\}$,\\ \qquad projection parameter $\theta$, privacy budget $\varepsilon_2$
\ENSURE
$\theta$-bounded node degree $\hat{d_i}$
\STATE R$_i$=[0] $\times$ $\hat{d_i}$ // Record which edges will be deleted
\STATE $d_i \leftarrow \sum_{j=1}^n b_{i,j}$
\IF{$d_i \geq \theta$}
\STATE Randomly select $(d_i - \theta)$ bits from R$_i$ and set '1'
\FOR{each r$_{ij}$ $\in$ R$_i$}
\STATE\[
r_{ij}\prime=\left \{
\begin{array}{ll}
r_{ij} & w.p.\ \frac{\theta}{d_i} \\
1-r_{ij} & w.p.\ \frac{d_i-\theta}{d_i}
\end{array}
\right.
\]
\ENDFOR
\ELSE
\FOR{each r$_{ij} \in$ R$_i$}
\IF{$\frac{d_i-\theta}{d_i} \leq \frac{e^{\varepsilon_2}-1}{e^{\varepsilon_2}-e^{-\varepsilon_2}}$ }
\STATE\[
r_{ij}\prime=\left \{
\begin{array}{ll}
r_{ij} & w.p.\ 1-\frac{e^{-\varepsilon_2}(d_i-\theta)}{d_i} \\
1-r_{ij} & w.p.\ \frac{e^{-\varepsilon_2}(d_i-\theta)}{d_i}
\end{array}
\right.
\]
\ELSE
\STATE\[
r_{ij}\prime=\left \{
\begin{array}{ll}
r_{ij} & w.p.\ \frac{e^{\varepsilon_2}\theta}{d_i} \\
1-r_{ij} & w.p.\ \frac{d_i-e^{\varepsilon_2}\theta}{d_i}
\end{array}
\right.
\]
\ENDIF
\ENDFOR
\ENDIF
\FOR{each r$_{ij}$ $\in$ R$_i$}
\IF{$r_{ij} =1$}
\STATE $b_{ij}=0$ and $b_{ji}=0$
\ENDIF
\ENDFOR
\RETURN $\hat{d_i}$
\end{algorithmic}
\end{algorithm}
\begin{equation}
\label{equation:edge_level}
\left \{
\begin{array}{ll}
e^{-\varepsilon_2} \leq \frac{x}{p} \leq e^{\varepsilon_2} \\
e^{-\varepsilon_2} \leq \frac{1-x}{1-p} \leq e^{\varepsilon_2}
\end{array}
\right. \nonumber
\end{equation}
Then, we can bound the scope of $x$ as follows:
\begin{equation}
\left \{
\begin{array}{cc}
pe^{-\varepsilon_2} \leq x \leq pe^{\varepsilon_2} \\
(p-1)e^{\varepsilon_2}+1 \leq x \leq (p-1)e^{-\varepsilon_2}+1
\end{array}
\right.\nonumber
\end{equation}
When $d_i \textless \theta$, we want to preserve more edges during projection, that is to say, the number of `1' in projection bit vector is as small as possible. Thus we have
\begin{equation}
x=\left \{
\begin{array}{lc}
pe^{-\varepsilon_2}, & pe^{-\varepsilon_2} \geq (p-1)e^{\varepsilon_2}+1 \\
(p-1)e^{\varepsilon_2}+1, & pe^{-\varepsilon_2} \textless (p-1)e^{\varepsilon_2}+1
\end{array}
\right.\nonumber
\end{equation}
After randomizing the bits of the projection bit vector, each user updates the adjacency bit vector according to randomized bit vector (Line 19).
Then, local users count the number of edges and obtain the bounded degree $\hat{d_i}$.
\section{Projection Parameter Selection}
\label{sec:projection_parameter}
\subsection{PureLDP Selection}
To find the optimal projection parameter $\theta$, we make use of the global view of the data collector.
After receiving the messages from local users, the data curator can help local users select the optimal parameter with the best utility from candidates {1,2,...,$K$}.
We design a utility loss function to evaluate each candidate parameter $k$. Our utility loss function has two parts, as shown in Equation \ref{equation:loss_function}, which includes projection loss and utility loss from publishing degree distribution.
The first part is projection loss $E_T=\sum_{i=1}^n\{d_i-k|v_i\in V, d_i>k\}$.
If these projection loss values are sent to the server directly, the data collector can easily deduce the sensitive information.
We guarantee each projection loss value under Node-LDP using the Laplace mechanism and sensitivity is $(n-1-k)$, as shown in Lemma \ref{lemma:sensitivity}.
The second part captures the adding noise while publishing the degree distribution.
We use the Laplace mechanism to guarantee differential privacy and the sensitivity is $2k$ in Node-LDP.
Since the bias of Laplace mechanism is zero, its MSE is equal to the variance, namely, $E_D=n.2(\frac{2k}{\varepsilon_3})^2=\frac{8nk^2}{\varepsilon_3^2}$.
\begin{equation}
\label{equation:loss_function}
F(k)=E_T+E_D,
\end{equation}
\begin{center}
$E_T=\sum_{i=1}^n|\{d_i-k|v_i\in V, d_i>k\}|$
\end{center}
\begin{center}
$E_D=n.2(\frac{2k}{\varepsilon_3})^2=\frac{8nk^2}{\varepsilon_3^2}$
\end{center}
\begin{lemma}
\label{lemma:sensitivity}
For any projection loss $|d_i-\hat{d_i}|$ and $|d_i-\hat{d_i}|^\prime$, we have
\begin{center}
$||d_i-\hat{d_i}|-|d_i-\hat{d_i}|^\prime|_1 \leq (n-1-k)$
\end{center}
\end{lemma}
\emph{Proof of Lemma \ref{lemma:sensitivity}:} Given the graph projection parameter is $k$, for each node degree $d_i$, if $d_i \leq k$, projected node degree $\hat{d_i}$ will remain the original value, namely, $\hat{d_i}=d_i$;
otherwise, $\hat{d_i}=k$. Thus, we have
\begin{equation}
|d_i-\hat{d_i}|=\left \{
\begin{array}{lc}
d_i-\theta, & d_i \textgreater k \\
0, & d_i \leq k
\end{array}
\right.\nonumber
\end{equation}
Since the maximum node degree is $(n-1)$, the projection loss value is bounded by ($n-1-k$).
\begin{algorithm}[t]
\caption{PureLDP parameter selection}
\label{alg:ldp_enhanced}
\begin{algorithmic}[1]
\REQUIRE
Adjacency bit vectors $\{B_1,...,B_n\}$, privacy budget $\varepsilon_1$
\ENSURE
Projection parameter $\theta$
\FOR{each integer $k \in \{1,2,...,K\}$}
\STATE /* User side. \qquad \qquad */
\FOR{each user $i\in \{1,2,...,n\}$}
\STATE $\hat{d_i} \leftarrow$ $\mathtt{LocalProjection}$($B_i$, $k$) // Sec. \ref{sec:local_projection}
\STATE $d_i \leftarrow \sum_{j=1}^n b_{i,j}$
\STATE $E_{T_{k,i}} \leftarrow |d_i-\hat{d_i}|$+Lap($\frac{n-1-k}{\varepsilon_1/K}$)
\STATE User $i$ sends $E_{T_{k,i}}$ to Curator
\ENDFOR \\
/* Curator side. \qquad \qquad */
\STATE $E_{T_{k}} \leftarrow \sum_{i=1}^n E_{T_{k,i}}$
\STATE $\theta \leftarrow k$ when $(E_{T_{k}}+E_D)$ is minimum
\ENDFOR \\
\RETURN $\theta$
\end{algorithmic}
\end{algorithm}
\textbf{Algorithm.} Algorithm \ref{alg:ldp_enhanced} presents the formal description of PureLDP parameter selection. It takes as input a graph $G$ that is represented as bit vectors $b_1,...b_n$, the privacy budget $\varepsilon_1$, and the size of candidate parameter $K$. For each candidate parameter $k$, the original graph is first projected to $k$-bounded graph using the local graph projection method (in Section \ref{sec:local_projection}).
Then, each user computes the projection loss and adds the Laplace noise with the sensitivity ($n-1-k$).
After collecting all noisy projection loss values, the data curator computes the sum of projection loss and error from publishing the degree distribution.
Finally, the optimal parameter $\theta$ is found when the overall utility loss is the minimum and sends this $\theta$ to every local user.
\textbf{Limitation.} Observing Algorithm \ref{alg:ldp_enhanced}, we can easily find that the accuracy of the PureLDP method depends on the parameter $K$. The $K$ is larger, and more noise will be added into the true projection loss values, which significantly the accuracy of the optimal projection parameter $\theta$.
For instance, two original projection loss values have the relationship: $E_{T_1}<E_{T_2}$.
But differential privacy noise may destroy this relationship, and the data curator may find that $E_{T_1}^\prime > E_{T_2}^\prime$.
\subsection{Crypto-assisted Selection}
Our goal is that each projection loss can be protected, and the relative relationship between different values can be preserved at the same time.
Order-preserving encryption (OPE) scheme \cite{agrawal2004order, boldyreva2011order} can achieve this idea that
the $i$-th data in the plaintext domain is transformed to the $i-$th data in the ciphertext domain, so the numerical order among plaintexts is preserved among ciphertexts.
Intuitively, if each projection loss value is encrypted by OPE scheme, the numerical order among different summation of encrypted projection loss values can be preserved.
At the same time, it will avoid adding much noise that makes a significant effect on the accuracy of results.
\textbf{OPE Schemes.}
\label{sec:ope}
There are many existing works related to OPE scheme.
For example, Popa et al. \cite{popa2013ideal} proposed an interactive OPE scheme between the client and the server, which allows the encrypted state to update over time as the new values are inserted.
The server organizes the encrypted values by maintaining a binary search tree, namely, OPE-tree.
To reduce the high cost of the encryption, Kerschbaum et al. \cite{kerschbaum2014optimal} designed a more efficient OPE scheme that uses a dictionary to keep the state and thus does not need to store too much data.
Roche et al. \cite{roche2016pope} proposed an alternative approach to optimize the heavy insertion of OPE schemes. It is very efficient at insertion and has a lower communication cost, but it provides only a partial order.
However, these works may not be directly used for our work.
On the one hand, existing OPE schemes are limited to one client and one server, which is totally different from our setting where there are many local users and one data collector.
To be specific, in OPE schemes, the client owns all data and knows the numerical order of the original data.
In contrast, in our local setting, there is no trusted server and each user just has the local view of the entire graph.
On the other hand, the data curator obtains the numerical order of different encrypted summations of projection loss;
at the same time, it knows the relative order of each local projection loss value.
This relative order will leak individual sensitive information, i.e., which is the high-degree node or which is a low-degree node.
\begin{algorithm}[t]
\caption{Crypto-assisted parameter selection}
\label{alg:ope_enhanced}
\begin{algorithmic}[1]
\REQUIRE
Adjacency bit vectors $\{B_1,...B_n\}$, \\
\qquad a security parameter $\lambda$
\ENSURE
Projection parameter $\theta$
\FOR{each integer $k \in \{1,2,...,K\}$}
\STATE /* User side. \qquad \qquad */
\FOR{each user $i\in \{1,2,...,n\}$}
\STATE $\hat{d_i} \leftarrow$ $\mathtt{LocalProjection}$($B_i$, $k$) // Sec. \ref{sec:local_projection}
\STATE $d_i \leftarrow \sum_{j=1}^n b_{i,j}$
\STATE $(a_i,b_i) \leftarrow \mathtt{KeyGeneration}$($\lambda$)
\STATE noise $\leftarrow randint(0,a-1)$
\STATE r $\leftarrow$ $\mathtt{PRG}$($seed$)
\STATE $mask=\sum_{j=i+1}^{n-1}r_{i,j}-\sum_{j=1}^{i-1}r_{i,j}$
\STATE $Enc_{T_{k,i}} \leftarrow a_i*|d_i-\hat{d_i}|+b_i+$noise+mask
\STATE User $i$ sends $Enc_{T_{k,i}}$ to Curator
\ENDFOR \\
/* Curator side. \qquad \qquad */
\STATE $Enc_{T_{k}} \leftarrow \sum_{i=1}^n Enc_{T_{k,i}}$
\STATE $\theta \leftarrow k$ when $(Enc_{T_{k}}+E_D)$ is minimum
\ENDFOR \\
\RETURN $\theta$
\end{algorithmic}
\end{algorithm}
To handle above challenges, we propose an Crypto-assisted parameter selection method, as presented in Algorithm \ref{alg:ope_enhanced}.
First, we choose a linear OPE scheme \cite{6253544} that may be extended to the multi-user setting, namely, $f(x)=a*|d_i-\hat{d_i}|+b+noise$.
Here the parameter $a$ and $b$ are kept secret from the data curator.
This linear OPE scheme consists of the secret key ($a,b$) and the randomly chosen noise $\in [0, a-1]$.
The security of this linear expression is not perfect since the secret coefficient $(a,b)$ is reused many times, which can be guessed by the attacker.
To avoid this, we generated the different secret key $(a_i,b_i)$ for each local user $i$ using the function KeyGeneration($\lambda$), where $\lambda$ is a security parameter, which can reduce the possibility of revealing the secret key.
Second, though the improved OPE scheme can be used for multi-user setting, the numerical order of the individual sensitive information is still revealed to the data curator.
Motivated by the secure aggregation \cite{bonawitz2017practical}, we add one mask into the encoding values of the OPE scheme.
An arbitrary user $i$ first determine the shared seeds with the rest other $n-1$ users by the Diffie-Hellman key exchange agreement \cite{bresson2002dynamic}.
Then the user generates the random numbers $r$ with the common seeds by the pseudorandom generator (PRG) \cite{blum1984generate}.
And the mask of each user's data can be aggregated (line 7).
Finally, the data curator collects all encrypted projection loss and aggregates them.
The padding masks can be canceled with each other but the actual projection loss value will not be revealed.
Since individual projection loss is added with random pairwise masks, the values that the data curator received look uniformly random.
That's to say, these pairwise masks protect all information about each user.
What's more, OPE scheme guarantees the security guarantees for their sum.
\section{Analysis and Discussions}
\label{sec:analysis discussion}
\textbf{Privacy Budget Allocation.}
As shown in Algorithm \ref{alg:degree_distribution}, there are three kinds of privacy budgets.
Our goal is to find the optimal privacy allocation scheme with the best accuracy.
Without loss of generality, we assume that the overall privacy budget is $\varepsilon$, $\varepsilon_3=\alpha\varepsilon$, and $\varepsilon_1+\varepsilon_2=(1-\alpha)\varepsilon$.
For inner privacy budget allocation of local graph projection, we distribute the same privacy budget for the projection parameter selection and executing the local graph projection, namely, $\varepsilon_1=\varepsilon_2$.
We find the optimal $\alpha$ with the least utility loss by conducting many experiments for different cases, as shown in Table \ref{tab:privacy allocation}.
And we use the optimal $\alpha$ for each case in the next experiments.
\textbf{Selection of Parameter $K$.}
In Algorithm \ref{alg:ldp_enhanced} and Algorithm~\ref{alg:ope_enhanced}, the parameter $K$, namely, the size of candidate pool, plays a significant role in the tradeoff between utility and privacy.
The size $K$ is larger, the more noise will be added by the PureLDP parameter selection and time overhead are higher.
Similarly, the running time of Crypto-assisted selection method will be higher.
But if the $K$ becomes smaller, the optimal projection parameter $\alpha$ is not covered possibly.
We conduct extensive experiments and find the optimal parameter $\alpha$ for every case, as shown in Table \ref{tab:hyper-parameter K}.
In our paper, we use $K=50$ that is ample to cover the optimal parameter $\alpha$ of different cases.
\begin{table}[t]
\caption{optimal privacy allocation scheme $\alpha$}
\centering
\label{tab:privacy allocation}
\setlength{\tabcolsep}{2mm}{
\begin{tabular}{lcccc}
\hline
$\varepsilon$ & Ca-HepPh & Cit-HepPh & Twitter & Com-DBLP \\\hline
0.5 & 0.895 & 0.927 & 0.945 & 0.945 \\
1 & 0.944 & 0.937 & 0.949 & 0.947 \\
1.5 & 0.901 & 0.940 & 0.944 & 0.948 \\
2 & 0.948 & 0.946 & 0.947 & 0.937 \\
2.5 & 0.944 & 0.922 & 0.948 & 0.943 \\
3 & 0.944 & 0.948 & 0.941 & 0.940 \\\hline
\end{tabular}}
\end{table}
\begin{table}[t]
\caption{optimal parameter $\theta$}
\centering
\label{tab:hyper-parameter K}
\setlength{\tabcolsep}{2mm}{
\begin{tabular}{lcccc}
\hline
$\varepsilon$ & Ca-HepPh & Cit-HepPh & Twitter & Com-DBLP \\\hline
0.5 & 3 & 4& 18& 13 \\
1 & 9 & 7& 31& 17 \\
1.5 & 15& 10& 41& 20 \\
2 & 19& 12& 43& 23 \\
2.5 & 24& 15& 45& 25 \\
3 & 26& 18& 48& 27 \\\hline
\end{tabular}}
\end{table}
\textbf{Time Complexity.}
As shown in Table \ref{tab:running time}, we conclude the running time complexity of different combinations theoretically, $|V|$ and $|E|$ represents the number of nodes and edges respectively.
Node-level local projection method transforms each node degree into $\theta$-bounded degree directly, which takes time $O(|V|)$.
In contrast, Edge-level local projection method needs to traverse each edge for each node, resulting an $O(|V|.|E|)$ running time.
PureLDP parameter selection method selects the optimal parameter $\theta$ from $K$ candidates and for each candidate $k$, each user has to compute the projection loss, which takes time at most $O(K.|V|)$.
By comparison, for each candidate parameter $k$ of Crypto-assisted selection method, each user has to communicate with the other $(|V|-1)$ users to determine the seed, resulting an $O(K.|V|^2)$ running time overhead.
\begin{table}[htb]
\caption{running time complexity}
\centering
\label{tab:running time}
\setlength{\tabcolsep}{1mm}{
\begin{tabular}{lcc}
\hline
& PureLDP & Crypto-assisted \\\hline
Node-level & \small$O(|V|+K.|V|)$ & \small$O(|V|+K.|V|^2)$ \\
Edge-level & \small$O(|V|.|E|+K.|V|)$ & \small$O(|V|.|E|+K.|V|^2)$ \\
\hline
\end{tabular}}
\end{table}
\begin{figure*}[t]
\centering
\subfigure{
\begin{minipage}[t]{\linewidth}
\centering
\includegraphics[width=0.7\linewidth]{figures/legend_line.pdf}
\end{minipage}
}%
\qquad
\subfigure[Ca-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mse_epsilon_caheph.pdf}
\label{fig:mse_epsilmse_epsilon_caheph}
\end{minipage}
}%
\subfigure[Cit-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mse_epsilon_citheph.pdf}
\label{fig:mse_epsilon_citheph}
\end{minipage}%
}
\subfigure[Twitter]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mse_epsilon_twitter.pdf}
\label{fig:mse_epsilon_twitter}
\end{minipage}%
}
\subfigure[Com-DBLP]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mse_epsilon_dblp.pdf}
\label{fig:mse_epsilon_dblp}
\end{minipage}%
}
\qquad
\subfigure[Ca-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mae_epsilon_caheph.pdf}
\label{fig:mae_epsilon_caheph}
\end{minipage}
}%
\subfigure[Cit-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mae_epsilon_citheph.pdf}
\label{fig:mae_epsilon_citheph}
\end{minipage}%
}
\subfigure[Twitter]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mae_epsilon_twitter.pdf}
\label{fig:mae_epsilon_twitter}
\end{minipage}%
}
\subfigure[Com-DBLP]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/mae_epsilon_dblp.pdf}
\label{fig:mae_epsilon_dblp}
\end{minipage}%
}
\centering
\caption{The MSE and MAE of algorithms on different graphs}
\label{fig:mse_mae}
\end{figure*}
\textbf{Security Analysis.} Publishing the degree distribution in Algorithm \ref{alg:degree_distribution} is under the following privacy guarantee.
\begin{lemma}
\label{lemma:privacy}
Publishing the degree distribution satisfies $(\varepsilon_1/K+\varepsilon_2+\varepsilon_3)$-Node-LDP.
\end{lemma}
\emph{Proof of Lemma \ref{lemma:privacy}:} In Algorithm \ref{alg:degree_distribution}, SelectParameter(.) (Line 1) uses the Laplace with privacy budget $\varepsilon_1/K$, $K$ is the number of candidate parameters.
Executing the local projection (Line 3) uses the Random Response mechanism and satisfies Node-LDP for $\varepsilon_2$.
And publishing the distribution with Laplace Mechanism using $\varepsilon_3$.
According to the post-processing theorem and composition property, Algorithm \ref{alg:degree_distribution} satisfies $(\varepsilon_1/K+\varepsilon_2+\varepsilon_3)$-Node-LDP.
\section{Experimental Evaluation}
\label{sec:experiment}
In this section, we would like to answer the following questions:
\begin{itemize}
\item What is the tradeoff between utility and privacy of our proposed methods?
\item How do parameters (i.e., $\varepsilon_1, \varepsilon_2, \varepsilon_3$ and $K$) affect the results?
What are the results of different privacy budget allocation schemes?
\item How much running time do our proposed algorithms take?
\end{itemize}
\subsection{ Datasets and Setting}
Our experiments run in python on a server with Intel Core i9-10920X CPU, 256GB RAM running Ubuntu 18.04 LTS. We use four real-world graph datasets from SNAP \cite{snapnets}, which are also used in \cite{day2016publishing,LF-GDPR}.
And we preprocess all graph datasets to be undirected and symmetric graphs.
Table \ref{tab:datasets} presents more details about every graph dataset, including the number of nodes $|V|$, the number of edges $|E|$, and the number of edges after preprocessing $|E^\prime|$ after preprocessing.
In all experiments, we vary the privacy budget $\varepsilon$ from 0.5 to 3. By default, we set hyper-parameter $K$=50 as we discussed above.
All of our experimental results are the average values computed from 20 runs.
We use `PureLDP', CryptoAssisted', `NodeProj' and `EdgeProj' to represent PureLDP parameter selection, Crypto-assisted parameter selection, Node-level local graph projection and Edge-level local graph projection respectively.
Thus we have four different combinations to publish the degree distribution.
\begin{table}[htb]
\caption{details of graph datasets}
\centering
\label{tab:datasets}
\setlength{\tabcolsep}{2mm}{
\begin{tabular}{lrrr}
\hline
Graph & $|V|$ & $ |E|$ & $ |E|$ $^\prime$ \\\hline
Ca-HepPh & 12,008 & 118,521 & 474,020 \\
Cit-HepPh & 34,546 & 421,578 & 843,156\\
Twitter & 81,306 & 1,768,149 & 3,536,298 \\
Com-DBLP & 317,080 & 1,049,866 & 2,099,732 \\\hline
\end{tabular}}
\end{table}
\begin{figure*}[t]
\centering
\subfigure[Ca-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/percent_epsilon_caheph.pdf}
\end{minipage}
}%
\subfigure[Cit-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/percent_epsilon_citheph.pdf}
\end{minipage}%
}
\subfigure[Twitter]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/percent_epsilon_twitter.pdf}
\end{minipage}%
}
\subfigure[Com-DBLP]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/percent_epsilon_dblp.pdf}
\end{minipage}%
}
\centering
\caption{The MSE on different graphs, varying $\alpha$}
\label{fig:alloaction}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure{
\begin{minipage}[t]{\linewidth}
\centering
\includegraphics[width=0.7\linewidth]{figures/legend_bar.pdf}
\end{minipage}
}%
\qquad
\subfigure[Ca-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/time_epsilon_caheph.pdf}
\label{fig:time_epsilon_caheph}
\end{minipage}
}%
\subfigure[Cit-HepPh]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/time_epsilon_citheph.pdf}
\label{fig:time_epsilon_citheph}
\end{minipage}%
}
\subfigure[Twitter]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/time_epsilon_twitter.pdf}
\label{fig:time_epsilon_twitter}
\end{minipage}%
}
\subfigure[Com-DBLP]{
\begin{minipage}[t]{0.238\linewidth}
\centering
\includegraphics[width=\linewidth]{figures/time_epsilon_dblp.pdf}
\label{fig:time_epsilon_dblp}
\end{minipage}%
}
\centering
\caption{The runtime on different graphs}
\label{fig:runtime}
\end{figure*}
\subsection{Relation between $\varepsilon$ and MSE, MAE}
As shown in Figure \ref{fig:mse_mae}, the utility of each combination method becomes better as the privacy budget $\varepsilon$ increases.
We can find that `CryptoAssisted+EdgeProj' method always performs the best in most cases, while the results of `PureLDP+NodeProj' method are always the worst.
To be specific, the MSE of `CryptoAssisted+EdgeProj' method is less than that of `PureLDP+NodeProj' by up to 87.2\% on Twitter when $\varepsilon=2.5$.
The MAE of `CryptoAssisted+NodeProj' method is larger than that of `CryptoAssisted+EdgeProj' method by up to 66.4\% in Twitter when $\varepsilon=3$.
The reason that `CryptoAssisted+EdgeProj' method sometimes performs not the best in terms of MAE when $\varepsilon=0.5$ is because our utility loss function uses the MSE as the evaluation metric, which makes a little influence on results of MAE, particularly when $\varepsilon$ is very small.
Also, the MAE of Edge-level local graph projection is much smaller than that of Node-level local graph projection in most cases, which may be a little different from our theoretical analysis in \ref{sec:node-level}.
The main reason is that our utility loss function (Equation \ref{equation:loss_function}) uses the MSE as the evaluation metric, which is more friendly to Edge-level method.
Overall, our proposed `CryptoAssisted+EdgeProj' method improves our baseline `PureLDP+NodeProj' approach for publishing the degree distribution under Node-LDP.
\subsection{Impact of privacy budget allocation}
To further estimate the influence of the privacy allocation scheme on the overall utility, we compare the best $\alpha$ with other three constant $\alpha$, including 0.3, 0.6, and 0.9.
We present the MSE results of different $\alpha$ on different graph datasets in Figure \ref{fig:alloaction}.
We can observe that the best $\alpha$ owns the lowest MSE against the other allocation schemes in most cases.
On the other hand, with the increase of the overall privacy budget $\varepsilon$, the MSE value is decreasing.
Thus surplus privacy budget can be allocated to the final publishing the degree distribution, which is roughly consistent with our best $\alpha$ in Table \ref{tab:privacy allocation}, namely, $\varepsilon_3$ for publishing degree distribution is approximately equal to the overall privacy budget $\varepsilon$.
\subsection{Analysis on running time}
Finally, we compare the running time overhead of our proposed methods, as shown in Figure \ref{fig:runtime}.
We can see that the running time of `CryptoAssisted+EdgeProj' method is much larger than that of `PureLDP+NodeProj' method.
This is mainly because Edge-level projection method needs to traverse each edge of every node and Crypto-assisted parameter selection method has $n$ users to communicate in pairs, which is in line with our theoretical analysis in Section \ref{sec:analysis discussion}.
The difference between `CryptoAssisted+EdgeProj' method and `PureLDP+NodeProj' method is larger on Twitter.
This is because Twitter has more edges than other graphs, as described in Table \ref{tab:datasets}, which results in higher computation and communication overhead.
\section{Related Work}
\label{sec:related work}
There are many existing works related to Node-LDP and Edge-LDP.
\textbf{Node-CDP.}
There have been many prior research works related to Node differential privacy (Node-DP).
For example, a handful of graph algorithms \cite{blocki2013differentially, kasiviswanathan2013analyzing,day2016publishing, raskhodnikova2016lipschitz} have been designed for publishing the degree distribution by proposing different graph projection methods.
For instance, the truncation method \cite{kasiviswanathan2013analyzing} removes all nodes with the degree over $\theta$.
Edge-removal approach \cite{blocki2013differentially} traverses all edges in an arbitrary order and removes each edge connected to a node with a degree more than $\theta$.
Edge-addition method \cite{day2016publishing} traverses the edges in a stable order and inserts each edge correlated to node with degree over $\theta$.
However, the existing projection methods are only designed for Node-CDP and are not viable in Node-LDP.
\textbf{Edge-LDP.} Since there is no need for a trusted server and a large amount of valuable information resides in a decentralized social network, LDP is becoming increasingly popular in privacy protection of graph analysis. Existing works focus on various graph statistics, such as degree distribution (or histogram)\cite{LF-GDPR}, subgraph counting (e.g., k-clique, k-star, k-triangle) \cite{sun2019analyzing,imola2020locally}, synthetic graph generation \cite{qin2017generating,zhang2018two}, publishing attributed graph\cite{AsgLDP,jorgensen2016publishing} etc.
For instance, Ye et al. \cite{LF-GDPR} proposes a LDP-enabled graph metric estimation framework for general graph analysis. In \cite{imola2020locally}, subgraph counting is protected locally by a more sophisticated algorithm that uses an additional round of interaction between individuals and data curator.
To strike a balance between noise added to satisfy LDP and information loss from a coarser granularity, Qin et al. \cite{qin2017generating} designs a novel multi-phase approach to synthetic decentralized social graph generation.
However, these existing works are all based on Edge-LDP which provides a weaker privacy guarantee than our work under Node-LDP.
\section{Conclusion}
\label{sec:conclusion}
To conclude, we first discuss the motivation of publishing the graph statistics under Node-LDP, and present the challenges of finishing the projection locally.
We design two methods for executing local graph projection: Node-level local projection and Edge-level local projection.
Also, we propose two methods for the projection parameter selection: PureLDP parameter selection and Crypto-assisted parameter selection.
Theoretical and experimental analysis verify the utility and privacy achieved by our proposed work.
|
1,116,691,500,514 | arxiv | \section{Introduction}
Recently, technological advances in the experimental control of ultracold gases \cite{ug1,ug2}, trapped ions \cite{tri}, and nitrogen-vacancy centres in diamonds \cite{nv}, allowed to probe the time evolution of isolated quantum systems. Since, in such systems, the time evolution is unitary, no information about the initial state is lost. However, most often, this information spreads over the whole system, so that, at long times, it is challenging to recollect it. The origin of this behavior lies in the Eigenstate Thermalization Hypothesis (ETH) \cite{eth1,eth2,eth3}, that, qualitatively speaking, states that, in the thermodynamic limit, the expectation value of local observables over any eigenstate with finite energy density can be well approximated by the average over a properly defined thermal density matrix. In this sense, most isolated systems (assuming ETH) thermalize. There are exceptions to this paradigm \cite{violation}. For example, in the Fibonacci chain describing Rydberg atoms\cite{fibo,fibo2}, most eigenstates do follow ETH, while some do not. This phenomenon opened the field of ``quantum many body scars" \cite{scars}. A stronger violation of ETH is provided by many-body systems that have an extensive amount of local or quasi-local conserved quantities. In this case, in fact, the local information stored in the initial wavefunction is preserved by the time evolution. Consequently, systems exhibiting such a behavior can have interesting applications in the field of quantum information \cite{mblquantuminfo}. A first class of systems with quasi-local conserved quantities are many-body localized systems \cite{mbl1,mbl2,mbl3,mbl4,mbl5}. In this case, the quasi-local integrals of motion arise due to real space localization and are robust with respect to weak perturbations. In these systems, there is no simple guideline for building a sensitive effective density matrix for the local observables. A second class of such systems is given by the so-called integrable models \cite{integrable}. In this case, while the violation of ETH is not stable with respect to generic perturbations \cite{thermalgge1,thermalgge2,thermalgge3}, it is indeed possible to build a statistical ensemble capturing the long time expectation value of the local observables. Such an ensemble is called generalized Gibbs Ensemble (GGE) \cite{gge1,gge2,gge3,gge4,gge5,gge6,gge7,gge8,gge9,gge10}. Conceptually speaking it is obtained by maximizing the entropy while taking into account the constraints posed by the local conserved quantities. From the formal point of view, the GGE density matrix could be an exceptionally useful tool, since concepts such as non-equilibrium phase transitions \cite{nept1,nept2,nept3,nept4,nept5,nept6,nept7,nept8,nept9,nept10,nept11} (non-analytical dependencies of long time expectation values as a function of the quench parameter) in integrable systems could be made universal, in this framework, in the very same way transitions in equilibrium are described within the canonical ensemble. However, GGE density matrices are in general difficult to obtain and no systematic link between them and non-equilibrium phase transitions has been performed.\\
\begin{figure}[h!]
\begin{center}
\includegraphics[width=\columnwidth]{Fig1.pdf}
\caption{Bottom: Density plot of $\bar{\mathcal{M}}$ as a function of $\delta_0$, $\delta_1$ for a sudden quench and the corresponding typical effective band structure $\xi_{\nu,k}$ (see text). Top: plot of $\bar{\mathcal{M}}$ as a function of $\delta_1$ for $\delta_0=2w$. Here, $\delta_\mu$ is in units $w$.}
\label{fig:1}
\end{center}
\end{figure}
In this article, we begin to address this issue in two paradigmatic cases: The Su-Schrieffer-Heeger (SSH) model \cite{ssh1,ssh2}, that represents a starting point for the study of topological phases of matter and the appearance of fractional charges in one dimension\cite{im1,im2,im3,im4,im5,im6,im7,im8,im9,im10,im11}, and the transverse field one-dimensional quantum Ising (QI) model, whose simplicity has opened the way to countless insights in the theory of quantum quenches\cite{calabreseefft,heyl,isi3}. Moreover, the QI model can be mapped onto the Kitaev chain and hence encodes the physics of the so called Majorana bound states\cite{majo}. First, we consider the steady state of a quenched Su-Schrieffer-Heeger (SSH) model. Quenches from an initial Hamiltonian with a hopping imbalance $\delta_0$ to a final one with $\delta_1$ are considered. In the thermodynamic limit, non-analyticities in observables such as the amount of dimerization $\bar{\mathcal{M}}$ (see below) can occur as a function of $\delta_\mu$ ($\mu=0,1$), signaling {non-equilibrium} quantum phase transitions (QPT). These transitions occur at the same parameter values of $\delta$ where also in equilibrium QPTs (non-analyticities in $\bar{\mathcal{M}}$ in dependence of the hopping imbalance) can be found, but there only at \emph{zero temperature}. In this regard this finding is remarkable, because the quench injects energy into the system~\cite{calabreseefft}, and is in accordance with the findings described in \onlinecite{Moessner}. As we will demonstrate, the GGE density matrix is equivalent to a {grand canonical density matrix} of free fermions, with effective Hamiltonian $\bar{H}$, at finite temperature. Interestingly, tuning the quench parameters, an effective metal-insulator transition (MIT) in $\bar{H}$ is achieved in concurrence with the non-equilibrium QPTs. The generic scenario is summarized in Fig.~\ref{fig:1}, which shows $\bar{\mathcal{M}}$ for a sudden quench $\delta_0\to\delta_1$ and the typical band structure of $\bar{H}$ occurring in each region of the parameters space spanned by the pre-quench and post-quench value of the gap $\delta_0$ and $\delta_1$, respectively. As we will show, the nature of this effective MIT explains the robustness of the non-equilibrium QPT against the initial preparation of the system and finite-duration quench protocols. The robustness of the non-analytic signatures and the effective MIT render this model a very promising candidate for experimental investigations. We also show that when the model is perturbed in such a way that the GGE does not provide a good description of the long time dynamics, the non-equilibrium QPT is washed out. With this respect, we analyse finite size systems and, in a model which is very similar to the SSH model, non-integrable interaction effects. We then consider quantum quenches in the transverse field QI model. Here, again, the entropy shows kinks as a function of the quench parameter in correspondence to the gapless points. As in the case of the SSH model, this non-equilibrium phase transition occurs together with an effective MIT in the GGE density matrix.\\
Our results suggest that the presence of an effective MIT in the GGE density matrix in connection to an equilibrium QPT leads to a non-equilibrium QPT.\\
The outline of the article is the following. In Sec.II, we inspect the quantum quench dynamics characterizing the SSH model and, in Sec.III, we discuss the same physics in the context of the transverse field QI model. Finally, in Sec.IV, we draw our conclusions.
\section{Quantum quench in the SSH model}
\subsection{Sudden quench}
The momentum space Hamiltonian for the quenched SSH model (on a finite ring of length $L$ with $\mathcal{N}$ unit cells and lattice constant set to one in the following) is given by \cite{ssh2} $H(t)=\sum_k \Psi^\dag_k \left\{\sigma_x \left[w+w\cos (k)+\delta(t)\right]+w\sigma_y \sin(k)\right\}\Psi_k$. Here, $\Psi^\dagger_k=(c^\dagger_{k,A},c^\dagger_{k,B})$ is a Fermi spinor, $A$ and $B$ the two sublattice labels and $k=2\pi j/\mathcal{N}$ with $|j|\leq\mathcal{N}$. Furthermore, $\sigma_{i}$ are Pauli matrices and $w$ is the hopping energy. The hopping imbalance term $\delta(t)$, which in equilibrium determines the gap, encodes the quench details: In most of the paper a sudden change $\delta(t)=\delta_0\theta(-t)+\delta_1\theta(t)$ - with $\theta(t)$ the Heaviside step function - is considered. The SSH Hamiltonian is diagonalized as
\begin{equation}
\label{eq:Hlat}
H_{\mu}=\sum_{k}\epsilon_{\mu,k}\left[d^{\dagger}_{\mu,c,k}d_{\mu,c,k}-d^{\dagger}_{\mu,v,k}d_{\mu,v,k}\right]
\end{equation}
with $\mu=0$ ($\mu=1$) for $t<0$ ($t>0$), $d_{\mu,\nu,k}$ fermionic operators for the $\nu=c,v$ bands and $\epsilon_{\mu,k}=\sqrt{\delta_\mu^2+2(w^2+w\delta_{\mu})[1+\cos(k)]}$. In the initial state ($t<0$) the system is prepared in the ground state $|G_0\rangle$ of $H_0$.\\
\noindent In the thermodynamic limit, the quantum average of local observables $\mathcal{O}(t)=\langle G_0| O(t)|G_0\rangle$ approaches a steady value $\bar{\mathcal{O}}=\mathcal O(t\to\infty)$ with a typical $\propto t^{-1}$ power-law decay (not shown). Since the system is integrable, this steady value can also be obtained as the trace $\bar{\mathcal{O}}=\langle\mathcal{O}\rangle\equiv\mathrm{Tr}\{O(0)\rho_{G}\}$ over the GGE density matrix \cite{gge1} constructed via the conserved charges $N_{\nu,k}=d^{\dagger}_{1,\nu,k}d_{1,\nu,k}$ as
\begin{equation}
\label{eq:rhog}
\!\!\!\rho_{G}=\frac{e^{-\sum_{\nu,k}\lambda_{\nu,k}N_{\nu,k}}}{Z_{G}} , Z_{G}=\mathrm{Tr}\left\{e^{-\sum_{\nu,k}\lambda_{\nu,k}N_{\nu,k}}\right\},
\end{equation}
where $\lambda_{\nu,k}=\log(n_{\bar{\nu},k}/n_{\nu,k})$, with $n_{\nu,k}=\langle G_0|N_{\nu,k}|G_0\rangle$ and $\bar{\nu}=v/c$ if $\nu=c/v$.
A physically interesting way of building the Lagrange multipliers giving the GGE density matrix in the case of sudden quench from the ground state is by the transformation $\mathcal{U}_{0,k}^{1}=e^{i\vec{\mathcal{D}}_k\cdot\vec{\sigma}}$ connecting post-quench Fermi operators $d_{1,\nu,k}$ to the pre-quench ones $d_{0,\nu,k}$, with $\vec{\sigma}$ the vector of Pauli matrices. The norm $|\vec{\mathcal{D}}_k|=\arctan\left\{\frac{\sqrt{4-(1-\Delta_k)^2}}{1-\Delta_k}\right\}$ plays a central role in defining the behavior of the quench. In fact, the function $ \Delta_{k}$ is such that~(cf \ref{SM:eq:Delta},~\ref{SM:eq:effbands})
\begin{equation}
\label{eq:popl}
n_{c/v,k}=\frac{1\pm\Delta_k}{2} , \quad \lambda_{c/v,k}=\pm\log\left(\frac{1-\Delta_k}{1+\Delta_k}\right),
\end{equation}
and thus directly controls the GGE. One finds $|\Delta_k|\leq 1$ with $0\leq|\vec{\mathcal{D}}_k|\leq\pi/2$. Furthermore, $|\Delta_k|=1$ {\em only} for $k=0,\pm\pi$: in particular one has
\begin{equation}
\label{eq:FP}
\Delta_0=-s(\delta_0+2)s(\delta_1+2) ,\, \Delta_{\pm\pi}=-s(\delta_0)s(\delta_1)\,,
\end{equation}
with $s(x)=|x|/x$ the sign function. When $\Delta_{k}=-1$, the transformation reduces to the identity $\mathcal{U}_{0,k}^{1}=\sigma_0$ and the quench does not affect the populations, while for $\Delta_{k}=1$ the transformation $\mathcal{U}_{0,{k}}^{1}=i\sigma_y$ induces a {swap} of the $c,v$ states. The value of $\Delta_k$ at $k=0,\pm\pi$ is constant and insensitive to variations of the quench parameters provided they remain within one of the nine regions bounded by the lines $\delta_\mu=0,-2w$ (see lower panel of Fig.~\ref{fig:1}). On the other hand, crossing one of the boundaries results in a sharp, non-analytical jump in $\Delta_k$. Thus, the center and edges of the Brillouin zone (BZ) act for the quench as fixed points, whose character is determined by the quench parameters. A detailed study of $ \Delta_k $~(see Appendix \ref{SM:sec:Delta}) allows us to identify four non-contiguous regions out of the nine defined above: Here, when $ \Delta_0\Delta_\pi <0 $, a non-trivial {\em inversion of population}~\cite{FN0}, characterized by $n_{\nu,0}n_{\bar{\nu},\pm\pi}=0$, takes place.
As mentioned, the transformation is directly linked to the GGE, since, in the sudden quench case from the ground state, one has $n_{\nu,k}^{-1}={1+e^{\xi_{\nu,k}\beta^{*}}}$
where $\xi_{\nu,k}=w\lambda_{\nu,k}$ and $\beta^{*}=w^{-1}$ is an inverse temperature. One can hence exactly rephrase the GGE density matrix as the Grancanonical ensemble of free fermions with Hamiltonian
\begin{equation}\bar{H}=\sum_{\nu,k}\xi_{\nu,k}d^\dag_{1,\nu,k}d_{1,\nu,k},
\end{equation}
inverse temperature $\beta^{*}$, and zero chemical potential. Combining the above analysis and Eq.~(\ref{eq:popl}) one can conclude that when a non-trivial inversion of population of the bands $\epsilon_{\nu,k}$ of the post-quench Hamiltonian is present, the effective bands $\xi_{\nu,k}$ cross zero energy and thus have a {metallic} character, while no crossing occurs in all other cases and the bands $\xi_{\nu,k}$ have an {insulating} character. When quench parameters cross one of the boundary lines described above, thus an effective MIT in $\bar{H}$ occurs. This is exemplified as black graphs in the nine different tiles of Fig.~\ref{fig:1} separated by $\delta_\mu=0,-2w$. We stress here that the MIT is an effective one showing up in the GGE. How this reflects in physically relevant (local) observables is a priori unclear, but in our case we will show explicitly in the following that its imprint is quite pronounced.
Next we analyse how the effective MIT influences observables of interest. The most intuitive one to investigate is the average level of dimerization $\bar{\mathcal{M}}$, given by the expectation value of $\mathcal{M}(x)=\Psi_{x}^{\dagger}\sigma_{x}\Psi_{x}$ with $\Psi_x=\sum_{k}e^{ikx}\Psi_{k}/\sqrt{L}$. Note that translational invariance implies that the expectation value is independent of the position. The main panel of Fig.~\ref{fig:1} shows a density plot of $\bar{\mathcal{M}}$ as a function of $\delta_\mu$. Crossing any of the transition lines $\delta_\mu=0,-2w$, a kink in $\bar{\mathcal{M}}$ is encountered. The top panel shows results for $\delta_0=2w>0$: the discontinuity in $\partial_{\delta_1}\bar{\mathcal{M}}$ at $\delta_1=0,-2w$ is evident. These kinks represent a signature of the occurrence of the effective MIT. Their origin is the non-analytic dependencies of the populations at $k=0,\pm\pi$ combined with the fact that, in the thermodynamic limit, the density of states of such points diverges as the curvature of $\epsilon_{1,k}$ vanishes at these points. Several other quantities show a similar behavior.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{Fig2.pdf}
\caption{(a) Plot of $\bar{\sigma}_0$ (units $w^2$); (b) Plot of $\bar{\sigma}$ (units $w^2$) ; (c) Plot of $\mathcal{S}$ as a function of $\delta_1$ (units $w$) for $\delta_0=2w$.}
\label{fig:2}
\end{center}
\end{figure}
Given the presence of the effective MIT in the GGE, we inspect the fluctuation of the space-averaged effective ``current" $J_0=\sum_{\nu,k}\left(\partial_{k}\xi_{\nu,k}\right)N_{\nu,k}$. Such fluctuation is defined using the phase velocity of the effective bands~\cite{mahan,FN2}, and in the steady state limit one has $\bar{\sigma}_0=\langle J_0^2\rangle$~(see Appendix \ref{SM:subsec:CurrFluc}). This quantity is shown in Fig.~\ref{fig:2}(a) for $\delta_0>0$. For $\delta_1=0$ (and $\delta_1=-2w$) it diverges $\propto|\delta_1|^{-1}$ (and $\propto|\delta_1+2|^{-1}$). Furthermore, fluctuations are larger in the effective metallic phase, while they tend to vanish in the insulating one, as one would expect \cite{mahan}. Although $\bar{\sigma}_0$ is not a directly accessible quantity, signatures of the effective MIT are present also in the steady state fluctuations $\bar{\sigma}=\langle J^2\rangle $ of the space-averaged physical current $J=\sum_{k}\left(\partial_{k}\epsilon_{\nu,k}\right)N_{\nu,k}$, shown in Fig.~\ref{fig:2}(b). In contrast to the current fluctuations in the effective picture though, here no marked differences in the magnitudes are found in the different phases. However, kinks occur at the boundaries between the phases. As a third example, Fig.~\ref{fig:2}(c) shows the thermodynamic entropy~(see Appendix \ref{SM:subsec:Entropy}) $\bar{\mathcal{S}}$ of the system for $\delta_0>0$: It is largest in the metallic phase and displays kinks for $\delta_1=0,-2w$. This quantity is particularly interesting, since it is intrinsic to thermodynamics.\\
\subsection{Robustness} As shown above, signatures of the effective MIT occur in a vast array of quantities. It is important to establish how robust the results are. We will consider $\bar{\mathcal{M}}$ as an example but the conclusions drawn below apply to all quantities discussed above.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{Fig3.pdf}
\caption{Plot of $\bar{\mathcal{M}}$ as a function of $\delta_1$ (in units of $w$) for different (a) number of lattice sites: solid the thermodynamic limit, dashed $\mathcal{N}=80$, dotted $\mathcal{N}=20$. The last two curves are averaged over a period - see text; (b) temperature of the initial state: solid $T=0$, dashed $T=5$, dotted $T=20$; (c) duration of the quench ramp: solid $\tau=0$, dashed $\tau=2$, dotted $\tau=10$; (d) Strength of the fermion-fermion interaction - see text: solid $U=0$, dashed $U=1$, dotted $U=2$, $T=0$. Here, $\delta_0=5w$ in panels (a-c) and $\delta_0=w$ in panel (d), $T$ is in units of $wk_B^{-1}$ with $k_B$ the Boltzmann constant, $\tau$ is in units of $\hbar w^{-1}$ and $U$ is in units of $w$.}
\label{fig:3}
\end{center}
\end{figure}
We begin by discussing deviations from the thermodynamic limit. With a finite number of lattice sites, averages do not converge to a steady value for long time but oscillate with a finite recursion time: the GGE hypothesis fails altogether. Figure~\ref{fig:3}(a) shows the quantum and time average over a period of $\mathcal{M}(t)$ near $\delta_1=0$. Dashed and dotted curves, calculated with a finite number of lattice sites, show that kinks are smoothed out as the number of sites decreases. This confirms the thermodynamic limit as a crucial ingredient for the non-analyticities to arise in the GGE predictions. Interestingly, though, even for $\mathcal{N}$ as small as $20$ one can still observe a distinct imprint of the non-analyticities found for $N\to\infty$ in the time-averaged $\bar{\mathcal{M}}$.
Furthermore, the features are robust in the case of a {thermal} preparation of the initial state~\cite{FN3}. Figure~\ref{fig:3}(b) shows $\bar{\mathcal{M}}$ obtained for an initial state at different temperatures $T$: although the curves are quantitatively different, with a global suppression of the dimerization, non-analyticities are always present. The origin of the robustness is that, for an initial temperature $T$, one has $n_{c,k}-n_{v,k}=f_{T,k}\Delta_k$ with $f_{T,k}=\sinh(\epsilon_{0,k}/k_BT)/[1+\cosh(\epsilon_{0,k}/k_BT)]>0$~(cf \ref{SM:eq:thermal}). This result means that the effective MIT occurs in the same parameter regions as in the $T=0$ case~\cite{FN4}. Note that the robustness of the non-analyticity with respect to temperature is particularly intriguing since it is not present in the equilibrium QPT characterizing the model.
We then consider the case of a quantum quench of finite time duration $\tau$, where the quench protocol is described by a linear ramp. Typical results are shown in Fig.~\ref{fig:3}(c). The non-analytic behavior persists, although results again differ quantitatively. This is due to the robustness of the effective MIT, that can be demonstrated by showing that the fixed points of the quench transformation only differ by an additional phase shift with respect to the case of sudden quench~(see Appendix \ref{SM:sec:Delta}).
Finally, we address the effects of static inter-particle interactions. We consider here a very similar model which - in the absence of interactions - displays the same qualitative behavior than the one discussed so far, but it is easier to simulate. The Hamiltonian is given by $H(t)=\sum_i^{N} w c_i^\dagger c_{i+1}+{\rm H.c.}+\delta(t)(-1)^i n_i+Un_{i}n_{i+1}$ and we consider $\bar{\mathcal{M}}=\left\langle n_0-1/2\right\rangle$ as an observable. $c_i^{(\dagger)}$ annihilates (creates) a spinless fermion on lattice site $i$. The model thus describes spinless fermions on a one-dimensional chain with staggered field $\delta(t)$ and nearest-neighbor interaction $U$. At time $t=0$ the staggered field is subject to the quench $\delta(t)=\delta_0\theta(-t)+\delta_1\theta(t)$, abruptly changing its value from $\delta_0$ to $\delta_1$. This model can be simulated with relative ease using standard density matrix renormalization group techniques based on matrix product states \cite{White92,Schollwock11,Kennes16}. The time scales which can be reached are bound within this approach by the entanglement growth of the system and, thus, the steady state behavior has to be read off at large but finite times. For $U=0$ strong oscillations in the dynamics after the quench render such an extrapolation difficult, but for this particular parameter value exact methods can be employed to extract the asymptotic behavior. At finite $U$ these oscillations are strongly damped out allowing for a straightforward extrapolation to long times~(see Appendix \ref{SM:sec:Interactions}). The inclusion of the interaction term makes the model non-integrable, which in turn is believed to destroy the GGE picture. Fig.~\ref{fig:3}(d) shows results for different values of the interaction strength: non-analyticities are washed out, as would be expected, by a thermal redistribution of the excitation energy in the long-time limit.\\
\section{Quantum quench in the Ising Model}
In this section, we consider the transverse field QI model. The Hamiltonian is
\begin{equation}
H_I(t)=-\sum_{j=1}^\mathcal{N}\frac{1}{2}\left[\sigma^x_{j+1}\sigma^x_j+h(t)\sigma^z_j\right]
\label{eq:c}\end{equation}
where $\sigma^\alpha_j$, $\alpha = x, y, z$, are the Pauli matrices at site $j$ of a chain of $\mathcal{N}$ sites with periodic boundary conditions, and $h(t)$ is the transverse field. We consider sudden quantum quenches, so that $h(t)=h_0\theta(-t)+h_1\theta(t)$, and we impose the system to be in the ground state $|0_I\rangle$ for $t<0$, and to evolve unitarily for $t>0$. Note that the state $|0_I\rangle$ is uniquely defined, even in the thermodynamic limit, since we consider $h_0>1$.\\
The Hamiltonian $H_I(t)$ can be diagonalized at any time by means of a Wigner-Jordan transformation onto spinless fermions, followed by a Bogoliubov transformation\cite{isi3}. In the even parity sector, relevant for the case inspected since we perform a quantum quench from the ground state at $h_0>1$, the diagonal forms of the pre($t<0$)/post$(t>0)$ quench Hamiltonians $H_I^{(i)}$ ($i=0/1$ respectively) read as
\begin{equation}
H_I^{(i)}=\sum_{k=-N}^{N-1}\xi^{(i)}_k
\left( b^{(i)\dagger}_kb^{(i)}_k-\frac{1}{2}\right),
\label{eq:cc}\end{equation}
with
\begin{equation}
\xi^{(i)}_k=\sqrt{[h_i-\cos (p_k)]^2+\sin^2(p_k)}.
\end{equation}
Here, $p_k=2\pi k/\mathcal{N}$ and $b^{(i)}_k$ are fermionic operators. Note that $b^{(0)}_k|0_I\rangle=0$, for every $k$. For the details of the transformation rewriting Eq.~\ref{eq:c} to Eq.~\ref{eq:cc}, see, for example, Ref.~\onlinecite{franchini}. The fermionic occupation numbers $N^{(I)}_k=b^{(1)\dagger}_kb^{(1)}_k$ and their averages ${n}^{(I)}_k=\langle 0_I|{N}^{(I)}_k|0_I\rangle$ allow to define, in the thermodynamic limit and for times $t\rightarrow\infty$, the post quench thermodynamic entropy $\mathcal{\bar{S}}_I=-\sum_k {n}^{(I)}_k\ln ({n}^{(I)}_k)+(1-{n}^{(I)}_k)\ln(1-{n}^{(I)}_k)$ and the GGE density matrix of the system. The latter quantity reads as
\begin{equation}
\rho_{G}^{(I)}=\frac{e^{-\sum_k \varepsilon^{(I)}_k N^{(I)}_k}}{Z_G^{(I)}},{Z_G^{(I)}}=\mathrm{Tr}\left\{e^{-\sum_k \varepsilon^{(I)}_k N^{(I)}_k}\right\},
\end{equation}
with $\varepsilon^{(I)}_k$ implicitly given by
\begin{equation}
{n}^{(I)}_k=\frac{1}{e^{\varepsilon^{(I)}_k}+1}.
\end{equation}
Again, we can interpret the GGE density matrix as a Grancanonical density matrix, at temperature set to unity and at zero chemical potential, for fermions with effective Hamiltonian
\begin{equation}
\bar{H}^{(I)}=\sum_k {\varepsilon^{(I)}_k}b^{(1)\dagger}_kb^{(1)}_k.
\end{equation}
As in the case of the SSH model, the entropy shows kinks as a function of the quench parameter, in correspondence to the gapless points of the dispersion relation signalling the equilibrium QPT between the paramagnetic and the ferromagnetic phase. Correspondingly the effective Hamiltonian $\bar{H}^{(I)}$ undergoes a metal insulator transition. The analogy to the behavior in the SSH model is hence complete. Examples are given in Fig.~\ref{fig:4}. In panel (a), the entropy $\mathcal{\bar{S}}_I$ is plotted as a function of $h_1$, for $h_0=10$. $\mathcal{\bar{S}}_I$ is shown to have non-analyticities in correspondence to the equilibrium QPTs occurring at $h_1=\pm 1$. In panel (b), the effective energies ${\varepsilon^{(I)}_k}$ are plotted, as a function of $k$, for $h_0=2$ and $h_1=5$ (red solid line), $h_1=1$ (green dashed line), and $h_1=0$ (blue dashed line). As in the case of the SSH model, these effective bands undergo an effective MIT in correspondence to the equilibrium QPT. In fact, for $h_1>1$ the dispersion does not cross the chemical potential (zero in this case), while for $h_1<1$ it does.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=\columnwidth]{Fig4.pdf}
\caption{(a) Plot of $\mathcal{\bar{S}}_I$ as a function of $h_1$, for $ h_0=0 $; (b) Plot of ${\varepsilon^{(I)}_k}$, as a function of $k$, for $h_0=2$ and $h_1=5$ (red solid line), $h_1=1$ (green dashed line), and $h_1=0$ (blue dashed line). The thin black line corresponds to the chemical potential.}
\label{fig:4}
\end{center}
\end{figure}
\section{Conclusions}
While in a previous work\cite{Moessner} the highly non-trivial relation between equilibrium and non-equilibrium QPT was inspected with reference to the topological nature of the equilibrium QPT, we have here adopted a different perspective, more suitable to generalizations in the context of integrable systems. We have observed that, in the paradigmatic cases of the SSH model and of the transverse field QI model, the non-equilibrium QPTs appear in connection to both an equilibrium QPT and an effective MIT in the GGE density matrix of the system. By direct inspection in the case of the SSH model, we have also shown that the non-equilibrium QPT is indeed robust with respect to those perturbations that do not spoil the validity of the GGE, and hence the presence of the effective MIT.
The phenomenology we describe appears general and should hold true also for higher dimensional systems. An interesting extension to our work includes the discussion of terms that break integrability only weakly. The results we report should carry over to the prethermal state reached in these situations.
\textit{Acknowledgements---}
D.M.K. acknowledges support by the Deutsche Forschungsgemeinschaft through the Emmy Noether program (KA 3360/2-1). Simulations were performed with computing resources granted by RWTH Aachen University under projects rwth0013 and prep0010. N.T.Z. acknowledges financial support by the DFG (SPP1666 and SFB1170 ToCoTronics),the Helmholtz Foundation (VITI), the ENB Graduate school on Topological Insulators. Interesting discussions with C. Fleckenstein, D. Hetterich, M. Serbyn, C. P{\'e}rez-Espigares and B. Trauzettel are also acknowledged.
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